Module: prescribedRot1DOF¶

Executive Summary¶

This module profiles a PrescribedMotionMsgPayload message for a specified 1 DOF rotational attitude maneuver for a secondary 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 are: r_FM_M, rPrime_FM_M, rPrimePrime_FM_M, omega_FM_F, omegaPrime_FM_F, and sigma_FM. Because this is a purely rotational profiler, the translational states r_FM_M, rPrime_FM_M, and rPrimePrime_FM_M are held constant in this module.

To use this module for prescribed motion purposes, it must be connected to the Module: prescribedMotionStateEffector dynamics module in order to profile the states of the secondary body. The required maneuver is determined from the user-specified scalar maximum angular acceleration for the attitude maneuver \(\alpha_{\text{max}}\), 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 maneuver and constant negatively for the second half of the maneuver. The resulting angular velocity of the prescribed body is linear, approaching a maximum magnitude halfway through the maneuver and ending with zero residual velocity. The corresponding angle 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.

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

Detailed Module Description¶

This 1 DOF rotational motion flight software 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 attitude maneuver \(\alpha_{\text{max}}\), 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 attitude maneuver.

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

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

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

During the first half of the attitude maneuver, 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 attitude maneuver. The switch time \(t_s\) is the simulation time halfway through the maneuver:

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

where the time required for the maneuver \(\Delta t\) 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 maneuver 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 attitude maneuver 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 attitude maneuver 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 maneuver 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 attitude maneuver 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 rotational attitude maneuver 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 attitude maneuver \(\alpha_{\text{max}}\), 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 PrescribedMotionMsgPayload. The first guidance message, describing the spinning body’s scalar states relative to the body-fixed mount frame can be directly connected to an attitude feedback control module. The second prescribed motion output message can be connected to the Module: prescribedMotionStateEffector dynamics module to directly profile a state effector’s rotational motion.

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:

PrescribedRot1DOFConfig = prescribedRot1DOF.PrescribedRot1DOFConfig()
PrescribedWrap = unitTestSim.setModelDataWrap(PrescribedRot1DOFConfig)
PrescribedWrap.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
PrescribedRot1DOFConfig.r_FM_M = np.array([1.0, 0.0, 0.0])
PrescribedRot1DOFConfig.rPrime_FM_M = np.array([0.0, 0.0, 0.0])
PrescribedRot1DOFConfig.rPrimePrime_FM_M = np.array([0.0, 0.0, 0.0])
PrescribedRot1DOFConfig.rotAxis_M = rotAxis_M
PrescribedRot1DOFConfig.thetaDDotMax = 0.01  # [rad/s^2]
PrescribedRot1DOFConfig.omega_FM_F = np.array([0.0, 0.0, 0.0])
PrescribedRot1DOFConfig.omegaPrime_FM_F = np.array([0.0, 0.0, 0.0])
PrescribedRot1DOFConfig.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, PrescribedWrap, PrescribedRot1DOFConfig)

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 r_FM_M[3]¶: [m] Position of the F frame origin with respect to the M frame origin in M frame components (fixed)

double rPrime_FM_M[3]¶: [m/s] B frame time derivative of r_FM_M in M frame components (fixed)

double rPrimePrime_FM_M[3]¶: [m/s^2] B frame time derivative of rPrime_FM_M in M frame components (fixed)

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.

PrescribedMotionMsg_C prescribedMotionOutMsg¶: Output msg for the spinning body prescribed states.