Classical Nonholonomic Constraints Research Articles

Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.

Quantizing conditionally variational nonholonomic systems

In this paper we present a quantization procedure for a class of nonholonomic systems—briefly, mechanical systems subject to nonintegrable constraints on the velocities—whose reduced mechanics is Hamiltonian. We illustrate the theory developed through several examples, one of which demonstrates how the classical nonholonomic constraints are sometimes manifested quantum mechanically as a shift in the ground state energy of the system.

Motion/force tracking control of nonholonomic mechanical systems via combining cascaded design and backstepping

This paper considers motion/force tracking control of a class of Lagrange mechanical systems with classical nonholonomic constraints. A tracking control method is proposed by combining cascaded methods and backstepping techniques. The main results of this paper include three parts: (1) error dynamics between the kinematic system and the desired paths are transformed into a cascaded system consisting of two subsystems and an interconnection function; (2) under the framework of cascaded methods, virtual controllers for the subsystems are designed to stabilize the error dynamics; (3) the tracking controller is designed for the overall mechanical systems using backstepping techniques.

Burst Oscillations in the Accelerating Bicycle

The purpose of this paper is to study the dynamics of the accelerating bicycle. It is shown that time-scale separation can be used to study the oscillatory characteristics of the accelerating machine using time-invariant models. These models are used to explain practically observed wobble-mode bursting oscillations that are associated most frequently with down-hill riding. If the vehicle is cornering under constant acceleration, at a fixed roll angle, it is shown that for low values of acceleration (and braking), it follows closely a logarithmic spiral shaped trajectory. The studies presented are facilitated by a novel adaptive control scheme that centers the machine’s trajectory on any arbitrary point in the ground plane. The influences of cambered road surfaces are also investigated. The bicycle model employed is an extension of that originally developed by Whipple, and comprises two road wheels and two laterally-symmetric frame assemblies that are free to rotate relative to each other along an inclined steering axis. For the most part, the front frame is treated as being flexible and the bicycle is fitted with force generating road tires, rather than classical nonholonomic rolling constraints. This research provides the ground work required for generating more complex dynamic models for high-performance motorcycle studies.

Adaptive motion tracking control of uncertain nonholonomic mechanical systems including actuator dynamics

The design of position tracking control of mechanical systems with classical nonholonomic constraints incorporating actuator dynamics is discussed. An adaptive controller is designed at the actuator level and the controller guarantees that the configuration state of the system tracks the desired trajectory. A detailed numerical example is presented to illustrate the developed method.

Robust adaptive motion/force tracking control of uncertain nonholonomic mechanical systems

The position/force tracking control of Lagrangian mechanical systems with classical nonholonomic constraints is addressed in this paper. The main feature of this paper is that 1) control strategy is developed at the dynamic level and can deal with model uncertainties in the mechanical systems; 2) the proposed control law ensures the desired trajectory tracking of the configuration state of the closed-loop system; 3) the tracking error of constraint force is bounded with a controllable bound; and 4) a global asymptotic stability result is obtained in the Lyapunov sense. A detailed numerical example is presented to illustrate the developed method.

ROBUST HYBRID CONTROL OF UNCERTAIN CONSTRAINED MECHANICAL SYSTEMS

Robust hybrid controller design is presented in this paper for motion/force control of mechanical systems subjected to a set of holonomic or classical nonholonomic constraints and in the presence of uncertainties about plant parameters. A unified and systematic procedure is employed to derive the controllers for both holonomic and nonholonomic constrained mechanical systems, respectively. The proposed controllers can guarantee the system motion asymptotically converges to the desired manifold, and the force tracking errors to be bounded. Numerical simulation has been done to show the effectiveness of the proposed controllers.

Switched mode feedback control laws for nonholonomic systems in extended power form

A class of nonholonomic control systems in extended power form is studied. It is demonstrated that under appropriate assumptions Lagrange's equations, including classical nonholonomic constraints, can be transformed into the extended power form. A switched mode feedback controller is used to obtain global convergence of the states of the extended power form to the origin. This feedback controller can be interpreted as a hybrid system consisting of a high level discrete event supervisor and a family of low level feedback controllers. The closed loop system exhibits finite-time responses.

Robust motion/force control of mechanical systems with classical nonholonomic constraints

The position/force control of mechanical systems subject to a set of classical nonholonomic constraints represents an important class of control problems. In this note, a reduced dynamic model, suitable for simultaneous independent motion and force control, is developed. Some properties of the dynamic model are exploited to facilitate the controller design. Based on the theory of guaranteed stability of uncertain systems, a robust control algorithm is derived, which guarantees the uniform ultimate boundedness of the tracking errors. A detailed numerical example is presented to illustrate the developed method. >