Combinatorial characterizations and algorithms for trajectory planning of an articulated robotic probe in three dimensions

Consider a three-dimensional workspace that contains n disjoint triangular obstacles and a destination point. We define the problem of trajectory planning for a two-segment articulated probe as the task of computing a feasible path that allows the probe to reach the destination while avoiding collisions with obstacles. The articulated probe is constrained to a sequence of movements – a straight-line insertion possibly followed by a rotation of the end segment. A feasible probe trajectory is referred to as extremal when the probe or its path is in immediate contact with obstacles, effectively defining the boundaries of the probe trajectory. We prove that if there exists a feasible probe trajectory, then a finite set of extremal feasible trajectories must be present. Through careful case analysis, we show that these extremal trajectories can be represented by $O\left(n^4\right)$ combinatorial events. We present a solution approach that enumerates and verifies these combinatorial events for feasibility in overall $O\left(n^{4+\epsilon }\right)$ time using $O\left(n^{4+\epsilon }\right)$ 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.