Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures
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Computing globally efficient solutions is a major challenge in optimal control of nonlinear dynamical systems. This work proposes a method combining local optimization and motion planning techniques based on exploiting inherent dynamical systems structures, such as symmetries and invariant manifolds. Prior to the optimal control, the dynamical system is analyzed for structural properties that can be used to compute pieces of trajectories that are stored in a motion planning library. In the context of mechanical systems, these motion planning candidates, termed primitives, are given by relative equilibria induced by symmetries and motions on stable or unstable manifolds of e.g. fixed points in the natural dynamics. The existence of controlled relative equilibria is studied through Lagrangian mechanics and symmetry reduction techniques. The proposed framework can be used to solve boundary value problems by performing a search in the space of sequences of motion primitives connected using optimized maneuvers. The optimal sequence can be used as an admissible initial guess for a post-optimization. The approach is illustrated by two numerical examples, the single and the double spherical pendula, which demonstrates its benefit compared to standard local optimization techniques.
KeywordsLagrangian mechanics Optimal control Symmetries Invariant manifolds
Mathematics Subject Classification37J15 49M37 70Q05
Jerry Marsden has been a great inspiration to us to work on this topic. We thank him for fruitful discussions and collaborations during the last years. This contribution was partly developed and published in the course of the Collaborative Research Centre 614 “Self-Optimizing Concepts and Structures in Mechanical Engineering” funded by the German Research Foundation (DFG) under grant number SFB 614.
M. Kobilarov was supported by the Keck Institute for Space Studies, Caltech.
- Abraham, R., Marsden, J.E.: Foundations of Mechanics. Addison-Wesley, Redwood City (1987) Google Scholar
- Ames, A.D., Sastry, S.: Hybrid Routhian reduction of Lagrangian hybrid systems. In: American Control Conference, 2006, June 2006. 6 pp. Google Scholar
- Binder, T., Blank, L., Bock, H.G., Bulirsch, R., Dahmen, W., Diehl, M., Kronseder, T., Marquardt, W., Schlöder, J.P., von Stryk, O.: Introduction to model based optimization of chemical processes on moving horizons. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds.) Online Optimization of Large Scale Systems: State of the Art, pp. 295–340. Springer, Berlin (2001) Google Scholar
- Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems. Texts in Applied Mathematics, vol. 49. Springer, New York (2004) Google Scholar
- Flaßkamp, K., Ober-Blöbaum, S.: Energy efficient control for mechanical systems based on inherent dynamical structures. In: American Control Conference (ACC), June 2012, Montréal, Canada, pp. 2609–2614 (2012) Google Scholar
- Kobilarov, M.: Discrete geometric motion control of autonomous vehicles. PhD thesis, University of Southern California, USA (2008) Google Scholar
- Kobilarov, M.: Cross-entropy randomized motion planning. In: Proceedings of Robotics: Science and Systems, Los Angeles, CA, USA, June 2011 Google Scholar
- Leyendecker, S., Ober-Blöbaum, S., Marsden, J.E., Ortiz, M.: Discrete mechanics and optimal control for constrained systems. In: Optimal Control, Applications and Methods, 2009 Google Scholar
- Marsden, J.E.: Lectures on Mechanics. London Mathematical Society Lecture Note Series, vol. 174. Cambridge University Press, Cambridge (1993) Google Scholar
- Marsden, J.E.: Geometric Mechanics, Stability, and Control, pp. 265–291. Springer, New York (1994) Google Scholar
- Marsden, J.E., Scheurle, J.: Lagrangian reduction and the double spherical pendulum. Z. Angew. Math. Phys. 44 (1993) Google Scholar
- McGehee, R.: Some homoclinic orbits for the restricted three-body problem. PhD thesis, University of Wisconsin (1969) Google Scholar