Reordering and partitioning Jacobian matrices using graph-spectral method

Abstract

An efficient spectral method is developed for reducing the Jacobian matrix to its block-triangular form in order to solve the inverse kinematics problem. Based on the kinematic structure matrix, the problem of reducing the Jacobian matrix to block-triangular form is transformed into reducing the bandwidth of the matrix. The second Laplacian eigenvector, associated with the bigraph of the structure matrix of the inverse Jacobian, is used to renumber the rows and columns of the Jacobian. The rearranged Jacobian can be divided into subsystems immediately according to the entry value of the Fiedler vector. This algorithm is applied in detail to kinematic analysis for a PUMA robot and T3 robot. Because of the algebraic nature of spectral algorithm, the algorithm can be implemented in a fairly straightforward manner. © 2009 Springer-Verlag Berlin Heidelberg.