# American Institute of Mathematical Sciences

September  2013, 3(3): 269-286. doi: 10.3934/mcrf.2013.3.269

## On the motion planning of the ball with a trailer

 1 Aix Marseille Université, CNRS, ENSAM, LSIS, UMR 7296, 13397 Marseille, France, France

Received  November 2012 Revised  March 2013 Published  September 2013

This paper is about motion planing for kinematic systems, and more particularly $\epsilon$-approximations of non-admissible trajectories by admissible ones. This is done in a certain optimal sense.
The resolution of this motion planing problem is showcased through the thorough treatment of the ball with a trailer kinematic system, which is a non-holonomic system with flag of type $(2,3,5,6)$.
Citation: Nicolas Boizot, Jean-Paul Gauthier. On the motion planning of the ball with a trailer. Mathematical Control & Related Fields, 2013, 3 (3) : 269-286. doi: 10.3934/mcrf.2013.3.269
##### References:
 [1] A. A. Agrachev, H. E. A. Chakir and J. P. Gauthier, Subriemannian metrics on $R^3$,, in Geometric Control and Nonholonomic Mechanics, 25 (1998), 29. Google Scholar [2] A. A. Agrachev and J. P. Gauthier, Subriemannian metrics and isoperimetric problems in the contact case,, in honor L. Pontriaguin, 103 (1999), 5. Google Scholar [3] A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric View Point,", Encyclopaedia of Mathematical Sciences, (2004). Google Scholar [4] A. M. Bloch, "Nonholonomic Mechanics and Control,", With the collaboration of J. Baillieul, (2003). doi: 10.1007/b97376. Google Scholar [5] N. Boizot and J-P. Gauthier, Motion planning for kinematic systems,, IEEE Transactions on Automatic Control, 58 (2013), 1430. doi: 10.1109/TAC.2012.2232376. Google Scholar [6] R. W. Brockett and L. Dai, Non-holonomic kinematics and the role of elliptic functions in constructive controllability,, in Z. Li and J. Canny (Eds), 192 (1993), 1. doi: 10.1007/978-1-4615-3176-0_1. Google Scholar [7] H. E. A. Chakir, J. P. Gauthier and I. A. K. Kupka, Small subriemannian balls on $R^3$,, Journal of Dynamical and Control Systems, 2 (1996), 359. doi: 10.1007/BF02269424. Google Scholar [8] J. Dixmier, Sur les représentations unitaires des groupes de lie nilpotents. II.,, (French) Bull. Soc. Math. France, 85 (1957), 325. Google Scholar [9] J. P. Gauthier, F. Monroy-Perez and C. Romero-Melendez, On complexity and motion planning for corank one subriemannian metrics,, ESAIM Control Optim. Calc. Var., 10 (2004), 634. doi: 10.1051/cocv:2004024. Google Scholar [10] J. P. Gauthier and V. Zakalyukin, On the codimension one motion planning problem,, J. Dyn. Control Syst., 11 (2005), 73. doi: 10.1007/s10883-005-0002-6. Google Scholar [11] J. P. Gauthier and V. Zakalyukin, On the One-Step-Bracket-Generating motion planning problem,, J. Dyn. Control Syst., 11 (2005), 215. doi: 10.1007/s10883-005-4171-0. Google Scholar [12] J. P. Gauthier and V. Zakalyukin, Robot motion planning, a wild case,, Proceedings of the Steklov Institute of Mathematics, 250 (2005), 56. Google Scholar [13] J. P. Gauthier and V. Zakalyukin, On the motion planning problem, complexity, entropy, and nonholonomic interpolation,, Journal of Dynamical and Control Systems, 12 (2006), 371. doi: 10.1007/s10450-006-0005-y. Google Scholar [14] J. P. Gauthier and V. Zakalyukin, Entropy estimations for motion planning problems in robotics,, Volume In honor of Dmitry Victorovich Anosov, 256 (2007), 62. doi: 10.1134/S008154380701004X. Google Scholar [15] J. P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls,, SIAM Journ. on Control and Optim., 48 (2010), 3433. doi: 10.1137/090761884. Google Scholar [16] M. Gromov, "Carnot Caratheodory Spaces Seen from Within,", Eds A. Bellaiche, (1996), 79. Google Scholar [17] F. Jean, Complexity of nonholonomic motion planning,, International Journal on Control, 74 (2001), 776. doi: 10.1080/00207170010017392. Google Scholar [18] F. Jean, Entropy and complexity of a path in subriemannian geometry,, ESAIM Control Optim. Calc. Var., 9 (2003), 485. doi: 10.1051/cocv:2003024. Google Scholar [19] F. Jean and E. Falbel, Measures and transverse paths in subriemannian geometry,, Journal d'Analyse Mathématique, 91 (2003), 231. doi: 10.1007/BF02788789. Google Scholar [20] V. Jurdjevic, The geometry of the plate-ball problem,, Archive for Rational Mechanics and Analysis, 124 (1993), 305. doi: 10.1007/BF00375605. Google Scholar [21] J. P. Laumond, (editor), "Robot Motion Planning and Control,", Lecture notes in Control and Information Sciences 229, (1998). doi: 10.1007/BFb0036069. Google Scholar [22] L. Pontryagin, V. Boltyanski, R. Gamkelidze and E. Michenko, "The Mathematical Theory of Optimal Processes,", Wiley, (1962). Google Scholar [23] D. A. Singer, Curves whose curvature depend on the distance from the origin,, the American mathematical Monthly, 106 (1999), 835. doi: 10.2307/2589616. Google Scholar [24] H. J. Sussmann and G. Lafferriere, Motion planning for controllable systems without drift,, In Proceedings of the IEEE Conference on Robotics and Automation, (1991), 109. Google Scholar [25] H. J. Sussmann and W. S. Liu, Lie Bracket extensions and averaging: The single bracket generating case,, in Nonholonomic Motion Planning, (1993), 109. Google Scholar [26]

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##### References:
 [1] A. A. Agrachev, H. E. A. Chakir and J. P. Gauthier, Subriemannian metrics on $R^3$,, in Geometric Control and Nonholonomic Mechanics, 25 (1998), 29. Google Scholar [2] A. A. Agrachev and J. P. Gauthier, Subriemannian metrics and isoperimetric problems in the contact case,, in honor L. Pontriaguin, 103 (1999), 5. Google Scholar [3] A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric View Point,", Encyclopaedia of Mathematical Sciences, (2004). Google Scholar [4] A. M. Bloch, "Nonholonomic Mechanics and Control,", With the collaboration of J. Baillieul, (2003). doi: 10.1007/b97376. Google Scholar [5] N. Boizot and J-P. Gauthier, Motion planning for kinematic systems,, IEEE Transactions on Automatic Control, 58 (2013), 1430. doi: 10.1109/TAC.2012.2232376. Google Scholar [6] R. W. Brockett and L. Dai, Non-holonomic kinematics and the role of elliptic functions in constructive controllability,, in Z. Li and J. Canny (Eds), 192 (1993), 1. doi: 10.1007/978-1-4615-3176-0_1. Google Scholar [7] H. E. A. Chakir, J. P. Gauthier and I. A. K. Kupka, Small subriemannian balls on $R^3$,, Journal of Dynamical and Control Systems, 2 (1996), 359. doi: 10.1007/BF02269424. Google Scholar [8] J. Dixmier, Sur les représentations unitaires des groupes de lie nilpotents. II.,, (French) Bull. Soc. Math. France, 85 (1957), 325. Google Scholar [9] J. P. Gauthier, F. Monroy-Perez and C. Romero-Melendez, On complexity and motion planning for corank one subriemannian metrics,, ESAIM Control Optim. Calc. Var., 10 (2004), 634. doi: 10.1051/cocv:2004024. Google Scholar [10] J. P. Gauthier and V. Zakalyukin, On the codimension one motion planning problem,, J. Dyn. Control Syst., 11 (2005), 73. doi: 10.1007/s10883-005-0002-6. Google Scholar [11] J. P. Gauthier and V. Zakalyukin, On the One-Step-Bracket-Generating motion planning problem,, J. Dyn. Control Syst., 11 (2005), 215. doi: 10.1007/s10883-005-4171-0. Google Scholar [12] J. P. Gauthier and V. Zakalyukin, Robot motion planning, a wild case,, Proceedings of the Steklov Institute of Mathematics, 250 (2005), 56. Google Scholar [13] J. P. Gauthier and V. Zakalyukin, On the motion planning problem, complexity, entropy, and nonholonomic interpolation,, Journal of Dynamical and Control Systems, 12 (2006), 371. doi: 10.1007/s10450-006-0005-y. Google Scholar [14] J. P. Gauthier and V. Zakalyukin, Entropy estimations for motion planning problems in robotics,, Volume In honor of Dmitry Victorovich Anosov, 256 (2007), 62. doi: 10.1134/S008154380701004X. Google Scholar [15] J. P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls,, SIAM Journ. on Control and Optim., 48 (2010), 3433. doi: 10.1137/090761884. Google Scholar [16] M. Gromov, "Carnot Caratheodory Spaces Seen from Within,", Eds A. Bellaiche, (1996), 79. Google Scholar [17] F. Jean, Complexity of nonholonomic motion planning,, International Journal on Control, 74 (2001), 776. doi: 10.1080/00207170010017392. Google Scholar [18] F. Jean, Entropy and complexity of a path in subriemannian geometry,, ESAIM Control Optim. Calc. Var., 9 (2003), 485. doi: 10.1051/cocv:2003024. Google Scholar [19] F. Jean and E. Falbel, Measures and transverse paths in subriemannian geometry,, Journal d'Analyse Mathématique, 91 (2003), 231. doi: 10.1007/BF02788789. Google Scholar [20] V. Jurdjevic, The geometry of the plate-ball problem,, Archive for Rational Mechanics and Analysis, 124 (1993), 305. doi: 10.1007/BF00375605. Google Scholar [21] J. P. Laumond, (editor), "Robot Motion Planning and Control,", Lecture notes in Control and Information Sciences 229, (1998). doi: 10.1007/BFb0036069. Google Scholar [22] L. Pontryagin, V. Boltyanski, R. Gamkelidze and E. Michenko, "The Mathematical Theory of Optimal Processes,", Wiley, (1962). Google Scholar [23] D. A. Singer, Curves whose curvature depend on the distance from the origin,, the American mathematical Monthly, 106 (1999), 835. doi: 10.2307/2589616. Google Scholar [24] H. J. Sussmann and G. Lafferriere, Motion planning for controllable systems without drift,, In Proceedings of the IEEE Conference on Robotics and Automation, (1991), 109. Google Scholar [25] H. J. Sussmann and W. S. Liu, Lie Bracket extensions and averaging: The single bracket generating case,, in Nonholonomic Motion Planning, (1993), 109. Google Scholar [26]
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