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Optimal Control of the Sphere Sn Rolling on En

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Abstract.

This paper investigates the optimal control problem concerning the sphere Sn rolling without slipping on n-dimensional Euclidean space En, n≥2. The differential equations governing the behaviour of the sphere constitute a sub-Riemannian distribution on the Lie group G=ℝn×SO n +1. Minimizing over the lengths of paths traced by the point of contact of the sphere yields an optimal control problem which is exploited using Noether’s Theorem to derive a family of integrals of motion for the system. This family is then employed to prove that all optimizing trajectories are projections of normal extremals. The Lax form of the extremal equations gives rise to an additional family of integrals which is reminiscent of the Manakov integrals of motion for the free rigid body problem. Both families are required to show complete integrability if n=4.

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Acknowledgments.

The author is deeply indebted to his dissertation supervisor, Vel Jurdjevic, for many helpful suggestions.

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  1.  , 1057 Shaw St., Toronto, ON, Canada, M6G 3N4

    Jason A. Zimmerman

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  1. Jason A. Zimmerman
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Correspondence to Jason A. Zimmerman.

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Zimmerman, J. Optimal Control of the Sphere Sn Rolling on En. Math. Control Signals Systems 17, 14–37 (2005). https://doi.org/10.1007/s00498-004-0143-2

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  • DOI: https://doi.org/10.1007/s00498-004-0143-2

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