Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

Author(s):  
Robin Chhabra ◽  
M. Reza Emami ◽  
Yael Karshon

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

2010 ◽  
Vol 1 (4) ◽  
Author(s):  
Vladimir G. Ivancevic ◽  
Tijana T. Ivancevic

AbstractThis paper reviews modern geometrical dynamics and control of humanoid robots. This general Lagrangian and Hamiltonian formalism starts with a proper definition of humanoid's configuration manifold, which is a set of all robot's active joint angles. Based on the ‘covariant force law’, the general humanoid's dynamics and control are developed. Autonomous Lagrangian dynamics is formulated on the associated ‘humanoid velocity phase space’, while autonomous Hamiltonian dynamics is formulated on the associated ‘humanoid momentum phase space’. Neural-like hierarchical humanoid control naturally follows this geometrical prescription. This purely rotational and autonomous dynamics and control is then generalized into the framework of modern non-autonomous biomechanics, defining the Hamiltonian fitness function. The paper concludes with several simulation examples.


1991 ◽  
Vol 15 (3) ◽  
pp. 235-256 ◽  
Author(s):  
X. Cyril ◽  
J. Angeles ◽  
A. Misra

In this paper the formulation and simulation of the dynamical equations of multibody mechanical systems comprising of both rigid and flexible-links are accomplished in two steps: in the first step, each link is considered as an unconstrained body and hence, its Euler-Lagrange (EL) equations are derived disregarding the kinematic couplings; in the second step, the individual-link equations, along with the associated constraint forces, are assembled to obtain the constrained dynamical equations of the multibody system. These constraint forces are then efficiently eliminated by simple matrix multiplication of the said equations by the transpose of the natural orthogonal complement of kinematic velocity constraints to obtain the independent dynamical equations. The equations of motion are solved for the generalized accelerations using the Cholesky decomposition method and integrated using Gear’s method for stiff differential equations. Finally, the dynamical behaviour of the Shuttle Remote Manipulator when performing a typical manoeuvre is determined using the above approach.


2020 ◽  
Vol 17 (06) ◽  
pp. 2050090 ◽  
Author(s):  
Jordi Gaset ◽  
Xavier Gràcia ◽  
Miguel C. Muñoz-Lecanda ◽  
Xavier Rivas ◽  
Narciso Román-Roy

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed.


1994 ◽  
Vol 04 (03) ◽  
pp. 521-551 ◽  
Author(s):  
MARTIN LESSER

This article provides an introduction to a technique for formulating nonlinear models of mechanical systems composed of interconnected and constrained rigid body systems such as those encountered in vehicle technology, biomechanics, spacecraft design and robotics. The approach is based on an algorithm developed by Kane to treat nonholonomic systems, for example systems with rolling constraints. The algorithm is interpreted geometrically in terms of tangent vectors to the instantaneous configuration manifold embedded in the space of nonconstrained motions for the system. The level and style of the presentation is intended to be understood by scientifically literate readers with minimal knowledge in mechanics beyond the introductory level. Examples also show how computer algebra can be used to reduce the effort required for treating complex systems. An annotated reference list, which includes a discussion of computer software, is also provided.


1992 ◽  
Vol 59 (2) ◽  
pp. 411-417 ◽  
Author(s):  
Steven Peterson

In this paper, screw theory is employed to develop a method for generating the dynamic equations of a system of rigid bodies. Exterior algebra is used to derive the structure of screw space from projective three space (homogeneous coordinate space). The dynamic equation formulation method is derived from the parametric form of the principle of least action, and it is shown that a set of screws exist which serves as a basis for the tangent space of the configuration manifold. Equations generated using this technique are analogs of Hamilton’s dynamical equations. The freedom screws defining the manifold’s tangent space are determined from the contact geometry of the joint using the virtual coefficient, which is developed from the principle of virtual work. This results in a method that eliminates all differentiation operations required by other virtual work techniques, producing a formulation method based solely on the geometry of the system of rigid bodies. The procedure is applied to the derivation of the dynamic equations for the first three links of the Stanford manipulator.


Author(s):  
Arun K. Banerjee ◽  
Mark Lemak

This paper deals with the motion of mechanical systems with non-ideal constraints, defined as constraints where the forces associated with the constraint do work. The first objective of the paper is to show that two newly published formulations of equations of motion of systems with such non-ideal constraints are unnecessarily complex for situations where the non-ideal constraint force does not depend on the ideal constraint force, because they introduce and then eliminate these non-working constraint forces. We point out that a method already exists for nonideal constraints, namely, Kane’s equations, which are simpler because, among other things, they are based on automatic elimination of non-working constraints. The examples considered in these recent publications are worked out with Kane’s method to show the applicability and simplicity of Kane’s method for non-ideal constraints. A second objective of the paper is to present an alternative form of equations for systems where the non-ideal constraint force depends on the ideal constraint force, as in the case of Coulomb friction. The formulation is shown to lend itself naturally to also analyzing impact dynamics. The method is applied to the dynamics of a slug moving against friction on a moving ellipsoidal surface. Such a crude model may simulate, in essence, propellant motion in a tank in zero-g, or during docking of a spacecraft.


2000 ◽  
Vol 68 (3) ◽  
pp. 462-467 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba

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.


1994 ◽  
Vol 116 (2) ◽  
pp. 437-444 ◽  
Author(s):  
H. T. Wu ◽  
N. K. Mani

Vibration normal modes and static correction modes have been previously used to model flexible bodies for dynamic analysis of mechanical systems. The efficiency and accuracy of using these modes to model a system depends on both the flexibility of each body and the applied loads. This paper develops a generalized method for the generation of a set of Ritz vectors to be used in addition to vibration normal modes to form the modal basis to model flexible bodies for dynamic analysis of multibody mechanical systems. The Ritz vectors are generated using special distribution of the D’Alembert force and the kinematic constraint forces due to gross-body motion of a flexible body. Combined with vibration normal modes, they form more efficient vector bases for the modeling of flexible bodies comparing to using vibration normal modes alone or using the combination of static correction modes and vibration normal modes. Ritz vectors can be regenerated when the system undergoes significant changes of its configuration and the regeneration procedure is inexpensive. The effectiveness of using the combination of vibration normal modes and the proposed Ritz vectors is demonstrated using a planar slider-crank mechanism.


1993 ◽  
Vol 60 (3) ◽  
pp. 662-668 ◽  
Author(s):  
R. E. Kalaba ◽  
F. E. Udwadia

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.


Author(s):  
J. ANGELES ◽  
SANGKOO LEE

A computationally efficient and systematic algorithm for the modelling of constrained mechanical systems is developed and implemented in this paper. With this algorithm, the governing equations of mechanical systems comprised of rigid bodies coupled by holonomic constraints are derived by means of an orthogonal complement of the matrix of the velocity-constraint equations. The procedure is applicable to all types of holonomic mechanical systems, and it can be extended to cases including simple nonholonomic constraints. Holonomic mechanical systems having a simple Kinematic-chain structure, such as single-loop linkages and serial-type robotic manipulators, are analysed regarding the derivation of the matrix of the constraint equations and its orthogonal complement, and the computation of the constraint forces.


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