Symmetries in vakonomic dynamics: applications to optimal control

Abstract Symmetries in vakonomic dynamics are discussed. Appropriate notions are introduced and their relationship with previous work on symmetries of singular Lagrangian systems is shown. Some Noether-type theorems are obtained. The results are applied to a class of general optimal control problems and to kinematic locomotion systems.

[1]  J. Cortés,et al.  Optimal control for nonholonomic systems with symmetry , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[2]  Marco Favretti,et al.  Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints , 1998 .

[3]  G. Prince.  A complete classification of dynamical symmetries in classical mechanics , 1985, Bulletin of the Australian Mathematical Society.

[4]  R. Bains,et al.  Methods of differential geometry in analytical mechanics , 1992 .

[5]  P. Krishnaprasad,et al.  Control problems on principal bundles and nonholonomic mechanics , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[6]  REDUCTION OF PRESYMPLECTIC MANIFOLDS WITH SYMMETRY , 1999, math-ph/9911008.

[7]  Sonia Marti nez,et al.  The geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems: The vakonomic bracket , 2000 .

[8]  A. Bloch,et al.  Nonholonomic and Vakonomic Control Systems on Riemannian Manifolds , 1993 .

[9]  Sonia Martínez,et al.  Geometric Description of Vakonomic and Nonholonomic Dynamics. Comparison of Solutions , 2002, SIAM J. Control. Optim..

[10]  P A Samuelson.  Law of conservation of the capital-output ratio. , 1970, Proceedings of the National Academy of Sciences of the United States of America.

[11]  G. Prince.  Toward a classification of dynamical symmetries in classical mechanics , 1983, Bulletin of the Australian Mathematical Society.

[12]  David Martín de Diego,et al.  Vakonomic mechanics versus non-holonomic mechanics: A unified geometrical approach , 2000 .

[13]  I. Neĭmark,et al.  Dynamics of Nonholonomic Systems , 1972 .

[14]  R. Montgomery.  Isoholonomic problems and some applications , 1990 .

[15]  R. Montgomery.  Abnormal Minimizers , 1994 .

[16]  J. M. Nester,et al.  Presymplectic lagrangian systems. II : the second-order equation problem , 1980 .

[17]  Ryuzo Sato.  Conservation Laws and Symmetry: Applications to Economics and Finance , 1990 .

[18]  J. Ostrowski.  The mechanics and control of undulatory robotic locomotion , 1995 .

[19]  Richard Montgomery.  Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations , 1984 .

[20]  Sonia Martinez,et al.  The geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems: The vakonomic bracket , 2000 .

[21]  A. D. Lewis,et al.  Variational Principles for Constrained Systems: Theory and Experiment , 1995 .

[22]  Mark J. Gotay,et al.  Presymplectic lagrangian systems. I : the constraint algorithm and the equivalence theorem , 1979 .

[23]  F. Wilczek,et al.  Geometry of self-propulsion at low Reynolds number , 1989, Journal of Fluid Mechanics.

[24]  Marco Favretti,et al.  On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints , 1996 .

[25]  J. Marsden,et al.  Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction , 1997 .

[26]  G. Bluman,et al.  Review: Peter J. Olver, Applications of Lie groups to differential equations , 1988 .

[27]  Anthony M. Bloch,et al.  Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[28]  A. Jamiołkowski.  Applications of Lie groups to differential equations , 1989 .

[29]  Richard M. Murray,et al.  Geometric phases and robotic locomotion , 1995, J. Field Robotics.

[30]  T. J. Willmore,et al.  TANGENT AND COTANGENT BUNDLES , 1975 .

[31]  Peter E. Crouch,et al.  Optimal control, optimization, and analytical mechanics , 1998 .

[32]  P. Krishnaprasad,et al.  Nonholonomic mechanical systems with symmetry , 1996 .

[33]  David Martín de Diego,et al.  Symmetries and constants of the motion for singular Lagrangian systems , 1996 .