Spatial Planning and Geometric Optimization: Combining Configuration Space and Energy Methods

  • Dmytro Chibisov
  • Ernst W. Mayr
  • Sergey Pankratov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3763)

Abstract

In this paper, we propose a symbolic-numerical algorithm for collision-free placement and motion of an object avoiding collisions with obstacles. The algorithm is based on the combination of configuration space and energy approaches. According to the configuration space approach, the position and orientation of the geometric object to be moved or placed is represented as an individual point in a configuration space, in which each coordinate represents a degree of freedom in the position or orientation of this object. The configurations which, due to the presence of obstacles, are forbidden to the object, can be characterized as regions in this configuration space called configuration space obstacles. As will be demonstrated, configuration space obstacles can be computed symbolically using quantifier elimination over the reals and represented by polynomial inequalities. We propose to use the functional representation of semi-algebraic point sets defined by such inequalities, so-called R-functions, to describe nonlinear geometric objects in the configuration space. The potential field defined by R-functions can be used to “move” objects in such a way as to avoid collisions. Introducing the additional function, which forces the object towards the goal position, we reduce the problem of finding collision free path to a solution of the Newton’s equations, which describes the motion of a body in the field produced by the superposition of “attractive” and “repulsive” forces. These equations can be solved iteratively in a computationally efficient manner. Furthermore, we investigate the differential properties of R-functions in order to construct a suitable superposition of attractive and repulsive potentials.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Basu, S., Pollack, R., Roy, M.-F.: Computing Roadmaps of Semi-algebraic Sets on a Variety. In: Cucker, F., Shub, M. (eds.) Foundations of Computational Mathematics, pp. 1–15. Springer, Heidelberg (1997)Google Scholar
  2. 2.
    Canny, J.: The Complexity of Robot Motion Planning. MIT Press, Cambridge (1987)Google Scholar
  3. 3.
    Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Heidelberg (1998)MATHGoogle Scholar
  4. 4.
    Collins, G.E.: Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  5. 5.
    Croft, T.H., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry. Springer, Heidelberg (1991)MATHGoogle Scholar
  6. 6.
    Khatib, O.: Real time obstacle avoidance for manipulators and mobile robots, Internat. J. Robotics 5, 90–99 (1986)CrossRefGoogle Scholar
  7. 7.
    Latombe, J.-C.: Robot Motion Planning. Kluwer Academic Publishers, Dordrecht (1991)Google Scholar
  8. 8.
    Lozano-Perez, T.: Spatial Planning: A Configuration Space Approach. IEEE Transactions on Computers C-32 (2), 108–120 (1983)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Reif, J.H.: Complexity of the Generalized Mover’s Problem. In: Schwartz, T., Sharir, M., Hopcroft, J. (eds.) Planing, Geometry and Complexity of Robot Motion, pp. 267–281. Ablex Publishing Corporation, Greenwich (1987)Google Scholar
  10. 10.
    Requicha, A.: Representations for Rigid Solids: Theory, Methods, and Systems. In: ACM Computing Surveys (CSUR) Archive, vol. 12(4), pp. 437–464. ACM Press, New York (1980)Google Scholar
  11. 11.
    Rimon, E., Koditschek, D.E.: The Construction of Analytic Diffeomorphisms for Exact Robot Navigation on Star Worlds. Transactions of the AMS 327(1), 71–116 (1991)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rvachov, V.L.: Methods of Logic Algebra in Mathematical Physics, Naukova Dumka, Kiev (1974) (in Russian)Google Scholar
  13. 13.
    Pasko, A., Okunev, O., Savchenko, V.: Minkowski sum of point sets defined by inequalities. Computers and Mathematics with Applications 45(10/11), 1479–1487 (2003)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Schwartz, J., Sharir, M.: On the Piano Movers Problem II. General Techniques to Computing Topological Properties of Real Algebraic Manifolds. Advances in Applied Mathematics 4, 298–351 (1983)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Schwartz, J., Yap, C.K.: Advances in Robotics. Lawrence Erlbaum associates, Hillside New Jersey (1986)Google Scholar
  16. 16.
    Shapiro, V.: Theory and Applications of R-Functions: A primer, Technical Report, Cornel University (1991)Google Scholar
  17. 17.
    Shapiro, V., Tsukanov, I.: Implicit Functions With Guaranteed Differential Properties. In: Proceedings of the Fifth Symposium on Solid Modeling SOLID MODELING 1999, Ann Abor, Michigan, pp. 258–269. ACM Press, New York (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Dmytro Chibisov
    • 1
  • Ernst W. Mayr
    • 1
  • Sergey Pankratov
    • 1
  1. 1.Institut für InformatikTechnische Universität MünchenGarchingGermany

Personalised recommendations