Time-optimal motion of two omnidirectional robots carrying a ladder under a velocity constraint

We consider the problem of computing a time-optimal motion for two omnidirectional robots carrying a ladder from an initial position to a final position in a plane without obstacles. At any moment during the motion, the distance between the robots remains unchanged and the speed of each robot must be either a given constant /spl upsi/, or O. A trivial lower bound on time for the robots to complete the motion is the time needed for the robot farther away from its destination to move to the destination along a straight line at a constant speed of /spl upsi/. This lower bound may or may not be achievable, however, since the other robot may not have sufficient time to complete the necessary rotation around the first robot (that is moving along a straight line at speed v) within the given time. We first derive, by solving an ordinary differential equation, a necessary and sufficient condition under which this lower bound is achievable. If the condition is satisfied, then a time-optimal motion of the robots is computed by solving another differential equation numerically. Next, we consider the case when this condition is not satisfied, and show that a time-optimal motion can be computed by taking the length of the trajectory of one of the robots as a functional and then applying the method of variational calculus. Several optimal paths that have been computed using the above methods are presented.