Non-Iterative MDS Method for Collaborative Network Localization With Sparse Range and Pointing Measurements

Abstract

Multi-agent localization is a basic requirement for many networked applications. The particular application to swarming unmanned aerial vehicles (UAVs) or munitions requires spatial coordination of agents, including the ability to assume and maintain a prescribed flight formation. An in-flight awareness of network morphology and node location is therefore needed. While global navigation satellite systems offer an attractive solution, signal occlusion, spoofing, and jamming present unacceptable vulnerabilities—particularly for mission-critical operations. Alternative network localization methods using inter-agent radio frequency ranging and Angle of Arrival have been well studied over the past 15 years, but existing algorithms are not well suited to fast-moving networks. Iterative methods tend to converge slowly. Faster noniterative multi-dimensional scaling (MDS) methods for range, bearing, or vector measurements have also been formulated. However, these MDS methods generally require the full pairwise inter-agent measurement matrix—placing a severe requirement on swarm connectivity and leading to low tolerance for missing or badly estimated measurements. Even vector-based MDS, which incorporates both range and direction constraints, is shown here to require 4-vertex connectivity to achieve perfect localization. Results from rigidity theory, however, suggest that a lower connectivity threshold should be sufficient to guarantee a unique configuration (up to translation and rotation). In contrast, our proposed “vertex resequencing” and “edge resequencing” techniques further lower the vertex-connectivity threshold to 3 and 2, respectively. These localization techniques, which extend vector-based MDS with Nyström approximation, prescribe a graph-based kernel sampling scheme and weighted coordinate reconstruction that suppress the effect of missing measurements.