Exponentially Stable First Order Control on Matrix Lie Groups
This provides a solution for precise control in robotics, particularly for Cartesian velocity control, but appears incremental as it builds on existing Lie group methods.
The paper tackles the problem of trajectory tracking for systems on matrix Lie groups, such as robot manipulators, by presenting a first-order controller that achieves global exponential tracking on specific groups like SO(n), SE(n), and GL(n, C), and local exponential tracking on all matrix Lie groups, with demonstrations on simulations and a 7-DOF Sawyer robot arm.
We present a novel first order controller for systems evolving on matrix Lie groups, a major use case of which is Cartesian velocity control on robot manipulators. This controller achieves global exponential trajectory tracking on a number of commonly used Lie groups including the Special Orthogonal Group SO(n), the Special Euclidean Group SE(n), and the General Linear Group over complex numbers GL(n, C). Additionally, this controller achieves local exponential trajectory tracking on all matrix Lie groups. We demonstrate the effectiveness of this controller in simulation on a number of different Lie groups as well as on hardware with a 7-DOF Sawyer robot arm.