Fast control using homotopy properties for obstacle-avoidance of systems with input constraints

For the optimal point-to-point control of a discrete-time linear system in a time varying non-convex environment, this paper investigates how such problems can be handled online. An illustrative example is that of steering a robotic end-effector from an initial to a final state, while executing the motion free of collision in the presence of a human worker. The challenge is to obtain a solution with low effort, to be online applicable, while maintaining a close to optimal solution. The proposed method consists of first computing a range of trajectories, homotopic to the optimal unconstrained solution offline. Then, upon detection of an obstacle which may block the currently executed trajectory, an offline synthesized controller steers the system quickly to a desired homotopic trajectory. The resulting trajectory is free of collision and is determined by a fast iterative procedure.