The problem of finding optimal trajectories is a basic problem in aerospace engineering with many different solution algorithms. One widely investigated branch is the problem of finding shortest paths under maximum curvature constraint. The shortest planar trajectories under curvature constraint are the well-known Dubins paths, which can either be arc-line-arc or arc-arc-arc combinations. In recent years few works dealt with 3D shortest paths under the curvature constraint. Theoretical works prove the existence of 3 types of spatial shortest trajectories: 3D arc-line-arc, 3D arc-arc-arc and helicoidal-arc. Some further works proposed algorithms for finding the first one (3D arc-line-arc trajectories), which is the shortest path when the start and the end point are far from each other.
This work first proposes an algorithm to calculate 3D arc-arc-arc type trajectories, (the solution for 3D arc-line-arc is already known), followed by an investigation of the helicoidal-arc type trajectory. In the case of helicoidal-arc the exact solution depends on a high order TPBVP (Two Point Boundary Value Problem), a difficult numerical problem. In this study we propose to approximate the helicoidal-arc with polynomials. The approximate polynomial solutions are compared to the optimal paths obtained from numerical integration and validated by GPOPS (General Purpose OPtimal Control Software). It is shown that the solutions based on the polynomials lead to good approximation of the optimal trajectories.
