Closed form parametrisation of 3D clothoids by arclength with both linear varying curvature and torsion

The extension from the planar case to three dimensions of the clothoid curve (Euler spiral) is herein presented, that is, a curve parametrised by arc length, whose curvature and torsion are linear (affine) functions of the arc length. The problem is modelled as a linear time variant system and its stability is studied with Lyapunov techniques. Closed form solutions in terms of the standard Fresnel integrals are for the first time presented and they are valid for a wide family of clothoids; for the remaining cases, numerical methods of order four, based on Lie algebra, Magnus Expansions and Commutator-Free Expansions are provided. These geometric integrators have been optimised to be easily implemented with few lines of code and require only matrix-vector products (avoiding explicit matrix exponentials at each iteration). The achieved computational efficiency has a dramatic impact for several real-time applications, especially for designing trajectories for flying vehicles, as previous approaches were fully numeric and could not take advantage of the present closed form solutions, which render the computation of the 3D clothoid as computationally expensive as its planar companion.