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 a mobile robot equipped with an omnidirectional camera. We propose a nonsmooth gradient algorithm for the problem of maximizing the area of the region visible to the observer in a nonself-intersecting 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 the LaSalle 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 illustrate numerically its performance.