Simple and Efficient Algorithms for Computing Smooth, Collision-Free Feedback Laws

This paper presents a novel approach to computing feedback laws in the presence of obstacles. Instead of computing a trajectory between a pair of initial and goal states, our algorithms compute a vector field over the entire state space; all trajectories obtained from following this vector field are guaranteed to asymptotically reach the goal state. As a result, the vector field globally solves the navigation problem and provides robustness to disturbances in sensing and control. In addition to being globally convergent, the vector field’s integral curves (system trajectories) are guaranteed to avoid obstacles and are C∞ smooth. We construct a vector field with these properties by using existing geometric algorithms to partition the space into simple cells; we then define local vector fields for each cell, and smoothly interpolate between them to obtain a global vector field that can be used as a feedback control for the robot. We present an algorithm that computes these feedback controls for a kinematic point robot in an arbitary dimensional space with piecewise linear boundary; the algorithm requires minimal preprocessing of the environment and is extremely fast during execution. For many practical applications in two-dimensional environments, full computation can be done in milliseconds; this means that our algorithms can be used for dynamic environments as well as static ones. We also present an algorithm for computing feedback laws over cylindrical algebraic decompositions, thereby solving a smooth feedback version of the generalized piano movers’ problem.