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On the motion planning of the ball with a trailer
The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion
1. | Institut de mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France |
2. | INRIA Sophia Antipolis Méditerranée, B.P. 93, route des Lucioles, 06902 Sophia Antipolis, France, France |
References:
[1] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Translated from the Russian by K. Vogtmann and A. Weinstein, (1989).
|
[2] |
L. Bates and F. Fassò, The conjugate locus for the euler top. I. The axisymmetric case,, Int. Math. Forum, 2 (2007), 2109.
|
[3] |
B. Bonnard, Contrôlabilité de systemes mécaniques sur les groupes de lie (French), [controllability of mechanical systems on lie groups],, SIAM J. Control Optim., 22 (1984), 711.
doi: 10.1137/0322045. |
[4] |
B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081.
doi: 10.1016/j.anihpc.2008.03.010. |
[5] |
B. Bonnard, O. Cots and N. Shcherbakova, Energy minimization problem in two-level dissipative quantum control: Meridian case,, J. Math. Sci., (2013). Google Scholar |
[6] |
M. do Carmo, "Riemannian Geometry,'' Translated from the second Portuguese edition by Francis Flaherty,, Mathematics: Theory & Applications. Birkhäuser Boston, (1992).
|
[7] |
P. Gurfil, A. Elipe, W. Tangren and M. Efroimsky, The Serret-Andoyer formalism in rigid-body dynamics. I. Symmetries and perturbations,, Regul. Chaotic Dyn., 12 (2007), 389.
doi: 10.1134/S156035470704003X. |
[8] |
J. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids,, Manuscripta Math., 114 (2004), 247.
doi: 10.1007/s00229-004-0455-z. |
[9] |
V. Jurdjevic, "Geometric Control Theory,'', Cambridge studies in advanced mathematics, (1997).
|
[10] |
M. Lara and S. Ferrer, Closed form integration of the Hitzl-Breakwell problem in action-angle variables,, IAA-AAS-DyCoSS1-01-02 (AAS 12-302), (): 1. Google Scholar |
[11] |
D. Lawden, "Elliptic Functions and Applications,'', Applied mathematical sciences, (1989).
|
[12] |
H. Poincaré, Sur les lignes géodésiques des surfaces convexes,, (French) [On the geodesic lines of convex surfaces] Trans. Amer. Math. Soc., 6 (1905), 237.
doi: 10.2307/1986219. |
[13] |
K. Shiohama, T. Shioya and M. Tanaka, "The Geometry of Total Curvature on Complete Open Surfaces,'', Cambridge tracts in mathematics, (2003).
doi: 10.1017/CBO9780511543159. |
[14] |
R. Sinclair and M. Tanaka, The cut locus of a two-sphere of revolution and toponogov's comparison theorem,, Tohoku Math. J. (2), 59 (2007), 379.
doi: 10.2748/tmj/1192117984. |
[15] |
A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions, and variational problems,, vol. 16 of Dynamical Systems VII, 16 (1994), 1. Google Scholar |
[16] |
H. Yuan, R. Zeier, N. Khaneja and S. Lloyd, Constructing two-qubit gates with minimal couplings,, Phys. Rev. A (3), 79 (2009).
doi: 10.1103/PhysRevA.79.042309. |
show all references
References:
[1] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Translated from the Russian by K. Vogtmann and A. Weinstein, (1989).
|
[2] |
L. Bates and F. Fassò, The conjugate locus for the euler top. I. The axisymmetric case,, Int. Math. Forum, 2 (2007), 2109.
|
[3] |
B. Bonnard, Contrôlabilité de systemes mécaniques sur les groupes de lie (French), [controllability of mechanical systems on lie groups],, SIAM J. Control Optim., 22 (1984), 711.
doi: 10.1137/0322045. |
[4] |
B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081.
doi: 10.1016/j.anihpc.2008.03.010. |
[5] |
B. Bonnard, O. Cots and N. Shcherbakova, Energy minimization problem in two-level dissipative quantum control: Meridian case,, J. Math. Sci., (2013). Google Scholar |
[6] |
M. do Carmo, "Riemannian Geometry,'' Translated from the second Portuguese edition by Francis Flaherty,, Mathematics: Theory & Applications. Birkhäuser Boston, (1992).
|
[7] |
P. Gurfil, A. Elipe, W. Tangren and M. Efroimsky, The Serret-Andoyer formalism in rigid-body dynamics. I. Symmetries and perturbations,, Regul. Chaotic Dyn., 12 (2007), 389.
doi: 10.1134/S156035470704003X. |
[8] |
J. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids,, Manuscripta Math., 114 (2004), 247.
doi: 10.1007/s00229-004-0455-z. |
[9] |
V. Jurdjevic, "Geometric Control Theory,'', Cambridge studies in advanced mathematics, (1997).
|
[10] |
M. Lara and S. Ferrer, Closed form integration of the Hitzl-Breakwell problem in action-angle variables,, IAA-AAS-DyCoSS1-01-02 (AAS 12-302), (): 1. Google Scholar |
[11] |
D. Lawden, "Elliptic Functions and Applications,'', Applied mathematical sciences, (1989).
|
[12] |
H. Poincaré, Sur les lignes géodésiques des surfaces convexes,, (French) [On the geodesic lines of convex surfaces] Trans. Amer. Math. Soc., 6 (1905), 237.
doi: 10.2307/1986219. |
[13] |
K. Shiohama, T. Shioya and M. Tanaka, "The Geometry of Total Curvature on Complete Open Surfaces,'', Cambridge tracts in mathematics, (2003).
doi: 10.1017/CBO9780511543159. |
[14] |
R. Sinclair and M. Tanaka, The cut locus of a two-sphere of revolution and toponogov's comparison theorem,, Tohoku Math. J. (2), 59 (2007), 379.
doi: 10.2748/tmj/1192117984. |
[15] |
A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions, and variational problems,, vol. 16 of Dynamical Systems VII, 16 (1994), 1. Google Scholar |
[16] |
H. Yuan, R. Zeier, N. Khaneja and S. Lloyd, Constructing two-qubit gates with minimal couplings,, Phys. Rev. A (3), 79 (2009).
doi: 10.1103/PhysRevA.79.042309. |
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