On the motion planning of the ball with a trailer

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

On the motion planning of the ball with a trailer

Nicolas Boizot ^1, and Jean-Paul Gauthier ^1,

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

Received November 2012 Revised March 2013 Published September 2013

Abstract
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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 $(2,3,5,6)$.

Keywords: subriemannian geometry, optimal control., robotics, motion planing, Ball with a trailer.

Mathematics Subject Classification: Primary: 49N90, 53C17, 53C80, 70Q05; Secondary: 22E6.

Citation: Nicolas Boizot, Jean-Paul Gauthier. On the motion planning of the ball with a trailer. Mathematical Control and 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, Mexico City 1996, Proc. Can. Math. Soc., 25 (1998), 29-78.
[2]	A. A. Agrachev and J. P. Gauthier, Subriemannian metrics and isoperimetric problems in the contact case, in honor L. Pontriaguin, 90th birthday commemoration, Contemporary Maths, Tome 64, (1999), 5-48, (Russian); English version: Journal of Mathematical Sciences, 103, 639-663.
[3]	A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric View Point," Encyclopaedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer Verlag Berlin Heidelberg, 2004.
[4]	A. M. Bloch, "Nonholonomic Mechanics and Control," With the collaboration of J. Baillieul, P. Crouch and J. Marsden. With scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov. Interdisciplinary Applied Mathematics, 24. Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.
[5]	N. Boizot and J-P. Gauthier, Motion planning for kinematic systems, IEEE Transactions on Automatic Control, 58, (2013), 1430-1442. doi: 10.1109/TAC.2012.2232376.
[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), Nonholonomic Motion Planning, Springer, International Series in Engineering and Computer Science, 192 (1993), 1-22. doi: 10.1007/978-1-4615-3176-0_1.
[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), 359-421. doi: 10.1007/BF02269424.
[8]	J. Dixmier, Sur les représentations unitaires des groupes de lie nilpotents. II., (French) Bull. Soc. Math. France, 85 (1957), 325-388.
[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), 634-655. doi: 10.1051/cocv:2004024.
[10]	J. P. Gauthier and V. Zakalyukin, On the codimension one motion planning problem, J. Dyn. Control Syst., 11 (2005), 73-89. doi: 10.1007/s10883-005-0002-6.
[11]	J. P. Gauthier and V. Zakalyukin, On the One-Step-Bracket-Generating motion planning problem, J. Dyn. Control Syst., 11 (2005), 215-235. doi: 10.1007/s10883-005-4171-0.
[12]	J. P. Gauthier and V. Zakalyukin, Robot motion planning, a wild case, Proceedings of the Steklov Institute of Mathematics, 250 (2005), 56-69.
[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), 371-404. doi: 10.1007/s10450-006-0005-y.
[14]	J. P. Gauthier and V. Zakalyukin, Entropy estimations for motion planning problems in robotics, Volume In honor of Dmitry Victorovich Anosov, Proceedings of the Steklov Mathematical Institute, 256 (2007), 62-79. doi: 10.1134/S008154380701004X.
[15]	J. P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls, SIAM Journ. on Control and Optim., 48 (2010), 3433-3448. doi: 10.1137/090761884.
[16]	M. Gromov, "Carnot Caratheodory Spaces Seen from Within," Eds A. Bellaiche, J. J. Risler, Birkhauser, (1996), 79-323.
[17]	F. Jean, Complexity of nonholonomic motion planning, International Journal on Control, 74 (2001), 776-782. doi: 10.1080/00207170010017392.
[18]	F. Jean, Entropy and complexity of a path in subriemannian geometry, ESAIM Control Optim. Calc. Var., 9 (2003), 485-508. doi: 10.1051/cocv:2003024.
[19]	F. Jean and E. Falbel, Measures and transverse paths in subriemannian geometry, Journal d'Analyse Mathématique, 91 (2003), 231-246. doi: 10.1007/BF02788789.
[20]	V. Jurdjevic, The geometry of the plate-ball problem, Archive for Rational Mechanics and Analysis, 124 (1993), 305-328. doi: 10.1007/BF00375605.
[21]	J. P. Laumond, (editor), "Robot Motion Planning and Control," Lecture notes in Control and Information Sciences 229, Springer Verlag, 1998. doi: 10.1007/BFb0036069.
[22]	L. Pontryagin, V. Boltyanski, R. Gamkelidze and E. Michenko, "The Mathematical Theory of Optimal Processes," Wiley, New-York, 1962.
[23]	D. A. Singer, Curves whose curvature depend on the distance from the origin, the American mathematical Monthly, 106 (1999), 835-841. doi: 10.2307/2589616.
[24]	H. J. Sussmann and G. Lafferriere, Motion planning for controllable systems without drift, In Proceedings of the IEEE Conference on Robotics and Automation, Sacramento, CA, April 1991. IEEE Publications, NewYork, (1991), 109-148.
[25]	H. J. Sussmann and W. S. Liu, Lie Bracket extensions and averaging: The single bracket generating case, in Nonholonomic Motion Planning, Z. X. Li and J. F. Canny Eds., Kluwer Academic Publishers, Boston, (1993), 109-148.
[26]	, http://www.lsis.org/boizotn/KinematicVids/.

show all references

References:

[1]	A. A. Agrachev, H. E. A. Chakir and J. P. Gauthier, Subriemannian metrics on $R^3$, in Geometric Control and Nonholonomic Mechanics, Mexico City 1996, Proc. Can. Math. Soc., 25 (1998), 29-78.
[2]	A. A. Agrachev and J. P. Gauthier, Subriemannian metrics and isoperimetric problems in the contact case, in honor L. Pontriaguin, 90th birthday commemoration, Contemporary Maths, Tome 64, (1999), 5-48, (Russian); English version: Journal of Mathematical Sciences, 103, 639-663.
[3]	A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric View Point," Encyclopaedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer Verlag Berlin Heidelberg, 2004.
[4]	A. M. Bloch, "Nonholonomic Mechanics and Control," With the collaboration of J. Baillieul, P. Crouch and J. Marsden. With scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov. Interdisciplinary Applied Mathematics, 24. Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.
[5]	N. Boizot and J-P. Gauthier, Motion planning for kinematic systems, IEEE Transactions on Automatic Control, 58, (2013), 1430-1442. doi: 10.1109/TAC.2012.2232376.
[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), Nonholonomic Motion Planning, Springer, International Series in Engineering and Computer Science, 192 (1993), 1-22. doi: 10.1007/978-1-4615-3176-0_1.
[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), 359-421. doi: 10.1007/BF02269424.
[8]	J. Dixmier, Sur les représentations unitaires des groupes de lie nilpotents. II., (French) Bull. Soc. Math. France, 85 (1957), 325-388.
[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), 634-655. doi: 10.1051/cocv:2004024.
[10]	J. P. Gauthier and V. Zakalyukin, On the codimension one motion planning problem, J. Dyn. Control Syst., 11 (2005), 73-89. doi: 10.1007/s10883-005-0002-6.
[11]	J. P. Gauthier and V. Zakalyukin, On the One-Step-Bracket-Generating motion planning problem, J. Dyn. Control Syst., 11 (2005), 215-235. doi: 10.1007/s10883-005-4171-0.
[12]	J. P. Gauthier and V. Zakalyukin, Robot motion planning, a wild case, Proceedings of the Steklov Institute of Mathematics, 250 (2005), 56-69.
[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), 371-404. doi: 10.1007/s10450-006-0005-y.
[14]	J. P. Gauthier and V. Zakalyukin, Entropy estimations for motion planning problems in robotics, Volume In honor of Dmitry Victorovich Anosov, Proceedings of the Steklov Mathematical Institute, 256 (2007), 62-79. doi: 10.1134/S008154380701004X.
[15]	J. P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls, SIAM Journ. on Control and Optim., 48 (2010), 3433-3448. doi: 10.1137/090761884.
[16]	M. Gromov, "Carnot Caratheodory Spaces Seen from Within," Eds A. Bellaiche, J. J. Risler, Birkhauser, (1996), 79-323.
[17]	F. Jean, Complexity of nonholonomic motion planning, International Journal on Control, 74 (2001), 776-782. doi: 10.1080/00207170010017392.
[18]	F. Jean, Entropy and complexity of a path in subriemannian geometry, ESAIM Control Optim. Calc. Var., 9 (2003), 485-508. doi: 10.1051/cocv:2003024.
[19]	F. Jean and E. Falbel, Measures and transverse paths in subriemannian geometry, Journal d'Analyse Mathématique, 91 (2003), 231-246. doi: 10.1007/BF02788789.
[20]	V. Jurdjevic, The geometry of the plate-ball problem, Archive for Rational Mechanics and Analysis, 124 (1993), 305-328. doi: 10.1007/BF00375605.
[21]	J. P. Laumond, (editor), "Robot Motion Planning and Control," Lecture notes in Control and Information Sciences 229, Springer Verlag, 1998. doi: 10.1007/BFb0036069.
[22]	L. Pontryagin, V. Boltyanski, R. Gamkelidze and E. Michenko, "The Mathematical Theory of Optimal Processes," Wiley, New-York, 1962.
[23]	D. A. Singer, Curves whose curvature depend on the distance from the origin, the American mathematical Monthly, 106 (1999), 835-841. doi: 10.2307/2589616.
[24]	H. J. Sussmann and G. Lafferriere, Motion planning for controllable systems without drift, In Proceedings of the IEEE Conference on Robotics and Automation, Sacramento, CA, April 1991. IEEE Publications, NewYork, (1991), 109-148.
[25]	H. J. Sussmann and W. S. Liu, Lie Bracket extensions and averaging: The single bracket generating case, in Nonholonomic Motion Planning, Z. X. Li and J. F. Canny Eds., Kluwer Academic Publishers, Boston, (1993), 109-148.
[26]	, http://www.lsis.org/boizotn/KinematicVids/.

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