Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 526-548.

For a riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the riemannian exponential map and the Lie group exponential map do not coincide. The consequence is that the reduced equations look more complicated than the original ones. The main scope of this paper is to treat the reduction of second order variational problems (corresponding to geometric splines) on such semidirect products of Lie groups. Due to the semidirect structure, a number of extra terms appears in the reduction, terms that are calculated explicitely. The result is used to compute the necessary conditions of an optimal control problem for a simple mechanical control system having invariant lagrangian equal to the kinetic energy corresponding to the metric tensor. As an example, the case of a rigid body on the Special euclidean group is considered in detail.

DOI : https://doi.org/10.1051/cocv:2004018
Classification : 22F30,  70Q05,  93B29,  49J15,  70H30,  70H33
Mots clés : Lie group, semidirect product, second order variational problems, reduction, group symmetry, geometric splines, optimal control
@article{COCV_2004__10_4_526_0,
author = {Altafini, Claudio},
title = {Reduction by group symmetry of second order variational problems on a semidirect product of {Lie} groups with positive definite riemannian metric},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {526--548},
publisher = {EDP-Sciences},
volume = {10},
number = {4},
year = {2004},
doi = {10.1051/cocv:2004018},
zbl = {1072.49001},
mrnumber = {2111078},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2004018/}
}
TY  - JOUR
AU  - Altafini, Claudio
TI  - Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2004
DA  - 2004///
SP  - 526
EP  - 548
VL  - 10
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2004018/
UR  - https://zbmath.org/?q=an%3A1072.49001
UR  - https://www.ams.org/mathscinet-getitem?mr=2111078
UR  - https://doi.org/10.1051/cocv:2004018
DO  - 10.1051/cocv:2004018
LA  - en
ID  - COCV_2004__10_4_526_0
ER  - 
Altafini, Claudio. Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 526-548. doi : 10.1051/cocv:2004018. http://www.numdam.org/articles/10.1051/cocv:2004018/

[1] C. Altafini, Geometric motion control for a kinematically redundant robotic chain: Application to a holonomic mobile manipulator. J. Rob. Syst. 20 (2003) 211-227. | Zbl 1047.70024

[2] V.I. Arnold, Math. methods of Classical Mechanics. 2nd ed., Grad. Texts Math. 60 (1989).

[3] L. Berard-Bergery, Sur la courbure des métriques Riemanniennes invariantes des groupes de Lie et des espaces homogènes. Ann. Sci. Ecole National Superior 11 (1978) 543. | Numdam | MR 533067 | Zbl 0426.53038

[4] F. Bullo, N. Leonard and A. Lewis, Controllability and motion algorithms for underactuates Lagrangian systems on Lie groups. IEEE Trans. Autom. Control 45 (2000) 1437-1454. | MR 1797397 | Zbl 0990.70019

[5] F. Bullo and R. Murray, Tracking for fully actuated mechanical systems: a geometric framework. Automatica 35 (1999) 17-34. | MR 1827788 | Zbl 0941.93014

[6] M. Camarinha, F. Silva Leite and P. Crouch, Second order optimality conditions for an higher order variational problem on a Riemannian manifold, in Proc. 35th Conf. on Decision and Control. Kobe, Japan, December (1996) 1636-1641.

[7] E. Cartan, La géométrie des groupes de transformations, in Œuvres complètes 2, part I. Gauthier-Villars, Paris, France (1953) 673-792. | JFM 53.0388.01

[8] H. Cendra, D. Holm, J. Marsden and T. Ratiu, Lagrangian reduction, the Euler-Poincaré equations and semidirect products. Amer. Math. Soc. Transl. 186 (1998) 1-25. | MR 1732406 | Zbl 0989.37052

[9] M. Crampin and F. Pirani, Applicable differential geometry. London Mathematical Society Lecture notes. Cambridge University Press, Cambridge, UK (1986). | MR 892315 | Zbl 0606.53001

[10] P.E. Crouch and F. Silva Leite, The dynamic interpolation problem on Riemannian manifolds, Lie groups and symmetric spaces. J. Dynam. Control Syst. 1 (1995) 177-202. | MR 1333770 | Zbl 0946.58018

[11] M. Do Carmo, Riemannian geometry. Birkhäuser, Boston (1992). | MR 1138207 | Zbl 0752.53001

[12] L. Eisenhart, Riemannian geometry. Princeton University Press, Princeton (1966). | MR 35081 | Zbl 0174.53303

[13] V. Jurdjevic, Geometric Control Theory. Cambridge Stud. Adv. Math. Cambridge University Press, Cambridge, UK (1996). | MR 1425878 | Zbl 0940.93005

[14] S. Kobayashi and K. Nomizu, Foundations of differential geometry I and II. Interscience Publisher, New York (1963) and (1969). | MR 152974 | Zbl 0119.37502

[15] J. Lee, Riemannian manifolds. An introduction to curvature. Springer, New York, NY (1997). | MR 1468735 | Zbl 0905.53001

[16] A. Lewis and R. Murray, Configuration controllability of simple mechanical control systems. SIAM J. Control Optim. 35 (1997) 766-790. | MR 1444338 | Zbl 0870.53013

[17] A. Lewis and R. Murray, Decompositions for control systems on manifolds with an affine connection. Syst. Control Lett. 31 (1997) 199-205. | MR 1475667 | Zbl 0901.93014

[18] J. Marsden, Lectures on Mechanics. Cambridge University Press, Cambridge (1992). | MR 1171218 | Zbl 0744.70004

[19] J. Marsden and T. Ratiu, Introduction to mechanics and symmetry, Springer-Verlag, 2nd ed., Texts Appl. Math. 17 (1999). | MR 1723696 | Zbl 0933.70003

[20] J. Milnor, Curvature of left invariant metrics on Lie groups. Adv. Math. 21 (1976) 293-329. | MR 425012 | Zbl 0341.53030

[21] R. Murray, Z. Li and S. Sastry, A Mathematical Introduction to Robotic Manipulation. CRC Press (1994). | MR 1300410 | Zbl 0858.70001

[22] L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces. IMA J. Math. Control Inform. 12 (1989) 465-473. | MR 1036158 | Zbl 0698.58018

[23] K. Nomizu, Invariant affine connections on homogeneous spaces. Amer. J. Math. 76 (1954) 33-65. | MR 59050 | Zbl 0059.15805

[24] F. Park and B. Ravani, Bézier curves on Riemannian manifolds and Lie groups with kinematic applications. ASME J. Mech. design 117 (1995) 36-40.

[25] J. M. Selig, Geometrical methods in Robotics. Springer, New York, NY (1996). | MR 1411680 | Zbl 0861.93001

[26] M. Zefran, V. Kumar and C. Croke, On the generation of smooth three-dimensional rigid body motions. IEEE Trans. Robot. Automat. 14 (1998) 576-589.

Cité par Sources :