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 $O(n^{2+ϵ})$ time using $O(n^{2+ϵ})$ space, for any constant $ϵ>0$. Furthermore, we prove that all values r for which a feasible probe trajectory exists form $O(n^2)$ intervals, and can be computed in $O(n^{5/2})$ time using $O(n^{2+ϵ})$ 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 $O(n^2)$, which can be constructed in $O(n^2)$ time. To obtain our solutions, we design efficient data structures for a number of interesting variants of geometric intersection and emptiness query problems.

铰接探针在平面上建模为两个线段ab和bc，在b处连接，其中ab很长，而bc的长度很小r。我们调查涉及铰接两段探头，其中长度的轨迹规划问题[R的BC可定制。考虑一组总共有n个顶点的简单多边形障碍物P，一个位于自由空间中的目标点t（使t看不到无穷大）和一个以t包围的圆心SP。探针最初位于S外部，且ab和bc是共线的，并且仅限于以下移动顺序：直线将abc插入S中，然后bc围绕b旋转。目的是为探针计算可行的避障轨迹，以便在移动序列之后，c与t重合。

我们证明，对于n个线段障碍物，可以找到存在可行探测轨迹的最小长度r。$Ø（{ñ}^{\mathrm{2个}+ϵ}）$ 时间使用 $Ø（{ñ}^{\mathrm{2个}+ϵ}）$ 空间，对于任何常数 $ϵ>0$。此外，我们证明了存在可行探测轨迹的所有值r都形成了$Ø（{ñ}^{\mathrm{2个}}）$ 间隔，可以在 $Ø（{ñ}^{5/\mathrm{2个}}）$ 时间使用 $Ø（{ñ}^{\mathrm{2个}+ϵ}）$空间。我们还表明，对于给定的r，铰接式探针的可行轨迹空间可以通过复杂度的简单排列来表征$Ø（{ñ}^{\mathrm{2个}}）$，可以在 $Ø（{ñ}^{\mathrm{2个}}）$时间。为了获得我们的解决方案，我们为许多有趣的几何交集和空度查询问题的变体设计了有效的数据结构。