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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 $L\geq 0$ isfree, $s$ denotes the arclength parameter, $\kappa$ denotes the absolutecurvature 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, thespatial projections of sub-Riemannian geodesics do not exhibit cusps and theysolve problem $\mathbf{P_{curve}}$. We apply the Pontryagin Maximum Principle(PMP) and prove Liouville integrability of the Hamiltonian system. We deriveexplicit analytic formulas for such sub-Riemannian geodesics, relying on theco-adjoint orbit structure, an underlying Cartan connection, and the matrixrepresentation of $\operatorname{SE(3)}\nolimits$ arising in the Cartan-matrix.These formulas allow us to extract geometrical properties of the sub-Riemanniangeodesics with cuspless projection, such as planarity conditions, explicitbounds on their torsion, and their symmetries. Furthermore, they allow us toparameterize all admissible boundary conditions reachable by geodesics withcuspless spatial projection. Such projections lay in the upper half space. Weprove this for most cases, and the rest is checked numerically. Finally, weemploy the formulas to numerically solve the boundary value problem, andvisualize the set of admissible boundary conditions.

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