Spacecraft Nonlinear Attitude Tracking Control with Non-constant Rate Command

Abstract

A nonlinear tracking control algorithm based on an attitude error quaternion is developed and demonstrated within this paper. The solutions and the equilibrium points of the closed-loop system (which is time-varying and nonlinear) are obtained. In order to analyze the stability of the system and its tracking performance two different forms of perturbation dynamics with seven state variables are introduced. Local stability and performance analysis shows that the eigenvalues of the linearized perturbation dynamics are determined only by the gain matrices within the control algorithm and the inertia matrix. The existence of a globally stable tracking controller is proved using a Lyapunov function. Simulation results show that the spacecraft can track the commanded attitude and rate quickly even for a non-zero acceleration rate command. INTRODUCTION Attitude controllers for spacecraft have been studied for many years. In this paper a quaternion feedback regulator is studied for accomplishing spacecraft eigenaxis rotational maneuvers as an extension to Ref. [2]. The stability analysis of the control system based on the Lyapunov method is presented in this reference, but the control algorithm is only valid for rest-to-rest maneuvers. References 3 and 4 also discuss quaternion based attitude controllers and the problems encountered in their design. In Ref. [5], an adaptive tracking control law that does not require knowledge of the inertia or center of mass of the spacecraft is developed. The * Principal Systems Engineer, 100 University Dr., Fairmont, WV 26554. Email: zzhou@ivv.nasa.gov. Member AIAA. † Associate Professor, 2120 Learned Hall, Lawrence, KS 66045. Email: rcolgren@ku.edu. Associate Fellow AIAA. global stabilization of a specified inertial pointing direction is studied using the inverse optimality method in Ref. [6]. Chevalley’s exponential coordinates for a Lie group are used to represent the points in the space. In Ref. [1], a quaternion-based nonlinear tracking control law is studied. The control algorithm tracks the desired attitude command via instantaneous eigenaxis rotation. The control system consists of an inner velocity loop that tracks the desired rate command and an outer attitude loop that tracks the desired attitude command. Various nonlinear quaternion-based control laws are studied in Ref. [3]. A general analytic framework based on the coordinate independent vectorial algebra for the stability analysis is presented. However, the existence of the solutions and the equilibrium points of the closed-loop nonlinear control system are not investigated. In addition, an explicit stability proof is not given, and no simulation results are presented to show the tracking performance of the control law. The second section of this paper presents the spacecraft dynamics and the kinematics equation of the commanded attitude and attitude rate. These represent the dynamics of the system to be controlled. In the third section of this paper the solutions and the equilibrium points of the closed-loop nonlinear control system are investigated; two different forms of perturbation dynamics are introduced. Local stability and performance analysis with constant attitude rate command are conducted; global stability of the system is proved using a Lyapunov function. A design example and simulation results are presented to demonstrate the method. Conclusions are given in the final section. AIAA Guidance, Navigation, and Control Conference and Exhibit 16 19 August 2004, Providence, Rhode Island AIAA 2004-5128 Copyright © 2004 by . Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. BACKGROUND AND PROBLEM STATEMENT The dynamic equations of motion for spacecraft are described as follows : u J u J J + Ω = + × − = ω ω ω ω ) ( & (1) In this equation J is the inertia matrix, ( ) 3 2 1 , , ω ω ω ω = is the angular velocity vector, ( ) u u u u 3 2 1 , , = is the control torque vector, and Ω is a skew-symmetric matrix defined by           − − − − ≡ Ω 0 0 0