TIME-OPTIMAL MOTION OF 2 OMNIDIRECTIONAL ROBOTS CARRYING A LADDER UNDER A VELOCITY CONSTRAINT

We consider the problem of computing a time-optimal motion for two omn idirectional robots carrying a ladder from an initial position to a fi nal position in a plane without obstacles. At any moment during the mo tion, the distance between the robots remains unchanged and the speed of each robot must be either a given constant upsilon, or 0. 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 upsilon. Thi s lower bound may or may not be achievable, however, since the other r obot may not have sufficient time to complete the necessary rotation a round the first robot (that is moving along a straight line at speed u psilon) within the given time. We first derive, by solving an ordinary differential equation, a necessary and sufficient condition under whi ch this lower bound is achievable. If the condition is satisfied, then a time-optimal motion of the robots is computed by solving another di fferential 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.