American Control Conference Albuquerque NM June Exponentially Convergent Controllers for n Dimensional Nonholonomic Systems in Power Form

This paper introduces a new method for constructing ex ponentially convergent control laws for n dimensional nonholo nomic systems in power form The methodology is based on the construction of a series of invariant manifolds for the closed loop system under a linear control law A recursive algorithm is presented to derive a feedback controller which drives the system exponentially to the origin A numerical example illus trates the proposed theoretical developments Introduction Nonholonomic control systems commonly arise from me chanical systems when non integrable constraints are imposed on the motion e g velocity constraints which can not be in tegrated to generate constraints on the con guration space Examples include a rolling disk mobile robots and un deractuated symmetric rigid spacecraft One of the main reasons these systems have attracted much attention in the past few years is that they represent inherently nonlinear systems in a certain sense For example these systems are controllable but not stabilizable by a smooth static or dynamic state feedback control laws A number of approaches have been proposed to solve the stabilization problem for nonholonomic systems These methodologies can be broadly classi ed as discontinuous time invariant stabilization and time varying usually smooth sta bilization The non smoothness of time invariant feedback con trols is a consequence of the structural properties of the system Stabilization results using non smooth time invariant con trol laws have been proposed in References deal with the attitude stabilization of underactuated spacecraft by developing non smooth time invariant control laws Piece wise continuous stabilization controller have been reported in A nonsmooth transformation was used to develop time invariant exponential convergent controller in Samson in showed how to asymptotically stabilize a mobile robot to a point using time varying smooth state feedback Coron in proved that all controllable driftless systems could be sta bilized to an equilibrium point using smooth periodic time varying feedback References and deal with the construction of time varying control laws for several nonholo nomic systems Hybrid feedback time varying control laws are constructed for a class of cascade nonlinear systems in which could also be used for stabilizing a class of nonholonomic systems as well as for solving tracking problems References and develop time varying control laws of exponen This work was supported in part by the National Science Foundation under Grant CMS tial convergence with respect to homogeneous norms Finally develops nonsmooth time varying feedback control laws which guarantee global asymptotic stability with exponential convergence about an arbitrary con guration For a more com prehensive review of all the recent advances in the control of nonholonomic systems the interested reader may consult The analysis of dynamic systems is often simpli ed by the introduction of canonical or normal forms that is systems of equations which all systems in a given family are equiv alent to For nonholonomic systems there are two normal forms which have been used extensively in the past namely the chained form and the power form The mathematical model of an n dimensional nonholonomic system in power form with two inputs can be described as