Application of the Quaternion to Spacecraft's Large Angle Attitude Control

Abstract

In this paper, we study the stabilization problem of a spacecraft with three independent torque actuators via state feedback. We describe its attitude with the quaternion q_??_(q₁, q₂, q₃, q₄), ||q||²_??_q₁²+q₂²+q₃²+q₄²=1, instead of the rotation matrix R∈SO(3), which is known to have the manifold structure. The purpose of this paper is to derive a global static feedback law such that quaternion q converges to (0, 0, 0, 1).
First of all, we cover the manifold SO(3) with four local co-ordinates A: (q₁, q₂, q₃), B: (q₁, q₂, q₄-1), C: (q₁, q₂, q₄-1), D: (q₂, q₃, q₄-1). And, by means of the nonlinear control theory, we obtain four local control laws u=u_A(x), u=u_B(x), u=u_C(x), u=u_D(x) corresponding to them. But, each control law has singular points where the inputs diverge. If we adopt an idea that the control laws are switched according to the value of the quaternion in order to avoid the problem of the singular points, the control inputs are not smooth at the switching point. So, we synthesize a new control law u=q₄²u_A+q₃²u_B+q₂²u_C+q₁²u_D from local control laws. The new control law, of which each term is equal to zero at the singular points, is smooth and gives globally stable closed-loop system. Furthermore, its structure is much simpler than four local control laws. This method is applicable to both gas-jet actuators and reaction wheel actuators. The control law for gas-jet actuators is the special case of the control law derived in Ref. 5).