  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. Our
codebase can be found at https://github.com/chengine/catnips, and videos can be
found on our project page (https://chengine.github.io/catnips).
