C Module: meanOEFeedback

Summary

The primary purpose of this module is to calculate feedback control input for deputy spacecraft in terms of mean orbital element difference. This module uses Lyapunov control theory described in chapter 14 of Analytical Mechanics of Space Systems. In addition to classic orbital element set \((a,e,i,\Omega,\omega,M)\), this module can also deal with equinoctial orbital element \((a,e\sin{(\omega+\Omega)},e\cos{(\omega+\Omega)},\tan{(i/2)}\sin{(\Omega)},\tan{(i/2)}\cos{(\Omega)},\Omega+\omega+M)\) in order to avoid singularity. A control input \(\bf{u}\) used in this module is described in Eq. (1).

(1)\[\bf{u} = -[B(\bf{oe_d})]^T[K]\Delta \bf{oe}\]

Control matrix \(B(oe)\) derived from Gauss’s Planetary Equation is given in Eq.(14.213) of Analytical Mechanics of Space Systems for classic orbital element, or in Application of several control techniques for the ionospheric observation nanosatellite formation for equinoctial one. Depending on preference, you can change which orbital element set to use by setting \(oe_{type}\) parameter. In order to calculate mean oe, position and velocity input messages are converted to osculating oe. Then, osculating oe is converted to mean oe by Brower’s theory.

Message Connection Descriptions

The following diagram and table list all the module input and output messages. The module message connection is set by the user from Python. The message 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
chiefTransInMsg	NavTransMsgPayload	The name of the chief’s position and velocity input message
deputyTransInMsg	NavTransMsgPayload	The name of the deputy’s position and velocity input message
forceOutMsg	CmdForceInertialMsgPayload	Calculated Force to control orbital element difference output message

Module Assumptions and Limitations

This module assumes that the target orbital element difference is constant during simulation.
The mean/osculating conversion is a first-order J2 approximation for elliptic orbits, valid when the corrections are small and the inclination is away from the critical values where \(1 - 5\cos^2 i = 0\).
Use oeType = 1 for circular or prograde equatorial orbits. The classical control matrix remains singular at zero eccentricity or equatorial inclination; the equinoctial control matrix remains singular at retrograde equatorial inclination, \(i = \pi\).
Failed mean-element or equinoctial conversions raise a Basilisk error before a force message is written. For nonzero J2, the mean-element map rejects abs(1 - 5*cos(i)**2) <= sqrt(machine epsilon) as numerically singular. The prograde equinoctial map rejects cos(i/2) <= 8*machine epsilon. Accepted states must still satisfy the small-perturbation assumption.
J2 = 0 disables the mean/osculating correction, so the controller can also operate on two-body orbital elements.

User Guide

This module requires the following variables to be set as parameters:

oeType 0 for classic oe (default), 1 for equinoctial oe
mu gravitational constant for a central body in m^3/s^2
req equatorial radius of a central body in meters
J2 finite, nonnegative, dimensionless J2 coefficient of the central body; zero gives an identity mean/osculating conversion
targetDiffOeMean desired mean orbital element difference. Assumed oe is linked to oeType.
K control gain matrix

You must be careful about a meter unit used for mu and req. For targetDiffOeMean, normalized semi major axis must be used.

Functions

void SelfInit_meanOEFeedback(meanOEFeedbackConfig *configData, int64_t moduleID)

This method initializes the configData for this module. It checks to ensure that the inputs are sane and then creates the output message

Parameters:

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

void Update_meanOEFeedback(meanOEFeedbackConfig *configData, uint64_t callTime, int64_t moduleID)

Add a description of what this main Update() routine does for this module

Parameters:

configData – The configuration data associated with the module
callTime – The clock time at which the function was called (nanoseconds)
moduleID – The Basilisk module identifier

void Reset_meanOEFeedback(meanOEFeedbackConfig *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. The local copy of the message output buffer should be cleared.

Parameters:

configData – The configuration data associated with the module
callTime – The clock time at which the function was called (nanoseconds)
moduleID – The Basilisk module identifier

struct meanOEFeedbackConfig

#include <meanOEFeedback.h>

Top level structure for the sub-module routines.

Public Members

NavTransMsg_C chiefTransInMsg: chief orbit input message

NavTransMsg_C deputyTransInMsg: deputy orbit input message

CmdForceInertialMsg_C forceOutMsg: deputy control force output message

double K[36]: Lyapunov Gain (6*6).

double targetDiffOeMean[6]: target mean orbital element difference

uint8_t oeType: 0: classic (default), 1: equinoctial

double mu: [m^3/s^2] gravitational constant

double req: [m] equatorial planet radius

double J2: [] J2 planet oblateness parameter

BSKLogger *bskLogger: BSK Logging.