Formal theory of noisy sensor network localization

Graph theory has been used to characterize t he solvability of the sensor network localization problem. If sensors correspond to vertices and edges correspond to sensor pairs between which the distance is known, a significant result in the theory of range-based sensor network localization is that if the graph underlying the sensor network is generically globally rigid and there is a suitable set of anchors at known positions, then the network can be localized, i.e., a unique set of sensor positions can be determined that is consistent with the data. In particular, for planar problems, provided the sensor network has three or more noncollinear anchors at known points, all sensors are located at generic points, and the intersensor distances corresponding to the graph edges are precisely known rather than being subject to measurement noise, generic global rigidity of the graph is necessary and sufficient for the network to be localizable (in the absence of any further information). In practice, however, distance measurements will never be exact, and the equations whose solutions deliver sensor positions in the noiseless case in general no longer have a solution. This paper then argues that if the distance measurement errors are not too great and otherwise the associated graph is generically globally rigid and there are three or more noncollinear anchors, the network will be approximately localizable, in the sense that estimates can be found for the sensor positions which are near the correct values; in particular, a bound on the position errors can be found in terms of a bound on the distance errors. The sensor positions in this case can be found by minimizing a cost function which, although nonconvex, does have a global minimum.