Quaternion Sliding Variables in Manipulator Control
Brett T. Lopez, Jean-Jacques Slotine
TL;DR
This work addresses robust orientation control for a manipulator end-effector by introducing two quaternion-based sliding variables on $\mathbb{S}^3 \times \mathbb{R}^3$ that guarantee global exponential convergence and prevent unwinding. The approach handles both local and global representations of the angular velocity, using the error quaternion $q_e = q_d^* \otimes q$, and leads to a task-space torque controller that combines a quaternion sliding variable with a position sliding variable to achieve simultaneous pose and orientation tracking. Key contributions include explicit definitions of a local ($s_q$) and a global ($\mathfrak{s}_q$) sliding variable, their invariant error dynamics proofs, and an inverse kinematics torque control law that achieves global exponential convergence under non-singular Jacobians. The results have practical impact for safe and efficient robotic manipulation, enabling full-envelope orientation tracking without unwinding and without reliance on Euler-angle representations, with potential extensions to dual-quaternion formalisms.
Abstract
We present two quaternion-based sliding variables for controlling the orientation of a manipulator's end-effector. Both sliding variables are free of singularities and represent global exponentially convergent error dynamics that do not exhibit unwinding when used in feedback. The choice of sliding variable is dictated by whether the end-effector's angular velocity vector is expressed in a local or global frame, and is a matter of convenience. Using quaternions allows the end-effector to move in its full operational envelope, which is not possible with other representations, e.g., Euler angles, that introduce representation-specific singularities. Further, the presented stability results are global rather than almost global, where the latter is often the best one can achieve when using rotation matrices to represent orientation.