Dynamical systems and geometry

The study of vector fields, fields of torques, and their dynamics allows interesting applications in the field of air traffic management.

Topological entropy and complexity of traffic

Air traffic can be viewed as a vector field defined by the positions and velocities of the aircraft, and evolves over time. The study of this dynamic vector system enable to introduce a measure of the intrinsic complexity of traffic, based on the topological the entropy of Kolmogorov.

Navigation functions

The use of navigation functions is common in robotics. They are used to calculate the successive positions and velocities of a robot moving in an environment in which potential energy is associated. Typically, the destination will be the lowest point of potential, while avoiding obstacles have the highest potential.

A navigation function enable to calculate a path leading from the initial position to the final destination, along a descent direction (opposite to the gradient of the potential). The existence of this pathway is warranted for potential functions having the property of Morse.

To apply this method to the type of path planning of aircraft, it is essential to avoid a major drawback of conventional navigation functions. They are not used to constrain the speed and the radius of curvature of trajectories simultaneously. However, the speed of an airplane must remain in a fixed range, and the radius of curvature must remain bounded by a value depending directly on the speed of the aircraft.

To meet these constraints, we propose a new navigation function based on bi-harmonic function. This function allows for constant velocity trajectories while keeping the good properties of navigation functions: the existence of a field of navigation, obstacle avoidance, guaranteed path to the destination.