A new factorization of the mass matrix for optimal serial and parallel calculation of multibody dynamics

This paper describes a new factorization of the inverse of the joint-space inertia matrix M. In this factorization, M−1 is directly obtained as the product of a set of sparse matrices wherein, for a serial chain, only the inversion of a block-tridiagonal matrix is needed. In other words, this factorization reduces the inversion of a dense matrix to that of a block-tridiagonal one. As a result, this factorization leads to both an optimal serial and an optimal parallel algorithm, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors. The novel feature of this algorithm is that it first calculates the interbody forces. Once these forces are known, the accelerations are easily calculated. We discuss the extension of the algorithm to the task of calculating the forward dynamics of a kinematic tree consisting of a single main chain plus any number of short side branches. We also show that this new factorization of M−1 leads to a new factorization of the operational-space inverse inertia, Λ−1, in the form of a product involving sparse matrices. We show that this factorization can be exploited for optimal serial and parallel computation of Λ−1, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors.

[1]  Amir Fijany,et al.  Parallel O(log N) algorithms for computation of manipulator forward dynamics , 1994, IEEE Trans. Robotics Autom..

[2]  K. Anderson,et al.  A logarithmic complexity divide-and-conquer algorithm for multi-flexible articulated body dynamics , 2007 .

[3]  Katsu Yamane,et al.  Comparative Study on Serial and Parallel Forward Dynamics Algorithms for Kinematic Chains* , 2009, Int. J. Robotics Res..

[4]  Gene H. Golub,et al.  Matrix computations , 1983 .

[5]  Oussama Khatib,et al.  A unified approach for motion and force control of robot manipulators: The operational space formulation , 1987, IEEE J. Robotics Autom..

[6]  C. S. George Lee,et al.  Efficient Parallel Algorithm for Robot Inverse Dynamics Computation , 1986, IEEE Transactions on Systems, Man, and Cybernetics.

[7]  Kurt S. Anderson,et al.  Highly Parallelizable Low-Order Dynamics Simulation Algorithm for Multi-Rigid-Body Systems , 2000 .

[8]  Amir Fijany,et al.  A technique for analyzing constrained rigid-body systems, and its application to the constraint force algorithm , 1999, IEEE Trans. Robotics Autom..

[9]  Amir Fijany,et al.  Schur complement factorizations and parallel O(log N) algorithms for computation of operational space mass matrix and its inverse , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[10]  Abhinandan Jain,et al.  A Spatial Operator Algebra for Manipulator Modeling and Control , 1989, Proceedings, 1989 International Conference on Robotics and Automation.

[11]  Roy Featherstone,et al.  A Divide-and-Conquer Articulated-Body Algorithm for Parallel O(log(n)) Calculation of Rigid-Body Dynamics. Part 1: Basic Algorithm , 1999, Int. J. Robotics Res..

[12]  Oussama Khatib,et al.  Inertial Properties in Robotic Manipulation: An Object-Level Framework , 1995, Int. J. Robotics Res..

[13]  Richard H. Lathrop,et al.  Parallelism in Manipulator Dynamics , 1985 .

[14]  Guillermo Rodríguez-Ortiz,et al.  Spatial operator factorization and inversion of the manipulator mass matrix , 1992, IEEE Trans. Robotics Autom..

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

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

[17]  Ming C. Lin,et al.  Adaptive dynamics of articulated bodies , 2005, SIGGRAPH 2005.

[18]  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).

[19]  Roy Featherstone,et al.  Rigid Body Dynamics Algorithms , 2007 .

[20]  Roy Featherstone,et al.  A Divide-and-Conquer Articulated-Body Algorithm for Parallel O(log(n)) Calculation of Rigid-Body Dynamics. Part 2: Trees, Loops, and Accuracy , 1999, Int. J. Robotics Res..