Rigid Components Identification and Rigidity Enforcement in Bearing-Only Localization using the Graph Cycle Basis
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Bearing-only localization can be formulated in terms of optimal graph embedding: one has to assign a 2-D or 3-D position to each node in a graph while satisfying as close as possible all the bearing-only constraints on the edges. If the graph is parallel rigid, this can be done via spectral methods. When the graph is not rigid the reconstruction is ambiguous, as different subsets of vertices can be scaled differently. It is therefore important to first identify a partition of the problem into maximal rigid components. In this paper we show that the cycle basis matrix of the graph not only translates into an algorithm to identify all rigid sub-graphs, but also provides a more intuitive way to look at graph rigidity, showing, for instance, why triangulated graphs are rigid and why graphs with long cycles may loose this property. Furthermore, it provides practical tools to enforce rigidity by adding a minimal number of measurements.