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This paper discusses how sparse local measurements of positions and surface normals may be used to identify and locate overlapping objects. The objects are modeled as polyhedra (or polygons) having up to six degrees of freedom relative to the sensors. The approach operates by examining all hypotheses about pairings between sensed data and object surfaces and efficiently discarding inconsistent ones by using local constraints on: distances between faces, angles between face normals, and angles (relative to the surface normals) of vectors between sensed points. The method described here is an extension of a method for recognition and localization of non-overlapping parts previously described in [Grimson & Lozano--Pérez 84] and [Gaston & Lozano-Pérez 84].