On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

Abstract

We consider the problem P curve of minimizing L 0 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 P curve on R 3 to a sub-Riemannian problem P mec 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 P curve . 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.

New exact and numerical solutions of the (convection–)diffusion kernels on SE(3)

“…Comparing the left and right figure, we note that the higher m, the higher the values for ρ where this branching occurs. Moreover, we have that Im(λ ρD 44 ) ∼ O(ρD 44 ).…”

“…Future work will include extensive analysis of the Monte-Carlo simulations, briefly described in Appendix E. Furthermore, we will put connections of hypo-elliptic diffusion (and Brownian bridges) on SE(3) and recently derived sub-Riemannian geodesics in SE(3) [44].…”

Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space

“…where A i are left-invariant vector fields on the group SE 3 , the controls u 3 , u 4 , u 5 are real-valued L ∞ (0, t 1 ) functions, the terminal time t 1 > 0 is free, e is the identity transformation of R 3 , g is a given element of SE 3 and ξ > 0 is a parameter that balances the influence of translation and rotations in R 3 on the length of the corresponding trajectory. Investigation of this problem was initiated in [7,12], where in particular it was shown that due to the scaling homothety, see [7,Remark 5], the general case ξ > 0 reduces to the case ξ = 1 by linear change of variables. Further in the article, we assume ξ = 1 without loss of generality.…”

“…Application of PMP to our problem gives an expression of extremal controls in terms of momenta (conjugate variables). Namely, the extremal controls u 3 , u 4 , u 5 coincide with three certain momenta, see [7,Section 3.1]. We denote the remaining three momenta by u 1 , u 2 and u 6 .…”

“…Further, being a little informal, we call the extremal control the entire vector u = (u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ) of the momenta. Application of PMP leads to a Hamiltonian system, the horizontal part of which is given by system (1.1) and determines SR geodesics, and the vertical part (on extremal controls) has the following form [7,Eq. (3.5)…”