Coordinated gradient descent: a case study of Lagrangian dynamics with projected gradient information

Luc Moreau, Ralf Bachmayer and Naomi Ehrich Leonard

The paper studies gradient descent algorithms for vehicle networks. Each vehicle within the network is modeled as a double integrator in the plane. For each individual vehicle, the control input enabling coordinated gradient descent consists of a gradient descent control term and additional inter-vehicle forcing terms. When each vehicle has enough sensors to measure the full gradient at its current position, then the closed-loop system becomes Lagrangian. We focus in the present paper upon the more practical situation where each vehicle has only one sensor with which to sample the environment. We take this into account by replacing the full gradient in the closed-loop equations by its projection on the direction of motion for each individual vehicle. This gives rise to a differential equation with discontinuous right-hand side. In order to avoid the (practical and theoretical) complications that arise as a consequence of these discontinuities, we modify the inter-vehicle forcing terms and represent the velocity of each vehicle by a magnitude and an angle, resulting in a set of smooth differential equations. We demonstrate our approach with simulations.