On sub-Riemannian geodesics in $SE(3)$ whose spatial projections do not have cusps

We consider the problem $\mathbf{P_{curve}}$ of minimizing $\int \limits_0^L \sqrt{\xi^2 + \kappa^2(s)} \, {\rm d}s$ for a curve $\mathbf{x}$ on $\mathbb R$ with fixed boundary points and directions. Here the total length $L\geq 0$ is free, $s$ denotes the arclength parameter, $\kappa$ denotes the absolute curvature of $\mathbf{x}$, and $\xi>0$ is constant. We lift problem $\mathbf{P_{curve}}$ on $\mathbb R^3$ to a sub-Riemannian problem $\mathbf{P_{mec}}$ on $\operatorname{SE(3)}\nolimits/(\{\mathbf{0}\}\times \operatorname{SO(2)}\nolimits)$. Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem $\mathbf{P_{curve}}$. 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 $\operatorname{SE(3)}\nolimits$ 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.