×

zbMATH — the first resource for mathematics

Rolling manifolds on space forms. (English) Zbl 1321.53021
The rolling problem without spinning nor slipping determines a dynamical system with complicated (fractal) properties. As a first step of the solution, the system is investigated at the level of kinematic motions using an appropriate affine connection. One says that the rolling problem is completely controllable, if for arbitrary two points in the configuration space there is a loop, along which the parallel transport joins the two states. In the paper, the question “if the rolling problem is controllable?” is reformulated in terms of curvature of the connection. This gives a sufficient condition of controllability and a description of the reachable set in a general case.

MSC:
53B21 Methods of local Riemannian geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alouges, F.; Chitour, Y.; Long, R., A motion planning algorithm for the rolling-body problem, IEEE trans. robot., 26, 5, (2010)
[2] A. Agrachev, Y. Sachkov, An intrinsic approach to the control of rolling bodies, in: Proceedings of the CDC, vol. 1, Phoenix, 1999, pp. 431-435.
[3] Agrachev, A.; Sachkov, Y., Control theory from the geometric viewpoint. control theory and optimization, II, Encyclopaedia math. sci., vol. 87, (2004), Springer-Verlag Berlin · Zbl 1062.93001
[4] Berger, M., Sur LES groupes dʼholonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. soc. math. France, 83, 279-330, (1955) · Zbl 0068.36002
[5] Bryant, R., Geometry of manifolds with special holonomy: “100 years of holonomy”, Contemp. math., vol. 395, (2006) · Zbl 1096.53027
[6] Bryant, R.; Hsu, L., Rigidity of integral curves of rank 2 distributions, Invent. math., 114, 2, 435-461, (1993) · Zbl 0807.58007
[7] Cartan, É., La géométrie des espaces de Riemann, Mémorial des sciences mathématiques, 9, 1-61, (1925) · JFM 51.0566.01
[8] Chelouah, A.; Chitour, Y., On the controllability and trajectories generation of rolling surfaces, Forum math., 15, 727-758, (2003) · Zbl 1044.93015
[9] Chitour, Y.; Godoy Molina, M.; Kokkonen, P., Extension of de Rham decomposition theorem to non Euclidean development · Zbl 1317.53069
[10] Chitour, Y.; Kokkonen, P., Rolling manifolds: intrinsic formulation and controllability, (2011)
[11] Grong, E., Controllability of rolling without twisting or slipping in higher dimensions, (2011) · Zbl 1257.37042
[12] Joyce, D.D., Riemannian holonomy groups and calibrated geometry, (2007), Oxford University Press · Zbl 1200.53003
[13] Jurdjevic, V.; Zimmerman, J., Rolling sphere problems on spaces of constant curvature, Math. proc. Cambridge philos. soc., 144, 729-747, (2008) · Zbl 1147.49037
[14] Kobayashi, S.; Nomizu, K., Foundations of differential geometry, vol. I, (1996), Wiley-Interscience · Zbl 0175.48504
[15] Lee, J., Introduction to smooth manifolds, Grad. texts in math., vol. 218, (2003), Springer-Verlag New York
[16] Marigo, A.; Bicchi, A., Rolling bodies with regular surface: controllability theory and applications, IEEE trans. automat. control, 45, 9, 1586-1599, (2000) · Zbl 0986.70002
[17] Molina, M.; Grong, E.; Markina, I.; Leite, F., An intrinsic formulation of the rolling manifolds problem, (2010) · Zbl 1261.37027
[18] Olmos, C., A geometric proof of the berger holonomy theorem, Ann. of math., 161, 579-588, (2005) · Zbl 1082.53048
[19] Petersen, P., Riemannian geometry, Grad. texts in math., vol. 171, (2006), Springer-Verlag New York · Zbl 1220.53002
[20] Sakai, T., Riemannian geometry, Transl. math. monogr., vol. 149, (1996), American Mathematical Society Providence, RI
[21] Sharpe, R.W., Differential geometry: cartanʼs generalization of kleinʼs erlangen program, Grad. texts in math., vol. 166, (1997), Springer-Verlag New York · Zbl 0876.53001
[22] Simons, J., On the transitivity of holonomy systems, Ann. of math. (2), 76, 2, 213-234, (1962) · Zbl 0106.15201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.