  Autonomous inspection tasks necessitate path-planning algorithms to
efficiently gather observations from points of interest (POI). However,
localization errors commonly encountered in urban environments can introduce
execution uncertainty, posing challenges to successfully completing such tasks.
Unfortunately, existing algorithms for inspection planning do not explicitly
account for execution uncertainty, which can hinder their performance. To
bridge this gap, we present IRIS-under uncertainty (IRIS-U^2), the first
inspection-planning algorithm that offers statistical guarantees regarding
coverage, path length, and collision probability. Our approach builds upon IRIS
-- our framework for deterministic inspection planning, which is highly
efficient and provably asymptotically-optimal. The extension to the much more
involved uncertain setting is achieved by a refined search procedure that
estimates POI coverage probabilities using Monte Carlo (MC) sampling. The
efficacy of IRIS-U^2 is demonstrated through a case study focusing on
structural inspections of bridges. Our approach exhibits improved expected
coverage, reduced collision probability, and yields increasingly precise
statistical guarantees as the number of MC samples grows. Furthermore, we
demonstrate the potential advantages of computing bounded sub-optimal solutions
to reduce computation time while maintaining statistical guarantees.
