# 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 table lists all the module input and output messages. The module msg variable name 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

chiefTransInMsg

The name of the chief’s position and velocity input message

deputyTransInMsg

The name of the deputy’s position and velocity input message

forceOutMsg

Calculated Force to control orbital element difference output message

## Module Assumptions and Limitations¶

• This module assumes that target orbital element differnce is constant during simulation.

## 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 J2 constant of a central body

• 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

Returns

void

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

Returns

void

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

Returns

void

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