We introduce a transformation of a Neural Radiance Field (NeRF) to an equivalent Poisson Point Process (PPP). This PPP transformation allows for rigorous quantification of uncertainty in NeRFs, in particular, for computing collision probabilities for a robot navigating through a NeRF environment. The PPP is a generalization of a probabilistic occupancy grid to the continuous volume and is fundamental to the volumetric ray-tracing model underlying radiance fields. Building upon this PPP representation, we present a chance-constrained trajectory optimization method for safe robot navigation in NeRFs. Our method relies on a voxel representation called the Probabilistic Unsafe Robot Region (PURR) that spatially fuses the chance constraint with the NeRF model to facilitate fast trajectory optimization. We then combine a graph-based search with a spline-based trajectory optimization to yield robot trajectories through the NeRF that are guaranteed to satisfy a user-specific collision probability. We validate our chance constrained planning method through simulations and hardware experiments, showing superior performance compared to prior works on trajectory planning in NeRF environments.

翻译：我们提出将神经辐射场(NeRF)转换为等效泊松点过程(PPP)的方法。这种PPP转换能够对NeRF中的不确定性进行严格量化，特别适用于计算机器人在NeRF环境中导航时的碰撞概率。PPP是概率占据网格在连续体积上的推广，也是构成辐射场基础的体积射线追踪模型的核心。基于该PPP表示，我们提出了一种机会约束轨迹优化方法，用于NeRF中的安全机器人导航。该方法依赖于称为概率不安全机器人区域(PURR)的体素表示，该表示将机会约束与NeRF模型在空间上融合，从而实现快速轨迹优化。随后，我们将基于图的搜索与基于样条的轨迹优化相结合，生成通过NeRF环境的机器人轨迹，该轨迹能够保证满足用户指定的碰撞概率。通过仿真和硬件实验验证了该机会约束规划方法的有效性，结果表明其在NeRF环境轨迹规划任务中优于先前工作。