  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.
