Abstract
Nonholonomic motion planning is best understood with some knowledge of the underlying geometry. In this chapter, we first introduce in Section 1 the basic notions of the geometry associated to control systems without drift. In the following sections, we present a detailed study of an example, the car with n trailers, then some general results on polynomial systems, which can be used to bound the complexity of the decision problem and of the motion planning for these systems.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
A. A. Agrachev and R. V. Gamkrelidze, “The exponential representation of flows and the chronological calculus,” Mat. Sbornik (N.S.), 107 (149), 467–532, 639, 1978. English transl.: Math. USSR Sbornik, 35, 727–785, 1979.
A. Bellaïche, “The tangent space in sub-Riemannian Geometry,” in Sub-Riemannian Geometry, A. Bellaïche and J.-J. Risler Ed., Progress in Mathematics, Birkhäuser, 1996.
A. Bellaïche, J.-P. Laumond and J. Jacobs, “Controllability of car-like robots and complexity of the motion planning problem,” in International Symposium on Intelligent Robotics, 322–337, Bangalore, India, 1991.
A. Bellaïche, J.-P. Laumond and J.-J. Risler, “Nilpotent infinitesimal approximations to a control Lie algebra,” in Proceedings of the IFAC Nonlinear Control Systems Design Symposium, 174–181, Bordeaux, France, June 1992.
R. W. Brockett, “Control theory and singular Riemannian geometry,” in New directions in applied mathematics, P. J. Hilton and G. J. Young Ed., Springer-Verlag, 1982.
J. F. Canny, The complexity of robot motion planning, MIT Press, 1998.
W. L. Chow, “Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung,” Math. Ann., 117, 98–115, 1940.
M. Fliess, J. Levine, P. Martin and P. Rouchon, “On differential flat nonlinear systems,” in Proceedings of the IFAC Non linear Control Systems Design Symposium, 408–412, Bordeaux, France, 1992.
A. Gabrielov, F. Jean and J.-J. Risler, Multiplicity of Polynomials on Trajectories of Polynomials Vector Fields in C 3, Preprint, Institut de Mathématiques, Univ. Paris VI, 1997.
A. Gabrielov, J.-M. Lion and R. Moussu, “Ordre de contact de courbes intégrales du plan,” CR Acad. Sci. Paris, 319, 219–221, 1994.
A. Gabrielov, “Multiplicities of Zeroes of Polynomials on Trajectories of Polynomial Vector Fields and Bounds on Degree of Nonholonomy,” Mathematical Research Letters, 2, 437–451, 1995.
A. Gabrielov, Multiplicity of a zero of an analytic function on a trajectory of a vector field, Preprint, Purdue University, March 1997.
C. Godbillon, Géométrie différentielle et Mécanique analytique, Hermann, Paris, 1969.
N. Goodman, Nilpotent Lie groups, Springer Lecture Notes in Mathematics, 562, 1976.
S. Grigoriev, “Complexity of deciding Tarski algebra,” J. Symb. Comp., 5, 37–64, 1988.
M. Gromov, “Carnot-Carathéodory spaces seen from within,” in Sub-Riemannian Geometry, A. Bellaïche and J.-J. Risler Ed., Progress in Mathematics, Birkhäuser, 1996.
J. Heinz, M.-F. Roy and P. Solerno, “Sur la complexité du principe de Tarski-Seidenberg,” Bull. Soc. Math. Fr., 118, 101–126, 1990.
H. Hermes, “Nilpotent and high-order approximations of vector field systems,” SIAM Review, 33 (2), 238–264.
F. Jean, “The car with N trailers: characterization of the singular configurations,” ESAIM: COCV, http://www.emath.fr/cocv/, 1, 241–266, 1996.
J.-P. Laumond, “Singularities and Topological Aspects in Nonholonomic Motion Planning,” in Nonholonomic Motion Planning, 149–199, Z. Li and J. Canny Ed., Klumer Academic Publishers, 1993.
J.-P. Laumond, “Controllability of a multibody mobile robot,” in Proceedings of the International Conference on Advanced robotics and Automation, 1033–1038, Pisa, Italy, 1991.
J.-P. Laumond, P. E. Jacobs, M. Taix and R. M. Murray, “A motion planner for nonholonomic mobile robot,” in LAAS-CNRS Report, Oct. 1992.
J.-P. Laumond and J.-J. Risler, “Nonholonomics systems: Controllability and complexity,” Theoretical Computer Science, 157, 101–114, 1996.
F. Luca and J.-J. Risler, “The maximum of the degree of nonholonomy for the car with n trailers,” 4th IFAC Symposium on Robot Control, 165–170, Capri, 1994.
R. M. Murray and S. S. Sastry, “Nonholonomic Motion Planning: Steering using sinusoids,” IEEE Transactions on Automatic Control, 38 (5), 700–716, May 1993.
A. Nagel, E. M. Stein and S. Wainger, “Metrics defined by vector fields,” Acta Math., 155, 103–147, 1985.
Y. V. Nesterenko, “Estimates for the number of zeros of certain functions,” New Advances in transcendance Theor, 263–269, A. Baker Ed., Cambridge, 1988.
P. K. Rashevsky, “Any two points of a totally nonholonomic space may be connected by an admissible line,” Uch. Zap. Ped. Inst. im. Liebknechta, Ser. Phys. Math., 2, 83–94, 1938 (Russian).
J.-J. Risler, “A bound for the degree of nonholonomy in the plane,” Theoretical Computer Science, 157, 129–136, 1996.
J. T. Schwartz and M. Sharir, “On the ‘Piano Movers’ problem II: general techniques for computing topological properties of real algebraic manifolds,” Advances in Applied Mathematics, 4, 298–351, 1983.
A. Seidenberg, “On the length of Hilbert ascending chain,” Proc. Amer. Math. Soc., 29, 443–450, 1971.
E. Sontag, “Some complexity questions regarding controllability,” Proc. of the 27-th IEEE Conf. on Decision and Control, 1326–1329, Austin, 1988.
O. J. Sørdalen, “Conversion of the kinematics of a car with n trailers into a chain from,” IEEE International Conference on Robotics and Automation, 1993.
O. J. Sørdalen, “On the global degree of non holonomy of a car with n trailers,” 4th IFAC Symposium on Robot Control, 343–348, Capri, 1994.
P. Stefan, “Accessible sets, orbits, and foliations with singularities,” Proc. London Math. Soc., 29 (3), 699–713, 1974.
H. J. Sussmann, “Orbits of families of vector fields and integrability of distributions,” Trans. Amer. Math. Soc., 180, 171–188, 1973.
H. J. Sussmann, “Lie brackets, Real Analyticity and Geometric control,” in Differential Geometric Control Theory, Brockett, Millman, Sussmann Ed., Birkäuser, 1–116, 1982.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag London Limited
About this chapter
Cite this chapter
Bellaïche, A., Jean, F., Risler, J.J. (1998). Geometry of nonholonomic systems. In: Laumond, J.P. (eds) Robot Motion Planning and Control. Lecture Notes in Control and Information Sciences, vol 229. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0036071
Download citation
DOI: https://doi.org/10.1007/BFb0036071
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-76219-5
Online ISBN: 978-3-540-40917-5
eBook Packages: Springer Book Archive