Computational kinematics of multibody systems: Two formulations for a modular approach based on natural coordinates

2019 ◽  
Vol 142 ◽  
pp. 103602 ◽  
Author(s):  
M. Saura ◽  
P. Segado ◽  
D. Dopico
Keyword(s):  
Multibody Systems ◽  
Modular Approach ◽  
Natural Coordinates ◽  
Computational Kinematics

Natural Coordinates in the Optimal Control of Multibody Systems

2011 ◽  
Author(s):  
Peter Betsch ◽  
Ralf Siebert ◽  
Nicolas Sa¨nger
Keyword(s):  
Optimal Control ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Open Loop ◽  
Algebraic Equations ◽  
Natural Coordinates ◽  
Joint Forces ◽  
New Energy

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Secondly, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

Natural Coordinates in the Optimal Control of Multibody Systems

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
Peter Betsch ◽  
Ralf Siebert ◽  
Nicolas Sänger
Keyword(s):  
Optimal Control ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Open Loop ◽  
Algebraic Equations ◽  
Natural Coordinates ◽  
Joint Forces ◽  
New Energy

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

Formulation of Three-Dimensional Rigid-Flexible Multibody Systems

2007 ◽  
Author(s):  
Daniel Garci´a-Vallejo ◽  
Jose´ L. Escalona ◽  
Juana M. Mayo ◽  
Jaime Domi´nguez
Keyword(s):  
Multibody Systems ◽  
Multibody System ◽  
Three Dimensional ◽  
Natural Coordinates ◽  
Flexible Multibody System ◽  
Absolute Nodal Coordinates ◽  
Constraint Equations ◽  
Flexible Multibody ◽  
The Matrix ◽  
Nodal Coordinates

Multibody systems generally contain solids the deformations of which are appreciable and which decisively influence the dynamics of the system. These solids have to be modeled by means of special formulations for flexible solids. At the same time, other solids are of such a high stiffness that they may be considered rigid, which simplifies their modeling. For these reasons, for a rigid-flexible multibody system, two types of formulations co-exist in the equations of the system. Among the different possibilities provided in bibliography on the material, the formulation in natural coordinates and the formulation in absolute nodal coordinates are utilized in this article to model the rigid and flexible solids, respectively. This article contains a mixed formulation based on the possibility of sharing coordinates between a rigid solid and a flexible solid. In addition, the fact that the matrix of the global mass of the system is shown to be constant and that many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated. In this work, the formulation presented is utilized to simulate a mechanism with both rigid and flexible components.

A Comparative Study on Some Different Formulations of the Dynamic Equations of Constrained Mechanical Systems

1987 ◽  
Vol 109 (4) ◽  
pp. 466-474 ◽  
Author(s):  
J. Unda ◽  
J. Garci´a de Jalo´n ◽  
F. Losantos ◽  
R. Enparantza
Keyword(s):  
Comparative Study ◽  
Relative Efficiency ◽  
Multibody Systems ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Dynamic Equations ◽  
Natural Coordinates ◽  
Constrained Mechanical Systems ◽  
Computer Codes

This paper presents a comparative theoretical and numerical study on the efficiency of several numerical methods for the dynamic analysis of constrained mechanical systems, also called in the literature multibody systems. This comparative study has been performed between methods based on the use of “reference point” coordinates and those based on the use of “natural” coordinates. This study embraces different possibilities to formulate the differential equations of motion. The relative efficiency of the resulting algorithms has been analyzed theoretically in terms of the number of multiplications needed to evaluate the mechanism accelerations. This efficiency has also been studied implementing the methods into computer codes and testing them with different examples. Conclusions on the relative efficiency of the methods are finally presented.

A New Sparse Solver for Kinematics Simulation of Multibody Systems in Natural Coordinates Based on Tearing Methods

2001 ◽  
Author(s):  
Francisco Javier Funes ◽  
José Manuel Jiménez ◽  
José Ignacio Rodríguez ◽  
Javier García de Jalón
Keyword(s):  
Multibody Systems ◽  
Linear Equations ◽  
New Method ◽  
Natural Coordinates ◽  
Open Chain ◽  
Triangular Form ◽  
Constraint Equations ◽  
Closed Loop Systems ◽  
Kinematics Simulation ◽  
The Matrix

Abstract This paper presents a new method for the factorization of the sparse system of linear equations arising from the kinematic simulation of multibody systems using natural coordinates. A special reordering of the jacobian matrix of the mechanism constraint equations will be described. This reordering depends only on the topology of the mechanism. In open-chain systems the matrix can be reordered to a block triangular form. For closed-loop systems this matrix can take a bordered block triangular form, with very few columns in the border. A modification of the Harwell’s implementation of the P5 algorithm from Erisman et al. [1] is used for reordering the rows and columns of the matrix. A new method of factorization is described. This method reduces the number of floating-point operations and the fill-ins. An efficient way for solving the least-squares problem arising from over-determined systems is explained.

Computer Analysis of Multibody Systems with »Natural Coordinates«

1986 ◽  
pp. 299-309 ◽  
Author(s):  
J. Unda ◽  
J. M. Jimenez ◽  
A. Avello ◽  
J. Garcia De Jalon
Keyword(s):  
Computer Analysis ◽  
Multibody Systems ◽  
Natural Coordinates

Natural Coordinates for Teaching Multibody Systems With Matlab

2007 ◽  
Author(s):  
Javier Garci´a de Jalo´n ◽  
Nobuyuki Shimizu ◽  
David Go´mez
Keyword(s):  
Multibody Systems ◽  
Engineering Students ◽  
Automatic Differentiation ◽  
Natural Coordinates ◽  
Practical Applications ◽  
Development Tool ◽  
Starting Point ◽  
Good Starting Point ◽  
High Level ◽  
Short Courses

This paper deals with teaching kinematic and dynamic analysis of 3-D multibody systems in a context of courses with severe time constraints and the objective of attaining practical abilities. This high course efficiency is intended by the use of a simple theoretical approach (the natural or fully Cartesian coordinates) and a high level programming language (the function rich and easy to learn development tool Matlab. The theoretical prerequisites for such a course can be kept to a minimum. This approach would allow the introduction of some lessons on multibody systems inside more general courses on machine dynamics. It can also be useful for short courses addressed to engineers in industry and for numerical analysis courses addressed to mechanical engineering students that are interested in practical applications of these numerical methods. In this paper the achievable theory level will be presented in detail by means of a practical but non trivial example: a closed-chain 3-D robot. Natural coordinates and Matlab are also a good starting point to present more advanced techniques such as numerical integration methods for ODEs and DAEs, or the not very well known automatic differentiation techniques. The latter is considered as a more advanced example in this paper.

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