Approximation of Curvature-Constrained Shortest Paths through a Sequence of Points

Let B be a point robot moving in the plane, whose path is constrained to forward motions with curvature at most 1, and let X denote a sequence of n points. Let s be the length of the shortest curvature-constrained path for B that visits the points of X in the given order. We show that if the points of X are given on-line and the robot has to respond to each point immediately, there is no strategy that guarantees apath whose length is at most f(n)s, for any finite function f(n). On the other hand, if all points are given at once, a path with length at most 5.03s can be computed in linear time. In the semi-online case, where the robot not only knows the next input point but is able to “see” the future input points included in the disk with radius R around the robot, a path of length (5.03 + O(1/R))s can be computed.