Second order sliding mode control using unit dual-quaternion dynamics with application to robotics

Abstract

This paper gives a new control framework for connected rigid bodies using unit dual quaternion dynamics. The equations of motion are obtained by direct differentiation of dual quaternion that represents the pose and position of the end effector. The novelty lies in the synthesis of torque vector in dual quaternion space. This is based entirely on the second derivative of the dual quaternion tracking error. The key contribution is in achieving finite time stability while explicitly using equations of acceleration dynamics of the underlying dual quaternion error, a concept that has not been explored before. The presented results utilize the method of modeling of serial link robots using an abstraction of inverted pendulum proposing a suitable controller for each link. A second order sliding mode control is utilized to enforce finite time stability of the error dual quaternion and its first temporal derivative. The utility of this method is demonstrated using a planar two link robot where individual generalized force vectors in dual quaternion space are synthesized with a discussion on the physical meaning of a torque defined in dual quaternion space.