This article describes a new statistical, model-based approach to buil
ding a contact-state observer for time-invariant constraints for a poi
nt in three dimensions. Three-dimensional constraint estimation and se
lection, and the application of these procedures to a planar, two-dime
nsional maze is described in detail. The observer uses measurements of
the contact force and position, and prior information about the task
encoded in a network, to determine the current location of the robot i
n the task-configuration space. Each node represents what the measurem
ents will look like in a small region of configuration space by starin
g a predictive, statistical, measurement model. Construction of the ta
sk network requires a model of both the grasped part and the environme
nt. The model makes the system robust to alignment errors, however, gr
oss errors can occur if the topology of the modeled configuration spac
e differs from the true topology. Arcs in the task network represent p
ossible transitions between models. Beam search is used to match the m
easurement history against possible paths through the model network to
estimate the most likely paths for the robot. The resulting approach
provides a new decision process that can be used as an observer for ev
ent-driven manipulation programming.