We formalize a novel interpretation of Neural Radiance Fields (NeRFs) as giving rise to a Poisson Point Process (PPP). This PPP interpretation allows for rigorous quantification of uncertainty in NeRFs, in particular, for computing collision probabilities for a robot navigating through a NeRF environment model. 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 model, 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, showing superior performance compared with two other methods for trajectory planning in NeRF environment models.

翻译：摘要：我们形式化了一种对神经辐射场（NeRF）的新解释，即将其视为生成泊松点过程（PPP）。这种PPP解释能够对NeRF中的不确定性进行严格量化，特别是用于计算机器人在NeRF环境模型中导航时的碰撞概率。PPP是将概率占用网格推广到连续体积的模型，也是支撑辐射场的体积光线追踪模型的基础。基于此PPP模型，我们提出了一种机会约束轨迹优化方法，用于在NeRF中实现安全的机器人导航。该方法依赖于一种称为概率不安全机器人区域（PURR）的体素表示，该表示在空间上将机会约束与NeRF模型融合，以促进快速轨迹优化。随后，我们结合基于图的搜索与样条轨迹优化，生成通过NeRF的机器人轨迹，这些轨迹保证满足用户指定的碰撞概率。通过仿真验证了我们的机会约束规划方法，结果表明与另外两种在NeRF环境模型中进行轨迹规划的方法相比，本方法性能更优。