Developed by: M. Sodano (2018).
Supervisor: prof. A. De Luca.
Achievement: Bachelor thesis (partial).
The aim of this project is the design of kinematic controllers based on the pseudo-inversion and on the Damped Least Squares (DLS) for the simulation of different trajectories. These trajectories are eventually validated on V-REP.
As it is well known, if an initial position and a velocity are assigned to the end-effector, the IDK determines the joint velocities that realize the goal. If the jacobian is square and full rank, this is achieved by the simple inversion. If not, the jacobian is not invertible, and alternative strategies are needed.
The pseudo-inversion is the solution of a constrained minimization problem: the joint velocities are minimized, and the differential kinematics must be satisfied. The solution is not unique but, if the jacobian is full rank, it is
where the geometric jacobian is used.
The DLS is the solution of an unconstrained minimization problem
in which
where the analytic jacobian is used.
Notice that the pseudo-inversion ensures that, if the task is realizable, no error will be obtained since it is enforced that the differential kinematics must be satisfied. The DLS, on the other hand, finds a solution that is a compromise between good error performances and low joint velocities. This means that, if a trajectory approaches a singularity and then drifts away from it, the pseudo-inversion is eventually able to recover the error while the DLS is not.
Simulation 1. The end-effector moves along a straight line parallel to the x-axis, starting in the zero-configuration. Singularities are not encountered. The pseudo-inversion gives good results.
Simulation 2. The end-effector moves along a straight line parallel to the y-axis, starting in the zero-configuration. A boundary singularity is encountered, and the DLS gives better results than the pseudo-inversion.
Simulation 3. The end-effector performs a circular trajectory, without changing the orientation. An internal singularity is encountered. The DLS avoids the saturation of the joint velocities, but terminates with an error of the order of centimeters.
Simulation 4. The end-effector performs a circular trajectory, changing the orientation. The situation is very similar to the previous case, with the difference that the trajectory ends in singularity, therefore the pseudo-inversion is not able to recover the error.
Images and video are reported in the folders.
The code is implemented in Matlab R2016c, and Simulink is also used for the control scheme. Simulation parameters and joint limits are defined in the script simulation_xxx_init.m that is present in every folder.
For the visualization, V-REP by Coppelia Robotics is used and it is interfaced (off-line) with Simulink.
Siciliano, B., Sciavicco, L., Villani, L., & Oriolo, G. (2010). Robotics: modelling, planning and control. Springer Science & Business Media.