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‎src/fswAlgorithms/attGuidance/hillPointCpp/hillPointCpp.rst

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@@ -5,9 +5,264 @@ This attitude guidance module computes the orbital Hill reference frame states.
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The orbit can be any type of Keplerian motion, including circular, elliptical or hyperbolic.
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The module
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:download:`PDF Description </../../src/fswAlgorithms/attGuidance/hillPoint/_Documentation/hillPoint.pdf>`
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contains further information on this module's function,
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how to run it, as well as testing.
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Module Input and Output
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=======================
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Table `1 <#tab:inputNavTable>`__ shows the input message from the
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navigation system.
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.. container::
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:name: tab:inputNavTable
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.. table:: Input Navigation Message
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+------------------+-----------+--------+----------------------+
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| Name | Type | Length | Description |
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+==================+===========+========+======================+
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| :math:`\boldsymbol{R}_S` | double [] | 3 | Position vector of |
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| | | | the spacecraft |
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| | | | body-point with |
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| | | | respect to the |
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| | | | inertial frame in |
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| | | | inertial frame |
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| | | | components |
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| | | | (:math:`\leftexp |
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| | | | p{N}{\boldsymbol{r}}_{B/N}`) |
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| | | | . |
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+------------------+-----------+--------+----------------------+
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| :math:`\boldsymbol{v}_S` | double [] | 3 | Velocity vector of |
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| | | | the spacecraft point |
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| | | | with respect to the |
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| | | | inertial frame in |
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| | | | inertial frame |
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| | | | components |
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| | | | (:math:`\leftexp |
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| | | | {N}{\boldsymbol{v}}_{B/N}`). |
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+------------------+-----------+--------+----------------------+
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Table `2 <#tab:inputCelTable>`__ shows the input message from Spice
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about the main celestial body.
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.. container::
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:name: tab:inputCelTable
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.. table:: Input Spice Planet Message
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+--------------------------+-----------+--------+----------------------+
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| Name | Type | Length | Description |
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+==========================+===========+========+======================+
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| :math:`\boldsymbol{R}_P` | double [] | 3 | Position vector of |
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| | | | the main celestial |
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| | | | object with respect |
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| | | | to the inertial |
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| | | | frame in inertial |
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| | | | frame components . |
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+--------------------------+-----------+--------+----------------------+
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| :math:`\boldsymbol{v}_P` | double [] | 3 | Velocity vector of |
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| | | | the main celestial |
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| | | | object with respect |
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| | | | to the inertial |
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| | | | frame in inertial |
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| | | | frame components . |
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+--------------------------+-----------+--------+----------------------+
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Table `3 <#tab:outputTable>`__ shows the Attitude Reference output
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message of the module Hill Point.
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.. container::
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:name: tab:outputTable
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.. table:: Output Attitude Reference Message
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+----------------------+-----------+--------+----------------------+
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| Name | Type | Length | Description |
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+======================+===========+========+======================+
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| :math:`\sigma_{R/N}` | double [] | 3 | MRP attitude set of |
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| | | | the reference frame |
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| | | | with respect to the |
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| | | | reference. |
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+----------------------+-----------+--------+----------------------+
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| :math:`\omega_{R/N}` | double [] | 3 | Angular rate vector |
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| | | | of the reference |
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| | | | frame with respect |
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| | | | to the inertial |
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| | | | expressed in |
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| | | | inertial frame |
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| | | | components. |
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+----------------------+-----------+--------+----------------------+
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| :math:`\dot{\omega}_{R/N}}` | double [] | 3 | Angular acceleration |
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| | | | vector of the |
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| | | | reference frame with |
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| | | | respect to the |
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| | | | inertial expressed |
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| | | | in inertial frame |
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| | | | components. |
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+-----------------------------+-----------+--------+----------------------+
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Hill Frame Definition
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=====================
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The Hill reference frame takes the spacecraft’s orbital plane as the
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principal one and has origin in the centre of the spacecraft. It is
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defined by the right-handed set of axes
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:math:`\mathcal{H}:\{ \hat{\boldsymbol\imath}_{r}, \hat{\boldsymbol\imath}_{\theta}, \hat{\boldsymbol\imath}_{h} \}`,
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where
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:math:`\hat {\boldsymbol\imath}_{r}` points radially outward in the direction
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that connects the center of the planet with the spacecraft.
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:math:`\hat {\boldsymbol\imath}_{h}` is defined normal to the orbital plane in
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the direction of the angular momentum.
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:math:`\hat {\boldsymbol\imath}_{\theta}` completes the right-handed triode.
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.. figure:: Figures/Fig2
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name: fig:Fig1
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Illustration of the Hill orbit frame
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:math:`\mathcal{H}:\{ \hat{\boldsymbol\imath}_{r}, \hat{\boldsymbol\imath}_{\theta}, \hat{\boldsymbol\imath}_{h} \}`,
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and the inertial frame
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:math:`\mathcal{N}:\{ \hat{\boldsymbol n}_{1}, \hat{\boldsymbol n}_{2}, \hat{\boldsymbol n}_{3} \}`.
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Introduction
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============
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In this module, the output reference frame :math:`\mathcal{R}` is to be
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aligned with the Hill reference frame :math:`\mathcal{H}`. Note that the
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presented technique does not require the planet-fixed frame to coincide
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with the inertial frame
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:math:`\mathcal{N}:\{ \hat{\boldsymbol n}_{1}, \hat{\boldsymbol n}_{2}, \hat{\boldsymbol n}_{3} \}`.
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Figure 1 illustrates the general situation in which :math:`\boldsymbol{R}_{s}`
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is the position vector of the spacecraft with respect to the inertial
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frame and :math:`\boldsymbol{R_{p}}` is the position vector of the celestial
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body with respect to the inertial frame as well. The relative position
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of the spacecraft with respect to the planet is obtained by simple
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subtraction:
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.. math::
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\label{eq:r}
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\boldsymbol r = \boldsymbol R_{s} - \boldsymbol R_{p}
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The same methodology is applied to compute the relative velocity vector:
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.. math::
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\label{eq:v}
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\boldsymbol v = \boldsymbol v_{s} - \boldsymbol v_{p}
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Note that the position and velocity vectors of the spacecraft and the
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celestial body, :math:`\boldsymbol{R}_S`, :math:`\boldsymbol{R}_P`, :math:`\boldsymbol{v}_S` and
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:math:`\boldsymbol{v}_P` are the only inputs that this module requires. Having
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:math:`\boldsymbol r` and :math:`\boldsymbol v`, the Hill frame orientation is
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completely defined:
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.. math::
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\begin{equation}
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\hat{\boldsymbol\imath}_{r} = \frac{\boldsymbol r}{r}
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\end{equation}
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\begin{equation}
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\hat{\boldsymbol\imath}_{h} = \frac{\boldsymbol{r}\times{\boldsymbol{v}}}{r v}
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\end{equation}
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\begin{equation}
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\hat{\boldsymbol\imath}_{\theta} = \hat{\boldsymbol\imath}_{h} \times \hat{\boldsymbol\imath}_{r}
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\end{equation}
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And the Direction Cosine Matrix to map from the reference frame to the
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inertial is obtained:
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.. math::
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= \begin{bmatrix}
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\hat{\boldsymbol\imath}_{r} \\
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\hat{\boldsymbol\imath}_{\theta} \\
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\hat{\boldsymbol\imath}_{h}
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\end{bmatrix}
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The corresponding MRP attitude set is computed using the following
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function from the Rigid Body Kinematics library of Reference :
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.. math:: [RN] = \textrm{C2MRP}(\boldsymbol\sigma_{R/N})
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Angular Velocity Descriptions
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=============================
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Let :math:`\mathcal{R}_{0}` reference the Hill orbit frame. The orbit
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frame angular rate and acceleration vectors are given by
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.. math::
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\label{eq:omega_R0}
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\boldsymbol\omega_{R_{0}/N} = \dot f \hat{\boldsymbol\imath}_{h}
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.. math::
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\label{eq:domega_R0}
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\dot{\boldsymbol\omega}_{R_{0}/N} = \ddot f \hat{\boldsymbol\imath}_{h}
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where :math:`f` is the true anomaly, whose variation is determined
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through the general standard astrodynamics relations:
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.. math::
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\begin{aligned}
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\dot f &= \frac{h}{r^{2}}
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\\
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\ddot f &= - 2 \frac{\boldsymbol v \cdot \hat{\boldsymbol\imath}_{r}}{r} \dot f
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\end{aligned}
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The angular rate :math:`\boldsymbol\omega_{R/N}` and acceleration
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:math:`\dot{\boldsymbol\omega}_{R/N}` of the output reference frame
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:math:`\mathcal{R}` still need to be computed. Since the desired
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attitude is a fixed-pointing one, :math:`\mathcal{R}` does not move
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relative to :math:`\mathcal{R}_{0}`. Thus, the angular velocity of the
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reference frame happens to be
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.. math::
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\label{eq:omega_R}
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\boldsymbol\omega_{R/N} = \boldsymbol\omega_{R/R_{0}} + \boldsymbol\omega_{R_{0}/N} = \dot{f} \hat{\boldsymbol\imath}_{h}
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Again, given that :math:`\hat{\boldsymbol\imath}_{h}` is fixed as seen by the
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reference frame :math:`R`, the acceleration vector of the reference
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frame expressed in the reference frame simply becomes:
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.. math::
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\label{eq:domega_R}
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\dot\omega_{R/N} = \ddot{f} \hat{\boldsymbol\imath}_{h}
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Both :math:`\boldsymbol\omega_{R/N}` and :math:`\dot\omega_{R/N}` need to be
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expressed in the inertial frame :math:`N`. Given
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.. math::
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\begin{equation}
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\leftexp{R}{\boldsymbol \omega_{R/N} } =
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\begin{bmatrix}
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0\\ 0 \\ \dot{f}
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\end{bmatrix}
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\end{equation}
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\begin{equation}
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\leftexp{R}{\boldsymbol\dot{\omega}_{R/N} } =
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\begin{bmatrix}
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0\\ 0 \\ \ddot{f}
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\end{bmatrix}
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\end{equation}
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Then,
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.. math::
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\begin{equation}
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\leftexp{N} {\boldsymbol{\omega}_{R/N}} = [NR] \textrm{ } \leftexp{R} {\boldsymbol\omega_{R/N} }
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\end{equation}
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\begin{equation}
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\leftexp{N} {\boldsymbol\dot{\omega}_{R/N} }=[NR] \textrm{ } \leftexp{R} {\boldsymbol\dot{\omega}_{R/N}}
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\end{equation}
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Where :math:`[NR] = [RN]^T`.
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Message Connection Descriptions
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-------------------------------

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