.. _prescribedTrans:

Module: prescribedTrans
=======================

Executive Summary
-----------------
This module profiles a :ref:`PrescribedMotionMsgPayload` message for a specified translational maneuver
for a secondary rigid body connected to a rigid spacecraft hub at a hub-fixed location, :math:`\mathcal{M}`. The body
frame for the prescribed body is designated by the frame :math:`\mathcal{F}`. Accordingly, the prescribed states for the
secondary body are written with respect to the mount frame, :math:`\mathcal{M}`. The prescribed states are: ``r_FM_M``,
``rPrime_FM_M``, ``rPrimePrime_FM_M``, ``omega_FM_F``, ``omegaPrime_FM_F``, and ``sigma_FM``. Because this is a
purely translational profiler, the states ``omega_FM_F``, ``omegaPrime_FM_F``, and ``sigma_FM`` are held
constant in this module.

To use this module for prescribed motion purposes, it must be connected to the :ref:`PrescribedMotionStateEffector`
dynamics module in order to profile the states of the secondary body. The required maneuver is determined from the
user-specified scalar maximum acceleration :math:`a_{\text{max}}`, the mount frame axis for the translational motion,
the prescribed body's initial position vector with respect to the mount frame :math:`\boldsymbol{r}_{F/M}(t_0)`, and
the reference position vector or the prescribed body with respect to the mount frame
:math:`\boldsymbol{r}_{F/M} (\text{ref})`.

The maximum scalar acceleration is applied constant and positively for the first half of the maneuver and
constant negatively for the second half of the maneuver. The resulting velocity of the prescribed body is
linear, approaching a maximum magnitude halfway through the maneuver and ending with zero residual velocity.
The corresponding translational trajectory the prescribed body moves through during the maneuver is parabolic in time.


Message Connection Descriptions
-------------------------------
The following table lists all the module input and output messages.  
The module msg connection is set by the user from python.  
The msg type contains a link to the message structure definition, while the description 
provides information on what this message is used for.

.. list-table:: Module I/O Messages
    :widths: 25 25 50
    :header-rows: 1

    * - Msg Variable Name
      - Msg Type
      - Description
    * - prescribedTransInMsg
      - :ref:`PrescribedTransMsgPayload`
      - input msg with the prescribed body reference states
    * - prescribedTransOutMsg
      - :ref:`PrescribedTransMsgPayload`
      - output message with the scalar prescribed body states
    * - prescribedMotionOutMsg
      - :ref:`PrescribedMotionMsgPayload`
      - output message with the prescribed body states



Detailed Module Description
---------------------------
This translational motion flight software module is written to profile a rigid body's motion with respect to a hub-fixed
mount frame. The inputs to the profiler are the scalar maximum acceleration for the maneuver :math:`a_{\text{max}}`,
the mount frame axis for the translational motion, the prescribed body's initial position vector with respect to
the mount frame :math:`\boldsymbol{r}_{F/M}(t_0)`, and the reference position vector or the prescribed body with respect
to the mount frame :math:`\boldsymbol{r}_{F/M} (\text{ref})`. The magnitudes of the initial and final position vectors
are denoted :math:`r_0` and :math:`r_{\text{ref}}`, respectively. The prescribed body is assumed to be at rest at the
beginning of the attitude maneuver.

Subtracting the initial position from the reference position vector gives the required relative position vector in the
direction of translation:

.. math::
    \Delta \boldsymbol{r} = \boldsymbol{r}_{F/M}(\text{ref}) - \boldsymbol{r}_{F/M}(t_0)

The magnitude of the determined relative position vector gives the required translational distance :math:`\Delta r`.
During the first half of the maneuver, the prescribed body is constantly accelerated with the given maximum acceleration.
The prescribed body's velocity increases linearly during the acceleration phase and reaches a maximum magnitude halfway
through the maneuver.

The switch time, :math:`t_s` is the simulation time halfway through the maneuver:

.. math::
    t_s = t_0 + \frac{\Delta t}{2}

The time required for the maneuver :math:`\Delta t` is determined using the inputs to the profiler:

.. math::
    \Delta t = \sqrt{\frac{4 r_{\text{ref}} - 8 r_0}{\ddot{a}_{\text{max}}}}

The resulting trajectory of the position vector :math:`r = || \boldsymbol{r}_{F/M} ||_2` magnitude during the first half of the
maneuver is parabolic. The profiled motion is concave upwards if the reference position magnitude :math:`r_{\text{ref}}`
is greater than the initial position magnitude :math:`r_0`. If the converse is true, the profiled motion is instead concave
downwards. The described motion during the first half of the maneuver is characterized by the expressions:

.. math::
    r^{''}_{F / M}(t) = a_{\text{max}}

.. math::
    r^{'}_{F / M}(t) = a_{\text{max}} (t - t_0)

.. math::
    r_{F / M}(t) = c_1 (t - t_0)^2  + r_0

where

.. math::
    c_1 = \frac{r_{\text{ref}} - r_0}{2(t_s - t_0)^2}


Similarly, the second half of the maneuver decelerates the prescribed body constantly until it reaches the desired
position with zero velocity. The prescribed body velocity decreases linearly from its maximum magnitude back to zero.
The trajectory during the second half of the maneuver is quadratic and concave downwards if the reference position
magnitude is greater than the initial position magnitude. If the converse is true, the profiled motion is instead
concave upwards. The described motion during the second half of the maneuver is characterized by the expressions:

.. math::
    r^{''}_{F / M}(t) = -a_{\text{max}}

.. math::
    r^{'}_{F / M}(t) = a_{\text{max}} (t - t_f)

.. math::
    r_{F / M}(t) = c_2 (t - t_f)^2  + r_{\text{ref}}

where

.. math::
    c_2 = \frac{r_{\text{ref}} - r_0}{2 (t_s - t_f)^2}

Module Testing
^^^^^^^^^^^^^^
This unit test for this module ensures that the profiled translational maneuver is properly computed for a series of
initial and reference positions and maximum accelerations. The final prescribed position magnitude ``r_FM_M_Final`` and
velocity magnitude ``rPrime_FM_M_Final`` are compared with the reference values ``r_FM_M_Ref`` and
``rPrime_FM_M_Ref``, respectively.

User Guide
----------
The user-configurable inputs to the profiler are the scalar maximum acceleration for the maneuver :math:`a_{\text{max}}`,
the mount frame axis for the translational motion, the prescribed body's initial position vector with respect to
the mount frame :math:`\boldsymbol{r}_{F/M}(t_0)`, and the reference position vector of the prescribed body with respect
to the mount frame :math:`\boldsymbol{r}_{F/M} (\text{ref})`.

This module provides two output messages in the form of :ref:`PrescribedTransMsgPayload` and
:ref:`PrescribedMotionMsgPayload`. The first guidance message, describing the prescribed body's scalar states relative to
the hub-fixed mount frame can be directly connected to a feedback control module. The second prescribed
motion output message can be connected to the :ref:`PrescribedMotionStateEffector` dynamics module to directly profile
a state effector's translational motion.

This section is to outline the steps needed to setup a prescribed translational module in python using Basilisk.

#. Import the prescribedTrans class::

    from Basilisk.fswAlgorithms import prescribedTrans

#. Create an instantiation of a prescribed translational C module and the associated C++ container::

    PrescribedTransConfig = prescribedTrans.PrescribedTransConfig()
    PrescribedTransWrap = unitTestSim.setModelDataWrap(PrescribedTransConfig)
    PrescribedTransWrap.ModelTag = "prescribedTrans"

#. Define all of the configuration data associated with the module. For example::

    PrescribedTransConfig.transAxis_M = np.array([1.0, 0.0, 0.0])
    PrescribedTransConfig.scalarAccelMax = 0.01  # [m/s^2]
    PrescribedTransConfig.r_FM_M = np.array([0.0, 0.0, 0.0])
    PrescribedTransConfig.rPrime_FM_M = np.array([0.0, 0.0, 0.0])
    PrescribedTransConfig.rPrimePrime_FM_M = np.array([0.0, 0.0, 0.0])
    PrescribedTransConfig.omega_FM_F = np.array([0.0, 0.0, 0.0])
    PrescribedTransConfig.omegaPrime_FM_F = np.array([0.0, 0.0, 0.0])
    PrescribedTransConfig.sigma_FM = np.array([0.0, 0.0, 0.0])

The user is required to set the above configuration data parameters, as they are not initialized in the module.

#. Make sure to connect the required messages for this module.

#. Add the module to the task list::

    unitTestSim.AddModelToTask(unitTaskName, PrescribedTransWrap, PrescribedTransConfig)



----

.. autodoxygenfile:: prescribedTrans.h
   :project: prescribedTrans