Time-optimal trajectories of a rod in the plane subject to velocity constraints

We consider the problem of moving a line segment (a “rod” or “ladder”) in the plane between two given placements when subject to the constraint that no point on the line segment may exceed a given velocity bound. Specifically, we consider those trajectories which minimize the total time between given initial and goal placements, and provide a complete characterization of all solution, together with explicit constructions for each of the various cases encountered. This research was supported by the National Science Foundation under Grant CCR-9207422, and by a grant from the James H. Zumberge Research Innovation Fund. This problem was initially posed in an open problem session of the Second DIMACS Workshop in Computational Geometry, held at Princeton University in October, 1989. A preliminary version of this paper was presented at the 9th Annual ACM Symposium on Computational Geometry [8].