Automatic conflict solving using biharmonic navigation functions
Résumé
Automatic conflict solving is an old standing problem within the field of ATM. Proposed algorithms fall into two categories : - Deterministic ones that have a provable property of collision avoidance. For all known algorithms, trajectories produced are generally not flyable because no bounds on speed and curvature can be imposed. - Stochastic methods that select an optimal sequence of manoeuvres. By design, trajectories are flyable, but no guarantee can be given on the fact that a collision-free planning can be found in finite time. It is highly desirable for a wide social acceptance of automated trajectory planning, even at a strategical level, that the algorithms in use have by-design the collision avoidance property and, at the same time, a mean of keeping the speed within a given interval. Navigation functions are common in the field of robotics but do not have the last property. We present a new approach based on biharmonic functions yielding a navigation field with constant speed. Such functions have been considered previously, but proof of collision avoidance is lacking: we address this problem in this work as summarized below. Navigation functions produce a speed field by taking the gradient of a potential function: if the obstacles to be avoided are at a higher potential than inner points of the domain (including destination), collision avoidance is guaranteed. If the potential has the Morse property (no critical point is degenerated) then there exists a descent direction at every point of the admissible domain, making the destination reachable. In the framework of biharmonic functions, a tensor field is produced instead of a vector one ; the Morse property is no longer relevant. We show here that all benefits of navigation functions can be recovered through the use of the bienergy density, with the ability to get constant speed fields.
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