Summary
Here we present the main lines of a theory we developed in a series of previous papers, about the motion planning problem in robotics. We illustrate the theory with a few academic examples.
Our theory, although at its starting point, looks promising even from the “constructive” point of view. It does not mean that we have precise general algorithms, but the theory contains this potentiality.
The robot is given under the guise of a set of linear kinematic constraints (a distribution). The cost is specified by a riemannian metric on the distribution. Given a non-admissible path for the robot, i.e. a path that does not satisfy the kinematic constraints), our theory allows to evaluate precisely and constructively the “metric complexity” and the “entropy” of the problem. This estimation of metric complexity provides methods for approximation of nonadmissible paths by admissible ones, while the estimation of entropy provides methods for interpolation of the nonadmissible path by admissible.
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References
. Agrachev AA, Chakir HEA, Gauthier JP (1998) Subriemannian Metrics on R3, in Geometric Control and Nonholonomic Mechanics. In: Mexico City 1996, 2976. Proc. Can. Math. Soc. 25.
. Agrachev AA, Gauthier JP (1999) Subriemannian Metrics and Isoperimetric Problems in the Contact Case. In: in honor L. Pontriaguin, 90th birthday commemoration. Contemporary Maths, 64:5–48 (Russian). English version: journal of Mathematical sciences, 103(6):639–663.
Charlot G (2002) Quasi Contact SR Metrics: Normal Form in R2n, Wave Front and Caustic in R4; Acta Appl. Math., 74(3):217–263.
Chakir HEA, Gauthier JP, Kupka IAK (1996) Small Subriemannian Balls on R3, Journal of Dynamical and Control Systems, 2(3):359–421.
. Clarke FH (1983) Optimization and nonsmooth analysis. John Wiley & Sons.
Gauthier JP, Monroy-Perez F, Romero-Melendez C (2004) On complexity and Motion Planning for Corank one SR Metrics. COCV 10:634–655.
Gauthier JP, Zakalyukin V (2005) On the codimension one Motion Planning Problem. JDCS, 11(1):73–89.
Gauthier JP, Zakalyukin V (2005) On the One-Step-Bracket-Generating Motion Planning Problem. JDCS, 11(2) 215–235.
Gauthier JP, Zakalyukin V (2005) Robot Motion Planning, a wild case. Proceedings of the Steklov Institute of Mathematics, 250:56–69.
. Gauthier JP, Zakalyukin V (2006) On the motion planning problem, complexity, entropy, and nonholonomic interpolation. Journal of dynamical and control systems, 12(3).
. Gauthier JP, Zakalyukin V (2007) Entropy estimations for motion planning problems in robotics. In: Volume In honor of Dmitry Victorovich Anosov, Proceedings of the Steklov Institute of Mathematics, 256(1):62–79.
. Gromov M (1996) Carnot Caratheodory Spaces Seen from Within. In: Bellaiche A, Risler J.J, 79–323. Birkhauser.
Jean F (2001) Complexity of Nonholonomic Motion Planning. International Journal on Control, 74(8):776–782. 210 Jean-Paul Gauthier and Vladimir Zakalyukin.
Jean F (2003) Entropy and Complexity of a Path in SR Geometry. COCV, 9:485–506.
Jean F, Falbel E (2003) Measures and transverse paths in SR geometry. Journal d’Analyse Mathématique, 91:231–246.
. Laumond JP (ed) (1998) Robot Motion Planning and Control, Lecture notes in Control and Information Sciences 229, Springer Verlag.
Love AEH (1944) A Treatise on the Mathematical Theory of Elasticity, forth edition. Dover, New-York.
Pontryagin L, Boltyanski V, Gamkelidze R, Michenko E (1962) The Mathematical theory of optimal processes. Wiley, New-York.
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Gauthier, JP., Zakalyukin, V. (2008). Nonholonomic Interpolation for Kinematic Problems, Entropy and Complexity. In: Sarychev, A., Shiryaev, A., Guerra, M., Grossinho, M.d.R. (eds) Mathematical Control Theory and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69532-5_11
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DOI: https://doi.org/10.1007/978-3-540-69532-5_11
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