N.H. McClamroch

Developments in nonholonomic control problems

Provides a summary of recent developments in control of nonholonomic systems. The published literature has grown enormously during the last six years, and it is now possible to give a tutorial presentation of many of these developments. The objective of this article is to provide a unified and accessible presentation, placing the various models, problem formulations, approaches, and results into a proper context. It is hoped that this overview will provide a good introduction to the subject for nonspecialists in the field, while perhaps providing specialists with a better perspective of the field as a whole. The paper is organized as follows: introduction to nonholonomic control systems and where they arise in applications, classification of models of nonholonomic control systems, control problem formulations, motion planning results, stabilization results, and current and future research topics.

Control and stabilization of nonholonomic dynamic systems

A class of inherently nonlinear control problems has been identified, the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system. Models are presented for mechanical systems with nonholonomic constraints represented both by differential-algebraic equations and by reduced state equations. Control issues for this class of systems are studied and a number of fundamental results are derived. Although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback, a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested. >

Feedback stabilization and tracking of constrained robots

Mathematical models for constrained robot dynamics, incorporating the effects of constraint force required to maintain satisfaction of the constraints, are used to develop explicit conditions for stabilization and tracking using feedback. The control structure allows feedback of generalized robot displacements, velocities, and the constraint forces. Global conditions for tracking, based on a modified computed-torque controller and local conditions for feedback stabilization, using a linear controller, are presented. The framework is also used to investigate the closed-loop properties if there are force disturbances, dynamics in the force feedback loops, or uncertainty in the constraint functions. >

Dynamics and control of a class of underactuated mechanical systems

This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable.

Rigid-Body Attitude Control

Rigid-body attitude control is motivated by aerospace applications that involve attitude maneuvers or attitude stabilization. The set of attitudes of a rigid body is the set of 3 X 3 orthogonal matrices whose determinant is one. This set is the configuration space of rigid-body attitude motion; however, this configuration space is not Euclidean. Since the set of attitudes is not a Euclidean space, attitude control is typically studied using various attitude parameterizations. Motivated by the desire to represent attitude both globally and uniquely in the analysis of rigid-body rotational motion, this article uses orthogonal matrices exclusively to represent attitude and to develop results on rigid-body attitude control. An advantage of using orthogonal matrices is that these control results, which include open-loop attitude control maneuvers and stabilization using continuous feedback control, do not require reinterpretation on the set of attitudes viewed as orthogonal matrices. The main objec tive of this article is to demonstrate how to characterize properties of attitude control systems for arbitrary attitude maneuvers without using attitude parameterizations.

Singular systems of differential equations as dynamic models for constrained robot systems

There are many robot system configurations where external contact forces on a robot play an important role in the system dynamics. Such contact forces arise due to constraints on the motion of the robot. Mathematical models of such robot system naturally give rise to a mathematical system of differential equations and algebraic equations, that can be viewed as a singular system of differential equations. Such singular models are developed for several robot system configurations. A general reduction approach is presented that can, in principle, form the basis for the development of planning and feedback control schemes for such robot system configurations.

Control of mechanical systems with classical nonholonomic constraints

A theoretical framework for the control of mechanical systems with m>or=1 classical nonholonomic constraints is established. In particular, the authors emphasize certain control properties for mechanical systems with nonholonomic constraints that have no counterpart in systems with holonomic constraints are emphasized. Conditions for smooth stabilization, of an m-dimensional equilibrium manifold are presented, and it is demonstrated that smooth stabilization of a single equilibrium solution is not possible. The development is illustrated using two physical examples: the control of a knife edge moving on a plane surface and the control of a wheel rolling without slipping on a plane surface. The physical significance of the theoretical results is described for these examples.<<ETX>>

Hybrid feedback laws for a class of cascade nonlinear control systems

A stabilization problem for a class of nonlinear control systems is considered. Systems in this class can be viewed as a cascade connection of a linear time-invariant subsystem, a nonlinear time-periodic static subsystem, and an integrator. Hybrid logic-based feedback controllers are constructed to globally stabilize these systems to the origin. The controllers operate by switching between various time-periodic control functions at discrete-time instants. As specific applications, we consider stabilization of nonholonomic control systems in power form to the origin and stabilization of trajectories for a class of nonlinear control systems. Numerical examples of global stabilization and tracking are reported.

Nonsmooth stabilization of an underactuated unstable two degrees of freedom mechanical system

This paper studies a specific mechanical example that is representative of a class of underactuated, weakly coupled, unstable mechanical systems that are exceptionally difficult to stabilize. In particular, systems in this class are not stabilizable using static smooth feedback but are stabilizable using nonsmooth feedback. Although similar purely theoretical developments have been previously presented, we emphasize the physical basis and physical implications of the theoretical conclusions in the context of a specific example. The development in this paper is limited to a specific physical example, but the approach is applicable to a wide class of underactuated, weakly coupled, unstable mechanical systems.

Asymptotic Smooth Stabilization of the Inverted 3-D Pendulum

The 3-D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. The 3-D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. The contribution of this paper is that two fundamental stabilization problems for the inverted 3-D pendulum are posed and solved. The first problem, asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold, is solved using smooth and globally defined feedback of angular velocity and attitude of the 3-D pendulum. The second problem, asymptotic stabilization of the inverted equilibrium manifold, is solved using smooth and globally defined feedback of angular velocity and a reduced attitude vector of the 3-D pendulum. These control problems for the 3-D pendulum exemplify attitude stabilization problems on the configuration manifold SO(3) in the presence of potential forces. Lyapunov analysis and nonlinear geometric methods are used to assess global closed-loop properties, yielding a characterization of the almost global domain of attraction for each case.