This paper investigates the connection between the kinematics of robots arms and the shortest paths for mobile robots. Lagrange multipliers are used to show that the shortest paths are equivalent to arms in configurations that balance an external force, while applying equal torques and forces at each joint. Analysis of the arm Jacobian yields a further geometric interpretations of optimal paths, constraining the locations of rotation centers and the directions of translations that may occur along optimal paths.

Analytical results describing the optimal trajectories for general classes of robot systems have proven elusive, in part because the optimal trajectories for a complex system may not exist, or may be computed only numerically from differential equations. This paper studies a simpler optimization problem: finding an optimal sequence and optimal durations of motion primitives (simple preprogrammed actions) to reach a goal. By adding a fixed cost for each switch between primitives, we ensure that optimal trajectories exist and are well-behaved. To demonstrate this approach, we prove some general results that geometrically characterize time-optimal trajectories for rigid bodies in the plane with costly switches (allowing comparison with previous analysis of optimal motion using Pontryagin's Maximum Principle), and also present a complete analytical solution for a problem of moving a heavy park bench by rotating the bench around each end point in sequence.

This paper presents first attempts at a method for searching for time-optimal trajectories for a general model of mobile robots that includes Dubins and Reeds-Shepp cars, differential-drive robots, and omnidirectional robots as special cases. The paper takes as a starting point recent results by the authors that describe necessary conditions on the trajectories, based on Pontryagin's Maximum Principle. These necessary conditions reduce the problem of finding an optimal trajectory between start and goal to a few one-dimensional search problems. This search is not formally guaranteed to find a near-optimal trajectory if the sampling of the search space is not fine enough, but comparison to existing analytical results for specific systems, and a complete numerical search over trajectories with only a few control switches, demonstrates effectiveness of the method.

What is the shortest or fastest path a mobile robot can follow between two configurations in the unobstructed plane? The answer to this fundamental question is only known analytically for a few planar mobile robots: the Dubins and Reeds-Shepp steered cars, the differential drive, and a particular omnidirectional robot. This paper explores the optimal trajectories for a general parameterized model of a mobile robot that includes each previously-studied vehicle as a special case. The model also allows characterization of the optimal trajectories for several other mobile robot designs for which the optimal trajectories have not been previously explored. The paper applies Pontryagin's Maximum Principle to the generalized robot to find necessary conditions that optimal trajectories must satisfy, and gives geometric interpretations of the conditions. We also present an algorithm that generates and classifies all optimal trajectories for a given design.