Geodesic
Asymptotics of extremal curves in the ball rolling problem on the plane
Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Mathematical Control Theory and Mechanics (Suzdal, July 3–7, 2009), Tome 42 (2011), pp. 158-165.

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In the present paper, we study an optimal sphere rolling problem on the plane (without slew and slip) with predefined boundary-value conditions. To solve it, we use methods from the optimal control theory. The controlled system for sphere orientation is represented via the rotation quaternion. Asymptotics of extremal paths on a sphere rolling along small-amplitude sine waves is found. 
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A. P. Mashtakov. Asymptotics of extremal curves in the ball rolling problem on the plane. Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Mathematical Control Theory and Mechanics (Suzdal, July 3–7, 2009), Tome 42 (2011), pp. 158-165. http://geodesic.mathdoc.fr/item/CMFD_2011_42_a14/

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