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Variational point-obstacle avoidance on Riemannian manifolds

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Abstract

In this paper, we study variational point-obstacle avoidance problems on complete Riemannian manifolds. The problem consists of minimizing an energy functional depending on the velocity, covariant acceleration and a repulsive potential function used to avoid an static obstacle given by a point in the manifold, among a set of admissible curves. We derive the dynamical equations for stationary paths of the variational problem, in particular on compact connected Lie groups and Riemannian symmetric spaces. Numerical examples are presented to illustrate the proposed method.

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Notes

  1. If \(R=Q\), \(\log (R^{T}Q)=0\) and this case is outside the problem formulation since we are in the obstacle.

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Acknowledgements

The research of A. Bloch was supported by NSF Grants DMS-1207693, DMS-1613819 and ASFOR. The research of M. Camarinha was partially supported by the Centre for Mathematics of the University of Coimbra—UID/MAT/00324/ 2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. L. Colombo was supported by La Caixa Foundation. The Project that gave rise to these results received the support of a fellowship from “la Caixa’ Foundation (ID 100010434). The fellowship code is LCF/BQ/PI19/11690016. L. Colombo was also partially supported by MINECO (Spain) Grant MTM2016-76072-P, I-Link Project (linkA20079) from CSIC, Severo Ochoa Program and Santander Universidades Fellowship.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

    Anthony Bloch

  2. Department of Mathematics, University of Coimbra, CMUC, 3001-501, Coimbra, Portugal

    Margarida Camarinha

  3. Instituto de Ciencias Matemáticas (CSIC-UAM-U CM-UC3M), Calle Nicolás Cabrera 13-15, Campus UAM, Cantoblanco, 28049, Madrid, Spain

    Leonardo Colombo

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  1. Anthony Bloch
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  2. Margarida Camarinha
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  3. Leonardo Colombo
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Correspondence to Leonardo Colombo.

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Bloch, A., Camarinha, M. & Colombo, L. Variational point-obstacle avoidance on Riemannian manifolds. Math. Control Signals Syst. 33, 109–121 (2021). https://doi.org/10.1007/s00498-021-00276-0

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