  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.
