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Minimal time splines on the sphere

  • Teresa Stuchi
  • Paula Balseiro
  • Alejandro Cabrera
  • Jair KoillerEmail author
Article
  • 78 Downloads

Abstract

An interesting problem in geometric control theory arises from robotics and space science: find a smooth curve, controlled by a bounded acceleration, connecting in minimum time two prescribed tangent vectors of a Riemannian manifold Q. The state equation is \( \nabla _{\dot{\gamma }} \dot{\gamma } = u \in TQ, |u| \le A\). Applying Pontryagin’s principle one gets a Hamiltonian system in \(T^*(TQ)\). We consider this problem in \(S^2(r)\). Seemingly, it has not been addressed before. Via the SO(3) symmetry, we reduce the four degrees of freedom system to the five variables \((a,v, M_1,M_2,M_3),\) where v is the scalar velocity, conjugated to a costate variable a and \((M_1,M_2,M_3)\) are costate variables that satisfy \(\{ M_i, M_j\} = \epsilon _{ijk} M_k\). We derive the reduced equations and find special analytical solutions, that are organizing centers for the dynamics. Reconstruction of the curve \(\gamma (t)\) is achieved by a time dependent linear system of ODEs for the orthogonal matrix R whose first column is the unit tangent vector of the curve and whose last column is the unit normal vector to the sphere.

Keywords

Riemannian splines Geometric control Reduction Reconstruction 

Mathematics Subject Classification

53D20 65D07 49J15 70H06 

Notes

Acknowledgements

This work was supported by CAPES/CNPq grants from the Science without Frontiers program PVE11/2012 and PVE089/2013. We thank Maria Soledad Aronna and Pierre Martinon for her help with the software BOCOP. The time minimal spline problem on the sphere was implemented by students from the Summer Undergraduate Program PIBIC at the Mathematics Department of UFMG. JK thanks the colleagues Raphael Campos Drumond, Mario Jorge Carneiro, Sylvie Kamphorst, Sonia Carvalho for the invitation, and to Matthew Perlmutter for sharing the classes. He also wishes to thank the Organizing Committee and the colleagues at the Instituto Superior Tecnico for a wonderful meeting in honor of Prof. Waldyr Oliva.

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Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2017

Authors and Affiliations

  • Teresa Stuchi
    • 1
  • Paula Balseiro
    • 2
  • Alejandro Cabrera
    • 3
  • Jair Koiller
    • 4
    Email author
  1. 1.Departamento de Física MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Departamento de Matemática AplicadaUniversidade Federal FluminenseNiteróiBrazil
  3. 3.Departamento de Matemática AplicadaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  4. 4.Departamento de MatemáticaUniversidade Federal de Juiz de Fora Campus UniversitárioJuiz de ForaBrazil

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