Mathematical Problems in Engineering

Volume 2015, Article ID 878724, 11 pages

http://dx.doi.org/10.1155/2015/878724

## Improved Path Planning and Attitude Control Method for Agile Maneuver Satellite with Double-Gimbal Control Moment Gyros

^{1}School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing 100191, China^{2}Science and Technology on Inertial Laboratory, Beijing 100191, China

Received 4 November 2014; Revised 2 February 2015; Accepted 8 February 2015

Academic Editor: Mohamed Abd El Aziz

Copyright © 2015 Peiling Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Double-gimbal control moment gyros can implement the satellite attitude maneuver efficiently. In order to reduce the energy consumption of double-gimbal control moment gyros and avoid the singularity state, an attitude maneuver path planning method is proposed by using the improved Fourier basis algorithm. Considering that the choice of the Fourier coefficients is important for the Fourier basis algorithm to converge quickly, a choosing method of the initial Fourier coefficients which can reduce the computational time of the path planning algorithm notably is proposed. Moreover, an attitude-tracking feedback controller based on the Fourier basis path planning algorithm is designed to acquire robustness. Simulation results show that the proposed path planning algorithm can implement attitude maneuver path planning in shortened time. Meanwhile, the feasibility of the attitude-tracking feedback controller which is based on the above path planning algorithm is verified in terms of the low energy consumption, high attitude-tracking precision, and the safe use of double-gimbal control moment gyros.

#### 1. Introduction

High-resolution earth observation demands the satellite to have the ability of rapid attitude maneuver. Therefore, control moment gyros (CMGs) which can output large and precise torque are ideal attitude control actuator for the agile maneuver satellite. The main problem of using CMGs is the existence of singularity state, at which CMGs cannot produce torque in a certain direction. According to the gimbal numbers, CMGs can be divided into two classes, including single-gimbal CMGs (SGCMGs) and double-gimbal CMGs (DGCMGs). The singularity avoidance steering law is very important for SGCMGs to fulfill the requirement of the rapid attitude maneuver. Compared with SGCMG, DGCMG has more than one degree of freedom, and its angular momentum envelop is nearly spherical. Therefore, it is believed that DGCMGs can implement the rapid attitude maneuver with high efficiency.

Existing attitude control logics with CMGs are composed of attitude control laws and CMG steering laws [1, 2]. According to attitude tracking errors, the desired attitude control torque is computed by the attitude control laws. The objective of the CMG steering laws is to avoid singularity states. A large-angle attitude maneuver control law based on nonlinear methods is proposed in [1], and a singularity robustness plus null motion steering law is designed to ensure DGCMGs against singularity. References [3, 4] investigate on the singularity avoidance steering law specially. A modified singular-direction avoidance steering law of SGCMGs is introduced in [3]. Kennel presents the vector distribution steering law for parallel-mounted DGCMGs in [4], which makes use of redundant degrees of freedom of DGCMGs.

However, when the CMG steering law is separated from the attitude control law, CMG singularity problem and energy consumption cannot be considered in the computing process of attitude control torque. In addition, there exist singularity states which cannot be avoided by CMG steering laws. To deal with the above problems, the attitude control and CMG steering law should be integrated together as a whole control system.

An effective method of integrating the attitude control law and CMG steering law together is to plan the attitude maneuver path according to the satellite maneuver characteristic. Recent path planning algorithms mainly include Legendre pseudospectral algorithm [5, 6], generic algorithm (GA) [7, 8], and Fourier basis algorithm (FBA) [9, 10]. By using an integral cost function consisting of energy consumption and time cost index, [5] applies the Legendre pseudospectral algorithm to the optimal path planning. Reference [6] reviews key theoretical results of the pseudospectral optimal method which are important for successful flights. An energy optimal attitude maneuver path is planned by using the improved GA [7].

The Legendre pseudospectral algorithm and generic algorithm have many optimal parameters, thus resulting in complex computation. Moreover, the two algorithms generate time-discrete control inputs, which need to be approximated by continuous functions after the path planning. The rapid attitude maneuver control systems of agile satellites are real-time systems. Therefore, the Legendre pseudospectral algorithm and generic algorithm are not very suitable for the rapid attitude maneuver control. Regarding the gimbal rates of SGCMGs as control inputs, [9, 10] proposes a continuous optimal path planning method based on the FBA. The idea of FBA is to approximate control inputs by applying a Fourier series. The path planning method based on FBA is advantageous in terms of the computational time, and the path planning results are continuous. However, the Fourier coefficient vector is initialized by some random process in [9, 10], and the algorithm cannot converge rapidly. The choosing method of the initial Fourier coefficient has not been introduced yet. Moreover, there exists no Fourier basis path planning method which is proposed to investigate the attitude control of agile maneuver satellite with DGCMGs. Besides, since the disturbances and attitude tracking errors have not been considered in the path planning, the attitude tracking controller which can achieve the system robustness needs to be improved [5, 9].

This paper fills the above gap in the existing literature. The path planning and attitude control law of the agile maneuver satellite with DGCMGs are integrated together. The energy consumption and terminal state errors of double-gimbal control moment gyroscopes are considered at the same time. An improved FBA is proposed for the attitude maneuver path planning for agile maneuver satellite with DGCMGs. In order to shorten the computational time of the path planning, a choosing method of the initial Fourier coefficient is introduced. Meanwhile, the attitude tracking feedback controller based on the path planning method is proposed by applying a sliding mode controller. Simulations are carried out to verify the availability of the introduced path planning method and the corresponding attitude tracking controller.

#### 2. Model of the Satellite Attitude Control System

As shown in Figure 1, the two parallel-mounted DGCMGs are used as the attitude control actuators. , and are unit vectors aligned along the axes of the satellite body frame coordinate. The orthonormal set of unit vectors is used to describe the orientation of the th DGCMG relative to the spacecraft body frame, where is fixed along the inner gimbal axis, is fixed along the outer gimbal axis, and is fixed along the wheel spin axis. and denote the inner gimbal angle and outer gimbal angle of the th DGCMG, respectively. and represent the inner gimbal rate and outer gimbal rate of the th DGCMG, respectively. Assuming that denote the wheel angular momentum of the DGCMG, the total angular momentum of two parallel-mounted DGCMGs expressed in the satellite body frame coordinate can be written as