Synthesis of Control of Spatial Motion of a Rigid Body Using Dual Quaternions

In this paper, we develop a new method of analytical design and control of the spatial motion of a rigid body (in particular, a spacecraft considered as a rigid body) in a nonlinear dynamic formulation using dual quaternions (Clifford biquaternions). The control provides the asymptotic stability in general of any selected programmed motion in the inertial coordinate system and the desired dynamics of the controlled motion of the body. To build control laws, new biquaternion differential equations of perturbed spatial motion of a rigid body are proposed, in which unnormalized biquaternions of finite displacements, biquaternions of angular and linear velocities of the body and accelerations with nonzero dual scalar parts are used. The concept of solving the inverse problems of dynamics, the feedback control principle, and the approach based on the reduction of the equations of perturbed body motion to linear stationary differential forms of the selected structure invariant with respect to any selected programmed motion due to the corresponding choice of dual nonlinear feedbacks in the proposed biquaternion control laws is presented. Analytical solutions of biquaternion differential equations are designed to describe the dynamics of controlling the spatial body motion using the proposed biquaternion control laws. The properties and patterns of such control are analyzed.