Precise Ranging: Modeling Bias and Variance of Double-Sided Two-Way Ranging with TDoA Extraction under Multipath and NLOS Effects

TL;DR

This paper models the bias and variance of Double-Sided Two-Way Ranging (DS-TWR) and TDoA extraction under multipath and NLOS effects, providing accurate predictions and insights for precise localization in challenging environments.

Contribution

It analytically derives bias and variance models for DS-TWR and DS-TDoA under NLOS and multipath, validated by experiments, improving understanding of their accuracy limits.

Findings

DS-TWR has half the variance of previous estimates.

DS-TDoA shows approximately five times higher variance.

Symmetric response times reduce variance and error.

Abstract

Location-based services such as autonomous vehicles, drones, and indoor positioning require precise and scalable distance estimates. The bias and variance of range estimators inherently influence the resulting localization quality. In this work, we revisit the well-established Double-Sided Two-Way-Ranging (DS-TWR) protocol and the extraction of timing differences (DS-TDoA) at devices overhearing DS-TWR. Under non-line-of-sight (NLOS) and multipath effects, we analytically derive their bias and variance. Our proposed model reveals that DS-TWR retains half the variance than anticipated while DS-TDoA comprises roughly a five-fold increase in variance. We conduct numerical simulations and experimental deployments using Ultra-Wideband (UWB) devices in a public testbed. Our results confirm the adequacy of our model, providing centimeter-accurate predictions based on the underlying timestamping noise with a median $R^2$ score of 77% (30% IQR). We find that both DS-TWR and DS-TDoA exhibit reduced variance when response times are symmetric. Our experimental results further show that double-sided variants exhibit less error and variance compared to Carrier Frequency Offset (CFO)-based single-sided methods.