Trajectory elongation strategies with minimum curvature discontinuities for a Dubins vehicle

In this paper, we present strategies for designing curvature-bounded trajectories of any desired length between any two given oriented points. The proposed trajectory is constructed by the concatenation of three circular arcs of varying radii. Such a trajectory guarantees a complete coverage of the maximum set of reachable lengths while minimising the number of changeover points in the trajectory to a maximum of two under all scenarios. Additionally, by using the notion of internally tangent circles, we expand the set of Circle-Circle-Circle trajectories to eight kinds, consisting of {LLL, LLR, LRR, LRL, RRL, RLL, RLR, RRR} paths. The paper presents a mathematical formulation of the proposed trajectory and the conditions for the existence and classification of each kind of trajectory. We also analyse the variation of the length of the trajectory using suitable elongation strategies and derive the set of reachable lengths for all pairs of oriented points. Finally, the results of this paper are illustrated using numerical simulations.