Angular velocity stabilization of underactuated rigid satellites based on energy shaping

Highlights

• A new angular velocity stabilization strategy based on energy shaping is proposed.

• A feasible sufficient condition for solving the matching condition is derived.

• Controllers are designed for three different inertia matrix structures.

• Three nonlinear control laws for underactuated rigid satellites are obtained.

• Global asymptotic stability of the closed-loop satellite systems is proved.

Abstract

Angular velocity stabilization problem for underactuated rigid satellites with only two independent control inputs is investigated, and a novel stabilization control strategy based on energy shaping is presented in this paper. Firstly, a desired closed-loop system with a specific structure is constructed for the control objective. The matching condition and controller structure are determined by equating the underactuated satellite system with the desired system. Moreover, a feasible sufficient condition satisfying the matching condition is derived. By solving the sufficient condition according to three different structures of the satellite inertia tensor matrix, three nonlinear smooth stabilization control laws are obtained, and global asymptotic stability of the related closed-loop system is rigorously proved. Simulation results show effectiveness of the proposed control method.

Introduction

The three-axis stable satellite is a typically full-actuated system with three independent control torques. However, if a part of satellite actuators fails such that complete three-axis control torques cannot be provided, the satellite becomes an underactuated system, such as the earth observation satellite UoSAT-12, the far ultraviolet spectroscopic explorer, and so on. In order to avoid out of control for the satellite at this time and improve reliability, redundancy and service life of the entire satellite system, many scholars and engineers in the control field have been studying and exploring angular velocity stabilization problem of underactuated satellites using a variety of methods.

Brockett first demonstrated that angular velocity of underactuated satellites can be asymptotically stabilized only by two independent control inputs in [1]. Aeyels, in [2], also obtained similar results and got a smooth control law that mainly relied on the central manifold theorem. The authors of [3] and [4] dealt with the issue of angular velocity stabilization for a satellite with only one control input by extending the conclusions in [1] and [2]. After that, most researchers focused their interest on angular velocity stabilization of underactuated satellites with two control inputs. For instance, in [5] and [6], a state feedback controller based on the Lyapunov theory and the LaSalle invariance principle was designed to stabilize the angular velocity. In [7] and [8], asymptotic stabilization of the angular velocity was realized via output feedback controllers. In [9], a robust stabilization controller was designed to realize semi-global asymptotic stabilization of the angular velocity. In [10], a line-of-sight control technique was applied to stabilize angular velocity under the assumption of zero angular momentum. In [11], a discontinuous control law was designed based on variable structure control theory such that the angular velocity converges to the origin within a finite time. In [12] and [13], asymptotic stabilization of the angular velocity with exponential convergence rate was realized by use of the terminal sliding mode theory and a discontinuous coordinate transformation, respectively. Besides, some scholars have studied this problem through newly proposed energy shaping method. For example, inspired by Blankenstein et al. [14], Ortega et al. [15], Chang et al. [16] and Bloch et al. [17], which constructively presented basic idea of energy shaping method and design steps to obtain controllers by solving matching conditions, the authors in [18] and [19] proposed an energy matching control strategy under the Hamilton framework to stabilize the angular velocity. In [20], the authors also obtained a stabilization controller with a similar method. They all constructed a desired controlled Hamilton system and derived some feasible matching conditions in order to obtain a feedback control law such that the closed-loop system can be asymptotically stabilized to the origin. But, in their control strategies, the partial differential equations in the matching condition must be solved, resulting in complex controller design process. In summary, among the existing methods mentioned above, most of them can only be applied to the situation where inertia tensor matrix of underactuated satellites is a diagonal matrix. However, when the inertial tensor matrix is a non-diagonal matrix, how to stabilize the angular velocity of the underactuated satellite with two control inputs is still a challenging problem.

In recent years, stabilization controller design methods proposed from energy perspective are aimed at underactuated Lagrange systems and Hamilton systems, which need both kinetic energy shaping and potential energy shaping, such as controlled Lagrangian method [21], [22], [23], interconnection and damping assignment passivity-based control [24], [25], distributed passivity-based control [26] and so on. But there is a little research on underactuated Euler systems. Thus, in this paper, considering an underactuated rigid satellite with only two independent control inputs, we propose a kinetic energy shaping strategy for stabilizing its angular velocity. Main contributions of this paper are listed as follows:

• A new controller design method based on kinetic energy shaping is presented. The underactuated satellite system is divided to three different situations according to structure of its inertia tensor matrix. For each situation, the process of obtaining the control law by solving the matching condition is given in detail and global asymptotic stability of the closed-loop system is rigorously proved. It is worth emphasizing that the proposed method is applicable to all possible structures of the inertia tensor matrix, while the methods in [2][5][6][19] can only be used when the inertia tensor matrix is a diagonal matrix.

• A desired closed-loop system with a specific structure is constructed and a feasible sufficient condition satisfying the matching condition is derived. The advantage of the sufficient condition is that it entirely consists of algebraic equations so that the difficulty of solving the matching condition is greatly reduced. Nevertheless, for the same energy-based approaches in [18], [19], [20], all obtained matching conditions include partial differential equations needed to be solved.

This paper is organized as follows. In Section 2, the dynamical model of an underactuated rigid satellite and control objective are given. The matching condition and controller structure are determined in Section 3. In Section 4 the derivation process of the controller and the stability proof of the closed-loop system are presented in detail. In Section 5 simulation results verify validity of the proposed controller, and finally Section 6 gives conclusions and future research direction.