On the Representation and Estimation of Spatial Uncertainty

TLDR

The frames may be known only indirectly through a series of spatial relationships, each with its associated error, arising from diverse causes such as positioning errors, measurement errors, or tolerances in part dimensions. The paper proposes a general method for estimating the nominal relationship and covariance between coordinate frames, enabling questions such as whether a robot‑mounted camera can see a reference object. The method generalizes to six degrees of freedom and estimates both the nominal position and orientation relationships among objects and their associated uncertainties. The estimates agree with independent Monte Carlo simulations, and the method allows early determination of whether an uncertain relationship suffices for a task and how much sensor improvement is needed.

Abstract

This paper describes a general method for estimating the nominal relationship and expected error (covariance) between coordinate frames representing the relative locations of ob jects. The frames may be known only indirectly through a series of spatial relationships, each with its associated error, arising from diverse causes, including positioning errors, measurement errors, or tolerances in part dimensions. This estimation method can be used to answer such questions as whether a camera attached to a robot is likely to have a particular reference object in its field of view. The calculated estimates agree well with those from an independent Monte Carlo simulation. The method makes it possible to decide in advance whether an uncertain relationship is known accu rately enough for some task and, if not, how much of an improvement in locational knowledge a proposed sensor will provide. The method presented can be generalized to six degrees offreedom and provides a practical means of esti mating the relationships ( position and orientation) among objects, as well as estimating the uncertainty associated with the relationships.