Symmetry Reduction of a Class of Hybrid Systems
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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.
- V. I. Arnold, K. Vogtmann, and A. Weinstein. Mathematical Methods of Classical Mechanics, 2nd edition. Springer-Verlag, 1989.
- 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.
- A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and T. S. Ratiu. The Euler-Poincare equations and double bracket dissipation. Comm. Math. Phys., 175(1):1–42, 1996. CrossRef
- J. C. P. Bus. The lagrange multiplier rule on manifolds and optimal control of nonlinear systems. SIAM J. Control and Optimization, 22(5):740–757, 1984. CrossRef
- M. P. de Carmo. Riemannian Geometry. Birkhäuser Boston, 1992.
- 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.
- A. Edelman, T. A. Arias, and S. T. Smith. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. and Appl., 20(2):303–353, 1998. CrossRef
- John Lygeros et al. Hybrid Systems: Modeling, Analysis and Control. ERL Memorandum No. UCB/ERLM99/34, Univ. of California at Berkeley, 1999.
- 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.
- 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.
- J. Hu and S. Sastry. Geodesics of manifolds with boundary: a case study. unpublished, 2002.
- J. E. Marsden and T.S. Ratiu. Introduction to Mechanics and Symmetry, 2nd edition. Springer-Verlag, 1994.
- 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.
- Symmetry Reduction of a Class of Hybrid Systems
- Book Title
- Hybrid Systems: Computation and Control
- Book Subtitle
- 5th International Workshop, HSCC 2002 Stanford, CA, USA, March 25–27, 2002 Proceedings
- pp 267-280
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
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- Editor Affiliations
- 4. Department of Aeronautics and Astronautics, Stanford University
- 5. Department of Computer Science, University of British Columbia
- Author Affiliations
- 6. Department of Electrical Engineering & Computer Sciences, University of California at Berkeley, USA
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