On the cuspless sub-Riemannian geodesics in $\mathbb{R}\rtimes S^2$

A. Ghosh, R. Duits, T.C.J. Dela Haije

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We study the cuspless curves in three dimensional Euclidean space that minimize the energy functional {formula omitted} defined for each spatial arc-length parameterized curve with curvature ¿ and length (free) L=0. To compute the stationary curves, we took the Lagrangian approach used by Bryant and Grifiths for similar minimization problems. We have also shown that two other approaches, a direct variational method in R3 and the Hamiltonian approach using the Pontryagin's Maximum Principle (PMP) give the same ODE's for the stationary curves (that are locally minimizing due to PMP). We show that using the spatial arc-length parameterizations is advantageous in our case over the sub-Riemannian arc-length parameterization. The Bryant and Griffiths approach leads us to explicit formulas for the stationary curves. The formulas allow us to extract geometrical properties of the cuspless sub-Riemannian geodesics, such as planarity conditions and explicit bounds on the torsion. Moreover, they allow a study of the symmetries of the associated exponential map, and they allow to numerically solve both the initial and the boundary value problem, and to numerically compute the range of the exponential map. As a result, we characterize necessary conditions on the boundary conditions for which our optimization is well-posed.
Original languageEnglish
Number of pages42
Publication statusPublished - 2013

Publication series

Volume1305.6061 [math.OC]


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