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Journal of Nonlinear Science

, Volume 22, Issue 4, pp 599–629 | Cite as

Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures

  • Kathrin Flaßkamp
  • Sina Ober-BlöbaumEmail author
  • Marin Kobilarov
Article

Abstract

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.

Keywords

Lagrangian mechanics Optimal control Symmetries Invariant manifolds 

Mathematics Subject Classification

37J15 49M37 70Q05 

Notes

Acknowledgements

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.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Kathrin Flaßkamp
    • 1
  • Sina Ober-Blöbaum
    • 1
    Email author
  • Marin Kobilarov
    • 2
  1. 1.Computational Dynamics and Optimal Control, Department of MathematicsUniversity of PaderbornPaderbornGermany
  2. 2.Control and Dynamical SystemsCalifornia Institute of TechnologyPasadenaUSA

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