Class Of Nonlinear Mechanical Systems Research Articles

An adaptive speed observer for a class of nonlinear mechanical systems: Theory and experiments

We propose a solution to the problem of adaptive speed observer for a class of mechanical systems containing cross terms in the velocity (momenta). We focused our attention on mechanical systems with unknown constant input disturbances and Coulomb friction matrix, which generate products of unmeasurable velocities and unknown friction parameters. The adaptive speed observer proposed is based on the well-known Immersion and Invariance technique. We ensure global convergence of the speed observer, while the friction parameters estimation remain bounded. Our approach is validated through simulations and experimental results.

Adaptive State-Feedback Shared Control for Constrained Uncertain Mechanical Systems

This study develops two kinds of adaptive state feedback shared-control algorithms for a class of nonlinear mechanical systems in the presence of output constraints described by a set of linear inequalities. The parametric uncertainties and unknown external disturbances in the dynamical model are compensated by the proposed adaptive control laws for the feedback controller. In particular, an adaptive fixed-time state-feedback control law is also developed after the classical robust adaptive state-feedback control law design. The properties of the closed-loop shared-control systems are studied by using Lyapunov method. Simulation results demonstrate the effectiveness of the proposed approach.

Passivity-based control of mechanical systems with linear damping identification

We propose a control approach for a class of nonlinear mechanical systems to stabilize the system under study while ensuring that the oscillations of the transient response are reduced. The approach is twofold: (i) we apply our technique for linear viscous damping identification of the system to improve the accuracy of the selected control technique, and (ii) we implement a passivity-based controller to stabilize and reduce the oscillations by selecting the control parameters properly in accordance with the identified damping. Moreover, we provide an analysis for a particular passivity-based control approach that has been shown successfully for reducing such oscillations. Also, we validate the methodology by implementing it experimentally in a planar manipulator.

Stability analysis and synthesis of stabilizing controls for a class of nonlinear mechanical systems

This paper is concerned with the problems of stability and stabilization for a class of nonlinear mechanical systems. It is assumed that considered systems are under the action of linear gyroscopic forces, nonlinear homogeneous positional forces and nonlinear homogeneous dissipative forces of positional–viscous friction. An approach to strict Lyapunov functions construction for such systems is proposed. With the aid of these functions, sufficient conditions of the asymptotic stability and estimates of the convergence rate of solutions are found. Moreover, systems with delay in the positional forces are studied, and new delay-independent stability conditions are derived. The obtained results are used for developing new approaches to the synthesis of stabilizing controls with delay in feedback law.

Global Sliding Mode Observers for Some Uncertain Mechanical Systems

This paper proposes a global sliding mode observer for a class of nonlinear mechanical systems with two degrees of freedom. For the observer design, besides the usual Coriolis and centrifugal forces, we consider (discontinuous) dry and viscous friction and non vanishing uncertainties/perturbations. Moreover, the system is not required to be bounded-input-bounded-state, rendering the observer design problem challenging. For this class of systems, a dissipativity-based sliding-mode observer, with theoretically exact global finite-time convergence to the actual velocities, is proposed. To illustrate the effectiveness of the proposed observer, experimental results on a cart-pendulum system are presented.

Use of the Adjoint Method for Controlling the Mechanical Vibrations of Nonlinear Systems

In this work, the analytical derivation and the computer implementation of the adjoint method are described. The adjoint method can be effectively used for solving the optimal control problem associated with a large class of nonlinear mechanical systems. As discussed in this investigation, the adjoint method represents a broad computational framework, rather than a single numerical algorithm, in which the control problem for nonlinear dynamical systems can be effectively formulated and implemented employing a set of advanced analytical methods as well as an array of well-established numerical procedures. A detailed theoretical derivation and a comprehensive description of the numerical algorithm suitable for the computer implementation of the methodology used for performing the adjoint analysis are provided in the paper. For this purpose, two important cases are analyzed in this work, namely the design of a feedforward control scheme and the development of a feedback control architecture. In this investigation, the control problem relative to the mechanical vibrations of a nonlinear oscillator characterized by a generalized Van der Pol damping model is considered in order to illustrate the effectiveness of the computational algorithm based on the adjoint method by means of numerical experiments.

An adaptive speed observers’ design for a class of nonlinear mechanical systems

An adaptive speed observer for mechanical systems, which are linear in the velocity (momenta) after a change of coordinates (Venkatraman et al. [2010]), was recently reported in the literature (Romero and Ortega [2015]). The velocity observer considers input disturbances and some unknown friction parameters in the design. In this note, we relax some assumptions imposed in that work so that, the class of systems is enlarged and quadratic terms in the velocity(momenta) and all the friction parameters are included in the design of the velocity observer. Simulation results using the robotic–leg system illustrate the performance of the proposed adaptive observer.

Control of nonlinear vibrations using the adjoint method

In this paper, a new methodology is proposed to address the problems of suppressing structural vibrations and attenuating contact forces in nonlinear mechanical systems. The computational algorithms developed in this work are based on the mathematical framework of the calculus of variation and take advantage of the numerical implementation of the adjoint method. To this end, the principal aspects of the optimal control theory are reviewed and employed to derive the adjoint equations which form a nonlinear differential-algebraic two-point boundary value problem that defines the mathematical solution of the optimal control problem. The adjoint equations are obtained and solved numerically for the optimal design of control strategies considering a twofold control structure: a feedforward (open-loop) control architecture and a feedback (closed-loop) control scheme. While the feedforward control strategy can be implemented using only the active control paradigm, the feedback control method can be realized employing both the active and the passive control approaches. For this purpose, two dual numerical procedures are developed to numerically compute a set of optimal control policies, namely the adjoint-based control optimization method for feedforward control actions and the adjoint-based parameter optimization method for feedback control actions. The computational methods developed in this work are suitable for controlling nonlinear nonautonomous dynamical systems and feature a broad scope of application. In particular, it is shown in this paper that by setting an appropriate mathematical form of the cost functional, the proposed methods allow for simultaneously solving the problems of suppressing vibrations and attenuating interaction forces for a general class of nonlinear mechanical systems. The numerical example described in the paper illustrates the key features of the adjoint method and demonstrates the feasibility and the effectiveness of the proposed adjoint-based computational procedures.

Design of a nonlinear observer for mechanical systems with unknown inputs

This paper deals with the design of nonlinear observers for a class of nonlinear mechanical systems in presence of uncertainties (or unknown inputs). Based on sliding-mode technique, a novel observer structure is developed in order to reconstruct the unmeasured velocity variable. The proposed observer guarantees an asymptotic velocity observation also in presence of uncertainties. Simulation results are included to show the effectiveness of our method.

On robust control of a class of nonlinear mechanical systems with parametric uncertainty

This article presents a method to design a robust controller for a class of nonlinear mechanical systems. The parameters of the system are assumed uncertain in the sense that they are known only within intervals of known lower and upper limits. The method involves using Kharitonov stability theory of interval polynomials to design a robust stabilising controller for the considered class of systems. To formulate the interval parameter problem in joint space, the dynamic equation, which is derived in Cartesian space for parallel kinematic machines, is transformed to joint space using the Jacobian matrix and its time derivative. The nonlinear joint space model is linearised yielding an interval linear model and a robust controller with interval parameter gains is then found. A simulation study on a dynamic model of Stewart platform based parallel kinematics machine demonstrates the design procedure and shows its effectiveness.

Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

This paper is an extension of the previous review, done by the same authors (Mikhlin, Y., and Avramov, K. V., 2010, “Nonlinear Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments,” ASME Appl. Mech. Rev., 63(6), p. 060802), and it is devoted to applications of nonlinear normal modes (NNMs) theory. NNMs are typical regimes of motions in wide classes of nonlinear mechanical systems. The significance of NNMs for mechanical engineering is determined by several important properties of these motions. Forced resonances motions of nonlinear systems occur close to NNMs. Nonlinear phenomena, such as nonlinear localization and transfer of energy, can be analyzed using NNMs. The NNMs analysis is an important step to study more complicated behavior of nonlinear mechanical systems.This review focuses on applications of Kauderer–Rosenberg and Shaw–Pierre concepts of nonlinear normal modes. The Kauderer–Rosenberg NNMs are applied for analysis of large amplitude dynamics of finite-degree-of-freedom nonlinear mechanical systems. Systems with cyclic symmetry, impact systems, mechanical systems with essentially nonlinear absorbers, and systems with nonlinear vibration isolation are studied using this concept. Applications of the Kauderer–Rosenberg NNMs for discretized structures are also discussed. The Shaw–Pierre NNMs are applied to analyze dynamics of finite-degree-of-freedom mechanical systems, such as floating offshore platforms, rotors, piece-wise linear systems. Studies of the Shaw–Pierre NNMs of beams, plates, and shallow shells are reviewed, too. Applications of Shaw–Pierre and King–Vakakis continuous nonlinear modes for beam structures are considered. Target energy transfer and localization of structures motions in light of NNMs theory are treated. Application of different asymptotic methods for NNMs analysis and NNMs based model reduction are reviewed.

On the asymptotic stability of equilibria of nonlinear mechanical systems with delay

We consider some classes of nonlinear mechanical systems with retarded argument. It is assumed that, in the absence of delay, the systems in question have asymptotically stable equilibria. We analyze how the delay affects the stability of these equilibria. The Lyapunov function method and Razumikhin’s approach are used to derive conditions under which asymptotic stability is preserved for arbitrary delay values. We suggest a method for stabilizing strongly nonlinear conservative systems by constructing a delay feedback control depending only on the generalized coordinates.

A Simple Intrinsic Reduced-Observer for Geodesic Flow $ $

Aghannan and Rouchon proposed a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. The (position and velocity) observer is based on the Riemannian structure of the configuration manifold endowed with the kinetic energy metric and is intrinsic. They proved local convergence. When the system is conservative, we propose an intrinsic reduced order (velocity) observer based on the Jacobi metric, which can be initialized such that it converges exponentially for any initial true velocity. For non-conservative systems the observer can be used as a complement to the one of Aghannan and Rouchon. More generally the reduced observer provides velocity estimation for geodesic flow with position measurements. Thus it can be (formally) used as a fluid flow soft sensor in the case of a perfect incompressible fluid. When the curvature is negative in all planes the geodesic flow is sensitive to initial conditions. Surprisingly in this case we have global exponential convergence and the more unstable the flow is, faster is the convergence.

The stability of the equilibrium positions of non-linear non-autonomous mechanical systems

The problem of the stability of the equilibrium positions for a certain class of non-linear mechanical systems under the action of time-dependent quasipotential and dissipative-accelerating forces is considered. A method is proposed for constructing Lyapunov functions for these systems. Sufficient conditions for the stability of an equilibrium position both with respect to all of the variables as well as with respect to some of the variables are determined using the direct Lyapunov method.

Nonholonomic Mechanics and Control

1R5. Nonholonomic Mechanics and Control. - AM Bloch (Dept of Math, Univ of Michigan, Ann Arbor MI 48109-1109). Springer-Verlag, New York. 2003. 483 pp. ISBN 0-387-95535-6. $69.95.Reviewed by B Brogliato (INRA, 655 Ave De L’Europe, Saint Ismier, 38334, France).This mathematically oriented book is dedicated to the modeling and control of a class of nonlinear mechanical systems, namely mechanical systems subject to nonholonomic (or non integrable) bilateral constraints. It is known that the control of such nonlinear systems requires specific tools, as they may not be stabilizable with continuous feedbacks, as a consequence of Brockett’s necessary conditions. Such problems have received considerable attention in the applied mathematics, mechanics, and systems and control communities, for many years. Therefore, this is a topic of major importance. The audience will mainly consist of graduate students, researchers, either working in this field or with a sufficiently advanced mathematical background (especially differential geometry tools). The book starts with an introductory chapter with many examples treated in detail. The next two chapters are devoted to presenting the mathematical tools needed to study such systems. This material will certainly be hard to follow for those who have no acquaintance in differential geometry and its applications to mechanics. However, in view of the literature on the subject, understanding the basics from geometry seems to be mandatory for someone who wants to understand the control of such mechanical systems. The fourth chapter is dedicated to nonlinear control theory, still with an emphasis on geometry. Controllability, stability and stabilization are reviewed, and the chapter ends with Hamiltonian and Lagrangian control systems. The fifth and sixth chapters are devoted to nonholonomic systems, their dynamics and their control. Stabilization techniques are explained, like time-varying and nonsmooth controllers. Here one can regret that the bibliography omits C Samson, one of the first contributors to the field and who introduced time-varying controllers for the control of chained systems in the paper Velocity and torque feedback control of a nonholonomic cart, in Advanced Robot Control, Proceedings of the International Workshop on Nonlinear and Adaptive Control: Issues in Robotics, Grenoble, France, November 21-23, 1990, Springer Verlag LNCIS 162. Samson’s paper Control of chained systems: Application to path following and time-varying point stabilization, IEEE Trans on Automatic Control, Volume 40, pp 64-77, 1995, would have also been welcome in the bibliography. Chapter 7 deals with optimal control and variational principles. The eighth chapter concerns stability analysis with energy arguments (generalization of the Lejeune-Dirichlet theorem on stability of fixed points). In a logical way the last chapter is devoted to energy-based control and stabilization. There are many other books that cover the topic of nonholonomic mechanical systems control in one or two chapters (for instance Sastry’s book in the same Springer’s series, or the book Theory of Robot Control, Springer CCE Series, 1996). But Nonholonomic Mechanics and Control is entirely dedicated to this topic and covers the aspects of modeling, analysis and control. It clearly belongs to the realm of geometry-oriented works in mechanics or control. The book is well organized, contains many examples and exercises, and can be recommended to all researchers working in the field (applied mathematics, mechanics or control).

An intrinsic observer for a class of lagrangian systems

We propose a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. Our main contribution is to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates. The observer dynamics equations, as the Euler-Lagrange equations, are intrinsic. The design method uses the Riemannian structure defined by the kinetic energy on the configuration manifold. The local convergence is proved by showing that the Jacobian of the observer dynamics is negative definite (contraction) for a particular metric defined on the state-space, a metric derived from the kinetic energy and the observer gains. From a practical point of view, such intrinsic observers can be approximated, when the estimated configuration is close to the true one, by an explicit set of differential equations involving the Riemannian curvature tensor. These equations can be automatically generated via symbolic differentiations of the metric and potential up to order two. Numerical simulations for the ball and beam system, an example where the scalar curvature is always negative, show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics.

Chattering‐free sliding mode control for a class of nonlinear mechanical systems

AbstractSecond‐order sliding mode control (2‐smc) and dynamic sliding mode control (dsmc) eliminate the disturbing characteristic of chattering in static sliding mode control under the assumption that the derivative of the sliding surface is available or complex inequalities at the acceleration level can be constructed. In this paper, passivity‐based adaptive and non‐adaptive chattering‐free sliding mode controllers are proposed assuming that the upper bound of the norm of the derivative of the sliding surface is available, a weaker and easy to implement assumption in comparison to those of 2‐smc and dsmc. The closed‐loop system accounts explicitly for the invariance condition without reaching phase, and therefore for a desired transient response with global exponential convergence of tracking errors. Preliminary experiments are presented. Copyright © 2001 John Wiley & Sons, Ltd.

Adaptive Realization of Linear Closed-Loop Tracking Dynamics in the Presence of Large System Model Errors

A novel adaptive feedback control approach for nonlinear mechanical systems is presented. The approach applies to nonlinear trajectory tracking and has the remarkable property that the tracking error dynamics asymptotically approach a specified linear PID response for the case where the external disturbances are constant. The methodology applies to a large class of nonlinear mechanical systems, however, it is illustrated for the case of nonlinear rigid body maneuvers subject to actuator saturation constraints and large uncertainty of the system mass and inertia properties. While the system mass or inertias are not identified in this approach, the external disturbances are accurately estimated if they are constant or slowly varying with respect to the adaptation rate. A benefit of this method is that it requires no a priori knowledge of the unknown system parameters or bounds thereof.

Hyper-stable Servo Controllers without Velocity Measurement for a Class of Nonlinear Mechanical Systems

Hyper-stable Servo Controllers without Velocity Measurement for a Class of Nonlinear Mechanical Systems

Robust adaptive control of a class of nonlinear mechanical systems with unbounded and fast-varying uncertainties

The class of system addressed in this paper consists of multibody mechanisms with open or closed kinematic chains. The constraints imposed by interbody connections are holonomic and scleronomic. Uncertainties result from unknown system parameters, external disturbances and some unmodeled dynamic effects. The plant model, available for a controller, represents only a few very general properties of multibody dynamic systems. The coefficients and terms of the plant equation may be potentially unbounded and fast varying. Such a general model includes a large class of unmodeled dynamics and any bounded external disturbances. Novel adaptive and control laws are presented. The stability and robustness analysis, performed by the Lyapunov method, has proven that the proposed controller keeps the tracking errors within a certain computable tolerance. As a particular case of the general theory, the tracking control of robotic manipulators has been discussed in detail.