Control Law Design Based on Geometric Algebra

Abstract

A position and attitude control law is developed using geometric algebra (GA). GA is a powerful representational and computational system for geometry. The rigid body motion can be represented by the versor product in GA framework. Using the kinematics of the motor (the versor which represents the rigid body motion in GA), the control law of the rigid body motion can be developed. This paper provides a GA-based position and attitude control law by using the negative feedback of the motor. The stability of the control law is validated by the Lyapunov theorem and the numerical simulation.

Position and Attitude Tracking Control Law Design Using Geometric Algebra

“…The proportional control law which transfers the rigid body to the equilibrium point can be formulated in GA as [7] 1…”

“…Suppose that the ideal position and attitude of a rigid body is d M , and the actual position and attitude of it is M , the error between them can then be written as (12) It is shown that the system is stable when using the negative feedback of the motor, i.e. [7]…”

“…Motived by Han and Wang, we have presented a GA-based proportional control law by using the kinematics of the motor [7]. The logarithm of the motor is utilized as the negative feedback.…”

“…However, in many ways, the equilibrium is not a point, but a trace. Therefore, the tracking control law is more practical than the one presented in [7]. As a subsequent work, this paper intends to build a GA-based tracking control law.…”