A note on Global Positioning System (GPS) and Euclidean distance matrices

TL;DR

This paper addresses a GPS-related problem of adjusting Euclidean distance matrices by finding the closest vector to a given one, introducing algorithms and a fault detection criterion for different matrix sizes.

Contribution

It proposes new algorithms and a fault detection method for augmenting Euclidean distance matrices in GPS applications, handling different matrix sizes.

Findings

Developed an algorithm for n=4 case.

Created two algorithms for n≥5 case.

Presented a fault detection criterion for GPS-related EDM adjustments.

Abstract

Let $D$ be an $n \times n$ Euclidean distance matrix (EDM) with embedding dimension $r$; and let $d \in R^n$ be a given vector. In this note, we consider the problem of finding a vector $y \in R^n$, that is closest to d in Euclidean norm, such that the augmented matrix $\left[ \begin{array}{cc} 0 & y^T \\ y & D \end{array}\right]$ is itself an EDM of embedding dimension $r$. This problem is motivated by applications in Global Positioning System (GPS). We present a fault detection criterion and three algorithms: one for the case $n=4$, and two for the case $n \geq 5$.