A direct method to find optimal trajectories for mobile robots using inverse kinematics

Robot motion planning has applications in a wide range of areas such as robot navigation and medical surgery. Among the topics that are related to motion planning, computing time-optimal trajectories is challenging and of great importance. We want to move a robot from the start to the goal in the shortest time. There are four different vehicle designs that are considered in our work: Dubins cars, Reeds-Shepp cars, differential drive cars and omni-directional vehicles. The objective is to help us better understand structures of optimal trajectories of some certain kinematic vehicles, as well as to form a good basis for comparison with the indirect method, which applies Pontryagin's principle to search rather than searching directly. In the proposed work, we build a direct method to explore time-optimal trajectories for mobile robots by using inverse kinematics: search free parameters (durations) to reach an intermediate configuration, then compute the last three durations from the intermediate configuration to the goal configuration by using inverse kinematics. The property of inverse kinematics guarantees the system to reach the goal precisely. Moreover, the inverse kinematics approach is a very simple algorithm for computing trajectories for systems with a small number of switches in their optimal trajectories. We have implemented the inverse kinematics solver for three-segment trajectories and a brute-force search planner in the absence of obstacles using naive uniform sampling algorithm. The results show that our method can compute optimal trajectories for Dubins cars and differential drive cars. It also produces good trajectories for omni-directional vehicles in most of our test cases. We also re-implemented the planner in C and applied optimization strategies. The search time turned out to be largely reduced.