Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T05:00:08.586Z Has data issue: false hasContentIssue false
  • English
  • Français

O(n) mass matrix inversion for serial manipulators and polypeptide chains using Lie derivatives

Published online by Cambridge University Press:  01 November 2007

Kiju Lee ,
Yunfeng Wang  and
Gregory S. Chirikjian
Kiju Lee
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA.
Yunfeng Wang
Affiliation:
Department of Mechanical Engineering, The College of New Jersey, Ewing, NJ 08628, USA.
Gregory S. Chirikjian*
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA.
*
*Corresponding author. E-mail: gregc@jhu.edu
Get access Rights & Permissions [Opens in a new window]

Summary

Over the past several decades, a number of O(n) methods for forward and inverse dynamics computations have been developed in the multibody dynamics and robotics literature. A method was developed by Fixman in 1974 for O(n) computation of the mass-matrix determinant for a serial polymer chain consisting of point masses. In other of our recent papers, we extended this method in order to compute the inverse of the mass matrix for serial chains consisting of point masses. In the present paper, we extend these ideas further and address the case of serial chains composed of rigid-bodies. This requires the use of relatively deep mathematics associated with the rotation group, SO(3), and the special Euclidean group, SE(3), and specifically, it requires that one differentiates real-valued functions of Lie-group-valued argument.

Keywords

Serial manipulatorsMass-matrix inversionForward dynamicsPolymer chainsPolypeptides
Type
Article
Information
Robotica , Volume 25 , Issue 6 , November 2007 , pp. 739 - 750
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Vereshchagin, A. F., “Gauss principle of least constraint for modeling the dynamics of automatic manipulators using a digital computer,” Sov. Phys.-Doklady 20 (1), pp. 33 (1975).Google Scholar
2.Luh, J. Y. S., Walker, M. W. and Paul, R. P., “On-line computational scheme for mechanical manipulators,” Trans. ASME J. Dy. Sys. Meas. Control 102: 69–76 (1980).CrossRefGoogle Scholar
3.Hollerbach, J. M., “A recursive Lagrangian formulation of manipulator dynamics and a comparative study of dynamics formulation complexity,” In: Tutorial on Robotics (Lee, C. S. G., Gonzalez, R. C. and Fu, K. S., ed.) (IEEE Computer Society Press, Silver Spring, Maryland, 1983), pp. 111–117.Google Scholar
4.Rodriguez, G., “Kalman Filtering, smoothing, and recursive robot arm forward and inverse dynamics',” IEEE J. Robot. Automat. RA-3 (6), 624–639 (1987).CrossRefGoogle Scholar
5.Rodriguez, G. and Keutz-Delgado, K., “Spatial operator factorization and inversion of the manipulator mass matrix,” IEEE Trans. Robot. Automat. 8 (1), 65–76 (1992).CrossRefGoogle Scholar
6.Jain, A. and Rodriguez, G., “Recursive flexible multibody system dynamics using spatial operators,” J. Guid. Control Dyn. 15 (6), 1453–1466 (1992).CrossRefGoogle Scholar
7.Saha, S. K., “A decomposition of the manipulator inertia matrix,” IEEE Trans. Robot. Automat 13 (2), 301–304 (1997).CrossRefGoogle Scholar
8.Saha, S. K., “Analytical expression for the inverted inertia matrix of serial robots,” Int. J. Robotics Research 18 (1), 116–124 (1999).Google Scholar
9.Angeles, J. and Ma, O., “Dynamic simulation of n-axis serial robotic manipulators using a natural orthogonal complement,” Int. J. Robot. Res. 7 (5), 32–47 (1998).CrossRefGoogle Scholar
10.Anderson, K. S. and Duan, S., “Highly Parallelizable Low Order Dynamics Algorithm for Complex Multi-Rigid-Body Systems,” AIAA J. Guid. Control Dy. 23 (2), 355–364 (2000).CrossRefGoogle Scholar
11.Anderson, K. S. and Duan, S., “A Hybrid Parallelizable Low Order Algorithm for Dynamics of Multi-Rigid-Body Systems: Part I, Chain Systems,” J. Math. Comput. Mod. 30, 193–215 (1999).CrossRefGoogle Scholar
12.Anderson, K. S. and Duan, S., “Parallel Implementation of a Low Order Algorithm for Dynamics of Multibody Systems on a Distributed Memory Computing System,” J. Eng. Comput. 16 (2), 96–108 (2000).Google Scholar
13.Anderson, K. S., “An Order-N Formulation for the Motion Simulation of General Constrained Multi-Rigid-Body Systems,” J. Comput. Str. 43 (3), 565–579 (1992).CrossRefGoogle Scholar
14.Anderson, K. S., “An Order-N Formulation for the Motion Simulation of General Multi-Rigid-Body Tree Systems,” J. Comput. Str. 46 (3), 547–559 (1991).CrossRefGoogle Scholar
15.Anderson, K. S. and Critchley, J. H., “Improved ‘Order-N’ performance algorithm for the simulation of constrained multi-rigid-body dynamic systems,” Multibody syst. dyn. 9, 185–212, (2003).CrossRefGoogle Scholar
16.Featherstone, R., “The calculation of robot dynamics using articulated-body inertia,” Int. J. Robot. Res. 2 (1), 13–30 (1983).CrossRefGoogle Scholar
17.Featherstone, R., “Efficient factorization of the joint space inertia matrix for branched kinematic trees,” submitted to Intl. Journal of Robotics Research.Google Scholar
18.Ploen, S. R. and Park, F. C., “Coordinate-Invariant Algorithms for Robot Dynamics,” IEEE Trans. Robot. Automat. 15 (6)1130–1135 (1999).CrossRefGoogle Scholar
19.Naudet, J. and Lefeber, D., “General Formulation of an Efficient Recursive Algorithm based on Canonical Momenta for Forward Dynamics of Closed-loop Multibody Systems,” Proceedings of IDETC/CIE, Long Beach, CA, USA (Sep. 2005) DETC 2005-84917.CrossRefGoogle Scholar
20.Naudet, J., Lafeber, D. and Terze, Z., “General Formulation of an Efficient Recursive Algorithm based on Canonical Momenta for Forward Dynamics of Open-loop Multibody Systems,” Multibody Dynamics, 2003.Google Scholar
21.Fixman, M., “Classical statistical mechanics of constraints: A theorem and application to polmers,” Proc. Nat. Acad. Sci. 71 (8), 3050–3053 (Aug. 1974).CrossRefGoogle Scholar
22.Lee, K. and Chirikjian, G. S., “A New Perspective on O(n) Mass-Matrix Inversion for Serial Revolute Manipulators,” IEEE International Conference on Robotics and Automation, Barcelona, Spain (Apr. 2005).Google Scholar
23.Lee, K., Chirikjian, G. S., “A New Method for O(n) Inversion of the Mass Matrix,” International Symposium on Multibody Systems and Mechatronics, Uberlandia, Brazil (Mar. 2005).Google Scholar
24.Wang, Y. and Chirikjian, G. S., “A New O(n) Method for Inverting the Mass-Matrix for Serial Chains Composed of Rigid Bodies,” Proceedings of IDETC/CIE, Long Beach, CA, USA (Sep. 2005) DETC 2005-84365.CrossRefGoogle Scholar
25.Fijani, A. and D'Eleuterio, G. M. T., “Parallel O(log N) Algorithms for Computation of Manipulator Forward Dynamics,” IEEE Trans. Robot. Automat., 11 (3), (1995) 389–400.CrossRefGoogle Scholar
26.Golub, G. H. and Vam Loan, C. F., Matrix Computations (Johns Hopkins University Press, Baltimore, MD, USA and London, UK, 1996).Google Scholar
27.Chirikjian, G. S. and Kyatkin, A. B., Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, Boca Raton, FL, 2001).Google Scholar
28.Craig, J. J., Introduction to Robotics, 2nd ed. (Addison-Wesley, Reading, MA, 1989).Google Scholar
29.Paul, R. P. and Zhang, H., “Computationally efficient kinematics for manipulators with spherical wrists,” International Journal of Robotic Research, 5 (2), (1986) 32–44.CrossRefGoogle Scholar
30.Pappu, R. V., Srinivasan, R. and Rose, G. D., “The Flory isolated-pair hypothesis is not valid for polypeptide chains: Implications for protein folding,” Biophysics 97 (23), pp. 12565–12570, (2000).Google Scholar
31.Naudet, J., “Forward Dynamics of Multibody Systems: A Recursive Hamiltonian Approach Ph.D. Thesis (Brussels, Belgium Vrije Universiteit Brussel, 2005).Google Scholar