# R. Duits, A. Ghosh, T. Dela Haije, A. Mashtakov. On sub-Riemannian geodesics in SE(3) whose spatial projections do not have cusps

Title: On sub-Riemannian geodesics in SE(3) whose spatial projections do not have cusps
Authors: R. Duits, A. Ghosh, T. Dela Haije, A. Mashtakov
Journal title: Journal of Dynamical and Control Systems
Year: 2016
Issue: 4
Volume: 22
Pages: 771–805
Citation:

R. Duits, A. Ghosh, T. Dela Haije, A. Mashtakov. On sub-Riemannian geodesics in SE(3) whose spatial projections do not have cusps // Journal of Dynamical and Control Systems, 22(4), pp. 771-805, 2016.

Abstract:

We consider the problem Pcurve of minimizing \int_0^L \sqrt{ξ^2 + κ^2(s)} ds for a curve x in R^3 with fixed boundary points and directions. Here, the total length L ≥ 0 is free, s denotes the arclength parameter, κ denotes the absolute curvature of x, and ξ > 0 is constant.We lift problem Pcurve on R^3 to a sub-Riemannian problem Pmec on SE(3)/({0} × SO(2)). Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem Pcurve. We apply the Pontryagin Maximum Principle (PMP) and prove Liouville integrability of the Hamiltonian system. We derive explicit analytic formulas for such sub-Riemannian geodesics, relying on the co-adjoint orbit structure, an underlying Cartan connection, and the matrix representation of SE(3) arising in the Cartan-matrix. These formulas allow us to extract geometrical properties of the sub-Riemannian geodesics with cuspless projection, such as planarity conditions, explicit bounds on their torsion, and their symmetries. Furthermore, they allow us to parameterize all admissible boundary conditions reachable by geodesics with cuspless spatial projection. Such projections lay in the upper half space. We prove this for most cases, and the rest is checked numerically. Finally, we employ the formulas to numerically solve the boundary value problem, and visualize the set of admissible boundary conditions.