We consider the problem of motion planning in the presence of uncertain obstacles, modeled as polytopes with Gaussian-distributed faces (PGDFs). A number of practical algorithms exist for motion planning in the presence of known obstacles by constructing a graph in configuration space, then efficiently searching the graph to find a collision-free path. We show that such an exact algorithm is unlikely to be practical in the domain with uncertain obstacles. In particular, we show that safe 2D motion planning among PGDF obstacles is $\mathrm{NP}$-hard with respect to the number of obstacles, and remains $\mathrm{NP}$-hard after being restricted to a graph. Our reduction is based on a path encoding of MAXQHORNSAT and uses the risk of collision with an obstacle to encode variable assignments and literal satisfactions. This implies that, unlike in the known case, planning under uncertainty is hard, even when given a graph containing the solution. We further show by reduction from $3$-SAT that both safe 3D motion planning among PGDF obstacles and the related minimum constraint removal problem remain $\mathrm{NP}$-hard even when restricted to cases where each obstacle overlaps with at most a constant number of other obstacles.

我们考虑存在不确定障碍物时的运动计划问题，该障碍物被建模为具有高斯分布面（PGDFs）的多面体。通过在配置空间中构造图形，然后有效地搜索图形以找到无碰撞的路径，存在许多在存在已知障碍物的情况下进行运动规划的实用算法。我们表明，这种精确的算法在具有不确定障碍的领域中不太可能实用。特别是，我们表明PGDF障碍物之间的安全2D运动计划是$\mathrm{NP}$-关于障碍物的数量很难，并且仍然存在 $\mathrm{NP}$-在被限制为图形之后很难。我们的减少是基于MAXQHORNSAT的路径编码，并使用与障碍物碰撞的风险来编码变量分配和文字满意度。这意味着，与已知情况不同，即使在给出包含解决方案的图形的情况下，不确定性下的计划也很难。我们进一步通过减少显示$3$-SAT，PGDF障碍物之间的安全3D运动计划以及相关的最小约束消除问题仍然存在 $\mathrm{NP}$-即使限于每个障碍物最多与一定数量的其他障碍物重叠的情况，也很难。