New Differential Geometric Methods in Nonholonomic Path Finding

  • Héctor J. Sussmann
Chapter
Part of the Progress in Systems and Control Theory book series (PSCT, volume 12)

Abstract

We outline three approaches for nonholonomic path finding —nilpotent approximation, highly oscillatory inputs and path deformation— that are based on the use of the techniques of modern geometric optimal control theory, as well as a more classical one —optimal control— where differential geometric methods are also beginning to play a significant role.

Keywords

Nonholonomic System Path Deformation Iterate Integral Smooth Vector Field Optimal Control Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Agrachev, A.A., and R.V. Gamkrelidze, “The exponential representation of flows and chronological calculus,” Matem. Sbornik 109 (149) (1978), pp. 467–532.Google Scholar
  2. [2]
    Brockett, R.W., “Asymptotic stability and feedback stabilization,” in Differential Geometric Control Theory, R.W. Brockett R.S. Millman and H.J. Sussmann Eds., Birkhâuser Boston Inc. (1983), pp. 1–115.Google Scholar
  3. [3]
    Brockett, R.W., and L. Dai, “Non-holonomic kinematics and the role of elliptic functions in constructive controllability,” in Proc. IEEE Int. Conf. Robotics and Automation, Sacramento, CA, April, 1991.Google Scholar
  4. [4]
    Fernandes, C., L. Gurvits and Z.X. Li, “Foundations of nonholonomic motion planning,” in Proc. Int. Conf. Robotics and Automation, Sacramento, CA, April, 1991.Google Scholar
  5. [5]
    Jacobs, P., J.-P. Laumond, and M. Taix, “A complete iterative motion planner for a car-like robot,” In Journées Géometrie Algorithmique, IN-RIA, 1990.Google Scholar
  6. [6]
    Lafferriere, G., “A general strategy for computing steering controls of systems without drift,” to appear in the Proceedings of the 1991 CDC.Google Scholar
  7. [7]
    Lafferriere, G., and H.J. Sussmann, “Motion planning for controllable systems without drift: A preliminary report,” Rutgers University Systems and Control Center Report SYCON-90–04, 1990.Google Scholar
  8. [8]
    Lafferriere, G., and H.J. Sussmann, “Motion planning for controllable systems without drift,” in Proc. Int. Conf. Robotics and Automation, Sacramento, CA, April, 1991, pp. 1148–1153.Google Scholar
  9. [9]
    Laumond, J.P., “Nonholonomic motion planning versus controllability via the multibody car system example,” Stanford University Technical Report STAN-CS-90–1345, December 1990.Google Scholar
  10. [10]
    Laumond, J.P., and T. Siméon, “Motion planning for a two degrees of freedom mobile robot with towing,”In IEEE International Conference on Control and Applications,1989.Google Scholar
  11. [11]
    R. Montgomery, “Geodesics which do not Satisfy the Geodesic Equa,- tions,” preprint, 1991.Google Scholar
  12. [12]
    Murray, R.M., and S.S. Sastry, “Grasping and manipulations using multi-fingered robot hands,” Memorandum No. UCB/ERL M90/24, Electronic Research Laboratory, Univ. of California at Berkeley, March, 1990.Google Scholar
  13. [13]
    Murray, R.M., and S.S. Sastry, “Steering controllable systems,” in Proc. 29th IEEE Conf. Dec. and Control, Honolulu, Hawaii, December, 1990.Google Scholar
  14. [14]
    Murray, R.M., and S.S. Sastry, “Steering nonholonomic systems using sinusoids,” Proc. IEEE Conference on Decision and Control, 1990.Google Scholar
  15. [15]
    Reeds, J.A. and L.A. Shepp, “Optimal paths for a car that goes both forwards and backwards,” Pacific J. Math., 145 (1990), pp 367–393.CrossRefGoogle Scholar
  16. [16]
    R. Strichartz, “Sub-Riemannian Geometry,” J. Dif. Geom. 24, 221–263, (1983).Google Scholar
  17. [17]
    R. Strichartz, “Corrections to ‘Sub-Riemannian Geometry’,” J. Dif. Geom. 30, no. 2, 595–596, (1989).Google Scholar
  18. [18]
    Sussmann, H.J., “A product expansion for the Chen-series,” in Theory and Applications of Nonlinear Control Systems, C.I. Byrnes, and A. Lindquist Eds., North-Holland, 1986, pp. 323–335.Google Scholar
  19. [19]
    Sussmann, H.J., and W. Liu, “Motion planning and approximate tracking for controllable systems without drift,” Proc. 25th Annual Conf. on Inf. Sciences and Systems, Johns Hopkins University, March 20–22, 1991, pp. 547–551.Google Scholar
  20. [20]
    Sussmann, H.J., and W. Liu, “Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories,” Rutgers Center for Systems and Control Technical Report 91–02, February 1991. To appear, in a slightly abridged form, in the Proceedings of the 30th IEEE Conf. Decision and Control, to be held in Brighton, UK, Dec. 1991.Google Scholar
  21. [21]
    Sussmann, H.J., and W. Liu, “Limiting behavior of highly oscillatory inputs, noncommutative formal power series, and the approximation of general paths by admissible trajectories,” in preparation.Google Scholar
  22. [22]
    Sussmann, H.J., and G. Tang, “Shortest paths for the Reeds-Shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control,” Rutgers Center for Systems and Control Technical Report 91–10, September 1991.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Héctor J. Sussmann
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickCanada

Personalised recommendations