diff --git a/Presentation iit/My Collection.bib b/Presentation iit/My Collection.bib deleted file mode 100755 index 56aa4c3..0000000 --- a/Presentation iit/My Collection.bib +++ /dev/null @@ -1,12 +0,0 @@ -@article{Lopes2009, -abstract = {We present a new class of gait generation and control algorithms based on the switching max-plus modeling framework that allows for the synchronization of multiple legs of walking robots. Transitions between stance and swing phases of each leg are modeled as discrete events on a system described by max-plus-linear state equations. Different gaits and gait parameters can be interleaved by using different system matrices. Switching in max-plus-linear systems offers a powerful collection of modeling, analysis, and control tools that, in particular, allow for safe transitions between different locomotion gaits that may involve breaking/enforcing synchronization or changing the order of leg lift off events. Experimental validation of the proposed algorithms is presented by the implementation of various horse gaits on a simple quadruped robot.}, -author = {Lopes, G. A D and Babu{\v{s}}ka, R. and {De Schutter}, B. and {Van Den Boom}, A. J J}, -doi = {10.1109/ROBIO.2009.5420626}, -file = {:media/octavio/Windows8{\_}OS/Documents and Settings/Tavo/Documents/Escuela/PhD/Papers Italy/09{\_}045.pdf:pdf}, -isbn = {9781424447756}, -journal = {2009 IEEE International Conference on Robotics and Biomimetics, ROBIO 2009}, -pages = {221--226}, -title = {{Switching max-plus models for legged locomotion}}, -volume = {19}, -year = {2009} -} diff --git a/Presentation iit/MyBib.bib b/Presentation iit/MyBib.bib deleted file mode 100755 index 58f1923..0000000 --- a/Presentation iit/MyBib.bib +++ /dev/null @@ -1,225 +0,0 @@ -@article{Cohen, -author = {Cohen, Guy}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/BCOQ-book.pdf:pdf}, -title = {{Synchronization and Linearity}} -} -@article{Turek2011, -author = {Turek, Michal and Turek, Richard}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/849-736-1-PB.pdf:pdf}, -keywords = {coordination,crossroads,max-plus algebra,modeling,transfer stop}, -number = {1}, -pages = {67--75}, -title = {{SOLVING SELECTED TRAFFIC PROBLEMS IN THE MAX-PLUS ALGEBRA}}, -volume = {9}, -year = {2011} -} -@article{Goverde2007, -author = {Goverde, Rob M P}, -doi = {10.1016/j.trb.2006.02.003}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/1-s2.0-S0191261506000208-main.pdf:pdf}, -keywords = {max-plus algebra,railway timetable,stability,timed event graph}, -pages = {179--201}, -title = {{Railway timetable stability analysis using max-plus system theory}}, -volume = {41}, -year = {2007} -} -@article{Winkler, -author = {Winkler, Alexander and Havoutis, Ioannis and Bazeille, Stephane and Ortiz, Jesus and Focchi, Michele and Caldwell, Darwin and Semini, Claudio and Robotics, Advanced and Italiano, Istituto}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/winkler14icra.pdf:pdf}, -title = {{Path planning with force-based foothold adaptation and virtual model control for torque controlled quadruped robots}} -} -@article{Winklera, -author = {Winkler, Alexander W and Mastalli, Carlos and Focchi, Michele and Caldwell, Darwin G and Havoutis, Ioannis}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/winkler15icra.pdf:pdf}, -title = {{Planning and Execution of Dynamic Whole-Body Locomotion for a Hydraulic Quadruped on Challenging Terrain}} -} -@article{Semini2010, -author = {Semini, Claudio}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/Semini2010PhD{\_}HyQ{\_}Robot.pdf:pdf}, -number = {April}, -title = {{HyQ - Design and Development of a Hydraulically Actuated Quadruped Robot}}, -year = {2010} -} -@article{Schutter2008, -author = {Schutter, B De and Boom, T Van Den}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/08{\_}007.pdf:pdf}, -title = {{Max-plus algebra and max-plus linear discrete event systems : An introduction}}, -volume = {19}, -year = {2008} -} -@article{Lopes2009, -abstract = {We present a new class of gait generation and control algorithms based on the switching max-plus modeling framework that allows for the synchronization of multiple legs of walking robots. Transitions between stance and swing phases of each leg are modeled as discrete events on a system described by max-plus-linear state equations. Different gaits and gait parameters can be interleaved by using different system matrices. Switching in max-plus-linear systems offers a powerful collection of modeling, analysis, and control tools that, in particular, allow for safe transitions between different locomotion gaits that may involve breaking/enforcing synchronization or changing the order of leg lift off events. Experimental validation of the proposed algorithms is presented by the implementation of various horse gaits on a simple quadruped robot.}, -author = {Lopes, G. A D and Babu{\v{s}}ka, R. and {De Schutter}, B. and {Van Den Boom}, A. J J}, -doi = {10.1109/ROBIO.2009.5420626}, -file = {:media/octavio/Windows8{\_}OS/Documents and Settings/Tavo/Documents/Escuela/PhD/Papers Italy/09{\_}045.pdf:pdf}, -isbn = {9781424447756}, -journal = {2009 IEEE International Conference on Robotics and Biomimetics, ROBIO 2009}, -pages = {221--226}, -title = {{Switching max-plus models for legged locomotion}}, -volume = {19}, -year = {2009} -} -} -@article{Mastalli, -author = {Mastalli, Carlos and Havoutis, Ioannis and Winkler, Alexander W and Caldwell, Darwin G and Semini, Claudio and Robotics, Advanced and Italiano, Istituto}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/mastalli15tepra.pdf:pdf}, -number = {Section II}, -title = {{On-line and On-board Planning and Perception for Quadrupedal Locomotion}} -} -@inproceedings{Lopes2010, -author = {Lopes, G A D and Boom, T J J Van Den and Schutter, B De and Babuˇ, R}, -booktitle = {Proceedings of the 10th International Workshop on Discrete Event Systems (WODES 2010)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/10{\_}038.pdf:pdf}, -keywords = {discrete event systems,legged locomotion,max-plus algebra,motion,robotics}, -pages = {6 pp.}, -title = {{Modeling and control of legged locomotion via switching max-plus systems}}, -year = {2010} -} -@article{Ijspeert, -author = {Ijspeert, Auke Jan and Crespi, Alessandro and Ryczko, Dimitri and Cabelguen, Jean-marie}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/1416.full.pdf:pdf}, -title = {{From Swimming to Walking with a Spinal Cord Model}} -} -@article{Ijspeert2008, -author = {Ijspeert, Auke Jan}, -doi = {10.1016/j.neunet.2008.03.014}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/Ijspeert08NeuralNEtworks.pdf:pdf}, -keywords = {central pattern generators}, -pages = {642--653}, -title = {{Central pattern generators for locomotion control in animals and robots : A review}}, -volume = {21}, -year = {2008} -} -@article{Holmes2006, -author = {Holmes, Philip and Koditschek, Daniel E and Guckenheimer, John}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/fulltext.pdf:pdf}, -number = {2}, -pages = {207--304}, -title = {{The Dynamics of Legged Locomotion : Models, Analyses, and Challenges}}, -volume = {48}, -year = {2006} -} -@article{Focchi2013, -author = {Focchi, Michele and Barasuol, Victor and Havoutis, Ioannis and Semini, Claudio and Caldwell, Darwin G}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/clawar13focchi.pdf:pdf}, -keywords = {control,quadrupedal locomotion,reflex}, -pages = {1--8}, -title = {{Local Reflex Generation for Obstacle Negotiation in Quadrupedal Locomotion}}, -year = {2013} -} -@article{Ekeberg1993, -author = {Ekeberg, Orjan}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/art{\%}3A10.1007{\%}2FBF00199436.pdf:pdf}, -pages = {363--374}, -title = {{Bkalog .,ai cyt mett 9}}, -volume = {374}, -year = 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{bio-inspiration,computational models,distributed motion control,neuron-based oscillators,robotics}, -number = {April}, -title = {{Central Pattern Generators for the control of robotic systems}}, -year = {2015} -} -@article{Barasuol, -author = {Barasuol, Victor and Focchi, Michele and Havoutis, Ioannis and Frigerio, Marco and Buchli, Jonas and Semini, Claudio and Caldwell, Darwin G}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/bazeille13tepra.pdf:pdf}, -title = {{Vision Enhanced Reactive Locomotion Control for Trotting on Rough Terrain}} -} -@article{Barasuola, -author = {Barasuol, Victor and Buchli, Jonas and Semini, Claudio and Frigerio, Marco and Pieri, Edson R De and Caldwell, Darwin G}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/barasuol13icra.pdf:pdf}, -title = {{A Reactive Controller Framework for Quadrupedal Locomotion on Challenging Terrain}} -} -@article{Barai, -author = {Barai, Ranjit Kumar and Nonami, Kenzo}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/47{\_}282.pdf:pdf}, -keywords = {gait,petri net,six-legged walking robot comet-iv,supervisory control}, -title = {{Petri Net Based Supervisory Control for Locomotion System of Six-Legged Walking Robot COMET-IV}} -} -@article{Wisse2004, -author = {Wisse, M and Schwab, A L and Helm, F C T Van Der}, -doi = {10.1017/S0263574704000475}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/WisSchHel04.pdf:pdf}, -keywords = {biped,passive,passive dynamic walking}, -pages = {681--688}, -title = {{Passive dynamic walking model with upper body}}, -volume = {22}, -year = {2004} -} -@article{Tsagarakis2013, -author = {Tsagarakis, N G and Morfey, S and Dallali, H and Caldwell, D G}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/06697155.pdf:pdf}, -isbn = {9781467363587}, -pages = {5512--5517}, -title = {{An Asymmetric Compliant Antagonistic Joint Design for High Performance Mobility}}, -year = {2013} -} -@article{Borovac2004, -author = {Borovac, Branislav}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/zmp-review.pdf:pdf}, -keywords = {biped locomotion,dynamically balanced gait,support,zero-moment point}, -number = {1}, -pages = {157--173}, -title = {{Zero-moment point — thirty five years of its life ´}}, -volume = {1}, -year = {2004} -} -@article{Aghili2005, -author = {Aghili, Farhad and Member, Associate}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/01512343.pdf:pdf}, -number = {5}, -pages = {834--849}, -title = {{A Unified Approach for Inverse and Direct Dynamics of Constrained Multibody Systems Based on Linear Projection Operator : Applications to Control and Simulation}}, -volume = {21}, -year = {2005} -} -@article{Geyer2006, -author = {Geyer, Hartmut and Seyfarth, Andre and Blickhan, Reinhard}, -doi = {10.1098/rspb.2006.3637}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/2861.full.pdf:pdf}, -keywords = {biomechanics,human gait,mass model,spring}, -number = {August}, -pages = {2861--2867}, -title = {{Compliant leg behaviour explains basic dynamics of walking and running}}, -year = {2006} -} -@article{Ketelaar, -author = {Ketelaar, J G and Visser, L C and Stramigioli, S and Carloni, R}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/ICRA13{\_}b.pdf:pdf}, -title = {{Controller Design for a Bipedal Walking Robot using Variable Stiffness Actuators}} -} -@ARTICLE{Quad, - author = {{Lee}, T. and {Leok}, M. and {Harris McClamroch}, N.}, - title = "{Control of Complex Maneuvers for a Quadrotor UAV using Geometric Methods on SE(3)}", - journal = {ArXiv e-prints}, -archivePrefix = "arXiv", - eprint = {1003.2005}, - primaryClass = "math.OC", - keywords = {Mathematics - Optimization and Control, Computer Science - Systems and Control}, - year = 2010, - month = mar, - adsurl = {http://adsabs.harvard.edu/abs/2010arXiv1003.2005L}, - adsnote = {Provided by the 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Transitions between stance and swing phases - of each leg are modeled as discrete events on a system described by - max-plus-linear state equations. Different gaits and gait parameters can be - interleaved by using different system matrices. Switching in max-plus-linear - systems offers a powerful collection of modeling, analysis, and control tools - that, in particular, allow for safe transitions between different locomotion - gaits that may involve breaking/enforcing synchronization or changing the - order of leg lift off events. Experimental validation of the proposed - algorithms is presented by the implementation of various horse gaits on a - simple quadruped robot.% - } - \verb{doi} - \verb 10.1109/ROBIO.2009.5420626 - \endverb - \field{isbn}{9781424447756} - \field{pages}{221\bibrangedash 226} - \field{title}{{Switching max-plus models for legged locomotion}} - \field{volume}{19} - \verb{file} - \verb :media/octavio/Windows8{\_}OS/Documents and Settings/Tavo/Documents/Esc - \verb uela/PhD/Papers Italy/09{\_}045.pdf:pdf - \endverb - \field{journaltitle}{2009 IEEE International Conference on Robotics and - Biomimetics, ROBIO 2009} - \field{year}{2009} -\endentry - -\entry{Lopes2010}{inproceedings}{} - \name{author}{4}{}{% - {{}% - {Lopes}{L.}% - {G~A~D}{G.~A.~D.}% - {}{}% - {}{}}% - {{}% - {Boom}{B.}% - {T~J J Van~Den}{T.~J. J. 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-\setbeamerfont{footnote}{size=\tiny} -\renewcommand{\cite}[1]{\footnote<.->[frame]{\fullcite{#1}}} - -\date{July 05, 2016} - -% Insert frame before each subsection (requires 2 latex runs) -\AtBeginSection { - \begin{frame}[noframenumbering] \frametitle{\titleSubsec} - \tableofcontents[currentsection] % Generation of the Table of Contents - \end{frame} -} - -% Define the title of each inserted pre-subsection frame -\newcommand*\titleSubsec{Outline} -% Define the title of the "Table of Contents" frame -\newcommand*\titleTOC{Outline} - -\setbeamertemplate{navigation symbols}{} % remove navigation symbols - -\graphicspath{ {./images/} } - -\begin{document} -%\begin{frame} -%\maketitle -%\end{frame} - -\section{Introduction} - -\begin{frame}{Introduction} - \begin{columns} - \begin{column}{0.6\textwidth} - Octavio A. Villarreal Maga\~na - \\ - \begin{itemize}\setlength\itemsep{2.5em} - \item MSc. Mechanical Engineering, track Control Engineering (TUDelft, The Netherlands) - \begin{itemize}\setlength\itemsep{1em} - \item [--] Control Methods for Robotics - \item [--] Robust Control - \end{itemize} - \item BSc. Mechatronic Engineering (UNAM, Mexico) - \begin{itemize}\setlength\itemsep{1em} - \item [--] Systems and Control - \item [--] Robotics - \end{itemize} - \end{itemize} - \end{column} - \begin{column}{0.4\textwidth} - \begin{figure}[ht] - \vspace{-30pt} - \includegraphics[width=1.5\textwidth]{images/yo.jpg} - \end{figure} - \end{column} - \end{columns} -\end{frame} - -\section{Relevant projects} - -%\subsection{Locomotion Control} - -\begin{frame}{Locomotion Control of Zebro} - \begin{columns} - \begin{column}{0.5\textwidth} - \begin{itemize} - \item Course project - \item Locomotion control of six-legged robot - \item Use of max-plus algebra\cite{Lopes2010} - \item Traversing unstructured terrain - \item Safe gait switching - \end{itemize} - \end{column} - \begin{column}{0.5\textwidth} - \includegraphics[height=0.55\textwidth]{recording.png} - \end{column} - \end{columns} -\end{frame} - -\begin{frame}{Implementation in real robot} - \begin{columns} - \begin{column}{0.5\textwidth} - \begin{itemize} - \item Commissioned by dutch artist - \item Max-plus algebra for reference generation - \item Independent controller for each leg - \item Construction of the robot - \item Implementation using ROS - \item Simple obstacle avoidance - \end{itemize} - \end{column} - \begin{column}{0.5\textwidth} - \includegraphics[height=0.7\textwidth]{real.png} - \end{column} - \end{columns} -\end{frame} - -%\subsection{Master thesis} - -\begin{frame}{Robust output-feedback control of 3D directional drilling systems} - \begin{columns} - \begin{column}{0.5\textwidth} - \begin{itemize} - \item MSc Thesis - \item In collaboration with the University of Minnesota - \item Complex dynamics (Delay Differential Equations) - \item Observer design to only use local measurements - \item Robust against parameter uncertainty - \end{itemize} - \end{column} - \begin{column}{0.5\textwidth} - \includegraphics[width = 1\textwidth]{drillingsystem.pdf} - \end{column} - \end{columns} -\end{frame} - -%\subsection{Control of robotic arm} - -\begin{frame}{Control of robotic arm} - \begin{columns} - \begin{column}{0.5\textwidth} - \begin{itemize}\setlength\itemsep{2.5em} - \item Control Methods for Robotics course assignment - \item Four rotational joints - \item Different control algorithms (Workspace and configuration space control) - \end{itemize} - \end{column} - \begin{column}{0.5\textwidth}\centering - \includegraphics[height = 0.55\textwidth]{Robot.png} \\ - \includegraphics[height=0.5\textwidth]{recording2.png} - \end{column} - \end{columns} -\end{frame} - -%\subsection{Control of quadcopter} - -\begin{frame}{Nonlinear geometric control of quadcopter} -\begin{columns} - \begin{column}{0.5\textwidth} - \begin{itemize}\setlength\itemsep{2.5em} - \item Control Methods for Robotics course assignment - \item Workspace and orientation nonlinear control - \item Capable of performing aggresive maneuvers\cite{Quad} - \end{itemize} - \end{column} - \begin{column}{0.5\textwidth} - \includegraphics[height=0.55\textwidth]{recording3.png} - \end{column} -\end{columns} -\end{frame} - -\section{Research proposal} - -%\subsection{Preliminaries} - -\begin{frame}{Objective of max-plus algebra in locomotion}\textbf{"Obtain synchronized time references for each of the leg's lift-off times (moment when a leg leaves the ground) and touch-down times (moment when a leg touches the ground) in a systematic way".} -\end{frame} - -\begin{frame}{Definition of max-plus algebra} - -"Tropical" algebra defined by: - - \begin{equation*} - (\Re_{max}, \oplus, \otimes, \varepsilon, e) - \end{equation*} - - where: - - \begin{align*} - \Re_{max} &:= \Re \cup \{-\infty\}, \nonumber\\ - x \oplus y &:= max(x,y), \nonumber\\ - x \otimes y &:= x + y, \nonumber\\ - \varepsilon &:= -\infty, \nonumber\\ - e &:= 0. \nonumber - \end{align*} -\end{frame} - -\begin{frame}{Definition of max-plus algebra} -\vspace{20pt} -Extension to matrices: -\begin{equation*} - (\Re_{max}^{n\times m}, \oplus, \otimes,\mathcal{E}, E) - \end{equation*} - - with: - - \begin{align*} - [A \oplus B]_{ij} &= a_{ij} \oplus b_{ij} := \max(a_{ij},b_{ij}), \nonumber \\ - [A \otimes C]_{ij} &= \underset{k=1}{\overset{m}{\mathlarger{\mathlarger\oplus}}}a_{ik} \otimes c_{kj} := \max\limits_{k=1,...,m}(a_{ik} + c_{kj}), \nonumber - \end{align*} - Absorbing and identity element: - \begin{align} - [\mathcal{E}]_{ij} &= \varepsilon, \nonumber \\ - [E]_{ij} &= \begin{cases} - e, & \text{if } i =j \\ - \varepsilon,& \text{otherwise.} - \end{cases} \nonumber - \end{align} - Powers: - \begin{equation*} - D^{\otimes k} := D \otimes D \otimes... \otimes D. - \end{equation*} -\end{frame} - -%\subsection{Gait generation} -\begin{frame}{Gait generation using max-plus algebra} -Gait Parameters\cite{Lopes2010}: -\begin{table}[ht] - \label{table:Parameters} - \begin{tabular}{ll} - \color{blue}$i$ & Leg index. \\ - \color{blue}$t_i (k)$ & Touchdown time of leg. \\ - \color{blue}$l_i (k)$ & Lift-off time. \\ - \color{blue}$\tau$ & Current time instant. \\ - \color{blue}$\tau_f$ & Flight time. \\ - \color{blue}$\tau_g$ & Ground time. \\ - \color{blue}$\tau_\Delta$ & Double stance time (adjustable). - \end{tabular} - \end{table} -\end{frame} - -\begin{frame}{Gait generation using max-plus algebra} - Gait cycle description for a biped: - \begin{align*} - t_1 (k+1) &= l_1 (k+1) + \tau_f \\ - l_1 (k+1) &= t_1 (k) + \tau_g \\ - t_2 (k+1) &= l_2 (k+1) + \tau_f \\ - l_2 (k+1) &= t_2 (k) + \tau_g - \end{align*} - Synchronization: - \begin{align*} - t_1 (k+1) &= l_1 (k+1) + \tau_f \\ - l_1 (k+1) &= \color{blue}\max (\color{black}t_1 (k) + \tau_g\color{blue},t_2(k) + \tau_\Delta)\\ - t_2 (k+1) &= l_2 (k+1) + \tau_f \\ - l_2 (k+1) &= \color{blue}\max (\color{black}t_2 (k) + \tau_g\color{blue},t_1(k+1) + \tau_\Delta) - \end{align*} -\end{frame} - -\begin{frame}{Gait generation using max-plus algebra} - Described as a max-plus linear system: - \begin{equation*} - \underbrace{ \begin{bmatrix} - t_1 (k+1) \\ - t_2 (k+1) \\ - l_1 (k+1) \\ - l_2 (k+1) - \end{bmatrix}}_{x(k+1)} = - \underbrace{\begin{bmatrix} - \tau_f \otimes \tau_g & \tau_f \otimes \tau_\Delta & \varepsilon & \varepsilon \\ - \tau_f^{\otimes 2} \otimes \tau_g \otimes \tau_\Delta & (\tau_f \otimes \tau_\Delta)^{\otimes 2} & \varepsilon & \varepsilon \\ - \tau_g & \tau_\Delta & \varepsilon & \varepsilon \\ - \tau_f \otimes \tau_g \otimes \tau_\Delta & \tau_f \otimes \tau_\Delta^{\otimes 2} & \varepsilon & \varepsilon - \end{bmatrix}}_{A} - \underbrace{\begin{bmatrix} - t_1 (k) \\ - t_2 (k) \\ - l_1 (k) \\ - l_2 (k) - \end{bmatrix}}_{x(k)} - \end{equation*} - Gait definition: - \begin{align*} - \mathcal{G}_{walk} &= \{1\} \prec \{2\} \\ - \mathcal{G}_{hop} &= \{1,2\} - \end{align*} -\end{frame} - -\begin{frame}{Advantages} - \begin{itemize} - \item Systematic gait generation - \item Number of legs does not increase complexity greatly - \item Safe gait switching \cite{Lopes2009} - \item Total cycle-time, steady-state and transient behavior analysis by studying properties of $A$ matrix - \end{itemize} -\end{frame} - -%\subsection{Locomotion control} - -\begin{frame}{Locomotion control of walking robots} - \includegraphics[width=1\textwidth]{images/Control.pdf} -\end{frame} - -\begin{frame}{Locomotion control of walking robots} - \begin{itemize}\setlength\itemsep{1.5em} - \item Supervisory control : Defines $A_j$ matrix ($\tau_g$, $\tau_f$, $\tau_\Delta$,etc.) - \begin{itemize} - \item [--] Uses sensor information - \end{itemize} - \item Gait scheduler: Defines lift-off and touch-down times for each leg (final and initial times for trajectory) - \item Continuous time scheduler: Defines trajectory for each of the legs based on scheduled times - \begin{itemize} - \item [--] For example, an ellipsoidal trajectory - \end{itemize} - \item Reference trajectory tracker: Follows the trajectory defined - \begin{itemize} - \item [--] e.g. workspace computed torque control - \end{itemize} - \end{itemize} -\end{frame} - -\section{Expected results} - -\begin{frame}{Expected results} - \begin{itemize}\setlength\itemsep{1.5em} - \item Systematic method to generate gaits - \item Methodology could provide safe gait switching - \item Could be applied to several topologies of multi-legged robots - \item Could provide robustness against unstructured scenarios - \end{itemize} -\end{frame} - -\end{document} \ No newline at end of file diff --git a/Presentation iit/document.toc b/Presentation iit/document.toc deleted file mode 100755 index 629c6a1..0000000 --- a/Presentation iit/document.toc +++ /dev/null @@ -1,13 +0,0 @@ -\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax -\defcounter {refsection}{0}\relax -\beamer@endinputifotherversion {3.24pt} -\defcounter {refsection}{0}\relax -\select@language {english} -\defcounter {refsection}{0}\relax -\beamer@sectionintoc {1}{Introduction}{1}{0}{1} -\defcounter {refsection}{0}\relax 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differ diff --git a/Presentation iit/vertical/MyBib.bib b/Presentation iit/vertical/MyBib.bib deleted file mode 100755 index 9a121bc..0000000 --- a/Presentation iit/vertical/MyBib.bib +++ /dev/null @@ -1,574 +0,0 @@ -@mastersthesis{Monsieurs2015, -author = {Monsieurs, F.H.A.}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/MasterThesis\_FrankMonsieurs\_19-07-2015\_comments.pdf:pdf}, -school = {Eindhoven University of Technology}, -title = {{Model-based decoupled control of a 3D directional drilling system}}, -year = {2015} -} - -@mastersthesis{Kremers2013, -author = {Kremers, N}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/MSc\_NiekKremers.pdf:pdf}, -school = {Eindhoven University of Technology}, -title = {{Model-based robust control of a directional drilling system}}, -year = {2013} -} - -@article{Downton2000, -abstract = {Rotary steerable drilling systems allow us to drill specialized well trajectories, including horizontal, extended-reach and other complex profiles, while avoiding the problems of sliding during conventional directional drilling. We explain how rotary steerable systems work and through field examples demonstrate the increase in penetration rate and improvement in reliability that are obtained with the latest generation of these rotary steerable tools.}, -author = {Downton, Geoff and Klausen, Trond Skei and Hendricks, Andy and Pafitis, Demos}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Control Papers/New directions on rotary steerable drilling.pdf:pdf}, -issn = {09231730}, -journal = {Oilfield Review}, -keywords = {Oilfield Review Spring 2000,Rotary Steerable Drilling}, -pages = {18--29}, -title = {{New Directions in Rotary Steerable Drilling}}, -url = {http://www.slb.com/~/media/Files/resources/oilfield\_review/ors00/spr00/p18\_29.pdf}, -year = {2000} -} - -@misc{Applications, - title = {{Applications of directional drilling systems}}, -howpublished = {\url{https://sites.google.com/site/directionaldrillingclub/}}, - note = {Accessed: 2015-10-10} -} - - -@misc{Applications2, - title = {{DIRECTIONAL DRILLING TECHNOLOGY}}, -howpublished = {\url{http://directionaldrilling.blogspot.nl/}}, - note = {Accessed: 2015-10-10} -} - -@article{Marck2015, -author = {Marck, Julien and Detournay, Emmanuel}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/SPE-173156-MS\_30.pdf:pdf}, -pages = {1--22}, -title = {{Spiraled Boreholes : An Expression of 3D Directional Instability of Drilling}}, -journal = {SPE Journal}, -year = {2015} -} - - - -@article{Perneder2012, -author = {Perneder, Luc and Detournay, Emmanuel and Earth, Csiro}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/SPE 151377.pdf:pdf}, -title = {{Anomalous Behaviors of a Propagating Borehole}}, -year = {2012}, -journal = {SPE 151377 } -} - -@article{Downton2011, -abstract = {A directionally controlled drilling mechanism has a lateral borehole propagation equation of state composed of spatially delayed feedback terms determined by the distribution of contact points between the borehole and drilling mechanism. A lateral borehole propagation equation is developed for a generic class of steering system. After simplification the resulting linear delay differential equation is partially validated against a Finite Element Model (FEM) of borehole propagation. The linear delay differential equations are then incorporated into various closed loop steering configurations to further demonstrate the efficacy of the approach. © 2011 IEEE.}, -author = {Downton, Gc and Ignova, M.}, -doi = {10.1109/CCA.2011.6044454}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Control papers downloaded by me/From bibliography of Kremers and Monsieurs/Stability and response of closed loop directional drilling system using linear delay...pdf:pdf}, -isbn = {9781457710636}, -journal = {Control Applications (CCA)}, -pages = {893--898}, -title = {{Stability and response of closed loop directional drilling system using linear delay differential equations}}, -year = {2011} -} - - -@article{JulienMarckEmmanuelDetournayAndreaKuesters2014, -author = {Marck, Julien and Detournay, Emmanuel and Kuesters, Andrea and Wingate, John}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/SPE-167992-PA.pdf:pdf}, -journal = {SPE Drilling \& Completion}, -number = {3}, -pages = {267 -- 278}, -title = {{Analysis of Spiraled-Borehole Data by Use of a Novel Directional-Drilling Model}}, -volume = {29}, -year = {2014} -} - - -@article{Marck2013, -author = {Marck, J and Detournay, E}, -file = {:C$\backslash$:/Users/Tavo/Desktop/SPE-174554-PA.pdf:pdf}, -pages = {1--10}, -number = {April}, -title = {{Influence of Rotary-Steerable-System Design on Borehole Spiraling}}, -journal = {SPE Journal}, -year = {2015} -} - - -@phdthesis{Perneder2013, -abstract = {The dynamical model governing the 3D kinematics of a drill bit is constructed for rotary drilling applications for which the bit is guided by a push-the-bit rotary steerable system. The evolution of the bit trajectory, and thus of the borehole geometry, is a consequence of the interaction between the borehole, a geometric object, and the drilling structure, a mechanical object. In this respect, the model describing this evolution consists of the association between (i) a model of the near-bit region of the drillstring, (ii) a model of the bit/rock interaction, and (iii) kinematic relationships relating the motion of the bit into the rock to geometric variables for the borehole evolution. The mathematical formulation of these three elements yields a set of functional differential equations with secular terms accounting for a delayed influence of the borehole geometry on the bit trajectory. The parameters entering these relations account for the loads and properties of the drilling structure and for the properties of the bit and rock formation. Three length scales are identified in the response of the directional drilling system; they correspond to short-, intermediate-, and long-range behaviors. The short-range response is associated with the dimensions of the bit, the small scale of the problem. It corresponds to fast variations of the bit orientation. On the intermediate-range, the wellbore trajectory converges to a quasi-constant curvature solution, if it is stable. On the long-range, the borehole curvature slowly varies and for appropriate set of drilling parameters the borehole converges toward a stationary helical path. Finally, the stability and rate of convergence on the intermediate and long range are investigated.}, -author = {Perneder, Luc}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/A Three-Dimensional Mathematical Model of Directional Drilling.pdf:pdf}, -pages = {125}, -school = {University of Minnesota}, -title = {{A three-dimensional Mathematical model of directional drilling}}, -url = {http://hdl.handle.net/11299/144996}, -year = {2013} -} - - -@article{Perneder2013a, -author = {Perneder, Luc and Detournay, Emmanuel}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/13SPEJ.pdf:pdf}, -issn = {1086055X}, -journal = {SPE Journal}, -keywords = {SPE 160335}, -number = {June}, -pages = {395--405}, -title = {{Equilibrium Inclinations of Straight Boreholes}}, -year = {2013} -} - -@article{Perneder2013b, -author = {Perneder, Luc and Detournay, Emmanuel}, -doi = {10.1016/j.ijsolstr.2012.12.011}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/Perneder2013.pdf:pdf}, -issn = {00207683}, -journal = {International Journal of Solids and Structures}, -keywords = {directional drilling}, -number = {9}, -pages = {1226--1240}, -title = {{Steady-state solutions of a propagating borehole}}, -url = {http://linkinghub.elsevier.com/retrieve/pii/S0020768312005227}, -volume = {50}, -year = {2013} -} - -@article{Panchal2012, -abstract = {An attitude control system is presented here, where the attitude of a drilling tool is represented by a unit vector and hence non-linearities of Euler angles are avoided. It is shown in simulation that the control system has a stable orbit about the desired attitude in the presence of drop and turn rate biases, and with a spatial delay due to the sensors. The attitude controller is embedded into a way-point tracking control system where it is demonstrated that the controller tracks the position and attitude of each way-point and minimizes the strain energy along the path.}, -author = {Panchal, N and Bayliss, M T and Whidborne, J F}, -doi = {10.3182/20120531-2-NO-4020.00025}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Control papers downloaded by me/From bibliography of Kremers and Monsieurs/Vector based kinematic closed loop attitude control system for directional drilling.pdf:pdf}, -isbn = {9783902661999}, -journal = {Proc.$\backslash$ 2012 IFAC Workshop on Automatic Control in Offshore Oil and Gas Production}, -keywords = {attitude control,directional drilling,nonlinear control systems,path tracking}, -number = {4020}, -pages = {78--83}, -title = {{Vector Based Kinematic Closed-Loop Attitude Control-System for Directional Drilling}}, -year = {2012} -} - -@article{Panchal2010, -abstract = {This paper describes the design of an inclination- and azimuth-hold controller for directional drilling tools typically used in the oil industry. A control input transformation that partially linearizes and decouples the plant dynamics is proposed. A pole-placement method is used to design the controller and an analysis of the stability robustness is performed using the small gain theorem. Results for a transient simulation of the proposed controller are also presented.}, -author = {Panchal, N and Bayliss, M T and Whidborne, J F}, -doi = {10.3182/20100802-3-ZA-2014.00022}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Control papers downloaded by me/From bibliography of Kremers and Monsieurs/Robust linear feedback control of attitude for directional drilling tools.pdf:pdf}, -isbn = {9783902661739}, -journal = {Journal of Modeling and Simulation}, -keywords = {attitude control,azimuth and inclination hold,feedback,linear,robust stability,rotary steerable system,rss,small gain theorem,trajectory control}, -pages = {41--47}, -title = {{Robust Linear Feedback Control of Attitude for Directional Drilling Tools}}, -volume = {4}, -year = {2010} -} - -@article{Panchal2011, -abstract = {Directional drilling has become increasingly important as a means for better exploitation of oil and gas reservoirs as well as part of a drive to increase the level of automation in the industry. Typically a well plan is created for a variety of reasons, such as maximizing productive drilling, minimizing well tortuosity, collision avoidance or navigating the tool to strategic points within a pay zone. Following a nominal well plan created to satisfy these requirements whilst drilling leads to a clear requirement for automated trajectory controlled drilling. This paper builds on the authors’ earlier work on generalized attitude control of directional drilling to develop a path following algorithm incorporating optimized geometric Hermite space curves as a means of generating the correction path from the instantaneous tool position to the well plan trajectory whilst minimizing the drill string strain energy. The proposed scheme is tested in simulation and shown to perform satisfactorily for a typical set of drilling operating parameters for converging towards a target position and attitude.}, -author = {Panchal, Neilkunal and Bayliss, M T and Whidborne, J F}, -doi = {10.1109/CCA.2011.6044507}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Control papers downloaded by me/From bibliography of Kremers and Monsieurs/Minimum Strain Energy Waypoint-following controller for directional drilling using OGH curves.pdf:pdf}, -isbn = {9781457710636}, -journal = {Interntional Conference on Control Applications, Denver, 28-30 September}, -pages = {887--892}, -title = {{Minimum strain energy waypoint-following controller for directional drilling using OGH curves}}, -year = {2011} -} - -@article{NeubertM.andHeisig1997, -author = {{Neubert, M. and Heisig}, G.}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Control Papers/Advanced trajectory simulation of drilling wellbores.pdf:pdf}, -number = {3}, -pages = {45--52}, -journal = {Energy Week Conference and Exhibition: incorporating ETCE}, -title = {{Advanced trajectory simulation of directional wellbores.}}, -url = {http://stbb.nichd.nih.gov/pdf/Mattiello97.pdf}, -year = {1997} -} - -@article{Matheus2014, -abstract = {Traditionally, trajectory control in the drilling industry has been performed by experienced directional drillers, using drilling tools such as mud motors, these tools are just iron components, that limits the efficiency of the drilling process and rate of penetration (ROP). The introduction of Rotary Steerable Systems (RSS) enabled faster and efficient well bore delivery. This tested the limits of human operators to assimilate the MWD data, compare the actual well trajectory with the planned and take corrective actions to get back to the desire well plan. A hybrid approach consisting of two levels is proposed to perform the trajectory control in the oil drilling industry using RSS. Level 1, attitude control, regulates inclination and azimuth downhole. Level 2, the outer loop, monitors the performance of the inner loop and controls the directional drilling commands issued to the tool to follow a predefined plan. A next generation nonlinear trajectory control algorithm was developed and tested. This algorithm can hold inclination and azimuth downhole, run below mud motors, and perform an automatic turn, while holding inclination with the ability to control the gains while drilling ahead. This novel algorithm has been applied in various locations worldwide to control inclination and azimuth and keep true vertical depth (TVD) within tight tolerances. Conclusions and lessons learned from the analysis will aid future applications of the hybrid approach and the new algorithm. Copyright 2014, Society of Petroleum Engineers.}, -author = {Matheus, J. and Ignova, M. and Hornblower, P.}, -doi = {10.3182/20120531-2-NO-4020.00008}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Control papers downloaded by me/From bibliography of Kremers and Monsieurs/A hybrid approach to closed-loop directional drilling control using rotary steerable systems.pdf:pdf}, -isbn = {9781632668882}, -issn = {14746670}, -journal = {SPE Latin American and Caribbean Petroleum Engineering Conference Proceedings}, -pages = {1996--2005}, -title = {{A hybrid approach to closed-loop directional drilling control using rotary steerable systems}}, -url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-84908495511\&partnerID=tZOtx3y1}, -volume = {3}, -year = {2014} -} - -@article{Chapman2012, -author = {Chapman, Clinton D and Ph, D and Luis, Jose and Flores, Sanchez and Perez, Roberto De Leon and Yu, Han}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Control papers downloaded by me/From bibliography of Kremers and Monsieurs/Automated closed-loop drilling with rop optimization algorithm significantly....pdf:pdf}, -number = {2006}, -journal = {Drilling Conference and Exhibition, 6-8 March}, -title = {{IADC / SPE 151736 Automated Closed-loop Drilling with ROP Optimization Algorithm Significantly Reduces Drilling Time and Improves Downhole Tool Reliability}}, -year = {2012} -} - -@article{Bayliss2015, -author = {Bayliss, Martin and Bogath, Chris and Whidborne, James}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Control papers downloaded by me/Others/MPC-Based feedback delay compensation scheme for directional drilling attitude control.pdf:pdf}, -journal = {Drilling Conference and Exhibition, 17-19 March}, -title = {{MPC-Based Feedback Delay Compensation Scheme for Directional Drilling Attitude Control}}, -year = {2015} -} - -@article{Sun2012, -author = {Sun, Hui and Li, Zhiyuan and Hovakimyan, Naira and Ba, Tamer}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Control papers downloaded by me/From bibliography of Kremers and Monsieurs/L1 adaptive control for directional drilling systems.pdf:pdf}, -keywords = {adaptive control,directional drilling,time-delay systems}, -journal = {Proceedings of the IFAC Workshop on Automic Control in Offshore and Gas Production, 6-11 July}, -pages = {72--77}, -title = {{L 1 Adaptive Control for Directional Drilling Systems}}, -year = {2012} -} - -@article{MarckJulien2014, -author = {Marck, Julien and Emmanuel, Detournay}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/ENOC2014\_latex\_template.pdf:pdf}, -journal = {ENOC, 6-11 July}, -title = {{Spiraling Tendency of Deep Drilling Systems CSIRO Earth Science and Resource Engineering , Australia}}, -year = {2014} -} - -@article{Felczak2012, -author = {Felczak, Edwin and Godwin, Neil D and Hawkins, Richard and Jones, Stephen and Slayden, Fred}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/Directional Drilling description/04\_hybrid\_rotary.pdf:pdf}, -keywords = {The Best of Both Worlds - A Hybrid Rotary Steerable System}, -pages = {36--44}, -journal = {Oilfield Review}, -title = {{The Best of Both Worlds — A Hybrid Rotary Steerable System}}, -year = {2012} -} - -@article{JulienMarckEmmanuelDetournayAndreaKuesters2014, -author = {{Julien Marck , Emmanuel Detournay, Andrea Kuesters}, John Wingate}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/First group of references sent by Nathan/Previous Thesis and base works/SPE-167992-PA.pdf:pdf}, -journal = {SPE Drilling \& Completion}, -number = {3}, -pages = {267 -- 278}, -title = {{Analysis of Spiraled-Borehole Data by Use of a Novel Directional-Drilling Model}}, -volume = {29}, -year = {2014} -} - -@misc{Bit, - title = {{Bit of best fit} }, -howpublished = {\url{http://www.drillingcontractor.org}}, - note = {Accessed: 2015-10-18} -} - -@inproceedings{Mill82, - Abstract = { This paper is the first reporting of a directional drilling simulation computer program that converts side force values, generated by a three-dimensional bottom-hole assembly algorithm, to actual displacements in the x and y planes. The program can be used to simulate directional wells program can be used to simulate directional wells that have been drilled, or to predict the actual trajectory for a given bottom-hole assembly, geology, and operating conditions. -The logic for the simulation program is presented, discussing how the bottom-hole assembly presented, discussing how the bottom-hole assembly program, and the bit and stabilizer side cutting data program, and the bit and stabilizer side cutting data were merged to form the basis of the program. Various inputs for a simulation or prediction, and the sensitivity of each input, are discussed. Mechanisms that are not included in the algorithm are cited, explaining how the lack of these mechanisms precluded the simulation of a certain variety of precluded the simulation of a certain variety of directional wells. -Results from the simulation of a directional well drilled in Holland are presented. Matching of actual inclination and direction data provided the basis of showing how the program could be used in a predictive mode for the selection and operation of predictive mode for the selection and operation of other bottom-hole assemblies for drilling another directional well in the same area. -A final section discusses some of the problems encountered in running simulations and how these problems equate to actual field conditions. problems equate to actual field conditions. Mechanisms that need to be added to the algorithm to increase the fidelity and the breath of the simulation are mentioned. -The existing program has been used successfully to match wellbore trajectories in areas where the geological effects are nil to moderate and the wellbore conditions can be accurately described. With further refinements and enhancements to the program, it will only be a matter of time before any directional or deviation control well can be simulated. This ultimately means a computer program can be used to aid the drilling person in making sound engineering decisions on how to drill a directional well. }, - Address = {Beijing, China}, - Author = {Millheim, K. K.}, - Booktitle = {International Petroleum Exhibition and Technical Symposium}, - Date-Modified = {2010-12-13 13:49:18 +0800}, - Doi = {10.2118/9990-MS}, - Keywords = {computer simulation; directional drilling process; directional drilling; computerized simulation; oil wells; drilling; simulation; wells}, - Local-Url = {PDFs/Mill82.pdf}, - Month = {March}, - Number = {SPE 9990}, - Pages = {483-496}, - Publisher = {Society of Petroleum Engineers}, - Title = {Computer Simulation of the Directional Drilling Process}, - Year = {1982}, - Bdsk-File-1 = {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}, - Bdsk-Url-1 = {http://dx.doi.org/10.2118/9990-MS}} - -@inproceedings{Rafi88, - Abstract = {BHA bending analysis and drilling ahead prediction are important topics because they can improve drilling efficiency and help prevent unplanned borehole deviation. Without them, there is extra risk of undesirable deviation which can cause severe drilling problems and greatly increase the cost of drilling. This paper presents the analytical basis and field applications of an integrated computer method in directional drilling. The method includes two-dimensional BRA bending analysis for designing BHAs and predicting their deviation tendencies, and a heuristic formation-bit interaction model for forecasting the future wellbore position while drilling. Using this computer method, parametric studies are performed on several single and multi-stabilizer bottom hole assemblies to demonstrate the effects of stabilizer placement and wear, WOB (weight on bit), hole angle, and borehole current curvature on the deviation tendency of these assemblies. Predicted wellbore positions from the drilling ahead model are compared with actual field data for two directional wells.}, - Address = {Dallas, Texas, U.S.A.}, - Author = {Rafie, S.}, - Booktitle = {IADC/SPE Drilling Conference}, - Date-Modified = {2011-08-11 14:00:26 +0800}, - Doi = {10.2118/17196-MS}, - Keywords = {mechanistic; approach; designing; BHA; forecasting; wellbore position; stabilizer}, - Local-Url = {PDFs/Rafi88.pdf}, - Month = {February-March}, - Pages = {161-170}, - Publisher = {Society of Petroleum Engineers}, - Title = {Mechanistic Approach in Designing {BHA}'s and Forecasting Welbore Position}, - Year = {1988}, - Bdsk-File-1 = {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}, - Bdsk-Url-1 = {http://dx.doi.org/10.2118/17196-MS}} -@article{Call81, - Abstract = {A method for predicting borehole trajectories is described and seven examples of results are considered which are based on field data. Computed results indicate good agreement with the field results.}, - Annote = {August}, - Author = {Callas, N. P.}, - Date-Added = {2009-07-15 15:53:46 +0800}, - Date-Modified = {2011-01-25 13:53:22 +0800}, - Issn = {0030-1388}, - Journal = {Oil and Gas Journal}, - Keywords = {prediting borehole trajectories; borehole; borehole trajectories}, - Local-Url = {PDFs/Call81.pdf}, - Number = {34}, - Pages = {129-133}, - Title = {Predicting Borehole Trajectories}, - Volume = {79}, - Year = {1981}, - Bdsk-File-1 = {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}} - -@inproceedings{Down07, - Abstract = {A directionally steered drilling system has a borehole-propagation transfer function composed of spatial delay exponentials corresponding to the system's touch points with the borehole. This insight enables frequency-domain methods to be applied to the problems of steering performance.}, - Address = {Singapore}, - Annote = {WeC06.3}, - Author = {Downton, G. C.}, - Booktitle = {16th IEEE International Conference on Control Applications}, - Date-Modified = {2011-08-10 11:47:57 +0800}, - Doi = {10.1109/CCA.2007.4389456}, - Isbn = {1085-1992}, - Keywords = {directional drilling; quasi-polynomial; stability}, - Local-Url = {PDFs/Down07.pdf}, - Month = {October}, - Pages = {1543-1550}, - Title = {Directional Drilling System Response and Stability}, - Year = {2007}, - Bdsk-File-1 = {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}, - Bdsk-Url-1 = {http://dx.doi.org/10.1109/CCA.2007.4389456}} - -@book{Neub97, - Author = {Neubert, M.}, - Date-Modified = {2009-04-16 14:05:32 +0800}, - Isbn = {3-920395-31-X}, - Keywords = {richtungsregelung beim tiefbohren}, - Local-Url = {PDFs/Neub97.pdf}, - Pages = {1-133}, - Title = {Richtungsregelung beim Tiefbohren}, - Year = {1997}, - Bdsk-File-1 = {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}} - -@article{Burke2005, -abstract = {Let f be a continuous function on {\$}\backslashRl{\^{}}n{\$}, and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but is a function whose gradient is easily computed where it is defined. We present a practical, robust algorithm to locally minimize such functions, based on gradient sampling. No subgradient information is required by the algorithm. When f is locally Lipschitz and has bounded level sets, and the sampling radius {\$}\backslasheps{\$} is fixed, we show that, with probability 1, the algorithm generates a sequence with a cluster point that is Clarke {\$}\backslasheps{\$}-stationary. Furthermore, we show that if f has a unique Clarke stationary point {\$}\backslashbar x{\$}, then the set of all cluster points generated by the algorithm converges to {\$}\backslashbar x{\$} as {\$}\backslasheps{\$} is reduced to zero. Numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts, including cases where f is not locally Lipschitz at minimizers. We report approximate local minimizers for functions in the applications literature which have not, to our knowledge, been obtained previously. When the termination criteria of the algorithm are satisfied, a precise statement about nearness to Clarke {\$}\backslasheps{\$}-stationarity is available. A MATLAB implementation of the algorithm is posted at http://www.cs.nyu.edu/overton/papers/gradsamp/alg.}, -author = {Burke, James V. and Lewis, Adrian S. and Overton, Michael L.}, -doi = {10.1137/030601296}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Papers and references/By chapter/Control/gradsamp.pdf:pdf}, -issn = {1052-6234}, -journal = {SIAM Journal on Optimization}, -keywords = {generalized gradient,gradient sampling,nonsmooth optimization,subgradient}, -number = {3}, -pages = {751--779}, -title = {{A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization}}, -volume = {15}, -year = {2005} -} - - -@article{Marck2015, -author = {Marck, Julien and Detournay, Emmanuel}, -file = {:C$\backslash$:/Users/Tavo/Desktop/MaDe15b.pdf:pdf}, -pages = {1--22}, -journal = {SPE/IADC Drilling Conference and Exhibition, 17-19 March}, -title = {{Spiraled Boreholes : An Expression of 3D Directional Instability of Drilling}}, -year = {2015} -} - -@book{Skogestad2007, -abstract = {Multivariable Feedback Control: Analysis and Design, Second Edition presents a rigorous, yet easily readable, introduction to the analysis and design of robust multivariable control systems. Focusing on practical feedback control and not on system theory in general, this book provides the reader with insights into the opportunities and limitations of feedback control.Taking into account the latest developments in the field, this fully revised and updated second edition:* features a new chapter devoted to the use of linear matrix inequalities (LMIs);* presents current results on fundamental performance limitations introduced by RHP-poles and RHP-zeros;* introduces updated material on the selection of controlled variables and self-optimizing control;* provides simple IMC tuning rules for PID control;* covers additional material including unstable plants, the feedback amplifier, the lower gain margin and a clear strategy for incorporating integral action into LQG control;* includes numerous worked examples, exercises and case studies, which make frequent use of Matlab and the new Robust Control toolbox.Multivariable Feedback Control: Analysis and Design, Second Edition is an excellent resource for advanced undergraduate and graduate courses studying multivariable control. It is also an invaluable tool for engineers who want to understand multivariable control, its limitations, and how it can be applied in practice. The analysis techniques and the material on control structure design should prove very useful in the new emerging area of systems biology.Reviews of the first edition:"Being rich in insights and practical tips on controller design, the book should also prove to be very beneficial to industrial control engineers, both as a reference book and as an educational tool." Applied Mechanics Reviews"In summary, this book can be strongly recommended not only as a basic text in multivariable control techniques for graduate and undergraduate students, but also as a valuable source of information for control engineers." International Journal of Adaptive Control and Signal Processing}, -author = {Skogestad, Sigurd and Postlethwaite, Ian}, -doi = {10.1002/(SICI)1099-1239(19981215)8:14<1237::AID-RNC377>3.0.CO;2-7}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Robust and Multivariable Control/Multivariable{\_}Feedback{\_}Control-Analysis{\_}and{\_}Design.pdf:pdf}, -isbn = {9780470011683}, -issn = {1049-8923}, -pages = {592}, -publisher = {John Wiley and Sons}, -title = {{Multivariable feedback control: analysis and design}}, -url = {http://www.amazon.com/Multivariable-Feedback-Control-Analysis-Design/dp/0470011688/ref=sr{\_}1{\_}1?s=books{\&}ie=UTF8{\&}qid=1401117789{\&}sr=1-1{\&}keywords=multivariable+feedback+control$\backslash$nhttp://alexandria.tue.nl/openaccess/Metis208895.pdf}, -year = {2007} -} - -@book{VandenBoom2013, -author = {van den Boom, Ton}, -booktitle = {Preprint}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Readers/Model Predictive Control SC4060.pdf:pdf}, -isbn = {0691769001}, -number = {January}, -pages = {264}, -publisher = {Delft University of Technology}, -title = {{Model predictive control}}, -year = {2013} -} - -@article{Pomet1992, -abstract = {A stabilizing adaptive controller for a nonlinear system depending affinely on some unknown parameters is presented. It is assumed that this system is feedback stabilizable. A key feature of the method is the use of the Lyapunov equation to design the adaptive law. A result on local stability, two different conditions for global stability, and a local result where the initial conditions of the state of the system only are restricted are given}, -author = {Pomet, J.-B. and Praly, L.}, -doi = {10.1109/9.256328}, -file = {:C$\backslash$:/Users/Tavo/Desktop/00256328.pdf:pdf}, -isbn = {0018-9286}, -issn = {00189286}, -journal = {IEEE Transactions on Automatic Control}, -keywords = {Adaptive control,Continuous-stirred tank reactor,Control systems,Linear feedback control systems,Lyapunov equation,Lyapunov methods,Manipulators,Nonlinear equations,Nonlinear systems,Programmable control,Robots,Stability,adaptive control,feedback,feedback stabilizable,global stability,local stability,nonlinear control systems,nonlinear system,stability,stabilizing adaptive controller}, -number = {6}, -pages = {729--740}, -title = {{Adaptive nonlinear regulation: estimation from the Lyapunov equation}}, -url = 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It will be recreated as -% required. -% -\begingroup -\makeatletter -\@ifundefined{ver@biblatex.sty} - {\@latex@error - {Missing 'biblatex' package} - {The bibliography requires the 'biblatex' package.} - \aftergroup\endinput} - {} -\endgroup - -\entry{Monsieurs2015}{thesis}{} - \name{author}{1}{}{% - {{}% - {Monsieurs}{M.}% - {F.H.A.}{F.}% - {}{}% - {}{}}% - } - \strng{namehash}{MF1} - \strng{fullhash}{MF1} - \field{labelyear}{2015} - \field{sortinit}{M} - \field{title}{{Model-based decoupled control of a 3D directional drilling - system}} - \list{institution}{1}{% - {Eindhoven University of Technology}% - } - \verb{file} - \verb :C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Paper - \verb s and references/First group of references sent by Nathan/Previous Thes - \verb is and base works/MasterThesis\_FrankMonsieurs\_19-07-2015\_comments.pd - \verb f:pdf - \endverb - \field{type}{mathesis} - \field{year}{2015} -\endentry - -\entry{Perneder2013}{thesis}{} - \name{author}{1}{}{% - {{}% - {Perneder}{P.}% - {Luc}{L.}% - {}{}% - {}{}}% - } - \strng{namehash}{PL1} - \strng{fullhash}{PL1} - \field{labelyear}{2013} - \field{sortinit}{P} - \field{abstract}{% - The dynamical model governing the 3D kinematics of a drill bit is constructed - for rotary drilling applications for which the bit is guided by a - push-the-bit rotary steerable system. The evolution of the bit trajectory, - and thus of the borehole geometry, is a consequence of the interaction - between the borehole, a geometric object, and the drilling structure, a - mechanical object. In this respect, the model describing this evolution - consists of the association between (i) a model of the near-bit region of the - drillstring, (ii) a model of the bit/rock interaction, and (iii) kinematic - relationships relating the motion of the bit into the rock to geometric - variables for the borehole evolution. The mathematical formulation of these - three elements yields a set of functional differential equations with secular - terms accounting for a delayed influence of the borehole geometry on the bit - trajectory. The parameters entering these relations account for the loads and - properties of the drilling structure and for the properties of the bit and - rock formation. Three length scales are identified in the response of the - directional drilling system; they correspond to short-, intermediate-, and - long-range behaviors. The short-range response is associated with the - dimensions of the bit, the small scale of the problem. It corresponds to fast - variations of the bit orientation. On the intermediate-range, the wellbore - trajectory converges to a quasi-constant curvature solution, if it is stable. - On the long-range, the borehole curvature slowly varies and for appropriate - set of drilling parameters the borehole converges toward a stationary helical - path. Finally, the stability and rate of convergence on the intermediate and - long range are investigated.% - } - \field{pages}{125} - \field{title}{{A three-dimensional Mathematical model of directional - drilling}} - \verb{url} - \verb http://hdl.handle.net/11299/144996 - \endverb - \list{institution}{1}{% - {University of Minnesota}% - } - \verb{file} - \verb :C$\backslash$:/Users/Tavo/Documents/Escuela/Master/Master Thesis/Paper - \verb s and references/First group of references sent by Nathan/Previous Thes - \verb is and base works/A Three-Dimensional Mathematical Model of Directional - \verb Drilling.pdf:pdf - \endverb - \field{type}{phdthesis} - \field{year}{2013} -\endentry - -\lossort -\endlossort - -\endinput diff --git a/Presentation iit/vertical/presentation.bcf b/Presentation iit/vertical/presentation.bcf deleted file mode 100755 index baccc21..0000000 --- a/Presentation iit/vertical/presentation.bcf +++ /dev/null @@ -1,1972 +0,0 @@ - - - - - - output_encoding - ascii - - - 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-\usepackage{calc} -\usepackage[absolute,overlay]{textpos} -\usepackage{graphicx} -\usepackage{subfig} -\usepackage{comment} -%\usepackage{tikz} -\usepackage{wasysym} -\usepackage{gensymb} -\usepackage{amssymb} -\usepackage{nccmath} -\usepackage{empheq} -\usepackage{xcolor} -\usepackage{relsize} -%\usepackage{apalike} - - -\newcommand*\widefbox[1]{\fbox{\hspace{1em}#1\hspace{1em}}} -%\useoutertheme{miniframes} - -%\usefonttheme[onlymath]{serif} - - - -\setbeamertemplate{navigation symbols}{} % remove navigation symbols -\mode{\usetheme{tud}} - - -%\ExecuteBibliographyOptions{parentracker=false} - - - -% BIB SETTINGS -\usepackage[backend=bibtex,firstinits=true,maxnames=30,maxcitenames=20,url=false,style=authoryear]{biblatex} -%authoryear -\bibliography{MyBib} -%\setcitestyle{square} -\setlength\bibitemsep{0.3cm} % space between entries in the reference list -\renewcommand{\bibfont}{\normalfont\scriptsize} -\setbeamerfont{footnote}{size=\tiny} -\makeatletter -%\renewcommand{\cite}[1]{\footnote<.->[frame][{\fullcite{#1}}]} -%\renewcommand{\cite}[1]{$[$\cite{#1}$]$} - - -% Insert frame before each subsection (requires 2 latex runs) -\AtBeginSection { - \begin{frame}[noframenumbering] \frametitle{\titleSubsec} - \tableofcontents[currentsection,currentsubsection] % Generation of the Table of Contents - \end{frame} -} - -% Define the title of each inserted pre-subsection frame -\newcommand*\titleSubsec{Outline} -% Define the title of the "Table of Contents" frame -\newcommand*\titleTOC{Outline} - -\graphicspath{ {./images/} } - -\title[Final MSc Presentation]{Robust output-feedback control of 3D directional drilling systems} -\institute[]{Delft University of Technology, The Netherlands \\ University of Minnesota, USA} -\author{Octavio A. Villarreal Maga\~na \vspace{10pt} \\ \scriptsize Committee: \\ Prof.Dr.Ir. Nathan van de Wouw (TUDelft, supervisor) \\ \vspace{1.5pt} \hspace{-11pt}Prof.Dr.Ir. Emmanuel Detournay (UMN, supervisor) \\ \vspace{-2.5pt} \hspace{-74pt}Dr.Ir. Manuel Mazo Jr. (TUDelft)} -\date{July 12, 2016} - -\begin{document} -{ -\setbeamertemplate{footline}{\usebeamertemplate*{minimal footline}} -\frame{\titlepage} -\begin{frame}\frametitle{\titleTOC} - \tableofcontents -\end{frame} -} - -{\setbeamertemplate{footline}{\usebeamertemplate*{minimal footline}} -} - -\section{General description of directional drilling systems} - -\begin{frame}{Applications of directional drilling} - \begin{columns} - \hspace{0.9cm}\begin{column}{0.4\textwidth} - \begin{figure}[t] - \centering - \begin{minipage}[t]{1\textwidth} - \centering - \includegraphics[width=1\linewidth]{images/appdd1.pdf}\\ - \end{minipage}%\\ - \\\begin{minipage}[t]{1\textwidth} - \centering - \includegraphics[width=1\linewidth]{images/appdd2.pdf}\\ - \end{minipage} - \begin{minipage}[t]{1\textwidth} - \centering - \includegraphics[width=1\linewidth]{images/appdd3.pdf}\\ - \end{minipage} - \end{figure} - \end{column} - \begin{column}{0.6\textwidth}\setlength{\leftmargini}{20pt} - \begin{itemize}\setlength\itemsep{2.5em} - \item Extract oil, mineral and thermal energy resources - \item Reach targets that need complex geometries such as: - \begin{itemize}\setlength\itemsep{1em} - \item Under a city or an ecosystem - \item Far from the drill rig - \item Relief for hazardous situations - \end{itemize} - \end{itemize} - \end{column} - \end{columns} -\end{frame} - -\begin{frame}\frametitle{General description of the system} - \begin{columns} - \hspace{1cm} \begin{column}{0.6\textwidth} - \begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/drillingsystem.pdf} - \end{figure} - \end{column} - \begin{column}{0.4\textwidth} - \centering - Rotary Steerable System - \vspace{-10pt} - \begin{figure}[ht] - \begin{minipage}[t]{1\textwidth} - \hspace{0.5cm} \includegraphics[width=0.5\textwidth]{images/RSS.pdf} - \end{minipage} - \begin{minipage}[b]{1\textwidth} - \hspace{1.1cm} \includegraphics[width=0.4\textwidth]{images/PDC-bit.pdf} - \end{minipage} - \end{figure} - \end{column} - \end{columns} -\end{frame} - - - -\begin{frame}{Context and challenges} - \begin{columns}[T] - \hspace{1cm} \begin{column}{0.5\textwidth}\setlength{\leftmargini}{0pt} - \vspace{1cm} - \begin{figure}[ht]\centering - \includegraphics[width=0.5\textwidth]{images/Spiraling.png} - \end{figure} - \end{column} - \begin{column}{0.6\textwidth}\setlength{\leftmargini}{0pt} - \begin{itemize} - \setlength\itemsep{3.5em} - \item State-of-practice: Constant RSS force - \item Negative effects: kinking, rippling and spiraling - \item Process consequences of negative effects: reduced ROP and accuracy - \end{itemize} - \end{column} - \end{columns} -\end{frame} - -\begin{frame}{Research goal} - \textbf{\Large Develop a control strategy for a 3D directional drilling system, that allows to drill boreholes with complex geometries.} -\end{frame} - -\begin{frame}{Previous works} - \begin{itemize}\setlength\itemsep{2.5em} - \item PD model of 3D directional drilling systems $[$\cite{Perneder2013}$]$ - \item Model-based decoupled control of a 3D directional drilling system $[$\cite{Monsieurs2015}$]$ - \begin{itemize}\setlength\itemsep{1em} - \item State-feedback controller - \item Relies on availability of measurements of the states (not possible) - \item Not "a priori" robustly stable - \end{itemize} - \end{itemize} -\end{frame} - -\begin{frame}{Subgoals} - \begin{itemize}\setlength\itemsep{2.5em} - \item \textbf{Control strategy that relies only in local measurements (observer design)} - \item \textbf{Robustness against parametric uncertainty} - \end{itemize} -\end{frame} - -\section{Mathematical model} - - -\begin{frame}\frametitle{Geometric description} - \begin{columns}[T] - \hspace{1cm}\begin{column}{0.5\textwidth} - \vspace{-20pt} - \begin{itemize} - \item <1|only@1> [] \vspace{3pt}\begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/geodesca.pdf} - \end{figure} - \item <2|only@2> [] \vspace{3pt}\begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/geodescb.pdf} - \end{figure} - \end{itemize} - \end{column} - \begin{column}{0.5\textwidth} - \begin{itemize} - \item <1|only@1> [] \begin{figure}\centering - \includegraphics[width=0.9\textwidth]{images/geodesc1a.pdf} - \end{figure} - $\Theta (L)$ and $\Phi (L)$: Describe borehole evolution - \item <2|only@2> [] \begin{figure}\centering - \includegraphics[width=0.9\textwidth]{images/geodesc2b.pdf} - \end{figure} - Only measurements of BHA - \end{itemize} - \end{column} - \end{columns} -\end{frame} - -\begin{frame}{RSS actuator} - \begin{columns}[T] - \begin{column}{0.5\textwidth} - \hspace{2cm} \includegraphics[width=0.5\textwidth]{images/RSSForce0.pdf} - \end{column} - \begin{column}{0.5\textwidth} - \includegraphics[width=0.9\textwidth]{images/RSSForce.pdf} - \end{column} - \end{columns} -\end{frame} - -\begin{frame}{Elements of the model} - \begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/modelinteraction.pdf} - \end{figure}\vspace{-20pt} - \begin{itemize}\setlength\itemsep{0.5em} - \item Nonlinear delay differential equations (Fit BHA in already drilled borehole) - \item Function of length $\xi$ - \end{itemize} -\end{frame} - - -\begin{frame}{System parameters} - \begin{columns}[T] - \begin{column}{0.45\textwidth} - \begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/Drawing_bitwalk.pdf} - \end{figure} - \begin{figure}[ht]\centering - \vspace{-1cm} \includegraphics[width=0.8\textwidth]{images/Overgauge.pdf} - \end{figure} - \end{column} - \begin{column}{0.5\textwidth}\setlength{\leftmargini}{0pt} - \vspace{1cm}\begin{itemize}\setlength\itemsep{2.5em} - \item Constant - \begin{itemize}\setlength\itemsep{1em} - \item Angular steering resistance $\eta$ - \item Lateral steering resistance $\chi$ - \end{itemize} - \item Uncertain - \begin{itemize} \setlength\itemsep{1em} - \item Active weight on bit $\Pi$ (Hookload, drag forces, bit sharpness). - \item Bit walk angle $\varpi$ (bit orientation, overgauging). - \end{itemize} - \end{itemize} - \end{column} - \end{columns} -\end{frame} - -\begin{frame}{Borehole evolution equations} - \begin{align*} - \Theta'(\xi) &= f_\Theta (\Theta,\Phi,\Gamma_2,\Gamma_3,\varpi,\Pi) \\ - \Phi'(\xi) &= f_\Phi (\Theta,\Phi,\Gamma_2,\Gamma_3,\varpi,\Pi) - \end{align*} - Non-neutral vs neutral bit walk tendency: - \begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/ModelComparison.pdf} - \end{figure} - \begin{columns}[T] - \begin{column}{0.47\textwidth}\centering - \LARGE$\varpi \ne 0$ - \end{column} - \begin{column}{0.47\textwidth}\centering - \LARGE$\varpi = 0$ - \end{column} - \end{columns} -\end{frame} - - -\begin{frame}{Neutral bit walk evolution equations}\setlength{\leftmargini}{0pt} - - - \begin{itemize} - \item <1|only@1> [] \vspace{-1.0cm} Two stabilizers case: \footnotesize - \begin{equation} - \begin{aligned} - \chi \Pi \Theta ' = &- \mathcal{M}_b(\Theta-\langle\Theta \rangle_1)-\frac{{{\mathcal{M}_b}{\mathcal{F}_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\Theta} \rangle_1} - {\langle{\Theta} \rangle_{2}}} \right) - \\ - &+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\Theta} - {{\Theta}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\Theta}} - {{\Theta}_1}}} - \frac{{{{\Theta}_1} - {{\Theta}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &- \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + (\eta \Pi - {\mathcal{F}_b}){\mathcal{M}_r}}}{{\eta \Pi }}{\Gamma _{\Theta}} - \frac{\chi }{\eta }{\mathcal{F}_r}\Gamma_{\!\Theta}' + W - \\ - \chi \Pi \Phi ' = &- {\mathcal{M}_b}\left( {\Phi - {\langle{\Phi} \rangle_1}} \right)-\frac{{{\mathcal{M}_b}{\mathcal{F}_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\Phi} \rangle_1} - {\langle{\Phi} \rangle_{2}}} \right) - \\ - &+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\Phi} - {{\Phi}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\Phi}} - {{\Phi}_1}}} - \frac{{{{\Phi}_1} - {{\Phi}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &+ \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta }}{{{{\left( {\sin \Theta } \right)}^2}}} - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi \sin \Theta }}} \right){\Gamma _{\Phi}} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}}}{{\sin \Theta }}\Gamma_{\!\Phi}' - \end{aligned} - \nonumber - \end{equation} - % - % - % - \item <2|only@2> [] \vspace{-0.4cm} Two stabilizers case: \footnotesize - \begin{equation} - \begin{aligned} - \color{gray}\chi \Pi \Theta ' = &\color{gray}- \color{black}\mathcal{M}_b\color{gray}(\Theta-\langle\Theta \rangle_1)-\frac{{{\color{black}\mathcal{M}_b\color{gray}}{\color{black}\mathcal{F}_1\color{gray}} + {\color{black}\mathcal{M}_1\color{gray}}\left( {\eta \Pi - {\color{black}\mathcal{F}_b\color{gray}}} \right)}}{{\eta \Pi }}\left( {{\langle{\Theta} \rangle_1} - {\langle{\Theta} \rangle_{2}}} \right) - \\ - &\color{gray}+\frac{\chi }{\eta }{\color{black}\mathcal{F}_b\color{gray}}\left( {{\Theta} - {{\Theta}_1}} \right) - - \frac{\chi }{\eta }{\color{black}\mathcal{F}_1\color{gray}}\left( {{{{{\Theta}} - {{\Theta}_1}}} - \frac{{{{\Theta}_1} - {{\Theta}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &\color{gray}- \frac{{{\color{black}\mathcal{M}_b\color{gray}}{\color{black}\mathcal{F}_r\color{gray}} + (\eta \Pi - {\color{black}\mathcal{F}_b\color{gray}}){\color{black}\mathcal{M}_r\color{gray}}}}{{\eta \Pi }}{\Gamma _{\Theta}} - \frac{\chi }{\eta }{\color{black}\mathcal{F}_r\color{gray}}\Gamma_{\!\Theta}' + W - \\ - \color{gray}\chi \Pi \Phi ' = &\color{gray}- {\color{black}\mathcal{M}_b\color{gray}}\left( {\Phi - {\langle{\Phi} \rangle_1}} \right)-\frac{{{\color{black}\mathcal{M}_b\color{gray}}{\color{black}\mathcal{F}_1\color{gray}} + {\color{black}\mathcal{M}_1\color{gray}}\left( {\eta \Pi - {\color{black}\mathcal{F}_b\color{gray}}} \right)}}{{\eta \Pi }}\left( {{\langle{\Phi} \rangle_1} - {\langle{\Phi} \rangle_{2}}} \right) - \\ - &\color{gray}+\frac{\chi }{\eta }{\color{black}\mathcal{F}_b\color{gray}}\left( {{\Phi} - {{\Phi}_1}} \right) - - \frac{\chi }{\eta }{\color{black}\mathcal{F}_1\color{gray}}\left( {{{{{\Phi}} - {{\Phi}_1}}} - \frac{{{{\Phi}_1} - {{\Phi}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &\color{gray}+ \left( {\frac{\chi }{\eta }\frac{{{\color{black}\mathcal{F}_r\color{gray}}\Theta '\cos \Theta }}{{{{\left( {\sin \Theta } \right)}^2}}} - \frac{{{\color{black}\mathcal{M}_b\color{gray}}{\color{black}\mathcal{F}_r\color{gray}} + {\color{black}\mathcal{M}_r\color{gray}}\left( {\eta \Pi - {\color{black}\mathcal{F}_b\color{gray}}} \right)}}{{\eta \Pi \sin \Theta }}} \right){\Gamma _{\Phi}} - \frac{\chi }{\eta }\frac{{{\color{black}\mathcal{F}_r\color{gray}}}}{{\sin \Theta }}\Gamma_{\!\Phi}' - \end{aligned} - \nonumber - \end{equation} - \\ - \begin{itemize}\setlength\itemsep{0.5em} - \item [] \normalsize \textbf{BHA configuration constant coefficients} - \end{itemize} - % - % - % - \item <3|only@3> [] \vspace{0.3cm} Two stabilizers case: \footnotesize - \begin{equation} - \begin{aligned} - \color{gray}\chi \Pi \Theta ' = &\color{gray}- \mathcal{M}_b(\Theta-\color{black}\langle\Theta \rangle_1\color{gray})-\frac{{{\mathcal{M}_b}{\mathcal{F}_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\color{black}\langle\Theta \rangle_1\color{gray}} - {\color{black}\langle\Theta \rangle_2\color{gray}}} \right) - \\ - &\color{gray}+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\Theta} - {{\Theta}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\Theta}} - {{\Theta}_1}}} - \frac{{{{\Theta}_1} - {{\Theta}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &\color{gray}- \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + (\eta \Pi - {\mathcal{F}_b}){\mathcal{M}_r}}}{{\eta \Pi }}{\Gamma _{\Theta}} - \frac{\chi }{\eta }{\mathcal{F}_r}\Gamma_{\!\Theta}' + W - \\ - \color{gray}\chi \Pi \Phi ' = &\color{gray}- {\mathcal{M}_b}\left( {\Phi - {\color{black}\langle{\Phi} \rangle_1\color{gray}}} \right)-\frac{{{\mathcal{M}_b}{\mathcal{F}_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\color{black}\langle{\Phi} \rangle_1\color{gray}} - {\color{black}\langle{\Phi} \rangle_2\color{gray}}} \right) - \\ - &\color{gray}+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\Phi} - {{\Phi}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\Phi}} - {{\Phi}_1}}} - \frac{{{{\Phi}_1} - {{\Phi}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &\color{gray}+ \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta }}{{{{\left( {\sin \Theta } \right)}^2}}} - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi \sin \Theta }}} \right){\Gamma _{\Phi}} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}}}{{\sin \Theta }}\Gamma_{\!\Phi}' - \end{aligned} - \nonumber - \end{equation}\\ - Averages (Distributed delays): - \begin{align*} - {{\rm{\langle\Theta\rangle }}_i}:= \frac{1}{\varkappa_i}\int \limits_{\xi_{i}}^{\xi_{i-1}} \Theta(\sigma)d\sigma, \qquad - {{\rm{\langle\Phi\rangle }}_i}:=\frac{1}{\varkappa_i}\int \limits_{\xi_{i}}^{\xi_{i-1}} \Phi(\sigma)d\sigma - \label{eq:definitionaveragestates} \\ - \varkappa_i: \text{Length between stabilizers}, \qquad \text{ for } i = 1,2 - \end{align*} - % - % - % - \item <4|only@4> [] Two stabilizers case: \footnotesize - \begin{equation} - \begin{aligned} - \color{gray}\chi \Pi \Theta ' = &\color{gray}- \mathcal{M}_b(\Theta-\langle\Theta \rangle_1)-\frac{{{\mathcal{M}_b}{\mathcal{F}_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\Theta} \rangle_1} - {\langle{\Theta} \rangle_{2}}} \right) - \\ - &\color{gray}+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\Theta} - {\color{black}{\Theta}_1 \color{gray}}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\Theta}} - {\color{black}{\Theta}_1\color{gray}}}} - \frac{{{\color{black}{\Theta}_1\color{gray}} - \color{black}{{\Theta}_{2}\color{gray}}}}{{{\varkappa _{2}}}}} \right) - \\ - &\color{gray}- \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + (\eta \Pi - {\mathcal{F}_b}){\mathcal{M}_r}}}{{\eta \Pi }}{\Gamma _{\Theta}} - \frac{\chi }{\eta }{\mathcal{F}_r}\Gamma_{\!\Theta}' + W - \\ - \color{gray}\chi \Pi \Phi ' = &\color{gray}- {\mathcal{M}_b}\left( {\Phi - {\langle{\Phi} \rangle_1}} \right)-\frac{{{\mathcal{M}_b}{\mathcal{F}_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\Phi} \rangle_1} - {\langle{\Phi} \rangle_{2}}} \right) - \\ - &\color{gray}+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\Phi} - {\color{black}{\Phi}_1\color{gray}}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\Phi}} - {\color{black}{\Phi}_1\color{gray}}}} - \frac{{{\color{black}{\Phi}_1\color{black}} - {\color{black}{\Phi}_{2}\color{gray}}}}{{{\varkappa _{2}}}}} \right) - \\ - &\color{gray}+ \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta }}{{{{\left( {\sin \Theta } \right)}^2}}} - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi \sin \Theta }}} \right){\Gamma _{\Phi}} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}}}{{\sin \Theta }}\Gamma_{\!\Phi}' - \end{aligned} - \nonumber - \end{equation}\\ - \textbf{Borehole orientation variables at the first and second stabilizers (point-wise delays)} - % - % - % - \item <5|only@5> [] Two stabilizers case: \footnotesize - \begin{equation} - \begin{aligned} - \color{gray}\chi \Pi \Theta ' = &\color{gray}- \mathcal{M}_b(\Theta-\langle\Theta \rangle_1)-\frac{{{\mathcal{M}_b}{F_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\Theta} \rangle_1} - {\langle{\Theta} \rangle_{2}}} \right) - \\ - &\color{gray}+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\Theta} - {{\Theta}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\Theta}} - {{\Theta}_1}}} - \frac{{{{\Theta}_1} - {{\Theta}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &\color{gray}- \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + (\eta \Pi - {F_b}){\mathcal{M}_r}}}{{\eta \Pi }}{\Gamma _{\Theta}} - \frac{\chi }{\eta }{\mathcal{F}_r}\Gamma_{\!\Theta}' + W - \\ - \color{gray}\chi \Pi \Phi ' = &\color{gray}- {\mathcal{M}_b}\left( {\Phi - {\langle{\Phi} \rangle_1}} \right)-\frac{{{\mathcal{M}_b}{F_1} + {\mathcal{M}_1}\left( {\eta \Pi - {F_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\Phi} \rangle_1} - {\langle{\Phi} \rangle_{2}}} \right) - \\ - &\color{gray}+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\Phi} - {{\Phi}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\Phi}} - {{\Phi}_1}}} - \frac{{{{\Phi}_1} - {{\Phi}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &+ \boxed{ \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta }}{{{{\left( {\sin \Theta } \right)}^2}}} - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi \sin \Theta }}} \right){\Gamma _{\Phi}} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}}}{{\sin \Theta }}\Gamma_{\!\Phi}'} - \\ - \\ - \end{aligned} - \nonumber - \end{equation} - \normalsize \centering \textbf{ \centering Coupling terms} - \end{itemize} -\end{frame} - -\begin{frame}{State definition} To have a set of first order DDE's with point-wise delays: - \small \begin{equation*} - x_\Theta = \begin{bmatrix} - \Theta \\ - \langle \Theta \rangle_1\\ - \langle\Theta \rangle_2 - \end{bmatrix} \qquad - x_\Phi = \begin{bmatrix} - \Phi \\ - \langle \Phi \rangle_1\\ - \langle\Phi \rangle_2 - \end{bmatrix} - \end{equation*} - Derivatives of the average states: - \begin{equation*} - {{\rm{\langle\Theta\rangle }}_i}'= \frac{1}{\varkappa_i}\left(\Theta_{i-1}-\Theta_{i}\right) \qquad - {{\rm{\langle\Phi\rangle }}_i}'=\frac{1}{\varkappa_i}\left(\Phi_{i-1}-\Phi_{i}\right) - \label{eq:derivativeaverage} - \end{equation*} - State-space description: - \small \begin{align*} - \begin{bmatrix} - x_\Theta' \\ - x_\Phi' - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 \\ - 0 & A_0 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi) \\ - x_\Phi(\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 \\ - 0 & A_1 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_1) \\ - x_\Phi(\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 \\ - 0 & A_2 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_2) \\ - x_\Phi(\xi_2) - \end{bmatrix} \nonumber\\ - +& - \begin{bmatrix} - B_{0\Theta} & 0\\ - 0 & \boxed{B_{0\Phi}} - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta \\ - \Gamma_\Phi - \end{bmatrix} + - \begin{bmatrix} - B_{1\Theta} & 0\\ - 0 & \boxed{B_{1\Phi}} - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta' \\ - \Gamma_\Phi' - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 - \end{bmatrix} - \end{align*} - \begin{equation*} - \xi_1 = \xi - \varkappa_1, \qquad \xi_2 = \xi - \varkappa_1 - \varkappa_2 - \end{equation*} -\end{frame} - -\begin{frame}{Output equations} -\begin{columns}[T] - \hspace{1cm}\begin{column}{0.45\textwidth} - \vspace{1.5cm} \begin{figure}[ht]\centering - \includegraphics[width=1.1\textwidth]{images/Sensors.pdf} - \end{figure} - \end{column} - \begin{column}{0.45\textwidth} - \begin{itemize}\setlength\itemsep{1.5em} - \item Only measurements of the BHA orientation - \item Equation for sensor location: - \begin{align*} - y_{\theta}&=[\theta_0(\xi, s_{m,1}),\; \theta_2(\xi, s_{m,2})]^T \\ - y_{\phi}&=[\phi_0(\xi, s_{m,1}),\; \phi_2(\xi, s_{m,2})]^T - \end{align*} - \item Output explicit expressions: - \begin{align*} - y_\Theta &= C_\Theta x_\Theta + D_\Theta \Gamma_\Theta + W_y \\ - y_\Phi &= C_\Phi x_\Phi + D_\Phi \frac{\Gamma_\Phi}{\sin \Theta} - \end{align*} - \end{itemize} - \end{column} -\end{columns} -\end{frame} - -\section{Controller design} - -\begin{frame}{Control objectives} - \begin{itemize}\setlength\itemsep{1.5em} - \item Track a desired reference trajectory corresponding to a complex borehole geometry - \item The response of the system should have favorable transient behavior (avoid kinking, rippling and spiraling) - \end{itemize} -\end{frame} - - - -\begin{frame}{Plant definition}\setlength{\leftmargini}{0pt} -\begin{itemize} - \item <1|only@1> [] \begin{figure}[ht]\centering - \includegraphics[width=0.7\textwidth]{images/Plant.pdf} - \end{figure} - \item <2|only@2> [] \begin{figure}[ht]\centering - \includegraphics[width=0.7\textwidth]{images/Plant2.pdf} - \end{figure} - \item <3|only@3> [] \begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/ControlStrategy0.pdf} - \end{figure} -\end{itemize} -\end{frame} - -\begin{frame}{Control structure} -Input decoupling transformation - -\begin{itemize} - - \item <1|only@1> [] \begin{figure} - \includegraphics[width=1\textwidth]{images/ControlStrategy1.pdf} - \end{figure} - \item <2|only@2> [] - \begin{itemize} - \item Coupling terms: - \footnotesize\begin{equation*} - \color{gray} \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\color{black}\mathlarger{\mathlarger{\boldsymbol{\Theta}}}\color{gray} '\cos \color{black}\mathlarger{\mathlarger{\boldsymbol{\Theta}}}\color{gray} }}{{{{\left( {\sin \color{black}\mathlarger{\mathlarger{\boldsymbol{\Theta}}}\color{gray} } \right)}^2}}} - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi \sin \color{black}\mathlarger{\mathlarger{\boldsymbol{\Theta}}}\color{gray} }}} \right){\color{blue}\mathlarger{\mathlarger{\boldsymbol{\Gamma _{\Phi}}}}}\color{gray} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}}}{{\sin \color{black}\mathlarger{\mathlarger{\boldsymbol{\Theta}}}\color{gray} }}\color{blue}\mathlarger{\mathlarger{\boldsymbol{\Gamma_{\Phi}'}}}\color{gray} - \end{equation*} - - \item \small Fully decouples system$[$\cite{Monsieurs2015}$]$: - \begin{equation*} - \begin{bmatrix} - \Gamma_{\!\Theta}^* \\ \Gamma_{\!\Phi}^* - \end{bmatrix} := \begin{bmatrix} 1 & 0 \\ 0 & \frac{1}{\sin \Theta}\end{bmatrix} \begin{bmatrix}\Gamma_{\!\Theta} \\ \Gamma_{\!\Phi} \end{bmatrix}, \qquad \text{for } \Theta\in (0,\; \pi). - \nonumber - \end{equation*} - \item Problem: no access to $\Theta$ - \begin{equation*} - \color{gray}\begin{bmatrix} - \Gamma_{\!\Theta}^* \\ \Gamma_{\!\Phi}^* - \end{bmatrix} := \begin{bmatrix} 1 & 0 \\ 0 & \frac{1}{\sin \color{black}\mathlarger{\mathlarger{\boldsymbol{\check{\Theta}}}}}\color{gray}\end{bmatrix} \begin{bmatrix}\Gamma_{\!\Theta} \\ \Gamma_{\!\Phi} \end{bmatrix}, \qquad \text{for } \check{\Theta}\in (0,\; \pi). - \nonumber - \end{equation*} - \item Decouples system providing proper observer - \end{itemize} -\end{itemize} -\end{frame} - -\begin{frame}{Control structure} -Input filter -\begin{itemize} -\item <1|only@1> [] \begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/ControlStrategy2.pdf} -\end{figure} -\item <2|only@2> []\begin{itemize} - \item Terms with derivative of the input - \begin{equation*} - \begin{bmatrix} - B_{0\Theta} & 0\\ - 0 & \boxed{B_{0\Phi}(\Theta,\Theta')} - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^* \\ - \Gamma_\Phi^* - \end{bmatrix} + - \begin{bmatrix} - B_{1\Theta} & 0\\ - 0 & \boxed{B_{1\Phi}(\Theta,\Theta')} - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^{*'} \\ - \Gamma_\Phi^{*'} - \end{bmatrix} -\end{equation*} -\item New control input for $i = \Theta, \Phi$ -\begin{equation*} -B u_i = B_{0i} \Gamma_i^* + B_{1i} \Gamma_i^{*'}, \qquad B=\begin{bmatrix} -1 & 0 & 0 -\end{bmatrix} -\end{equation*} -\item Solve ODE -\begin{equation*} -\Gamma_i^{*'} = B_{1i}^+ B u_i - B_{1i}^+ B_{0i} \Gamma_i^* -\end{equation*} -\item Problem $\Theta \ne \check{\Theta}$, include $\Gamma_i$ in states -\begin{equation*} - \begin{bmatrix} - x_\Theta(\xi) & - \Gamma_\Theta^{*}(\xi) & - x_\Phi(\xi) & - \Gamma_\Phi^{*} (\xi) - \end{bmatrix}^T -\end{equation*} -\end{itemize} - -\end{itemize} -\end{frame} - -\begin{frame}{Control structure} -Feedback + feedforward controller -\begin{itemize} -\item <1|only@1> [] -\begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/ControlStrategy3.pdf} -\end{figure} -\item <2|only@2> [] -\begin{itemize} - \item Control input: - \begin{equation*} - u_i = v_i + u_{ri}, \text{ for } i = \Theta,\Phi. - \end{equation*} - \item Feedforward: - \begin{equation*} - u_{ri} = B^T (x'_{ri}(\xi) - A_0 x_{ri}(\xi) - A_1 x_{ri}(\xi_1) - A_2 x_{ri}(\xi_2)). - \end{equation*} - \item Feedback (with $e_i := x_{ri} - x_i$): - \begin{align*} - z_{1i}' &= \zeta \begin{bmatrix} - k_{1i} & 0 & 0 - \end{bmatrix}e_i \\ - z_{2i}' &= -\gamma z_{2i} + \gamma (z_{1i} + K_i e_i)\\ - v_i &= z_{2i}. - \end{align*} - \item Integral ($\zeta$) and low-pass filter ($\gamma$) gains fixed -\end{itemize} -\end{itemize} -\end{frame} - - -\begin{frame}{Control structure} -Observer -\begin{itemize}\setlength{\leftmargini}{0pt} -\item <1|only@1> [] -\begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/ControlStrategy4.pdf} -\end{figure} -\item <2|only@2> [] -\begin{itemize}\setlength\itemsep{3em} - \item Only available measurements of the BHA orientation - \item Model-based observer with output-injection part - \item Gravity-related terms in the output rejected using integral action: - \begin{equation*} - q_i' = \zeta[l_1,l_2](y_i - \check{y}_i), \qquad \text{ for } i=\Theta,\Phi. - \end{equation*} -\end{itemize} -\item <3|only@3> [] -\footnotesize\begin{align*} - \begin{bmatrix} - \check{x}_\Theta' \\ - q_\Theta' \\ - \check{x}_\Phi'\\ - q_\Phi' - \end{bmatrix} &= - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_0 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi) \\ - q_\Theta(\xi) \\ - \check{x}_\Phi(\xi) \\ - q_\Phi (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi_1) \\ - q_\Theta(\xi_1) \\ - \check{x}_\Phi(\xi_1) \\ - q_\Phi(\xi_1) \\ - \end{bmatrix} \nonumber\\ - \nonumber\\ - &+ - \begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi_2) \\ - q_\Theta(\xi_2) \\ - \check{x}_\Phi(\xi_2) \\ - q_\Phi(\xi_2) - \end{bmatrix} + - \begin{bmatrix} - L_\Theta(y_{\Theta} - \check{y}_{\Theta})\\ - \zeta[l_{1\Theta},l_{2\Theta}](y_\Theta - \check{y}_\Theta) \\ - L_{\Phi}(y_{\Phi} - \check{y}_{\Phi}) \\ - \zeta[l_{1\Phi},l_{2{\Phi}}](y_\Phi - \check{y}_\Phi) - \end{bmatrix} \\ - \\ - &+ - \begin{bmatrix} - Bq_\Theta \\ - 0 \\ - Bq_\Phi \\ - 0 - \end{bmatrix} + - \begin{bmatrix} - B(u_{r\Theta} + v_\Theta)\\ - 0 \\ - B(u_{r\Phi} + v_\Phi) \\ - 0 - \end{bmatrix},\label{eq:Observer} -\end{align*} -\end{itemize} -\end{frame} - - -\begin{frame}{Control approach} - \begin{itemize}\setlength\itemsep{1.5em} - \item Tune $K_i$ and $L_i$, such that the reference trajectory is the solution of the closed-loop dynamics - \item Tracking error dynamics and observer error dynamics: - \begin{equation*} - e_i := x_{ri} - x{i}, \qquad \delta_i := x_i - \hat{x}_i, \text{ for } i=\Theta,\Phi. - \end{equation*} - \item \textbf{Linearize system around}: - \begin{equation*} - e_i = 0, \qquad \delta_i = 0. - \end{equation*} - \item Linearized closed-loop system: \textbf{$\xi$-dependent} - \end{itemize} -\end{frame} - - - -\begin{frame}{Controller synthesis} -\vspace{-1cm} -Analysis of the structure of closed-loop matrices: -\\ -\vspace{0.5cm}\begin{itemize}\setlength\itemsep{3.5em} - \item $K_i$ and $L_i$ separation principle for both $\Theta$ and $\Phi$ - \item $\xi$-dependent terms present in coupling terms - \item Isolated dynamics of $e_i$ and $\delta_i$ are $\xi$-independent (bounded) -\end{itemize} -\end{frame} - -\begin{frame}{Controller synthesis} -\begin{itemize}\setlength\itemsep{1.5em} - \item Synthesize $K_i$ and $L_i$ for isolated systems - \item Favorable transients, with proper order of convergence - \item Due to infinite number of poles in delay systems, optimize location of right-most pole over $K_i$ and $L_i$ -\end{itemize} - \begin{figure}[ht]\centering - \includegraphics[width=0.7\textwidth]{images/ISS.pdf} - \end{figure} -\end{frame} - - -\section{Simulation results} - -\begin{frame}{Reference trajectory} -\begin{figure}[t] - \centering - \begin{minipage}[b]{0.5\textwidth} - \centering - \includegraphics[height=2.3in]{ReferenceTrajectoryThetaPhi.pdf} - \label{fig:ReferenceTrajectoryThetaPhi} - \end{minipage} - \begin{minipage}[b]{0.45\textwidth} - \centering - \includegraphics[height=2.3in]{ReferenceTrajectory.pdf} - \label{fig:ReferenceTrajectoryV} - \end{minipage} -\end{figure} -\end{frame} - -\begin{frame}{"Time" domain simulation} -\begin{figure}[H] - \includegraphics[height=3.25in]{ErrorsNeutral.pdf} -\end{figure} -\end{frame} - - -\begin{frame}{Robust stability analysis, uncertainty in $\Pi$} - Closed-loop right-most pole of the system for $\Pi \in [0.5\bar{\Pi},1.5\bar{\Pi}]$ - \begin{columns}[T] - \hspace{0.5cm}\begin{column}{0.45\textwidth} - \begin{figure}[ht] - \includegraphics[width=1\textwidth]{images/RobustTestNominalInclination.pdf} - \end{figure} - \centering\LARGE Inclination - \end{column} - \hspace{0.5cm}\begin{column}{0.45\textwidth} - \begin{figure}[ht] - \includegraphics[width=1\textwidth]{images/RobustTestNominalAzimuth.pdf} - \end{figure} - \centering \LARGE Azimuth - \end{column} - \end{columns} -\end{frame} - - -\begin{frame}{"Time" domain simulation, $\Pi = 0.5 \bar{\Pi}$} - \begin{figure}[ht]\centering - \includegraphics[width=0.7\textwidth]{images/ErrorsNeutralNominal.pdf} - \end{figure} -\end{frame} - -\begin{frame}{Robust controller synthesis} - \begin{itemize}\setlength\itemsep{3.5em} - \item Optimize location of right-most pole of closed-loop system for a grid of values of $\Pi \in [0.5 \bar{\Pi}, 1.5 \bar{\Pi}]$ - \item Interval of $\Pi$ - \begin{equation} - \delta \Pi_i = -\delta \Pi_{max} + \frac{i - 1}{m - 1} 2\delta \Pi_{max}, \qquad \text{ for } i\in \{1,2,...,m\}. \nonumber - \end{equation} (this case m = 7) - \begin{equation*} - \Pi = \bar{\Pi} + \delta \Pi - \end{equation*} - \end{itemize} -\end{frame} - -\begin{frame}{Robustness comparison} - Closed-loop right-most pole of the system for $\Pi \in [0.5\bar{\Pi},1.5\bar{\Pi}]$ - \begin{columns}[T] - \hspace{0.5cm}\begin{column}{0.44\textwidth} - \begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/RobustTestComparisonInclination.pdf} - \end{figure} - \centering\LARGE Inclination - \end{column} - \hspace{0.5cm}\begin{column}{0.44\textwidth} - \begin{figure}[ht]\centering - \includegraphics[width=1\textwidth]{images/RobustTestComparisonAzimuth.pdf} - \end{figure} - \centering \LARGE Azimuth - \end{column} - \end{columns} -\end{frame} - - -\begin{frame}{"Time" domain simulation for $\Pi = 0.5\bar{\Pi}$} - \begin{figure}[ht]\centering - \includegraphics[width=0.7\textwidth]{images/ErrorsNeutralRobust.pdf} - \end{figure} -\end{frame} - -\section{Conclusions and recommendations} - - -\begin{frame}{Conclusions} - \begin{itemize}\setlength\itemsep{3em} - \item A controller that only relies on local measurements of the BHA orientation was developed by means of an observer - \item The strategy yields robustness against uncertainty in the active weight on bit $\Pi$ - \item Negative effects are greatly reduced even under extreme conditions of low weight on bit - \end{itemize} -\end{frame} - -\begin{frame}{Recommendations} - \begin{itemize}\setlength\itemsep{3em} - \item Extend current design to non-neutral bit walk tendency - \item Study alternative angle parameterization - \item Analyze use of the developed strategy for models with non-linear effects (bit-tilt saturation, non-ideal stabilizer contact) - \end{itemize} -\end{frame} - -\begin{frame} -Thank you for your attention. Any questions? -\end{frame} - - -\end{document} diff --git a/Presentation iit/vertical/presentation.toc b/Presentation iit/vertical/presentation.toc deleted file mode 100755 index bbe761d..0000000 --- a/Presentation iit/vertical/presentation.toc +++ /dev/null @@ -1,15 +0,0 @@ -\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax -\defcounter {refsection}{0}\relax -\beamer@endinputifotherversion {3.24pt} -\defcounter {refsection}{0}\relax -\select@language {english} -\defcounter {refsection}{0}\relax -\beamer@sectionintoc {1}{General description of directional drilling systems}{3}{0}{1} -\defcounter {refsection}{0}\relax -\beamer@sectionintoc {2}{Mathematical model}{10}{0}{2} -\defcounter {refsection}{0}\relax -\beamer@sectionintoc {3}{Controller design}{24}{0}{3} -\defcounter {refsection}{0}\relax -\beamer@sectionintoc {4}{Simulation results}{41}{0}{4} -\defcounter {refsection}{0}\relax -\beamer@sectionintoc {5}{Conclusions and recommendations}{49}{0}{5} diff --git a/Presentation iit/vertical/texput.log b/Presentation iit/vertical/texput.log deleted file mode 100755 index 79a590d..0000000 --- a/Presentation iit/vertical/texput.log +++ /dev/null @@ -1,21 +0,0 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2016.4.20) 6 JUL 2016 11:03 -entering extended mode - restricted \write18 enabled. - %&-line parsing enabled. -**beamerouterthemetud.tex - -! Emergency stop. -<*> beamerouterthemetud.tex - -*** (job aborted, file error in nonstop mode) - - -Here is how much of TeX's memory you used: - 3 strings out of 493304 - 120 string characters out of 6139871 - 51880 words of memory out of 5000000 - 3564 multiletter control sequences out of 15000+600000 - 3640 words of font info for 14 fonts, out of 8000000 for 9000 - 957 hyphenation exceptions out of 8191 - 0i,0n,0p,1b,6s stack positions out of 5000i,500n,10000p,200000b,80000s -! ==> Fatal error occurred, no output PDF file produced! diff --git a/Presentation iit/vertical/tud_white.pdf b/Presentation iit/vertical/tud_white.pdf deleted file mode 100755 index 535ccd4..0000000 Binary files a/Presentation iit/vertical/tud_white.pdf and /dev/null differ diff --git a/README.md b/README.md new file mode 100644 index 0000000..57498c2 --- /dev/null +++ b/README.md @@ -0,0 +1,5 @@ +# Personal + +Shi lo estoy cambiando shi + +Lo cambio de nuevo diff --git a/Research Proposal IIT/Research proposal/Assignment1.synctex.gz b/Research Proposal IIT/Research proposal/Assignment1.synctex.gz deleted file mode 100755 index d80722d..0000000 Binary files a/Research Proposal IIT/Research proposal/Assignment1.synctex.gz and /dev/null differ diff --git a/Research Proposal IIT/Research proposal/My Bib.bib b/Research Proposal IIT/Research proposal/My Bib.bib deleted file mode 100755 index 372f96f..0000000 --- a/Research Proposal IIT/Research proposal/My Bib.bib +++ /dev/null @@ -1,5 +0,0 @@ -@article{Ketelaar, -author = {Ketelaar, J G and Visser, L C and Stramigioli, S and Carloni, R}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/ICRA13{\_}b.pdf:pdf}, -title = {{Controller Design for a Bipedal Walking Robot using Variable Stiffness Actuators}} -} diff --git a/Research Proposal IIT/Research proposal/My Bib.log b/Research Proposal IIT/Research proposal/My Bib.log deleted file mode 100755 index e69de29..0000000 diff --git a/Research Proposal IIT/Research proposal/MyBib.bib b/Research Proposal IIT/Research proposal/MyBib.bib deleted file mode 100755 index e0add49..0000000 --- a/Research Proposal IIT/Research proposal/MyBib.bib +++ /dev/null @@ -1,206 +0,0 @@ -@article{Cohen, -author = {Cohen, Guy}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/BCOQ-book.pdf:pdf}, -title = {{Synchronization and Linearity}} -} -@article{Turek2011, -author = {Turek, Michal and Turek, Richard}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/849-736-1-PB.pdf:pdf}, -keywords = {coordination,crossroads,max-plus algebra,modeling,transfer stop}, -number = {1}, -pages = {67--75}, -title = {{SOLVING SELECTED TRAFFIC PROBLEMS IN THE MAX-PLUS ALGEBRA}}, -volume = {9}, -year = {2011} -} -@article{Goverde2007, -author = {Goverde, Rob M P}, -doi = {10.1016/j.trb.2006.02.003}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/1-s2.0-S0191261506000208-main.pdf:pdf}, -keywords = {max-plus algebra,railway timetable,stability,timed event graph}, -pages = {179--201}, -title = {{Railway timetable stability analysis using max-plus system theory}}, -volume = {41}, -year = {2007} -} -@article{Winkler, -author = {Winkler, Alexander and Havoutis, Ioannis and Bazeille, Stephane and Ortiz, Jesus and Focchi, Michele and Caldwell, Darwin and Semini, Claudio and Robotics, Advanced and Italiano, Istituto}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/winkler14icra.pdf:pdf}, -title = {{Path planning with force-based foothold adaptation and virtual model control for torque controlled quadruped robots}} -} -@article{Winklera, -author = {Winkler, Alexander W and Mastalli, Carlos and Focchi, Michele and Caldwell, Darwin G and Havoutis, Ioannis}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/winkler15icra.pdf:pdf}, -title = {{Planning and Execution of Dynamic Whole-Body Locomotion for a Hydraulic Quadruped on Challenging Terrain}} -} -@article{Semini2010, -author = {Semini, Claudio}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/Semini2010PhD{\_}HyQ{\_}Robot.pdf:pdf}, -number = {April}, -title = {{HyQ - Design and Development of a Hydraulically Actuated Quadruped Robot}}, -year = {2010} -} -@article{Schutter2008, -author = {Schutter, B De and Boom, T Van Den}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/08{\_}007.pdf:pdf}, -title = {{Max-plus algebra and max-plus linear discrete event systems : An introduction}}, -volume = {19}, -year = {2008} -} -@article{Lopes2009, -author = {Schutter, B De and Boom, A J J Van Den}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/09{\_}045.pdf:pdf}, -title = {{Switching max-plus models for legged locomotion}}, -volume = {19}, -year = {2009} -} -@article{Mastalli, -author = {Mastalli, Carlos and Havoutis, Ioannis and Winkler, Alexander W and Caldwell, Darwin G and Semini, Claudio and Robotics, Advanced and Italiano, Istituto}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/mastalli15tepra.pdf:pdf}, -number = {Section II}, -title = {{On-line and On-board Planning and Perception for Quadrupedal Locomotion}} -} -@inproceedings{Lopes2010, -author = {Lopes, G A D and Boom, T J J Van Den and Schutter, B De and Babuˇ, R}, -booktitle = {Proceedings of the 10th International Workshop on Discrete Event Systems (WODES 2010)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/10{\_}038.pdf:pdf}, -keywords = {discrete event systems,legged locomotion,max-plus algebra,motion,robotics}, -pages = {6 pp.}, -title = {{Modeling and control of legged locomotion via switching max-plus systems}}, -year = {2010} -} -@article{Ijspeert, -author = {Ijspeert, Auke Jan and Crespi, Alessandro and Ryczko, Dimitri and Cabelguen, Jean-marie}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/1416.full.pdf:pdf}, -title = {{From Swimming to Walking with a Spinal Cord Model}} -} -@article{Ijspeert2008, -author = {Ijspeert, Auke Jan}, -doi = {10.1016/j.neunet.2008.03.014}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/Ijspeert08NeuralNEtworks.pdf:pdf}, -keywords = {central pattern generators}, -pages = {642--653}, -title = {{Central pattern generators for locomotion control in animals and robots : A review}}, -volume = {21}, -year = {2008} -} -@article{Holmes2006, -author = {Holmes, Philip and Koditschek, Daniel E and Guckenheimer, John}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/fulltext.pdf:pdf}, -number = {2}, -pages = {207--304}, -title = {{The Dynamics of Legged Locomotion : Models, Analyses, and Challenges}}, -volume = {48}, -year = {2006} -} -@article{Focchi2013, -author = {Focchi, Michele and Barasuol, Victor and Havoutis, Ioannis and Semini, Claudio and Caldwell, Darwin G}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/clawar13focchi.pdf:pdf}, -keywords = {control,quadrupedal locomotion,reflex}, -pages = {1--8}, -title = {{Local Reflex Generation for Obstacle Negotiation in Quadrupedal Locomotion}}, -year = {2013} -} -@article{Ekeberg1993, -author = {Ekeberg, Orjan}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/art{\%}3A10.1007{\%}2FBF00199436.pdf:pdf}, -pages = {363--374}, -title = {{Bkalog .,ai cyt mett 9}}, -volume = {374}, -year = {1993} -} -@article{Crespi1985, -author = {Crespi, Alessandro and Lachat, Daisy and Pasquier, Ariane and Ijspeert, Auke Jan}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/crespi08b.pdf:pdf}, -title = {{Controlling swimming and crawling in a fish robot using a central pattern generator The fish robot BoxyBot}}, -year = {1985} -} -@article{Crespi2013, -author = {Crespi, Alessandro and Karakasiliotis, Konstantinos and Ijspeert, Auke Jan}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/06416074.pdf:pdf}, -number = {2}, -pages = {308--320}, -title = {{Salamandra Robotica II : An Amphibious Robot to Study Salamander-Like Swimming and Walking Gaits}}, -volume = {29}, -year = {2013} -} -@article{Carlos2015, -archivePrefix = {arXiv}, -arxivId = {arXiv:1509.02417v1}, -author = {Carlos, Author and Aura, G Arcia}, -eprint = {arXiv:1509.02417v1}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/1509.02417.pdf:pdf}, -keywords = {bio-inspiration,computational models,distributed motion control,neuron-based oscillators,robotics}, -number = {April}, -title = {{Central Pattern Generators for the control of robotic systems}}, -year = {2015} -} -@article{Barasuol, -author = {Barasuol, Victor and Focchi, Michele and Havoutis, Ioannis and Frigerio, Marco and Buchli, Jonas and Semini, Claudio and Caldwell, Darwin G}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/bazeille13tepra.pdf:pdf}, -title = {{Vision Enhanced Reactive Locomotion Control for Trotting on Rough Terrain}} -} -@article{Barasuola, -author = {Barasuol, Victor and Buchli, Jonas and Semini, Claudio and Frigerio, Marco and Pieri, Edson R De and Caldwell, Darwin G}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/barasuol13icra.pdf:pdf}, -title = {{A Reactive Controller Framework for Quadrupedal Locomotion on Challenging Terrain}} -} -@article{Barai, -author = {Barai, Ranjit Kumar and Nonami, Kenzo}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/47{\_}282.pdf:pdf}, -keywords = {gait,petri net,six-legged walking robot comet-iv,supervisory control}, -title = {{Petri Net Based Supervisory Control for Locomotion System of Six-Legged Walking Robot COMET-IV}} -} -@article{Wisse2004, -author = {Wisse, M and Schwab, A L and Helm, F C T Van Der}, -doi = {10.1017/S0263574704000475}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/WisSchHel04.pdf:pdf}, -keywords = {biped,passive,passive dynamic walking}, -pages = {681--688}, -title = {{Passive dynamic walking model with upper body}}, -volume = {22}, -year = {2004} -} -@article{Tsagarakis2013, -author = {Tsagarakis, N G and Morfey, S and Dallali, H and Caldwell, D G}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/06697155.pdf:pdf}, -isbn = {9781467363587}, -pages = {5512--5517}, -title = {{An Asymmetric Compliant Antagonistic Joint Design for High Performance Mobility}}, -year = {2013} -} -@article{Borovac2004, -author = {Borovac, Branislav}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/zmp-review.pdf:pdf}, -keywords = {biped locomotion,dynamically balanced gait,support,zero-moment point}, -number = {1}, -pages = {157--173}, -title = {{Zero-moment point — thirty five years of its life ´}}, -volume = {1}, -year = {2004} -} -@article{Aghili2005, -author = {Aghili, Farhad and Member, Associate}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/01512343.pdf:pdf}, -number = {5}, -pages = {834--849}, -title = {{A Unified Approach for Inverse and Direct Dynamics of Constrained Multibody Systems Based on Linear Projection Operator : Applications to Control and Simulation}}, -volume = {21}, -year = {2005} -} -@article{Geyer2006, -author = {Geyer, Hartmut and Seyfarth, Andre and Blickhan, Reinhard}, -doi = {10.1098/rspb.2006.3637}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/2861.full.pdf:pdf}, -keywords = {biomechanics,human gait,mass model,spring}, -number = {August}, -pages = {2861--2867}, -title = {{Compliant leg behaviour explains basic dynamics of walking and running}}, -year = {2006} -} -@article{Ketelaar, -author = {Ketelaar, J G and Visser, L C and Stramigioli, S and Carloni, R}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/ICRA13{\_}b.pdf:pdf}, -title = {{Controller Design for a Bipedal Walking Robot using Variable Stiffness Actuators}} -} - diff --git a/Research Proposal IIT/Research proposal/MyBib.log b/Research Proposal IIT/Research proposal/MyBib.log deleted file mode 100755 index e69de29..0000000 diff --git a/Research Proposal IIT/Research proposal/MyBib2.bib b/Research Proposal IIT/Research proposal/MyBib2.bib deleted file mode 100755 index 8af875d..0000000 --- a/Research Proposal IIT/Research proposal/MyBib2.bib +++ /dev/null @@ -1,256 +0,0 @@ -@article{Aghili2005, -abstract = {—This paper presents a unified approach for inverseand direct dynamics of constrained multibody systems that can serve as a basis for analysis, simulation, and control. The main advantage of the dynamics formulation is that it does not require the constraint equations to be linearly independent. Thus, a simulation may proceed even in the presence of redundant constraints or singular configurations, and a controller does not need to change its structure whenever the mechanical system changes its topology or number of degrees of freedom. A motion-control scheme is proposed based on a projected inverse-dynamics scheme which proves to be stable and minimizes the weighted Euclidean norm of the actuation force. The projection-based control scheme is further developed for constrained systems, e.g., parallel manipulators, which have some joints with no actuators (passive joints). This is complemented by the development of constraint force control. A condition on the inertia matrix resulting in a decoupled mechanical system is analytically derived that simplifies the implementation of the force control. Finally, numerical and experimental results obtained from dynamic simulation and control of constrained mechanical systems, based on the proposed inverse and direct dynamics formulations, are documented. Index}, -author = {Aghili, Farhad}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/01512343.pdf:pdf}, -journal = {IEEE Transactions on robotics}, -keywords = {Constrained multibody systems,constraint mo-,tion}, -number = {5}, -pages = {834--849}, -title = {{A Unified Approach for Inverse and Direct Dynamics of Constrained Multibody Systems Based on Linear Projection Operator : Applications to Control and Simulation}}, -volume = {21}, -year = {2005} -} -@inproceedings{Barai, -author = {Barai, Ranjit Kumar and Nonami, Kenzo}, -booktitle = {Proceedings of the Japan Joint Automatic Control Conference}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/47{\_}282.pdf:pdf}, -keywords = {gait,petri net,six-legged walking robot comet-iv,supervisory control}, -pages = {282--282}, -title = {{Petri Net Based Supervisory Control for Locomotion System of Six-Legged Walking Robot COMET-IV}}, -year = {2004} -} -@inproceedings{Barasuola, -author = {Barasuol, Victor and Buchli, Jonas and Semini, Claudio and Frigerio, Marco and Pieri, Edson R De and Caldwell, Darwin G}, -booktitle = {Conference on Robotics and Automation (ICRA), 2013 IEEE International}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/barasuol13icra.pdf:pdf}, -pages = {2554--2561}, -title = {{A Reactive Controller Framework for Quadrupedal Locomotion on Challenging Terrain}}, -year = {2013} -} -@inproceedings{Barasuol, -author = {Bazeille, St{\'{e}}phane and Barasuol, Victor and Focchi, Michele and Havoutis, Ioannis and Frigerio, Marco and Buchli, Jonas and Semini, Claudio and Caldwell, Darwin G}, -booktitle = {Conference on Technologies for Practical Robot Applications (TePRA), 2013 IEEE International}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/bazeille13tepra.pdf:pdf}, -pages = {1--6}, -title = {{Vision Enhanced Reactive Locomotion Control for Trotting on Rough Terrain}}, -year = {2013} -} -@article{Borovac2004, -author = {Miomir, Vukobratovi{\'{c}} and Borovac, Branislav}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/zmp-review.pdf:pdf}, -journal = {International Journal of Humanoid Robotics}, -keywords = {biped locomotion,dynamically balanced gait,support,zero-moment point}, -number = {1}, -pages = {157--173}, -title = {{Zero-moment point — thirty five years of its life ´}}, -volume = {1}, -year = {2004} -} -@techreport{Carlos2015, -archivePrefix = {arXiv}, -arxivId = {arXiv:1509.02417v1}, -author = {Garc{\'{i}}a-Saura, Carlos and Shanahan, Murray}, -eprint = {arXiv:1509.02417v1}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/1509.02417.pdf:pdf}, -keywords = {bio-inspiration,computational models,distributed motion control,neuron-based oscillators,robotics}, -number = {April}, -title = {{Central Pattern Generators for the control of robotic systems}}, -year = {2015} -} -@book{Cohen, -author = {Bacceli, Fran{\c{c}}ois and Cohen, Guy and {Jan Olsder}, Geert and Quadrat, Jean-Pierre}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/BCOQ-book.pdf:pdf}, -title = {{Synchronization and Linearity, An Algebra for Discrete Event Systems}}, -year = {2001} -} -@article{Crespi2013, -author = {Crespi, Alessandro and Karakasiliotis, Konstantinos and Guignard, Andr{\'{e}} and Ijspeert, Auke Jan}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/06416074.pdf:pdf}, -journal = {IEEE Transactions on robotics}, -number = {2}, -pages = {308--320}, -title = {{Salamandra Robotica II : An Amphibious Robot to Study Salamander-Like Swimming and Walking Gaits}}, -volume = {29}, -year = {2013} -} -@article{Crespi1985, -abstract = {Online trajectory generation for robots with multiple degrees of freedom is still a difficult and unsolved problem, in particular for non-steady state locomotion, that is, when the robot has to move in a complex environment with continuous variations of the speed, direction, and type of locomotor behavior. In this article we address the problem of controlling the non-steady state swimming and crawling of a novel fish robot. For this, we have designed a control architecture based on a central pattern generator (CPG) implemented as a system of coupled nonlinear oscillators. The CPG, like its biological counterpart, can produce coordinated patterns of rhythmic activity while being modulated by simple control parameters. To test our controller, we designed BoxyBot, a simple fish robot with three actuated fins capable of swimming in water and crawling on firm ground. Using the CPG model, the robot is capable of performing and switching between a variety of different locomotor behaviors such as swimming forwards, swimming backwards, turning, rolling, moving upwards/downwards, and crawling. These behaviors are triggered and modulated by sensory input provided by light, water, and touch sensors. Results are presented demonstrating the agility of the robot and interesting properties of a CPG-based control approach such as stability of the rhythmic patterns due to limit cycle behavior, and the production of smooth trajectories despite abrupt changes of control parameters. The robot is currently used in a temporary 20-month long exhibition at the EPFL. We present the hardware setup that was designed for the exhibition, and the type of interactions with the control system that allow visitors to influence the behavior of the robot. The exhibition is useful to test the robustness of the robot for long term use, and to demonstrate the suitability of the CPG-based approach for interactive control with a human in the loop. This article is an extended version of an article presented at BioRob2006 the first IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics.}, -author = {Crespi, Alessandro and Lachat, Daisy and Pasquier, Ariane and Ijspeert, Auke Jan}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/crespi08b.pdf:pdf}, -journal = {Autonomous robots}, -number = {1-2}, -pages = {3--13}, -title = {{Controlling swimming and crawling in a fish robot using a central pattern generator}}, -volume = {25}, -year = {2008} -} -@article{Ekeberg1993, -author = {Ekeberg, Orjan}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/art{\%}3A10.1007{\%}2FBF00199436.pdf:pdf}, -journal = {Biological Cybernetics}, -pages = {363--374}, -title = {{Bkalog .,ai cyt mett 9}}, -volume = {374}, -year = {1993} -} -@inproceedings{Focchi2013, -author = {Focchi, Michele and Barasuol, Victor and Havoutis, Ioannis and Semini, Claudio and Caldwell, Darwin G}, -booktitle = {16th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/clawar13focchi.pdf:pdf}, -keywords = {control,quadrupedal locomotion,reflex}, -pages = {443--450}, -title = {{Local Reflex Generation for Obstacle Negotiation in Quadrupedal Locomotion}}, -year = {2013} -} -@inproceedings{Geyer2006, -author = {Geyer, Hartmut and Seyfarth, Andre and Blickhan, Reinhard}, -booktitle = {Proceedings of the Royal Society B: Biological Sciences.}, -doi = {10.1098/rspb.2006.3637}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/2861.full.pdf:pdf}, -keywords = {biomechanics,human gait,mass model,spring}, -number = {August}, -pages = {2861--2867}, -title = {{Compliant leg behaviour explains basic dynamics of walking and running}}, -year = {2006} -} -@article{Goverde2007, -author = {Goverde, Rob M.P.}, -doi = {10.1016/j.trb.2006.02.003}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/1-s2.0-S0191261506000208-main.pdf:pdf}, -journal = {Transportation Research Part B: Methodological}, -keywords = {max-plus algebra,railway timetable,stability,timed event graph}, -number = {2}, -pages = {179--201}, -title = {{Railway timetable stability analysis using max-plus system theory}}, -volume = {41}, -year = {2007} -} -@article{Holmes2006, -author = {Holmes, Philip and Full, Robert J. and Koditschek, Daniel E and Guckenheimer, John}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/fulltext.pdf:pdf}, -journal = {SIAM review}, -number = {2}, -pages = {207--304}, -title = {{The Dynamics of Legged Locomotion : Models, Analyses, and Challenges}}, -volume = {48}, -year = {2006} -} -@article{Ijspeert2008, -author = {Ijspeert, Auke Jan}, -doi = {10.1016/j.neunet.2008.03.014}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/Ijspeert08NeuralNEtworks.pdf:pdf}, -journal = {Neural Networks}, -keywords = {central pattern generators}, -number = {4}, -pages = {642--653}, -title = {{Central pattern generators for locomotion control in animals and robots : A review}}, -volume = {21}, -year = {2008} -} -@article{Ijspeert, -author = {Ijspeert, Auke Jan and Crespi, Alessandro and Ryczko, Dimitri and Cabelguen, Jean-marie}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/1416.full.pdf:pdf}, -journal = {Science}, -number = {5817}, -pages = {1416--1420}, -title = {{From Swimming to Walking with a Spinal Cord Model}}, -volume = {315}, -year = {2007} -} -@inproceedings{Ketelaar, -author = {Ketelaar, J G and Visser, L C and Stramigioli, S and Carloni, R}, -booktitle = {2013 IEEE International Conference on Robotics and Automation (ICRA)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/ICRA13{\_}b.pdf:pdf}, -pages = {5650--5655}, -title = {{Controller Design for a Bipedal Walking Robot using Variable Stiffness Actuators}}, -year = {2013} -} -@inproceedings{Lopes2010, -author = {Lopes, G.A.D. and van den Boom, T.J.J. and {De Schutter}, B. and Babu{\v{s}}ka, R.}, -booktitle = {Proceedings of the 10th International Workshop on Discrete Event Systems (WODES 2010)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/10{\_}038.pdf:pdf}, -keywords = {discrete event systems,legged locomotion,max-plus algebra,motion,robotics}, -pages = {6 pp.}, -title = {{Modeling and control of legged locomotion via switching max-plus systems}}, -year = {2010} -} -@inproceedings{Mastalli, -author = {Mastalli, Carlos and Havoutis, Ioannis and Winkler, Alexander W and Caldwell, Darwin G and Semini, Claudio}, -booktitle = {2015 IEEE International Conference on Technologies for Practical Robot Applications (TePRA)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/mastalli15tepra.pdf:pdf}, -pages = {1--7}, -title = {{On-line and On-board Planning and Perception for Quadrupedal Locomotion}}, -year = {2015} -} -@inproceedings{Schutter2009, -author = {Lopes, G.A.D. and Babu{\v{s}}ka, R. and {De Schutter}, B. and van den Boom, T.J.J.}, -booktitle = {Proceedings of the 2009 IEEE Interna- tional Conference on Robotics and Biomimetics (ROBIO 2009)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/09{\_}045.pdf:pdf}, -pages = {221--226}, -title = {{Switching max-plus models for legged locomotion}}, -volume = {19}, -year = {2009} -} -@inproceedings{Schutter2008, -author = {{De Schutter}, B. and van den Boom, T.J.J.}, -booktitle = {Proceedings of the 9th International Workshop on Discrete Event Systems (WODES'08)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/08{\_}007.pdf:pdf}, -pages = {36--42}, -title = {{Max-plus algebra and max-plus linear discrete event systems : An introduction}}, -volume = {19}, -year = {2008} -} -@phdthesis{Semini2010, -author = {Semini, Claudio}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/Semini2010PhD{\_}HyQ{\_}Robot.pdf:pdf}, -number = {April}, -title = {{HyQ - Design and Development of a Hydraulically Actuated Quadruped Robot}}, -year = {2010} -} -@inproceedings{Tsagarakis2013, -author = {Tsagarakis, N G and Morfey, S and Dallali, H and Caldwell, D G}, -booktitle = {2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/06697155.pdf:pdf}, -isbn = {9781467363587}, -pages = {5512--5517}, -title = {{An Asymmetric Compliant Antagonistic Joint Design for High Performance Mobility}}, -year = {2013} -} -@article{Turek2011, -author = {Turek, Michal and Turek, Richard}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/849-736-1-PB.pdf:pdf}, -journal = {Journal of Information, Control and Management Systems}, -keywords = {coordination,crossroads,max-plus algebra,modeling,transfer stop}, -number = {1}, -pages = {67--75}, -title = {{Solving Selected Traffic Problems in the Max-Plus Algebra}}, -volume = {9}, -year = {2011} -} -@inproceedings{Winklera, -author = {Winkler, Alexander W and Mastalli, Carlos and Havoutis, Ioannis and Focchi, Michele and Caldwell, Darwin G}, -booktitle = {IEEE International Conference on Robotics and Automation (ICRA), 2015.}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/winkler15icra.pdf:pdf}, -pages = {5148--5154}, -title = {{Planning and Execution of Dynamic Whole-Body Locomotion for a Hydraulic Quadruped on Challenging Terrain}}, -year = {2015} -} -@inproceedings{Winkler, -author = {Winkler, Alexander and Havoutis, Ioannis and Bazeille, Stephane and Ortiz, Jesus and Focchi, Michele and Caldwell, Darwin and Semini, Claudio and Robotics, Advanced and Italiano, Istituto}, -booktitle = {2014 IEEE International Conference on Robotics and Automation (ICRA)}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/winkler14icra.pdf:pdf}, -pages = {6476--6482}, -title = {{Path planning with force-based foothold adaptation and virtual model control for torque controlled quadruped robots}}, -year = {2014} -} -@article{Wisse2004, -author = {Wisse, M and Schwab, A L and Helm, F C T Van Der}, -doi = {10.1017/S0263574704000475}, -file = {:C$\backslash$:/Users/Tavo/Documents/Escuela/PhD/Papers Italy/WisSchHel04.pdf:pdf}, -journal = {Robotica}, -keywords = {biped,passive,passive dynamic walking}, -pages = {681--688}, -title = {{Passive dynamic walking model with upper body}}, -volume = {22}, -year = {2004} -} diff --git a/Research Proposal IIT/Research proposal/dcsclogo.png b/Research Proposal IIT/Research proposal/dcsclogo.png deleted file mode 100755 index a0e0596..0000000 Binary files a/Research Proposal IIT/Research proposal/dcsclogo.png and /dev/null differ diff --git a/Research Proposal IIT/Research proposal/img/BHAmodel.pdf b/Research Proposal IIT/Research proposal/img/BHAmodel.pdf deleted file mode 100755 index 58238de..0000000 Binary files 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0ad82b3..0000000 --- a/Research Proposal IIT/Research proposal/main.bbl +++ /dev/null @@ -1,102 +0,0 @@ -\begin{thebibliography}{10} - -\bibitem{Lopes2010} -G.~Lopes, T.~van~den Boom, B.~{De Schutter}, and R.~Babu{\v{s}}ka, ``{Modeling - and control of legged locomotion via switching max-plus systems},'' in {\em - Proceedings of the 10th International Workshop on Discrete Event Systems - (WODES 2010)}, p.~6 pp., 2010. - -\bibitem{Barai} -R.~K. Barai and K.~Nonami, ``{Petri Net Based Supervisory Control for - Locomotion System of Six-Legged Walking Robot COMET-IV},'' in {\em - Proceedings of the Japan Joint Automatic Control Conference}, pp.~282--282, - 2004. - -\bibitem{Barasuola} -V.~Barasuol, J.~Buchli, C.~Semini, M.~Frigerio, E.~R.~D. Pieri, and D.~G. - Caldwell, ``{A Reactive Controller Framework for Quadrupedal Locomotion on - Challenging Terrain},'' in {\em Conference on Robotics and Automation (ICRA), - 2013 IEEE International}, pp.~2554--2561, 2013. - -\bibitem{Holmes2006} -P.~Holmes, R.~J. Full, D.~E. Koditschek, and J.~Guckenheimer, ``{The Dynamics - of Legged Locomotion : Models, Analyses, and Challenges},'' {\em SIAM - review}, vol.~48, no.~2, pp.~207--304, 2006. - -\bibitem{Wisse2004} -M.~Wisse, A.~L. Schwab, and F.~C. T. V.~D. Helm, ``{Passive dynamic walking - model with upper body},'' {\em Robotica}, vol.~22, pp.~681--688, 2004. - -\bibitem{Tsagarakis2013} -N.~G. Tsagarakis, S.~Morfey, H.~Dallali, and D.~G. Caldwell, ``{An Asymmetric - Compliant Antagonistic Joint Design for High Performance Mobility},'' in {\em - 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems - (IROS)}, pp.~5512--5517, 2013. - -\bibitem{Borovac2004} -V.~Miomir and B.~Borovac, ``{Zero-moment point — thirty five years of its - life ´},'' {\em International Journal of Humanoid Robotics}, vol.~1, no.~1, - pp.~157--173, 2004. - -\bibitem{Aghili2005} -F.~Aghili, ``{A Unified Approach for Inverse and Direct Dynamics of Constrained - Multibody Systems Based on Linear Projection Operator : Applications to - Control and Simulation},'' {\em IEEE Transactions on robotics}, vol.~21, - no.~5, pp.~834--849, 2005. - -\bibitem{Geyer2006} -H.~Geyer, A.~Seyfarth, and R.~Blickhan, ``{Compliant leg behaviour explains - basic dynamics of walking and running},'' in {\em Proceedings of the Royal - Society B: Biological Sciences.}, no.~August, pp.~2861--2867, 2006. - -\bibitem{Ketelaar} -J.~G. Ketelaar, L.~C. Visser, S.~Stramigioli, and R.~Carloni, ``{Controller - Design for a Bipedal Walking Robot using Variable Stiffness Actuators},'' in - {\em 2013 IEEE International Conference on Robotics and Automation (ICRA)}, - pp.~5650--5655, 2013. - -\bibitem{Crespi1985} -A.~Crespi, D.~Lachat, A.~Pasquier, and A.~J. Ijspeert, ``{Controlling swimming - and crawling in a fish robot using a central pattern generator},'' {\em - Autonomous robots}, vol.~25, no.~1-2, pp.~3--13, 2008. - -\bibitem{Ijspeert2008} -A.~J. Ijspeert, ``{Central pattern generators for locomotion control in animals - and robots : A review},'' {\em Neural Networks}, vol.~21, no.~4, - pp.~642--653, 2008. - -\bibitem{Ijspeert} -A.~J. Ijspeert, A.~Crespi, D.~Ryczko, and J.-m. Cabelguen, ``{From Swimming to - Walking with a Spinal Cord Model},'' {\em Science}, vol.~315, no.~5817, - pp.~1416--1420, 2007. - -\bibitem{Barasuol} -S.~Bazeille, V.~Barasuol, M.~Focchi, I.~Havoutis, M.~Frigerio, J.~Buchli, - C.~Semini, and D.~G. Caldwell, ``{Vision Enhanced Reactive Locomotion Control - for Trotting on Rough Terrain},'' in {\em Conference on Technologies for - Practical Robot Applications (TePRA), 2013 IEEE International}, pp.~1--6, - 2013. - -\bibitem{Schutter2009} -G.~Lopes, R.~Babu{\v{s}}ka, B.~{De Schutter}, and T.~van~den Boom, ``{Switching - max-plus models for legged locomotion},'' in {\em Proceedings of the 2009 - IEEE Interna- tional Conference on Robotics and Biomimetics (ROBIO 2009)}, - vol.~19, pp.~221--226, 2009. - -\bibitem{Schutter2008} -B.~{De Schutter} and T.~van~den Boom, ``{Max-plus algebra and max-plus linear - discrete event systems : An introduction},'' in {\em Proceedings of the 9th - International Workshop on Discrete Event Systems (WODES'08)}, vol.~19, - pp.~36--42, 2008. - -\bibitem{Goverde2007} -R.~M. Goverde, ``{Railway timetable stability analysis using max-plus system - theory},'' {\em Transportation Research Part B: Methodological}, vol.~41, - no.~2, pp.~179--201, 2007. - -\bibitem{Turek2011} -M.~Turek and R.~Turek, ``{Solving Selected Traffic Problems in the Max-Plus - Algebra},'' {\em Journal of Information, Control and Management Systems}, - vol.~9, no.~1, pp.~67--75, 2011. - -\end{thebibliography} diff --git a/Research Proposal IIT/Research proposal/main.blg b/Research Proposal IIT/Research proposal/main.blg deleted file mode 100755 index d4cac55..0000000 --- a/Research Proposal IIT/Research proposal/main.blg +++ /dev/null @@ -1,49 +0,0 @@ -This is BibTeX, Version 0.99d (TeX Live 2013/Debian) -Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 -The top-level auxiliary file: main.aux -The style file: ieeetr.bst -Database file #1: MyBib2.bib -Warning--I didn't find a database entry for "Lopes2009" -Warning--there's a number but no series in Geyer2006 -You've used 18 entries, - 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-\usepackage[english]{babel} -\usepackage[utf8]{inputenc} -\usepackage{amsmath} -\usepackage{graphicx} -\usepackage[colorinlistoftodos]{todonotes} -\usepackage{float} -\usepackage{listings} -\usepackage{color} -\usepackage{fullpage} -\usepackage{amsmath, amssymb, graphics, setspace} -\usepackage{subcaption} -\usepackage{subfiles} -\usepackage{mathtools} -%\usepackage[framed,numbered,autolinebreaks,useliterate]{mcode} -\usepackage{nomencl} -\usepackage[export]{adjustbox} -\usepackage{etoolbox} -\usepackage[nottoc,notlot,notlof]{tocbibind} -\usepackage{url} -\usepackage{multicol} -\usepackage{subcaption} -\usepackage{dcolumn} -\usepackage{amsmath} -\usepackage{amsfonts} -\usepackage{amssymb} -\usepackage{arydshln} -\usepackage{graphicx} -\usepackage{amsmath, amssymb, graphics, setspace} -\usepackage{subcaption} -\usepackage{enumerate} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{textcomp} -\usepackage{gensymb} -\usepackage{amssymb,amsmath} -\usepackage{tikz} -\usepackage{relsize} - - - -\newcommand{\circleland}{ - \mathbin{ - \mathchoice - {\buildcircleland{\displaystyle}} - {\buildcircleland{\textstyle}} - {\buildcircleland{\scriptstyle}} - {\buildcircleland{\scriptscriptstyle}} - } -} - -\newcommand\buildcircleland[1]{% - \begin{tikzpicture}[baseline=(X.base), inner sep=0, outer sep=0] - \node[draw,circle] (X) {$#1\land$}; - \end{tikzpicture}% -} - -%\renewcommand{\familydefault}{\sfdefault} -%\usepackage{helvet,sfmath} - -%\usepackage{lmodern} -%\usepackage{bm} -%\usepackage{sansmath} -%\usepackage{cmbright} - -\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]} -\let\endchangemargin=\endlist - -\makeatletter -\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{% - \hskip -\arraycolsep - \let\@ifnextchar\new@ifnextchar - \array{#1}} -\makeatother - -\title{Leg locomotion control using max-plus algebra systems} - -\author{O.A. Villarreal Maga\~na} - -\date{\today} - -\graphicspath{ {./img/} } - -\begin{document} - -\maketitle -\thispagestyle{empty} -%\clearpage -\setcounter{page}{1} -% To prevent a page number on the bottom of the page -%\newpage -%\tableofcontents % To automatically create a table of contents from the \section and \subsection commands. -%\newpage - - -\thispagestyle{empty} -\thispagestyle{empty} -%\newpage - - - - -% How to use the nomenclature - - -%\newpage -%\section*{Overview} -%\subfile{sections/overview.tex} - -%\section*{Planning} -%\subfile{sections/planning.tex} - -%\section*{Introduction} -\subfile{sections/Introduction.tex} - -%\newpage -\section{Motivation and rationale} -\subfile{sections/Motivation.tex} - -%\newpage -\section{State of the art} -\subfile{sections/Stateoftheart.tex} - -%\newpage -\section{Objectives} -\subfile{sections/Objectives.tex} - -%\newpage -\section{Methodology} -\subfile{sections/Methodology.tex} - -%\newpage -\section{Tentative workplan} -\subfile{sections/Workplan.tex} - -%\newpage -\section{Expected results} -\subfile{sections/Results.tex} - -%\newpage -%\section{Questions} -%\subfile{sections/questions.tex} - - - -%========================== Back matter ====================================== - -% -% Bibliography -\newpage -\bibliographystyle{ieeetr} -%\printbib{MyBib} -\bibliography{MyBib2}{} -%\bibliographystyle{plain} -% -%\include{sections/glossary} -% -% Index -\cleardoublepage - - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/main.toc b/Research Proposal IIT/Research proposal/main.toc deleted file mode 100755 index edb2bbd..0000000 --- a/Research Proposal IIT/Research proposal/main.toc +++ /dev/null @@ -1,8 +0,0 @@ -\select@language {english} -\contentsline {section}{\numberline {1}Motivation and rationale}{2} -\contentsline {section}{\numberline {2}State of the art}{3} -\contentsline {section}{\numberline {3}Objectives}{5} -\contentsline {section}{\numberline {4}Methodology}{5} -\contentsline {section}{\numberline {5}Tentative workplan}{5} -\contentsline {section}{\numberline {6}Expected results}{5} -\contentsline {section}{References}{6} diff --git a/Research Proposal IIT/Research proposal/mcode.sty b/Research Proposal IIT/Research proposal/mcode.sty deleted file mode 100755 index 2384bb4..0000000 --- a/Research Proposal IIT/Research proposal/mcode.sty +++ /dev/null @@ -1,291 +0,0 @@ -%% -%% This is file `mcode.sty' -%% -%% It is supposed to help you easily include MATLAB source code -%% into LaTeX document, but have it nicely highlighted, using -%% the great listings package. -%% -%% PLEASE NOTE that this package does nothing but save you from -%% figuring out some configurations in setting up the LISTINGS -%% package. ALL the work is done by that package! Thus, please -%% refer your questions to the listings package documentation. -%% -%% Usage: You have three ways of including your MATLAB code. As -%% environment, as inline object and directly from an external -%% file. -%% -%% 1) Environment: -%% -%% \begin{lstlisting} -%% YOUR CODE HERE -%% \end{lstlisting} -%% -%% -%% 2) Inline object*: -%% -%% Bla bla \mcode{CODEFRAGMENT} bla bla. -%% -%% -%% 3) Include external file (in environment form) -%% -%% \lstinputlisting{YOUR-FILE.m} -%% -%% -%% For your convenience this package has the following options: -%% -%% - bw if you intend to print the document (highlighting done -%% via text formatting (bold, italic) and shades of gray) -%% -%% - numbered if you want line numbers -%% -%% - autolinebreaks if you want the package to automatically -%% wrap your code. This is buggy as it may well break -%% break syntax and it doesn't work well with comments. -%% You REALLY should wrap your code manually. -%% -%% - useliterate if you want some characters / relations in -%% your code to be replace with something more readable. -%% Example: ~= becomes $\neq$, >= becomes $\geq$, delta -%% becomes $\delta$ and so on. -%% -%% - framed if you want a frame around the source code blocks -%% -%% - final if you have ``gloablly'' set the draft option, the -%% listings package will not output the code at all. to -%% force it to do so anyway, load this package with the -%% final option (passes the ``final'' on to listings). -%% -%% For example, you may use \usepackage[numbered,framed]{mcode} -%% in your document preamble. -%% -%% * If you want to place some inline code in a footnote, use -%% \mcodefn{} instead (this will reduce the font size a bit). -%% -%% Note: Inside code blocks you can escape to LaTeX text mode -%% using §...§. For ex. §text and some math: $x^2$§, which is -%% especially useful in comments for putting nicely typeset -%% equations etc. To get the same colour/style as in the rest -%% of the comment use \mcommentfont, i.e. §\mcommentfont $x^2$§ -%% -%% To change the font used, edit the first line in the "custo- -%% mise below" section. And feel free to edit other things as -%% well. Refer to the documentation of the listings package to -%% see what else you could do. If an extra small font is re- -%% quired, use {\fontfamily{pcr}\fontsize{3}{4.6}\selectfont} -%% in the definition of \lstbasicfont. -%% -%% Author: -%% Florian Knorn | florian@knorn.org | www.florian-knorn.com -%% -%% Version history: -%% 2.5 -- Renamed internal variables (thx S. Kranenbarg!) -%% 2.4 -- Added \mcodefn{} command (thx Tony Almeida!) -%% 2.3 -- More keywords (thx Dominik Wild!) -%% 2.2 -- Bugfix (thx Willi Gerbig!) -%% 2.1 -- Finally automatic detection between end and end -%% 2.0 -- New options for line breaking and literate prog. -%% 1.8 -- Fixed typo in documentation regarding §...§ -%% 1.7 -- Added MATLAB block comment syntax %{ ...... %} -%% 1.6 -- Added some infos, dealing with keyword ``end'' -%% 1.5 -- Tweaked check to see wether textcomp is loaded -%% 1.4 -- Fixed misconfig (mathescape now set to false) -%% 1.3 -- Purely cosmetic (tabs replaced by spaces) -%% 1.2 -- Added \lstset{showstringspaces=false} -%% 1.1 -- Added \mcode command and [final] option -%% 1.0 -- Release - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% D O N ' T T O U C H T H I S % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\def\fileversion{2.5} -\def\filedate{2014/03/06} - -\typeout{-- Package: `mcode' \fileversion\space <\filedate> --} -\NeedsTeXFormat{LaTeX2e} -\ProvidesPackage{mcode}[\filedate\space\fileversion] - -% for bw-option -\newif\ifmcode@bw -\DeclareOption{bw}{\mcode@bwtrue} - -% numbered option -\newif\ifmcode@numbered -\DeclareOption{numbered}{\mcode@numberedtrue} - -% final option -\newif\ifmcode@final -\DeclareOption{final}{\mcode@finaltrue} - -% autolinebreaks option -\newif\ifmcode@autolinebreaks -\DeclareOption{autolinebreaks}{\mcode@autolinebreakstrue} - -% literate programming (replace certain characters/relations -\newif\ifmcode@useliterate -\DeclareOption{useliterate}{\mcode@useliteratetrue} - -% framed option -\newif\ifmcode@framed -\DeclareOption{framed}{\mcode@framedtrue} - -\DeclareOption*{% default - \PackageWarning{mcode}{Unknown option `\CurrentOption' !}% -} -\ProcessOptions - -\ifmcode@bw\typeout{ - settings optimized for printing (bw formating)} -\else\typeout{ - settings optimized for display (colour formating)}\fi -\ifmcode@numbered\typeout{ - line numbering enabled}\else\fi -\ifmcode@useliterate\typeout{ - literate programming (character replacements) enabled}\else\fi -\ifmcode@autolinebreaks\typeout{ - automatic line breaking enabled (careful, buggy!)}\else\fi -\ifmcode@framed\typeout{ - framed listings}\else\fi - -% This command allows you to typeset syntax highlighted Matlab -% code ``inline''. The font size \small seems to look best... -\newcommand{\mcode}[1]{\lstinline[basicstyle=\lstbasicfont\small]|#1|} - -% Same, but for footnotes -\newcommand{\mcodefn}[1]{\lstinline[basicstyle=\lstbasicfont\footnotesize]|#1|} - -% check if color command exists -\ifx\color\undefined% - \RequirePackage{xcolor}% -\fi - -% check if listings has been loaded -\ifx\lstset\undefined% - \ifmcode@final - \RequirePackage[final]{listings} - \else - \RequirePackage{listings} - \fi -\fi - -% Check if textcomp has been loaded (this package is needed for -% upright quotes '' (instead of typographic ones `´)... -\ifx\textquotesingle\undefined% - \RequirePackage{textcomp}% -\fi - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% C U S T O M I S E B E L O W % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -% --------------------------------------------------------------------------------- -% default font -\def\lstbasicfont{\fontfamily{pcr}\selectfont\footnotesize} - -% --------------------------------------------------------------------------------- -% matlat languate definition -\lstdefinelanguage{matlabfloz}{% - alsoletter={...},% - morekeywords={% % keywords - break,case,catch,continue,elseif,else,end,% - for,function,global,if,otherwise,persistent,% - return,switch,try,while,methods,properties,% - events,classdef,...},% - comment=[l]\%, % comments - morecomment=[l]..., % comments - morecomment=[s]{\%\{}{\%\}}, % block comments - morestring=[m]' % strings -}[keywords,comments,strings]% - -% --------------------------------------------------------------------------------- -% general definitions -\lstset{% - basicstyle={\lstbasicfont}, % set font - showstringspaces=false, % do not emphasize spaces in strings - tabsize=4, % number of spaces of a TAB - mathescape=false,escapechar=§, % escape to latex with §...§ - upquote=true, % upright quotes - aboveskip={1.5\baselineskip}, % a bit of space above listings - columns=fixed % nice spacing -} - -% --------------------------------------------------------------------------------- -% define colours and styles -\ifmcode@bw % use font formating and gray 'colors' - \def\mcommentfont{\color[gray]{.75}\itshape} %comments light gray and italic - \lstset{language=matlabfloz, % use our version of highlighting - keywordstyle=\bfseries, % keywords in bold - commentstyle=\mcommentfont, % comments - stringstyle=\color[gray]{0.5} % strings darker gray - } -\else% notbw => use colors : ) - \def\mcommentfont{\color[rgb]{.133,.545,.133}} %comments in green - \lstset{language=matlabfloz, % use our version of highlighting - keywordstyle=\color[rgb]{0,0,1}, % keywords in blue - commentstyle=\mcommentfont, % comments - stringstyle=\color[rgb]{.627,.126,.941} % strings in purple - } -\fi%bw - -% --------------------------------------------------------------------------------- -% automatic line breaking --- warning, this is buggy and -% doesn't break comments correctly! -\ifmcode@autolinebreaks - \newsavebox{\lbreakdots}\sbox{\lbreakdots}{\lstbasicfont\mcommentfont...} - \lstset{breaklines=true,breakatwhitespace=true,prebreak=\usebox{\lbreakdots}} -\fi - -% --------------------------------------------------------------------------------- -% literate replacements -% the following is for replacing some matlab relations like >= or ~= -% by the corresponding LaTeX symbols, which are much easier to read ... -\ifmcode@useliterate - \lstset{% - literate=% - {~}{{$\neg$}}1 % \neg - {<=}{{\tiny$\leq$}}1 % \leq - {>=}{{\tiny$\geq$}}1 % \geq - {~=}{{\tiny$\neq$}}1 % \neq - {delta}{{\tiny$\Delta$}}1 % \Delta - {(end)}{\lstbasicfont (end)}{5} % black ``end'' when indexing last vector element - {({ }end)}{\lstbasicfont ({ }end)}{6} - {(end{ })}{\lstbasicfont (end{ })}{6} - {({ }end{ })}{\lstbasicfont ({ }end{ })}{7} - {:end}{\lstbasicfont :end}{4} - {:{ }end}{\lstbasicfont :{ }end}{5} - {end:}{\lstbasicfont end:}{4} - {end{ }:}{\lstbasicfont end{ }:}{5} - {,end}{\lstbasicfont ,end}{4} - {,{ }end}{\lstbasicfont ,{ }end}{5} - } -\else - \lstset{% - literate=% - {(end)}{\lstbasicfont (end)}{5} % black ``end'' when indexing last vector element - {({ }end)}{\lstbasicfont ({ }end)}{6} - {(end{ })}{\lstbasicfont (end{ })}{6} - {({ }end{ })}{\lstbasicfont ({ }end{ })}{7} - {:end}{\lstbasicfont :end}{4} - {:{ }end}{\lstbasicfont :{ }end}{5} - {end:}{\lstbasicfont end:}{4} - {end{ }:}{\lstbasicfont end{ }:}{5} - {,end}{\lstbasicfont ,end}{4} - {,{ }end}{\lstbasicfont ,{ }end}{5} - } -\fi%literates - -% --------------------------------------------------------------------------------- -% line numbering -\ifmcode@numbered% numbered option - \lstset{% - numbersep=3mm, numbers=left, numberstyle=\tiny, % number style - } -\fi - -\ifmcode@framed% framed option - \lstset{% - frame=single, % frame - } - \ifmcode@numbered% - \lstset{% - framexleftmargin=6mm, xleftmargin=6mm % tweak margins - } - \fi -\fi - -\endinput -%% End of file `mcode.sty'. \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/missfont.log b/Research Proposal IIT/Research proposal/missfont.log deleted file mode 100755 index 565621f..0000000 --- a/Research Proposal IIT/Research proposal/missfont.log +++ /dev/null @@ -1,10 +0,0 @@ -mktextfm pcrr7t -mktextfm pcrr7t -mktextfm pcrr7t -mktextfm pcrr7t -mktextfm pcrr7t -mktextfm pcrr7t -mktextfm pcrr7t -mktextfm pcrr7t -mktextfm pcrr7t -mktextfm pcrr7t diff --git a/Research Proposal IIT/Research proposal/sections (copy)/big matrix.tex b/Research Proposal IIT/Research proposal/sections (copy)/big matrix.tex deleted file mode 100755 index cc14348..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/big matrix.tex +++ /dev/null @@ -1,34 +0,0 @@ - where: - - \begin{changemargin}{-2cm}{1cm} - \begingroup - \renewcommand*{\arraystretch}{0.1} - \begin{equation} - \bar{A}_{0cl} = \begin{bmatrix}[cccc:cc|cccc:cc] - A_0 & 0 & 0 & B & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & -b_0 & 0 & -b1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ - % - \zeta \begin{bmatrix}k_1 , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_1 , 0 , 0\end{bmatrix} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ - % - \gamma K & 0 & \gamma & -\gamma & -\gamma K & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_0 - LC & -B & 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & \zeta \begin{bmatrix}l_1 , l_2 \end{bmatrix} C & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline - % - 0 & 0 & 0 & 0 & -BB^Tu_{r\Phi}\bigg( \frac{\cos \Theta_d}{\sin \Theta_d}\bigg) & 0 & A_0 & 0 & 0 & B & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 & 0 & -b_0 & 0 & -b_1 & 0 & 0 \\ - % - 0 & 0 & 0 & 0 & 0 & 0 & \zeta \begin{bmatrix}k_1 , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_1 , 0 , 0\end{bmatrix} & 0 \\ - % - 0 & 0 & 0 & 0 & 0 & 0 & \gamma K & 0 & \gamma & -\gamma & -\gamma K & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & LDB^T\Gamma_{\Phi d}^*\big(\frac{\cos \Theta_d}{\sin \Theta_d}\big)-BB^Tu_{r\Phi}\bigg( \frac{\cos \Theta_d}{\sin \Theta_d}\bigg) & 0 & 0 & 0 & 0 & 0 & A_0 - LC & -B \\ - % - 0 & 0 & 0 & 0 & \zeta \begin{bmatrix}l_1 , l_2\end{bmatrix} D B^T \Gamma_{\Phi d}^* & 0 & 0 & 0 & 0 & 0 & \zeta \begin{bmatrix}l_1 , l_2\end{bmatrix}C & 0 - \end{bmatrix}. - \end{equation} - \endgroup - \end{changemargin} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/citations.log b/Research Proposal IIT/Research proposal/sections (copy)/citations.log deleted file mode 100755 index 38841d1..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/citations.log +++ /dev/null @@ -1,50 +0,0 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9) (preloaded format=pdflatex 2015.9.7) 17 MAR 2016 10:00 -entering extended mode -**citations.tex - -("C:\Users\Tavo\Documents\Escuela\Master\Master Thesis\Thesis Reports\Template -progress reports\sections\citations.tex" -LaTeX2e <2011/06/27> -Babel and hyphenation patterns for english, afrikaans, ancientgreek, ar -abic, armenian, assamese, basque, bengali, bokmal, bulgarian, catalan, coptic, -croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga -lician, german, german-x-2013-05-26, greek, gujarati, hindi, hungarian, iceland -ic, indonesian, interlingua, irish, italian, kannada, kurmanji, latin, latvian, - lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n -german-x-2013-05-26, nynorsk, oriya, panjabi, pinyin, polish, portuguese, roman -ian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, swissgerm -an, tamil, telugu, turkish, turkmen, ukenglish, ukrainian, uppersorbian, usengl -ishmax, welsh, loaded. -("C:\Program Files (x86)\MiKTeX 2.9\tex\latex\subfiles\subfiles.cls" -Document Class: subfiles 2012/05/23 Federico Garcia -("C:\Program Files (x86)\MiKTeX 2.9\tex\latex\tools\verbatim.sty" -Package: verbatim 2003/08/22 v1.5q LaTeX2e package for verbatim enhancements -\every@verbatim=\toks14 -\verbatim@line=\toks15 -\verbatim@in@stream=\read1 -) -Preamble taken from file `main.tex' - -! LaTeX Error: File `main.tex' not found. - -Type X to quit or to proceed, -or enter new name. (Default extension: tex) - -Enter file name: -! Emergency stop. - - -l.40 \input{\preamble@file} - -*** (cannot \read from terminal in nonstop modes) - - -Here is how much of TeX's memory you used: - 53 strings out of 493921 - 1304 string characters out of 3144880 - 49031 words of memory out of 3000000 - 3436 multiletter control sequences out of 15000+200000 - 3640 words of font info for 14 fonts, out of 3000000 for 9000 - 841 hyphenation exceptions out of 8191 - 18i,0n,17p,168b,16s stack positions out of 5000i,500n,10000p,200000b,50000s -! ==> Fatal error occurred, no output PDF file produced! diff --git a/Research Proposal IIT/Research proposal/sections (copy)/conclusions.tex b/Research Proposal IIT/Research proposal/sections (copy)/conclusions.tex deleted file mode 100755 index 13cb720..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/conclusions.tex +++ /dev/null @@ -1,24 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\vspace{20pt} - -The conclusions for this project can be divided into several parts. First of all, modeling and system identification proved to be a difficult and time consuming task, due to the fact that while performing them, it is required to decide the moment when a model is good for control. This is related to make the correct choices of identification signals, experiment duration and the initial guesses for the nonlinear least-squares parameter estimation. Using the subspace identification method was considered along the identification process, but discarded because this method is mainly used for system from which it is little known about the inner dynamics of the system. - - - -Implementation of the State-Feedback Pole Placement controller on the real setup brought unforeseen problems with it. As noted before in Section \ref{Encoders} and \ref{sec:Control} Encoder 2 has an offset when it reaches $\pi$. Also the controller was really sensitive to small offsets when it had to be tuned to stabilize the system at $(0,0)$. Both these problems were overcome by implementing an automatized calibration before each experiment. This was done by balancing the pendulum manually or holding it in place on the unstable equilibrium for 2 seconds, and then taking the mean of the outputs and subtracting that from the during the experiment. This way the controller could start each experiment from a new zero point; decreasing its initial control action when the experiment started caused by the offset. \\ -The deadzone of motor as identified in Section \ref{sec:Deadzone} did create as much problems as anticipated. Linear control was able to get good results. -The noise in the real setup also caused for a lot of problems, until the Kalman filter was properly tuned. In the beginning it was difficult to differentiate if the bad performance of the closed loop system was due to a poorly tuned Kalman filter or due to a poorly tuned controller. This took some practice and tuning experience to overcome. When the time was taken to properly tune the Kalman filter in the $(\pi,0)$ position, the response of the closed-loop system became much smoother. \\ -The tuning of the Pole Placement controller was made a lot easier by the scripted computation of the pole locations. This way a requested behaviour could be given, and the required poles would be computed automatically. The stabilization of $(\pi,\pi)$ with the pole placement controller was the first real accomplishment. After that the next hurdle was stabilizing the $(0,0)$ position. This took a few days of tuning both the controller and the Kalman filter. \\ -For the $H_{\infty}$ controller, tuning was surprisingly less hard than expected at the $(\pi,\pi)$ position, since the input and performance weights could be deduced by using guidelines set by \cite{Skogestad}. The performance obtained from start with the first design was unexpected, because the experience using this framework was limited. Moreover, the controller allowed the system to be able to track a sine signal and reject disturbances. -A great part of the success of the initial controller, could have been due to the use of the filtered estimates and not the direct output of the system. Since model uncertainty was not introduced to the design process and no D-K iterations were performed, robustness could not be ensured by using the output signal directly Nevertheless, when applying the $H_\infty$ for the upright position the results were not as successful as for the previous position. - -There could be two main reasons for this result. First of all the implementation of the $H_\infty$ controller on discrete time just by using \mcode{c2d}. By doing some research there are specific methods and routines in MATLAB\textsuperscript{\textregistered} for $H_\infty$ in discrete time(such as \mcode{sdhinfsyn}). Secondly, having a refined function for the input weight could have influenced the performance, because the size of the input differs greatly if the system has small and high frequent oscillations. A possible choice could have been a transfer function with one double zero at low frequencies and one double pole at high frequencies. The position of the poles a zeroes should be based on the closed-loop bandwidth. A possible subsequent step could be obtaining the closed-loop bandwidth of the system with the pole-placement controller and provide a multivariable design using the $H_\infty$ controller. - -Finally, overall the control of the Rotational Inverted Pendulum proved to be a challenging task because of the involved dynamics of the system, the filtering task and the controller design and tuning. This project gave great insight about the problems regarding real life implementation. Elements that may be seen as simple during the process (such as the direction of the angles) can cause great delays in the control design process. This specific setup, is also hard to control at the upright position because from the theoretical point of view, both links are on an unstable equilibrium. - - - -\end{document} diff --git a/Research Proposal IIT/Research proposal/sections (copy)/controller_design.log b/Research Proposal IIT/Research proposal/sections (copy)/controller_design.log deleted file mode 100755 index 5cbd386..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/controller_design.log +++ /dev/null @@ -1,50 +0,0 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9) (preloaded format=pdflatex 2015.9.7) 29 MAR 2016 11:21 -entering extended mode -**controller_design.tex - -("C:\Users\Tavo\Documents\Escuela\Master\Master Thesis\Thesis Reports\Template -progress reports\sections\controller_design.tex" -LaTeX2e <2011/06/27> -Babel and hyphenation patterns for english, afrikaans, ancientgreek, ar -abic, armenian, assamese, basque, bengali, bokmal, bulgarian, catalan, coptic, -croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga -lician, german, german-x-2013-05-26, greek, gujarati, hindi, hungarian, iceland -ic, indonesian, interlingua, irish, italian, kannada, kurmanji, latin, latvian, - lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n -german-x-2013-05-26, nynorsk, oriya, panjabi, pinyin, polish, portuguese, roman -ian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, swissgerm -an, tamil, telugu, turkish, turkmen, ukenglish, ukrainian, uppersorbian, usengl -ishmax, welsh, loaded. -("C:\Program Files (x86)\MiKTeX 2.9\tex\latex\subfiles\subfiles.cls" -Document Class: subfiles 2012/05/23 Federico Garcia -("C:\Program Files (x86)\MiKTeX 2.9\tex\latex\tools\verbatim.sty" -Package: verbatim 2003/08/22 v1.5q LaTeX2e package for verbatim enhancements -\every@verbatim=\toks14 -\verbatim@line=\toks15 -\verbatim@in@stream=\read1 -) -Preamble taken from file `main.tex' - -! LaTeX Error: File `main.tex' not found. - -Type X to quit or to proceed, -or enter new name. (Default extension: tex) - -Enter file name: -! Emergency stop. - - -l.40 \input{\preamble@file} - -*** (cannot \read from terminal in nonstop modes) - - -Here is how much of TeX's memory you used: - 53 strings out of 493921 - 1344 string characters out of 3144880 - 49031 words of memory out of 3000000 - 3436 multiletter control sequences out of 15000+200000 - 3640 words of font info for 14 fonts, out of 3000000 for 9000 - 841 hyphenation exceptions out of 8191 - 18i,0n,17p,176b,16s stack positions out of 5000i,500n,10000p,200000b,50000s -! ==> Fatal error occurred, no output PDF file produced! diff --git a/Research Proposal IIT/Research proposal/sections (copy)/controller_design.tex b/Research Proposal IIT/Research proposal/sections (copy)/controller_design.tex deleted file mode 100755 index ba8a4be..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/controller_design.tex +++ /dev/null @@ -1,309 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - - - -In order to propose an output-based controller design for the system, the matrices of equation \eqref{eq:totaldynamicslinears} are analyzed. - -Initially, it can be noticed that the main diagonal of $\bar{A}_{ocl}$, $\bar{A}_{1cl}$ and $\bar{A}_{2cl}$ have the same structure for both the inclination and the azimuth. Matrices $A_{0c}$, $A_{1c}$ and $A_{2c}$ couple the azimuth dynamics with the inclination dynamics, but this coupling is only present in terms of the inclination observer error and the integral action of the inclination observer. This fact can be used advantageously, since the inclination observer error dynamics does not depend on the azimuth dynamics. Figure \ref{fig:ISS} depicts a cascaded structure showing how the error dynamics and the observer error dynamics are interconnected. - -\begin{figure}[h]\centering - \includegraphics[width=0.5\textwidth]{ISS.pdf} - \caption{Cascaded structure for stability of the system. - \label{fig:ISS} } -\end{figure} - -As mentioned before, the inclination dynamics are independent from the azimuth dynamics. It is important to stress this fact since this unilateral coupling is the key element for the stability analysis in this neutral bit walk case. Considering the inclination, the error and the observer error dynamics are in a series interconnection, and this interconnection is through constant ($\xi$-independent) terms, allowing to design controller and observer separately due to its linear time-invariant nature, complying with the separation principle. - -It is important to analyze the signal coming out of the inclination observer error dynamics, since it perturbs the rest of the system, as depicted in Figure \ref{fig:ISS}. As stated before, the design for the observer gain $L_\Theta$ can be done independently and this dynamics are guaranteed to be asymptotically stable if the poles of $\delta_\Theta$ are located on the left-hand side of the complex plane. On the other hand, the way this signal is interconnected with the blocks $\delta_\Phi$ and $e_\Phi$ is through $\xi$-dependent terms (namely, the "$p(\xi)$" terms). This may render controller design difficult, since this coefficients change along the trajectory, which would not allow to take an eigenvalue approach to design the azimuth observer and controller. Despite this, the $\xi$-dependent terms are entirely related to the designed reference trajectory, meaning that this terms remain bounded according to the reference trajectory. - -If then the combination of the signal coming from the $\delta_\Theta$ block and the $\xi$-dependent terms is considered, it is known that $\delta_\Theta$ is asymptotically stable (and eventually equal to zero) and that the "$p(\xi)$" terms are bounded, making the perturbation signals coming into the $\delta_\Phi$ and $e_\Phi$ blocks stable. Another remark is that there is also coupling through a $\xi$-dependent term in $\delta_\Phi$ and $e_\Phi$, related to the state $q_\Theta$, for which the same argument of it being asymptotically stable (since it corresponds to the observer integral action of the inclination, which should be stable by design) and the boundedness of its coupling term also holds. - -In order to guarantee stability of the whole system, it only remains to check the internal stability of the $\delta_\Phi$ and $e_\Phi$ blocks in Figure \ref{fig:ISS}. Considering that the perturbation signal going into these blocks is asymptotically stable, the stability of the error and observer error dynamics can be analysed without considering them. This leads to the same series interconnection from $\delta_\Phi$ to $e_\Phi$, allowing to design once again, controller and observer gains separately, without compromising stability of the whole system. An important observation as well, is that the linearized system stability is independent from the trajectory, as this only influences the interconnection terms. - -After this analysis, it can be concluded that the controller synthesis of gains $K_i$ and $L_i$ can be done using an eigenvalue approach as in \cite{Kremers2013} and \cite{Monsieurs2015}, for the following isolated system dynamics: - -\begin{align} - \begin{bmatrix} - e_\Theta'(\xi) \\ - \Delta \bar{\Gamma}_\Theta^{*'}(\xi) \\ - \Delta z_{1\Theta}'(\xi) \\ - \Delta z_{2\Theta}'(\xi) \\ - \end{bmatrix} =& - \begin{bmatrix}[cccc] - A_0 & 0 & 0 & B\\ - % - 0 & -b_0 & 0 & -b_1\\ - \zeta \begin{bmatrix}k_{1\Theta} , 0 , 0\end{bmatrix} & 0 & 0 & 0\\ - % - \gamma K_\Theta & 0 & \gamma & -\gamma - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi) \\ - \Delta \bar{\Gamma}_\Theta^{*}(\xi) \\ - \Delta z_{1\Theta}(\xi) \\ - \Delta z_{2\Theta}(\xi) \\ - \end{bmatrix} + - \begin{bmatrix}[cccc] - A_1 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_1) \\ - \Delta \bar{\Gamma}_\Theta^{*}(\xi_1) \\ - \Delta z_{1\Theta}(\xi_1) \\ - \Delta z_{2\Theta}(\xi_1) \\ - \end{bmatrix} \nonumber\\ - &+ - \begin{bmatrix}[cccc] - A_2 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_2) \\ - \Delta \bar{\Gamma}_\Theta^{*}(\xi_2) \\ - \Delta z_{1\Theta}(\xi_2) \\ - \Delta z_{2\Theta}(\xi_2) \\ - \end{bmatrix} \label{eq:SynthesisConInc}\\ - % - % - % - \nonumber\\ - \begin{bmatrix} - e_\Phi'(\xi) \\ - \Delta \Gamma_\Phi^{*'}(\xi) \\ - z_{1\Phi}'(\xi) \\ - z_{2\Phi}'(\xi) \\ - \end{bmatrix} =& - \begin{bmatrix}[cccc] - A_0 & 0 & 0 & B\\ - % - 0 & -b_0 & 0 & -b_1\\ - \zeta \begin{bmatrix}k_{1\Phi} , 0 , 0\end{bmatrix} & 0 & 0 & 0\\ - % - \gamma K_\Phi & 0 & \gamma & -\gamma - \end{bmatrix} - \begin{bmatrix} - e_\Phi(\xi) \\ - \Delta \Gamma_\Phi^{*}(\xi) \\ - z_{1\Phi}(\xi) \\ - z_{2\Phi}(\xi) \\ - \end{bmatrix} + - \begin{bmatrix}[cccc] - A_1 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Phi(\xi_1) \\ - \Delta \Gamma_\Phi^{*}(\xi_1) \\ - z_{1\Phi}(\xi_1) \\ - z_{2\Phi}(\xi_1) \\ - \end{bmatrix} \nonumber\\ - &+ - \begin{bmatrix}[cccc] - A_2 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Phi(\xi_2) \\ - \Delta \Gamma_\Phi^{*}(\xi_2) \\ - z_{1\Phi}(\xi_2) \\ - z_{2\Phi}(\xi_2) \\ - \end{bmatrix} \label{eq:SynthesisConAzi}\\ - % - % - % - \nonumber\\ - \begin{bmatrix} - \delta_\Theta '(\xi) \\ - \Delta q_\Theta'(\xi) - \end{bmatrix} =& - \begin{bmatrix}[cc] - A_0 - L_\Theta C_\Theta & -B\\ - % - \zeta \begin{bmatrix}l_{1\Theta} , l_{2\Theta} \end{bmatrix} C_\Theta & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta (\xi)\\ - \Delta q_\Theta (\xi) - \end{bmatrix} + - \begin{bmatrix}[cc] - A_1 & 0\\ - % - 0 & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta (\xi_1)\\ - \Delta q_\Theta (\xi_1) - \end{bmatrix} + - \begin{bmatrix}[cc] - A_2 & 0\\ - % - 0 & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta (\xi_2)\\ - \Delta q_\Theta (\xi_2) - \end{bmatrix} \label{eq:SynthesisObsInc}\\ - % - % - % - \nonumber \\ - \begin{bmatrix} - \delta_\Phi '(\xi) \\ - q_\Phi'(\xi) - \end{bmatrix} =& - \begin{bmatrix}[cc] - A_0 - L_\Phi C_\Phi & -B\\ - % - \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi} \end{bmatrix} C_\Phi & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Phi (\xi)\\ - q_\Phi (\xi) - \end{bmatrix} + - \begin{bmatrix}[cc] - A_1 & 0\\ - % - 0 & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Phi (\xi_1)\\ - q_\Phi (\xi_1) - \end{bmatrix} + - \begin{bmatrix}[cc] - A_2 & 0\\ - % - 0 & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Phi (\xi_2)\\ - q_\Phi (\xi_2) - \end{bmatrix} \label{eq:SynthesisObsAzi} -\end{align} - -Stability can be achieved if gains $K_i$ and $L_i$ are chosen in such way that the poles of \eqref{eq:SynthesisConInc}, \eqref{eq:SynthesisConAzi}, \eqref{eq:SynthesisObsInc} and \eqref{eq:SynthesisObsAzi} are on the left half of the complex plane. Despite that stability can be ensure, the performance of the system could not be adequate, since the initial error in estimation could cause a high level of transient oscillations (borehole spiraling) and if it is above certain limit, this could generate a steady state error in the final response of the azimuth dynamics, since the system is performing far away from the linearization. - -In order to cope with this initial estimation error, it is proposed to use the available measurements of the BHA orientation close to the bit as initial value for the observer dynamics. These measurements are given by output equations in \eqref{eq:output1} and \eqref{eq:output2}. - -\subsection{Controller and observer gain design} - -Although the structure of the controller has already been decided, it remains to choose the values of the controller, observer, integral-action and low-pass filter gains. The stability of the system is guaranteed as long as the closed-loop poles are located on the left-hand side of the imaginary axis in the complex plane. In other words, for the system to be stable, the following conditions have to be met (for corresponding isolated systems \eqref{eq:SynthesisConInc}, \eqref{eq:SynthesisConAzi}, \eqref{eq:SynthesisObsInc} and \eqref{eq:SynthesisObsAzi}): - -\begin{align} - \max_{j\in[1,2,...,\infty]} \{\Re(\lambda_{jK\Theta}(K_{\Theta},\zeta,\gamma)) \} < 0, - \label{eq:condition1} \\ - \max_{j\in[1,2,...,\infty]} \{\Re(\lambda_{jK\Phi}(K_{\Phi},\zeta,\gamma)) \} < 0, - \label{eq:condition2} \\ - \max_{j\in[1,2,...,\infty]} \{\Re(\lambda_{jL\Theta}(L_{\Theta},\zeta)) \} < 0, - \label{eq:condition3} \\ - \max_{j\in[1,2,...,\infty]} \{\Re(\lambda_{jL\Phi}(L_{\Phi},\zeta)) \} < 0, - \label{eq:condition4} -\end{align} - -where $\lambda_{jKi}(K_{i},\zeta,\gamma)$ and $\lambda_{jLi}(L_{i},\zeta)$ for $i = \Theta, \Phi$ represent closed-loop pole $j$ of the isolated systems corresponding to the controller and observer dynamics, respectively. Furthermore, the settling time of the isolated system decreases as the poles are located further to the left. In other to obtain specific values for the controller and observer gains, based on the conditions given by \eqref{eq:condition1}, \eqref{eq:condition2}, \eqref{eq:condition3} and \eqref{eq:condition4}we can formulate an optimization problem consisting on minimizing the following objective functions: - -\begin{align} -\varLambda_1 &= \max_{j\in[1,2,...,\infty]} \{\Re(\lambda_{jK\Theta}(K_{\Theta},\zeta,\gamma)) \}, -\label{eq:op1} \\ -\varLambda_2 &= \max_{j\in[1,2,...,\infty]} \{\Re(\lambda_{jK\Phi}(K_{\Phi},\zeta,\gamma)) \} , -\label{eq:op2} \\ -\varLambda_3 &= \max_{j\in[1,2,...,\infty]} \{\Re(\lambda_{jL\Theta}(L_{\Theta},\zeta)) \}, -\label{eq:op3} \\ -\varLambda_4 &= \max_{j\in[1,2,...,\infty]} \{\Re(\lambda_{jL\Phi}(L_{\Phi},\zeta)) \}, -\label{eq:op4} -\end{align} - -corresponding to isolated systems \eqref{eq:SynthesisConInc}, \eqref{eq:SynthesisConAzi}, \eqref{eq:SynthesisObsInc} and \eqref{eq:SynthesisObsAzi}, respectively. This non-smooth optimization problem can be solved using a gradient-sampling algorithm \cite{Kremers2013}. This algorithm can be implemented via the Matlab toolbox \cite{Michiels:2007:SST:1355294}, which also allows to choose the location of the real part of the right-most pole. Despite this, the objective functions proposed can be computationally expensive. In order to reduce the effort, the gains of the integral action $\zeta$ and the low-pass filter $\gamma$ are fixed for each iteration. This can be done since the control objectives that these gains pursue are on a different "length" scale as the control objectives that the state-feedback and observer gains \cite{Monsieurs2015}. The selection of these gains will be done in an iterative process until the desired performance is reached. - -It has to be kept in mind that, since the inclination observer dynamics have to converge faster than the rest of the dynamics of the system, the location of its right-most pole has to be chosen further than for the rest of the rest of the dynamics of the system, this will be explained in the next section. - - -\subsection{Controller synthesis} - -In order to propose the values for the right-most poles for the optimization problem to compute the controller and observer gains, a time ('length') scale for the convergence of $\delta_i$ and $e_i$ based on the cascaded structure of Figure \ref{fig:ISS} is introduced. The determining dynamics of the cascaded structure are given by the $\delta_\Theta$ block. Because of this, it has been decided to set this as $fast$ converging dynamics. As long as the rest of the elements in the structure have a slower convergence, the inclination observer will not lead to large transients in the other dynamics (regarding $\delta_\Phi$, $e_\Theta$ and $e_\Phi$) which would invalidate the assumptions motivating the stability analysis based on linearization. On the other hand, $\delta_\Phi$ has to be faster than $e_\Phi$, so it is placed in a $medium$ scale. For the error dynamics $e_\Theta$ and $e_\Phi$, the choice can be made to set $e_\Theta$ in the same $medium$ time scale as $\delta_\Phi$ and to set $e_\Phi$ in a $slow$ time scale (since inclination does not depend on the azimuth). It has been decided to keep $e_\Theta$ and $e_\Phi$ on the same $slow$ time scale to have a similar behavior for both $\Theta$ and $\Phi$. - -Table \ref{table:controllervalues} shows the values chosen for the low-pass filter and integral part fixed for the optimization, as well as the value of the real part of the right-most pole and its corresponding computed feedback gains. - -\begin{table}[h] -\centering -\caption{Parameters for controller gains.} -\label{table:controllervalues} -\begin{tabular}{|l|l|l|l|l|} -\hline -Gain & Objective function right-most pole location & Feedback values & $\gamma$ & $\zeta$ \\ \hline -$e_\Theta$ & -0.6 & $K_\Theta = \begin{bmatrix} -6310 & -2571 & 1288 \end{bmatrix}$ & 0.8 & 0.5\\ \hline -$\delta_\Theta$ & -0.9 & $L_\Theta = \begin{bmatrix} 1014 & 3299 \\ 0 & 0 \\ 0 & 0\end{bmatrix}$ & 0 & 0.5\\ \hline -$e_\Phi$ & -0.6 & $K_\Phi = \begin{bmatrix} -7058 & -72.1 & 1021 \end{bmatrix}$ & 0.8 & 0.3\\ \hline -$\delta_\Phi$ & -0.7 & $L_\Phi = \begin{bmatrix} -48.5 & 2555 \\ 0 & 0 \\ 0 & 0\end{bmatrix}$ & 0 & 0.3 \\\hline -\end{tabular} -\end{table} - -The most critical values to be designed are the ones corresponding to the inclination observer error $\delta_\Theta$, since all of the dynamics are in a series interconnection with its states. It can be seen that the real part of the right-most pole is chosen to be the furthest from the imaginary axis, in order to have a faster convergence. It has to be considered as well, that if the poles are pushed too much to the left, this could result in high observer gains, which could potentially amplify the effect of disturbances such as noise or model uncertainty. The integrator value is chosen to be the same as the one for the error dynamics, since it is also counteracting the effect of gravity related terms. It has been chosen to not implement a low-pass filter in both observers, since its function is to avoid fast changes in the response, which is not a problem in the estimation error. The low-pass filter gains are kept the same for both the inclination and azimuth tracking error dynamics. As for the integral part of the azimuth error $e_\Phi$ and the observer error $\delta_\Phi$, despite the fact that there are no gravity terms affecting the these dynamics, the initial error of $\delta_\Theta$ could act as a disturbance for a short period of time. This could be analogous to an impulse-like disturbance, which can be effectively rejected by implementing integral action. This is implemented in both the controller an observer of the azimuth, although with different gains, since the disturbance to be rejected is of a different type. - -Figure \ref{fig:ClosedLoopPolesNeutral} shows the union of the closed-loop poles of the isolated systems \eqref{eq:SynthesisConInc}, \eqref{eq:SynthesisConAzi}, \eqref{eq:SynthesisObsInc} and \eqref{eq:SynthesisObsAzi} after implementing the controller gains of table \ref{table:controllervalues}. The right-most pole (poles at the origin are not considered since they are introduced by the state description) is at -0.6195, which is above the chosen maximum value for the right-most poles of the error dynamics. - -\begin{figure}[h]\centering - \includegraphics[width=0.5\textwidth]{ClosedLoopPolesNeutral.png} - \caption{Closed-loop poles for the neutral bit walk system for $\eta \Pi = 0.261$. - \label{fig:ClosedLoopPolesNeutral} } -\end{figure} - - -To implement the controller the desired trajectory to follow is the same as in \cite{Monsieurs2015}. The controller will be tested for several initial conditions, since its behavior could be sensitive to initial errors in the observer. The first measurement of the sensor is being taken into account to improve the transient response. Figure \ref{fig:Resoponse} shows the inclination and azimuth response for each set of initial conditions. - -\begin{figure}[H]\centering - \includegraphics[height=3.5in]{Response.png} - \caption{Inclination and azimuth responses of the system. - \label{fig:Resoponse} } -\end{figure} - - -Despite the fact that the initial condition does affect the transient at the beginning of the response, the controller is able to track the trajectory close to the reference. It also has to be considered the influence of the initial condition in the applied RSS force, this is depicted in Figure \ref{fig:RSSForces}. - -\begin{figure}[H]\centering - \includegraphics[height=3.5in]{RSSInputs.png} - \caption{RSS Forces applied to the system. - \label{fig:RSSForces} } -\end{figure} - -The influence of the initial condition is clearly seen as well when the RSS begins to be applied, for both the inclination and the azimuth. This has to do with the fact that although the initial condition is taken from the measurement, as you start further away from the reference, the RSS has to apply more force in order to compensate for that initial error. Finally, the system and observer errors are shown in Figures \ref{fig:ErrorNeutral} and \ref{fig:ObserverErrors}, up to $\xi = 25$ since the error already converged to zero. - - -\begin{figure}[H]\centering - \includegraphics[height=3.5in]{ErrorsNeutral.png} - \caption{Errors for different initial conditions. - \label{fig:ErrorNeutral} } -\end{figure} - -\begin{figure}[H]\centering - \includegraphics[height=3.5in]{ObserverErrors.png} - \caption{Observer errors for different initial conditions. - \label{fig:ObserverErrors} } -\end{figure} - -The error for both the inclination and the azimuth reaches the value of zero at approximately $\xi = 10$, this result complies with a fast enough response in order to drill a complex borehole geometry. The behavior of the observer is as expected in the case of the inclination, since the initial condition is further from the reference, and the difference between the measurement and the value of $\Theta$ starts increasing as the absolute value of $\Theta_0$ increases, if we consider Equation \eqref{eq:output12}. - -On the other hand, the observer error of the azimuth shows a particular behavior. It can be seen, that in the case of the initial condition $\Theta_0 = 20 [^{\circ}]$ and $\Phi_0 = 80[^{\circ}]$, the error is much bigger in comparison with the other two sets of initial conditions. This could be explained, since the system was linearized to work around $e_i = 0$ and $\delta_i = 0$, and the nonlinear terms were all related to the difference between $\Theta$ and $\check{\Theta}$. If there is a big enough difference between this two variables, the linearized system will no longer be valid and this could generate estimation errors. Furthermore, besides having to deal with its own initial estimation error, the error in $\delta_\Theta$ also affects the azimuth observer until it reaches zero as shown in the nonlinear dynamic equations \eqref{eq:totaldynamics}. - - - -\end{document} - diff --git a/Research Proposal IIT/Research proposal/sections (copy)/equations.tex b/Research Proposal IIT/Research proposal/sections (copy)/equations.tex deleted file mode 100755 index d470738..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/equations.tex +++ /dev/null @@ -1,47 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\subsection*{Equations for the Linearized Model}\label{AppB:LinearizedModel} -\begin{equation*} -\dot{x} = -\begin{bmatrix} -0 & 0 & 1 & 0 \\ -0 & 0 & 0 & 1 \\ -a_{31} & a_{32} & a_{33} & a_{34} \\ -a_{41} & a_{42} & a_{43} & a_{44} \\ -\end{bmatrix} x + -\begin{bmatrix} -0 \\ -0 \\ -b_3 \\ -b_4 -\end{bmatrix} u -\end{equation*} - -\begin{equation*} -y = -\begin{bmatrix} -1 & 0 & 0 & 0 \\ -0 & 1 & 0 & 0 \\ -\end{bmatrix}x -\end{equation*} - -where: - -\begin{eqnarray*} -a_{31} &=& \frac{(g (c_1 m_1 m_2 c_2^2 + I_2 c_1 m_1 + I_2 l_1 m_2))}{(m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -a_{32} &=& \frac{ - ( c_2^2 g l_1 m_2^2 )}{( m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -a_{33} &=& \frac{- ( d_1 ( m_2 c_2^2 + I_2 ) )}{( m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -a_{34} &=& \frac{( d_2 ( m_2 c_2^2 + l_1 m_2 c_2 + I_2 ) )}{( m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -a_{41} &=& \frac{- ( g ( - m_1 m_2 c_1^2 c_2 + m_1 m_2 c_1 c_2^2 + l_1 m_1 m_2 c_1 c_2 + I_2 m_1 c_1 - I_1 m_2 c_2 + I_2 l_1 m_2 ) )}{( m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -a_{42} &=& \frac{(c_2 g m_2 ( m_1 c_1^2 + m_2 l_1^2 + c_2 m_2 l_1 + I_1))}{(m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -a_{43} &=& \frac{(d_1 (m_2 c_2^2 + l_1 m_2 c_2 + I_2))}{(m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -a_{44} &=& \frac{ - ( d_2 ( m_1 c_1^2 + m_2 c_2^2 + 2 m_2 c_2 l_1 + m_2 l_1^2 + I_1 + I_2))}{(m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -b_{3} &=& \frac{ - ( G ( m_2 c_2^2 + I_2 ) )}{ ( m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -b_{4} &=& \frac{( G ( m_2 c_2^2 + l_1 m_2 c_2 + I_2 ) )}{( m_1 m_2 c_1^2 c_2^2 + I_2 m_1 c_1^2 + I_1 m_2 c_2^2 + I_2 m_2 l_1^2 + I_1 I_2)} \\ -\end{eqnarray*} - - - -\end{document} diff --git a/Research Proposal IIT/Research proposal/sections (copy)/errorlin.tex b/Research Proposal IIT/Research proposal/sections (copy)/errorlin.tex deleted file mode 100755 index dfaf5b9..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/errorlin.tex +++ /dev/null @@ -1,1019 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - - - \subsection{Error dynamics} - - The system equations for the neutral bit walk model are given by: - - \begin{equation} - \begin{aligned} - \chi \Pi \Theta ' = &- \mathcal{M}_b(\Theta-\langle\Theta \rangle_1)-\frac{{{\mathcal{M}_b}{F_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\rm{\Theta }} \rangle_1} - {\langle{\rm{\Theta }} \rangle_{2}}} \right) - \\ - &+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\rm{\Theta }} - {{\rm{\Theta }}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\rm{\Theta }}} - {{\rm{\Theta }}_1}}} - \frac{{{{\rm{\Theta }}_1} - {{\rm{\Theta }}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &- \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + (\eta \Pi - {F_b}){\mathcal{M}_r}}}{{\eta \Pi }}{\Gamma _{\Theta}} - \frac{\chi }{\eta }{\mathcal{F}_r}\Gamma_{\!\Theta}' + W, - \\ - \chi \Pi \Phi ' = &- {\mathcal{M}_b}\left( {\Phi - {\langle{\rm{\Phi }} \rangle_1}} \right)-\frac{{{\mathcal{M}_b}{F_1} + {\mathcal{M}_1}\left( {\eta \Pi - {F_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\rm{\Phi }} \rangle_1} - {\langle{\rm{\Phi }} \rangle_{2}}} \right) - \\ - &+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\rm{\Phi }} - {{\rm{\Phi }}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\rm{\Phi }}} - {{\rm{\Phi }}_1}}} - \frac{{{{\rm{\Phi }}_1} - {{\rm{\Phi }}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &+ \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta }}{{{{\left( {\sin \Theta } \right)}^2}}} - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi \sin \Theta }}} \right){\Gamma _{\Phi}} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}}}{{\sin \Theta }}\Gamma_{\!\Phi}', - \end{aligned} - \label{eq:neutralbitwalkmodelfordesign} - \end{equation} - - - where the term related to the weight of the BHA is included in the first equation and is given by $W = -\frac{\mathcal{M}_b \mathcal{F}_w + (\eta \Pi - \mathcal{F}_b)\mathcal{M}_w}{\eta \Pi}\Upsilon \sin \langle\Theta\rangle_1 -\frac{\chi}{\eta}\mathcal{F}_w (\Theta - \Theta_1)\Upsilon \cos \langle\Theta\rangle_1$ and $\Upsilon$ is the normalized weight of the BHA. - - From Equation \eqref{eq:neutralbitwalkmodelfordesign}, the states of the system can be defined as: - - \begin{equation} - x_\Theta = \begin{bmatrix} - \Theta \\ - \langle \Theta \rangle_1\\ - \langle\Theta \rangle_2 - \end{bmatrix}, \qquad - x_\Phi = \begin{bmatrix} - \Phi \\ - \langle \Phi \rangle_1\\ - \langle\Phi \rangle_2 - \end{bmatrix}. - \end{equation} - - The system's output equations are given corresponding to measurements not exactly at the bit, but as the orientation of the BHA at locations between the stabilizers and given by: - - \begin{align} - y_\Theta &= C_\Theta x_\Theta + D_\Theta \Gamma_\Theta + E \Upsilon \sin \langle \Theta \rangle_1 \label{eq:output1}\\ - y_\Phi &= C_\Phi x_\Phi + D_\Phi \frac{\Gamma_\Phi}{\sin \Theta}.\label{eq:output2} - \end{align} - - If it is assumed that both the inclination and the azimuth sensors are at the same location then $C_\Theta = C_\Phi$ and $D_\Theta = D_\Phi$. - - The input transformation on Equation \eqref{eq:decouplinginputtransformation2} is proposed in order to achieve decoupling of the system, it has to be noticed that the transformation on $\Gamma_\Phi$ depends on the estimate $\check{\Theta}$ of the inclination, if there is no access to measurements of the orientation of the borehole at the bit. - - \begin{equation} - \begin{bmatrix} - \Gamma_{\!\Theta}^* \\ \Gamma_{\!\Phi}^* - \end{bmatrix} := \begin{bmatrix} 1 & 0 \\ 0 & \frac{1}{\sin \check{\Theta}}\end{bmatrix} \begin{bmatrix}\Gamma_{\!\Theta} \\ \Gamma_{\!\Phi} \end{bmatrix}, \qquad \text{for } \Theta\in (0,\; \pi), - \label{eq:decouplinginputtransformation2} - \end{equation} - - Since the input transformation does not change the structure of the first equation in \eqref{eq:neutralbitwalkmodelfordesign}, only the second equation is shown and is given by: - - \begin{equation} - \begin{aligned} - \chi \Pi \Phi ' = &- {\mathcal{M}_b}\left( {\Phi - {\langle{\rm{\Phi }} \rangle_1}} \right)-\frac{{{\mathcal{M}_b}{F_1} + {\mathcal{M}_1}\left( {\eta \Pi - {F_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\rm{\Phi }} \rangle_1} - {\langle{\rm{\Phi }} \rangle_{2}}} \right) - \\ - &+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\rm{\Phi }} - {{\rm{\Phi }}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\rm{\Phi }}} - {{\rm{\Phi }}_1}}} - \frac{{{{\rm{\Phi }}_1} - {{\rm{\Phi }}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &+ \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta \sin \check{\Theta} }}{{{{\left( {\sin \Theta } \right)}^2}}} - - \frac{{{(\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}}) \right)\sin\check{\Theta}}}{{\eta \Pi \sin \Theta }} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\check{\Theta} '\cos \check{\Theta}}}{{{{\left( {\sin \Theta } \right)}}}}} \right){\Gamma _{\Phi}^*} - - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r \sin \check{\Theta}}}}{{\sin \Theta }}\Gamma_{\!\Phi}^{* '}. - \end{aligned} - \label{neutralbitwalkphi} - \end{equation} - - - and the system equations can be rewritten as: - - \begin{align} - \begin{bmatrix} - x_\Theta' \\ - x_\Phi' - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 \\ - 0 & A_0 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi) \\ - x_\Phi(\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 \\ - 0 & A_1 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_1) \\ - x_\Phi(\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 \\ - 0 & A_2 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_2) \\ - x_\Phi(\xi_2) - \end{bmatrix} \nonumber\\ - +& - \begin{bmatrix} - B_{0\Theta} & 0\\ - 0 & B_{0\Phi} - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^* \\ - \Gamma_\Phi^* - \end{bmatrix} + - \begin{bmatrix} - B_{1\Theta} & 0\\ - 0 & B_{1\Phi} - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^{*'} \\ - \Gamma_\Phi^{*'} - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 - \end{bmatrix}, - \label{eq:systemstatespace} - \end{align} - - where the matrices and vectors $A_0$, $A_1$, $A_2$, $B$, $B_{0i}$ and $B_{1i}$, for $i = \Theta, \Phi$, are given by: - - \begin{align} - A_{0}&= \frac{1}{\chi\Pi}\begin{bmatrix} - -\mathcal{M}_b+ \frac{\chi}{\eta}(\mathcal{F}_b-\mathcal{F}_1) & \mathcal{M}_b-\mathcal{M}_1+\frac{\mathcal{F}_b\mathcal{M}_1-\mathcal{F}_1\mathcal{M}_b}{\eta\Pi} & \mathcal{M}_1+\frac{(-\mathcal{F}_b\mathcal{M}_1+\mathcal{F}_1\mathcal{M}_b)}{\eta\Pi} \\ - \chi\Pi & 0 & 0 \\ - 0 & 0 & 0 \\ - \end{bmatrix}, \nonumber \\\nonumber - \\ \nonumber - A_{1}&= \frac{1}{\chi\Pi}\begin{bmatrix} - \frac{\chi}{\eta}(\mathcal{F}_1+\frac{\mathcal{F}_1}{\varkappa_2}-{\mathcal{F}_b}) & 0 & 0 \\ - -\chi\Pi & 0 & 0 \\ - \frac{\chi\Pi}{\varkappa_2} & 0 & 0 \\ - \end{bmatrix}, \qquad - A_{2}= \frac{1}{\chi\Pi}\begin{bmatrix} - -\frac{\chi \mathcal{F}_1}{\eta\varkappa_2} & 0 & 0 \\ - 0 & 0 & 0 \\ - -\frac{\chi\Pi}{\varkappa_2} & 0 & 0 \\ - \end{bmatrix},\nonumber - \\\nonumber - \\ \nonumber - B_{0\Theta}&= \frac{1}{\chi\Pi}\begin{bmatrix} - - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + (\eta \Pi - {F_b}){\mathcal{M}_r}}}{{\eta \Pi }}, & 0, & 0 \\ - \end{bmatrix}^{T} , \qquad - B_{1\Theta}= \frac{1}{\chi\Pi}\begin{bmatrix} - -\frac{\chi}{\eta}\mathcal{F}_r , & 0, & 0 \\ - \end{bmatrix}^{T}, - \\\nonumber - \\ \nonumber - B_{0\Phi}&= \frac{1}{\chi\Pi}\begin{bmatrix} - {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta \sin \check{\Theta} }}{{{{\left( {\sin \Theta } \right)}^2}}} - - \frac{{{(\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}}) \right)\sin\check{\Theta}}}{{\eta \Pi \sin \Theta }} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\check{\Theta} '\cos \check{\Theta}}}{{{{\left( {\sin \Theta } \right)}}}}}, & 0, & 0 \\ - \end{bmatrix}^{T} \nonumber \\ \nonumber - \\ \nonumber - B_{1\Phi}&= \frac{1}{\chi\Pi}\begin{bmatrix} - -\frac{\chi}{\eta}\mathcal{F}_r\frac{\sin \check{\Theta}}{\sin \Theta} , & 0, & 0 \\ - \end{bmatrix}^{T}, \qquad B = \begin{bmatrix}1 & 0 & 0\end{bmatrix}^T. - \end{align} - - After the input transformation, the output equations are given by: - - \begin{align} - y_\Theta &= C_\Theta x_\Theta + D_\Theta \Gamma_\Theta^* + E W_y, \label{eq:output12}\\ - y_\Phi &= C_\Phi x_\Phi + D_\Phi \Gamma_\Phi^* \frac{\sin \check{\Theta}}{\sin \Theta} .\label{eq:output22}, - \end{align} - - where the gravity related term in $y_\Theta$ has been renamed to $E W_y$, with $W_y = \Upsilon \sin \langle \Theta \rangle_1$ to simplify notation, and take them also as a quasi-constant perturbation. It can be seen that the system equations are complex, especially because of the fact that the terms related to the derivatives of the inclination and its estimate are affecting the azimuth dynamics. This is mainly due to the difference between $\Theta$ and $\check{\Theta}$. For the zero bit walk case (as considered here), this complexity vanishes when $\check{\Theta} = \Theta$. Assuming that we will be able to design an observer that indeed provides (asymptotically) such correct estimate of the inclination, the structure of the controller will be designed for the case when this condition is met. - - \begin{figure}[h]\centering - \includegraphics[width=0.65\textwidth]{ControlStrategy.pdf} - \caption{Control strategy. - \label{fig:ControlStrat} } - \end{figure} - - - The total control scheme proposed here is depicted schematically in Figure \ref{fig:ControlStrat}. In order to combine the (transformed) RSS force inputs $\Gamma_i^*$ for $i = \Theta, \Phi$ and their derivatives, the following transformation is proposed: - - \begin{equation} - B u_i = B_{0i}\Gamma_i^* + B_{1i}\Gamma_i^{*'}, \label{eq:inputs1}\\ - \end{equation} - - where $B = \begin{bmatrix} 1 & 0 & 0\end{bmatrix}^T$ and vectors $B_{0i}$ and $B_{1i}$ correspond to the 3 by 1 vectors related to the inputs of the azimuth and the inclination. - - This would fully replace the terms related to $\Gamma_i^*$ and their derivative in the dynamics given by \eqref{eq:systemstatespace} with $Bu$. However, it is not possible to use the input filter as in \eqref{eq:inputs1} here, since vectors $B_{0\Phi}$ and $B_{1\Phi}$ depend on $\Theta$, which is not known to the controller, since it can not be measured. To overcome this, the design of this input filter will be done for the case when $\check{\Theta} = \Theta$, i.e. - - \begin{align} - \Gamma_{i}^{*'} = -\frac{\mathcal{M}_b \mathcal{F}_r + (\eta \Pi - \mathcal{F}_b)\mathcal{M}_r}{\chi \Pi \mathcal{F}_r}\Gamma_{i}^{*} - \frac{\eta \Pi}{\mathcal{F}_r}u_i. \label{eq:filter1} - \end{align} - - Herein, $u_i$ is a new control input, comprised of the sum of the feedforward and feedback inputs given by $u_i = v_i + u_{ri}$. The stability of the filter is guaranteed as long as $\frac{B_{0i}}{B_{1i}} > 0$. It can be seen, that considering only uncertainty for the weight on bit, this condition is met up to a certain value of $\Pi$. It was proven in \cite{Kremers2013} that this value corresponds to the situation where the system is minimum-phase, this is as long as $\Pi < \Pi_{nmp}$, which is the focus of this work. In \eqref{eq:filter1}, the feedforward input $u_{ri}$ is defined based on the inverse dynamics of the system for a reference trajectory: - - \begin{equation} - u_{ri} = B^T (x'_{ri}(\xi) - A_0 x_{ri}(\xi) - A_1 x_{ri}(\xi_1) - A_2 x_{ri}(\xi_2)). - \label{eq:feedforward} - \end{equation} - - It is important to mention that the gravity related term in the inclination dynamics (W in \eqref{eq:neutralbitwalkmodelfordesign}), is omitted from the feedforward design since it can be considered as a quasi-constant(unknown) disturbance, which can be dealt with by implementing integral action in the control structure. Furthermore, this feedforward is designed again for the case when $\check{\Theta} = \Theta$. Due to this simplification of the feedforward design, a (transient) error is introduced to the system, which will be taken into account explicitly in the resulting error dynamics later. - - - If the input filters are substituted in Equation \eqref{eq:systemstatespace}, then the system dynamics are given by: - - \begin{align} - \begin{bmatrix} - x_\Theta' \\ - x_\Phi' - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 \\ - 0 & A_0 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi) \\ - x_\Phi(\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 \\ - 0 & A_1 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_1) \\ - x_\Phi(\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 \\ - 0 & A_2 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_2) \\ - x_\Phi(\xi_2) - \end{bmatrix} \nonumber\\ - +& \begin{bmatrix} - 0 & 0 \\ - 0 & B\frac{\mathcal{F}_r}{\eta \Pi \sin \Theta} \bigg( \frac{\Theta' \cos \Theta \sin \check{\Theta}}{\sin \Theta} - \check{\Theta}' \cos \check{\Theta}\bigg) - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^* \\ - \Gamma_\Phi^* - \end{bmatrix} + \begin{bmatrix} - B & 0 \\ - 0 & B\frac{\sin \check{\Theta}}{\sin \Theta} - \end{bmatrix} - \begin{bmatrix} - u_\Theta \\ - u_\Phi - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 - \end{bmatrix}. - \label{eq:systemstatespace2} - \end{align} - - - In the case of the inclination dynamics, the input filter successfully makes the terms related to $\Gamma_\Theta^*$ disappear from the equation. On the other hand, in the equation related to the azimuth dynamics, one term related to $\Gamma_\Phi^*$ remains due to the mismatch between $\check{\Theta}$ and $\Theta$. Moreover, in \eqref{eq:systemstatespace2} there is a nonlinear factor multiplied by input $u_\Phi$. If the input filters in \eqref{eq:filter1} are included in the equation, the system dynamics can be rewritten as follows: - - \begin{align} - \begin{bmatrix} - x_\Theta' \\ - \Gamma_\Theta^{*'} \\ - x_\Phi' \\ - \Gamma_\Phi^{*'} - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & -b_0 & 0 & 0 \\ - 0 & 0 & A_0 & B\alpha(\Theta,\check{\Theta},\Theta',\check{\Theta}') \\ - 0 & 0 & 0 & -b_0 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi) \\ - \Gamma_\Theta^{*}(\xi) \\ - x_\Phi(\xi) \\ - \Gamma_\Phi^{*} (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_1) \\ - \Gamma_\Theta^{*}(\xi_1) \\ - x_\Phi(\xi_1) \\ - \Gamma_\Phi^{*} (\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 & 0 & 0 \\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_2) \\ - \Gamma_\Theta^{*}(\xi_2) \\ - x_\Phi(\xi_2) \\ - \Gamma_\Phi^{*} (\xi_2) - \end{bmatrix}\nonumber \\ - &+ - \begin{bmatrix} - B & 0 \\ - -b_1 & 0 \\ - 0 & B \frac{\sin \check{\Theta}}{\sin \Theta} \\ - 0 & -b_1 - \end{bmatrix} - \begin{bmatrix} - u_\Theta \\ - u_\Phi - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 \\ - 0 \\ - 0 - \end{bmatrix}. - \end{align} - - where: - - \begin{equation} - b_0 = \frac{\mathcal{M}_b \mathcal{F}_r + (\eta \Pi - \mathcal{F}_b)\mathcal{M}_r}{\chi \Pi \mathcal{F}_r}, \quad b_1 = \frac{\eta \Pi}{\mathcal{F}_r}, \quad \alpha (\Theta,\check{\Theta},\Theta',\check{\Theta}') = \frac{\mathcal{F}_r}{\eta \Pi \sin \Theta} \bigg( \frac{\Theta' \cos \Theta \sin \check{\Theta}}{\sin \Theta} - \check{\Theta}' \cos \check{\Theta}\bigg). \label{eq:Constants} - \end{equation} - - It has to be noticed that $\alpha$ depends on the states $\Theta$ and $\check{\Theta}$, and their derivatives. To simplify notation, $\alpha$ will be written instead of $\alpha(\Theta,\check{\Theta}, \Theta', \check{\Theta}')$. From these equations, the error dynamics of the system (with the error defined as $e_i = x_i - x_{ri}$) are obtained in \eqref{eq:errordynamics}, using for the feedforward the expression given in \eqref{eq:feedforward} and the control input decomposition $u_i = u_{ri} + v_i$: - - \begin{align} - \begin{bmatrix} - e_\Theta' \\ - \Gamma_\Theta^{*'} \\ - e_\Phi' \\ - \Gamma_\Phi^{*'} - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & -b_0 & 0 & 0 \\ - 0 & 0 & A_0 & B \alpha \\ - 0 & 0 & 0 & -b_0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi) \\ - \Gamma_\Theta^{*} (\xi) \\ - e_\Phi(\xi) \\ - \Gamma_\Phi^{*} (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_1) \\ - \Gamma_\Theta^{*} (\xi_1) \\ - e_\Phi(\xi_1) \\ - \Gamma_\Phi^{*} (\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_2) \\ - \Gamma_\Theta^{*} (\xi_2) \\ - e_\Phi(\xi_2) \\ - \Gamma_\Phi^{*} (\xi_2) - \end{bmatrix} \nonumber\\ - &+ \begin{bmatrix} - B & 0 \\ - -b_1 & 0 \\ - 0 & B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta}\bigg) \\ - 0 & -b_1 - \end{bmatrix} - \begin{bmatrix} - v_\Theta \\ - v_\Phi - \end{bmatrix} + - \begin{bmatrix} - 0 \\ - -b_1 u_{r\Theta} \\ - B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1 \bigg) u_{r\Phi} \\ - -b_1 u_{r\Phi} - \end{bmatrix}+ - \begin{bmatrix} - BW \\ - 0 \\ - 0 \\ - 0 - \end{bmatrix}. - \label{eq:errordynamics} - \end{align} - - In a similar way, if an input filter as in equation \eqref{eq:filter1} is designed based on the feedforward input, the following equation can be obtained: - - \begin{equation} - \Gamma_{id}^{*'} = -\frac{\mathcal{M}_b \mathcal{F}_r + (\eta \Pi - \mathcal{F}_b)}{\chi \Pi \mathcal{F}_r}\Gamma_{id}^{*} - \frac{\eta \Pi}{\mathcal{F}_r}u_{ri}, \label{eq:filter2} - \end{equation} - - where $\Gamma_{id}^*$ is a desired input of the RSS corresponding to the feedforward input. Then, an error coordinate for the transformed input $\Gamma_i^*$ can be defined as $\Delta \Gamma^*_{i} = \Gamma_i^* - \Gamma_{id}^*$. Then equation \eqref{eq:errordynamics} can be rewritten as: - - \begin{align} - \begin{bmatrix} - e_\Theta' \\ - \Delta \Gamma_\Theta^{*'} \\ - e_\Phi' \\ - \Delta \Gamma_\Phi^{*'} - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & -b_0 & 0 & 0 \\ - 0 & 0 & A_0 & B \alpha \\ - 0 & 0 & 0 & -b_0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi) \\ - \Delta \Gamma_\Theta^{*} (\xi) \\ - e_\Phi(\xi) \\ - \Delta \Gamma_\Phi^{*} (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_1) \\ - \Delta \Gamma_\Theta^{*} (\xi_1) \\ - e_\Phi(\xi_1) \\ - \Delta \Gamma_\Phi^{*} (\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_2) \\ - \Delta \Gamma_\Theta^{*} (\xi_2) \\ - e_\Phi(\xi_2) \\ - \Delta \Gamma_\Phi^{*} (\xi_2) - \end{bmatrix} \nonumber\\ - &+ \begin{bmatrix} - B & 0 \\ - -b_1 & 0 \\ - 0 & B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta}\bigg) \\ - 0 & -b_1 - \end{bmatrix} - \begin{bmatrix} - v_\Theta \\ - v_\Phi - \end{bmatrix} + - \begin{bmatrix} - 0 \\ - 0 \\ - B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1 \bigg) u_{r\Phi} \\ - 0 - \end{bmatrix} - \label{eq:errordynamics2}+ - \begin{bmatrix} - BW \\ - 0 \\ - B\alpha \Gamma_{\Phi d}^* \\ - 0 - \end{bmatrix}. - \end{align} - - After obtaining the error dynamics, the state feedback controller corresponding to the input $v_i$ is defined as: - - \begin{align} - z_{1i}' &= \zeta \begin{bmatrix} - k_{1i} & 0 & 0 - \end{bmatrix}e_i \label{eq:controllerlowpass}\\ - z_{2i}' &= -\gamma z_{2i} + \gamma (z_{1i} + K_i e_i) \label{eq:controllerintegral}\\ - v_i &= z_{2i}\label{eq:controllerfeedback}. - \end{align} - - This controller consists of a static state (error $e_i$) feedback part, a low-pass filter and integral action. Considering the fact that the states can not be measured directly, an observer needs to be designed in order to support the implementation of the controller in \eqref{eq:controllerlowpass}-\eqref{eq:controllerfeedback}. Since the weight of the BHA is taken into account as a quasi-constant disturbance, integral action is also added to the observer design. The integral action of the observer is embedded through the integral filter: - - \begin{equation} - q_i' = \zeta[l_1,l_2](y_i - \check{y}_i), \qquad \text{ for } i=\Theta,\Phi. \label{eq:observerintegral} - \end{equation} - - The observer consists of a model-based (predictor) part and an output-injection part. The predictor part of the observer is designed by again considering the condition where $\Theta = \check{\Theta}$. In total, the dynamics of the observer with integral action are given by: - - \begin{align} - \begin{bmatrix} - \check{x}_\Theta' \\ - q_\Theta' \\ - \check{x}_\Phi'\\ - q_\Phi' - \end{bmatrix} &= - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_0 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi) \\ - q_\Theta(\xi) \\ - \check{x}_\Phi(\xi) \\ - q_\Phi (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi_1) \\ - q_\Theta(\xi_1) \\ - \check{x}_\Phi(\xi_1) \\ - q_\Phi(\xi_1) \\ - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi_2) \\ - q_\Theta(\xi_2) \\ - \check{x}_\Phi(\xi_2) \\ - q_\Phi(\xi_2) - \end{bmatrix} \nonumber\\ - &+ - \begin{bmatrix} - L_\Theta(y_{\Theta} - \check{y}_{\Theta})\\ - \zeta[l_{1\Theta},l_{2\Theta}](y_\Theta - \check{y}_\Theta) \\ - L_{\Phi}(y_{\Phi} - \check{y}_{\Phi}) \\ - \zeta[l_{1\Phi},l_{2{\Phi}}](y_\Phi - \check{y}_\Phi) - \end{bmatrix} + - \begin{bmatrix} - Bq_\Theta \\ - 0 \\ - Bq_\Phi \\ - 0 - \end{bmatrix} + - \begin{bmatrix} - B(u_{r\Theta} + v_\Theta)\\ - 0 \\ - B(u_{r\Phi} + v_\Phi) \\ - 0 - \end{bmatrix}, - \end{align} - - where $L_i$ is defined as: - - \begin{equation} - L_i = \begin{bmatrix} - l_{1i} & l_{2i} \\ - 0 & 0 \\ - 0 & 0 - \end{bmatrix}, - \end{equation} - - and with the output observer equations (taking into account the ideal input decoupling transformation) given by: - - \begin{align} - \check{y}_\Theta &= C_\Theta \check{x}_\Theta + D_\Theta \Gamma_\Theta^* \\ - \check{y}_\Phi &= C_\Phi \check{x}_\Phi + D_\Phi \Gamma_\Phi^* - \end{align} - - Computing the observer error dynamics (with $\delta_i = x_i - \check{x}_i$) and including the integral action results in the following equations: - - - \begin{align} - \begin{bmatrix} - \delta_\Theta' \\ - q_\Theta' \\ - \delta_\Phi' \\ - q_\Phi' - \end{bmatrix} =& - \begin{bmatrix} - A_0 - L_\Theta C_\Theta & -B & 0 & 0\\ - \zeta [l_{1\Theta},l_{2\Theta}]C_\Theta & 0 & 0 & 0 \\ - 0 & 0 & A_0 - L_\Phi C_\Phi & -B \\ - 0 & 0 & \zeta [l_{1\Phi},l_{2\Phi}]C_\Phi &0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta(\xi) \\ - q_\Theta (\xi) \\ - \delta_\Phi(\xi) \\ - q_\Phi(\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_1 & 0\\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta(\xi_1) \\ - q_\Theta (\xi_1) \\ - \delta_\Phi(\xi_1) \\ - q_\Phi (\xi_1) - \end{bmatrix} \nonumber \\ - +&\begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_2 & 0\\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta(\xi_2) \\ - q_\Theta (\xi_2) \\ - \delta_\Phi(\xi_2) \\ - q_\Phi (\xi_2) - \end{bmatrix} + - \begin{bmatrix} - 0 \\ - 0 \\ - B(u_{r\Phi} + v_\Phi)\big(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\big) \\ - 0 - \end{bmatrix} - + - \begin{bmatrix} - BW - L_{\Theta}EW_y\\ - 0 \\ - \big(B\alpha - L_\Phi D_\Phi\big(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\big)\big)\Gamma_\Phi^*\\ - \zeta[l_{1\Phi},l_{2\Phi}]D_\Phi (\frac{\sin \check{\Theta}}{\sin \Theta} - 1)\Gamma_\Phi^* - \end{bmatrix}. - \end{align} - - If now, the observer is combined with the state-feedback controller, the error dynamics can be obtained, first applying the controller in the following observer-based feedback form: - - \begin{align} - z_{1i}' &= \zeta \begin{bmatrix} - k_{1i} & 0 & 0 - \end{bmatrix}(\check{x}_i - x_{ri}) \label{eq:controllerlowpasserror}\\ - z_{2i}' &= -\gamma z_{2i} + \gamma (z_{1i} + K_i(\check{x}_i - x_{ri})) \label{eq:controllerintegralerror}\\ - v_i &= z_{2i} \label{eq:controllerfeedbackerror}, - \end{align} - - and using the fact that $\check{x}_i - x_{ri} = e_i - \delta_i$ and $\Gamma_i^* = \Delta \Gamma_i^* + \Gamma_{id}^*$, the total error dynamics of the state vector: - - \begin{equation} - X(\xi) = \begin{bmatrix} - e_{\Theta}^T(\xi) \text{ }\Delta \Gamma_\Theta^{*}(\xi) \text{ }z_{1\Theta}(\xi) \text{ } z_{2\Theta}(\xi) \text{ }\delta_{\Theta}^T(\xi) \text{ } q_\Theta(\xi) \text{ }e_{\Phi}^T(\xi) \text{ }\Delta \Gamma_\Phi^*(\xi) \text{ }z_{1\Phi}(\xi) \text{ }z_{2\Phi}(\xi) \text{ }\delta_{\Phi}^T(\xi) \text{ }q_\Phi(\xi) \nonumber - \end{bmatrix}^T. - \end{equation} - - are given by: - - \begin{equation} - X'(\xi) = A_{0cl}X(\xi) + A_{1cl}X(\xi_1) + A_{2cl} X(\xi_2) + P_{cl}(u_{r\Phi},\Gamma_{\Phi d}^*,\alpha, \Theta,\check{\Theta},W),\\ - \label{eq:totaldynamics} - \end{equation} - - where: - - \begin{equation} - A_{0cl} = - \begin{bmatrix} - A_{0\Theta} & 0 \\ - 0 & A_{0\Phi} - \end{bmatrix}, \qquad - A_{1cl} = - \begin{bmatrix} - A_{1\Theta} & 0 \\ - 0 & A_{1\Phi} - \end{bmatrix}, \qquad - A_{2cl} = - \begin{bmatrix} - A_{2\Theta} & 0 \\ - 0 & A_{2\Phi} - \end{bmatrix}, - \label{eq:ClosedLoopMatrices} - \end{equation} - - where the system matrices in \eqref{eq:ClosedLoopMatrices} and the vector $P_{cl}$ are given by: - - \begin{align*} - A_{0\Theta} &= - \begin{bmatrix}[cccc:cc] - A_0 & 0 & 0 & B & 0 & 0\\ - % - 0 & -b_0 & 0 & -b_1 & 0 & 0\\ - \zeta \begin{bmatrix}k_{1\Theta} , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_{1\Theta} , 0 , 0\end{bmatrix} & 0\\ - % - \gamma K_\Theta & 0 & \gamma & -\gamma & -\gamma K_\Theta & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_0 - L_\Theta C_\Theta & -B\\ - % - 0 & 0 & 0 & 0 & \zeta \begin{bmatrix}l_{1\Theta} , l_{2\Theta} \end{bmatrix} C_\Theta & 0 - \end{bmatrix},\\ - \nonumber \\ - A_{0\Phi} &= - \begin{bmatrix}[cccc:cc] - A_0 & B\alpha & 0 & B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta}\bigg) & 0 & 0\\ - % - 0 & -b_0 & 0 & -b_1 & 0 & 0 \\ - % - \zeta \begin{bmatrix}k_{1\Phi} , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_{1\Phi} , 0 , 0\end{bmatrix} & 0 \\ - % - \gamma K_{\Phi} & 0 & \gamma & -\gamma & -\gamma K_{\Phi} & 0\\ \hdashline - % - 0 & B\alpha -L_\Phi D_{\Phi}\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg) & 0 & B\big(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\big) & A_0 - L_\Phi C_{\Phi} & -B \\ - % - 0 & \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix} D_\Phi\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg) & 0 & 0 & \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix}C_\Phi & 0 - \end{bmatrix},\nonumber \\ - \nonumber \\ - A_{1i} &= - \begin{bmatrix}[cccc:cc] - A_1 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_1 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix}, \quad - A_{2i} = - \begin{bmatrix}[cccc:cc] - A_2 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_2 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix}, \nonumber \\ - \nonumber \\ - P_{cl}&(u_{r\Phi},\Gamma_{\Phi d}^*,\alpha, \Theta,\check{\Theta},W) = \begin{bmatrix} - BW \\ - 0 \\ - 0 \\ - 0 \\ \hdashline - BW - L_\Theta E W_y\\ - 0 \\ \hline - B \bigg(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg) u_{r\Phi} + B\alpha\Gamma_{\Phi d}^*\\ - 0 \\ - 0 \\ - 0 \\ \hdashline - B \bigg(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg) u_{r\Phi} + B\alpha\Gamma_{\Phi d}^* - LD\bigg( \frac{\sin \check{\Theta}}{\sin \Theta}-1\bigg)\Gamma_{\Phi d}^*\\ - \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix} D_\Phi\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg)\Gamma_{\Phi d}^* \\ - \end{bmatrix}. - \\\nonumber - \end{align*} - - It can be seen that the vector $P_{cl}$ contains the gravity related terms, which were not included in the feedforward design. Nevertheless, the integral part of the controller will take care of the rejection of this perturbation. As well, the other perturbation terms in $P_{cl}$ vanish if $\check{\Theta} = \Theta$ (i.e. if the inclination observer error is zero). - - \newpage - - \section{Linearization} - - This system will be linearized around $e_i = 0$ and $\delta_i = 0$. Initially the inclination-related errors heve been defined as: - - - \begin{equation} - e_\Theta = \begin{bmatrix} - e_{\Theta_0} \\ - e_{\langle \Theta \rangle_1} \\ - e_{\langle \Theta \rangle_2} - \end{bmatrix} := \begin{bmatrix} - \Theta - \Theta_d \\ - \langle \Theta \rangle_1 - \langle \Theta \rangle_{1d} \\ - \langle \Theta \rangle_2 - \langle \Theta \rangle_{2d} - \end{bmatrix}. - \end{equation} - - Then from this expression $\Theta$, $\langle \Theta \rangle_1$ and $\langle \Theta \rangle_2 $ can be obtained as: - - \begin{align} - \Theta &= e_{\Theta_0} + \Theta_d,\quad \langle \Theta \rangle_1 = e_{\langle \Theta \rangle_1} + \langle \Theta \rangle_{1d},\quad \langle \Theta \rangle_2 = e_{\langle \Theta \rangle_2} + \langle \Theta \rangle_{2d}. - \label{eq:theta} - \end{align} - - In a similar fashion, the observer errors related to the inclination have been defined as: - \begin{equation} - \delta_\Theta = \begin{bmatrix} - \delta_{\Theta_0} \\ - \delta_{\langle \Theta \rangle_1} \\ - \delta_{\langle \Theta \rangle_2} - \end{bmatrix} := \begin{bmatrix} - \Theta - \check{\Theta} \\ - \langle \Theta \rangle_1 - \langle \check{\Theta} \rangle_{1} \\ - \langle \Theta \rangle_2 - \langle \check{\Theta} \rangle_{2} - \end{bmatrix}. - \label{eq:observererror} - \end{equation} - - Then $\check{\Theta}$, $\langle \check{\Theta} \rangle_{1}$ and $\langle \check{\Theta} \rangle_{2}$ can be obtained as: - - \begin{align} - \check{\Theta} &= e_{\Theta_0} + \Theta_d - \delta_{\Theta_0}, \quad \langle \check{\Theta} \rangle_{1} = e_{\langle \Theta \rangle_1} + \langle \Theta \rangle_{1d} - \delta_{\langle \check{\Theta} \rangle_{1}}, \quad \langle \check{\Theta} \rangle_{2} = e_{\langle \Theta \rangle_2} + \langle \Theta \rangle_{2d} - \delta_{\langle \check{\Theta} \rangle_{2}} - \label{eq:esttheta}. - \end{align} - - In order to obtain the nominal values for the rest of the states, the values of $e_i = 0$ and $\delta_i = 0$ are substituted into equation \eqref{eq:totaldynamics}. When the observer errors $\delta_i$ are zero, the condition $\check{\Theta} = \Theta$ is met, and by subsequently setting the derivatives to zero we obtain the following set of equilibrium equations: - - \begin{align} - 0 &= BW + Bz_{2\Theta} \nonumber\\ - %1 - 0 &= -b_0 \Delta \Gamma_\Theta^{*} - b_1 z_{2\Theta} \nonumber\\ - %3 - 0 &= -\gamma z_{2\Theta} + \gamma z_{1\Theta} \nonumber\\ - %4 - 0 &= BW - Bq_\theta \nonumber - L_\Theta E W_y \\ - %6 - 0 &= Bz_{2\Phi}\nonumber \\ - %7 - 0 &= -b_0 \Delta \Gamma_\Phi^{*} - b_1 z_{2\Phi} \nonumber\\ - %9 - 0 &= -\gamma z_{2\Phi} + \gamma z_{1\Phi} \nonumber\\ - %10 - 0 &= Bq_\Phi \nonumber\\ - \label{eq:linpoint} - \end{align} - - From the set of equations \eqref{eq:linpoint}, the nominal values for $z_{1i}$, $z_{2i}$, $\Delta \Gamma_i^*$ and $q_i$ can be obtained. With these remarks, the equilibrium values for all the states are given by: - - \begin{align} - e_i &= 0\nonumber\\ - \delta_i &= 0 \nonumber\\ - z_{1\Theta} &= -W\nonumber\\ - z_{2\Theta} &= -W\nonumber\\ - q_\Theta &= W - [l_{1\Theta}, l_{2\Theta}]E W_y \nonumber\\ - z_{1\Phi} &= 0\nonumber\\ - z_{2\Phi} &= 0\nonumber\\ - q_\Phi &= 0 \nonumber\\ - \Delta \Gamma_\Theta^* &= \frac{b_1}{b_0}W \nonumber\\ - \Delta \Gamma_\Phi^* &= 0 - \end{align} - - We can then introduce the perturbed versions of the states where the equilibrium is not at a value of zero as follows: - - \begin{align} - \Delta z_{1\Theta} &= z_{1\Theta} - (-W) \nonumber\\ - \Delta z_{2\Theta} &= z_{2\Theta} - (-W) \nonumber\\ - \Delta q_{\Theta} &= q_{\Theta} - (W - [l_{1\Theta}, l_{2\Theta}]E W_y ) \nonumber\\ - \Delta \bar{\Gamma}_{\Theta}^* &= \Delta \Gamma_{\Theta}^* - \frac{b_1}{b_0}W - \end{align} - - - - In order to perform the linearization, the equations are expressed in terms of the error dynamics and the observer error dynamics. Furthermore, the terms related to $\Theta'$ and $\check{\Theta}'$ in $\alpha$ as defined in \eqref{eq:Constants}, were substituted by the dynamics in \eqref{eq:neutralbitwalkmodelfordesign}. Due to the complexity of the computations involved this was done using the symbolic toolbox of Matlab according to the following equation: - - \begin{equation} - \bar{X}'(\xi) = \frac{\partial X'(\xi)}{\partial X(\xi)} \bigg|_N \bar{X}(\xi) - + \frac{\partial X'(\xi)}{\partial X(\xi_1)} \bigg|_N \bar{X}(\xi_1) + \frac{\partial X'(\xi)}{\partial X(\xi_2)} \bigg|_N \bar{X}(\xi_2) - \label{eq:linearization} - \end{equation} - - - where the perturbation state vector $\bar{X}(\xi)$ is given by: - - \begin{equation} - \bar{X}(\xi) = \begin{bmatrix} - e_{\Theta}(\xi) \text{ }\Delta \bar{\Gamma}_\Theta^{*}(\xi) \text{ }\Delta z_{1\Theta}(\xi) \text{ }\Delta z_{2\Theta}(\xi) \text{ }\delta_{\Theta}(\xi) \text{ }\Delta q_\Theta(\xi) \text{ }e_{\Phi}(\xi) \text{ }\Delta \Gamma_\Phi^*(\xi) \text{ }z_{1\Phi}(\xi) \text{ }z_{2\Phi}(\xi) \text{ }\delta_{\Phi}(\xi) \text{ }q_\Phi(\xi) \nonumber - \end{bmatrix}^T. - \end{equation} - - After linearizing, the system equations are: - - \begin{equation} - \bar{X}'(\xi) = \bar{A}_{0cl}\bar{X}(\xi) + \bar{A}_{1cl}\bar{X}(\xi_1) + \bar{A}_{2cl} \bar{X}(\xi_2),\\ - \label{eq:totaldynamicslinears} - \end{equation} - - where the linearized matrices $\bar{A}_{0cl}$, $\bar{A}_{1cl}$ and $\bar{A}_{2cl}$, are given by: - - \begin{equation} - \bar{A}_{0cl} = - \begin{bmatrix} - A_{0\Theta} & 0 \\ - A_{0c} & A_{0\Phi} - \end{bmatrix}, \qquad - \bar{A}_{1cl} = - \begin{bmatrix} - A_{1\Theta} & 0 \\ - A_{1c} & A_{1\Phi} - \end{bmatrix}, \qquad - \bar{A}_{2cl} = - \begin{bmatrix} - A_{2\Theta} & 0 \\ - A_{2c} & A_{2\Phi} - \end{bmatrix}, \label{eq:LinearMatrices} - \end{equation} - - and sub-matrices defined as: - - \begin{align} - A_{0i} &= - \begin{bmatrix}[cccc:cc] - A_0 & 0 & 0 & B & 0 & 0\\ - % - 0 & -b_0 & 0 & -b_1 & 0 & 0\\ - \zeta \begin{bmatrix}k_{1i} , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_{1i} , 0 , 0\end{bmatrix} & 0\\ - % - \gamma K_i & 0 & \gamma & -\gamma & -\gamma K_i & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_0 - L_iC_i & -B\\ - % - 0 & 0 & 0 & 0 & \zeta \begin{bmatrix}l_{1i} , l_{2i} \end{bmatrix} C_i & 0 - \end{bmatrix}, \qquad - A_{0c} = - \begin{bmatrix}[cccc:cc] - 0 & 0 & 0 & 0 & p_{0e1}(\xi) & p_{0e2}(\xi) \\ - 0 & 0 & 0 & 0 & 0 & 0 \\ - 0 & 0 & 0 & 0 & 0 & 0 \\ - 0 & 0 & 0 & 0 & 0 & 0 \\ \hdashline - 0 & 0 & 0 & 0 & p_{0\delta 1}(\xi) & p_{0\delta 2}(\xi) \\ - 0 & 0 & 0 & 0 & p_{0q}(\xi) & 0 - \end{bmatrix} \nonumber\\ - \nonumber\\ - A_{1i} &= - \begin{bmatrix}[cccc:cc] - A_1 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_1 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix}, \qquad - A_{1c} = \begin{bmatrix}[cccc:cc] - 0 & 0 & 0 & 0 & p_{1e}(\xi) & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - 0 & 0 & 0 & 0 & p_{1\delta}(\xi) & 0\\ - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix} \nonumber\\ - \nonumber\\ - A_{2i} &= - \begin{bmatrix}[cccc:cc] - A_2 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_2 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix}, \qquad - A_{2c} = \begin{bmatrix}[cccc:cc] - 0 & 0 & 0 & 0 & p_{2e}(\xi) & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - 0 & 0 & 0 & 0 & p_{2\delta}(\xi) & 0\\ - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix} \nonumber - \end{align} - - and with coefficients given by: - - \begin{align} - p_{0e1}(\xi) =& -B\frac{\mathcal{F}_r \Gamma_{\Phi d}^*}{\eta \Pi \sin \Theta_d} \begin{bmatrix} \begin{aligned} - &\beta_1 \Theta_d + \beta_2 \langle \Theta \rangle_{1d} + \beta_3 \langle \Theta \rangle_{2d} + \beta_4 \Theta_{1d} + \beta_5 \Theta_{2d} \bigg(\frac{1}{\sin \Theta_d}\bigg) +...\\ - &\kern2cm (-\beta_1 + \beta_2 \langle \Theta \rangle_{1d} + \beta_3 \langle \Theta \rangle_{2d} + \beta_4 \Theta_{1d} + \beta_5 \Theta_{2d})\cos \Theta_d \\ - &-\beta_2 \cos \Theta_d \\ - &-\beta_3 \cos \Theta_d\end{aligned}\end{bmatrix}^T - B\begin{bmatrix}\frac{\cos \Theta_d}{\sin \Theta_d}u_{r\Phi} \\\\0 \\0 \end{bmatrix}^T, \nonumber \\ - \nonumber\\ - p_{0e2}(\xi) =& -B\frac{\mathcal{F}_r \Gamma_{\Phi d}^*}{\eta \Pi \sin \Theta_d}\cos \Theta_d, \nonumber \\ - \nonumber \\ - p_{0\delta 1}(\xi) =& -B\frac{\mathcal{F}_r \Gamma_{\Phi d}^*}{\eta \Pi \sin \Theta_d} \begin{bmatrix} \begin{aligned} - &\beta_1 \Theta_d + \beta_2 \langle \Theta \rangle_{1d} + \beta_3 \langle \Theta \rangle_{2d} + \beta_4 \Theta_{1d} + \beta_5 \Theta_{2d} \bigg(\frac{1}{\sin \Theta_d}\bigg) +...\\ - &\kern2cm (-\beta_1 + \beta_2 \langle \Theta \rangle_{1d} + \beta_3 \langle \Theta \rangle_{2d} + \beta_4 \Theta_{1d} + \beta_5 \Theta_{2d})\cos \Theta_d \\ - &-\beta_2 \cos \Theta_d \\ - &-\beta_3 \cos \Theta_d\end{aligned}\end{bmatrix}^T - B\begin{bmatrix}\frac{\cos \Theta_d}{\sin \Theta_d}u_{r\Phi} \\\\0\\0 \end{bmatrix}^T \nonumber \\ - &+ L_\Phi D_\Phi\begin{bmatrix}\Gamma_{\Phi d}^*\frac{\cos \Theta_d}{\sin \Theta_d} \\\\0\\0\end{bmatrix}^T, \nonumber \\ - \nonumber \\ - p_{0\delta 2}(\xi) =& -B\frac{\mathcal{F}_r \Gamma_{\Phi d}^*}{\eta \Pi \sin \Theta_d}\cos \Theta_d, \quad - p_{0q}(\xi)= -\zeta \Gamma_{\Phi d}^* [l_{1\Phi},l_{2\Phi}]D_\Phi\frac{\cos \Theta_d}{\sin \Theta_d}, \quad p_{1e}(\xi) = B\frac{\mathcal{F}_r \Gamma_{\Phi d}^* \cos \Theta_d}{\eta \Pi \sin \Theta_d}\begin{bmatrix}\beta_4 & 0 & 0\end{bmatrix}, \nonumber \\ - \nonumber \\ - p_{1\delta}(\xi) =& B\frac{\mathcal{F}_r \Gamma_{\Phi d}^* \cos \Theta_d}{\eta \Pi \sin \Theta_d}\begin{bmatrix}\beta_4& 0 & 0\end{bmatrix}, \quad p_{2e}(\xi) = B\frac{\mathcal{F}_r \Gamma_{\Phi d}^* \cos \Theta_d}{\eta \Pi \sin \Theta_d}\begin{bmatrix}\beta_5& 0 & 0\end{bmatrix}, \quad p_{2\delta}(\xi) = B\frac{\mathcal{F}_r \Gamma_{\Phi d}^* \cos \Theta_d}{\eta \Pi \sin \Theta_d}\begin{bmatrix}\beta_5& 0 & 0\end{bmatrix}, \nonumber - \end{align} - - where elements $\beta_j$, for $j \in \{1,2...,5\}$ correspond to the following elements of the matrices $A_0$, $A_1$ and $A_2$: - - \begin{equation*} - \beta_1 = A_0(1,1), \quad \beta_2 = A_0(1,2), \quad \beta_3 = A_0(1,3), \quad \beta_4 = A_1(1,1), \quad \beta_5 = A_2(1,1). - \end{equation*} - - It can be noticed that, after the linearization the nonlinear perturbation vector $P_{cl}$ vanishes. The nonlinear elements in this vector depend on $\Theta$ and $\check{\Theta}$. As a result of expressing these two states in terms of $e_i$ and $\delta_i$, after linearizing their effects become present inside the $A_{0c}$, $A_{1c}$ and $A_{2c}$ matrices. Moreover, the partial derivatives of these terms with respect to $e_\Theta$ are zero, so only coefficients in this matrices corresponding to the state $\delta_\Theta$ appear in the equation. - - - - - \newpage - -\end{document} - \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/identification.tex b/Research Proposal IIT/Research proposal/sections (copy)/identification.tex deleted file mode 100755 index 9ec838f..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/identification.tex +++ /dev/null @@ -1,136 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\vspace{20pt} -The Rotational Pendulum is open loop unstable in three of its four equilibrium points; linear system identication is not possible at those points in open loop. An approach is taken where first the parameters of the nonlinear model as derived in \ref{sec:Mathematical} will be estimated by the use of a nonlinear parameter estimation at the stable equilibrium, subsequently linearizations can be made at the unstable equilibrium points. Initial guesses for all of the parameters in \ref{eq:eqom} are needed in order to start the parameter estimation. Table \ref{tab:parameters} gives initial guesses for most of the parameters except for three: $d_1$, $d_2$ and the lumped motor gain $K_u$, so initial guesses for those parameters have to be found first. The parameter estimation also needs the input-output data from the real system to have certain properties in order to work. This section will start the system identification steps that are necessary to gather this data, it will then continue with the nonlinear parameter estimation. - -\subsection{System Identification} -The system identification will follow the guidelines as set in Chapter 10 of M. Verhaegen and M. Verdult \cite{Filtering}. -\subsubsection{Choice of Sampling Frequency and Experiment Duration} -In order to properly identify the system it has to be excited in the frequency band of interest defined as $\left[0,\omega_I\right]$ \nomenclature{$\omega_I$}{Frequency band of interest [rad/s]}. A rule of thumb set by \cite{Filtering} is to take $\omega_I=5\omega_B$, where $\omega_B$ is the bandwidth of the system. Subsequently if one then takes the sampling frequency as by Shannon's sampling theorem \cite{Kamen}, the sampling frequency is given as $\omega_S = 10\omega_I$. \nomenclature{$\omega_S$}{Sampling frequency [rad/s]}\nomenclature{$\omega_B$}{Bandwidth of the system [rad/s]} \\ Since -\begin{equation} -\omega_B=\frac{1}{TC} -\end{equation} -with $TC$\nomenclature{TC}{Timeconstant of system [s]} the timeconstant of the system, the bandwidth can be determined by a impulse-response of the system. Figure \ref{fig:impulse2} shows $\theta_2(t)$. -From this impulse response the timeconstant can be determined as -\begin{equation} -TC= \frac{\kappa t_{cycle}}{4} -\end{equation} -Where $\kappa$ is the number of cycles before steady state is reached and $t_{cycle}$ the time of the first cycle. After inspection: $\kappa\approx 12$ and $t_{cycle}\approx 0.55s$. This leads to $TC=1.65s$ and thus $\omega_B\approx0.61\frac{rad}{s}$. Finally the sampling frequency for the input-output data is found to be $6.1\frac{rad}{s}$. The experiment duration must be larger then $10\times TC \approx 16.5s$, again following a rule of thumb from \cite{Filtering}. - -\subsubsection{Choice of Input Signal} -The input-output data must contain all the important system dynamics. The input signal must therefor excite the system in a way that it extracts all the important information. A way to measure the ability of an input signal to do this is by the notion of persistency of excitation\cite{persistancy}. -A choice was made to use multi-sines as an input, since it is straightforward to get the input signal in the right frequency band and with sufficient persistency of excitation. The values of the frequencies chosen for the sine signals are shown in table \ref{tabel:sines} - - -\begin{table}[h] -\centering -\caption{Sine signals for system identification.} -\label{tabel:sines} -\begin{tabular}{|l|l|l|l|l|} -\hline - & Sine 1 & Sine 2 & Sine 3 & Sine 4 \\ \hline -Amplitude & 0.3 & 0.1 & 0.2 & 0.2 \\ \hline -Phase & 2 & 0 & 0 & 0 \\ \hline -Frequency & $\frac{100}{2\pi}$ & $20\sqrt{4}\pi$ & $e^2(\frac{50}{2\pi})$ & $\sqrt{20}$ \\ \hline -\end{tabular} -\end{table} - -Additionally, the plot of the input signal is shown in figure \ref{fig:FirstFitInput}. - - -\begin{figure}[ht] -\centering -\includegraphics[width=0.7\textwidth]{FirstFitInput.png} -\caption{\label{fig:FirstFitInput} Input signal used for system identification.} -\end{figure} - -\subsection{Parameter Estimation} -First the damping coefficients $d_1$ and $d_2$ can be computed from step and free responses of the system as shown in Figure \ref{fig:impulse1} and \ref{fig:impulse2}. The step response of $\theta_1$ is actually the motor stopping, the input was a pulse of 0.1 s. The signal for $\theta_1$ was filtered in order to better show the overshoot. Since $\theta_2$ is not directly actuated, a step response could not be realized. Therefor suspending it manually and then letting it gave a free response. \ref{fig:impulse2}. -\begin{figure}[ht] -\centering -\includegraphics[width=0.8\textwidth]{Theta1Damping.png} -\caption{\label{fig:impulse1} Step response of $\theta_1$} -\end{figure} -\begin{figure}[ht] -\centering -\includegraphics[width=0.8\textwidth]{Theta2Damping.png} -\caption{\label{fig:impulse2} Free response of $\theta_2$} -\end{figure} -Since only an initial guess of the damping coefficients is necessary, the assumption is made that for both $\theta_1$ and $\theta_2$ the following relation for underdamped systems holds for a settling time $T_s$ to within $2\%$ -\begin{align*} - T_s = -\frac{\ln(0.02)}{\zeta \omega_n}\approx\frac{3.9}{\zeta \omega_n} \text{with} \ \zeta = \frac{d}{d_c} \ \text{and} \ \ d_c=2m\omega_n \ \text{leads to} \ d = \frac{3.9\times 2 m}{T_s} -\end{align*} -And since the values for $T_s$ can be read from the responses as $T_{s_1}=0.2s$, $T_{s_2}=8.3s$ and $m_1$ and $m_2$ are given, we get: -\begin{align*} -d_1 &= \frac{3.9\times 2(0.18+0.06)}{0.2}\approx 9.3 \ \ \text{and} \ \ -d_2 = \frac{3.9\times 2(0.06)}{8.3}\approx 0.056 -\end{align*} -The only parameter left without an initial guess is $K_u$. To get an estimate of $K_u$ it is varied in the nonlinear model until the output of the model has the same order of magnitude as the real system for the same input. This gave an initial guess for $K_u$ as -20$\frac{N}{m}$ -Table \ref{tab:initialguesses} shows the summary of all the initial guesses for the parameter estimation. -\begin{table}[H] -\caption{Initial guesses of the system parameters} -\centering -\begin{tabular}{|l|l|l|} -\hline -Symbol &Parameter &Guess \\ \hline -$l_1$ &Length of first link &0.1 [m] \\ -$l_2$ &Length of second link &0.1 [m]\\ -$m_1$ &Mass of first link &0.18 [kg]\\ -$m_2$ &Mass of second link &0.06 [kg]\\ -$c_1$ &Center of mass of first link &0.06 [m]\\ -$c_2$ &Center of mass of second link &0.045 [m]\\ -$I_1$ &Inertia of first link &0.037 [kgm$^2$]\\ -$I_2$ &Inertia of second link &0.00011 [kgm$^2$]\\ -$d_1$ &Damping of first link &9.3 [$\frac{Ns}{m}$]\\ -$d_2$ &Damping of second link &0.056 [$\frac{Ns}{m}$]\\ -$K_u$ &Lumped motor constant &-20 [$\frac{N}{m}$]\\ -$g$ & gravitation constant & 9.81 [$\frac{m}{s^2}$]\\ \hline -\end{tabular} -\label{tab:initialguesses} -\end{table} -\subsubsection{Nonlinear Parameter Estimation with SIMULINK\textsuperscript{\textregistered} and MATLAB\textsuperscript{\textregistered}} -The nonlinear parameter estimation was done by using the nonlinear mathematical model derived in \ref{sec:Mathematical} as a grey box model. All parameters that have to estimated will be represented by the vector $x$. The same inputs are fed into the grey box model and the real system to generated $\hat{y}(k,x)$ and $y(k)$ respectively, and MATLAB\textsuperscript{\textregistered}'s \mcode{lsqnonlin} is used to optimize $x$ for the following cost function: $e=y(k)-\hat{y}(k,x)$. The \mcode{lsqnonlin} function is set by default to use the large-scale algorithm to solve the nonlinear least-squares problem for each iteration. This algorithm is a subspace trust region method and is based on the interior-reflective Newton method described in \cite{interior}. The SIMULINK\textsuperscript{\textregistered} schematic used to generated the output $\hat{y}(k,x)$ can be seen in Figure \ref{fig:parametersimulink}. -\begin{figure}[ht] -\centering -\includegraphics[width=0.6\textwidth]{SimulinkNonlinearident.png} -\caption{\label{fig:parametersimulink} SIMULINK\textsuperscript{\textregistered} Schematic of the parameter estimation} -\end{figure} -\subsubsection{Results of the Nonlinear Parameter Estimation} - -After performing the parameter estimation, the plots for the result of the optimization are shown in figure \ref{fig:FirstFit}. - -\begin{figure}[H] -\centering -\includegraphics[width=0.8\textwidth]{FirstFit.png} -\caption{\label{fig:FirstFit} Fit of the first parameter estimation.} -\end{figure} - -As it can be seen directly from the plot, the estimated parameters give a good fit to the identification data, which was expected. This can also be confirmed by the values of the VAF, which are $VAF_{\theta_1}=94.2176$ and $VAF_{\theta_2}=89.1879$. The optimized estimates of the parameters are shown in table \ref{tab:estimatedparam}. - -\begin{table}[H] -\caption{Estimated Parameters.} -\centering -\begin{tabular}{|l|l|l|} -\hline -Symbol &Parameter &Value \\ \hline -$l_1$ &Length of first link &0.1 [m] \\ -$l_2$ &Length of second link &0.1 [m]\\ -$m_1$ &Mass of first link &0.3838 [kg]\\ -$m_2$ &Mass of second link &0.1419 [kg]\\ -$c_1$ &Center of mass of first link &-0.0626 [m]\\ -$c_2$ &Center of mass of second link &0.0590 [m]\\ -$I_1$ &Inertia of first link &0.0976 [kgm$^2$]\\ -$I_2$ &Inertia of second link &0.0002 [kgm$^2$]\\ -$d_1$ &Damping of first link &7.8345 [$\frac{Ns}{m}$]\\ -$d_2$ &Damping of second link &0.0003[$\frac{Ns}{m}$]\\ -$K_u$ &Lumped motor constant &-41.9288 [$\frac{N}{m}$]\\ -$g$ & gravitation constant & 9.81 [$\frac{m}{s^2}$]\\ \hline -\end{tabular} -\label{tab:estimatedparam} -\end{table} - - - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/initial_analysis.log b/Research Proposal IIT/Research proposal/sections (copy)/initial_analysis.log deleted file mode 100755 index 6e45b41..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/initial_analysis.log +++ /dev/null @@ -1,50 +0,0 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9) (preloaded format=pdflatex 2015.9.7) 16 MAR 2016 09:49 -entering extended mode -**initial_analysis.tex - -("C:\Users\Tavo\Documents\Escuela\Master\Master Thesis\Thesis Reports\Template -progress reports\sections\initial_analysis.tex" -LaTeX2e <2011/06/27> -Babel and hyphenation patterns for english, afrikaans, ancientgreek, ar -abic, armenian, assamese, basque, bengali, bokmal, bulgarian, catalan, coptic, -croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga -lician, german, german-x-2013-05-26, greek, gujarati, hindi, hungarian, iceland -ic, indonesian, interlingua, irish, italian, kannada, kurmanji, latin, latvian, - lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n -german-x-2013-05-26, nynorsk, oriya, panjabi, pinyin, polish, portuguese, roman -ian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, swissgerm -an, tamil, telugu, turkish, turkmen, ukenglish, ukrainian, uppersorbian, usengl -ishmax, welsh, loaded. -("C:\Program Files (x86)\MiKTeX 2.9\tex\latex\subfiles\subfiles.cls" -Document Class: subfiles 2012/05/23 Federico Garcia -("C:\Program Files (x86)\MiKTeX 2.9\tex\latex\tools\verbatim.sty" -Package: verbatim 2003/08/22 v1.5q LaTeX2e package for verbatim enhancements -\every@verbatim=\toks14 -\verbatim@line=\toks15 -\verbatim@in@stream=\read1 -) -Preamble taken from file `main.tex' - -! 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Emergency stop. - - -l.40 \input{\preamble@file} - -*** (cannot \read from terminal in nonstop modes) - - -Here is how much of TeX's memory you used: - 53 strings out of 493921 - 1339 string characters out of 3144880 - 49031 words of memory out of 3000000 - 3436 multiletter control sequences out of 15000+200000 - 3640 words of font info for 14 fonts, out of 3000000 for 9000 - 841 hyphenation exceptions out of 8191 - 18i,0n,17p,175b,16s stack positions out of 5000i,500n,10000p,200000b,50000s -! ==> Fatal error occurred, no output PDF file produced! diff --git a/Research Proposal IIT/Research proposal/sections (copy)/intro_objectives.tex b/Research Proposal IIT/Research proposal/sections (copy)/intro_objectives.tex deleted file mode 100755 index 992d29d..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/intro_objectives.tex +++ /dev/null @@ -1,1030 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - - - \subsection{Error dynamics} - - The system equations for the neutral bit walk model are given by: - - \begin{equation} - \begin{aligned} - \chi \Pi \Theta ' = &- \mathcal{M}_b(\Theta-\langle\Theta \rangle_1)-\frac{{{\mathcal{M}_b}{F_1} + {\mathcal{M}_1}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\rm{\Theta }} \rangle_1} - {\langle{\rm{\Theta }} \rangle_{2}}} \right) - \\ - &+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\rm{\Theta }} - {{\rm{\Theta }}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\rm{\Theta }}} - {{\rm{\Theta }}_1}}} - \frac{{{{\rm{\Theta }}_1} - {{\rm{\Theta }}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &- \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + (\eta \Pi - {F_b}){\mathcal{M}_r}}}{{\eta \Pi }}{\Gamma _{\Theta}} - \frac{\chi }{\eta }{\mathcal{F}_r}\Gamma_{\!\Theta}' + W, - \\ - \chi \Pi \Phi ' = &- {\mathcal{M}_b}\left( {\Phi - {\langle{\rm{\Phi }} \rangle_1}} \right)-\frac{{{\mathcal{M}_b}{F_1} + {\mathcal{M}_1}\left( {\eta \Pi - {F_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\rm{\Phi }} \rangle_1} - {\langle{\rm{\Phi }} \rangle_{2}}} \right) - \\ - &+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\rm{\Phi }} - {{\rm{\Phi }}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\rm{\Phi }}} - {{\rm{\Phi }}_1}}} - \frac{{{{\rm{\Phi }}_1} - {{\rm{\Phi }}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &+ \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta }}{{{{\left( {\sin \Theta } \right)}^2}}} - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}} \right)}}{{\eta \Pi \sin \Theta }}} \right){\Gamma _{\Phi}} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}}}{{\sin \Theta }}\Gamma_{\!\Phi}', - \end{aligned} - \label{eq:neutralbitwalkmodelfordesign} - \end{equation} - - - where the term related to the weight of the BHA is included in the first equation and is given by $W = -\frac{\mathcal{M}_b \mathcal{F}_w + (\eta \Pi - \mathcal{F}_b)\mathcal{M}_w}{\eta \Pi}\Upsilon \sin \langle\Theta\rangle_1 -\frac{\chi}{\eta}\mathcal{F}_w (\Theta - \Theta_1)\Upsilon \cos \langle\Theta\rangle_1$ and $\Upsilon$ is the normalized weight of the BHA. - - From Equation \eqref{eq:neutralbitwalkmodelfordesign}, the states of the system can be defined as: - - \begin{equation} - x_\Theta = \begin{bmatrix} - \Theta \\ - \langle \Theta \rangle_1\\ - \langle\Theta \rangle_2 - \end{bmatrix}, \qquad - x_\Phi = \begin{bmatrix} - \Phi \\ - \langle \Phi \rangle_1\\ - \langle\Phi \rangle_2 - \end{bmatrix}. - \end{equation} - - The system's output equations are given corresponding to measurements not exactly at the bit, but as the orientation of the BHA at locations between the stabilizers and given by: - - \begin{align} - y_\Theta &= C_\Theta x_\Theta + D_\Theta \Gamma_\Theta + E \Upsilon \sin \langle \Theta \rangle_1\\ - y_\Phi &= C_\Phi x_\Phi + D_\Phi \frac{\Gamma_\Phi}{\sin \Theta}. - \end{align} - - If it is assumed that both the inclination and the azimuth sensors are at the same location then $C_\Theta = C_\Phi$ and $D_\Theta = D_\Phi$. - - The input transformation on Equation \eqref{eq:decouplinginputtransformation2} is proposed in order to achieve decoupling of the system, it has to be noticed that the transformation on $\Gamma_\Phi$ depends on the estimate $\check{\Theta}$ of the inclination, if there is no access to measurements of the orientation of the borehole at the bit. - - \begin{equation} - \begin{bmatrix} - \Gamma_{\!\Theta}^* \\ \Gamma_{\!\Phi}^* - \end{bmatrix} := \begin{bmatrix} 1 & 0 \\ 0 & \frac{1}{\sin \check{\Theta}}\end{bmatrix} \begin{bmatrix}\Gamma_{\!\Theta} \\ \Gamma_{\!\Phi} \end{bmatrix}, \qquad \text{for } \Theta\in (0,\; \pi), - \label{eq:decouplinginputtransformation2} - \end{equation} - - Since the input transformation does not change the structure of the first equation in \eqref{eq:neutralbitwalkmodelfordesign}, only the second equation is shown and is given by: - - \begin{equation} - \begin{aligned} - \chi \Pi \Phi ' = &- {\mathcal{M}_b}\left( {\Phi - {\langle{\rm{\Phi }} \rangle_1}} \right)-\frac{{{\mathcal{M}_b}{F_1} + {\mathcal{M}_1}\left( {\eta \Pi - {F_b}} \right)}}{{\eta \Pi }}\left( {{\langle{\rm{\Phi }} \rangle_1} - {\langle{\rm{\Phi }} \rangle_{2}}} \right) - \\ - &+\frac{\chi }{\eta }{\mathcal{F}_b}\left( {{\rm{\Phi }} - {{\rm{\Phi }}_1}} \right) - - \frac{\chi }{\eta }{\mathcal{F}_1}\left( {{{{{\rm{\Phi }}} - {{\rm{\Phi }}_1}}} - \frac{{{{\rm{\Phi }}_1} - {{\rm{\Phi }}_{2}}}}{{{\varkappa _{2}}}}} \right) - \\ - &+ \left( {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta \sin \check{\Theta} }}{{{{\left( {\sin \Theta } \right)}^2}}} - - \frac{{{(\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}}) \right)\sin\check{\Theta}}}{{\eta \Pi \sin \Theta }} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\check{\Theta} '\cos \check{\Theta}}}{{{{\left( {\sin \Theta } \right)}}}}} \right){\Gamma _{\Phi}^*} - - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r \sin \check{\Theta}}}}{{\sin \Theta }}\Gamma_{\!\Phi}^{* '}. - \end{aligned} - \label{neutralbitwalkphi} - \end{equation} - - - and the system equations can be rewritten as: - - \begin{align} - \begin{bmatrix} - x_\Theta' \\ - x_\Phi' - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 \\ - 0 & A_0 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi) \\ - x_\Phi(\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 \\ - 0 & A_1 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_1) \\ - x_\Phi(\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 \\ - 0 & A_2 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_2) \\ - x_\Phi(\xi_2) - \end{bmatrix} \nonumber\\ - +& - \begin{bmatrix} - B_{0\Theta} & 0\\ - 0 & B_{0\Phi} - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^* \\ - \Gamma_\Phi^* - \end{bmatrix} + - \begin{bmatrix} - B_{1\Theta} & 0\\ - 0 & B_{1\Phi} - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^{*'} \\ - \Gamma_\Phi^{*'} - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 - \end{bmatrix}, - \label{eq:systemstatespace} - \end{align} - - where the matrices and vectors $A_0$, $A_1$, $A_2$, $B$, $B_{0i}$ and $B_{1i}$, for $i = \Theta, \Phi$, are given by: - - \begin{align} - A_{0}&= \frac{1}{\chi\Pi}\begin{bmatrix} - -\mathcal{M}_b+ \frac{\chi}{\eta}(\mathcal{F}_b-\mathcal{F}_1) & \mathcal{M}_b-\mathcal{M}_1+\frac{\mathcal{F}_b\mathcal{M}_1-\mathcal{F}_1\mathcal{M}_b}{\eta\Pi} & \mathcal{M}_1+\frac{(-\mathcal{F}_b\mathcal{M}_1+\mathcal{F}_1\mathcal{M}_b)}{\eta\Pi} \\ - \chi\Pi & 0 & 0 \\ - 0 & 0 & 0 \\ - \end{bmatrix}, \nonumber \\\nonumber - \\ \nonumber - A_{1}&= \frac{1}{\chi\Pi}\begin{bmatrix} - \frac{\chi}{\eta}(\mathcal{F}_1+\frac{\mathcal{F}_1}{\varkappa_2}-{\mathcal{F}_b}) & 0 & 0 \\ - -\chi\Pi & 0 & 0 \\ - \frac{\chi\Pi}{\varkappa_2} & 0 & 0 \\ - \end{bmatrix}, \qquad - A_{2}= \frac{1}{\chi\Pi}\begin{bmatrix} - -\frac{\chi \mathcal{F}_1}{\eta\varkappa_2} & 0 & 0 \\ - 0 & 0 & 0 \\ - -\frac{\chi\Pi}{\varkappa_2} & 0 & 0 \\ - \end{bmatrix},\nonumber - \\\nonumber - \\ \nonumber - B_{0\Theta}&= \frac{1}{\chi\Pi}\begin{bmatrix} - - \frac{{{\mathcal{M}_b}{\mathcal{F}_r} + (\eta \Pi - {F_b}){\mathcal{M}_r}}}{{\eta \Pi }}, & 0, & 0 \\ - \end{bmatrix}^{T} , \qquad - B_{1\Theta}= \frac{1}{\chi\Pi}\begin{bmatrix} - -\frac{\chi}{\eta}\mathcal{F}_r , & 0, & 0 \\ - \end{bmatrix}^{T}, - \\\nonumber - \\ \nonumber - B_{0\Phi}&= \frac{1}{\chi\Pi}\begin{bmatrix} - {\frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\Theta '\cos \Theta \sin \check{\Theta} }}{{{{\left( {\sin \Theta } \right)}^2}}} - - \frac{{{(\mathcal{M}_b}{\mathcal{F}_r} + {\mathcal{M}_r}\left( {\eta \Pi - {\mathcal{F}_b}}) \right)\sin\check{\Theta}}}{{\eta \Pi \sin \Theta }} - \frac{\chi }{\eta }\frac{{{\mathcal{F}_r}\check{\Theta} '\cos \check{\Theta}}}{{{{\left( {\sin \Theta } \right)}}}}}, & 0, & 0 \\ - \end{bmatrix}^{T} \nonumber \\ \nonumber - \\ \nonumber - B_{1\Phi}&= \frac{1}{\chi\Pi}\begin{bmatrix} - -\frac{\chi}{\eta}\mathcal{F}_r\frac{\sin \check{\Theta}}{\sin \Theta} , & 0, & 0 \\ - \end{bmatrix}^{T}, \qquad B = \begin{bmatrix}1 & 0 & 0\end{bmatrix}^T. - \end{align} - - After the input transformation, the output equations are given by: - - \begin{align} - y_\Theta &= C_\Theta x_\Theta + D_\Theta \Gamma_\Theta^*, \\ - y_\Phi &= C_\Phi x_\Phi + D_\Phi \Gamma_\Phi^* \frac{\sin \check{\Theta}}{\sin \Theta} . - \end{align} - - It can be seen that the system equations are complex, especially because of the fact that the terms related to the derivatives of the inclination and its estimate are affecting the azimuth dynamics. This is mainly due to the difference between $\Theta$ and $\check{\Theta}$. For the zero bit walk case (as considered here), this complexity vanishes when $\check{\Theta} = \Theta$. Assuming that we will be able to design an observer that indeed provides (asymptotically) such correct estimate of the inclination, the structure of the controller will be designed for the case when this condition is met. - - \begin{figure}[h]\centering - \includegraphics[width=0.65\textwidth]{ControlStrat.png} - \caption{Control strategy. - \label{fig:ControlStrat} } - \end{figure} - - - The total control scheme proposed here is depicted schematically in Figure \ref{fig:ControlStrat}. In order to combine the (transformed) RSS force inputs $\Gamma_i^*$ for $i = \Theta, \Phi$ and their derivatives, the following transformation is proposed: - - \begin{equation} - B u_i = B_{0i}\Gamma_i^* + B_{1i}\Gamma_i^{*'}, \label{eq:inputs1}\\ - \end{equation} - - where $B = \begin{bmatrix} 1 & 0 & 0\end{bmatrix}^T$ and vectors $B_{0i}$ and $B_{1i}$ correspond to the 3 by 1 vectors related to the inputs of the azimuth and the inclination. - - This would fully replace the terms related to $\Gamma_i^*$ and their derivative in the dynamics by $Bu$. However, it is not possible to use the input filter as in \eqref{eq:inputs1} here, since vectors $B_{0\Phi}$ and $B_{1\Phi}$ depend on $\Theta$, which is not known to the controller, since it can not be measured. To overcome this, the design of this input filter will be done for the case when $\check{\Theta} = \Theta$, i.e. - - \begin{align} - \Gamma_{i}^{*'} = -\frac{\mathcal{M}_b \mathcal{F}_r + (\eta \Pi - \mathcal{F}_b)}{\chi \Pi \mathcal{F}_r}\Gamma_{i}^{*} - \frac{\eta \Pi}{\mathcal{F}_r}u_i. \label{eq:filter1} - \end{align} - - Herein, $u_i$ is a new control input, comprised of the sum of the feedforward and feedback inputs given by $u_i = v_i + u_{ri}$. The stability of the filter is guaranteed as long as $\frac{B_{0i}}{B_{1i}} > 0$. It can be seen, that considering only uncertainty for the weight on bit, this condition is met up to a certain value of $\Pi$. It was proven in \cite{Kremers2013} that this value corresponds to the situation where the system is minimum-phase, this is as long as $\Pi < \Pi_{nmp}$, which is the focus of this work. In \eqref{eq:filter1}, the feedforward input $u_{ri}$ is defined based on the inverse dynamics of the system for a reference trajectory: - - \begin{equation} - u_{ri} = B^T (x'_{ri}(\xi) - A_0 x_{ri}(\xi) - A_1 x_{ri}(\xi_1) - A_2 x_{ri}(\xi_2)). - \label{eq:feedforward} - \end{equation} - - It is important to mention that the gravity related term in the inclination dynamics (W in \eqref{eq:neutralbitwalkmodelfordesign}), is omitted from the feedforward design since it can be considered as a quasi-constant(unknown) disturbance, which can be dealt with by implementing integral action in the control structure. Furthermore, this feedforward is designed again for the case when $\check{\Theta} = \Theta$. Due to this simplification of the feedforward design, transient error is introduced to the system, which will be explained in the resulting error dynamics later. - - - If the input filters are substituted in Equation \eqref{eq:systemstatespace}, then the system dynamics are given by: - - \begin{align} - \begin{bmatrix} - x_\Theta' \\ - x_\Phi' - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 \\ - 0 & A_0 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi) \\ - x_\Phi(\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 \\ - 0 & A_1 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_1) \\ - x_\Phi(\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 \\ - 0 & A_2 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_2) \\ - x_\Phi(\xi_2) - \end{bmatrix} \nonumber\\ - +& \begin{bmatrix} - 0 & 0 \\ - 0 & B\frac{\mathcal{F}_r}{\eta \Pi \sin \Theta} \bigg( \frac{\Theta' \cos \Theta \sin \check{\Theta}}{\sin \Theta} - \check{\Theta}' \cos \check{\Theta}\bigg) - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^* \\ - \Gamma_\Phi^* - \end{bmatrix} + \begin{bmatrix} - B & 0 \\ - 0 & B\frac{\sin \check{\Theta}}{\sin \Theta} - \end{bmatrix} - \begin{bmatrix} - u_\Theta \\ - u_\Phi - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 - \end{bmatrix}. - \label{eq:systemstatespace2} - \end{align} - - - In the case of the inclination dynamics, the input filter successfully makes the terms related to $\Gamma_\Theta^*$ disappear from the equation. On the other hand, in the equation related to the azimuth dynamics, one term related to $\Gamma_\Phi^*$ remains due to the mismatch between $\check{\Theta}$ and $\Theta$. Moreover, in \eqref{eq:systemstatespace2} there is a nonlinear factor multiplied by input $u_\Phi$. If the input filters in \eqref{eq:filter1} are included in the equation, the system dynamics can be rewritten as follows: - - \begin{align} - \begin{bmatrix} - x_\Theta' \\ - \Gamma_\Theta^{*'} \\ - x_\Phi' \\ - \Gamma_\Phi^{*'} - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & -b_0 & 0 & 0 \\ - 0 & 0 & A_0 & B\alpha \\ - 0 & 0 & 0 & -b_0 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi) \\ - \Gamma_\Theta^{*}(\xi) \\ - x_\Phi(\xi) \\ - \Gamma_\Phi^{*} (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_1) \\ - \Gamma_\Theta^{*}(\xi_1) \\ - x_\Phi(\xi_1) \\ - \Gamma_\Phi^{*} (\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 & 0 & 0 \\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_2) \\ - \Gamma_\Theta^{*}(\xi_2) \\ - x_\Phi(\xi_2) \\ - \Gamma_\Phi^{*} (\xi_2) - \end{bmatrix}\nonumber \\ - &+ - \begin{bmatrix} - B & 0 \\ - -b_1 & 0 \\ - 0 & B \frac{\sin \check{\Theta}}{\sin \Theta} \\ - 0 & -b_1 - \end{bmatrix} - \begin{bmatrix} - u_\Theta \\ - u_\Phi - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 \\ - 0 \\ - 0 - \end{bmatrix}. - \end{align} - - where: - - \begin{equation} - b_0 = \frac{\mathcal{M}_b \mathcal{F}_r + (\eta \Pi - \mathcal{F}_b)}{\chi \Pi \mathcal{F}_r}, \quad b_1 = \frac{\eta \Pi}{\mathcal{F}_r}, \quad \alpha (\Theta,\check{\Theta},\Theta',\check{\Theta}') = \frac{\mathcal{F}_r}{\eta \Pi \sin \Theta} \bigg( \frac{\Theta' \cos \Theta \sin \check{\Theta}}{\sin \Theta} - \check{\Theta}' \cos \check{\Theta}\bigg). \nonumber - \end{equation} - - It has to be notice that $\alpha$ depends on the states $\Theta$, $\check{\Theta}$, $\Theta'$, and $\check{\Theta}$' and was introduced in order to simplify notation. From these equations, the error dynamics of the system (with the error defined as $e_i = x_i - x_{ri}$) are obtained in \eqref{eq:errordynamics}, using for the feedforward the expression given in \eqref{eq:feedforward} and the control input decomposition $u_i = u_{ri} + v_i$: - - \begin{align} - \begin{bmatrix} - e_\Theta' \\ - \Gamma_\Theta^{*'} \\ - e_\Phi' \\ - \Gamma_\Phi^{*'} - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & -b_0 & 0 & 0 \\ - 0 & 0 & A_0 & B \alpha \\ - 0 & 0 & 0 & -b_0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi) \\ - \Gamma_\Theta^{*} (\xi) \\ - e_\Phi(\xi) \\ - \Gamma_\Phi^{*} (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_1) \\ - \Gamma_\Theta^{*} (\xi_1) \\ - e_\Phi(\xi_1) \\ - \Gamma_\Phi^{*} (\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_2) \\ - \Gamma_\Theta^{*} (\xi_2) \\ - e_\Phi(\xi_2) \\ - \Gamma_\Phi^{*} (\xi_2) - \end{bmatrix} \nonumber\\ - &+ \begin{bmatrix} - B & 0 \\ - -b_1 & 0 \\ - 0 & B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta}\bigg) \\ - 0 & -b_1 - \end{bmatrix} - \begin{bmatrix} - v_\Theta \\ - v_\Phi - \end{bmatrix} + - \begin{bmatrix} - 0 \\ - -b_1 u_{r\Theta} \\ - B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1 \bigg) u_{r\Phi} \\ - -b_1 u_{r\Phi} - \end{bmatrix}+ - \begin{bmatrix} - BW \\ - 0 \\ - 0 \\ - 0 - \end{bmatrix}. - \label{eq:errordynamics} - \end{align} - - In a similar way, if an input filter as in equation \eqref{eq:filter1} is designed based on the feedforward input, the following equation can be obtained: - - \begin{equation} - \Gamma_{id}^{*'} = -\frac{\mathcal{M}_b \mathcal{F}_r + (\eta \Pi - \mathcal{F}_b)}{\chi \Pi \mathcal{F}_r}\Gamma_{id}^{*} - \frac{\eta \Pi}{\mathcal{F}_r}u_{ri}, \label{eq:filter2} - \end{equation} - - where $\Gamma_{id}^*$ is a desired input of the RSS corresponding to the feedforward input. Then, an error coordinate for the transformed input $\Gamma_i^*$ can be defined as $\Delta \Gamma^*_{i} = \Gamma_i^* - \Gamma_{id}^*$. Then equation \eqref{eq:errordynamics} can be rewritten as: - - \begin{align} - \begin{bmatrix} - e_\Theta' \\ - \Delta \Gamma_\Theta^{*'} \\ - e_\Phi' \\ - \Delta \Gamma_\Phi^{*'} - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & -b_0 & 0 & 0 \\ - 0 & 0 & A_0 & B \alpha \\ - 0 & 0 & 0 & -b_0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi) \\ - \Delta \Gamma_\Theta^{*} (\xi) \\ - e_\Phi(\xi) \\ - \Delta \Gamma_\Phi^{*} (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_1) \\ - \Delta \Gamma_\Theta^{*} (\xi_1) \\ - e_\Phi(\xi_1) \\ - \Delta \Gamma_\Phi^{*} (\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - e_\Theta(\xi_2) \\ - \Delta \Gamma_\Theta^{*} (\xi_2) \\ - e_\Phi(\xi_2) \\ - \Delta \Gamma_\Phi^{*} (\xi_2) - \end{bmatrix} \nonumber\\ - &+ \begin{bmatrix} - B & 0 \\ - -b_1 & 0 \\ - 0 & B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta}\bigg) \\ - 0 & -b_1 - \end{bmatrix} - \begin{bmatrix} - v_\Theta \\ - v_\Phi - \end{bmatrix} + - \begin{bmatrix} - 0 \\ - 0 \\ - B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1 \bigg) u_{r\Phi} \\ - 0 - \end{bmatrix} - \label{eq:errordynamics2}+ - \begin{bmatrix} - BW \\ - 0 \\ - B\alpha \Gamma_{\Phi d}^* \\ - 0 - \end{bmatrix}. - \end{align} - - After obtaining the error dynamics, the state feedback controller corresponding to the input $v_i$ is defined as: - - \begin{align} - z_{1i}' &= \zeta \begin{bmatrix} - k_{1i} & 0 & 0 - \end{bmatrix}e_i \label{eq:controllerlowpass}\\ - z_{2i}' &= -\gamma z_{2i} + \gamma (z_{1i} + K_i e_i) \label{eq:controllerintegral}\\ - v_i &= z_{2i}\label{eq:controllerfeedback}. - \end{align} - - This controller consists of a static state (error $e_i$) feedback part, a low-pass filter and integral action. Considering the fact that the states can not be measured directly, an observer needs to be designed in order to support the implementation of the controller in \eqref{eq:controllerlowpass}-\eqref{eq:controllerfeedback}. Since the weight of the BHA is taken into account as a quasi-constant disturbance, integral action is also added to the observer design. The integral action of the observer is embedded through the integral filter: - - \begin{equation} - q_i' = \zeta[l_1,l_2](y_i - \check{y}_i), \qquad \text{ for } i=\Theta,\Phi. - \end{equation} - - The observer consists of a model-based (predictor) part and an output-injection part. The predictor part of the observer is designed by again considering the condition where $\Theta = \check{\Theta}$. In total, the dynamics of the observer with integral action are given by: - - \begin{align} - \begin{bmatrix} - \check{x}_\Theta' \\ - q_\Theta' \\ - \check{x}_\Phi'\\ - q_\Phi' - \end{bmatrix} &= - \begin{bmatrix} - A_0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_0 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi) \\ - q_\Theta(\xi) \\ - \check{x}_\Phi(\xi) \\ - q_\Phi (\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_1 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi_1) \\ - q_\Theta(\xi_1) \\ - \check{x}_\Phi(\xi_1) \\ - q_\Phi(\xi_1) \\ - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0\\ - 0 & 0 & A_2 & 0 \\ - 0 & 0 & 0 & 0 \\ - \end{bmatrix} - \begin{bmatrix} - \check{x}_\Theta(\xi_2) \\ - q_\Theta(\xi_2) \\ - \check{x}_\Phi(\xi_2) \\ - q_\Phi(\xi_2) - \end{bmatrix} \nonumber\\ - &+ - \begin{bmatrix} - L_\Theta(y_{\Theta} - \check{y}_{\Theta})\\ - \zeta[l_{1\Theta},l_{2\Theta}](y_\Theta - \check{y}_\Theta) \\ - L_{\Phi}(y_{\Phi} - \check{y}_{\Phi}) \\ - \zeta[l_{1\Phi},l_{2{\Phi}}](y_\Phi - \check{y}_\Phi) - \end{bmatrix} + - \begin{bmatrix} - Bq_\Theta \\ - 0 \\ - Bq_\Phi \\ - 0 - \end{bmatrix} + - \begin{bmatrix} - B(u_{r\Theta} + v_\Theta) \\ - 0 \\ - B(u_{r\Phi} + v_\Phi) \\ - 0 - \end{bmatrix}, - \end{align} - - where $L_i$ is defined as: - - \begin{equation} - L_i = \begin{bmatrix} - l_{1i} & l_{2i} \\ - 0 & 0 \\ - 0 & 0 - \end{bmatrix}, - \end{equation} - - and with the output observer equations (taking into account the ideal input decoupling transformation) given by: - - \begin{align} - \check{y}_\Theta &= C_\Theta \check{x}_\Theta + D_\Theta \Gamma_\Theta^* \\ - \check{y}_\Phi &= C_\Phi \check{x}_\Phi + D_\Phi \Gamma_\Phi^* - \end{align} - - Computing the observer error dynamics (with $\delta_i = x_i - \check{x}_i$) and including the integral action results in the following equations: - - - \begin{align} - \begin{bmatrix} - \delta_\Theta' \\ - q_\Theta' \\ - \delta_\Phi' \\ - q_\Phi' - \end{bmatrix} =& - \begin{bmatrix} - A_0 - L_\Theta C_\Theta & -B & 0 & 0\\ - \zeta [l_{1\Theta},l_{2\Theta}]C_\Theta & 0 & 0 & 0 \\ - 0 & 0 & A_0 - L_\Phi C_\Phi & -B \\ - 0 & 0 & \zeta [l_{1\Phi},l_{2\Phi}]C_\Phi &0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta(\xi) \\ - q_\Theta (\xi) \\ - \delta_\Phi(\xi) \\ - q_\Phi(\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_1 & 0\\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta(\xi_1) \\ - q_\Theta (\xi_1) \\ - \delta_\Phi(\xi_1) \\ - q_\Phi (\xi_1) - \end{bmatrix} \nonumber \\ - +&\begin{bmatrix} - A_2 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 \\ - 0 & 0 & A_2 & 0\\ - 0 & 0 & 0 & 0 - \end{bmatrix} - \begin{bmatrix} - \delta_\Theta(\xi_2) \\ - q_\Theta (\xi_2) \\ - \delta_\Phi(\xi_2) \\ - q_\Phi (\xi_2) - \end{bmatrix} + - \begin{bmatrix} - 0 \\ - 0 \\ - B(u_{r\Phi} + v_\Phi)\big(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\big) \\ - 0 - \end{bmatrix} - + - \begin{bmatrix} - 0\\ - 0 \\ - \big(B\alpha - L_\Phi D_\Phi\big(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\big)\big)\Gamma_\Phi^*\\ - \zeta[l_{1\Phi},l_{2\Phi}]D_\Phi (\frac{\sin \check{\Theta}}{\sin \Theta} - 1)\Gamma_\Phi^* - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 \\ - 0 \\ - 0 - \end{bmatrix}. - \end{align} - - If now, the observer is combined with the state-feedback controller, the error dynamics can be obtained, first applying the controller in the following observer-based feedback form: - - \begin{align} - z_{1i}' &= \zeta \begin{bmatrix} - k_{1i} & 0 & 0 - \end{bmatrix}(\check{x}_i - x_{ri}) \\ - z_{2i}' &= -\gamma z_{2i} + \gamma (z_{1i} + K_i(\check{x}_i - x_{ri}))\\ - v_i &= z_{2i}, - \end{align} - - and using the fact that $\check{x}_i - x_{ri} = e_i - \delta_i$ and $\Gamma_i^* = \Delta \Gamma_i^* + \Gamma_{id}^*$, the total error dynamics of the state vector: - - \begin{equation} - X(\xi) = \begin{bmatrix} - e_{\Theta}^T(\xi) \text{ }\Delta \Gamma_\Theta^{*}(\xi) \text{ }z_{1\Theta}(\xi) \text{ } z_{2\Theta}(\xi) \text{ }\delta_{\Theta}^T(\xi) \text{ } q_\Theta(\xi) \text{ }e_{\Phi}^T(\xi) \text{ }\Delta \Gamma_\Phi^*(\xi) \text{ }z_{1\Phi}(\xi) \text{ }z_{2\Phi}(\xi) \text{ }\delta_{\Phi}^T(\xi) \text{ }q_\Phi(\xi) \nonumber - \end{bmatrix}^T. - \end{equation} - - are given by: - - \begin{equation} - X'(\xi) = A_{0cl}X(\xi) + A_{1cl}X(\xi_1) + A_{2cl} X(\xi_2) + P_{cl}(u_{r\Phi},\Gamma_{\Phi d}^*,\alpha, \Theta,\check{\Theta},W),\\ - \label{eq:totaldynamics} - \end{equation} - - where: - - \begin{equation*} - A_{0cl} = - \begin{bmatrix} - A_{0\Theta} & 0 \\ - 0 & A_{0\Phi} - \end{bmatrix}, \qquad - A_{1cl} = - \begin{bmatrix} - A_{1\Theta} & 0 \\ - 0 & A_{1\Phi} - \end{bmatrix}, \qquad - A_{2cl} = - \begin{bmatrix} - A_{2\Theta} & 0 \\ - 0 & A_{2\Phi} - \end{bmatrix}, - \end{equation*} - - where the system matrices and the vector $B_{cl}$ are: - - \begin{align*} - A_{0\Theta} &= - \begin{bmatrix}[cccc:cc] - A_0 & 0 & 0 & B & 0 & 0\\ - % - 0 & -b_0 & 0 & -b_1 & 0 & 0\\ - \zeta \begin{bmatrix}k_{1\Theta} , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_{1\Theta} , 0 , 0\end{bmatrix} & 0\\ - % - \gamma K_\Theta & 0 & \gamma & -\gamma & -\gamma K_\Theta & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_0 - L_\Theta C_\Theta & -B\\ - % - 0 & 0 & 0 & 0 & \zeta \begin{bmatrix}l_{1\Theta} , l_{2\Theta} \end{bmatrix} C_\Theta & 0 - \end{bmatrix},\\ - \nonumber \\ - A_{0\Phi} &= - \begin{bmatrix}[cccc:cc] - A_0 & B\alpha & 0 & B\bigg( \frac{\sin \check{\Theta}}{\sin \Theta}\bigg) & 0 & 0\\ - % - 0 & -b_0 & 0 & -b_1 & 0 & 0 \\ - % - \zeta \begin{bmatrix}k_{1\Phi} , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_{1\Phi} , 0 , 0\end{bmatrix} & 0 \\ - % - \gamma K_{\Phi} & 0 & \gamma & -\gamma & -\gamma K_{\Phi} & 0\\ \hdashline - % - 0 & B\alpha -L_\Phi D_{\Phi}\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg) & 0 & B\big(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\big) & A_0 - L_\Phi C_{\Phi} & -B \\ - % - 0 & \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix} D_\Phi\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg) & 0 & 0 & \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix}C_\Phi & 0 - \end{bmatrix},\nonumber \\ - \nonumber \\ - A_{1i} &= - \begin{bmatrix}[cccc:cc] - A_1 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_1 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix}, \quad - A_{2i} = - \begin{bmatrix}[cccc:cc] - A_2 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_2 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix}, \nonumber \\ - \nonumber \\ - P_{cl}&(u_{r\Phi},\Gamma_{\Phi d}^*,\alpha, \Theta,\check{\Theta},W) = \begin{bmatrix} - BW \\ - 0 \\ - 0 \\ - 0 \\ \hdashline - BW \\ - 0 \\ \hline - B \bigg(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg) u_{r\Phi} + B\alpha\Gamma_{\Phi d}^*\\ - 0 \\ - 0 \\ - 0 \\ \hdashline - B \bigg(\frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg) u_{r\Phi} + B\alpha\Gamma_{\Phi d}^* - LD\bigg( \frac{\sin \check{\Theta}}{\sin \Theta}-1\bigg)\Gamma_{\Phi d}^*\\ - \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix} D_\Phi\bigg( \frac{\sin \check{\Theta}}{\sin \Theta} - 1\bigg)\Gamma_{\Phi d}^* \\ - \end{bmatrix}. - \\\nonumber - \end{align*} - - It can be seen that the vector $P_{cl}$ contains the gravity related terms, which were not included in the feedforward design. Nevertheless, the integral part of the controller will take care of this perturbation. - - \newpage - - \section{Linearization} - - This system will be linearized around $e_i = 0$ and $\delta_i = 0$. Initially the inclination-related errors heve been defined as: - - - \begin{equation} - e_\Theta = \begin{bmatrix} - e_{\Theta_0} \\ - e_{\langle \Theta \rangle_1} \\ - e_{\langle \Theta \rangle_2} - \end{bmatrix} := \begin{bmatrix} - \Theta - \Theta_d \\ - \langle \Theta \rangle_1 - \langle \Theta \rangle_{1d} \\ - \langle \Theta \rangle_2 - \langle \Theta \rangle_{2d} - \end{bmatrix}. - \end{equation} - - Then from this expression $\Theta$, $\langle \Theta \rangle_1$ and $\langle \Theta \rangle_2 $ can be obtained as: - - \begin{align} - \Theta &= e_{\Theta_0} + \Theta_d,\quad \langle \Theta \rangle_1 = e_{\langle \Theta \rangle_1} + \langle \Theta \rangle_{1d},\quad \langle \Theta \rangle_2 = e_{\langle \Theta \rangle_2} + \langle \Theta \rangle_{2d}. - \label{eq:theta} - \end{align} - - In a similar fashion, the observer errors related to the inclination have been defined as: - \begin{equation} - \delta_\Theta = \begin{bmatrix} - \delta_{\Theta_0} \\ - \delta_{\langle \Theta \rangle_1} \\ - \delta_{\langle \Theta \rangle_2} - \end{bmatrix} = \begin{bmatrix} - \Theta - \check{\Theta} \\ - \langle \Theta \rangle_1 - \langle \check{\Theta} \rangle_{1} \\ - \langle \Theta \rangle_2 - \langle \check{\Theta} \rangle_{2} - \end{bmatrix}. - \label{eq:observererror} - \end{equation} - - Then $\check{\Theta}$, $\langle \check{\Theta} \rangle_{1}$ and $\langle \check{\Theta} \rangle_{2}$ can be obtained as: - - \begin{align} - \check{\Theta} &= e_{\Theta_0} + \Theta_d - \delta_{\Theta_0}, \quad \langle \check{\Theta} \rangle_{1} = e_{\langle \Theta \rangle_1} + \langle \Theta \rangle_{1d} - \delta_{\langle \check{\Theta} \rangle_{1}}, \quad \langle \check{\Theta} \rangle_{2} = e_{\langle \Theta \rangle_2} + \langle \Theta \rangle_{2d} - \delta_{\langle \check{\Theta} \rangle_{2}} - \label{eq:esttheta}. - \end{align} - - In order to obtain the nominal values for the rest of the states, the values of $e_i = 0$ and $\delta_i = 0$ are substituted into equation \eqref{eq:totaldynamics}. When the error dynamics and the observer error dynamics are zero, the condition $\check{\Theta} = \Theta$ is met, and by subsequently setting the derivatives to zero we obtain the following set of equilibrium equations: - - \begin{align} - 0 &= BW + Bz_{2\Theta} \nonumber\\ - %1 - 0 &= -b_0 \Delta \Gamma_\Theta^{*} - b_1 z_{2\Theta} \nonumber\\ - %3 - 0 &= -\gamma z_{2\Theta} + \gamma z_{1\Theta} \nonumber\\ - %4 - 0 &= BW - Bq_\theta \nonumber \\ - %6 - 0 &= Bz_{2\Phi}\nonumber \\ - %7 - 0 &= -b_0 \Delta \Gamma_\Phi^{*} - b_1 z_{2\Phi} \nonumber\\ - %9 - 0 &= -\gamma z_{2\Phi} + \gamma z_{1\Phi} \nonumber\\ - %10 - 0 &= Bq_\Phi \nonumber\\ - \label{eq:linpoint} - \end{align} - - From the set of equations \eqref{eq:linpoint}, the nominal values for $z_{1i}$, $z_{2i}$, $\Delta \Gamma_i^*$ and $q_i$ can be obtained. With these remarks, the equilibrium values for all the states are given by: - - \begin{align} - e_i &= 0\nonumber\\ - \delta_i &= 0 \nonumber\\ - z_{1\Theta} &= -W\nonumber\\ - z_{2\Theta} &= -W\nonumber\\ - q_\Theta &= W \nonumber\\ - z_{1\Phi} &= 0\nonumber\\ - z_{2\Phi} &= 0\nonumber\\ - q_\Phi &= 0 \nonumber\\ - \Delta \Gamma_\Theta^* &= \frac{b_1}{b_0}W \nonumber\\ - \Delta \Gamma_\Phi^* &= 0 - \end{align} - - We can then introduce the perturbed versions of the states where the equilibrium is not at a value of zero as follows: - - \begin{align} - \Delta z_{1\Theta} &= z_{1\Theta} - (-W) \nonumber\\ - \Delta z_{2\Theta} &= z_{2\Theta} - (-W) \nonumber\\ - \Delta q_{\Theta} &= q_{\Theta} - W \nonumber\\ - \Delta \bar{\Gamma}_{\Theta}^* &= \Delta \Gamma_{\Theta}^* - \frac{b_1}{b_0}W - \end{align} - - - - In order to perform the linearization, the equations are expressed in terms of the error dynamics and the observer error dynamics. Furthermore, the terms related to $\Theta'$ and $\check{\Theta}'$ in $\alpha$ were substituted by the dynamics in \eqref{eq:neutralbitwalkmodelfordesign}. Due to the complexity of the computations involved this was done using the symbolic toolbox of Matlab according to the following equation: - - \begin{equation} - \bar{X}'(\xi) = \frac{\partial X'(\xi)}{\partial X(\xi)} \bigg|_N \bar{X}(\xi) - + \frac{\partial X'(\xi)}{\partial X(\xi_1)} \bigg|_N \bar{X}(\xi_1) + \frac{\partial X'(\xi)}{\partial X(\xi_2)} \bigg|_N \bar{X}(\xi_2) - \label{eq:linearization} - \end{equation} - - - where the state vector $\bar{X}(\xi)$ is given by: - - \begin{equation} - \bar{X}(\xi) = \begin{bmatrix} - e_{\Theta}(\xi) \text{ }\Delta \bar{\Gamma}_\Theta^{*}(\xi) \text{ }\Delta z_{1\Theta}(\xi) \text{ }\Delta z_{2\Theta}(\xi) \text{ }\delta_{\Theta}(\xi) \text{ }\Delta q_\Theta(\xi) \text{ }e_{\Phi}(\xi) \text{ }\Delta \Gamma_\Phi^*(\xi) \text{ }z_{1\Phi}(\xi) \text{ }z_{2\Phi}(\xi) \text{ }\delta_{\Phi}(\xi) \text{ }q_\Phi(\xi) \nonumber - \end{bmatrix}^T. - \end{equation} - - After linearizing, the system equations are: - - \begin{equation} - \bar{X}'(\xi) = \bar{A}_{0cl}\bar{X}(\xi) + \bar{A_{1cl}}\bar{X}(\xi_1) + \bar{A_{2cl}} \bar{X}(\xi_2),\\ - \label{eq:totaldynamicslinears} - \end{equation} - - where the linearized matrices $\bar{A}_{0cl}$, $\bar{A}_{1cl}$ and $\bar{A}_{2cl}$, are given by: - - \begin{equation*} - \bar{A}_{0cl} = - \begin{bmatrix} - A_{0\Theta} & 0 \\ - A_{0c} & A_{0\Phi} - \end{bmatrix}, \qquad - \bar{A}_{1cl} = - \begin{bmatrix} - A_{1\Theta} & 0 \\ - A_{1c} & A_{1\Phi} - \end{bmatrix}, \qquad - \bar{A}_{2cl} = - \begin{bmatrix} - A_{2\Theta} & 0 \\ - A_{2c} & A_{2\Phi} - \end{bmatrix}, - \end{equation*} - - and block matrices: - - \begin{align} - A_{0i} &= - \begin{bmatrix}[cccc:cc] - A_0 & 0 & 0 & B & 0 & 0\\ - % - 0 & -b_0 & 0 & -b_1 & 0 & 0\\ - \zeta \begin{bmatrix}k_{1i} , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_{1i} , 0 , 0\end{bmatrix} & 0\\ - % - \gamma K_i & 0 & \gamma & -\gamma & -\gamma K_i & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_0 - L_iC_i & -B\\ - % - 0 & 0 & 0 & 0 & \zeta \begin{bmatrix}l_{1i} , l_{2i} \end{bmatrix} C_i & 0 - \end{bmatrix}, \qquad - A_{0c} = - \begin{bmatrix}[cccc:cc] - 0 & 0 & 0 & 0 & p_{0e1}(\xi) & p_{0e2}(\xi) \\ - 0 & 0 & 0 & 0 & 0 & 0 \\ - 0 & 0 & 0 & 0 & 0 & 0 \\ - 0 & 0 & 0 & 0 & 0 & 0 \\ \hdashline - 0 & 0 & 0 & 0 & p_{0\delta 1}(\xi) & p_{0\delta 2}(\xi) \\ - 0 & 0 & 0 & 0 & p_{0q}(\xi) & 0 - \end{bmatrix} \nonumber\\ - \nonumber\\ - A_{1i} &= - \begin{bmatrix}[cccc:cc] - A_1 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_1 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix}, \qquad - A_{1c} = \begin{bmatrix}[cccc:cc] - 0 & 0 & 0 & 0 & p_{1e}(\xi) & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - 0 & 0 & 0 & 0 & p_{1\delta}(\xi) & 0\\ - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix} \nonumber\\ - \nonumber\\ - A_{2i} &= - \begin{bmatrix}[cccc:cc] - A_2 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - % - 0 & 0 & 0 & 0 & A_2 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix}, \qquad - A_{2c} = \begin{bmatrix}[cccc:cc] - 0 & 0 & 0 & 0 & p_{2e}(\xi) & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - 0 & 0 & 0 & 0 & 0 & 0\\ - % - 0 & 0 & 0 & 0 & 0 & 0\\ \hdashline - 0 & 0 & 0 & 0 & p_{2\delta}(\xi) & 0\\ - 0 & 0 & 0 & 0 & 0 & 0 - \end{bmatrix} \nonumber - \end{align} - - and with coefficients given by: - - \begin{align} - p_{0e1}(\xi) =& -B\frac{\mathcal{F}_r \Gamma_{\Phi d}^*}{\eta \Pi \sin \Theta_d} \begin{bmatrix} \begin{aligned} - &\beta_1 \Theta_d + \beta_2 \langle \Theta \rangle_{1d} + \beta_3 \langle \Theta \rangle_{2d} + \beta_4 \Theta_{1d} + \beta_5 \Theta_{2d} \bigg(\frac{1}{\sin \Theta_d}\bigg) +...\\ - &\kern2cm (-\beta_1 + \beta_2 \langle \Theta \rangle_{1d} + \beta_3 \langle \Theta \rangle_{2d} + \beta_4 \Theta_{1d} + \beta_5 \Theta_{2d})\cos \Theta_d \\ - &-\beta_2 \cos \Theta_d \\ - &-\beta_3 \cos \Theta_d\end{aligned}\end{bmatrix}^T - B\begin{bmatrix}\frac{\cos \Theta_d}{\sin \Theta_d}u_{r\Phi} \\\\0 \\0 \end{bmatrix}^T, \nonumber \\ - \nonumber\\ - p_{0e2}(\xi) =& -B\frac{\mathcal{F}_r \Gamma_{\Phi d}^*}{\eta \Pi \sin \Theta_d}\cos \Theta_d, \nonumber \\ - \nonumber \\ - p_{0\delta 1}(\xi) =& -B\frac{\mathcal{F}_r \Gamma_{\Phi d}^*}{\eta \Pi \sin \Theta_d} \begin{bmatrix} \begin{aligned} - &\beta_1 \Theta_d + \beta_2 \langle \Theta \rangle_{1d} + \beta_3 \langle \Theta \rangle_{2d} + \beta_4 \Theta_{1d} + \beta_5 \Theta_{2d} \bigg(\frac{1}{\sin \Theta_d}\bigg) +...\\ - &\kern2cm (-\beta_1 + \beta_2 \langle \Theta \rangle_{1d} + \beta_3 \langle \Theta \rangle_{2d} + \beta_4 \Theta_{1d} + \beta_5 \Theta_{2d})\cos \Theta_d \\ - &-\beta_2 \cos \Theta_d \\ - &-\beta_3 \cos \Theta_d\end{aligned}\end{bmatrix}^T - B\begin{bmatrix}\frac{\cos \Theta_d}{\sin \Theta_d}u_{r\Phi} \\\\0\\0 \end{bmatrix}^T \nonumber \\ - &+ L_\Phi D_\Phi\begin{bmatrix}\Gamma_{\Phi d}^*\frac{\cos \Theta_d}{\sin \Theta_d} \\\\0\\0\end{bmatrix}^T, \nonumber \\ - \nonumber \\ - p_{0\delta 2}(\xi) =& -B\frac{\mathcal{F}_r \Gamma_{\Phi d}^*}{\eta \Pi \sin \Theta_d}\cos \Theta_d, \quad - p_{0q}(\xi)= -\zeta \Gamma_{\Phi d}^* [l_{1\Phi},l_{2\Phi}]D_\Phi\frac{\cos \Theta_d}{\sin \Theta_d}, \quad p_{1e}(\xi) = B\frac{\mathcal{F}_r \Gamma_{\Phi d}^* \cos \Theta_d}{\eta \Pi \sin \Theta_d}\begin{bmatrix}\beta_4 & 0 & 0\end{bmatrix}, \nonumber \\ - \nonumber \\ - p_{1\delta}(\xi) =& B\frac{\mathcal{F}_r \Gamma_{\Phi d}^* \cos \Theta_d}{\eta \Pi \sin \Theta_d}\begin{bmatrix}\beta_4& 0 & 0\end{bmatrix}, \quad p_{2e}(\xi) = B\frac{\mathcal{F}_r \Gamma_{\Phi d}^* \cos \Theta_d}{\eta \Pi \sin \Theta_d}\begin{bmatrix}\beta_5& 0 & 0\end{bmatrix}, \quad p_{2\delta}(\xi) = B\frac{\mathcal{F}_r \Gamma_{\Phi d}^* \cos \Theta_d}{\eta \Pi \sin \Theta_d}\begin{bmatrix}\beta_5& 0 & 0\end{bmatrix}, \nonumber - \end{align} - - where elements $\beta_j$, for $j = [1,5]$ correspond to the following elements of the matrices $A_0$, $A_1$ and $A_2$: - - \begin{equation*} - \beta_1 = A_0(1,1), \quad \beta_2 = A_0(1,2), \quad \beta_3 = A_0(1,3), \quad \beta_4 = A_1(1,1), \quad \beta_5 = A_2(1,1). - \end{equation*} - - It can be noticed that, after the linearization the nonlinear perturbation terms related to matrix $P_{cl}$ vanish, due to the fact that the effect of these terms appears in the equation as coefficients related to the observer error of the inclination $\delta_\Theta$. - - \newpage - \section{Proposal for controller synthesis} - - By looking at the structure of the linearized matrices, it can be seen that not all of the gains need to be synthesized at the same time. As expected, the inclination dynamics matrices are decoupled from the azimuth dynamics for this neutral bit walk case. Because of this, both the controller gains and observer can be synthesized separately from the azimuth dynamics. In the case of the azimuth dynamics, this is not possible due to the presence of the $A_{kc}$ (for $k = [0,2]$) block matrices. This matrices depend on the inclination dynamics, in particular with the observer error of the inclination (namely $\delta_\Theta$). - - The controller and observer gains of the azimuth dynamics could be synthesized after substituting the previously obtained feedback gains, despite the fact that the optimization would have to be done using an objective function that includes the entire system dynamics. Figure (diagram pending), shows the structure to be followed to synthesize the controller gains of the system. - - - - \newpage - -\end{document} - \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/introduction.tex b/Research Proposal IIT/Research proposal/sections (copy)/introduction.tex deleted file mode 100755 index 2bdfeaf..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/introduction.tex +++ /dev/null @@ -1,137 +0,0 @@ -% -\begin{document} - % Introduction - -\chapter{Introduction} \label{chap::intro} - -The exhaustion of conventional supplies of mineral and energy resources has become a rising and well known problem in the past decades. This has lead to the development of new techniques which provide the possibility to exploit more difficult-to-reach reservoirs of these resources, as well as preventing and reducing environmental costs. An important breakthrough in this matter is directional drilling, a technology that allows to drill wells with complex geometry in order to access unconventional reservoirs of energy and mineral. - -Directional drilling has proven to be a successful and cost-efficient process in important applications, such as: - -\begin{enumerate}[(a)] - \item Drilling to reservoirs, which are inaccessible using conventional techniques, such as resource deposits under a city or a natural area, providing a smaller environmental and social impact (Figure~\ref{fig:app1}). - \item Drilling relief wells to release pressure in hazardous situations when control over a well is lost. This was demonstrated at the gulf of Mexico Deepwater Horizon oil spill in 2010, when two reliefs wells where drilled in order to seal the failure (Figure~\ref{fig:app2}). - \item If a reservoir is close to a geological fault, directional drilling can be used to drill away from the fault plane avoiding damage to the well caused by fault slippage (Figure~\ref{fig:app3}). - \item Multiple wells can be drilled from the same drill rig, reducing economic and environmental costs compared to drilling processes that require to drill separate boreholes from a different rig (Figure~\ref{fig:app4}). -\end{enumerate} - - -\begin{figure}[ht] - \centering - \begin{subfigure}[b]{0.45\textwidth} - \includegraphics[width=0.9\textwidth]{img/appdd1.pdf} - \caption{\label{fig:app1}} - \end{subfigure} - \begin{subfigure}[b]{0.45\textwidth} - \includegraphics[width=0.9\textwidth]{img/appdd2.pdf} - \caption{\label{fig:app2}} - \end{subfigure} - \begin{subfigure}[b]{0.45\textwidth} - \includegraphics[width=0.9\textwidth]{img/appdd3.pdf} - \caption{\label{fig:app3}} - \end{subfigure} - \begin{subfigure}[b]{0.45\textwidth} - \includegraphics[width=0.9\textwidth]{img/appdd4.pdf} - \caption{\label{fig:app4}} - \end{subfigure} - \caption{\label{fig:apps}Applications of directional drilling.} -\end{figure} - -Figure~\ref{fig:apps} shows the different applications of directional drilling. - -In order to steer the drilling mechanism, different methods have been implemented~\cite{Downton2000}. - -Conventionally, steerable motors were used to change direction during the drilling process. This motors have two modes: rotating and sliding. In the first mode, the whole drill rotates as in conventional drilling in a straight direction. In order to change direction, the drillstring is halted and a downhole motor is used, which is powered by drilling muds in order to point the bit in the desired drilling direction, the two modes are shown in Figure~\ref{fig:drillingmodes}. This method proved to be very challenging, requiring a lot of skill from the drill operators and as the complexity of boreholes was increased the quality diminished due to the crookedness induced by the rotating mode. - - - -\begin{figure}[ht]\centering - \begin{subfigure}[b]{0.15\textwidth} - \includegraphics[width=0.9\textwidth]{img/drillingmode1.pdf} - \caption{\label{fig:drillingmode1}Sliding.} - \end{subfigure} - \begin{subfigure}[b]{0.15\textwidth} - \includegraphics[width=0.9\textwidth]{img/drillingmode2.pdf} - \caption{\label{fig:drillingmode2}Rotating.} - \end{subfigure} - \caption{\label{fig:drillingmodes}Drilling modes of a steerable motor.} -\end{figure} - -Because of the previously mentioned issues, a new method to steer the direction of the drill was developed, namely the \acf{RSS}. This type of technology allows the drilling system to change direction while drilling, reducing borehole tortuosity, drilling time as well as increasing the \acf{ROP}. - -\section{General description of the directional drilling system} -%Read ~\cite{Monsieurs2015}, ~\cite{Kremers2013}, ~\cite{Perneder2013} introductions and ~\cite{Downton2000} - -A directional drilling system consists of the elements depicted in Figure~\ref{fig:directionaldrilling}. The drillsting (generally of a length in the order of kilometers) is the assembly of components from surface to bottom. Along this, power is transmitted to provide axial force and rotational speed imposed at the drill rig. For the upper part of the drillstring, the axial force (commonly known as hookload) along with its own weight, provide tension in the drillstring to prevent buckling. The \acf{BHA} constitutes the lower part of the drillstring and is usually in the order of hundreds meters long. The \acs{BHA} is in compression, in contrast with the upper part of the drillstring, to induce an axial force transmitted to the drill bit, called weight on bit. The \acs{BHA} is commonly equipped with 3 to 5 stabilizers. - -\begin{figure}[ht]\centering - \includegraphics[width=0.4\textwidth]{img/drillingsystem.pdf} - \caption{\label{fig:directionaldrilling}General description of a directional drilling system.} -\end{figure} - -Between the first stabilizer and the bit, the previously mentioned \acs{RSS} is placed. It is equipped with mud actuators and sensors to provide inclination and azimuth control of the borehole. There exist two main types of \acs{RSS}: push-the-bit and point-the-bit. The first one applies lateral force, pushing the drillstring against the borehole using pads actuated by mud diverted with a disk valve. In point-the-bit systems, the bit is tilted relative to the rest of the tool to achieve the desired trajectory~\cite{Downton2000}. In recent years, a new type of \acs{RSS} is being researched on~\cite{Felczak2012}. This type of mechanism (called hybrid \acs{RSS}) combines the principles of point-the-bit and push-the-bit systems. Despite this, the most widely implemented type of \acs{RSS} is the push-the-bit type and its functioning is demonstrated in Figure~\ref{fig:RSS}. - -\begin{figure} - \centering - \begin{minipage}[t]{.5\textwidth} - \centering - \includegraphics[width=.35\linewidth]{img/RSS.pdf} - \captionof{figure}{RSS push-the-bit system.} - \label{fig:RSS} - \end{minipage}% - \begin{minipage}[t]{.5\textwidth} - \centering - \includegraphics[width=.6\linewidth]{img/PDC-bit.pdf} - \captionof{figure}{PDC drill bit~\cite{Bit}.} - \label{fig:PDC} - \end{minipage} -\end{figure} - - -In most cases for directional drilling, \acf{PDC} bits are used, an example of these is shown in Figure~\ref{fig:PDC}. This type of bit rotates using mud injected from the rig and it is, in particular, laterally aggressive while drilling a borehole. This last characteristic, among other causes, can produce a commonly known phenomenon called bit walk tendency, which is the natural behavior of the bit to deviate from its original path due to the lateral forces present while drilling. This effect induces the formation of spiraling patterns along the borehole walls~\cite{JulienMarckEmmanuelDetournayAndreaKuesters2014} which can be harmful to the drilling process causing: - -\begin{enumerate}[(a)] - \item An increase of frictional resistance of the borehole, which reduces the \acs{ROP}, considerably affecting drilling times. - \item Reduced target-reach accuracy. - \item Difficulty in the insertion of the casing after the borehole has been drilled. - \item In a severe case, instability of the borehole evolution. -\end{enumerate} - -Reducing this borehole spiraling effect is one of the main challenges in the directional drilling industry in recent years. - -\begin{figure}[ht]\centering - \includegraphics[width=0.3\textwidth]{img/boreholespiraling.pdf} - \caption{\label{fig:boreholespiraling}Borehole spiraling mechanism~\cite{Marck2013}.} -\end{figure} - -In addition to the \acs{RSS}, directional drilling systems nowadays are equipped with \acf{MWD} systems, that allow the driller to receive information from the downhole elements to the surface at the same time as the drilling process takes place. Current MWD systems use the fluid flowing from inside the borehole, propagating it up to the top of the drillstring, which in practice generates a slow transmission of information (at a rate of the order of 10 bits per second~\cite{Perneder2013}). - -The control process of the \acs{RSS} is generally conducted by three control loops: - -\begin{enumerate} - \item The main control loop, which sends commands for the actuation forces of the \acs{RSS} in order to steer the borehole in a desired orientation using parameterized angles. - \item An inner control loop which translates the incoming commands of the main loop into the \acs{RSS} valve system. - \item An outer control loop which is in charge of correcting possible deviations in Cartesian coordinates, which are generally corrected by a human operator. -\end{enumerate} - -A schematic block diagram of the three control loops is shown in Figure~\ref{fig:controlloops}. In present time, the most common practice for \acs{RSS} type of systems is to apply constant force, expecting to obtain a constant curvature, which is generally not the case. The focus of this survey will be on the main control loop, as this is the most involved level of control when steering the systems and, furthermore, it is believed that borehole spiraling can be corrected at this loop by implementing an RSS orientation controller. - -There is not much research on control of directional drilling with a \acs{RSS}, in part because it is relatively new to the industry~\cite{Downton2000}. As a consequence of this novelty, there are still discrepancies between the models that describe it. In most situations, developed controllers are designed for two-dimensional models, for which the bit walk tendency is not taken into account. This is greatly disadvantageous, since this effect is one of the main causes of borehole spiraling. - -\begin{figure}[ht]\centering - \includegraphics[width=0.8\textwidth]{img/controlloops.pdf} - \caption{\label{fig:controlloops}Control loops of a directional drilling system.} -\end{figure} - -\section{Outline of the literature survey} -The main goal of this literature survey is to provide an overview of previous works on the control of directional drilling, to ultimately assess the main challenges for this type of process and to propose a direction for the upcoming research in the author's MSc project. Since the models on which controllers are applied differ greatly from one to another, Chapter 2 will focus on the different numerical and mathematical descriptions of directional drilling. It is important to notice, that the notation for the different models varies significantly from one representation to the other. It has been chosen to keep the notation given by the authors so it can be compared to the original papers. - - Chapter 3 will go deeper into the type of controllers that have been developed for the different models, analyzing their qualities in order to choose a starting point for the upcoming research work. It is important to notice that the main challenges of directional drilling for \acs{RSS} are related to borehole quality which is, in turn, directly related to borehole spiraling. This chapter analyzes how each of the controllers tackles this problem. As before, that notation is kept the same as the one given by the authors. - - Chapter 4 will draw the conclusions derived from the literature survey performed in order to provide a first initial control approach for the remaining work of the thesis. Ultimately the conclusion of this survey has to be a definite proposal for a control strategy that deals with the unconsidered or inconclusive issues in previous works. - -\end{document} - - - - diff --git a/Research Proposal IIT/Research proposal/sections (copy)/linearizing.tex b/Research Proposal IIT/Research proposal/sections (copy)/linearizing.tex deleted file mode 100755 index b3958fb..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/linearizing.tex +++ /dev/null @@ -1,79 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\vspace{20pt} - -In this section a linearization of the previously derived model is performed, this step is needed since the controllers that are going to be implemented are linear. Linearization is based on the Taylor Series, using only the first term to approximate the function. This means that the behavior of the nonlinear system will be similar to the linear one in a region close to the point where the function is evaluated. The point on which the system is linearized is based on the equilibrium points of the system, which are the regions where the derived controllers work. - -Considering only one full rotation of the system (since $\pi=2\pi=...=n\pi$ in the setup), The Rotational Pendulum has four equilibrium points, three unstable and one stable, which are: - -\begin{eqnarray} -SE &=& \begin{bmatrix} -\pi & 0 & 0 & 0 -\end{bmatrix} \nonumber \\ -UE_1 &=& \begin{bmatrix} -0 & \pi & 0 & 0 -\end{bmatrix} \nonumber \\ -UE_2 &=& \begin{bmatrix} -0 & 0 & 0 & 0 -\end{bmatrix} \nonumber \\ -UE_3 &=& \begin{bmatrix} -\pi & \pi & 0 & 0 -\end{bmatrix} \nonumber \\ -\end{eqnarray} - -$SE$ \nomenclature{$SE$}{Stable equilibrium.} stands for the stable equilibrium and $UE_i$ \nomenclature{$UE_i$}{Unstable equilibrium i.} stands for the unstable equilibria \textit{i}. It has to be taken into account, that due to the fact that in the setup the first link presents high friction, the equilibrium point $UE_1$, can also be consider as stable, despite the fact that in theory is unstable. In order to perform the linearization, a value to the input must be given. In this case the input $u=0$ will be used. The obtained equations of motion from section \ref{sec:Mathematical} will be linearized according to the following equation: - -\begin{equation} -\ddot{q_l} = \ddot{q}(0) + - \frac{\partial \ddot{q}}{\partial q}\bigg\vert_{q = q_0}(q-q_0) + - \frac{\partial \ddot{q}}{\partial u}\bigg\vert_{u = 0}(u) -\end{equation} - -where $q_0$ \nomenclature{$q_0$}{Point at which the linearization will be performed.} is the point at which the linearization will be performed and $\ddot{q_l}$ \nomenclature{$\ddot{q_l}$}{Linearized equation for the angular acceleration.} is the linearized equation of the angular acceleration. Furthermore, after the linearization, the system is brought to state-space form using the following change of variables: - -\begin{eqnarray} -x_1 &=& \theta_1 = q_1 \nonumber \\ -x_2 &=& \theta_2 = q_2\nonumber \\ -x_3 &=& \dot{x_1} = \dot{\theta_1} = \dot{q_1} \nonumber \\ -x_4 &=& \dot{x_2} = \dot{\theta_2} = \dot{q_2} \nonumber \\ -\end{eqnarray} - -where $x_i$ \nomenclature{$x_i$}{State "i" of the system.} represents the state "i" of the system. As it can be seen, the system corresponds to fourth order. As before, the linearization was performed using the Symbolic Math Toolbox\textsuperscript{\textregistered}. An expression in terms of the parameters of the system was obtained, which can be found completely on appendix \ref{AppB:LinearizedModel}. The system has the following form: - -\begin{equation} -\begin{bmatrix} -\dot{x_1} \\ -\dot{x_2} \\ -\dot{x_3} \\ -\dot{x_4} \\ -\end{bmatrix} = -A -\begin{bmatrix} -x_1 \\ -x_2 \\ -x_3 \\ -x_4 \\ -\end{bmatrix} + -B u -\end{equation} - -Since the system gives measurements for $\theta_1$ and $\theta_2$, these two will be taken as outputs of the system, and direct feedthrough is not considered in the system ($D=0$) the resulting equation is the following: - -\begin{equation} -y = -\begin{bmatrix} -1 & 0 & 0 & 0 \\ -0 & 1 & 0 & 0 \\ -\end{bmatrix} -\begin{bmatrix} -x_1 \\ -x_2 \\ -x_3 \\ -x_4 \\ -\end{bmatrix} -\end{equation} - - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/modeling.tex b/Research Proposal IIT/Research proposal/sections (copy)/modeling.tex deleted file mode 100755 index d72b95a..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/modeling.tex +++ /dev/null @@ -1,192 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\vspace{20pt} -In this section the mathematical model of the system is computed, due to the fact that this is one of the main tools required for control design. In future steps, having a proper model can be crucial for controller synthesis and tuning. Since the mechanical properties of the system are mostly known, a first principles model will be obtained based on Lagrangian mechanics to derive the equations of motion for the system. The computation of the model was done using the MATLAB\textsuperscript{\textregistered} Symbolic Math Toolbox\textsuperscript{\textregistered}. - -\subsection{Kinematics}\label{kinematics} - -Firstly, the geometrical relation between the variables of the system is obtained based on the simplified schematic drawing of the system depicted in figure \ref{fig:Schematic}: - -\begin{figure}[H] - \centering - \begin{subfigure}[b]{0.4\textwidth} - \includegraphics[width=0.6\textwidth]{SchematicModel.png} -\caption{\label{fig:Schematic} Schematic of the Rotational Pendulum.} - \end{subfigure} - \begin{subfigure}[b]{0.4\textwidth} - \includegraphics[width=0.8\textwidth]{Velocity.png} -\caption{\label{fig:Velocity} Components of linear velocity on link 2.} - \end{subfigure} -\end{figure} - -In figure \ref{fig:Schematic} it can be seen that the angle of the first link with respect to the vertical axis is represented by the variable \nomenclature{$\theta_1$}{Angle of the first link with respect to the vertical axis.}$\theta_1$, and the angle between the second link and the first link is represented by \nomenclature{$\theta_2$}{Angle of the second link with respect to the first link.}$\theta_2$, this angles, as mentioned in section \ref{Encoders}, have positive rotation in the clockwise direction. The schematic drawing also shows that the torque $\tau$ \nomenclature{$\tau$}{Torque applied to first link.}is being applied to the first link of the system. The lengths of the links are represented by the variables \nomenclature{$l_1$}{Length of the first link.}$l_1$ and \nomenclature{$l_2$}{Length of the second link.}$l_1$. The linear velocities of the links \nomenclature{$v_1$}{Linear velocity of the first link.}$v_1$ and \nomenclature{$v_2$}{Linear velocity of the second link.}$v_2$ are represented as the product of the angular velocities \nomenclature{$\dot{\theta}_1$}{Angular velocity of the first link.}$\dot{\theta}_1$ and \nomenclature{$\dot{\theta}_2$}{Angular velocity of the second link.}$\dot{\theta}_2$ and the distance to their center of mass \nomenclature{$c_1$}{Distance to center of mass of the first link.}$c_1$ and \nomenclature{$c_2$}{Distance to center of mass of the second link.}$c_2$. Nevertheless, the total linear velocity of link two is the sum of the velocity caused by the first link and its own angular velocity. An schematic drawing of the sum of the velocities is shown in figure \ref{fig:Velocity}. - -From the schematic drawing, the total velocity of the second link can be expressed by: - -\begin{equation} - v_2^2=(l_1\dot{\theta_1}cos(\theta_2)+c_2\dot{\theta_2})^2+(l_1\dot{\theta_1}sin(\theta_2))^2 -\end{equation} - -\begin{equation} v_2^2=l_1^2\dot{\theta_1}^2cos^2(\theta_2)+2l_1c_2\dot{\theta_1}\dot{\theta_2}cos(\theta_2)+c_2^2\dot{\theta_2}^2+l_1^2\dot{\theta_1}^2sin^2(\theta_2) -\end{equation} - -since: - -\begin{equation*} -sin^2(\theta)+cos^2(\theta)=1 -\end{equation*} - -Then the final expression for $v_2^2$ is given by: - -\begin{equation} -v_2^2=l_1^2\dot{\theta_1}^2+c_2^2\dot{\theta_2}^2+2l_1c_2\dot{\theta_1}\dot{\theta_2}cos(\theta_2) -\end{equation} - -This velocity will be kept as a squared term, due to the fact that the expression for the kinetic energy is given by the squared values of the velocities. In order to simplify the computation of the potential and kinetic energy of the system, the components of the positions of the centers of mass $c_1$ and $c_2$ with respect to the horizontal and vertical axis are computed as: - - -\begin{eqnarray} -x_1 &=& c_1 sin(\theta_1) \nonumber\\ -x_2 &=& l_1 sin(\theta_1) + c_1 sin(\theta_2 + \theta_1) \nonumber\\ -y_1 &=& c_1 cos(\theta_1) \nonumber\\ -y_2 &=& l_1 cos(\theta_1) + c_2 cos(\theta_2 + \theta_1) -\end{eqnarray} - -as well as their time derivatives, shown in the set of equations \ref{eqn:TimeDerivativesXY}. - -\begin{eqnarray}\label{eqn:TimeDerivativesXY} -\dot{x_1} &=& c_1 cos(\theta_1) \dot{\theta_1}; \nonumber\\ -\dot{x_2} &=& l_1 cos(\theta_1) \dot{\theta_1} + c_2 cos(\theta_2 + \theta_1) \dot{\theta_2} + c_2 cos(\theta_2 + \theta_1)\dot{\theta_1} \nonumber\\ -\dot{y_1} &=& -c_1 sin(\theta_1) \dot{\theta_1} \nonumber\\ -\dot{y_2} &=& -l_1 sin(\theta_1) \dot{\theta_1} - c_2 sin(\theta_2 + \theta_1)*\dot{\theta_2} - c_2 sin(\theta_2 + \theta_1)\dot{\theta_1} -\end{eqnarray} - -\subsection{Dynamics}\label{math} - - -A proper framework to generate the equations of motion, for a mechanical system such as the rotational pendulum is Lagrangian Mechanics. The Euler-Lagrange equations are shown in \ref{eqn:EulerLagrange}. - -\begin{equation}\label{eqn:EulerLagrange} -\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = F -\end{equation} - -Where $q$ \nomenclature{$q$}{Vector of generalized coordinates.} stands for the vector of generalized coordinates of the system, which in this case are $\theta_1$ and $\theta_2$; $F$ \nomenclature{$F$}{External inputs to the system in Lagrangian Mechanics framework.}stands for external inputs into the system, which in this case is the torque $\tau$; and $L$ \nomenclature{$L$}{Lagrangian from Euler-Lagrange equations.} which is the Lagrangian obtained by the difference of the kinetic energy $T$ \nomenclature{$T$}{Kinetic energy.}and the potential energy $V$ \nomenclature{$V$}{Potential energy.}as shown in equation \ref{eqn:Lagrangian}. - -\begin{equation}\label{eqn:Lagrangian} -L=T-V -\end{equation} - -The total kinetic energy is represented by the following equation: - -\begin{equation} -T= \frac{1}{2} m_1 v_1^2 + - \frac{1}{2} m_2 v_2^2 + - \frac{1}{2} I_1 \dot{\theta_1}^2 + - \frac{1}{2} I_2 \dot{\theta_2}^2 -\end{equation} - -where $I_1$ and $I_2$ correspond to the inertias of link 1 and link 2 respectively. If the expressions for the components on the $x$ and $y$ axes are substituted the resulting equation is: - -\begin{equation} -T= \frac{1}{2} m_1 (\dot{x_1}^2 + \dot{y_1}^2) + - \frac{1}{2} m_2 (\dot{x_2}^2 + \dot{y_2}^2) + - \frac{1}{2} I_1 \dot{\theta_1}^2 + - \frac{1}{2} I_2 \dot{\theta_2}^2 -\end{equation} - - -moreover, the same expression in terms of $\theta_1$ and $\theta_2$ results in: - -\begin{equation} - T= \frac{1}{2}m_1 c_1 \dot{\theta_1}^2 + - \frac{1}{2}m_2 l_1^2\dot{\theta_1}^2+c_2^2\dot{\theta_2}^2 + - 2l_1c_2\dot{\theta_1}\dot{\theta_2}cos(\theta_2) + - \frac{1}{2}I_1\dot{\theta_1} + - \frac{1}{2}I_2\dot{\theta_2} -\end{equation} - - -As it can be seen, the total kinetic energy is the sum of the energy supplied to the system due to both angular and linear velocities. - -In the case of the potential energy, the drawing in figure \ref{fig:PotentialEnergy} is used as guidance. - -\begin{figure}[h] -\centering -\includegraphics[width=0.5\textwidth]{PotentialEnergy.png} -\caption{\label{fig:PotentialEnergy} Schematic drawing for the potential energy.} -\end{figure} - - -From the drawing, $g$ \nomenclature{$g$}{Gravitational acceleration.} represents the gravitational acceleration and its value will be taken as $g=9.81 [\frac{m}{s_2}]$. The potential energy can be derived in terms of the $y$ components as: - -\begin{equation} -V = m_1 g y_1 + m_2 g y_2 -\end{equation} - -as done with the kinetic energy, the potential energy is expressed in terms of $\theta_1$ and $\theta_2$ as: - -\begin{equation} -V = m_1 g c_1 cos(\theta_1) + m_2 g l_1 cos(\theta_1) + m_2 g c_2cos(\theta_2 + \theta_1); -\end{equation} - -In order to have a more accurate model of the system, the dissipated energy will be included into equation \ref{eqn:EulerLagrange} using a damping coefficient for the joints of both links $d_1$ \nomenclature{$d_1$}{Damping coefficient of link 1.} and $d_2$ \nomenclature{$d_2$}{Damping coefficient of link 2.}. The dissipated energy $D$ \nomenclature{$D$}{Dissipated energy.}then can be expressed as: - -\begin{equation} -D =\frac{1}{2} d_1 \dot{\theta_1}^2 + \frac{1}{2}d_2\dot{\theta_2}^2; -\end{equation} - -Furthermore, since the potential energy does not depend on velocities, the derivative with respect to $\dot{q}$ does not need to be computed for this term. Finally with all these considerations, the final expression of equation \ref{eqn:EulerLagrange} is: - -\begin{equation}\label{eqn:EulerLagrange2} -\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right) - \frac{\partial T}{\partial q} - \frac{\partial V}{\partial q} + \frac{\partial D}{\partial \dot{q}}= F -\end{equation} - -Then the previously computed expressions for the energy are plugged into equation \ref{eqn:EulerLagrange2} and expressing the angles $\theta_1$ and $\theta_2$ as generalized coordinates $q_1$ and $q_2$ the resulting motion equation is: - -\begin{equation} -M(q)\ddot{q} + C(q,\ddot{q}) \dot{q} + G(q)= -\begin{bmatrix} -\tau \\ -0 -\end{bmatrix} -\end{equation} - -where $M(q)$ \nomenclature{$M$}{Generalized mass matrix.} is the generalized mass matrix, $C(q,\dot{q})$ \nomenclature{$C$}{Coriolis matrix.} is the Coriolis matrix and $G(q)$ \nomenclature{$G$}{Matrix of gravitational terms.} is the matrix of gravitational terms. The matrices for this system are defined as: - -\begin{equation*} -M(q)= -\begin{bmatrix} -m_1 c_1^2 + m_2 c_2^2 + 2 m_2 cos(q_2) c_2 l_1 + m_2 l_1^2 + I_1 + I_2 & m_2 c_2^2 + l1*m2*cos(q_2) c_2 + I_2 \\ -m_2 c_2^2 + l_1 c_2 m_2 cos(q_2) + I_2 & m_2 c_2^2 + I_2 -\end{bmatrix} -\end{equation*} - -\begin{equation*} -C(q,\dot{q})= -\begin{bmatrix} -d_1 & -c_2 l_1 m_2 sin(q_2)(2 \dot{q_1} + \dot{q_2} ) \\ -c_2 l_1 m_2 sin(q_2) \dot{q_1} & d_2 -\end{bmatrix} -\end{equation*} - -\begin{equation*} -G(q)= -\begin{bmatrix} --g m_2 (c_2 sin(q_2 + q_1) + l_1 sin(q_1)) - c_1 g m_1 sin(q_1) \\ --c_2 g m_2 sin(q_2 + q_1) -\end{bmatrix} -\end{equation*} - -Finally, if the accelerations of the system $\ddot{\theta_1}$ and $\ddot{\theta_2}$ are isolated, the equations can be expressed as: - -\begin{equation}\label{eq:eqom} -\ddot{q} = M^{-1}(q)\left(\begin{bmatrix} -\tau \\ 0 -\end{bmatrix} - C(q,\dot{q})- G(q)\right) -\end{equation} - - - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/observer.tex b/Research Proposal IIT/Research proposal/sections (copy)/observer.tex deleted file mode 100755 index d32cda7..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/observer.tex +++ /dev/null @@ -1,52 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\vspace{20pt} -In similar fashion as for controlability, the observability of the system can be tested by checking the observability matrix as given in \cite{Controlsystemdesign} for rank. Using the MATLAB\textsuperscript{\textregistered} commands \mcode{obsv} and \mcode{rank} all four equilibria are tested to be observable, thus an observer can be implemented. \\ -An implementation of a deterministic observer was tried, but yielded unsatisfactory results; the estimates of the angular velocities were very unreliable. Since the noise in the measurement signal is white it contains high-frequent components which will cause for large spikes if the derivative of this signal is taken. In order to get better estimates of the states, a Kalman filter is designed. - -\subsection{Kalman Filter} -The Kalman Filter is implemented in Simulink\textsuperscript{\textregistered} as can be seen in Figure \ref{fig:KalmanFilter}. The Kalman Filter Block contains the user defined function below, which is an implementation of the conventional Kalman filter as given in\cite{Filtering}. -\begin{lstlisting} -function [x, P, K] = fcn(u, y_real, Q, R, x_old, P_old, A, B, C) - -% Measurement update -K = (P_old * C') / (C*P_old*C'+R); -x = x_old + K * (y_real - C*x_old); -P = (eye(size(A))-K*C)*P_old; - -% Time update -x = A * x + B * u; -P = A * P * A' + Q; -end -\end{lstlisting} -\begin{figure}[H] -\centering -\includegraphics[width=0.4\textwidth]{KalmanFilter2.png} -\caption{\label{fig:KalmanFilter} Schematic of the Kalman Filter} -\end{figure} -The Kalman filter is sensitive to changes in the Q and R matrices, which therefor have to be chosen carefully. Since the R matrix represents the covariance of the measurement noise, an initial guess for R is chosen as an diagonal matrix with entries in the same order of magnitude as covariance data from \ref{Encoders}. The assumption was made that the variance of the process is smaller than that of the measurement noise and that cross-covariance is zero. This yields again diagonal matrix of size 4 with entries smaller than that in R. Through mostly trial and error, the best results were found for the following Q and R -\begin{align*} -R = -\begin{bmatrix} -1\times10^{-2} & 0 \\ -0 & 3\times10^{-2} -\end{bmatrix} -& \ \ \ -Q = -\begin{bmatrix} -1\times10^{-6} &0&0&0 \\ -0&1\times10^{-5}&0&0\\ -0&0&1\times10^{-6}&0\\ -0&0&0&1\times10^{-5} -\end{bmatrix} -\end{align*} - -Figure \ref{fig:KalmanFilterP} shows the performance of the Kalman filter. The Kalman filter is seen to perform sufficiently; it predicts the states, has no bias and it filters out the noise. However the performance could still be improved, the Kalman estimated $\theta_2$ has a smaller amplitude then the real system, also there is some phase lead on the real angles. Since the performance of the Kalman filter has a big influence on the performance of the overall closed loop system, further investigation in a more systematic tuning of the Kalman matrices Q and R is desirable. -\begin{figure}[H] -\centering -\includegraphics[width=0.6\textwidth]{KalmanEstimates.png} -\caption{\label{fig:KalmanFilterP} Real versus Kalman estimates of $\theta_1$ and $\theta_2$} -\end{figure} -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/overview.tex b/Research Proposal IIT/Research proposal/sections (copy)/overview.tex deleted file mode 100755 index 5a7cee4..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/overview.tex +++ /dev/null @@ -1,15 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -This report will go through the following topics: - -\begin{enumerate} - \item \textbf{Derivation of the closed loop error dynamics for the 3D directional drilling system using output-feedback controller}. Sections 1 and 2 of the document are the corrected version of the error dynamics and the linearization. They are the same as in the previous report, but with the changes made based on the remarks from our last meeting. - \item \textbf{Controller Design}. This section shows the results of the proposed controller design with the changes made on the remarks on comments from our last meeting. - \item \textbf{Robust stability analysis} This section starts with the derivation of the error dynamics for the system including parameter uncertainty. The analysis and results is not complete yet. - \item \textbf{Questions} The last part of the document contains two questions that I ran into while working. -\end{enumerate} - - -\end{document} diff --git a/Research Proposal IIT/Research proposal/sections (copy)/planning.tex b/Research Proposal IIT/Research proposal/sections (copy)/planning.tex deleted file mode 100755 index c87a2d7..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/planning.tex +++ /dev/null @@ -1,178 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\vspace{20pt} -\subsection{Planning} - -In order to set up a plan considering the amount of time that it will take to complete the thesis project, initially a tentative structure for the final document is proposed. - -\begin{enumerate} - \item Abstract - \item Introduction - \item 3D Directional drilling model - \item Open-loop dynamics - \item Controller Design - \item Conclusions and recommendations -\end{enumerate} - -With this structure in mind, the following table for the planning is proposed: - -\begin{table}[H] -\caption{Planning.} -\centering -\begin{tabular}{|l|l|l|} -\hline -Task &Time to fulfill task& Date of completion \\ \hline -Closed-loop error dynamics of neutral bit walk model & 5 days & 21/03/2016\\ -Controller synthesis neutral bit walk & 3 days & 24/03/2016\\ -Stability analysis and simulations of neutral bit walk model&4 days & 28/03/2016\\ -Controller simulations neutral bit walk& 2 days & 30/03/2016\\ -Write results of controller & 5 days & 04/04/2016\\ -Robustness analysis of neutral bit walk model controller &1 week & 11/04/2016\\ -Write results of robustness analysis & 5 days & 16/04/2016\\ -Closed-loop error dynamics of non-neutral bit walk model &2 weeks & 30/04/2016\\ -Controller synthesis of non-neutral bit walk model &3 days & 03/05/2016 \\ -Stability analysis and simulations of non-neutral bit walk model &4 days & 07/05/2016 \\ -Controller simulations of non-neutral bit walk model &2 days & 09/05/2016 \\ -Writing results of non-neutral bit walk controller &5 days & 14/05/2016 \\ -Robustness analysis of non-neutral bit walk controller & 1 week & 21/05/2016\\ -Writing results of robustness analysis & 5 days & 26/05/2016\\ -Write model and open loop dynamics section & 3 days & 29/05/2016 \\ -Writing conclusions, recommendations, abstract and introduction & 5 days & 03/06/2016\\ \hline -\end{tabular} -\label{tab:Planning} -\end{table} - -A tentative date to have a draft of the report is the 3rd of June. - -\begin{table}[H] - \caption{Daily planning.} - \centering - \begin{tabular}{|l|l||} - \hline - Task & Date\\ \hline - Develop Robust error dynamics & 04/09\\ - Analyze matrices & 04/10 \\ - Synthesize controller and simulations & 04/11\\ - Write down results of robustness and correct previous report & 04/12 - 04/15\\ \hline - \end{tabular} - \label{tab:Planning} -\end{table} - - - -\newpage -\section{Colloquia, workshops and presentations} - -In this section the complete list of colloquia and workshops attended to will be given, a total of 10 colloquia out of 15 is considered, as well as 5 workshops out of 7. It has also to be taken into account that I have already given my literature colloquium, remaining the final thesis presentation to be given and 2 workshop presentations. - -\subsection{Colloquia} - -\begin{enumerate} - \item Date: October 9th, 2015. \\ - Location: TUDelft.\\ - Topic: Model predictive control for efficiency improvement of a gas to liquids pilot plant.\\ - Presenter: Leon Kuiper. - - \item Date: November 11th, 2015.\\ - Location: TUDelft.\\ - Topic: GPU implementation of D-SABRE, a spline based wavefront reconstruction method for large-scale AO systems.\\ - Presenter: Niels Tielen. - - \item Date: January 22nd, 2016.\\ - Location: TUDelft.\\ - Topic: Robust urban traffic control.\\ - Presenter: Dik Jansen. - - \item Date: January 22nd, 2016.\\ - Location: TUDelft.\\ - Topic: Hybrid human in the loop model predictive control for urban drainage systems.\\ - Presenter: Franka Veltman. - - \item Date: January, 27th 2016.\\ - Location: TUDelft.\\ - Topic: Stochastic Distributed Coordination of Energy Balance Smart Thermal Grids.\\ - Presenter: Wayan Wicak Ananduta. - - \item Date: March 11th, 2016.\\ - Location: TUDelft.\\ - Topic: Propulsion control assesment by a system modelling approach\\ - Presenter: Max van de Leijgraaf. - - \item Date: March 11th, 2016.\\ - Location: TUDelft.\\ - Topic: Dynamic wind farm control for a sparse quasi LPV descriptor model.\\ - Presenter: Joeri Frederik. - - \item Date: October 14th, 2015.\\ - Location: TU Eindhoven.\\ - Topic: Model-based decoupled control of a 3D directional drilling system.\\ - Presenter: Frank Monsieurs. - - \item Date: December 4th, 2015.\\ - Location: University of Minnesota.\\ - Topic: A nonlinear dynamical model of borehole spiraling.\\ - Presenter: Julien Marck. - - \item Date: December 8th, 2015.\\ - Location: University of Minnesota\\ - Topic: Nonlinear model of a 2D directional drilling system.\\ - Presenter: Bart Heijke. - -\end{enumerate} - -\subsection{Workshops} - -\begin{enumerate} - \item Date: September 30th, 2015.\\ - Location: TUDelft.\\ - Topics and presenters: - \subitem Oscar Sondermeijer. Regression-based inverter control for power flow and voltage regulation. - \subitem Frank Warffemius. Iterative learning control:feasibility and implementation for a reticle stage. - \subitem Matteo Ciocca. Simple models for robot interaction. - \subitem Pascal Zijlstra. Control-based cryptography for a networked control system. - \subitem Filip Paszkiewicz. How to model a generic short stroke actuator? - \subitem Lex Blenkers. Railway disruption management. - - \item Date: October 30th, 2015.\\ - Location: TUDelft.\\ - Topics and presenters: - \subitem Filip Paszkiewicz. Control of generic short stroke actuator. - \subitem Raziel Gonzalez. Collision avoidance with haptic feedback for telepresence. - \subitem Pascal Zijlstra. Control based cryptography for a networked control system. - \subitem Nelleke van der Steen. Ship motion predcition for the ampelmann system. - \subitem Lex Blenkers. Railway disruption management. - \subitem Alfons Schure. End effector dynamics control of parallel manipulators for haptic feedback. - - \item Date: January 29th, 2016.\\ - Location: TUDelft.\\ - Topics and presenters: - \subitem Jelle Munk. Learning a state representation. - \subitem Ruben Burger. Dynamic obstacle avoidance of a 6DoF robotic manipulator. - \subitem Bender Botterman. - \subitem Guangshuo Xin. Parameter estimation of a human-like cybernetic model. - \subitem Frank de Winkel. Engineering the distributed robotcis lab v2.0. - \subitem Matteo Ciocca. Simple models for robot interaction. - - \item Date: February 24th, 2016.\\ - Location: TUDelft.\\ - Topics and presenters: - \subitem Jessica Bautista. Use of the nuclear norm for system identification. - \subitem Frank de Winkel. Distributed Formation Control using consensus and spatial predictive control. - \subitem Wour Feitsma. Design of an above water position determination system for a ship's hull maintenance robot. - \subitem Jelle Munk. Computing in the cloud. - \subitem Danny Hammeeteman. Convey information through non-verbal communication in robotics. - \subitem Niels Tielen. Implementation of a distributed spline based wavefront reconstruction method for extremely large telescopes on graphics processing units. - - \item Date: December 11th, 2015.\\ - Location: University of Minnesota.\\ - Topics and presenters: - \subitem Anna Liakou. Linear complementarity problem. - -\end{enumerate} - - - - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/questions.tex b/Research Proposal IIT/Research proposal/sections (copy)/questions.tex deleted file mode 100755 index b838ba4..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/questions.tex +++ /dev/null @@ -1,11 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - - -As for now, I have already performed the linearization of the system in matlab, although I have some questions regarding the stability analysis and the controller synthesis. - -1.- After linearizing, I checked how the pole locations changed for several values of $\Pi$ in the linearized system. It seems that the controller already designed, as long as you are close to the linearization point, it is already pretty robust (I checked values from $\Pi = 6000$ to $\Pi = 20000$, nominal being $\Pi = 14000$). I will write down that part of the report during the weekend and show it to you during the meeting to check if what I did was correct. - -2.- Regarding controller synthesis, the difference with the nominal case is that terms appear related to the error, the input and the integral action in the observer. But, if we do same analysis as before (all the perturbed states will go to zero if we are close to the linearization point), can we then design a controller for a grid of values of $\Pi$? -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/robustness_analysis.tex b/Research Proposal IIT/Research proposal/sections (copy)/robustness_analysis.tex deleted file mode 100755 index f41c870..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/robustness_analysis.tex +++ /dev/null @@ -1,464 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -An important aspect to analyze when doing controller design, is the robustness of the strategy against variations of the parameters that determine the model. In the case of a directional drilling system, there are two main parameters that can be considered to be uncertain, the active weight on bit $\Pi$ and the bit walk angle $\varpi$. In the case of $\Pi$, this uncertainty is related to different factors. In particular, changes in the applied hook-load, the variation of the interaction of the rock with the drillsting and the bit, and the decrease of sharpness of the bit as the trajectory evolves, are some of the factors that may affect the value of this parameter. It has to be mentioned, that all of this changes are usually considered to influence the fluctuation of $\Pi$ slowly with respect to overall evolution of the borehole. Regarding the bit walk angle $\varpi$, this parameter is present in the process due to its 3D nature, being one of the main factors that causes undesired behaviors such as borehole spiraling. The effect of this parameter will be analyzed and dealt with in subsequent sections, since the terms related to this parameter add nonlinear effects and coupling to the already complex dynamics of the system. - -Considering that the previous controller design was based on a nominal value of weight on bit $\Pi$, to test if the designed strategy is able to cope with the uncertainty of this parameter, it will be considered as $\Pi = \bar{\Pi} + \delta\Pi$, where $\bar{\Pi}$ represents the nominal value of weight on bit. - -\subsection{Error dynamics for robust stability analysis and controller design} - -In this section, the error dynamics of the system are derived once again, since the change of parameter $\Pi$ affects the structure of the system. Recalling Equation \eqref{eq:systemstatespace}, the states of the system are given by: - -\begin{align} - \begin{bmatrix} - x_\Theta' \\ - x_\Phi' - \end{bmatrix} =& - \begin{bmatrix} - A_0 & 0 \\ - 0 & A_0 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi) \\ - x_\Phi(\xi) - \end{bmatrix} + - \begin{bmatrix} - A_1 & 0 \\ - 0 & A_1 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_1) \\ - x_\Phi(\xi_1) - \end{bmatrix} + - \begin{bmatrix} - A_2 & 0 \\ - 0 & A_2 - \end{bmatrix} - \begin{bmatrix} - x_\Theta(\xi_2) \\ - x_\Phi(\xi_2) - \end{bmatrix} \nonumber\\ - +& - \begin{bmatrix} - B_{0\Theta} & 0\\ - 0 & B_{0\Phi}(\Theta,\check{\Theta},\Theta',\check{\Theta}') - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^* \\ - \Gamma_\Phi^* - \end{bmatrix} + - \begin{bmatrix} - B_{1\Theta} & 0\\ - 0 & B_{1\Phi}(\Theta,\check{\Theta}) - \end{bmatrix} - \begin{bmatrix} - \Gamma_\Theta^{*'} \\ - \Gamma_\Phi^{*'} - \end{bmatrix} + - \begin{bmatrix} - BW \\ - 0 - \end{bmatrix}, - \label{eq:systemstatespacerobust} -\end{align} - -and with output equations given by: - - \begin{align} - y_\Theta &= C_\Theta x_\Theta + D_\Theta \Gamma_\Theta^* + E W_y, \label{eq:output12robust}\\ - y_\Phi &= C_\Phi x_\Phi + D_\Phi \Gamma_\Phi^* \frac{\sin \check{\Theta}}{\sin \Theta} .\label{eq:output22robust} - \end{align} - -Herein matrices $A_0$, $A_1$, $A_2$, $B_{0i}$, $B_{1i}$, $C_i$ and $D_i$ for $i = \Theta, \Phi$ defined the same as in \eqref{eq:systemstatespace}, evaluated at $\Pi = \bar{\Pi} + \delta \Pi$. It has to be note that matrices $B_{0\Phi}$ and $B_{1\Phi}$ are kept (i.e. not introducing the $\alpha$ term) and have a dependency with respect to $\Theta$, $\check{\Theta}$, $\Theta'$ and $\check{\Theta}'$. From here on the dependency is not made explicit. - -As before, an input filter is introduced as: - -\begin{equation} - \Gamma_i^{*'} = -\bar{b}_0\Gamma_i^* - \bar{b}_1 u_i, -\end{equation} - -where $\bar{b}_0$ and $\bar{b}_0$ are defined as in \eqref{eq:Constants} and evaluated at $\bar{\Pi}$. An important remark is, that this input filter will not be able to get rid of the $\Gamma_i^*$-related terms (not even in the case of the inclination dynamics), due to the difference between nominal matrices $\bar{B}_{0i}$ and $\bar{B}_{1i}$ and their real versions. The system dynamics of the system can be derived (using the actual versions of the state matrices), applying the input filter and including it as a state of the system, as following: - - -\begin{align} -\begin{bmatrix} -x_\Theta' \\ -\Gamma_\Theta^{*'} \\ -x_\Phi' \\ -\Gamma_\Phi^{*'} -\end{bmatrix} =& -\begin{bmatrix} -A_0 & (B_{0\Theta} - B_{1\Theta} \bar{b}_0) & 0 & 0\\ -0 & -\bar{b}_0 & 0 & 0 \\ -0 & 0 & A_0 & (B_{0\Phi} - B_{1\Phi} \bar{b}_0)\\ -0 & 0 & 0 & -\bar{b}_0 -\end{bmatrix} -\begin{bmatrix} -x_\Theta(\xi) \\ -\Gamma_\Theta^{*}(\xi) \\ -x_\Phi(\xi) \\ -\Gamma_\Phi^{*} (\xi) -\end{bmatrix} + -\begin{bmatrix} -A_1 & 0 & 0 & 0\\ -0 & 0 & 0 & 0 \\ -0 & 0 & A_1 & 0 \\ -0 & 0 & 0 & 0 \\ -\end{bmatrix} -\begin{bmatrix} -x_\Theta(\xi_1) \\ -\Gamma_\Theta^{*}(\xi_1) \\ -x_\Phi(\xi_1) \\ -\Gamma_\Phi^{*} (\xi_1) -\end{bmatrix} \nonumber\\ -&+\begin{bmatrix} -A_2 & 0 & 0 & 0 \\ -0 & 0 & 0 & 0 \\ -0 & 0 & A_2 & 0 \\ -0 & 0 & 0 & 0 \\ -\end{bmatrix} -\begin{bmatrix} -x_\Theta(\xi_2) \\ -\Gamma_\Theta^{*}(\xi_2) \\ -x_\Phi(\xi_2) \\ -\Gamma_\Phi^{*} (\xi_2) -\end{bmatrix} + -\begin{bmatrix} --B_{1\Theta}\bar{b}_1 & 0 \\ --\bar{b}_1 & 0 \\ -0 & -B_{1\Phi}\bar{b}_1 \\ -0 & -\bar{b}_1 -\end{bmatrix} -\begin{bmatrix} -u_\Theta \\ -u_\Phi -\end{bmatrix} + -\begin{bmatrix} -BW \\ -0 \\ -0 \\ -0 -\end{bmatrix}. -\end{align} - - The feedforward input can be now defined as: - - \begin{equation} - u_{ri} = B^T (x'_{ri}(\xi) - \bar{A}_0 x_{ri}(\xi) - \bar{A}_1 x_{ri}(\xi_1) - \bar{A}_2 x_{ri}(\xi_2)). - \label{eq:feedforwardrobust} - \end{equation} - -This feedforward input can be utilized to define once again a desired input $\Gamma_{id}^*$ as in Equation \eqref{eq:filter2} and define an error coordinate $\Delta \Gamma_i^*$. Then the error dynamics can be defined as in \eqref{eq:errordynamics2robust}, using the fact that $x_i = e_i + x_{ri}$. - - -\begin{align} -\begin{bmatrix} -e_\Theta' \\ -\Delta \Gamma_\Theta^{*'} \\ -e_\Phi' \\ -\Delta \Gamma_\Phi^{*'} -\end{bmatrix} =& -\begin{bmatrix} -A_0 & (B_{0\Theta} - B_{1\Theta} \bar{b}_1) & 0 & 0\\ -0 & -b_0 & 0 & 0 \\ -0 & 0 & A_0 & (B_{0\Phi} - B_{1\Phi} \bar{b}_1) \\ -0 & 0 & 0 & -b_0 -\end{bmatrix} -\begin{bmatrix} -e_\Theta(\xi) \\ -\Delta \Gamma_\Theta^{*} (\xi) \\ -e_\Phi(\xi) \\ -\Delta \Gamma_\Phi^{*} (\xi) -\end{bmatrix} + -\begin{bmatrix} -A_1 & 0 & 0 & 0\\ -0 & 0 & 0 & 0 \\ -0 & 0 & A_1 & 0 \\ -0 & 0 & 0 & 0 -\end{bmatrix} -\begin{bmatrix} -e_\Theta(\xi_1) \\ -\Delta \Gamma_\Theta^{*} (\xi_1) \\ -e_\Phi(\xi_1) \\ -\Delta \Gamma_\Phi^{*} (\xi_1) -\end{bmatrix} \nonumber\\ -&+ -\begin{bmatrix} -A_2 & 0 & 0 & 0\\ -0 & 0 & 0 & 0 \\ -0 & 0 & A_2 & 0 \\ -0 & 0 & 0 & 0 -\end{bmatrix} -\begin{bmatrix} -e_\Theta(\xi_2) \\ -\Delta \Gamma_\Theta^{*} (\xi_2) \\ -e_\Phi(\xi_2) \\ -\Delta \Gamma_\Phi^{*} (\xi_2) -\end{bmatrix} + \begin{bmatrix} --B_{1\Theta} \bar{b}_1 & 0 \\ --\bar{b}_1 & 0 \\ -0 & -B_{1\Phi} \bar{b}_1 \\ -0 & -\bar{b}_1 -\end{bmatrix} -\begin{bmatrix} -v_\Theta \\ -v_\Phi -\end{bmatrix} + -\begin{bmatrix} -(-B - B_{1\Theta}\bar{b}_1) \\ -0 \\ -(-B - B_{1\Phi}\bar{b}_1) \\ -0 -\end{bmatrix} -\begin{bmatrix} -u_{r\Theta} \\ -u_{r\Phi} -\end{bmatrix} -\label{eq:errordynamics2robust}+ -\begin{bmatrix} -F_{p\Theta} \\ -0 \\ -F_{p\Phi} \\ -0 -\end{bmatrix}, -\end{align} - -where the perturbation terms $F_{p\Theta}$ and $F_{p\Phi}$ are defined as: - -\begin{align} - F_{p\Theta} &= (B_{0\Theta} - B_{1\Theta} \bar{b}_0)\Gamma_{\Theta d}^* + dA_0 x_{r\Theta} (\xi) + dA_1 x_{r\Theta} (\xi_1) + dA_2 x_{r\Theta} (\xi_2) + BW \label{eq:PerturbationForces1}\\ - F_{p\Phi} &= (B_{0\Phi} - B_{1\Phi} \bar{b}_0)\Gamma_{\Phi d}^* + dA_0 x_{r\Theta} (\xi) + dA_1 x_{r\Theta} (\xi_1) + dA_2 x_{r\Theta} (\xi_2) \label{eq:PerturbationForces2}\\ -\end{align} - -where $\Delta A_t = A_t - \bar{A}_t$ for $t = 0,1,2$. Then, the state feedback controller $v_i$ is implemented in the same way as in \eqref{eq:controllerlowpasserror}, \eqref{eq:controllerintegralerror} and \eqref{eq:controllerfeedbackerror} i.e. - - \begin{align} - z_{1i}' &= \zeta \begin{bmatrix} - k_{1i} & 0 & 0 - \end{bmatrix}(\check{x}_i - x_{ri}) \\ - z_{2i}' &= -\gamma z_{2i} + \gamma (z_{1i} + K_i(\check{x}_i - x_{ri})) \\ - v_i &= z_{2i}, - \end{align} - -Accounting for the observer design, the same integral action will be included and defined as in \eqref{eq:observerintegral}, namely, - - \begin{equation} - q_i' = \zeta[l_{1i},l_{2i}](y_i - \check{y}_i). \label{eq:observerintegral2} \nonumber - \end{equation} - -Then the observer dynamics can be defined as: - -\begin{align} -\begin{bmatrix} -\check{x}_\Theta' \\ -q_\Theta' \\ -\check{x}_\Phi'\\ -q_\Phi' -\end{bmatrix} &= -\begin{bmatrix} -\bar{A}_0 & 0 & 0 & 0\\ -0 & 0 & 0 & 0\\ -0 & 0 & \bar{A}_0 & 0 \\ -0 & 0 & 0 & 0 \\ -\end{bmatrix} -\begin{bmatrix} -\check{x}_\Theta(\xi) \\ -q_\Theta(\xi) \\ -\check{x}_\Phi(\xi) \\ -q_\Phi (\xi) -\end{bmatrix} + -\begin{bmatrix} -\bar{A}_1 & 0 & 0 & 0\\ -0 & 0 & 0 & 0\\ -0 & 0 & \bar{A}_1 & 0 \\ -0 & 0 & 0 & 0 \\ -\end{bmatrix} -\begin{bmatrix} -\check{x}_\Theta(\xi_1) \\ -q_\Theta(\xi_1) \\ -\check{x}_\Phi(\xi_1) \\ -q_\Phi(\xi_1) \\ -\end{bmatrix} + -\begin{bmatrix} -\bar{A}_2 & 0 & 0 & 0\\ -0 & 0 & 0 & 0\\ -0 & 0 & \bar{A}_2 & 0 \\ -0 & 0 & 0 & 0 \\ -\end{bmatrix} -\begin{bmatrix} -\check{x}_\Theta(\xi_2) \\ -q_\Theta(\xi_2) \\ -\check{x}_\Phi(\xi_2) \\ -q_\Phi(\xi_2) -\end{bmatrix} \nonumber\\ -&+ -\begin{bmatrix} -L_\Theta(y_{\Theta} - \check{y}_{\Theta})\\ -\zeta[l_{1\Theta},l_{2\Theta}](y_\Theta - \check{y}_\Theta) \\ -L_{\Phi}(y_{\Phi} - \check{y}_{\Phi}) \\ -\zeta[l_{1\Phi},l_{2{\Phi}}](y_\Phi - \check{y}_\Phi) -\end{bmatrix} + -\begin{bmatrix} -Bq_\Theta \\ -0 \\ -Bq_\Phi \\ -0 -\end{bmatrix} + -\begin{bmatrix} -B(u_{r\Theta} + v_\Theta)\\ -0 \\ -B(u_{r\Phi} + v_\Phi) \\ -0 -\end{bmatrix}, -\end{align} - -and with corresponding output equations - -\begin{align} - \check{y}_\Theta &= \bar{C}_\Theta \check{x}_\Theta + \bar{D}_\Theta \Gamma_\Theta^* \\ - \check{y}_\Phi &= \bar{C}_\Phi \check{x}_\Phi + \bar{D}_\Phi \Gamma_\Phi^* -\end{align} - -Since matrices $C_i$ and $D_i$ differ from their nominal versions, the description for the derivatives of the states of the integral action for both inclination and azimuth are defined by (after substituting the output equations and considering the fact that $\check{x}_i = e_i + x_{ri} - \delta_i$): - -\begin{align} - q_\Theta' &= \zeta[l_{1\Theta},l_{2\Theta}](\Delta C_\Theta e_\Theta + \Delta C_\Theta x_{r\Theta} + \Delta D_\Theta \Delta \Gamma_\Theta^* + \Delta D_\Theta \Gamma_{\Theta d}^* + \bar{C}_\Theta \delta_\Theta + EW_y), \\ - q_\Phi' &= \zeta[l_{1\Phi},l_{2\Phi}](\Delta C_\Phi e_\Phi + \Delta C_\Phi x_{r\Phi} + D_\Phi (\Delta \Gamma_{\Phi} + \Gamma_{\Phi d}^*)^*\frac{\sin \check{\Theta}}{\sin \Theta} - \bar{D}_\Phi (\Delta \Gamma_\Phi^* + \Gamma_{\Phi d}^*) + \bar{C}_\Phi \delta_\Phi), -\end{align} - -where $\Delta D_i = D_i-\bar{D}_i$ and $\Delta C_i = C_i - \bar{C}_i$. Afterwards, the observer error dynamics can be obtained. The main difference is that in this case, due to model uncertainty, the observer dynamics are not decoupled from the system dynamics (not even for the inclination). The complete system closed-loop error dynamics (for state vector $X(\xi)$ defined as in \eqref{eq:totaldynamics}) are given by - -\begin{align} - X'(\xi) =& A_{0cl}X(\xi) + A_{1cl}X(\xi_1) + A_{2cl} X(\xi_2) + \nonumber\\ &P_{cl}(u_{r\Theta},u_{r\Phi},\Gamma_{\Theta d}^*,\Gamma_{\Phi d}^*,x_{r\Theta}(\xi),x_{r\Theta}(\xi_1), x_{r\Theta}(\xi_2),x_{r\Phi}(\xi),x_{r\Phi}(\xi_1),x_{r\Phi}(\xi_2), \Theta,\check{\Theta},W,W_y), - \label{eq:totaldynamicsrobust} -\end{align} - -where: - -\begin{equation} -A_{0cl} = -\begin{bmatrix} -A_{0\Theta} & 0 \\ -0 & A_{0\Phi} -\end{bmatrix}, \qquad -A_{1cl} = -\begin{bmatrix} -A_{1\Theta} & 0 \\ -0 & A_{1\Phi} -\end{bmatrix}, \qquad -A_{2cl} = -\begin{bmatrix} -A_{2\Theta} & 0 \\ -0 & A_{2\Phi} -\end{bmatrix}, -\label{eq:ClosedLoopMatricesRobust} -\end{equation} - -where the system matrices in \eqref{eq:ClosedLoopMatricesRobust} and the vector $P_{cl}(\xi,\xi_1,\xi_2, \Theta,\check{\Theta},W,W_y)$ (where all the dependencies on terms related to trajectory have been substituted by $\xi$, $\xi_1$ and $\xi_2$) are given by: -\begingroup -\renewcommand*{\arraystretch}{1.5} -\begin{align*} -A_{0\Theta} &= -\begin{bmatrix}[cccc:cc] -A_0 & (B_{0\Theta} - B_{1\Theta} \bar{b}_0) & 0 & -B_{1\Theta}\bar{b} _1 & 0 & 0\\ -% -0 & -\bar{b}_0 & 0 & -\bar{b}_1 & 0 & 0\\ -\zeta \begin{bmatrix}k_{1\Theta} , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_{1\Theta} , 0 , 0\end{bmatrix} & 0\\ -% -\gamma K_\Theta & 0 & \gamma & -\gamma & -\gamma K_\Theta & 0\\ \hdashline -% -(\Delta A_0 - L_\Theta \Delta C_\Theta) & (B_{0\Theta} - B_{1\Theta} \bar{b}_0 - L_\Theta \Delta D_\Theta) & 0 & (-B -B_{1\Theta}\bar{b}_1) & \bar{A}_0 - L_\Theta \bar{C}_\Theta & -B\\ -% -\zeta \begin{bmatrix}l_{1\Theta} , l_{2\Theta} \end{bmatrix} \Delta C_\Theta & \zeta \begin{bmatrix}l_{1\Theta} , l_{2\Theta} \end{bmatrix} \Delta D_\Theta & 0 & 0 & \zeta \begin{bmatrix}l_{1\Theta} , l_{2\Theta} \end{bmatrix} \bar{C}_\Theta & 0 -\end{bmatrix},\\ -\nonumber \\ -% -% -% -A_{0\Phi} &= -\begin{bmatrix}[cccc:cc] -A_0 &(B_{0\Phi} - B_{1\Phi} \bar{b}_0) & 0 & -B_{1\Phi}\bar{b} _1\\ -% -0 & -\bar{b}_0 & 0 & -\bar{b}_1 & 0 & 0 \\ -% -\zeta \begin{bmatrix}k_{1\Phi} , 0 , 0\end{bmatrix} & 0 & 0 & 0 & -\zeta\begin{bmatrix}k_{1\Phi} , 0 , 0\end{bmatrix} & 0 \\ -% -\gamma K_{\Phi} & 0 & \gamma & -\gamma & -\gamma K_{\Phi} & 0\\ \hdashline -% -(\Delta A_0 - L_\Phi \Delta C_\Phi) & (B_{0\Phi} - B_{1\Phi} \bar{b}_0 - L_\Phi (D_\Phi \frac{\sin \check{\Theta}}{\sin \Theta} - \bar{D}_\Phi)) & 0 & (-B -B_{1\Phi}\bar{b}_1) & \bar{A}_0 - L_\Phi \bar{C}_{\Phi} & -B \\ -% -\zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi} \end{bmatrix} \Delta C_\Phi & \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix} (D_\Phi \frac{\sin \check{\Theta}}{\sin \Theta} - \bar{D}_\Phi) & 0 & 0 & \zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix}\bar{C}_\Phi & 0 -\end{bmatrix},\nonumber \\ -\nonumber \\ -% -% -% -A_{1i} &= -\begin{bmatrix}[cccc:cc] -A_1 & 0 & 0 & 0 & 0 & 0\\ -% -0 & 0 & 0 & 0 & 0 & 0\\ -0 & 0 & 0 & 0 & 0 & 0\\ -% -0 & 0 & 0 & 0 & 0 & 0\\ \hdashline -% -\Delta A_1 & 0 & 0 & 0 & \bar{A}_1 & 0\\ -% -0 & 0 & 0 & 0 & 0 & 0 -\end{bmatrix}, \quad -% -% -% -A_{2i} = -\begin{bmatrix}[cccc:cc] -A_2 & 0 & 0 & 0 & 0 & 0\\ -% -0 & 0 & 0 & 0 & 0 & 0\\ -0 & 0 & 0 & 0 & 0 & 0\\ -% -0 & 0 & 0 & 0 & 0 & 0\\ \hdashline -% -\Delta A_2 & 0 & 0 & 0 & \bar{A}_2 & 0\\ -% -0 & 0 & 0 & 0 & 0 & 0 -\end{bmatrix}, \nonumber \\ -\nonumber \\ -P_{cl}&(\xi,\xi_1,\xi_2, \Theta,\check{\Theta},W,W_y) = \begin{bmatrix} -F_{p\Theta} + (-B - B_{1\Theta}\bar{b}_1)u_{r\Theta} \\ -0 \\ -0 \\ -0 \\ \hdashline -F_{r\Theta} + (-B - B_{1\Theta}\bar{b}_1)u_{r\Theta}\\ -\zeta \begin{bmatrix}l_{1\Theta} , l_{2\Theta}\end{bmatrix} (\Delta C_\Theta x_{r\Theta} + \Delta D_\Theta \Gamma_{\Theta d}^* + EW_y) \\ \hline -F_{p\Phi} + (-B - B_{1\Phi}\bar{b}_1)u_{r\Phi}\\ -0 \\ -0 \\ -0 \\ \hdashline -F_{r\Phi} + (-B - B_{1\Phi}\bar{b}_1)u_{r\Phi}\\ -\zeta \begin{bmatrix}l_{1\Phi} , l_{2\Phi}\end{bmatrix} (\Delta C_\Phi x_{r\Phi} + (D_\Phi\frac{\sin \check{\Theta}}{\sin \Theta} - \bar{D}_\Phi)\Gamma_{\Phi d}^*) \\ -\end{bmatrix}. -\\\nonumber -\end{align*} -\endgroup - -The non-defined terms $F_{r\Theta}$ and $F_{r\Phi}$ are given by: - -\begin{align} -F_{r\Theta} &= (B_{0\Theta} - B_{1\Theta} \bar{b}_0 - L_\Theta \Delta D_\Theta)\Gamma_{\Theta d}^* + (\Delta A_0 - L_\Theta \Delta C_\Theta) x_{r\Theta} (\xi) + \Delta A_1 x_{r\Theta} (\xi_1) + \Delta A_2 x_{r\Theta} (\xi_2) + BW - L_\Theta EW_y \label{eq:PerturbationForces1obs},\\ -F_{r\Phi} &= \bigg(B_{0\Phi} - B_{1\Phi} \bar{b}_0 - L_\Phi \bigg(D_\Phi\frac{\sin \check{\Theta}}{\sin \Theta} - \bar{D}_\Phi \bigg)\bigg)\Gamma_{\Phi d}^* + (\Delta A_0 - L_\Phi \Delta C_\Phi) x_{r\Phi} (\xi) + \Delta A_1 x_{r\Phi} (\xi_1) + \Delta A_2 x_{r\Phi} (\xi_2) \label{eq:PerturbationForces2obs}, -\end{align} - -where $\Delta A_0 = A_0 - \bar{A}_0$, $\Delta A_1 = A_1 - \bar{A}_1$ and $\Delta A_1 = A_1 - \bar{A}_1$. - -\subsection{Linearization of robust system} - - - - - - - - - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/unstable_control.tex b/Research Proposal IIT/Research proposal/sections (copy)/unstable_control.tex deleted file mode 100755 index ab485d9..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/unstable_control.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\vspace{20pt} - -controller - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections (copy)/validation.tex b/Research Proposal IIT/Research proposal/sections (copy)/validation.tex deleted file mode 100755 index 118ba88..0000000 --- a/Research Proposal IIT/Research proposal/sections (copy)/validation.tex +++ /dev/null @@ -1,77 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - -\vspace{20pt} - -In order to perform a validation of the derived model and the estimated parameters, a measurement for the quality of its relation with the actual setup can be done using the Variance Accounted For (VAF)\cite{Filtering} which gives a measurement of how well the measured data fits the model. \\ -For the set of obtained parameters using the nonlinear least-squares, the nonlinear model and the linearized model were tested using 6 different data sets for different inputs. The plots of the results with input and output data are shown in figure \ref{fig:VAF1}. - -\begin{figure}[H] -\centering -\includegraphics[width=1\textwidth]{FirstFitVAF.png} -\caption{\label{fig:VAF1}Comparison between nonlinear, linear and measured data.} -\end{figure} - -The corresponding values for the VAF are shown in table \ref{tab:VAF_1} -\begin{table}[h] -\centering -\caption{VAF values for the different data sets.} -\label{tab:VAF_1} -\begin{tabular}{|l|l|l|l|l|l|l|l|} -\hline - & & $\begin{matrix}Data & 1 \\\pi & 0 \end{matrix}$ & $\begin{matrix}Data & 2 \\\pi & 0 \\ \end{matrix}$ & $\begin{matrix}Data & 3 \\\pi & 0 \\ \end{matrix}$ & $\begin{matrix}Data & 4 \\\pi & 0 \\ \end{matrix}$ & $\begin{matrix}Data & 5\\\pi & 0 \\ \end{matrix}$ & $\begin{matrix}Data &6\\0 & \pi \\ \end{matrix}$ \\ \hline -Nonlinear Model & $\begin{matrix}\theta_1 \\\theta_2 \end{matrix}$ & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix} 10.97\\\\ 51.27\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}90.13\\\\ 66.9279\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}86.72\\\\ 62.2466\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}93.33\\\\ 8.0216\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}98.01\\\\ 94.2972\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}96.96\\\\ 94.89\end{bmatrix}$\end{tabular} \\ \hline -Linear Model & $\begin{matrix}\theta_1 \\\theta_2 \end{matrix}$ & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}10.86\\\\ 41.81\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}90.10\\\\ 57.70\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}86.74\\\\ 51.02\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix} 93.39\\\\ 0\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix} 97.74\\\\ 91.33\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}97.06\\\\ 89.62\end{bmatrix}$\end{tabular} \\ \hline -\end{tabular} -\end{table} - -As it can be seen, the values for the fit of the model with respect to the measured data are mostly below 70\%. Because of this, a new parameter estimation will be performed. An educated guess is made by looking at the response that the damping in link 1 might be too high, and therefor the $K_u$ is too high as well. The initial guesses for $K_u$ and $d_1$ for the next parameter estimation will be $d_1$=3.5 and $K_u$=-20, taking half for both values. Finally the bounds of some of the parameters during the optimization were changed. The final estimated parameters are shown in table \ref{tab:FinalPar} - -\begin{table}[H] -\caption{Final parameter estimation.} -\centering -\begin{tabular}{|l|l|l|} -\hline -Symbol &Parameter &Value \\ \hline -$l_1$ &Length of first link &0.1 [m] \\ -$l_2$ &Length of second link &0.1 [m]\\ -$m_1$ &Mass of first link &0.6448 [kg]\\ -$m_2$ &Mass of second link &0.0966[kg]\\ -$c_1$ &Center of mass of first link &0.537 [m]\\ -$c_2$ &Center of mass of second link &0.0536 [m]\\ -$I_1$ &Inertia of first link &0.0366 [kgm$^2$]\\ -$I_2$ &Inertia of second link &0.0001 [kgm$^2$]\\ -$d_1$ &Damping of first link &3.5117 [$\frac{Ns}{m}$]\\ -$d_2$ &Damping of second link &0.0001 [$\frac{Ns}{m}$]\\ -$K_u$ &Lumped motor constant &-18.7850 [$\frac{N}{m}$]\\ -$g$ & gravitation constant & 9.81 [$\frac{m}{s^2}$]\\ \hline -\end{tabular} -\label{tab:FinalPar} -\end{table} - -The plots for the fits corresponding to the same data sets are shown in figure \ref{fig:FinalFitVAF} - -\begin{figure}[H] -\centering -\includegraphics[width=1\textwidth]{FinalFitVAF.png} -\caption{\label{fig:FinalFitVAF}Comparison between nonlinear, linear and measured data for final estimation.} -\end{figure} - -And the corresponding values for the VAF are shown in table - -\begin{table}[h] -\centering -\caption{VAF values for the different data sets.} -\label{tab:VAFl} -\begin{tabular}{|l|l|l|l|l|l|l|l|} -\hline - & & $\begin{matrix}Data & 1 \\\pi & 0 \end{matrix}$ & $\begin{matrix}Data & 2 \\\pi & 0 \\ \end{matrix}$ & $\begin{matrix}Data & 3 \\\pi & 0 \\ \end{matrix}$ & $\begin{matrix}Data & 4 \\\pi & 0 \\ \end{matrix}$ & $\begin{matrix}Data & 5\\\pi & 0 \\ \end{matrix}$ & $\begin{matrix}Data &6\\0 & \pi \\ \end{matrix}$ \\ \hline -Nonlinear Model & $\begin{matrix}\theta_1 \\\theta_2 \end{matrix}$ & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix} 34.13\\\\ 54.56\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}92.69\\\\ 68.37\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}75.22\\\\ 62.1107\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}79.49\\\\ 85.70\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}98.16\\\\ 94.68\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}82.67\\\\ 84.91\end{bmatrix}$\end{tabular} \\ \hline -Linear Model & $\begin{matrix}\theta_1 \\\theta_2 \end{matrix}$ & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}10.86\\\\ 41.81\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}90.10\\\\ 57.70\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}86.74\\\\ 51.02\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix} 78.72\\\\ 0\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix} 96.24\\\\ 90.2128\end{bmatrix}$\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}76.24\\\\ 73.65\end{bmatrix}$\end{tabular} \\ \hline -\end{tabular} -\end{table} - -This final result gives a much better value of the VAF for the nonlinear model. Despite the fact that the linear model fit seems to be less accurate, this low value may be due to the fact that in some of the signals the output is far from the linearization point - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections/.fuse_hidden00000e2700000001 b/Research Proposal IIT/Research proposal/sections/.fuse_hidden00000e2700000001 deleted file mode 100755 index eeaf494..0000000 Binary files a/Research Proposal IIT/Research proposal/sections/.fuse_hidden00000e2700000001 and /dev/null differ diff --git a/Research Proposal IIT/Research proposal/sections/.fuse_hidden00000e2800000002 b/Research Proposal IIT/Research proposal/sections/.fuse_hidden00000e2800000002 deleted file mode 100755 index edba897..0000000 Binary files a/Research Proposal IIT/Research proposal/sections/.fuse_hidden00000e2800000002 and /dev/null differ diff --git a/Research Proposal IIT/Research proposal/sections/Introduction.tex b/Research Proposal IIT/Research proposal/sections/Introduction.tex deleted file mode 100755 index 7e7bddb..0000000 --- a/Research Proposal IIT/Research proposal/sections/Introduction.tex +++ /dev/null @@ -1,10 +0,0 @@ -\documentclass[../main.tex]{subfiles} -\begin{document} - -This document aims to describe a research proposal for the PhD program in Bioengineering and Robotics, for the curriculum in Advanced and Humanoid Robotics. It is based on the topic "Locomotion planning for multi-legged robotic platforms for traversing unstructured terrains", included in the list of research themes found on the website of the PhD school of the Italian Institute of Technology. - -\end{document} - - - - diff --git a/Research Proposal IIT/Research proposal/sections/Methodology.tex b/Research Proposal IIT/Research proposal/sections/Methodology.tex deleted file mode 100755 index 72ac107..0000000 --- a/Research Proposal IIT/Research proposal/sections/Methodology.tex +++ /dev/null @@ -1,126 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - - As it has already been mentioned, the goal of this proposal is to use max-plus algebra to execute the locomotion planning task. As an initial step, the basics of this framework applied to locomotion are explained. - - Consider the following algebra: - - \begin{equation} - (\Re_{max}, \oplus, \otimes, \varepsilon, e) - \end{equation} - - where: - - \begin{align} - \Re_{max} &:= \Re \cup \{-\infty\}, \nonumber\\ - x \oplus y &:= max(x,y), \nonumber\\ - x \otimes y &:= x + y, \nonumber\\ - \varepsilon &:= -\infty, \nonumber\\ - e &:= 0. \nonumber - \end{align} - - In place, this algebra is defined by the set of real numbers including the negative infinity; the two binary operations: the maximum of two numbers and the addition; the absorbing element $\varepsilon$ given by $-\infty$ and the identity element given by $0$. Subsequently, max-plus algebra can be extended to matrices defined by the following structure: - - \begin{equation} - (\Re_{max}^{n\times m}, \oplus, \otimes,\mathcal{E}, E) - \end{equation} - - with: - - \begin{align} - [A \oplus B]_{ij} &= a_{ij} \oplus b_{ij} := \max(a_{ij},b_{ij}), \nonumber \\ - [A \otimes C]_{ij} &= \underset{k=1}{\overset{m}{\mathlarger{\mathlarger\oplus}}}a_{ik} \otimes c_{kj} := \max\limits_{k=1,...,m}(a_{ik} + c_{kj}), \nonumber - \end{align} - where $A,B \in \Re_{max}^{n\times m}$, $C \in \Re_{max}^{m\times p}$, and the $i,j$ element of $A$ is denoted by $a_{ij} = [A]_{ij}$, and identity and zero matrices defined by: - - \begin{align} - [\mathcal{E}]_{ij} &= \varepsilon, \nonumber \\ - [E]_{ij} &= \begin{cases} - e, & \text{if } i =j \\ - \varepsilon,& \text{otherwise.} - \end{cases} \nonumber - \end{align} - - - Powers of matrices can also be defined as: - - \begin{equation} - D^{\otimes k} := D \otimes D \otimes... \otimes D. - \end{equation} - - The max-plus algebra structure corresponds to a commutative idempotent semiring. Gaits can be represented and controlled via switching max-plus linear systems of the form: - - \begin{equation} - x (k+1) = A \otimes x(k) - \end{equation} - - In the case of locomotion the parameters and states involved in the process are given in Table \ref{table:Parameters}. - - \begin{table}[ht] - \centering - \caption{Parameters for locomotion using max-plus algebra systems\cite{Lopes2009}.} - \label{table:Parameters} - \begin{tabular}{ll} - Symbol & Parameter \\ - $i$ & Index to indicate each of the legs. \\ - $t_i (k)$ & Touchdown time of leg $i$. \\ - $l_i (k)$ & Lift-off time of leg $i$. \\ - $\tau$ & Current time instant. \\ - $\tau_f$ & Time that a leg spends in flight (swing). \\ - $\tau_g$ & Time that leg spends on the ground (stance). \\ - $\tau_\Delta$ & Double stance time (this can be adjusted depending on the gait that it is desired to achieve). - \end{tabular} - \end{table} - - Using this parameters one could write the equations of a leg cycle as: - - \begin{align} - t_i (k+1) &= l_i (k+1) + \tau_f \\ - l_i (k+1) &= t_i (k) + \tau_g \label{eq:liftoff} - \end{align} - - Furthermore, in order to achieve synchronization, one could enforce a leg ($i$) to lift-off only until some time after ($\tau_\Delta$) other leg ($j$) has touched the ground. Then Equation \eqref{eq:liftoff} can be rewritten as: - - \begin{equation} - l_i (k+1) = \max (t_i (k) + \tau_g , t_j (k-1) + \tau_\Delta)\label{eq:liftoffsync} - \end{equation} - - Then using the max-plus algebra notation: - - \begin{equation} - l_i (k+1) =\begin{bmatrix} - \tau_g & \tau_\Delta - \end{bmatrix} \otimes \begin{bmatrix} - t_i (k) \\ - t_j (k) - \end{bmatrix} - \end{equation} - - There exist several examples of gaits modeled using petri nets or timed event graphs. One could easily go from this kind of representation to a max-plus algebra system in a systematic manner \cite{Schutter2009} by defining a state vector that contains the touchdown and lift-off times of all the legs at the current time instant. Explaining this method would go beyond the purpose of this proposal. - - This max-plus algebra representation, could be used to generate, for each time instant $k$, the time reference at which each of the legs should leave and touch the ground, depending on the gait parameters. In order to implement these time references, a continuous position reference generator has to be designed. A tentative method to implement this position reference generator could be as follows: - - \begin{enumerate} - \item As an initial step, one could design a polynomial velocity profile for the leg based on initial (lift-off time $l_i$) and final (touchdown time $t_i$) time. - \item A trajectory for each leg could be designed via parametric equations, expressing the position as a function of time. This trajectory could be modeled as an ellipse, in accordance to the current reactive framework. - \item Finally, setting this trajectory as the reference to be tracked, one could implement various type of control strategies. One option is to use workspace PD control, via the transpose geometric Jacobian. Another algorithm that could be used, could be workspace computed torque control, either using the transpose geometric Jacobian (if the number of actuators of each leg is equal to the dimension of the workspace) or by using the pseudo-inverse of the geometric Jacobian. - \end{enumerate} - - This would be an initial proposal for locomotion controller design. One could simulate and design this strategy via several tools. Initially, to design this controller one could use MATLAB in combination with Simulink in order to simulate the responses and effects of the modeled system. Furthermore after simulating the results in MATLAB, one could go one step further and implement a graphic simulation via V-REP. One powerful element that could simplify the task to take the leap to a 3D environment, is making use of the Robot Operating System (ROS). One could write the control algorithms in MATLAB as ROS nodes, in order to publish the data into an specific topic. On the V-REP side, there are several libraries to connect this software to ROS, and building a subscriber node, which could be used to read the data from the previously mentioned topic. Furthermore, one could use virtual sensors and actuators to simulate the real life conditions of the robot. - - Other programming softwares could be use to implement the control strategy before. Python is one attractive alternative, due to its similarity with MATLAB, and the flexibility that provides as an open-source developing language. There is a vast number of libraries and it can be used to work in connection with ROS. - - After extensive testing of the locomotion algorithm, one could make use of the modular structure of ROS, to implement the same algorithms in the HyQ robotic platform. If work has not been performed in building the necessary nodes for the sensors and actuators of the robot, these could be build again by using the ROS framework. This process also involves getting familiar with the platform, understanding the previous work in depth, and adapting the possible new strategies to be implemented. - \iffalse - \begin{enumerate} - \item Design max-plus algebra system for the four-legged robot HyQ, design the scheduling for different types of gaits (adapt the framework). - \item Design the reference generator for each of the legs based on trajectory design for each leg using their dynamic model (screw theory) - \item Implementation of supervisory control based on sensor data to select the proper gait depending on the situation where the robot is - \item Analyze robustness of the designed algorithm based on the theory and properties of the max-plus algebra system - \item Perform simulations in software (V-REP) and ROS - \item After simulations start testing in the robot - \end{enumerate} - \fi - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections/Motivation.tex b/Research Proposal IIT/Research proposal/sections/Motivation.tex deleted file mode 100755 index e5c4ea4..0000000 --- a/Research Proposal IIT/Research proposal/sections/Motivation.tex +++ /dev/null @@ -1,29 +0,0 @@ -\documentclass[../main.tex]{subfiles} - -\begin{document} - -There is no doubt, that there has been a growing interest in robotics in the last decades. Catastrophic events, such as the nuclear disaster in Fukushima have made evident, that despite the technological advances achieved in recent years, both theory and practice are still far from developing robots that could face situations and environments which are hazardous to humans, with the same flexibility and adaptability that a human would display. Although robotic platforms can surpass humans and animals on certain tasks (fast computation and decision making, force demanding tasks such as lifting objects), there are some that biological beings perform with extraordinary easiness and elegance. Among these lies the problem of locomotion. - -It has been observed that the performance of legged robots surpasses that of wheeled platforms while traversing rough or unstructured terrains. This feature comes at the cost of more difficult ways to model and control the dynamics of the agents involved in the process. This complexity has pushed research to develop advanced techniques in order to tackle the problem of robot locomotion. - -Several approaches have been taken to produce locomotion on legged robots. Some take advantage of the underlying mechanic properties of walking, others try to force stability by analyzing quasi-static solutions of the problem online, and the most advanced ones, have taken inspiration from biological systems to emulate walking patterns in robots. From the previously mentioned alternatives, the latter one, has proven to be the most robust against disturbances, and unstructured terrain, by effectively changing gait parameters depending on the situation that the robot is facing. Despite this, there are just a few examples where the agent is able to entirely switch the gait in order to perform a specific task. - -All the previously mentioned strategies are based on continuous time models of locomotion. An alternative to this continuous models is the use of Discrete Event Systems (DES), an approach that has been taken in \cite{Lopes2010}, \cite{Barai}. In particular in \cite{Lopes2010}, the authors make use of a framework based on max-plus algebra systems. This strategy provides several advantages with respect to its continuous counterpart. One of the most notable is that the resulting description is a max-plus linear system, for which linear control techniques can be implemented in order to provide safe switching between different gaits. Due to this max-plus linear property, the computational effort is also reduced, allowing faster reaction times. At current time no paper related to this topic that involves gaits where the robot spends longer times with no support (for example gallop) has been published, although it is mentioned to be studied in the future. - -This document proposes the implementation of a locomotion planning strategy based on Discrete Event Systems (DES) on the HyQ robotic platform. This is done by making use of a framework based on max-plus algebra, which offers the possibility to analyze and implement in a max-plus linear way, different gaits for several configurations of legged robots. The main purpose would be to adapt the max-plus algebra framework to generate time references for each leg, using them to design the trajectory of each leg. It is believed that this locomotion planning strategy can be implemented in the current reactive controller framework present in the HyQ robot \cite{Barasuola}, in substitution of or in combination with the Central Pattern Generator (CPG) already present in the system. Furthermore, other elements of the current framework could be improved, such as the torque control used for each of the joint angles in order to cope with uncertainty. One of the suggested strategies, is to investigate the influence on the performance if adaptive computed torque control is used. - -\iffalse -\begin{itemize} - \item Importance of robot locomotion - \item Advantages of legged robots over wheeled robots - \item Nonlinear descriptions of CPG - \item Current locomotion planning techniques do not perform with robustness, reliability and flexibility it is needed - \item Gait switching problems - \item importance of investigating new planning strategies to analyze the problem from a differente perspective - \item why max-plus? Gait switching, variation of parameters more intuitive, less computationally expensive - \item disadvantages, no work has been done on aerial phases -\end{itemize} -\fi - -\end{document} - \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections/Objectives.tex b/Research Proposal IIT/Research proposal/sections/Objectives.tex deleted file mode 100755 index 0aaea62..0000000 --- a/Research Proposal IIT/Research proposal/sections/Objectives.tex +++ /dev/null @@ -1,22 +0,0 @@ -\documentclass[../main.tex]{subfiles} - -\begin{document} - -The analysis performed on the state of the art of the locomotion problem, has made clear that there are several open questions in the field. This has given important guidelines in order to propose the objectives that can be pursued in the research project on this topic. First of all, the robustness and flexibility of the current strategies has to be improved. Several alternatives have been studied in the previous section, but one that shows systematic implementation, is to use discrete event systems described by making use of max-plus algebra. One of the initial goals, is to prove if this framework can be implemented in a practical manner to robots with topologies different than the RHex inspired platforms. - -One subtask of this first objective, is to investigate if this locomotion strategy can be implemented in the reactive framework of the HyQ robot proposed in \cite{Barasuola}. This can be done either in tandem with the already implemented CPG trajectory generator, or by implementing a unique discrete max-plus gait scheduler. It is believed that the rest of the elements in this reactive framework, can be used in the same way as it is used making use only of a CPG. - -A second objective, would be to test if the use of max-plus algebra systems, indeed provides the alternative to perform gait changes, along with modifying the parameters of the gaits. A subtask of this objective may include the implementation of a supervisory controller, which uses the information from the sensors of the robot, in order to decide the most adequate gait and parameters to face each situation (in a similar way as in \cite{Barasuol}). - -Furthermore, it is believed that the currently implemented joint controller could be improved if an adaptive scheme is added, which could provide online estimations of the model parameters. This could be helpful to improve robustness, since in most dynamical system the parameters that describe the model could change due to several factors (unmodelled or neglected dynamics, disturbances). - -This objectives have to be tested on both simulation and the real system. - -\iffalse -\begin{itemize} - \item Describe objectives with respect to robust locomotion generation (sub-objectives) - \item Describe objectives with respect to implementation of max-plus in current system (sub-objectives) -\end{itemize} -\fi - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections/Results.aux b/Research Proposal IIT/Research proposal/sections/Results.aux deleted file mode 100755 index 0f4106e..0000000 --- a/Research Proposal IIT/Research proposal/sections/Results.aux +++ /dev/null @@ -1,6 +0,0 @@ -\relax -\providecommand \oddpage@label [2]{} -\select@language{english} -\@writefile{toc}{\select@language{english}} -\@writefile{lof}{\select@language{english}} -\@writefile{lot}{\select@language{english}} diff --git a/Research Proposal IIT/Research proposal/sections/Results.log b/Research Proposal IIT/Research proposal/sections/Results.log deleted file mode 100755 index 05de920..0000000 --- a/Research Proposal IIT/Research proposal/sections/Results.log +++ /dev/null @@ -1,1185 +0,0 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9) (preloaded format=pdflatex 2015.9.7) 5 MAY 2016 16:46 -entering extended mode -**Results.tex - 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-\begin{document} - - - The first expected outcome of this research would be, a novel locomotion control strategy that could be adapted to the current reactive framework of the HyQ robot. This in place, is expected to improve robustness and flexibility thanks to its systematic design approach. - - On a larger scale, this research could provide, a strategy to systematically design and implement locomotion controllers to a variety of multi-legged robots. The proposed strategy, if proven to be effective for this project, could be easily adapted to several platforms that can design gaits via discrete event systems. - - On the practical side, examples where max-plus algebra systems are used for locomotion are scarce. Furthermore, it was not found an example where this framework is implemented for robots with multiple joints in the legs. This would be a first experience using this strategy for this type of legs. This in place, would provide the analytical tools to reproduce this framework. In addition, software tools to implement discrete event systems in locomotion would be a natural consequence of the research process. - - It is clear, that if this strategy can manage to provide safe gait switching and parameter variation, robots would be able to overcome more difficult tasks while facing unstructured environments. In an ideal scenario, this strategy may represent a step towards successfully overcoming some of the greatest obstacles while encountering hazardous situations. - - -\end{document} \ No newline at end of file diff --git a/Research Proposal IIT/Research proposal/sections/Stateoftheart.aux b/Research Proposal IIT/Research proposal/sections/Stateoftheart.aux deleted file mode 100755 index 8270a89..0000000 --- a/Research Proposal IIT/Research proposal/sections/Stateoftheart.aux +++ /dev/null @@ -1,15 +0,0 @@ -\relax -\citation{Holmes2006} -\citation{Ijspeert} -\citation{Barasuol} -\citation{Ijspeert} -\citation{Schutter2009} -\citation{Schutter2008} -\providecommand \oddpage@label [2]{} -\select@language{english} -\@writefile{toc}{\select@language{english}} -\@writefile{lof}{\select@language{english}} -\@writefile{lot}{\select@language{english}} -\citation{Goverde2007} -\citation{Turek2011} -\citation{Lopes2009} diff --git a/Research Proposal IIT/Research proposal/sections/Stateoftheart.bbl b/Research Proposal IIT/Research proposal/sections/Stateoftheart.bbl deleted file mode 100755 index e69de29..0000000 diff --git a/Research Proposal IIT/Research proposal/sections/Stateoftheart.blg b/Research Proposal IIT/Research proposal/sections/Stateoftheart.blg deleted file mode 100755 index f9de855..0000000 --- a/Research Proposal IIT/Research proposal/sections/Stateoftheart.blg +++ /dev/null @@ -1,47 +0,0 @@ -This is BibTeX, Version 0.99d (TeX Live 2013/Debian) -Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 -The top-level auxiliary file: Stateoftheart.aux -I found no \bibdata command---while reading file Stateoftheart.aux -I found no \bibstyle command---while reading file Stateoftheart.aux -You've used 5 entries, - 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-\begin{document} - - In robotics, the problem of legged locomotion has proven to be a challenging task. There are three particular areas where most of the issues are encountered: control, mechanics and electronics. Regarding control, the dynamics of a walking robot can be seen as a hybrid system, due to the fact that it involves continuous control of each of the walking limbs, but in coordination with the event triggered movements of the other limbs. More importantly, in most cases, the action of balancing (which humans and animals perform naturally) represents an intrinsically unstable equilibrium point \cite{Holmes2006}. Considering the mechanics, apart from the underlying properties of the different robotic platforms, modeling and providing proper alternatives to cope with the interactions of the platform with the environment (such as impact and slipping) is still difficult. Finally, performing gaits is a demanding task by itself, which demands high amounts of power, which could limit the autonomy of the robot. This problem is intimately related to the evolution of the mechanics in robotic systems, since the problem has been tackled from the point of view where passive-walking can be achieved without the use of any external source of energy \cite{Wisse2004}, or the development of actuators that provide enough force and torque while optimizing power consumption \cite{Tsagarakis2013}. - - Several approaches to achieve locomotion have been researched. One of the first steps towards accomplishing successful walking, was by using the concept of Zero-Moment Point (ZMP), which is defined as "\textit{the point on the ground at which the net moment of the inertial forces and the gravity forces has no component along the horizontal axes}" \cite{Borovac2004}. Using this concept, traditionally one could design a controller that successfully moves the end effector while maintaining the ZMP inside the support foot, by means of tools like inverse kinematics (which yields several disadvantages such as singularity points and multiple solutions) or by implementing an operational space formulation with constraints \cite{Aghili2005}. This framework has been applied to some robotic platforms such as Honda's Asimo. Nevertheless, in reality it yields several problems, as it does not represent a natural way of walking, and due to the fact that the resulting equations for the dynamics of the robot can be very complex, it may require a lot of computational power and may not be robust to large perturbations. Furthermore, it considers a quasi-static movement, so running yields a high level of complexity as well. Finally, as the number of legs increases, the problem becomes less tractable. - - A different way of tackling the problem is to model the body of the human or animal as a point mass located at the hip joint with two massless legs, equipped with telescopic springs with constant stiffness and length. This model is called the Spring Loaded Inverted Pendulum (SLIP) and it has been shown that it represents both walking and running reliably \cite{Geyer2006} and it can reproduce the ground reaction forces and hip trajectories observed in gaits. Moreover, it can encode various gaits by defining different model parameters (length of the leg, stiffness of the spring, angle of attack of the foot), and can execute energy efficient passive gaits (in the case of bipeds). A step has been taken further by proposing a Variable Spring Loaded Inverted Pendulum (V-SLIP) model \cite{Ketelaar}. This is similar to the way humans control the actuation of their limbs via the muscular system, changing their stiffness in order to achieve movement. This strategy makes use of energy-based Hamiltonian mechanics to design a controller to achieve stable walking using different gaits. These two models have been proven to be very suitable to describe walking and running in humans and animals. Despite this, they face the same problem of complexity when it comes to controller design. - - One of the most widely utilized tools in robot locomotion, are the so-called CPGs. The problem of animal and human locomotion had been widely studied long before robotics were a relevant subject. The interest on this topic initially arose from the field of biology, achieving great success in understanding how animals such as mammals and insects manage to coordinate in a natural and effortless way the movement of their legs in order to perform specific gaits corresponding to the conditions they face on a daily basis. It was discovered, that complex networks of neurons are in charge of generating the signals to send to each walking limb, using advantageously the periodic nature of the process. CPGs do not actuate each limb independently or are entirely based on the senses of animals/humans. This is because in most of the scenarios, walking constitutes a periodic activity, where the legs move following the same patterns with phase differences. Depending on the situation that the agent is facing, it naturally changes the parameters of the gait (such as the time that the legs remain on the ground or the air or the phase differences between each leg) or even changes the gait completely (such as going from walking to running). - - Roboticists around the world, have payed close attention to these neuronal arranges in order to apply them in robot locomotion. This type of structures have been emulated by using simplified models of neurons and their interactions. Using this kind of framework, various types of gaits can be achieved. The way that CPGs are used, differs among all different platforms, and has proven to be efficient to generate different gait motions for several types of robots \cite{Crespi1985}, \cite{Ijspeert2008}, \cite{Barasuola}. Some examples \cite{Ijspeert}, do not use any kind of feedback control in order to generate gaits, although the main purpose of this kind of study is to analyze the animal behavior using robotic models to simulate gait patterns. On the other hand, CPGs can be used to effectively generate angle references for each of the legs in robotic platforms \cite{Barasuol}, to subsequently be tracked via control strategies (the most widely used in humanoid and animal-like platforms are computed-torque control and feedback linearization techniques). - - Despite the elegance that locomotion generation using CPGs displays, roboticists are facing big challenges regarding robustness against unstructured terrain and dealing with disturbances. In most cases, when encountering unfavorable situations, proprioceptive (information coming from the agent itself) and exteroceptive (information coming from the environment) sensory information, is used to change the parameters of the gait such as coupling variables, ground and flight times and velocities of each leg. This framework has proven to efficiently deal with complex types of terrain. In some specific cases, gait switching has been achieved \cite{Ijspeert}, although no control is implemented in the locomotion planning routines. It can be seen that even in the simplest case, a set of nonlinear differential oscillator equations describes the evolution in time of each limb. This is due to the intrinsic periodicity of the process. This provides an extra difficulty in order to change from one gait to another, especially as the number of legs increases. - - All previously described strategies, model the locomotion problem as a continuous-time system. Another way to approach the locomotion planning problem, is to model it via Discrete Event Systems \cite{Schutter2009}. In recent years the use of tools such as petri nets and max-plus algebra systems have been tested and proved to be able to generate locomotion patterns. Max-plus algebra systems have provided not only a simpler and systematic way to generate gait motions but also, due to its linear (in max-plus) properties, provides tools to analyze transitions and velocity of the generated planning. - - Max-plus algebra is a type of "tropical algebra" that complies with the axioms of an algebraic semi-ring \cite{Schutter2008}. Max-plus algebra represents a very useful tool to model timed events such as railway scheduling \cite{Goverde2007} or traffic control \cite{Turek2011}. Robot locomotion can be modeled as well as a discrete event system, making use of a subclass of petri nets called timed event graphs \cite{Schutter2009}. Subsequently, one can represent the evolution equations for a subset of timed event graphs as max-plus algebra linear systems. - - Using this kind of framework, one could generate time references for each of the leg's touchdown times (time instant when the leg touches the ground) and lift-off times (time instant when the leg leaves the ground). Subsequently, with these time references for each leg, a trajectory can be computed and achieved by means of a controller. It has been proven that this strategy can safely switch between gaits, minimizing the variation in velocity of all the legs touching the ground. It yields the disadvantage that it has been implemented in robots, with rather simple dynamics (RHex-inspired platform), although in theory, complexity in the mechanics of the platform would only represent a difference in the generation of references to be followed by the controller for each of the legs. - - In summary, robust and flexible locomotion planning remains an open question in the robotic community. Finding the right balance between complexity and tractability of the studied strategies represents one of the most important trade-offs in the field. The introduction of the concept of ZMP, gave a first glance at the complexity of the locomotion problem, making evident that novel strategies should be developed in order to render the problem tractable. On one hand, accurate models of both human and animal locomotion have been found throughout the past decades, which capture the essential dynamics of the process (such as the SLIP/V-SLIP models). These models have been widely utilized for biped locomotion, achieving great success. But, as the number of legs increases, the underlying non-linearities and the combinatorial space for coordination, render the task more difficult. The use of CPGs has aid regarding coordination of the legs, elegantly modeling the phase differences of each limb in order to achieve different types of gaits. Nevertheless, it faces the same problem regarding non-linearity and combinatorial "gait space". Finally, modeling the problem as a discrete event system using max-plus algebra presents itself as a simple and systematic way to coordinate the legs in order to achieve gaits and, furthermore, can provide safe gait switching. It remains to investigate if this framework can be applied to mammal-like walking limbs, made of several joints. - - - -\iffalse -\begin{itemize} - \item Explain HyQ framework broadly - \item Explain CPG - \item Introduction to max-plus - \item Max-plus in locomotion - \item Advantages and disadvantages of max-plus - \item Discussion -\end{itemize} - - -\begin{itemize} - \item How is locomotion done at this time? - \item Continuous models (Central pattern generators) - \item Discrete time-event system (Max-plus algebra systems) - \item Introduction max-plus - \item Max-plus in locomotion - \item HyQ decription -\end{itemize} - -\fi - - -\end{document} - diff --git a/Research Proposal IIT/Research proposal/sections/Workplan.tex b/Research Proposal IIT/Research proposal/sections/Workplan.tex deleted file mode 100755 index 760cbd2..0000000 --- a/Research Proposal IIT/Research proposal/sections/Workplan.tex +++ /dev/null @@ -1,38 +0,0 @@ -\documentclass[main.tex]{subfiles} - -\begin{document} - In order to achieve the tasks proposed in the methodology a tentative work plan is designed. A brief summary of the plan is shown in Table \ref{tab:Planning}. - - \begin{table}[H] - \caption{Tentative plan of work.} - \centering - \begin{tabular}{|l|l|} - \hline - Task &Time to fulfill task\\ \hline - Literature research & 2-3 months\\ - Analysis of current implementation and platform & 4-6 months\\ - Locomotion controller design & 7-9 months\\ - Simulation testing & 4-6 months\\ - Physical implementation & 5-7 months\\ - Unstructured scenario design and testing & 3-5 months\\ - \hline - \end{tabular} - \label{tab:Planning} - \end{table} - - Initially, an extensive literature research has to be performed to strongly justify the methods and strategies proposed to achieved the desired objectives. All the suggested time frames include documentation of each of the steps taken, this meaning that the time to write down the results is being considered. Based on previous experiences, a literature research con take between 2 and 3 months, depending on how the topic has been studied. For this topic, the time frame seems reasonable, to both analyze and document the previous research. - - Afterwards, an analysis of the previous work performed on the HyQ platform is in place. This analysis includes several subtasks. These can be divided in two main groups, theoretical and practical. Regarding the theoretical tasks, a crucial component is getting a deep understanding of the mechanical modeling of the robot. Furthermore, it is also needed to analyze deeply the current locomotion framework, to give solid reasons to proceed with the research. One should be able to reproduce all previous results. Also, during this part of the research, it has to be considered the need to learn the current tools being used to implement the locomotion controller design, i.e. unfamiliar programming languages and software. It is considered that this task can take between 4 and 6 months. - - The next step, is to proceed with the design of the new locomotion strategy based on the previously mentioned methodology. This is one of the most demanding and important tasks of the project. Analyzing the previous section, the max-plus algebra discrete time scheduler has already been implemented in several languages and softwares (python, MATLAB, V-REP) and could be easily adapted to other. The most demanding tasks would be adapting the time scheduler to the configuration of the HyQ robot. Based on the proposal in this last regard, it is considered that this task could take between 5 and 7 months. At this moment, taking into account the information gathered in the literature research and the locomotion strategy, it should be considered if there is time available to improve the motion controller, (i.e., implementing adaptive computed torque control in workspace), or if the current controller will be retained. - - Once the controller design has been achieved, the strategy has to be tested via simulation. Several tools can be used. Simulation can be done easily via Simulink and MATLAB, this being convenient since they will be useful as well for controller design. On the other hand, a step further would be to provide with 3D simulations of the robot executing tasks using the previously designed locomotion control. As mentioned in the methodology, V-REP and ROS could be use for this task. Also learning other tools such as Gazebo could be considered. This task could take between 4 and 6 months. - - Physical implementation is to be done after testing in simulation. It is considered that this task could be very demanding as well, since it also involves learning how to use the sensors and actuators of the platform. It is considered that this task can take from 5 to 7 months. - - Finally, as the end goal of this project is to design and implement a strategy which provides robustness to unstructured terrain as well as disturbances, it is important to carefully design the experiments that are going to be performed in order to fulfill this goal. it is considered that this task can take from 3 to 5 months. - - It is considered that the project could be completed in a time frame of 25 to 36 months. If there is time left, additional tasks could be considered. - - -\end{document} diff --git a/Research Proposal IIT/Research proposal/sections/citations.log b/Research Proposal IIT/Research proposal/sections/citations.log deleted file mode 100755 index 38841d1..0000000 --- a/Research Proposal IIT/Research proposal/sections/citations.log +++ /dev/null @@ -1,50 +0,0 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9) (preloaded format=pdflatex 2015.9.7) 17 MAR 2016 10:00 -entering extended mode -**citations.tex - -("C:\Users\Tavo\Documents\Escuela\Master\Master Thesis\Thesis Reports\Template -progress reports\sections\citations.tex" -LaTeX2e <2011/06/27> -Babel and hyphenation patterns for english, afrikaans, ancientgreek, ar -abic, armenian, assamese, basque, bengali, bokmal, bulgarian, catalan, coptic, -croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga -lician, german, german-x-2013-05-26, greek, gujarati, hindi, hungarian, iceland -ic, indonesian, interlingua, irish, italian, kannada, kurmanji, latin, latvian, - lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n -german-x-2013-05-26, nynorsk, oriya, panjabi, pinyin, polish, portuguese, roman -ian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, swissgerm -an, tamil, telugu, turkish, turkmen, ukenglish, ukrainian, uppersorbian, usengl -ishmax, welsh, loaded. -("C:\Program Files (x86)\MiKTeX 2.9\tex\latex\subfiles\subfiles.cls" -Document Class: subfiles 2012/05/23 Federico Garcia -("C:\Program Files (x86)\MiKTeX 2.9\tex\latex\tools\verbatim.sty" -Package: verbatim 2003/08/22 v1.5q LaTeX2e package for verbatim enhancements -\every@verbatim=\toks14 -\verbatim@line=\toks15 -\verbatim@in@stream=\read1 -) -Preamble taken from file `main.tex' - -! 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