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.

scholarly journals A Note on Full-Rank Factorization of Matrix

2019 ◽  
Vol 15 (2) ◽  
pp. 152-154
Author(s):  
Gyan Bahadur Thapa ◽  
J. López-Bonilla ◽  
R. López-Vázquez
Keyword(s):  
Singular Value Decomposition ◽  
Full Rank ◽  
Singular Value ◽  
The Matrix ◽  
Rank Factorization ◽  
Value Decomposition

We exhibit that the Singular Value Decomposition of a matrix Anxm implies a natural full-rank factorization of the matrix.

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.

Application of Singular Value Decomposition for Analysis of Mechanical System Dynamics

1985 ◽  
Vol 107 (1) ◽  
pp. 82-87 ◽  
Author(s):  
N. K. Mani ◽  
E. J. Haug ◽  
K. E. Atkinson
Keyword(s):  
Singular Value Decomposition ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Singular Value ◽  
Algebraic Equations ◽  
Generalized Coordinates ◽  
Singular Value Decomposition Method ◽  
System Information ◽  
Value Decomposition ◽  
Algebraic Constraint

A singular value decomposition method for efficient solution of mixed differential-algebraic equations of motion of mechanical systems is developed. Differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates that are related through nonlinear algebraic constraint equations. Singular value decomposition of the constraint Jacobian matrix is used to define a new set of generalized coordinates that are partitioned into optimal independent and dependent sets. Integration of only independent generalized coordinates generates all system information. A numerical example is presented to demonstrate effectiveness of the method.

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.

scholarly journals Alternative methods of calculation of the pseudo inverse of a non full-rank matrix

2008 ◽  
Vol 6 (03) ◽  
Author(s):  
M. A. Murray-Lasso
Keyword(s):  
Singular Value Decomposition ◽  
Full Rank ◽  
Singular Value ◽  
Diagonal Matrix ◽  
Alternative Methods ◽  
Minimum Length ◽  
Original Matrix ◽  
Numerical Work ◽  
Value Decomposition ◽  
Pseudo Inverse

The calculation of the pseudo inverse of a matrix is intimately related to the singular value decomposition which applies to any matrix be it singular or not and square or not. The matrices involved in the singular value decomposition of a matrix A are formed with the orthogonal eigen vectors of the symmetric matrices ATA and AAT associated with their nonzero eigenvalues which forms a diagonal matrix. If instead of using the eigenvectors, which are difficult to calculate, we use any set of vectors that span the same spaces, which are easier to obtain, we can get simpler expressions for calculating the pseudoinverse, although the diagonal matrix of eigenvalues is filled. All numerical work to obtain the pseudo inverse whose components are rational numbers when the original matrix is also rational reduces to elementary row operations. We can, thus, generalize the least-squares/ minimum-length normal equations for full-rank matrices and solve said problems and obtain the pseudo inverse in terms of A and AT. without solving any eigen problems or factoring matrices.

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