Module: prescribedRot2DOF

Executive Summary

This module profiles a PrescribedRotationMsgPayload message for a specified 2 DOF rotation 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 profiled in this module are: omega_FM_F, omegaPrime_FM_F, and sigma_FM.

It should be noted that although the inputs to this module are two consecutive rotation angles and axes, the resulting rotation that is profiled is a 1 DOF rotation. The module converts two given reference angles and their corresponding rotation axes for the rotation to a single 1 DOF rotation for the rotation. Simple Principal Rotation Vector (PRV) addition is used on the two given reference PRVs to determine the single PRV required for the rotation.

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, \(\alpha_{\text{max}}\), the spinning body’s initial attitude with respect to the mount frame as the Principal Rotation Vector prv_F0M \((\Phi_0, \hat{\boldsymbol{e}}_0)\), and two reference Principal Rotation Vectors for the rotation, prv_F1F0 \((\Phi_{1a}, \hat{\boldsymbol{e}}_{1a})\) and prv_F2F1 \((\Phi_{1b}, \hat{\boldsymbol{e}}_{1b})\).

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.

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
spinningBodyRef1InMsg	HingedRigidBodyMsgPayload	input msg with the scalar spinning body rotational reference states for the first rotation
spinningBodyRef2InMsg	HingedRigidBodyMsgPayload	input msg with the scalar spinning body rotational reference states for the second rotation
prescribedMotionOutMsg	PrescribedRotationMsgPayload	output message with the prescribed spinning body rotational states

Detailed Module Description

This 2 DOF rotational motion kinematic profiler module converts a given 2 DOF rotation to a single 1 DOF rotation and profiles the required spinning body rotational motion with respect to a body-fixed mount frame for the rotation. The inputs to the profiler are the maximum angular acceleration for the rotation, \(\alpha_{\text{max}}\), the spinning body’s initial attitude with respect to the mount frame as the Principal Rotation Vector prv_F0M \((\Phi_0, \hat{\boldsymbol{e}}_0)\), and two reference Principal Rotation Vectors for the rotation, prv_F1F0 \((\Phi_{1a}, \hat{\boldsymbol{e}}_{1a})\) and prv_F2F1 \((\Phi_{1b}, \hat{\boldsymbol{e}}_{1b})\).

The module first converts the two given reference PRVs to a single PRV, prv_F2M that represents the final spinning body attitude with respect to the body-fixed mount frame:

\[\Phi_2 = 2 \cos^{-1} \left ( \cos \frac{\Phi_{1a}}{2} \cos \frac{\Phi_{1b}}{2} - \sin \frac{\Phi_{1a}}{2} \sin \frac {\Phi_{1b}}{2} \ \hat{\boldsymbol{e}}_{1a} \cdot \hat{\boldsymbol{e}}_{1b} \right )\]

\[\hat{\boldsymbol{e}}_2 = \frac{\cos \frac{\Phi_{1b}}{2} \sin \frac{\Phi_{1a}}{2} \ \hat{\boldsymbol{e}}_{1a} + \cos \frac{\Phi_{1a}}{2} \sin \frac{\Phi_{1b}}{2} \ \boldsymbol{e}_{1b} + \sin \frac{\Phi_{1a}}{2} \sin \frac{\Phi_{1b}}{2} \ \hat{\boldsymbol{e}}_{1a} \times \hat{\boldsymbol{e}}_{1b} }{\sin \frac{\Phi_2}{2}}\]

Subtracting the initial Principal Rotation Vector prv_F0M from the found reference PRV prv_F2M gives the required PRV for the rotation, prv_F2F0:

\[\Phi_{\text{ref}} = \Delta \Phi = 2 \cos^{-1} \left ( \cos \frac{\Phi_2}{2} \cos \frac{\Phi_0}{2} + \sin \frac{\Phi_2}{2} \sin \frac {\Phi_0}{2} \ \hat{\boldsymbol{e}}_2 \cdot \hat{\boldsymbol{e}}_0 \right )\]

\[\hat{\boldsymbol{e}}_3 = \frac{\cos \frac{\Phi_0}{2} \sin \frac{\Phi_2}{2} \ \hat{\boldsymbol{e}}_2 - \cos \frac{\Phi_2}{2} \sin \frac{\Phi_0}{2} \ \hat{\boldsymbol{e}}_0 + \sin \frac{\Phi_2}{2} \sin \frac{\Phi_0}{2} \ \hat{\boldsymbol{e}}_2 \times \hat{\boldsymbol{e}}_0 }{\sin \frac{\Delta \Phi}{2}}\]

Note that the initial PRV angle, \(\Phi_0\) is reset to zero for consecutive rotations so that the reference PRV angle, \(\Phi_{\text{ref}}\) is always taken as the full angle to be swept during the rotation.

During the first half of the rotation, the spinning body is constantly accelerated with the given maximum angular acceleration. The spinning 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 found PRV angle for the rotation:

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

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

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

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

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

where

\[a = \frac{ \frac{1}{2} \Phi_{\text{ref}}}{(t_s - t_0)^2}\]

Similarly, the second half of the rotation decelerates the spinning body constantly until it reaches a non-rotating state. The spinning body’s 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 greater than zero. If the reference angle is negative, the profiled motion is instead concave upwards. The described motion during the second half of the rotation is characterized by the expressions:

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

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

\[\Phi(t) = b (t - t_f)^2 + \Phi_{\text{ref}}\]

where

\[b = \frac{ \frac{1}{2} \Phi_{\text{ref}}}{(t_s - t_f)^2}\]

Module Testing

The unit test for this module simulates TWO consecutive 2 DOF rotational attitude maneuvers for a secondary rigid body connected to a rigid spacecraft hub. Two maneuvers are simulated to ensure that the module correctly updates the required relative PRV attitude when a new attitude reference message is written. The unit test checks that the prescribed body’s MRP attitude converges to both reference attitudes for a series of initial and reference attitudes and maximum angular accelerations. (sigma_FM_Final1 is checked to converge to sigma_FM_Ref1, and sigma_FM_Final2 is checked to converge to sigma_FM_Ref2). Additionally, the prescribed body’s final angular velocity magnitude thetaDot_Final is checked for convergence to the reference angular velocity magnitude, thetaDot_Ref.

User Guide

The user-configurable inputs to the profiler are the maximum angular acceleration for the rotation, \(\alpha_{\text{max}}\), the spinning body’s initial attitude with respect to the mount frame as the Principal Rotation Vector prv_F0M \((\Phi_0, \hat{\boldsymbol{e}}_0)\), and two reference Principal Rotation Vectors for the rotation, prv_F1F0 \((\Phi_{1a}, \hat{\boldsymbol{e}}_{1a})\) and prv_F2F1 \((\Phi_{1b}, \hat{\boldsymbol{e}}_{1b})\).

This module provides a single output message in the form of PrescribedRotationMsgPayload. This prescribed 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 2 DOF rotational module in python using Basilisk.

1. Import the prescribedRot1DOF class:

   from Basilisk.fswAlgorithms import prescribedRot2DOF
   

2. Create an instantiation of a prescribed rotational 2 DOF C module and the associated C++ container:

   PrescribedRot2DOF = prescribedRot2DOF.prescribedRot2DOF()
   PrescribedRot2DOF.ModelTag = "PrescribedRot2DOF"
   

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

   rotAxis1_M = np.array([0.0, 1.0, 0.0])                                      # Rotation axis for the first reference rotation angle, thetaRef1a
   rotAxis2_F1 = np.array([0.0, 0.0, 1.0])                                     # Rotation axis for the second reference rotation angle, thetaRef2a
   PrescribedRot2DOF.rotAxis1_M = rotAxis1_M
   PrescribedRot2DOF.rotAxis2_F1 = rotAxis2_F1
   PrescribedRot2DOF.phiDDotMax = phiDDotMax
   PrescribedRot2DOF.omega_FM_F = np.array([0.0, 0.0, 0.0])              # [rad/s] Angular velocity of frame F relative to frame M in F frame components
   PrescribedRot2DOF.omegaPrime_FM_F = np.array([0.0, 0.0, 0.0])         # [rad/s^2] B frame time derivative of omega_FB_F in F frame components
   PrescribedRot2DOF.sigma_FM = np.array([0.0, 0.0, 0.0])                # MRP attitude of frame F relative to frame M
   

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

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

2. Add the module to the task list:

   unitTestSim.AddModelToTask(unitTaskName, PrescribedRot2DOF)
   

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Functions

void SelfInit_prescribedRot2DOF(PrescribedRot2DOFConfig *configData, int64_t moduleID)

This method initializes the output message for this module.

Parameters

• configData – The configuration data associated with this module

• moduleID – The module identifier

Returns

void

void Reset_prescribedRot2DOF(PrescribedRot2DOFConfig *configData, uint64_t callTime, int64_t moduleID)

This method performs a complete reset of the module. Local module variables that retain time varying states between function calls are reset to their default values. A check is also performed to ensure the user sets the configurable module variables.

Parameters

• configData – The configuration data associated with the module

• callTime – [ns] Time the method is called

• moduleID – The module identifier

Returns

void

void Update_prescribedRot2DOF(PrescribedRot2DOFConfig *configData, uint64_t callTime, int64_t moduleID)

This method profiles a 1DOF rotational trajectory given two rotation angles and rotation axes. The prescribed states are updated in this routine as a function of time and written to the prescribedMotion 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 PrescribedRot2DOFConfig

#include <prescribedRot2DOF.h>

Top level structure for the sub-module routines.

Include the required files.

Public Members

double phiDDotMax

[rad/s^2] Maximum angular acceleration of the spinning body

double rotAxis1_M[3]

M frame rotation axis for the first rotation.

double rotAxis2_F1[3]

F1 frame intermediate rotation axis for the second rotation.

double omega_FM_F[3]

[rad/s] angular velocity of frame F relative to frame M in F frame components

double omegaPrime_FM_F[3]

[rad/s^2] B frame time derivative of omega_FB_F in F frame components

double sigma_FM[3]

MRP attitude of frame F relative to frame M.

bool isManeuverComplete

Boolean variable is true when the attitude maneuver is complete.

double maneuverStartTime

[s] Simulation time at the start of the attitude maneuver

double rotAxis_M[3]

Reference PRV axis expressed in M frame components.

double phiRef

[rad] Reference PRV angle (The positive short rotation is chosen)

double phiDotRef

[rad/s] Reference PRV angle rate

double phi

[rad] Current PRV angle

double phiRefAccum

[rad] This variable logs the accumulated reference PRV angles

double phiAccum

[rad] This variable logs the accumulated current PRV angle

double maneuverSwitchTime

[s] Simulation time halfway through the attitude maneuver (switch time)

double maneuverEndTime

[s] Simulation time when the maneuver is complete

double a

Parabolic constant for the first half of the attitude maneuver.

double b

Parabolic constant for the second half of the attitude maneuver.

double dcm_F0M[3][3]

DCM from the M frame to the spinning body body frame at the beginning of the maneuver.

HingedRigidBodyMsg_C spinningBodyRef1InMsg

Input msg for the first reference angle and angle rate.

HingedRigidBodyMsg_C spinningBodyRef2InMsg

Input msg for the second reference angles and angle rate.

PrescribedRotationMsg_C prescribedRotationOutMsg

Output msg for the profiled prescribed rotational states.

BSKLogger *bskLogger

BSK Logging.