smallBodyWaypointFeedback#

Executive Summary#

This module is provides a feedback control law for waypoint-to-waypoint control about a small body. The waypoints are defined in the Hill frame of the body.

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
navTransInMsg	NavTransMsgPayload	translational navigation input message
navAttInMsg	NavAttMsgPayload	attitude navigation input message
asteroidEphemerisInMsg	EphemerisMsgPayload	asteroid ephemeris input message
sunEphemerisInMsg	EphemerisMsgPayload	sun ephemeris input message
forceOutMsg	CmdForceBodyMsgPayload	force command output
forceOutMsgC	CmdForceBodyMsgPayload	C-wrapped force output message

Detailed Module Description#

General Function#

The smallBodyWaypointFeedback() module provides a solution for waypoint-to-waypoint control about a small body. The feedback control law is similar to the cartesian coordinate continuous feedback control law in Chapter 14 of Analytical Mechanics of Space Systems. A cannonball SRP model, third body perturbations from the sun, and point-mass gravity are utilized. The state inputs are messages written out by simpleNav and planetNav modules or an estimator that provides the same input messages.

Algorithm#

The state vector is defined as follows:

(1)#\[\begin{split}\mathbf{X} = \begin{bmatrix} \mathbf{x}_1\\ \mathbf{x}_2\\ \end{bmatrix}= \begin{bmatrix} {}^O\mathbf{r}_{B/O} \\ {}^O\dot{\mathbf{r}}_{B/O} \\ \end{bmatrix}\end{split}\]

The associated frame definitions may be found in the following table.

Frame Definitions#
Frame Description	Frame Definition
Small Body Hill Frame	\(O: \{\hat{\mathbf{o}}_1, \hat{\mathbf{o}}_2, \hat{\mathbf{o}}_3\}\)
Spacecraft Body Frame	\(B: \{\hat{\mathbf{b}}_1, \hat{\mathbf{b}}_2, \hat{\mathbf{b}}_3\}\)

The derivation of the control law is skipped here for brevity. The thrust, however, is computed as follows:

(2)#\[\begin{equation} \mathbf{u} = -(f(\mathbf{x}) - f(\mathbf{x}_{ref})) - [K_1]\Delta\mathbf{x}_1 - [K_2]\Delta\mathbf{x}_2 \end{equation}\]

The relative velocity dynamics are described in detail by Takahashi and Scheeres.

(3)#\[\begin{split}\begin{split} f(\mathbf{x}) = ^O\ddot{\mathbf{r}}_{S/O} = -\ddot{F}[\tilde{\hat{\mathbf{o}}}_3]\mathbf{x}_1 - 2\dot{F}[\tilde{\hat{\mathbf{o}}}_3]\mathbf{x}_2 - \dot{F}^2[\tilde{\hat{\mathbf{o}}}_3][\tilde{\hat{\mathbf{o}}}_3]\mathbf{x}_1- \dfrac{\mu_a \mathbf{x}_1}{||\mathbf{x}_1||^3} + \dfrac{\mu_s(3{}^O\hat{\mathbf{d}}{}^O\hat{\mathbf{d}}^T-[I_{3 \times 3}])\mathbf{x}_1}{d^3} \\ + C_{SRP}\dfrac{P_0(1+\rho)A_{sc}}{M_{sc}}\dfrac{(1\text{AU})^2}{d^2}\hat{\mathbf{o}}_1 + \sum_i^I\dfrac{{}^O\mathbf{F}_i}{M_{sc}} + \sum_j^J\dfrac{{}^O\mathbf{F}_j}{M_{sc}} \end{split}\end{split}\]

User Guide#

A detailed example of the module is provided in scenarioSmallBodyFeedbackControl. However, the initialization of the module is also shown here. The module is first initialized as follows:

waypointFeedback = smallBodyWaypointFeedback.SmallBodyWaypointFeedback()

The asteroid ephemeris input message is then connected. In this example, we use the planetNav module.

waypointFeedback.asteroidEphemerisInMsg.subscribeTo(planetNavMeas.ephemerisOutMsg)

A standalone message is created for the sun ephemeris message.

sunEphemerisMsgData = messaging.EphemerisMsgPayload()
sunEphemerisMsg = messaging.EphemerisMsg()
sunEphemerisMsg.write(sunEphemerisMsgData)
waypointFeedback.sunEphemerisInMsg.subscribeTo(sunEphemerisMsg)

The navigation attitude and translation messages are then subscribed to

waypointFeedback.navAttInMsg.subscribeTo(simpleNavMeas.attOutMsg)
waypointFeedback.navTransInMsg.subscribeTo(simpleNavMeas.transOutMsg)

Finally, the area, mass, inertia, and gravitational parameter of the asteroid are initialized

waypointFeedback.A_sc = 1.  # Surface area of the spacecraft, m^2
waypointFeedback.M_sc = mass  # Mass of the spacecraft, kg
waypointFeedback.IHubPntC_B = unitTestSupport.np2EigenMatrix3d(I)  # sc inertia
waypointFeedback.mu_ast = mu  # Gravitational constant of the asteroid

The reference states are then defined:

waypointFeedback.x1_ref = [-2000., 0., 0.]
waypointFeedback.x2_ref = [0.0, 0.0, 0.0]

Finally, the feedback gains are set:

waypointFeedback.K1 = unitTestSupport.np2EigenMatrix3d([5e-4, 0e-5, 0e-5, 0e-5, 5e-4, 0e-5, 0e-5, 0e-5, 5e-4])
waypointFeedback.K2 = unitTestSupport.np2EigenMatrix3d([1., 0., 0., 0., 1., 0., 0., 0., 1.])

Class SmallBodyWaypointFeedback#

class SmallBodyWaypointFeedback : public SysModel#

This module is provides a Lyapunov feedback control law for waypoint to waypoint guidance and control about a small body. The waypoints are defined in the Hill frame of the body.

Public Functions

SmallBodyWaypointFeedback()#: This is the constructor for the module class. It sets default variable values and initializes the various parts of the model

void reset(uint64_t currentSimNanos)#

This method is used to reset the module and checks that required input messages are connect.

Returns:: void

void updateState(uint64_t currentSimNanos)#

This is the main method that gets called every time the module is updated. Provide an appropriate description.

Returns:: void

void readMessages()#

This method reads the input messages each call of updateState

Returns:: void

void computeControl(uint64_t currentSimNanos)#

This method computes the control using a Lyapunov feedback law

Returns:: void

void writeMessages(uint64_t currentSimNanos)#