User Manual 4.14 Attitude Profile : Différence entre versions

Version du 2 septembre 2024 à 07:44

Introduction

Scope

An attitude profile is an attitude law providing the computation of an instantaneous attitude without any reference to the satellite orbit, nor to its attitude laws sequence. The purpose of this section is to present the attitude profiles available through the Patrius library.

Javadoc

The guidance models and classes are available in the attitude profiles package in the Patrius library.

Links

None as of now.

Useful Documents

None as of now.

Package Overview

Features Description

PATRIUS handles different guidance profiles types:

• Profile expressed as quaternions represented as polynomials by means of UnivariateFunction: QuaternionPolynomialProfile
• Profile expressed as quaternions represented as polynomials by means of DatePolynomialFunctionInterface: QuaternionDatePolynomialProfile
• Profile expressed as quaternions represented as harmonics: QuaternionHarmonicProfile
• Profile expressed as angular velocities represented as polynomials: AngularVelocitiesPolynomialProfile
• Profile expressed as angular velocities represented as harmonics: AngularVelocitiesHarmonicProfile

Harmonic guidance profiles

The functions representing the angular velocity of the satellite and the quaternion (or the Euler angles) expressing its orientation can be represented by an harmonic profile. The harmonic profile is a [MAT_TPOL_Home#HFourierSeries Fourier decomposition] of a periodic function:

[math] S_n\left(f(x)\right) = a_0(f) + \sum_{k=1}^n\left(a_k(f)\cos\left(k x {2\pi\over T}\right) + b_k(f)\sin\left(k x {2\pi\over T}\right)\right) [/math]

The period T is generally computed knowing the nodes crossing periods.

The Fourier coefficients (n > 0) of [math]f[/math] (the angular velocity / quaternions / Euler angles function) are given by :

[math] a_0(f) = {1\over T}\int_{-T/2}^{T/2} \! f(t) \, \mathrm{d}t [/math]

[math] b_0(f) = 0 [/math]

[math] a_n(f) = {2\over T}\int_{-T/2}^{T/2} \! f(t) \cos\left(n t {2\pi\over T}\right) \, \mathrm{d}t [/math]

[math] b_n(f) = {2\over T}\int_{-T/2}^{T/2} \! f(t) \sin\left(n t {2\pi\over T}\right) \, \mathrm{d}t [/math]

Polynomial guidance profiles

A polynomial guidance profile is generally composed by several segments (a segment corresponds to a time interval). Each segment contains the coefficients for the computation of a polynomial law representing the quaternion (or the Euler angles), or the angular velocity of the satellite; this polynomial law is valid only inside the time interval associated to the segment.

For a quaternion, the polynomials are defined as :

[math]Qc_k(t) = \begin{pmatrix} \displaystyle\sum_{i=0}^n {a_0}_k^i \left( t-t_k \over {\Delta t} \right) \\ \displaystyle\sum_{i=0}^n {a_1}_k^i \left( t-t_k \over {\Delta t} \right) \\ \displaystyle\sum_{i=0}^n {a_2}_k^i \left( t-t_k \over {\Delta t} \right) \\ \displaystyle\sum_{i=0}^n {a_3}_k^i \left( t-t_k \over {\Delta t} \right) \end{pmatrix}[/math]

And for the angular velocity :

[math]\Omega c_k(t) = \begin{pmatrix} \displaystyle\sum_{i=0}^n {a_x}_k^i \left( \frac{t-t_k}{\Delta t} \right) \\ \displaystyle\sum_{i=0}^n {a_y}_k^i \left( \frac{t-t_k}{\Delta t} \right) \\ \displaystyle\sum_{i=0}^n {a_z}_k^i \left( \frac{t-t_k}{\Delta t} \right) \end{pmatrix}[/math]

Getting Started

Build an harmonic profile from an attitude law

Given :

• the profile start date origin,
• the frame of expression frame,
• the Fourier series for each quaternion q0, q1, q2 and q3,
• the profile time boundaries timeInterval,

one can build an angular velocities guidance profile using the code snippet below :

profile = new QuaternionHarmonicProfile(origin, frame, q0, q1, q2, q3, timeInterval);

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