Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-30T23:41:15.874Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction by group symmetry of second order variational problems on asemidirect product of Lie groups with positive definite Riemannian metric

Published online by Cambridge University Press:  15 October 2004

Claudio Altafini
Claudio Altafini*
Affiliation:
SISSA-ISAS, International School for Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy; [email protected].
Get access

Abstract

For a Riemannian structure on asemidirect product of Lie groups, the variational problems can bereduced using the group symmetry.Choosing the Levi-Civita connection of a positive definitemetric tensor,instead of any of the canonical connections for the Lie group,simplifies the reduction of the variations but complicates theexpression for the Lie algebra valued covariant derivatives.The origin of the discrepancy is in the semidirect productstructure, which implies that the Riemannianexponential map and the Lie group exponential map do not coincide.The consequence is that the reduced equations look more complicated thanthe 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 Euclideangroup is considered in detail.

Keywords

Lie groupsemidirect productsecond ordervariational problemsreductiongroup symmetrygeometric splinesoptimal control.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 4 , October 2004 , pp. 526 - 548
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Altafini, C., Geometric motion control for a kinematically redundant robotic chain: Application to a holonomic mobile manipulator. J. Rob. Syst. 20 (2003) 211-227. CrossRef
V.I. Arnold, Math. methods of Classical Mechanics. 2nd ed., Grad. Texts Math. 60 (1989).
Berard-Bergery, L., Sur la courbure des métriques Riemanniennes invariantes des groupes de Lie et des espaces homogènes. Ann. Sci. Ecole National Superior 11 (1978) 543.
Bullo, F., Leonard, N. and Lewis, A., Controllability and motion algorithms for underactuates Lagrangian systems on Lie groups. IEEE Trans. Autom. Control 45 (2000) 1437-1454. CrossRef
Bullo, F. and Murray, R., Tracking for fully actuated mechanical systems: a geometric framework. Automatica 35 (1999) 17-34. CrossRef
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.
E. Cartan, La géométrie des groupes de transformations, in Œuvres complètes 2, part I. Gauthier-Villars, Paris, France (1953) 673-792.
Cendra, H., Holm, D., Marsden, J. and Ratiu, T., Lagrangian reduction, the Euler-Poincaré equations and semidirect products. Amer. Math. Soc. Transl. 186 (1998) 1-25.
M. Crampin and F. Pirani, Applicable differential geometry. London Mathematical Society Lecture notes. Cambridge University Press, Cambridge, UK (1986).
Crouch, P.E. and Silva Le, F.ite, The dynamic interpolation problem on Riemannian manifolds, Lie groups and symmetric spaces. J. Dynam. Control Syst. 1 (1995) 177-202. CrossRef
M. do Carmo, Riemannian geometry. Birkhäuser, Boston (1992).
L. Eisenhart, Riemannian geometry. Princeton University Press, Princeton (1966).
V. Jurdjevic, Geometric Control Theory. Cambridge Stud. Adv. Math. Cambridge University Press, Cambridge, UK (1996).
S. Kobayashi and K. Nomizu, Foundations of differential geometry I and II. Interscience Publisher, New York (1963) and (1969).
J. Lee, Riemannian manifolds. An introduction to curvature. Springer, New York, NY (1997).
Lewis, A. and Murray, R., Configuration controllability of simple mechanical control systems. SIAM J. Control Optim. 35 (1997) 766-790. CrossRef
Lewis, A. and Murray, R., Decompositions for control systems on manifolds with an affine connection. Syst. Control Lett. 31 (1997) 199-205. CrossRef
J. Marsden, Lectures on Mechanics. Cambridge University Press, Cambridge (1992).
J. Marsden and T. Ratiu, Introduction to mechanics and symmetry, Springer-Verlag, 2nd ed., Texts Appl. Math. 17 (1999).
Milnor, J., Curvature of left invariant metrics on Lie groups. Adv. Math. 21 (1976) 293-329. CrossRef
R. Murray, Z. Li and S. Sastry, A Mathematical Introduction to Robotic Manipulation. CRC Press (1994).
Noakes, L., Heinzinger, G. and Cubic, B. Paden splines on curved spaces. IMA J. Math. Control Inform. 12 (1989) 465-473. CrossRef
Nomizu, K., Invariant affine connections on homogeneous spaces. Amer. J. Math. 76 (1954) 33-65. CrossRef
Park, F. and Ravani, B., Bézier curves on Riemannian manifolds and Lie groups with kinematic applications. ASME J. Mech. design 117 (1995) 36-40. CrossRef
J. M. Selig, Geometrical methods in Robotics. Springer, New York, NY (1996).
Zefran, M., Kumar, V. and Croke, C., On the generation of smooth three-dimensional rigid body motions. IEEE Trans. Robot. Automat. 14 (1998) 576-589. CrossRef