Time-optimal paths for a Dubins airplane

We consider finding a time-optimal trajectory for an airplane from some starting point and orientation to some final point and orientation. Our model extends the Dubins car by L.E. Dubins (1957) to have altitude, which leads to
Dubins

airplane
. We assume that the system has independent bounded control over the altitude velocity as well as the turning rate in the plane. Through the use of the Pontryagin Maximum Principle, we characterize the time-optimal trajectories for the system. They are composed of turns with minimum radius, straight line segments, and pieces of planar elastica. One motivation for determining these elementary pieces is for use as motion primitives in modern planning and control algorithms that consider obstacles.