This article studies the following question: "When is it possible to decide
, on the basis of images of point-features observed by an imprecisely model
ed two-camera stereo vision system, whether or not a prescribed robot posit
ioning task has been precisely accomplished?" Results are shown for three c
amera model classes: injective cameras, weakly calibrated projective camera
s, and uncalibrated projective cameras. In particular, given a weakly calib
rated stereo pair, it is shown that a positioning task can be precisely acc
omplished if and only if the task specification is invariant to projective
transformations. It is shown that injective and uncalibrated projective cam
eras can accomplish fewer tasks, but are still able to accomplish tasks inv
olving point coincidences.
The same formal framework is applied to the problem of determining the set
of tasks which can be precisely accomplished with the well-known position-b
ased control architecture. It is shown that, for any class of camera models
, the set of tasks which can be precisely accomplished using a position-bas
ed control architecture is a subset of the complete set of tasks which can
be decided on the set, but includes all positioning tasks based on point co
incidences. Two ways of extending the idea of position-based control to acc
omplish more tasks are also presented.