Feedback Control of a Class of Nonholonomic Hamiltonian Systems

Abstract

Feedback control of nonholonomic systems has always been problematic due to the nonholonomic constraints that limit the space of possible system velocities. This property is very basic, and Brockett proved that a nonholonomic system cannot be asymptotically stabilized by a time-invariant smooth feedback. This thesis presents a novel way of controlling a special class of nonholonomic Hamiltonian systems. The basic idea is to split the configuration coordinates in two; a primary part that we wish to asymptotically stabilize, and a secondary part that not necessarily has to be stabilized, but is useful when controlling the primary part. The secondary part is introduces as the integral of so-called kinematic inputs. The kinematic inputs have the property that they cannot change the amount of energy in the system, i.e., the Hamiltonian function is invariant with respect to the kinematic inputs. The resulting nonholonomic Hamiltonian system with kinematic inputs shares many of the properties of the classical Hamiltonian system, and some of the methods involved in controlling classical systems are proved to also apply to the augmented system. The extra degree of freedom provided by the kinematic inputs turns out to be useful when stabilizing the nonholonomic system. If the system is properly actuated it is possible to asymptotically stabilize the primary part of the configuration coordinates via a passive energy shaping and damping injecting feedback. The feedback is smooth and time-invariant, but since it does not asymptotically stabilize the secondary part of the configuration coordinates, it does not violate Brockett’s obstruction. The results from the general class of nonholonomic Hamiltonian systems with kinematic inputs are applied to a real implementation of a four wheel steered, four wheel driven nonholonomic robotic vehicle, where the velocity of the steering motors are assumed to satisfy the conditions of proper kinematic inputs. The proposed controller is general enough to achieve both global asymptotic stabilization and path tracking for the robot. To improve the operation of the closed loop system some extensions are provided: integral action for asymptotic stabilization under the influence of disturbances, and an adaptive damping scheme ensuring that the robot travels at a predefined speed when tracking a path. Both of these extensions are defined in the framework of Hamiltonian systems.