Exponentially Stable First Order Control on Matrix Lie Groups

TL;DR

This paper introduces a new first order control method for systems on matrix Lie groups, enabling exponential trajectory tracking globally and locally, demonstrated through simulations and real robot experiments.

Contribution

A novel first order controller for matrix Lie groups that guarantees exponential trajectory tracking both globally and locally, applicable to robotic manipulators.

Findings

Achieves global exponential trajectory tracking on SO(n), SE(n), and GL(n,C).

Demonstrates effectiveness in simulation and on a 7-DOF Sawyer robot.

Provides local exponential tracking on all matrix Lie groups.

Abstract

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.