Constrained Motion of Mechanical Systems and Tracking Control of Nonlinear Systems: Connections and Closed-form Results (original) (raw)

Constrained Motion of Mechanical Systems and Tracking Control of Nonlinear Systems

Springer Proceedings in Mathematics & Statistics, 2014

This paper aims to expose the connections between the determination of the equations of motion of constrained systems and the problem of tracking control of nonlinear mechanical systems. The duality between the imposition of constraints on a mechanical system and the trajectory requirements for tracking control is exposed through the use of a simple example. It is shown that given a set of constraints, d'Alembert's principle corresponds to the problem of finding the optimal tracking control of a mechanical system for a specific control cost function that Nature seems to choose. Furthermore, the general equations for constrained motion of mechanical systems that do not obey d'Alembert's principle yield, through this duality, the entire set of continuous controllers that permit exact tracking of the trajectory requirements. The way Nature seems to handle the tracking control problem of highly nonlinear systems suggests ways in which we can develop new control methods that do not make any approximations and/or linearizations related to the nonlinear system dynamics, or its controllers. More general control costs are used and Nature's approach is thereby extended to general control problems.

On general nonlinear constrained mechanical systems

Numerical Algebra, Control and Optimization, 2013

This paper develops a new, simple, general, and explicit form of the equations of motion for general constrained mechanical systems that can have holonomic and/or nonholonomic constraints that may or may not be ideal, and that may contain either positive semi-definite or positive definite mass matrices. This is done through the replacement of the actual unconstrained mechanical system, which may have a positive semi-definite mass matrix, with an unconstrained auxiliary system whose mass matrix is positive definite and which is subjected to the same holonomic and/or nonholonomic constraints as those applied to the actual unconstrained mechanical system. A simple, unified fundamental equation that gives in closed-form both the acceleration of the constrained mechanical system and the constraint force is obtained. The results herein provide deeper insights into the behavior of constrained motion and open up new approaches to modeling complex, constrained mechanical systems, such as those encountered in multi-body dynamics.

ON GENERAL NONLINEAR CONSTRAINED MECHANICAL SYSTEM

EQUATIONS OF MOTION FOR CONSTRAINED MECHANICAL SYSTEMS AND THE EXTENDED D'ALEMBERT'S PRINCIPLE

Starting from the principle of virtual work, this paper states and establishes an extended version of D'Alembert's Principle. Using this extended principle and elementary linear algebra, it develops, from first principles, the explicit equation of motion for constrained mechanical systems. The results are compared with the authors' previous results. The approach points to new ways of extending these results.

Constrained Dynamics and Tracking Control of Structural and Mechanical Systems

2000

Based on recent results from analytical dynamics, this paper develops a class of tracking controllers for controlling general, non-linear, structural and mechanical systems. Unlike most control methods that perform some kind of linearization and/or cancellation, the methodology developed herein obtains a class of control forces that 'exactly' maintain the nonlinear system along a certain trajectory, which, in general, may be

Optimal tracking control of nonlinear dynamical systems

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008

This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems...

Explicit Equations of Motion for Mechanical Systems With Nonideal Constraints

Journal of Applied Mechanics, 2001

Since its inception about 200 years ago, Lagrangian mechanics has been based upon the Principle of D'Alembert. There are, however, many physical situations where this confining principle is not suitable, and the constraint forces do work. To date, such situations are excluded from general Lagrangian formulations. This paper releases Lagrangian mechanics from this confinement, by generalizing D'Alembert's principle, and presents the explicit equations of motion for constrained mechanical systems in which the constraints are nonideal. These equations lead to a simple and new fundamental view of Lagrangian mechanics. They provide a geometrical understanding of constrained motion, and they highlight the simplicity with which Nature seems to operate.

Equation of Motion of the Nonlinear Robotic Systems with Constraints

2012

Optimal Trajectory Tracking of Nonholonomic Mechanical Systems: a geometric approach

2019 American Control Conference (ACC), 2019

We study the tracking of a trajectory for a nonholonomic system by recasting the problem as an optimal control problem. The cost function is chosen to minimize the error in positions and velocities between the trajectory of a nonholonomic system and the desired reference trajectory evolving on the distribution which defines the nonholonomic constraints. We prepose a geometric framework since it describes the class of nonlinear systems under study in a coordinate-free framework. Necessary conditions for the existence of extrema are determined by the Pontryagin Minimum Principle. A nonholonomic fully actuated particle is used as a benchmark example to show how the proposed method is applied.

On classical mechanical systems with non-linear constraints

Journal of Geometry and Physics, 2004

In the present work, we analyze classical mechanical systems with non-linear constraints in the velocities. We prove that the d'Alembert-Chetaev trajectories of a constrained mechanical system satisfy both Gauss' principle of least constraint and Hölder's principle. In the case of a free mechanics, they also satisfy Hertz's principle of least curvature if the constraint manifold is a cone. We show that the Gibbs-Maggi-Appell (GMA) vector field (i.e. the second-order vector field which defines the d'Alembert-Chetaev trajectories) conserves energy for any potential energy if, and only if, the constraint is homogeneous (i.e. if the Liouville vector field is tangent to the constraint manifold). We introduce the Jacobi-Carathéodory metric tensor and prove Jacobi-Carathéodory's theorem assuming that the constraint manifold is a cone. Finally, we present a version of Liouville's theorem on the conservation of volume for the flow of the GMA vector field.

Constrained Motion of Mechanical Systems and Tracking Control of Nonlinear Systems: Connections and Closed-form Results (original) (raw)

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