In this paper we consider under-actuated, drift-free, invariant systems on matrix Lie groups and show how motion control results for nilpotent systems can be extended to invariant systems on non-nilpotent Lie groups by applying the method of nilpotentization. An algorithm for computing nilpotent approximations for invariant systems on Lie groups is presented. These approximations are used to construct a nilpotent model system for nilpotentization of a generic system in the class defined above. Using this method the corresponding feedback and state transformations can be computed in cases to which none of the previously known sufficient conditions apply. New applications of this feedback equivalence are introduced by showing how approximate tracking control laws and exponentially stabilizing feedback control laws can be improved by making use of nilpotentization. The proposed methods are illustrated with a kinematic model of an under-actuated rigid body.
Journal Paper (postcript, 481k, 28 pages)
This paper develops a method for constructing nilpotent approximations for local representations of invariant systems on matrix Lie groups via a simple operation on the structure constants of the associated Lie algebra. The crucial role such nilpotent approximations play for the problem of feedback nilpotentization is discussed. The presented ideas are illustrated with an example modeling the kinematics of an under-actuated rigid body.
Conference Paper (postcript, 208k, 6 pages)
We consider under-actuated, drift-free, invariant systems on matrix Lie groups and show how motion control results for nilpotent systems can be extended to invariant systems on non-nilpotent Lie groups by applying the technique of nilpotentization. A constructive procedure is presented to compute the required transformations for local representations of systems on three-dimensional Lie groups. Special emphasis is put on the choice of coordinates which accounts for simple transformation equations and a maximal domain of validity. It is shown how exponentially stabilizing feedback control laws can be constructed using these transformations.
Conference Paper (postcript, 175k, 4 pages)
In this dissertation we study the control of nonholonomic systems defined by invariant vector fields on matrix Lie groups. We make use of canonical constructions of coordinates and other mathematical tools provided by the Lie group setting. An approximate tracking control law is derived for so-called chained form systems which arise as local representations of systems on a certain nilpotent matrix group. After studying the technique of nilpotentization in the setting of systems on matrix Lie groups we show how motion control laws derived for nilpotent systems can be extended to nilpotentizable systems using feedback and state transformations. The proposed control laws exhibit highly oscillatory components both for tracking and feedback stabilization of local representations of nonholonomic systems on Lie groups. Applications to the control and analysis of the kinematics of mechanical systems are discussed and numerical simulations are presented.
PhD thesis (postcript, 1070k, 121 pages)
In this paper we extend the concept of approximate tracking in the high-frequency limit to non-nilpotent three-dimensional matrix Lie groups by making use of feedback nilpotentization for the local representations of these systems. Further it is shown how to convert these tracking controls involving a state-feedback to an open-loop control law, which can be interpreted as an approximate inverse of the original system.
Conference paper (postscript, 335k, 9 pages)
This paper studies open-loop tracking and feedback stabilization for nonholonomic systems. Using the concept of approximate inversion, results for drift-free, left-invariant systems on specific matrix Lie groups are presented.
Conference paper (postscript, 460k, 5 pages)
A wide range of dynamical systems from fields as diverse as mechanics, electrical networks and molecular chemistry can be modeled by invariant systems on matrix Lie groups. This paper introduces control systems on matrix Lie groups along with their exponential representations and studies open-loop tracking and feedback stabilization for these systems in the presence of nonholonomic constraints. Using the concept of approximate inversion, results for drift-free, left-invariant systems on specific matrix Lie groups are presented.
Technical Report (postscript, 700k, 23 pages)
Latest Update: March 23, 1998