Projection continuation for minimal coordinate set formulation and singularity detection of redundantly constrained system dynamics

Abstract

The formulation of (possibly redundantly) constrained system dynamics using coordinate projection onto a subspace locally tangent to the constraint manifold is revisited using the QR factorization of the constraint Jacobian matrix, using column pivoting to identify a suitable subspace, possibly detect any singular configurations that may arise, and extract it. The evolution of the QR factorization is integrated along with that of the constraint Jacobian matrix as the solution evolves, generalizing to redundant constraints a recently proposed true continuation algorithm that tracks the evolution of the subspace of independent coordinates. The resulting subspace does not visibly affect the quality of the solution, as it is merely a recombination of that resulting from the blind application of the QR factorization but avoids the artificial algorithmic irregularities or discontinuities in the generalized velocities that could otherwise result from arbitrary reparameterizations of the coordinate set, and identifies and discriminates any further possible motions that arise at singular configurations. The characteristics of the proposed subspace evolution approach are exemplified by solving simple problems with incremental levels of redundancy and singularity orders.