Performance Comparison of Closed-Form Least Squares Algorithms for Hyperbolic 3-D Positioning

An accurate 3-D wireless local positioning system (LPS) is a highly demanded tool for increasing safety in, e.g., emergency response and security operations. An LPS is an attractive approach that can meet stringent requirements and can achieve acceptable accuracies for a long time during extended operations in global navigation satellite system (GNSS)-denied environments. In this work, three closed-form (CF) least squares (LS) algorithms were considered, where two of them were adapted to exploit the knowledge about nuisance parameters for accurate 3-D positioning based on time difference of arrival (TDoA) measurements. The algorithms utilize the single set (SS) of the TDoA measurements, an extended SS (ExSS) of the TDoA measurements, or the full set (FS) of the TDoA measurements, and were denoted, respectively, as the CFSSLS, CFExSSLS, and CFFSLS solutions. The performance of the algorithms was investigated with simulations and real-world measurements, where the wireless system transmitters were placed in a quasi-coplanar arrangement. At moderate to high signal-to-noise ratio (SNR) levels, the CFSSLS solution has the best performance, followed by the CFExSSLS solution and then by the CFFSLS solution. At low SNR levels, the CFFSLS algorithm outperformed the other two algorithms. Both the CFSSLS and CFFSLS solutions estimate nuisance parameters that are utilized in refining the vertical position estimate of the receiver. The CFFSLS solution delivers more accurate refined vertical position estimates since it utilizes more nuisance parameters, i.e., more information. The experimental results confirmed the simulation study in which the CFFSLS algorithm outperformed the other two algorithms, where the experimental environment was dominated by total non-line-of-sight (NLoS) conditions and low SNR levels at the receiver to be located. Therefore, it is recommended to use the FS TDoA measurements for 3-D positioning in bad signal conditions, such as high noise levels and NLoS propagation.