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

1995 ◽  
Vol 62 (1) ◽  
pp. 216-222 ◽  
Author(s):  
T. A. Loduha ◽  
B. Ravani
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Numerical Implementation ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
First Order ◽  
Holonomic Systems ◽  
Constrained Dynamical Systems ◽  
Generalized Coordinate

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.

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

1993 ◽  
Author(s):  
Timothy A. Loduha ◽  
Bahram Ravani
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Relaxation Method ◽  
Nonholonomic Systems ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
First Order ◽  
Constraint Relaxation ◽  
Constrained Dynamical Systems ◽  
Generalized Coordinate

Abstract In this paper we present a method for obtaining first-order decoupled equations of motion for multi-rigid 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 more complex dynamical systems, where the appropriate congruency transformation may be difficult to obtain, we present a constraint relaxation method based on the use of orthogonal complements. The results are illustrated using several examples. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

A Novel Approach for Decoupling of Dynamical Equations of Flexible Manipulators

1999 ◽  
Author(s):  
Ali Meghdari ◽  
Farbod Fahimi
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Flexible Manipulator ◽  
Flexible Manipulators ◽  
Dynamical Equations ◽  
First Order ◽  
Novel Approach ◽  
Elastic Multibody Systems ◽  
Two Degree Of Freedom ◽  
Generalized Coordinate

Abstract Recent advances in the study of dynamics of elastic multibody systems, specially the flexible manipulators, indicate the need and importance of decoupling the equations of motion. In this paper, an improved method for deriving elastic generalized coordinates is presented. In this regard, the Kane’s equations of motion for elastic multibody systems are considered. These equations are in the generalized form and may be applied to any desired holonomic system. Flexibility in choosing generalized speeds in terms of generalized coordinate derivatives in Kane’s method is used. It is shown that proper choice of a congruency transformation between generalized coordinate derivatives and generalized speeds leads to a series of first order decoupled equations of motion for holonomic elastic multibody systems. Furthermore, numerical implementation of the decoupling technique using congruency transformation is discussed and presented via simulation of a two degree of freedom flexible manipulator.

Singular Value Decomposition for Constrained Dynamical Systems

1985 ◽  
Vol 52 (4) ◽  
pp. 943-948 ◽  
Author(s):  
R. P. Singh ◽  
P. W. Likins
Keyword(s):  
Dynamical Systems ◽  
Singular Value Decomposition ◽  
Gaussian Elimination ◽  
Equations Of Motion ◽  
Full Rank ◽  
Individual Case ◽  
Singular Value ◽  
Dynamical Equations ◽  
Constrained Dynamical Systems ◽  
Value Decomposition

The method of singular value decomposition is shown to have useful application to the problem of reducing the equations of motion for a class of constrained dynamical systems to their minimum dimension. This method is shown to be superior to classical Gaussian elimination for several reasons: (i) The resulting equations of motion are assured to be of full rank. (ii) The process is more amenable to automation, as may be appropriate in the development of a computer program for application to a generic class of systems. (iii) The analyst is spared the responsibility for the selection of specific coordinates to be eliminated by substitution in each individual case, a selection that has no physical justification but presents abundant risk of mathematical contradiction. This approach is shown to be very efficient when the governing dynamical equations are derived via Kane’s method.

Equations of motion with the identity mass matrix for holonomic systems

2004 ◽  
Vol 218 (7) ◽  
pp. 731-736
Author(s):  
P Herman
Keyword(s):  
Differential Equations ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Interesting Insight ◽  
Holonomic Systems ◽  
Insight Into ◽  
First Order Differential Equations

Some consequences concerning holonomic systems described in terms of the inertial quasi-velocities (IQV) are discussed in this note. Introducing the IQV vector into Lagrange's formulation leads to first-order equations with the identity mass matrix of the system. The first-order differential equations give an interesting insight into dynamics and some important properties. The two examples that are provided use the dynamical equations in terms of IQV.

Conservation Principles for Systems With a Varying Number of Degrees-of-Freedom

1998 ◽  
Vol 65 (3) ◽  
pp. 719-726 ◽  
Author(s):  
S. Djerassi
Keyword(s):  
Angular Momentum ◽  
Degrees Of Freedom ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Linear Momentum ◽  
Dynamical Equations ◽  
The Third ◽  
Conservation Principles

This paper is the third in a trilogy dealing with simple, nonholonomic systems which, while in motion, change their number of degrees-of-freedom (defined as the number of independent generalized speeds required to describe the motion in question). The first of the trilogy introduced the theory underlying the dynamical equations of motion of such systems. The second dealt with the evaluation of noncontributing forces and of noncontributing impulses during such motion. This paper deals with the linear momentum, angular momentum, and mechanical energy of these systems. Specifically, expressions for changes in these quantities during imposition and removal of constraints are formulated in terms of the associated changes in the generalized speeds.

Comparison of Kane’s and Lagrange’s Methods in Analysis of Constrained Dynamical Systems

Robotica ◽  
2020 ◽  
Vol 38 (12) ◽  
pp. 2138-2150
Author(s):  
Amin Talaeizadeh ◽  
Mahmoodreza Forootan ◽  
Mehdi Zabihi ◽  
Hossein Nejat Pishkenari
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Constrained Systems ◽  
Robot Dynamics ◽  
Energy Error ◽  
Kane’S Method ◽  
Kane's Method ◽  
Constrained Dynamical Systems ◽  
Error Energy

SUMMARYDynamic modeling is a fundamental step in analyzing the movement of any mechanical system. Methods for dynamical modeling of constrained systems have been widely developed to improve the accuracy and minimize computational cost during simulations. The necessity to satisfy constraint equations as well as the equations of motion makes it more critical to use numerical techniques that are successful in decreasing the number of computational operations and numerical errors for complex dynamical systems. In this study, performance of a variant of Kane’s method compared to six different techniques based on the Lagrange’s equations is shown. To evaluate the performance of the mentioned methods, snake-like robot dynamics is considered and different aspects such as the number of the most time-consuming computational operations, constraint error, energy error, and CPU time assigned to each method are compared. The simulation results demonstrate the superiority of the variant of Kane’s method concerning the other ones.

Dynamics of a Basketball Rolling Around the Rim

2005 ◽  
Vol 128 (2) ◽  
pp. 359-364
Author(s):  
C. Q. Liu ◽  
Fang Li ◽  
R. L. Huston
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Coaching Strategies ◽  
Ball Motion

Governing dynamical equations of motion for a basketball rolling on the rim of a basket are developed and presented. These equations form a system of five first-order, ordinary differential equations. Given suitable initial conditions, these equations are readily integrated numerically. The results of these integrations predict the success (into the basket) or failure (off the outside of the rim) of the basketball shot. A series of examples are presented. The examples show that minor changes in the initial conditions can produce major changes in the subsequent ball motion. Shooting and coaching strategies are recommended.

A Discrete Quasi-Coordinate Formulation for the Dynamics of Elastic Bodies

2006 ◽  
Vol 74 (2) ◽  
pp. 231-239 ◽  
Author(s):  
G. M. T. D’Eleuterio ◽  
T. D. Barfoot
Keyword(s):  
Equations Of Motion ◽  
Computational Cost ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
Order Form ◽  
Elastic Bodies ◽  
Alternative Approach ◽  
Elastic Systems ◽  
Computationally Intensive ◽  
Derivatives Of

The discretized equations of motion for elastic systems are typically displayed in second-order form. That is, the elastic displacements are represented by a set of discretized (generalized) coordinates, such as those used in a finite-element method, and the elastic rates are simply taken to be the time-derivatives of these displacements. Unfortunately, this approach leads to unpleasant and computationally intensive inertial terms when rigid rotations of a body must be taken into account, as is so often the case in multibody dynamics. An alternative approach, presented here, assumes the elastic rates to be discretized independently of the elastic displacements. The resulting dynamical equations of motion are simplified in form, and the computational cost is correspondingly lessened. However, a slightly more complex kinematical relation between the rate coordinates and the displacement coordinates is required. This tack leads to what may be described as a discrete quasi-coordinate formulation.

scholarly journals Equations of motion in Poincaré-Četaev variables with constraint multipliers

1976 ◽  
Vol 14 (3) ◽  
pp. 359-369 ◽  
Author(s):  
Q.K. Ghori
Keyword(s):  
Dynamical Systems ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Holonomic Systems

Suslev's constraint multipliers are used to derive the equations of motion of dynamical systems (holonomic or nonholonomic) in the form of Poincaré-Četaev equations and in the canonical form. For holonomic systems defined by redundant variables, the constraint multipliers occuring in the canonical equations are determined and a modification of the Hamilton-Jacobi Theorem for integrating the canonical equations is presented.

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