Approximation of Jacobian inverse kinematics algorithms

Krzysztof Tchoń; Joanna Karpińska; Mariusz Janiak

International Journal of Applied Mathematics and Computer Science (2009)

  • Volume: 19, Issue: 4, page 519-531
  • ISSN: 1641-876X

Abstract

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This paper addresses the synthesis problem of Jacobian inverse kinematics algorithms for stationary manipulators and mobile robots. Special attention is paid to the design of extended Jacobian algorithms that approximate the Jacobian pseudoinverse algorithm. Two approaches to the approximation problem are developed: one relies on variational calculus, the other is differential geometric. Example designs of the extended Jacobian inverse kinematics algorithm for 3DOF manipulators as well as for the unicycle mobile robot illustrate the theoretical concepts.

How to cite

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Krzysztof Tchoń, Joanna Karpińska, and Mariusz Janiak. "Approximation of Jacobian inverse kinematics algorithms." International Journal of Applied Mathematics and Computer Science 19.4 (2009): 519-531. <http://eudml.org/doc/207952>.

@article{KrzysztofTchoń2009,
abstract = {This paper addresses the synthesis problem of Jacobian inverse kinematics algorithms for stationary manipulators and mobile robots. Special attention is paid to the design of extended Jacobian algorithms that approximate the Jacobian pseudoinverse algorithm. Two approaches to the approximation problem are developed: one relies on variational calculus, the other is differential geometric. Example designs of the extended Jacobian inverse kinematics algorithm for 3DOF manipulators as well as for the unicycle mobile robot illustrate the theoretical concepts.},
author = {Krzysztof Tchoń, Joanna Karpińska, Mariusz Janiak},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {robot inverse kinematics; extended Jacobian; Jacobian pseudoinverse; approximation},
language = {eng},
number = {4},
pages = {519-531},
title = {Approximation of Jacobian inverse kinematics algorithms},
url = {http://eudml.org/doc/207952},
volume = {19},
year = {2009},
}

TY - JOUR
AU - Krzysztof Tchoń
AU - Joanna Karpińska
AU - Mariusz Janiak
TI - Approximation of Jacobian inverse kinematics algorithms
JO - International Journal of Applied Mathematics and Computer Science
PY - 2009
VL - 19
IS - 4
SP - 519
EP - 531
AB - This paper addresses the synthesis problem of Jacobian inverse kinematics algorithms for stationary manipulators and mobile robots. Special attention is paid to the design of extended Jacobian algorithms that approximate the Jacobian pseudoinverse algorithm. Two approaches to the approximation problem are developed: one relies on variational calculus, the other is differential geometric. Example designs of the extended Jacobian inverse kinematics algorithm for 3DOF manipulators as well as for the unicycle mobile robot illustrate the theoretical concepts.
LA - eng
KW - robot inverse kinematics; extended Jacobian; Jacobian pseudoinverse; approximation
UR - http://eudml.org/doc/207952
ER -

References

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  1. Baillieul, J. (1985). Kinematic programming alternatives for redundant manipulators, Proceedings of the 1985 IEEE International Conference on Robotics and Automation, St. Louis, LO, USA, pp. 722-728. 
  2. Chitour, Y. and and Sussmann, H. J. (1998). Motion planning using the continuation method, in J. Baillieul, S. S. Sastry and H. J. Sussmann (Eds), Essays on Mathematical Robotics, Springer-Verlag, New York, NY, pp. 91-125. Zbl0946.70003
  3. Gelfand, I. M. and Fomin, S. V. (1963). Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ. Zbl0127.05402
  4. Janiak, M. and Tchoń, K. (2008). Extended Jacobian inverse kinematics and approximation of distributions, in J. Lenarcic and Ph. Wenger (Eds), Advances in Robot Kinematics, Springer Science+Business Media, Berlin, pp. 137-146. 
  5. Klein, Ch. A. and Huang, C. (1983). Review of the pseudoinverse control for use with kinematically redundant manipulators, IEEE Transactions on Systems, Man and Cybernetics 13(3): 245-250. 
  6. Klein, Ch. A., Chu-Jenq, C. and Ahmed, Sh. (1995). A new formulation of the extended Jacobian method and its use in mapping algorithmic singularities for kinematically redundant manipulators, IEEE Transactions on Robotics and Automation 11(1): 50-55. 
  7. Roberts, R. G. and Maciejewski, A. A. (1992). Nearest optimal repeatable control strategies for kinematically redundant manipulators, IEEE Transactions on Robotics and Automation 8(3): 327-337. 
  8. Roberts, R. G. and Maciejewski, A. A. (1993). Repeatable generalized inverse control strategies for kinematically redundant manipulators, IEEE Transactions on Automatic Control 38(5): 689-699. Zbl0800.93537
  9. Roberts, R. G. and Maciejewski, A. A. (1993). Singularities, stable surfaces, and the repeatable behavior of kinematically redundant manipulators, International Journal of Robotics Research 13(1): 207-213. 
  10. Shamir, T. and Yomdin, Y. (1988). Repeatability of redundant manipulators: Mathematical solution of the problem, IEEE Transactions on Automatic Control 33(11): 1004-1009. Zbl0672.93055
  11. Sluis, W. M., Banaszuk, A., Hauser, J. and Murray, R. M.(1996). A homotopy algorithm for approximating geometric distributions by integrable systems, Systems & Control Letters 27(5): 285-291. Zbl0866.93021
  12. Tchoń, K. (2002). Repeatability of inverse kinematics algorithms for mobile manipulators, IEEE Transactions on Automatic Control 47(8): 1376-1380. 
  13. Tchoń, K. (2007). Continuation method in robotics, Proceedings of the 7th Conference on Computer Methods and Systems, Cracow, Poland, pp. 17-24. 
  14. Tchoń, K. (2008). Optimal extended Jacobian inverse kinematics algorithms for robotic manipulators, IEEE Transactions on Robotics 28(6): 1440-1445. 
  15. Tchoń, K. and Jakubiak, J. (2006). Extended Jacobian inverse kinematics algorithm for non-holonomic mobile robots, International Journal of Control 79(8): 895-909. Zbl1114.70006

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