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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.

In many robotic applications the need to rep resent and reason about spatial relationships is of great importance. However, our knowledge of par ticular spatial relationships is inherently uncertain. The most used method for handling the uncertainty is to "pre-engineer" the problem away, by structur ing the working environment and using specially suited high-precision equipment. In some advanced robotic research domains, however, such as auto matic task planning, off-line robot programming, 267 and autonomous vehicle operation, prior structur ing will not be possible, because of dynamically changing environments, or because of the demand for greater reasoning flexibility. Spatial reasoning is further complicated because relationships are often not described explicitly, but are given by uncertain relative information. This is particularly true when many different frames of reference are used, produc ing a network of uncertain relationships. Rather than treat spatial uncertainty as a side issue in geo metrical reasoning, we believe it must be an intrin sic part of spatial representations. In this paper, we describe a representation for spatial informa tion, called the stochastic map, and associated pro cedures for building it, reading information from it, and revising it incrementally as new information is obtained. The map always contains the best esti mates of relationships among objects in the map, and their uncertainties. The procedures provide a general solution to the problem of estimating un certain relative spatial relationships. The estimates are probabilistic in nature, an advance over the pre vious, very conservative, worst-case approaches to the problem. Finally, the procedures are developed in the context of state-estimation and filtering the ory, which provides a solid basis for numerous ex tensions.

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