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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 alarge sphere $S$ of radius $R$ centered at $t$. An articulated probe is modeledas 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 $r \leqR$. The probe is initially located outside $S$, assuming an unarticulatedconfiguration, in which $ab$ and $bc$ are collinear and $b \in ac$. The goal isto find a feasible (obstacle-avoiding) probe trajectory to reach $t$, with thecondition that the probe is constrained by the following sequence of moves -- astraight-line insertion of the unarticulated probe into $S$, possibly followedby a rotation of $bc$ at $b$ for at most $\pi/2$ radians, so that $c$ coincideswith $t$.  We prove that if there exists a feasible probe trajectory, then a set ofextremal 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 andverifies 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 thateach combinatorial event can be generated and verified for feasibilityindependently of the others. In the process of deriving our solution, we designthe first data structure for addressing a special instance of circular sectoremptiness queries among polyhedral obstacles in three dimensional space, andprovide a simplified data structure for the corresponding emptiness queryproblem in two dimensions.

Computing Feasible Trajectories for an Articulated Probe in Three Dimensions