Singular Value Decomposition for Constrained Dynamical Systems

Abstract

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. (Hi) 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.

A direct violation correction method in numerical simulation of constrained multibody systems

“…However, the independent base vectors of the tangent space of the constraint surface must be calculated. Kim and Vanderploeg (1986) used QR decomposition technique, Singh and Likins (1985) used Singular Value Decomposition method, Liang and Lance (1987) applied Gramm-Schmidt orthonormalization method to generate these independent base vectors [9±11]. It will be comparatively expensive if the number of the generalized coordinates is large and these vectors are not unique.…”

Constraint Gradient Projective Method for Stabilized Dynamic Simulation of Constrained Multibody Systems

“…It can be integrated in time to obtain kinetic motion of the system as well as constraint reactions. Although constraints at the acceleration level will be immanently satisfied since (15) is included in mathematical model (13) and will be explicitly solved during integration, the numerical non-stability of (15) can induce constraints violation at the both position and velocity level [9].…”

“…If, beside holonomic constrains, the additional nonholonomic constraints given in Pfaffian form (10) are imposed on the system, the procedures [9,11], similar to those described above, allow for shaping of mathematical models given by (13), (17), (18).…”

On-Line Symbolic Constraint Embedding for Simulation of Hybrid Dynamical Systems

“…In [12,38] Design II makes use of numerical projection methods. The n unconstrained equations (9) are projected onto the p ∆ = n−m dimensional constraint manifold by pre-multiplying with a p × n projection matrix R, which is produced numerically during solution from the constraint Jacobian B via an SVD [34], QR decomposition [19], or Gauss triangularization [33]. The projection matrix R satisfies the equation…”