On Shortest Arc-To-Arc Dubins Path.

For a given set of orbits, the Orbiting Dubins Traveling Salesman Problem (ODTSP) involves finding Dubins tour that is tangential to each orbit at some point. We consider a shortest Arc-to-Arc Dubins (ATAD) path problem that arrives in solving lower bound to the ODTSP. Given an initial and a final arc, the objective of ATAD is to find the shortest Dubins path such that the initial and final point lie on the given two arcs, and the path is tangential to the arcs. We analyze the six Dubins modes and the degenerate cases to find local minima. We present the optimal solution for the ATAD, along with an algorithm that uses this solution to compute tight lower bounds for the ODTSP. We test the lower bounding algorithm on several random instances and report the results. Using this algorithm, we show that the percent gap between upper and lower bounds is less than 10% for most instances.