Dynamics Of Constrained Systems Research Articles

The Embedded Projection Method: A general index reduction procedure for constrained system dynamics

The ‘Embedded Projection Method’ (EPM) is an attempt to eliminate most of the difficulties related to the solution of high-index differential-algebraic equations (DAEs), as those typical of the dynamics of constrained systems. The intimate coupling between the algebraic and the differential parts lies at the origin of these difficulties, therefore we uncouple these components, introduce an unconstrained, modified state variable, and complement the resulting ordinary differential equation (ODE) by additional algebraic variables. As a result, a general, consistent index reduction from arbitrarily high values to one is obtained, allowing the use of any suitable ODE integration algorithm, without the need for a specialized DAE solver. The EPM solution is intrinsically endowed with greater accuracy and stability than conventional counterparts, thus balancing the higher complexity of the procedure. We illustrate the EPM including some applications of holonomic and non-holonomic mechanical systems.

A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework

The paper deals with controlled mechanical systems in which the number of control inputs is equal to the number of desired system outputs, and is smaller than the number of degrees of freedom of the system. The determination of control input strategy that force the underactuated system to complete the partly specified motion is a challenging problem. In the present formulation, the outputs (performance goals), expressed in terms of system states, are treated as constraints on the system—called control or program constraints as distinct from contact constraints in the classical sense, and a mathematical resemblance of the inverse control problem to the constrained system dynamics is exploited. However, while the reactions of contact constraints act in the directions orthogonal to the respective constraint manifold, the available control reactions may have arbitrary directions with respect to the program constraint manifold, and in the extreme may be tangent. A specific methodology must then be developed to find the solution of such “singular” problems, related to a class of control tracking problems such as position control of elastic joint robots, control of cranes, and aircraft control in prescribed trajectory flight. The governing equations of the problem arise as a set of differential-algebraic equations (DAEs), and an effective method for solving the DAEs, based on backward Euler method, is proposed. The open-loop control formulation obtained this way is then extended by a closed-loop control law to provide stable tracking of the required reference trajectories in the presence of perturbations. Some examples of applications of the theory and results of numerical simulations are reported.

Order-(n+m) direct differentiation determination of design sensitivity for constrained multibody dynamic systems

With the complexity and large dimensionality of many modern multibody dynamic applications, the efficiency of the sensitivity evaluation methods used can greatly impact the overall computation cost and as such can greatly limit the usefulness of the sensitivity information. Most current direct differentiation approaches suffer from prohibitive computational cost, which may be as great as O(n4+n2m2+nm3) (for systems with n generalized coordinates and m algebraic constraints). This paper presents a concise and computationally efficient sensitivity analysis scheme to facilitate such sensitivity calculations. A unique feature of this scheme is its use of recursive procedures to directly embed the algebraic constraint relations in forming and simultaneously solving a minimal set of equations. This results in far fewer operations than more traditional, or so-called O(n), counterparts. The algorithm determines the derivatives of generalized accelerations in O(n+m) operations overall. The resulting equations are “exact” to integration accuracy and enforce constraints exactly at both the velocity and acceleration levels.

Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; for Engineers, Physicists, and Mathematicians

Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; for Engineers, Physicists, and Mathematicians

Predictive Interfaces for Teleoperation of Constrained Dynamical Systems via the Internet

In this paper we present a predictive strategy for teleoperating, through communication channels with unknown and possibly unbounded time-delay, a remotely located pre-compensated plant subject to input and state-related pointwise-in-time constraints. The method can be specialized to deal either with bounded time-delays, as typically encountered in space and underwater applications, or with random and possibly unbounded round-trip time-delays, as in the Internet case.

Multi-Multiplier Ambient-Space Formulations of Constrained Dynamical Systems, with an Application to Elastodynamics

Various formulations of the equations of motion for both finite- and infinite-dimensional constrained Lagrangian dynamical systems are studied. The different formulations correspond to different ways of enforcing constraints through multiplier fields. All the formulations considered are posed on ambient spaces whose members are unrestricted by the need to satisfy constraint equations, but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied. The stability properties of the invariant set within its ambient space are shown to be different in each case. We use the specific model problem of linearized incompressible elastodynamics to compare properties of three different ambient-space formulations. We establish the well-posedness of one formulation in the particular case of a homogeneous, isotropic body subject to specified tractions on its boundary.

An Efficient Newton-Type Iteration for the Numerical Solution of Highly Oscillatory Constrained Multibody Dynamic Systems

In this paper we present a coordinate-split (CS) technique for the numerical solution of the equations of motion of constrained multibody dynamic systems. We show how the CS technique can be implemented within the context of commonly used solution methods, for increased efficiency and reliability. A particularly challenging problem for multibody dynamics is the numerical solution of highly oscillatory nonlinear mechanical systems. Highly stable implicit integration methods with large stepsizes can be used to damp the oscillation, if it is of small amplitude. However, the standard Newton iteration is known to experience severe convergence difficulties which force a restriction of the stepsize. We introduce a modified coordinate-split (CM) iteration which overcomes these problems. Convergence analysis explains the improved convergence for nonlinear oscillatory systems, and numerical experiments illustrate the effectiveness of the new method.

Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems

A pseudospectral method for generating optimal trajectories of linear and nonlinear constrained dynamic systems is proposed. The method consists of representing the solution of the optimal control problem by an mth degree interpolating polynomial, using Chebyshev nodes, and then discretizing the problem using a cell-averaging technique. The optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the proposed method avoids many of the numerical difficulties typically encountered in solving standard optimal control problems. Furthermore, for discontinuous optimal control problems, we develop and implement a Chebyshev smoothing procedure which extracts the piecewise smooth solution from the oscillatory solution near the points of discontinuities. Numerical examples are provided, which confirm the convergence of the proposed method. Moreover, a comparison is made with optimal solutions obtained by closed-form analysis and/or other numerical methods in the literature.

A Geometric Unification of Constrained System Dynamics

A unified geometric formulation of the methods used for solving constrained system problems is given. Both holonomic and nonholonomic systems are treated in like manner, and the dynamic equations are expressible in either generalized velocities or quasi-velocities. Moreover, a wide range of ’unconstrained‘ systems are uniformly regarded as generalized particles in the multi-dimensional metric spaces relating to their configuration. The derivation is grounded on the tensor calculus formalism and appropriate geometric interpretations are reported. In its useful matrix form, the formulation turns out short, elementary and general. This unified geometric approach to constrained system dynamics may deserve to become a generally accepted method inacademic and engineering applications.

On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems

In this paper we present a method for obtaining first-order decoupled equations of motion for multirigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or holonomic systems with unreduced configuration coordinates, we incorporate an orthogonal complement in conjunction with the congruency transformation. A pair of examples illustrate the results. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

Dynamics of large constrained nonlinear systems-a taxonomy theory [power system stability

This paper provides an overview of the taxonomy theory which has been proposed as a fundamental platform for solving practical stability related problems in large constrained nonlinear systems such as the electric power system. The theory reveals a two-level intertwined cellular nature of the constrained system dynamics which serves as a unifying structure, a taxonomy, for analyzing nonlinear phenomena in large system models. These broadly divide into the state space aspects (related to dynamic stability issues among others) and the parameter space aspects (connected with bifurcation phenomena among others). In the state-space formulation, the boundary of the region of attraction for the operating point is shown (under certain Morse-Smale like assumptions) to be composed of stable manifolds of certain anchors and portions of the singularity surface. Such boundary characterization provides the foundation for rigorous Lyapunov theoretic transient stability methods. In the parameter space analysis, the feasibility region which is bounded by the feasibility boundary provides a safe opening region for guaranteeing local stability at the equilibrium under slow parametric variations. The feasibility boundary where the operating point undergoes loss of local stability is characterized in the form of three principal bifurcations including a new bifurcation called the singularity induced bifurcation.

Equations of Motion for Nonholonomic, Constrained Dynamical Systems via Gauss’s Principle

In this paper we develop an analytical set of equations to describe the motion of discrete dynamical systems subjected to holonomic and/or nonholonomic Pfaffian equality constraints. These equations are obtained by using Gauss’s Principle to recast the problem of the constrained motion of dynamical systems in the form of a quadratic programming problem. The closed-form solution to this programming problem then explicitly yields the equations that describe the time evolution of constrained linear and nonlinear mechanical systems. The direct approach used here does not require the use of any Lagrange multipliers, and the resulting equations are expressed in terms of two different classes of generalized inverses—the first class pertinent to the constraints, the second to the dynamics of the motion. These equations can be numerically solved using any of the standard numerical techniques for solving differential equations. A closed-form analytical expression for the constraint forces required for a given mechanical system to satisfy a specific set of nonholonomic constraints is also provided. An example dealing with the position tracking control of a nonlinear system shows the power of the analytical results and provides new insights into application areas such as robotics, and the control of structural and mechanical systems.

Canonical ensembles from chaos II: Constrained dynamical systems

We present dynamical equations of motion to compute thermodynamic properties of constrained dynamical systems. In particular, we examine systems that can be described by generators of Lie algebras. In this method, additional (noncompact) degrees of freedom are added to the compact phase space to mock the effects of a heat bath. The equations of motion in this extended space are ergodic, and canonical ensemble averages reduce to time averages over the classical trajactory. We compute explicitly the thermodynamic properties of several simple systems, in particular, Hamiltonians with SU(2) and SU(3) symmetry.

Observation and Estimation in Linear Descriptor Systems With Application to Constrained Dynamical Systems

This paper considers the problems of simultaneous observation or estimation of the positions, velocities, and contact forces in a constrained dynamical system. The equations of such systems are not ordinary differential equations, but descriptor equations, i.e., differential equations where the coefficient of the highest order derivative is singular. An asymptotic observer in descriptor form based on pole assignment techniques is used in the time-invariant case to reconstruct the positions, velocities, and contact forces. For time invariant constrained dynamical systems subject to random disturbances, an optimal estimator in descriptor form is designed based on Wiener-Hopf theory. Constrained dynamical systems yield descriptor systems that are uncontrollable and unobservable at infinity. As a consequence, the observer and estimator may not change the infinite eigenstructure of the system. Examples are given to illustrate the use of our method.

Design Sensitivity Analysis of Large-Scale Constrained Dynamic Mechanical Systems

A method for computer-aided design sensitivity analysis of large-scale constrained dynamic systems is presented. A generalized coordinate partitioning method is used for assembling and solving sets of mixed differential-algebraic equations of motion and adjoint equations required for calculation of derivatives of dynamic response measures with respect to design variables. The reduction in dimension of the equations of motion and associated adjoint equations obtained through use of generalized coordinate partitioning significantly reduces the computational burden, as compared to methods previously employed. Use of predictor-corrector numerical integration algorithms, rather than an implicit integration algorithm used in the past is shown to greatly simplify the equations that must be formulated and solved. Two examples are presented to illustrate accuracy of the design sensitivity analysis method developed.

Two applications of a general method for studying constrained dynamical systems

We continue the investigation of a method for studying the canonical formalism for constrained systems by developing two examples (the free relativistic scalar particle and a model of relativistic interaction between two scalar particles).

On Optimal Control Laws for a Class of Constrained Dynamical Systems (With Application to Control of Bipedal Locomotion)

The physical and dynamical constraints of a constrained dynamical system are related to system controllability and stability. Proper formulation of these inequality constraints and treatment of the active ones leads to stabilizing controls with relatively smooth control efforts—in all cases, control laws. These approaches were useful in a study of the biomechanics of climbing and descending gaits by mathematical synthesis techniques, necessitated by the increased importance of terrain and lower extremity kinematics and incomplete specification of the tasks. The general criteria entail no uniqueness requirements on system motions and controls, although for the most common (and probably most desirable) condition of the constraints (fewer active constraints than system degrees of freedom) an “optimal” control law can be derived. Two examples are presented, and some general discussion is given relating mainly to the control of biped locomotion.