Orthogonal Square Root Eigenfactor Parameterization of Mass Matrices

Abstract

An improved method is presented to parameterize a smoothly time varying, symmetric, positive dee nite system mass matrix M(t) in terms of the instantaneous eigenfactors, namely, the eigenvalues and eigenvectors of M(t). Differential equations are desired whose solutions generate the instantaneous spectral decomposition of M(t). The derivationmakesuseofthefactthattheeigenvectormatrixisorthogonaland,thus,evolvesanalogously to ahigherdimensional rotation matrix. Careful attention is given to cases where some eigenvalues and/or their derivatives are equal or near equal. A robust method is presented to approximate the corresponding eigenvector derivatives in these cases, which ensures that the resulting eigenvectors still diagonalize the instantaneous M(t) matrix. This method is also capable of handling the rare case of discontinuous eigenvectors, which may only occur if both the corresponding eigenvalues and their derivatives are equal. I. Introduction M OST multibody dynamical systems such as multilink robots have cone guration-dependent mass matrices. This dependency makes the mass matrix vary with time. Solving such dynamical systems involves performing an inverse of the system mass matrix at each integration step. Finding this inverse is computationally dife cult and expensive for large systems. Furthermore, standard inverse and linear equations solution techniques are not easily parallelizable and, therefore, cannot take full advantage of modern parallel computing systems. Amethodisintroducedthatparameterizesthesymmetric,positive dee nite mass matrix in terms of its eigenfactors, i.e., eigenvectors and eigenvalues. Instead of forward integrating the original mass matrix differential equation directly, only the eigenfactors themselves are forward integrated. The resulting formulation is one that could be easily implemented on a massively parallel computer system. A paper by Oshman and Bar-Itzhack 1 introduces an important orthogonal square root eigenfactor parameterization to solve the differential matrix Riccati equation. However, some details of their treatmentofequalornear-equaleigenvalues werefoundtobeincorrect, and the case of discontinuous eigenvectors was not accounted for. This paper provides an approximate treatment of the near-equal eigenvalue case and an associated error analysis where the symmetric,positive dee nite matrix parameterized is a state-dependent mass matrix M.x;t/. Eigenfactor derivatives have been discussed in the literature for quite some time, but they are mostly used to establish modal sensitivities and not to derive eigenfactor differential equations. 2;3 A majority of the engineering literature on eigenfactor derivatives is for the general structural eigenvalue problem .K i ¸iM/vi D 0. This paper deals with the problem where we need to e nd the eigenfactors and their derivatives for given continuous matrices M.x;t/ and P M.x; P