Reordering and Partitioning Jacobian Matrices Using Graph-Spectral Method

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.

[1]  K. H. Hunt.  Robot Kinematics—A Compact Analytic Inverse Solution for Velocities , 1987 .

[2]  Marcelo Gattass,et al.  Node and element resequencing using the laplacian of a finite element graph: part i---general concep , 1994 .

[3]  Bruno Siciliano,et al.  Modeling and Control of Robot Manipulators , 1995 .

[4]  Yoshihiko Nakamura,et al.  Inverse kinematic solutions with singularity robustness for robot manipulator control , 1986 .

[5]  Hamid Jafarkhani,et al.  in Electrical and Computer Engineering , 2005 .

[6]  Ali Kaveh,et al.  A multi-level finite element nodal ordering using algebraic graph theory , 2001 .

[7]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[8]  K. Waldron,et al.  A Study of the Jacobian Matrix of Serial Manipulators , 1985 .

[9]  Craig R. Tischler,et al.  Real-Time Solution of the Inverse Kinematic-Rate Problem , 2000, Int. J. Robotics Res..

[10]  Daniel Martins,et al.  Hierarchical kinematic analysis of robots , 2003 .

[11]  Shih-Liang Wang,et al.  A Study of the Singular Configurations of Serial Manipulators , 1987 .

[12]  L. Siciliano.  Modelling and Control of Robot Manipulators , 2000 .

[13]  Albert Y. Zomaya,et al.  Computing robot Jacobians on meshes with multiple buses , 1999, Microprocess. Microsystems.

[14]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[15]  Alex Pothen,et al.  PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS* , 1990 .

[16]  Michael Holzrichter,et al.  A Graph Based Method for Generating the Fiedler Vector of Irregular Problems , 1999, IPPS/SPDP Workshops.