-
Previous Article
The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion
- MCRF Home
- This Issue
-
Next Article
Estimates on trajectories in a closed set with corners for $(t,x)$ dependent data
On the motion planning of the ball with a trailer
1. | Aix Marseille Université, CNRS, ENSAM, LSIS, UMR 7296, 13397 Marseille, France, France |
  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 $(2,3,5,6)$.
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.
|
[3] |
A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric View Point,", Encyclopaedia of Mathematical Sciences, (2004).
|
[4] |
A. M. Bloch, "Nonholonomic Mechanics and Control,", With the collaboration of J. Baillieul, (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.
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), 192 (1993), 1.
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
M. Gromov, "Carnot Caratheodory Spaces Seen from Within,", Eds A. Bellaiche, (1996), 79.
|
[17] |
F. Jean, Complexity of nonholonomic motion planning,, International Journal on Control, 74 (2001), 776.
doi: 10.1080/00207170010017392. |
[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. |
[19] |
F. Jean and E. Falbel, Measures and transverse paths in subriemannian geometry,, Journal d'Analyse Mathématique, 91 (2003), 231.
doi: 10.1007/BF02788789. |
[20] |
V. Jurdjevic, The geometry of the plate-ball problem,, Archive for Rational Mechanics and Analysis, 124 (1993), 305.
doi: 10.1007/BF00375605. |
[21] |
J. P. Laumond, (editor), "Robot Motion Planning and Control,", Lecture notes in Control and Information Sciences 229, (1998).
doi: 10.1007/BFb0036069. |
[22] |
L. Pontryagin, V. Boltyanski, R. Gamkelidze and E. Michenko, "The Mathematical Theory of Optimal Processes,", Wiley, (1962).
|
[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. |
[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] |
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, 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.
|
[3] |
A. A. Agrachev and Y. Sachkov, "Control Theory from the Geometric View Point,", Encyclopaedia of Mathematical Sciences, (2004).
|
[4] |
A. M. Bloch, "Nonholonomic Mechanics and Control,", With the collaboration of J. Baillieul, (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.
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), 192 (1993), 1.
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
M. Gromov, "Carnot Caratheodory Spaces Seen from Within,", Eds A. Bellaiche, (1996), 79.
|
[17] |
F. Jean, Complexity of nonholonomic motion planning,, International Journal on Control, 74 (2001), 776.
doi: 10.1080/00207170010017392. |
[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. |
[19] |
F. Jean and E. Falbel, Measures and transverse paths in subriemannian geometry,, Journal d'Analyse Mathématique, 91 (2003), 231.
doi: 10.1007/BF02788789. |
[20] |
V. Jurdjevic, The geometry of the plate-ball problem,, Archive for Rational Mechanics and Analysis, 124 (1993), 305.
doi: 10.1007/BF00375605. |
[21] |
J. P. Laumond, (editor), "Robot Motion Planning and Control,", Lecture notes in Control and Information Sciences 229, (1998).
doi: 10.1007/BFb0036069. |
[22] |
L. Pontryagin, V. Boltyanski, R. Gamkelidze and E. Michenko, "The Mathematical Theory of Optimal Processes,", Wiley, (1962).
|
[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. |
[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] |
[1] |
Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239 |
[2] |
M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223 |
[3] |
Dingjun Yao, Rongming Wang, Lin Xu. Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission. Journal of Industrial & Management Optimization, 2015, 11 (2) : 461-478. doi: 10.3934/jimo.2015.11.461 |
[4] |
Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078 |
[5] |
Francesca Biagini, Thilo Meyer-Brandis, Bernt Øksendal, Krzysztof Paczka. Optimal control with delayed information flow of systems driven by G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 8-. doi: 10.1186/s41546-018-0033-z |
[6] |
Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014 |
[7] |
Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. Impulsive control of a symmetric ball rolling without sliding or spinning. Journal of Geometric Mechanics, 2010, 2 (4) : 321-342. doi: 10.3934/jgm.2010.2.321 |
[8] |
Ching-Lung Lin, Gunther Uhlmann, Jenn-Nan Wang. Optimal three-ball inequalities and quantitative uniqueness for the Stokes system. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1273-1290. doi: 10.3934/dcds.2010.28.1273 |
[9] |
Robert J. McCann. A glimpse into the differential topology and geometry of optimal transport. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605 |
[10] |
Thalya Burden, Jon Ernstberger, K. Renee Fister. Optimal control applied to immunotherapy. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 135-146. doi: 10.3934/dcdsb.2004.4.135 |
[11] |
Ellina Grigorieva, Evgenii Khailov. Optimal control of pollution stock. Conference Publications, 2011, 2011 (Special) : 578-588. doi: 10.3934/proc.2011.2011.578 |
[12] |
Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 |
[13] |
Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058 |
[14] |
Alexander Shmyrov, Vasily Shmyrov. The optimal stabilization of orbital motion in a neighborhood of collinear libration point. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 185-189. doi: 10.3934/naco.2017012 |
[15] |
Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275 |
[16] |
Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021021 |
[17] |
Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021 |
[18] |
Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 |
[19] |
C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial & Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435 |
[20] |
Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]