Maximizing visibility in nonconvex polygons: nonsmooth analysis and gradient algorithm design

This paper presents a motion control algorithm for a planar mobile observer such as, e.g., a mobile robot equipped with an omni-directional camera. We propose a nonsmooth gradient algorithm for the problem of maximizing the area of the region visible to the observer in a simple nonconvex polygon. First, we show that the visible area is almost everywhere a locally Lipschitz function of the observer location. Second, we provide a novel version of LaSaIIe invariance principle for discontinuous vector fields and Lyapunov functions with a finite number of discontinuities. Finally, we establish the asymptotic convergence properties of the nonsmooth gradient algorithm and we illustrate numerically its performance.