Using Lie Group Symmetries for Fast Corrective Motion Planning

  • Konstantin Seiler
  • Surya P. N. Singh
  • Hugh Durrant-Whyte
Part of the Springer Tracts in Advanced Robotics book series (STAR, volume 68)

Abstract

For a mechanical system it often arises that its planned motion will need to be corrected either to refine an approximate plan or to deal with disturbances. This paper develops an algorithmic framework allowing for fast and elegant path correction for nonholonomic underactuated systems with Lie group symmetries, which operates without the explicit need for control strategies. These systems occur frequently in robotics, particularly in locomotion, be it ground, underwater, airborne, or surgical domains. Instead of reintegrating an entire trajectory, the method alters small segments of an initial trajectory in a consistent way so as to transform it via symmetry operations. This approach is demonstrated for the cases of a kinematic car and for flexible bevel tip needle steering, showing a prudent and simple, yet computationally tractable, trajectory correction.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bloch, A.M.: Nonholonomic Mechanics and Control. In: Interdisciplinary Applied Mathematics, vol. 24. Springer, New York (2003)Google Scholar
  2. 2.
    Byrd, R.H., Nocedal, J., Waltz, R.A.: Knitro: An integrated package for nonlinear optimization. In: Large Scale Nonlinear Optimization, pp. 35–59. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Canny, J.F.: The complexity of robot motion planning. MIT Press, Cambridge (1988)Google Scholar
  4. 4.
    Cheng, P., Frazzoli, E., LaValle, S.M.: Improving the performance of sampling-based motion planning with symmetry-based gap reduction. IEEE Transactions on Robotics 24(2), 488–494 (2008)CrossRefGoogle Scholar
  5. 5.
    Kanayama, Y., Kimura, Y., Miyazaki, F., Noguchi, T.: A stable tracking control method for an autonomous robot. In: Proceedings of the IEEE International Conference on Robotics and Automation, vol. 1, pp. 384–389 (1990)Google Scholar
  6. 6.
    Kelly, A., Nagy, B.: Reactive nonholonomic trajectory generation via parametric optimal control. The International Journal of Robotics Research 22(7-8), 583–601 (2003)CrossRefGoogle Scholar
  7. 7.
    Kobilarov, M., Desbrun, M., Marsden, J.E., Sukhatme, G.S.: A discrete geometric optimal control framework for systems with symmetries. In: Proceedings of Robotics: Science and Systems, Atlanta, GA, USA (2007)Google Scholar
  8. 8.
    Koon, W.S., Marsden, J.E.: Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction. SIAM Journal on Control and Optimization 35(3), 901–929 (1995)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Lamiraux, F., Bonnafous, D., Lefebvre, O.: Reactive path deformation for nonholonomic mobile robots. IEEE Transactions on Robotics 20(6), 967–977 (2004)CrossRefGoogle Scholar
  10. 10.
    Latombe, J.C.: Robot Motion Planning. Kluwer, Boston (1991)Google Scholar
  11. 11.
    LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006)MATHCrossRefGoogle Scholar
  12. 12.
    Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994)MATHGoogle Scholar
  13. 13.
    Ollero, A., Heredia, G.: Stability analysis of mobile robot path tracking. In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 461–466 (1995)Google Scholar
  14. 14.
    Ostrowski, J.P.: Computing reduced equations for robotic systems with constraintsand symmetries. IEEE Transactions on Robotics and Automation 15(1), 111–123 (1999)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Park, W., Reed, K.B., Chirikjian, G.S.: Estimation of model parameters for steerable needles. In: IEEE International Conference on Robotics and Automation, pp. 3703–3708 (2010)Google Scholar
  16. 16.
    Shewchuk, J.R.: An introduction to the conjugate gradient method without the agonizing pain. Tech. rep., Carnegie Mellon University Pittsburgh (1994)Google Scholar
  17. 17.
    Webster III, R.J., Cowan, N.J., Chirikjian, G.S., Okamura, A.M.: Nonholonomic modeling of needle steering. In: Proc. 9th International Symposium on Experimental Robotics (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Konstantin Seiler
    • 1
  • Surya P. N. Singh
    • 1
  • Hugh Durrant-Whyte
    • 1
  1. 1.Australian Centre for Field RoboticsUniversity of SydneyAustralia

Personalised recommendations