Efficient Factorization of the Joint-Space Inertia Matrix for Branched Kinematic Trees

This paper describes new factorization algorithms that exploit branch-induced sparsity in the joint-space inertia matrix (JSIM) of a kinematic tree. It also presents new formulae that show how the cost of calculating and factorizing the JSIM vary with the topology of the tree. These formulae show that the cost of calculating forward dynamics for a branched tree can be considerably less than the cost for an unbranched tree of the same size. Branches can also reduce complexity; some examples are presented of kinematic trees for which the complexity of calculating and factorizing the JSIM are less than O(n2) and O(n3) , respectively. Finally, a cost comparison is made between an O(n) algorithm and an O(n3) algorithm, the latter incorporating one of the new factorization algorithms. It is shown that the O(n3) algorithm is only 15% slower than the O(n) algorithm when applied to a 30-degrees-of-freedom humanoid, but is 2.6 times slower when applied to an equivalent unbranched chain. This is due mainly to the O(n3) algorithm running about 2.2 times faster on the humanoid than on the chain.

[1]  M. A. Chace,et al.  A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 1 , 1977 .

[2]  N. Orlandea,et al.  A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 2 , 1977 .

[3]  Alan George,et al.  Computer Solution of Large Sparse Positive Definite , 1981 .

[4]  David E. Orin,et al.  Efficient Dynamic Computer Simulation of Robotic Mechanisms , 1982 .

[5]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[6]  R. Featherstone.  The Calculation of Robot Dynamics Using Articulated-Body Inertias , 1983 .

[7]  Martin Otter,et al.  A very efficient algorithm for the simulation of robots and similar multibody systems without invers , 1986 .

[8]  Giuseppe Rodriguez,et al.  Kalman filtering, smoothing, and recursive robot arm forward and inverse dynamics , 1987, IEEE Journal on Robotics and Automation.

[9]  Roy Featherstone,et al.  Robot Dynamics Algorithms , 1987 .

[10]  Edward J. Haug,et al.  A Recursive Formation for Constrained Mechanical Systems Dynamics: Part I, Open Loop Systems , 1987 .

[11]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[12]  E. Haug,et al.  A Recursive Formulation for Constrained Mechanical System Dynamics: Part II. Closed Loop Systems , 1987 .

[13]  Rajnikant V. Patel,et al.  Efficient modeling and computation of manipulator dynamics using orthogonal Cartesian tensors , 1988, IEEE J. Robotics Autom..

[14]  K. Kreutz,et al.  A Spatial Operator Algebra For Manipulator Modeling And Control , 1988, Other Conferences.

[15]  Jorge Angeles,et al.  Dynamic Simulation of n-Axis Serial Robotic Manipulators Using a Natural Orthogonal Complement , 1988, Int. J. Robotics Res..

[16]  Rajnikant V. Patel,et al.  Efficient computation of manipulator inertia matrices and the direct dynamics problem , 1989, IEEE Trans. Syst. Man Cybern..

[17]  D. E. Rosenthal.  An Order n Formulation for Robotic Ststems , 1990 .

[18]  David E. Orin,et al.  Alternate Formulations for the Manipulator Inertia Matrix , 1991, Int. J. Robotics Res..

[19]  Scott McMillan,et al.  Efficient computation of articulated-body inertias using successive axial screws , 1995, IEEE Trans. Robotics Autom..

[20]  Scott McMillan,et al.  Efficient dynamic simulation of an underwater vehicle with a robotic manipulator , 1995, IEEE Trans. Syst. Man Cybern..

[21]  Michael Valášek,et al.  Kinematics and Dynamics of Machinery , 1996 .

[22]  David Baraff,et al.  Linear-time dynamics using Lagrange multipliers , 1996, SIGGRAPH.

[23]  Subir Kumar Saha,et al.  A decomposition of the manipulator inertia matrix , 1997, IEEE Trans. Robotics Autom..

[24]  S. Saha.  Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices , 1999 .

[25]  David E. Orin,et al.  Robot dynamics: equations and algorithms , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[26]  Wisama Khalil,et al.  Modeling, Identification and Control of Robots , 2003 .