Symmetry Reduction of a Class of Hybrid Systems
The optimal control problem for a class of hybrid systems (switched Lagrangian systems) is studied. Some necessary conditions of the optimal solutions of such a system are derived based on the assumption that there is a group of symmetries acting uniformly on the domains of different discrete modes, such that the Lagrangian functions, the guards, and the reset maps are all invariant under the action. Lagrangian reduction approach is adopted to establish the conservation law of certain quantities for the optimal solutions. Some examples are presented. In particular, the problems of optimal collision avoidance (OCA) and optimal formation switching (OFS) of multiple agents moving on a Riemannian manifold are studied in some details.
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- V. I. Arnold, K. Vogtmann, and A. Weinstein. Mathematical Methods of Classical Mechanics, 2nd edition. Springer-Verlag, 1989.Google Scholar
- A. Bicchi and L. Pallottino. Optimal planning for coordinated vehicles with bounded curvature. In Proc. Work. Algorithmic Foundation of Robotics (WAFR’2000), Dartmouth, Hanover, NH, 2000.Google Scholar
- M. P. de Carmo. Riemannian Geometry. Birkhäuser Boston, 1992.Google Scholar
- J. P. Desai and V. Kumar. Nonholonomic motion planning for multiple mobile manipulators. In Proc. IEEE Int. Conf. on Robotics and Automation, volume 4, ages 20–25, Albuquerque, NM, 1997.Google Scholar
- John Lygeros et al. Hybrid Systems: Modeling, Analysis and Control. ERL Memorandum No. UCB/ERLM99/34, Univ. of California at Berkeley, 1999.Google Scholar
- J. Hu, M. Prandini, and S. Sastry. Hybrid geodesics as optimal solutions to the collision-free motion planning problem. In Proc. Hybrid Systems: Computation and Control, 4th Int. Workshop (HSCC 2001), pages 305–318, Rome, Italy, 2001.Google Scholar
- J. Hu and S. Sastry. Optimal collision avoidance and formation switching on Riemannian manifolds. In Proc. 40th IEEE Int. Conf. on Decision and Control, Orlando, Florida, 2001.Google Scholar
- J. Hu and S. Sastry. Geodesics of manifolds with boundary: a case study. unpublished, 2002.Google Scholar
- J. E. Marsden and T.S. Ratiu. Introduction to Mechanics and Symmetry, 2nd edition. Springer-Verlag, 1994.Google Scholar
- H. J. Sussmann. A maximum principle for hybrid optimal control problems. In Proc. 38th IEEE Int. Conf. on Decision and Control, volume 1, pages 425–430, Phoenix, AZ, 1999.Google Scholar