### Abstract

We consider the problem P_{curve} of minimizing (Formula presented.) for a curve x in (Formula presented.) 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 (Formula presented.) 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.

Original language | English |
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Pages (from-to) | 771-805 |

Number of pages | 35 |

Journal | Journal of Dynamical and Control Systems |

Volume | 22 |

Issue number | 4 |

DOIs | |

Publication status | Published - Oct 2016 |

### Keywords

- Geodesics
- Pontryagin Maximum Principle
- Special Euclidean motion group
- Sub-Riemannian geometry

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## Cite this

*Journal of Dynamical and Control Systems*,

*22*(4), 771-805. https://doi.org/10.1007/s10883-016-9329-4