Time optimal robot navigation via the slack set method

The slack set technique, an algorithm for determining a minimum-time obstacle avoidance trajectory for a robot in a known environment, is presented. For time-optimal trajectories with constrained acceleration and velocity, the shortest time of motion may be different for each joint or axis of the system. Thus some delay of a joint other than the slowest will not necessarily affect the time of motion for the entire system. This natural redundancy for obstacle avoidance is exploited in order to simplify the trajectory search algorithm by at least one order of magnitude. By neglecting the presence of all obstacles and assigning maximum control to each actuator (bang-bang control), a lower-bound estimate of the time needed to complete a task (T_task) is calculated. The A* heuristic search is used to search what is named the slack set, namely, a subset of the state space that contains only those states that are members of a trajectory with a task time equal to T _task. If no trajectory is found during the initial search, the subset of the state-space being examined is sequentially increased until a valid trajectory is found. The slack set technique is guaranteed to find a feasible monotonic trajectory if such a trajectory exists in the slack set. Since, in general, the minimum-time obstacle avoidance trajectory is not unique, secondary constraints such as minimum distance, minimum distance in the state space, and others can also be satisfied. The method is demonstrated via a planar mobile robot