Solving Relative Measurements on Finite Graphs
Titan Yuan, Kristofer S. J. Pister · 2024
Sensor networks often rely on relative, or differential, measurements between sensing nodes due to the absence of an absolute reference at each node. This distributed measurement model can be represented as a graph, where each node denotes an absolute “potential” to be estimated and each edge corresponds to a measurement of the relative potential difference between the two incident nodes. Solving for the absolute node potentials can be formulated as a matrix equation involving the graph's Laplacian matrix. For finite graphs where all of the relative measurement processing is handled by a single node, we propose a modified iterative Jacobi algorithm with a priority queue to solve for the node potentials. Using the eigendecomposition of the Laplacian matrix, we then derive an expression for the error variance of the estimated node potentials and show how it can be adapted for different boundary conditions.