# 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 (2010)

- Volume: 10, Issue: 4, page 526-548
- ISSN: 1292-8119

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topAltafini, 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 10.4 (2010): 526-548. <http://eudml.org/doc/90741>.

@article{Altafini2010,

abstract = {
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.
},

author = {Altafini, Claudio},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {
Lie group; semidirect product; second order
variational problems; reduction; group symmetry; geometric splines; optimal control.; Lie group; second order variational problems; optimal control},

language = {eng},

month = {3},

number = {4},

pages = {526-548},

publisher = {EDP Sciences},

title = {Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric},

url = {http://eudml.org/doc/90741},

volume = {10},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 10

IS - 4

SP - 526

EP - 548

AB -
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.

LA - eng

KW -
Lie group; semidirect product; second order
variational problems; reduction; group symmetry; geometric splines; optimal control.; Lie group; second order variational problems; optimal control

UR - http://eudml.org/doc/90741

ER -

## References

top- C. Altafini, Geometric motion control for a kinematically redundant robotic chain: Application to a holonomic mobile manipulator. J. Rob. Syst.20 (2003) 211-227.
- V.I. Arnold, Math. methods of Classical Mechanics. 2nd ed., Grad. Texts Math.60 (1989).
- L. Berard-Bergery, Sur la courbure des métriques Riemanniennes invariantes des groupes de Lie et des espaces homogènes. Ann. Sci. Ecole National Superior11 (1978) 543.
- F. Bullo, N. Leonard and A. Lewis, Controllability and motion algorithms for underactuates Lagrangian systems on Lie groups. IEEE Trans. Autom. Control45 (2000) 1437-1454.
- F. Bullo and R. Murray, Tracking for fully actuated mechanical systems: a geometric framework. Automatica35 (1999) 17-34.
- 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 complètes2, part I. Gauthier-Villars, Paris, France (1953) 673-792.
- 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.
- M. Crampin and F. Pirani, Applicable differential geometry. London Mathematical Society Lecture notes. Cambridge University Press, Cambridge, UK (1986).
- 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.
- 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 geometryI and II. Interscience Publisher, New York (1963) and (1969).
- J. Lee, Riemannian manifolds. An introduction to curvature. Springer, New York, NY (1997).
- A. Lewis and R. Murray, Configuration controllability of simple mechanical control systems. SIAM J. Control Optim.35 (1997) 766-790.
- A. Lewis and R. Murray, Decompositions for control systems on manifolds with an affine connection. Syst. Control Lett.31 (1997) 199-205.
- 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).
- J. Milnor, Curvature of left invariant metrics on Lie groups. Adv. Math.21 (1976) 293-329.
- R. Murray, Z. Li and S. Sastry, A Mathematical Introduction to Robotic Manipulation. CRC Press (1994).
- L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces. IMA J. Math. Control Inform.12 (1989) 465-473.
- K. Nomizu, Invariant affine connections on homogeneous spaces. Amer. J. Math.76 (1954) 33-65.
- F. Park and B. Ravani, Bézier curves on Riemannian manifolds and Lie groups with kinematic applications. ASME J. Mech. design117 (1995) 36-40.
- J. M. Selig, Geometrical methods in Robotics. Springer, New York, NY (1996).
- 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.

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