Dynamics of Nonholonomic Mechanical Systems Using a Natural Orthogonal Complement

1991 ◽  
Vol 58 (1) ◽  
pp. 238-243 ◽  
Author(s):  
Subir Kumar Saha ◽  
Jorge Angeles
Keyword(s):  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Rigid Bodies ◽  
Orthogonal Complement ◽  
Constraint Equations ◽  
Nonholonomic Mechanical Systems ◽  
Holonomic And Nonholonomic Constraints ◽  
Automatic Guided Vehicle ◽  
The Matrix ◽  
Velocity Constraint

The dynamics equations governing the motion of mechanical systems composed of rigid bodies coupled by holonomic and nonholonomic constraints are derived. The underlying method is based on a natural orthogonal complement of the matrix associated with the velocity constraint equations written in linear homogeneous form. The method is applied to the classical example of a rolling disk and an application to a 2-dof Automatic Guided Vehicle is outlined.

THE MODELLING OF HOLONOMIC MECHANICAL SYSTEMS USING A NATURAL ORTHOGONAL COMPLEMENT

1989 ◽  
Vol 13 (4) ◽  
pp. 81-89 ◽  
Author(s):  
J. ANGELES ◽  
SANGKOO LEE
Keyword(s):  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Kinematic Chain ◽  
Chain Structure ◽  
Rigid Bodies ◽  
Orthogonal Complement ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Constraint Equations ◽  
The Matrix

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.

Cayley kinematics and the Cayley form of dynamic equations

2005 ◽  
Vol 461 (2055) ◽  
pp. 761-781 ◽  
Author(s):  
Andrew J. Sinclair ◽  
John E. Hurtado
Keyword(s):  
Euler Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Matrix Form ◽  
Dynamic Equations ◽  
Cayley Transform ◽  
Higher Dimensional ◽  
The Matrix ◽  
General Motion

The Cayley transform and the Cayley–transform kinematic relationships are an important and fascinating set of results that have relevance in N –dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N –dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher–dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N –dimensional body as pure rotations of an ( N + 1)–dimensional body.

A geometric approach to the transpositional relations in dynamics of nonholonomic mechanical systems

2018 ◽  
Vol 15 (07) ◽  
pp. 1850112 ◽  
Author(s):  
Mahdi Khajeh Salehani
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Geometric Approach ◽  
Nonholonomic Mechanical Systems ◽  
Nonholonomic Dynamics ◽  
Geometric Objects

Exploring the geometry of mechanical systems subject to nonholonomic constraints and using various bundle and variational structures intrinsically present in the nonholonomic setting, we study the structure of the equations of motion in a way that aids the analysis and helps to isolate the important geometric objects that govern the motion of such systems. Furthermore, we show that considering different sets of transpositional relations corresponding to different transitivity choices provides different variational structures associated with nonholonomic dynamics, but the derived equations (being referred to as the generalized Hamel–Voronets equations) are equivalent to the Lagrange–d’Alembert equations. To illustrate results of this work and as some applications of the generalized Hamel–Voronets formalisms discussed in this paper, we conclude with considering the balanced Tennessee racer, as well as its modification being referred to as the generalized nonholonomic cart, and an [Formula: see text]-snake with three wheeled planar platforms whose snake-like motion is induced by shape variations of the system.

Simulation of Constrained Mechanical Systems—Part II: Explicit Numerical Integration

2012 ◽  
Vol 79 (4) ◽  
Author(s):  
David J. Braun ◽  
Michael Goldfarb
Keyword(s):  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Direct Consequence ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Constrained Motion ◽  
Constrained Mechanical Systems ◽  
Holonomic And Nonholonomic Constraints ◽  
Long Time ◽  
Constraint Correction

This paper presents an explicit to integrate differential algebraic equations (DAEs) method for simulations of constrained mechanical systems modeled with holonomic and nonholonomic constraints. The proposed DAE integrator is based on the equation of constrained motion developed in Part I of this work, which is discretized here using explicit ordinary differential equation schemes and applied to solve two nontrivial examples. The obtained results show that this integrator allows one to precisely solve constrained mechanical systems through long time periods. Unlike many other implicit DAE solvers which utilize iterative constraint correction, the presented DAE integrator is explicit, and it does not use any iteration. As a direct consequence, the present formulation is simple to implement, and is also well suited for real-time applications.

scholarly journals A new method for determining the reactions of mechanical constraints

2006 ◽  
Vol 28 (1) ◽  
pp. 35-42
Author(s):  
Do Sanh ◽  
Do Dang Khoa
Keyword(s):  
Mechanical System ◽  
Nonholonomic Constraints ◽  
Inverse Matrix ◽  
Algebraic Equations ◽  
Closed Set ◽  
Mechanical Constraints ◽  
Large Size ◽  
Holonomic And Nonholonomic Constraints ◽  
The Matrix ◽  
Lagrange’S Multipliers

In the present paper it is introduced the method for determining the reactions of mechanical constraints (holonomic and nonholonomic constraints).As is known, for studying dynamical characters of a mechanical system it is necessary to determine the constraint reactions acting on the system. Up to now, the reactions are calculated through Lagrange's multipliers. By such a way the reactions are determined only indirectly. In the [3, 4], two methods of determining directly the reactions are discussed. However, for applying these methods, it is necessary to compute the inverse matrix of the matrix of inertia. This thing in general is not convenient, specially when the matrix of inertial is of large size and dense.In the present paper it is represented the method for determining the constraint reactions, by which it is possible to avoid inertia the computation of the inverse matrix of the matrix of inertia is avoided. For this in the paper it is used the middle variables by which we obtain a closed set of algebraic equations for directly determining reactions.

On the Lagrangian Formalism of Nonholonomic Mechanical Systems

2005 ◽  
Author(s):  
Hiroaki Yoshimura
Keyword(s):  
Mechanical System ◽  
Tangent Bundle ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Lagrangian System ◽  
Lagrangian Formalism ◽  
Nonholonomic Mechanical Systems ◽  
Points Of View ◽  
Configuration Manifold

The paper illustrates the Lagrangian formalism of mechanical systems with nonholonomic constraints using the ideas of geometric mechanics. We first review a Lagrangian system for a conservative mechanical system in the context of variational principle of Hamilton, and we investigate the case that a given Lagrangian is hyperregular, which can be illustrated in the context of the symplectic structure on the tangent bundle of a configuration space by using the Legendre transformation. The Lagrangian system is denoted by the second order vector field and the Lagrangian one- and two-forms associated with a given hyperregular Lagrangian. Then, we demonstrate that a mechanical system with nonholonomic constraints can be formulated on the tangent bundle of a configuration manifold by using Lagrange multipliers. To do this, we investigate the Lagrange-d’Alembert principle from geometric points of view and we also show the intrinsic expression of the Lagrange-d’Alembert equations of motion for nonholonomic mechanical systems with nonconservative force fields.

scholarly journals A new form of equations of motion of nonholonomic mechanical systems

1997 ◽  
Vol 19 (3) ◽  
pp. 59-64
Author(s):  
Do Sanh
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Matrix Form ◽  
Inverse Matrix ◽  
Nonholonomic Mechanical Systems ◽  
Solution Of Equations ◽  
The Matrix ◽  
New Form

In the paper it is given a new form of equations of motion of nonholonomic mechanical systems. The obtained equations are written in a matrix form and in order to establish them it is unnecessary to calculate the inverse matrix of the matrix of inertia. It is important that it is possible to obtain the relation between the error of realizing constraints and that of computing the solution of equations of motion of the system under consideration.

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