Motion Planning for Small Formations of Autonomous Vehicles Navigating on Gradient Fields

In this paper, we present a motion planning scheme for navigation of a contour-like formation of autonomous underwater vehicles on gradient fields and subsequent convergence to desired isoclines, inspired by evolution of closed planar curves. The basic evolution behaviour is modified to include moving boundary points and incorporate safety constraints on formation parameters. Also, the whole process is decomposed into a sequence of well-behaving states. As opposed to the basic model, the regularized solution is characterized by the maximum allowable curvature rather than balance of forces determined by fixed coefficients. Nevertheless, the proposed framework subsumes the original model. Blocking states and fairness are briefly discussed.