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Mathematics of Control, Signals, and Systems

, Volume 31, Issue 4, pp 455–485 | Cite as

Symmetry and motion primitives in model predictive control

  • Kathrin FlaßkampEmail author
  • Sina Ober-Blöbaum
  • Karl Worthmann
Original Article
  • 55 Downloads

Abstract

Symmetries, e.g.  rotational and translational invariances for the class of mechanical systems, allow to characterize solution trajectories of nonlinear dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g.  by using the concept of motion primitives, may be considered as a quantization of the system. Symmetry exploitation is well established in both motion planning and control. However, the linkage between the respective techniques to optimal control is not yet fully explored. In this manuscript, we want to lay the foundation for the usage of symmetries in Model Predictive Control (MPC). To this end, we investigate a mobile robot example in detail where our contribution is twofold: Firstly, we establish asymptotic stability of a desired set point w.r.t. the MPC closed loop, which is also demonstrated numerically by using motion primitives applied to the parallel parking scenario. Secondly, if the optimization criterion is not consistent with the symmetry action, we provide guidelines to rigorously derive stability guarantees based on symmetry exploitation.

Keywords

Model predictive control Geometric control Motion primitives Optimal control Symmetry 

Mathematics Subject Classification

49M37 93D15 34H15 

Notes

Acknowledgements

K. Flaßkamp thanks L. Lüttgens and S. Roy for helpful discussions on the mobile robot example, in particular for the derivation of the Lie algebra representation used to derive the trim primitives. K. Worthmann thanks F. Rußwurm for helpful discussions on the mobile robot example.

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Industrial MathematicsUniversity of BremenBremenGermany
  2. 2.Department of Engineering ScienceUniversity of OxfordOxfordUK
  3. 3.Institut für MathematikTechnische Universität IlmenauIlmenauGermany

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