Computing Feasible Trajectories for an Articulated Probe in Three Dimensions

Consider an input consisting of a set of $n$ disjoint triangular obstacles in $\mathbb{R}^3$ and a target point $t$ in the free space, all enclosed by a large sphere $S$ of radius $R$ centered at $t$. An articulated probe is modeled as two line segments $ab$ and $bc$ connected at point $b$. The length of $ab$ can be equal to or greater than $R$, while $bc$ is of a given length $r \leq R$. The probe is initially located outside $S$, assuming an unarticulated configuration, in which $ab$ and $bc$ are collinear and $b \in ac$. The goal is to find a feasible (obstacle-avoiding) probe trajectory to reach $t$, with the condition that the probe is constrained by the following sequence of moves -- a straight-line insertion of the unarticulated probe into $S$, possibly followed by a rotation of $bc$ at $b$ for at most $\pi/2$ radians, so that $c$ coincides with $t$. We prove that if there exists a feasible probe trajectory, then a set of extremal feasible trajectories must be present. Through careful case analysis, we show that these extremal trajectories can be represented by $O(n^4)$ combinatorial events. We present a solution approach that enumerates and verifies these combinatorial events for feasibility in overall $O(n^{4+\epsilon})$ time using $O(n^{4+\epsilon})$ space, for any constant $\epsilon > 0$. The enumeration algorithm is highly parallel, considering that each combinatorial event can be generated and verified for feasibility independently of the others. In the process of deriving our solution, we design the first data structure for addressing a special instance of circular sector emptiness queries among polyhedral obstacles in three dimensional space, and provide a simplified data structure for the corresponding emptiness query problem in two dimensions.