Combining Range and Direction for Improved Localization

Abstract

Self-localization of nodes in a sensor network is typically achieved using either range or direction measurements; in this paper, we show that a constructive combination of both improves the estimation. We propose two localization algorithms that make use of the differences between the sensors' coordinates, or edge vectors; these can be calculated from measured distances and angles. Our first method improves the existing edge-multidimensional scaling algorithm (E-MDS) by introducing additional constraints that enforce geometric consistency between the edge vectors. On the other hand, our second method decomposes the edge vectors onto 1-dimensional spaces and introduces the concept of coordinate difference matrices (CDMs) to independently regularize each projection. This solution is optimal when Gaussian noise is added to the edge vectors. We demonstrate in numerical simulations that both algorithms outperform state-of-the-art solutions.

Realizability of Planar Point Embeddings from Angle Measurements

“…Various methods have been proposed that incorporate both distance and angle information in point recovery algorithms [9,10,11,12,13,14]. For instance, the concept of edge kernels was introduced in [10], and the authors show that the angle and distance information can be estimated through matrix factorization.…”

“…Although elegant, this concept does not address realizability. In fact, it was shown in [11] that the recovered angles cannot in general be realized, and additional linear constraints are added to the optimization problem to remedy this. Other methods have shown improved performance over distance-based SDPs by adding hyperplane-based constraints [12] or constraints based on the cosine law [13].…”

Multi-Modal Probabilistic Indoor Localization on a Smartphone

“…When distance measurements are obtained between the sensor nodes, Euclidean distance matrices (EDMs) provide a tool to denoise and complete missing measurements [9]. These ideas have also been extended to include both distance and angle measurements [2], [19].…”