Quaternion Domain Super MDS for 3D Localization

TL;DR

This paper addresses accurate 3D localization in wireless sensor networks with low computational complexity by recasting the classical SMDS into the quaternion domain. It introduces QD-SMDS, which represents 3D coordinates as quaternions and constructs a rank-1 quaternion-domain edge kernel $\bm{K}=\bm{\nu}\bm{\nu}^{\mathsf{H}}$ that fuses distance and angular information; localization is performed via QSVD to obtain $\hat{\bm{\nu}}=\sqrt{\lambda}\,\mathbf{u}$ and reconstructs the coordinates with Procrustes alignment. The authors derive quaternion-edge relationships that underpin the GEK construction and propose two practical GEK-construction scenarios depending on available measurements, including azimuth/elevation angles from planar antennas. Numerical results show improved accuracy over conventional SMDS in scenarios with substantial angle errors, highlighting the method's robustness and potential for low-complexity 3D localization in indoor WSNs. Overall, the QD-SMDS approach offers a principled, geometry-aware framework that leverages quaternion calculus and low-rank noise reduction to enhance 3D localization performance when angular information is imperfect or partially available.

Abstract

We propose a novel low-complexity three-dimensional (3D) localization algorithm for wireless sensor networks, termed quaternion-domain super multidimensional scaling (QD-SMDS). This algorithm reformulates the conventional SMDS, which was originally developed in the real domain, into the quaternion domain. By representing 3D coordinates as quaternions, the method enables the construction of a rank-1 Gram edge kernel (GEK) matrix that integrates both relative distance and angular (phase) information between nodes, maximizing the noise reduction effect achieved through low-rank truncation via singular value decomposition (SVD). The simulation results indicate that the proposed method demonstrates a notable enhancement in localization accuracy relative to the conventional SMDS algorithm, particularly in scenarios characterized by substantial measurement errors.

Figures (4)

• Figure 1: Illustration of the parameters required to construct quaternion-domain GEK matrix $\bm{K}$.
• Figure 2: Parameters that can be obtained using a planar antenna.
• Figure 3: Comparison of localization accuracy between the SMDS and QD-SMDS algorithms in Scenario I, where only mutual distance and ADoA between nodes are available.
• Figure 4: Comparison of localization accuracy between the SMDS and QD-SMDS algorithms in Scenario II, where all data constituting the quaternion-domain GEK matrix in \ref{['eq:q_kernel']} is accessible.