Minimal time splines on the sphere

  • Teresa Stuchi
  • Paula Balseiro
  • Alejandro Cabrera
  • Jair KoillerEmail author


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.


Riemannian splines Geometric control Reduction Reconstruction 

Mathematics Subject Classification

53D20 65D07 49J15 70H06 



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.


  1. 1.
    Francis, B., Maggiore, M.: Flocking and Rendezvous in Distributed Robotics. Springer, New York (2016)CrossRefzbMATHGoogle Scholar
  2. 2.
    Smith, S., Broucke, M., Francis, B.: Curve shortening and the rendezvous problem for mobile autonomous robots. IEEE Trans. Autom. Control 52(6), 1154 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lin, Z., Francis, B., Maggiore, M.: Getting mobile autonomous robots to rendezvous. In: Francis, B.A., Smith, M.C., Willems, J.C. (eds.) Control of Uncertain Systems: Modelling, Approximation, and Design. Lecture Notes in Control and Information Sciences, vol. 329. Springer-Verlag, Berlin, Heidelberg (2006)Google Scholar
  4. 4.
    Weyr, A.: The Martian. Broadway Books, Portland (2014)Google Scholar
  5. 5.
    Lewis, A.D.: Aspects of geometric mechanics and control of mechanical systems. Ph.D. thesis, Caltech. (1995)
  6. 6.
    Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems. Texts in Applied Mathematics, vol. 49. Springer, New York (2005)CrossRefGoogle Scholar
  7. 7.
    Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inf. 6(4), 465 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Crouch, P., Leite, F.S.: Geometry and the dynamic interpolation problem. In: Proceedings of the 1991 American Control Conference, pp. 1131–1136 (1991)Google Scholar
  9. 9.
    Younes, L.: Shapes and Diffeomorphisms. Applied Mathematical Sciences, vol. 171. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Singh, N., Niethammer, M.: Splines for diffeomorphic image regression. In: Golland, P., Hata, N., Barillot, C., Hornegger, J., Howe, R. (eds.) Medical Image Computing and Computer-Assisted Intervention–MICCAI 2014. Lecture Notes in Computer Science, vol. 8674, pp. 121–129. Springer (2014).
  11. 11.
    Fiot, J.B., Raguet, H., Risser, L., Cohen, L.D., Fripp, J., Vialard, F.X.: Longitudinal deformation models, spatial regularizations and learning strategies to quantify Alzheimer’s disease progression. NeuroImage Clin. 4, 718 (2014).
  12. 12.
    Durrleman, S., Fletcher, T., Gerig, G., Niethammer, M., Pennec, X.: Spatio-Temporal Image Analysis for Longitudinal and Time-Series Image Data. Lecture Notes in Computer Science, vol. 8682. Springer, Cambridge (2015). Google Scholar
  13. 13.
    Kaya, C.Y., Noakes, J.L.: Finding interpolating curves minimizing \(L^\infty \) acceleration in the Euclidean space via optimal control theory. SIAM J. Control Optim. 51(1), 442 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Noakes, L.: Minimum \(L^\infty \) accelerations in Riemannian manifolds. Adv. Comput. Math. 40(4), 839 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Castro, A.L., Koiller, J.: On the dynamic Markov-Dubins problem: from path planning in robotics and biolocomotion to computational anatomy. Regul. Chaotic Dyn. 18(1–2), 1 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Balseiro, P., Stuchi, T., Cabrera, A., Koiller, J.: About simple variational splines from the Hamiltonian viewpoint. J. Geom. Mech. 9(3), 257–290 (2017).
  17. 17.
    Popiel, T.: Mathematics of control. Signals Syst. 19(3), 235 (2007). MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hinkle, J., Muralidharan, P., Fletcher, P., Joshi, S.: Polynomial regression on Riemannian Manifolds. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) Computer Vision—ECCV 2012. Lecture Notes in Computer Science, vol. 7574, pp. 1–14. Springer, Berlin (2012).
  19. 19.
    Hinkle, J., Fletcher, P., Joshi, S.: Intrinsic polynomials for regression on Riemannian manifolds. J. Math. Imaging Vis. 50(1–2), 32 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Burnett, C.L., Holm, D.D., Meier, D.M.: Inexact trajectory planning and inverse problems in the Hamilton–Pontryagin framework. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. (2013).
  21. 21.
    Gay-Balmaz, F., Holm, D.D., Meier, D.M., Ratiu, T.S., Vialard, F.X.: Invariant higher-order variational problems. Commun. Math. Phys. 309(2), 413 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gay-Balmaz, F., Holm, D.D., Meier, D.M., Ratiu, T.S., Vialard, F.X.: Invariant higher-order variational problems II. J. Nonlinear Sci. 22(4), 553 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Niethammer, M., Huang, Y., Vialard, F.X.: Geodesic regression for image time-series. In: Fichtinger, G., Martel, A., Peters, T. (eds.) Medical Image Computing and Computer-Assisted Intervention—MICCAI 2011. Lecture Notes in Computer Science, vol. 6892, pp. 655–662. Springer, Berlin (2011).
  24. 24.
    Steinke, F., Hein, M., Schölkopf, B.: Nonparametric regression between general Riemannian manifolds. SIAM J. Imaging Sci. 3(3), 527 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Desai, N., Ploskonka, S., Goodman, L.R., Austin, C., Goldberg, J., Falcone, T.: Analysis of embryo morphokinetics, multinucleation and cleavage anomalies using continuous time-lapse monitoring in blastocyst transfer cycles. Reproduct. Biol. Endocrinol. RB&E 12, 54 (2014). CrossRefGoogle Scholar
  26. 26.
    Chang, D.E.: A simple proof of the Pontryagin maximum principle on manifolds. Automatica 47(3), 630 (2011).
  27. 27.
    Crouch, P., Leite, F.: The dynamic interpolation problem: on Riemannian manifolds, Lie groups, and symmetric spaces. J. Dyn. Control Syst. 1(2), 177 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Intriligator, M.D.: Mathematical Optimization and Economic Theory. Classics in Applied Mathematics, vol. 39. SIAM, Philadelphia (2002)CrossRefGoogle Scholar
  29. 29.
    Pauley, M., Noakes, L.: Cubics and negative curvature. Differ. Geom. Appl. 30(6), 694 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Johnson, S.D.: Computing minimum time paths with bounded acceleration. arXiv:1310.5905 [math.NA] (2013)
  31. 31.
    Carozza, D., Johnson, S., Morgan, F.: Baserunner’s optimal path. Math. Intell. 32(1), 10 (2010). CrossRefzbMATHGoogle Scholar
  32. 32.
    Venkatraman, A., Bhat, S.P.: Optimal planar turns under acceleration constraints. In: Proceedings of the 45th IEEE Conference on Decision and Control, pp. 235–240 (2006).

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

Personalised recommendations