The optimal rolling of a sphere, with twisting but without slipping

Beschatnyi, I Yu

doi:10.1070/sm2014v205n02abeh004370

Cited by 8 publications

(6 citation statements)

References 16 publications

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“…Below we will identify the Lie algebra m with the dual space m * via the basis V 1 , V 2 , V 3 , e 1 , e 2 . The vertical subsystem of Pontryagin's Hamiltonian system is as follows [25]:…”

Section: Homogeneous Geodesicsmentioning

confidence: 99%

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Homogeneous geodesics in sub-Riemannian geometry

Podobryaev¹

2022

Preprint

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We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step 1 and 2. Finally, we get a broad condition for existence of at least one homogeneous geodesic.

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“…Below we will identify the Lie algebra m with the dual space m * via the basis V 1 , V 2 , V 3 , e 1 , e 2 . The vertical subsystem of Pontryagin's Hamiltonian system is as follows [25]:…”

Section: Homogeneous Geodesicsmentioning

confidence: 99%

“…Yu. Beschastnyi [25] obtained geodesics as products of two one-parametric subgroups m(t) = exp t(p + (0)) exp (−tp).…”

Section: Homogeneous Geodesicsmentioning

confidence: 99%

Homogeneous geodesics in sub-Riemannian geometry

Podobryaev¹

2022

Preprint

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“…Thus, their created trajectories were not easy to be realized by dynamical systems [24] and there were singularities in different regions of spherical manifold due to the locality of the solutions. Beschastnyi [25] studied the optimality problem of the spin-rolling sphere. Extremal trajectories were parameterized, and their cut times were estimated for optimality.…”

Section: Dexterous Manipulationsmentioning

confidence: 99%

“…We plan the sphere to reach its final configuration C 0 → C 1 with a straight trajectory on L s in the given t f time. This planning problem can be looked as the optimal solution since the shortest trajectory on the plane L s is always straight [25].…”

Section: Problem Statementmentioning

confidence: 99%

Motion Planning of a Spin-Rolling Sphere on a Plane

Tafrishi¹,

Yamamoto²,

Hirata³

2021

Preprint

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The paper deals with motion planning for a spin-rolling sphere when the sphere follows a straight path on a plane. Since the motion of the sphere is constrained by the straight line, the control of the sphere's spin motion is essential to converge to a desired configuration of the sphere. In this paper, we show a new geometric-based planning approach that is based on a full-state description of this nonlinear system. First, the problem statement of the motion planning is posed. Next, we develop a geometric controller implemented as a virtual surface by using the Darboux frame kinematics. This virtual surface generates arc-length-based inputs for controlling the trajectories of the sphere. Then, an iterative algorithm is designed to tune these inputs for the desired configurations. The feasibility of the proposed approach is verified by simulations.

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“…Figure 1 with elasticas was taken from Euler's book [16]. Geodesics in the problem of rolling with spinning were investigated in [10].…”

Section: T *mentioning

confidence: 99%