Characterization and computation of feasible trajectories for an articulated probe with a variable-length end segment

Abstract

An articulated probe is modeled in the plane as two line segments, ab and bc, joined at b, with ab being very long, and bc of some small length r. We investigate a trajectory planning problem involving the articulated two-segment probe where the length r of bc can be customized. Consider a set P of simple polygonal obstacles with a total of n vertices, a target point t located in the free space such that t cannot see to infinity, and a circle S centered at t enclosing P. The probe initially resides outside S, with ab and bc being collinear, and is restricted to the following sequence of moves: a straight line insertion of abc into S followed by a rotation of bc around b. The goal is to compute a feasible obstacle-avoiding trajectory for the probe so that, after the sequence of moves, c coincides with t.We prove that, for n line segment obstacles, the smallest length r for which there exists a feasible probe trajectory can be found in O(n2+ϵ) time using O(n2+ϵ) space, for any constant ϵ>0. Furthermore, we prove that all values r for which a feasible probe trajectory exists form O(n2) intervals, and can be computed in O(n5/2) time using O(n2+ϵ) space. We also show that, for a given r, the feasible trajectory space of the articulated probe can be characterized by a simple arrangement of complexity O(n2), which can be constructed in O(n2) time. To obtain our solutions, we design efficient data structures for a number of interesting variants of geometric intersection and emptiness query problems.