Reordering and Partitioning Jacobian Matrices Using Graph-Spectral Method

Authors:
Xuelin Wang;Yujin Hu
Affiliations:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, P.R. China 430074;School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, P.R. China 430074
Venue:
ICIRA '09 Proceedings of the 2nd International Conference on Intelligent Robotics and Applications
Year:
2009

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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.