Robust Path Planning and Control For Polygonal Environments via Linear Programming

We propose a novel approach for navigating in polygonal environments by synthesizing controllers that take as input relative displacement measurements with respect to a set of landmarks. Our algorithm is based on solving a sequence of robust min-max Linear Programming problems on the elements of a cell decomposition of the environment. The optimization problems are formulated using linear Control Lyapunov Function (CLF) and Control Barrier Function (CBF) constraints, to provide stability and safety guarantees, respectively. The inner maximization problem ensures that these constraints are met by all the points in each cell, while the outer minimization problem balances the different constraints in a robust way. We show that the min-max optimization problems can be solved efficiently by transforming it into regular linear programming via the dualization of the inner maximization problem. We test our algorithm to agents with first and second order integrator dynamics, although our approach is in principle applicable to any system with piecewise linear dynamics. Through our theoretical results and simulations, we show that the resulting controllers: are optimal (with respect to the criterion used in the formulation), are applicable to linear systems of any order, are robust to changes to the start location (since they do not rely on a single nominal path), and to significant deformations of the environment.