Module: prescribedRot1DOF

Executive Summary

This module profiles a PrescribedRotationMsgPayload message for a specified 1 DOF rotation for a secondary prescribed rigid body connected to a rigid spacecraft hub at a hub-fixed location, \(\mathcal{M}\). The body frame for the prescribed body is designated by the frame \(\mathcal{F}\). Accordingly, the prescribed states for the secondary body are written with respect to the mount frame, \(\mathcal{M}\). The prescribed states profiled in this module are: omega_FM_F, omegaPrime_FM_F, and sigma_FM.

To use this module for prescribed motion, it must be connected to the Module: prescribedMotionStateEffector dynamics module in order to profile the rotational states of the secondary body. A second kinematic profiler module must also be connected to the prescribed motion dynamics module to profile the translational states of the prescribed body. The required rotation is determined from the user-specified scalar maximum angular acceleration for the rotation \(\alpha_{\text{max}}\), prescribed body’s initial attitude with respect to the mount frame as the Principal Rotation Vector prv_F0M \((\Phi_0, \hat{\textbf{{e}}}_0)\), and the prescribed body’s reference attitude with respect to the mount frame as the Principal Rotation Vector prv_F1M \((\Phi_1, \hat{\textbf{{e}}}_1)\).

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

Warning

This module is now deprecated. See the Module: prescribedRotation1DOF module that replaces this module.

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.

Module I/O Messages
Msg Variable Name	Msg Type	Description
spinningBodyInMsg	HingedRigidBodyMsgPayload	input msg with the scalar spinning body rotational reference states
spinningBodyOutMsg	HingedRigidBodyMsgPayload	output message with the profiled scalar spinning body rotational states
prescribedMotionOutMsg	PrescribedRotationMsgPayload	output message with the profiled prescribed spinning body rotational states

Detailed Module Description

This 1 DOF rotational motion kinematic profiler module is written to profile spinning body motion with respect to a body-fixed mount frame. The inputs to the profiler are the scalar maximum angular acceleration for the rotation \(\alpha_{\text{max}}\), the prescribed body’s initial attitude with respect to the mount frame as the Principal Rotation Vector prv_F0M \((\Phi_0, \hat{\textbf{{e}}}_0)\), and the prescribed body’s reference attitude with respect to the mount frame as the Principal Rotation Vector prv_F1M \((\Phi_1, \hat{\textbf{{e}}}_1)\). The prescribed body is assumed to be non-rotating at the beginning of the rotation.

Subtracting the initial principal rotation vector from the reference principal rotation vector gives the required rotation angle and axis for the rotation:

\[\Phi_{\text{ref}} = 2 \cos^{-1} \left ( \cos \frac{\Phi_1}{2} \cos \frac{\Phi_0}{2} + \sin \frac{\Phi_1}{2} \sin \frac {\Phi_0}{2} \hat{\textbf{{e}}}_1 \cdot \hat{\textbf{{e}}}_0 \right )\]

\[\hat{\textbf{{e}}} = \frac{\cos \frac{\Phi_0}{2} \sin \frac{\Phi_1}{2} \hat{\textbf{{e}}}_1 - \cos \frac{\Phi_1}{2} \sin \frac{\Phi_0}{2} \hat{\textbf{{e}}}_0 + \sin \frac{\Phi_1}{2} \sin \frac{\Phi_0}{2} \hat{\textbf{{e}}}_1 \times \hat{\textbf{{e}}}_0 }{\sin \frac{\Phi_{\text{ref}}}{2}}\]

During the first half of the rotation, the prescribed body is constantly accelerated with the given maximum angular acceleration. The prescribed body’s angular velocity increases linearly during the acceleration phase and reaches a maximum magnitude halfway through the rotation. The switch time \(t_s\) is the simulation time halfway through the rotation:

\[t_s = t_0 + \frac{\Delta t}{2}\]

where the time required for the rotation \(\Delta t\) is determined using the inputs to the profiler:

\[\Delta t = t_f - t_0 = 2 \sqrt{ \Phi_{\text{ref}} / \ddot{\Phi}_{\text{max}}}\]

The resulting trajectory of the angle \(\Phi\) swept during the first half of the rotation is parabolic. The profiled motion is concave upwards if the reference angle \(\Phi_{\text{ref}}\) is greater than zero. If the converse is true, the profiled motion is instead concave downwards. The described motion during the first half of the rotation is characterized by the expressions:

\[\omega_{\mathcal{F} / \mathcal{M}}(t) = \alpha_{\text{max}}\]

\[\dot{\Phi}(t) = \alpha_{\text{max}} (t - t_0)\]

\[\Phi(t) = c_1 (t - t_0)^2\]

where

\[c_1 = \frac{\Phi_{\text{ref}}}{2(t_s - t_0)^2}\]

Similarly, the second half of the rotation decelerates the prescribed body constantly until it reaches a non-rotating state. The prescribed body angular velocity decreases linearly from its maximum magnitude back to zero. The trajectory swept during the second half of the rotation is quadratic and concave downwards if the reference angle \(\Phi_{\text{ref}}\) is positive. If \(\Phi_{\text{ref}}\) is negative, the profiled motion is instead concave upwards. The described motion during the second half of the rotation is characterized by the expressions:

\[\ddot{\Phi}(t) = -\alpha_{\text{max}}\]

\[\dot{\Phi}(t) = \alpha_{\text{max}} (t - t_f)\]

\[ \begin{align}\begin{aligned}\Phi(t) = c_2 (t - t_f)^2 + \Phi_{\text{ref}}\\where\end{aligned}\end{align} \]

\[c_2 = \frac{\Phi_{\text{ref}}}{2(t_s - t_f)^2}\]

Module Testing

The unit test for this module ensures that the profiled 1 DOF rotation is properly computed for a series of initial and reference PRV angles and maximum angular accelerations. The final prescribed angle theta_FM_Final and angular velocity magnitude thetaDot_Final are compared with the reference values theta_Ref and thetaDot_Ref, respectively.

User Guide

The user-configurable inputs to the profiler are the scalar maximum angular acceleration for the rotation \(\alpha_{\text{max}}\), the prescribed body’s initial attitude with respect to the mount frame as the Principal Rotation Vector prv_F0M \((\Phi_0, \hat{\textbf{{e}}}_0)\), and the prescribed body’s reference attitude with respect to the mount frame as the Principal Rotation Vector prv_F1M \((\Phi_1, \hat{\textbf{{e}}}_1)\).

This module provides two output messages in the form of HingedRigidBodyMsgPayload and PrescribedRotationMsgPayload. The first message describes the spinning body’s scalar rotational states relative to the body-fixed mount frame. The second prescribed rotational motion output message can be connected to the Module: prescribedMotionStateEffector dynamics module to directly profile a state effector’s rotational motion. Note that a separate translational profiler module must also be connected to the prescribed motion dynamics module to fully define the kinematic motion of the prescribed body.

This section is to outline the steps needed to setup a prescribed 1 DOF rotational module in python using Basilisk.

Import the prescribedRot1DOF class:

from Basilisk.fswAlgorithms import prescribedRot1DOF

Create an instantiation of a prescribed rotational 1 DOF C module and the associated C++ container:

PrescribedRot1DOF = prescribedRot1DOF.prescribedRot1DOF()
PrescribedRot1DOF.ModelTag = "prescribedRot1DOF"

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

thetaInit = 0.0  # [rad]
rotAxis_M = np.array([1.0, 0.0, 0.0])
prvInit_FM = thetaInit * rotAxisM
PrescribedRot1DOF.rotAxis_M = rotAxis_M
PrescribedRot1DOF.thetaDDotMax = 0.01  # [rad/s^2]
PrescribedRot1DOF.omega_FM_F = np.array([0.0, 0.0, 0.0])
PrescribedRot1DOF.omegaPrime_FM_F = np.array([0.0, 0.0, 0.0])
PrescribedRot1DOF.sigma_FM = rbk.PRV2MRP(prvInit_FM)

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, PrescribedRot1DOF)

Functions

void SelfInit_prescribedRot1DOF(PrescribedRot1DOFConfig *configData, int64_t moduleID)

Method for module initialization.

This method initializes the output messages for this module.

Parameters:

configData – The configuration data associated with this module
moduleID – The module identifier

Returns:

void

void Reset_prescribedRot1DOF(PrescribedRot1DOFConfig *configData, uint64_t callTime, int64_t moduleID)

Method for module reset.

This method performs a complete reset of the module. The input messages are checked to ensure they are linked.

Parameters:

configData – The configuration data associated with the module
callTime – [ns] Time the method is called
moduleID – The module identifier

Returns:

void

void Update_prescribedRot1DOF(PrescribedRot1DOFConfig *configData, uint64_t callTime, int64_t moduleID)

Method for module time update.

This method profiles the prescribed trajectory and updates the prescribed states as a function of time. The prescribed states are then written to the output message.

Parameters:

configData – The configuration data associated with the module
callTime – [ns] Time the method is called
moduleID – The module identifier

Returns:

void

struct PrescribedRot1DOFConfig

#include <prescribedRot1DOF.h>

Top level structure for the sub-module routines.

Public Members

double thetaDDotMax: [rad/s^2] Maximum angular acceleration of spinning body

double rotAxis_M[3]: Rotation axis for the maneuver in M frame components.

double omega_FM_F[3]: [rad/s] Angular velocity of frame F wrt frame M in F frame components

double omegaPrime_FM_F[3]: [rad/s^2] B frame time derivative of omega_FM_F in F frame components

double sigma_FM[3]: MRP attitude of frame F with respect to frame M.

bool convergence: Boolean variable is true when the maneuver is complete.

double tInit: [s] Simulation time at the beginning of the maneuver

double thetaInit: [rad] Initial spinning body angle from frame M to frame F about rotAxis_M

double thetaDotInit: [rad/s] Initial spinning body angle rate between frame M to frame F

double thetaRef: [rad] Reference angle from frame M to frame F about rotAxis_M

double thetaDotRef: [rad/s] Reference angle rate between frame M to frame F

double ts: [s] The simulation time halfway through the maneuver (switch time for ang accel)

double tf: [s] Simulation time when the maneuver is finished

double a: Parabolic constant for the first half of the maneuver.

double b: Parabolic constant for the second half of the maneuver.

BSKLogger *bskLogger: BSK Logging.

HingedRigidBodyMsg_C spinningBodyInMsg: Input msg for the spinning body reference angle and angle rate.

HingedRigidBodyMsg_C spinningBodyOutMsg: Output msg for the spinning body angle and angle rate.

PrescribedRotationMsg_C prescribedRotationOutMsg: Output msg for the spinning body prescribed rotational states.