The structure and control of nonholonomic
systems with symmetry are studied through study of the bicycle
and the rolling disk. Control of the bicycle is a rich problem
offering a number of considerable challenges of current research
interest in the area of mechanics and nonlinear control. It is
an underactuated, nonminimum phase system subject to nonholonomic
contact constraints associated with the rolling constraints on
the front and rear wheels. Like the rolling disk it is also, when
considered to traverse flat ground, a system subject to symmetries;
its Lagrangian and constraints are invariant with respect to translations
and rotations in the ground plane. We have succeeded in developing
controllers which, using steering and rear-wheel torque, cause
a simple model of a riderless bicycle to recover its balance from
a near fall as well as converge to a time parameterized path in
the ground plane. We hope to extend and refine these results while
gaining further insight into the nature of nonholonomic and underactuated
systems.
Neil H. Getz, "Control of Nonholonomic Systems with Dynamically
Decoupled Actuators", 32nd Conference
on Decision and Control, 15-17 December 1993. San Antonio.
A compensator is derived for a class of nonholonomic systems incorporating both dynamically coupled and uncoupled actuators. The compensator affords exponential convergence to smooth trajectories in a reduced state space.
Neil H. Getz, "Control of Balance for a Nonlinear Nonholonomic
Non-Minimum Phase Model of a Bicycle",
American Control Conference, July 1994 BaltimoreA feedback control law is derived that causes a nonlinear, nonholonomic, non-minimum phase model of a riderless powered two-wheeled bicycle to stably track arbitrary smooth trajectories of roll-angle and non-zero rear-wheel velocity.
Neil H. Getz, "Path Tracking with Balance for a Nonlinear
Nonholonomic Non-Minimum Phase Model of a Bicycle",
34th IEEE Conference on Decision and Control, December
1995 New Orleans.A control is derived whereby a nonholonomic nonlinear nonminimum phase model of a bicycle can be made to track time-parameterized trajectories in the plane while maintaining balance. The derivation of the controller utilizes recently introduced theories of dynamic inversion and a new method of output regulation for nonminimum phase nonlinear systems. A desired path tracking error-dynamics for a kinematic model of a bicycle is is first chosen. There exists a roll-angle manifold, a graph over the tracking error phase space, such that stabilization of the roll-angle to the image of the error state in this manifold provides path tracking with internal stability (balance) for the dynamic bicycle. Dynamic inversion is incorporated into the controller in order to provide a continuous estimation of the image of the path-tracking error state on the roll-angle manifold. Previous results on roll-angle tracking for the bicycle then allow stabization of the image point.
Neil H. Getz and Jerrold
E. Marsden, "Control for an
Autonomous Bicycle", IEEE
International Conference on Robotics and Automation, 21-27
May 1995 Nagoya, Japan.The control of nonholonomic and underactuated systems with symmetry is illustrated by the problem of controlling a bicycle. We derive a controller which, using steering and rear-wheel torque, causes a model of a riderless bicycle to recover its balance from a near fall as well as converge to a time parameterized path in the ground plane. Our construction utilizes new results for both the derivation of equations of motion for nonholonomic systems with symmetry, as well as the control of underactuated robotic systems.