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

Abstract

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.

Evaluation of Time Difference of Arrival (TDOA) Networks Performance for Launcher Vehicles and Spacecraft Tracking

“…The calculations are proposed by using a linear least square (LLS) problem for the TDOA solution, as representative for a conservative estimation of the performance of each analyzed network. Indeed, not only has LLS been shown to reach poorer performance, when compared to other, iterative, methods [14,31], but the implementation of a simple method needing low calculus power is perfectly representative with the future perspective of implementing low-cost SDR-based networks, as already presented in this paragraph. The estimation of MDOP has thus to be considered a worst case estimation of the potential positioning accuracy of the analyzed network.…”

“…The target position is calculated by evaluating the intersection of the hyperboloids that mark a constant difference in time of arrival between each pair of stations. The method is often used for terminal aircraft tracking [8][9][10], indoor positioning [11][12][13][14] and other applications. TDOA could be easily applicable for supporting orbit determination and spacecraft tracking, as all the operational potential targets down-link frequencies (e.g., of the beacon or telemetry channels) are well-known and often concentrated in a limited number of bands [15][16][17].…”

Special Issue: Localization in Wireless Sensor Networks

“…Published works in the Special Issue cover a wide range of topics, namely development of a prototype of a new cooperative unmanned aerial vehicles (UAV) swarm localization system using low-cost sensors and cameras [1], lightweight distributed localization schemes for energy-restricted WSNs [2], fabrication of additional (angle) information for aiding tracking performance [3], target localization using combined radio measurements in the presence of synchronization errors [4], performance comparison of existing algorithms for hyperbolic 3-dimensional localization [5], and characterization of the log-normal model for received signal strength (RSS) measurements [6]. A short overview of each of the above-mentioned works is provided in the following.…”

“…In [5], Khalaf-Allah considered three closed-form LS algorithms, where two of them were adapted to exploit the knowledge about nuisance parameters for accurate three-dimensional localization based on time difference of arrival (TDOA) measurements. The algorithms utilized a single set an extended single set and a full set of the TDOA measurements.…”

Particle Filtering for Three-Dimensional TDoA-Based Positioning Using Four Anchor Nodes

“…The localization of a target node using time difference of arrival (TDoA) measurements received and is still receiving considerable attention. TDoA-based positioning algorithms [ 1 , 2 , 3 , 4 , 5 , 6 , 7 ] are widely applied in sensor networks [ 1 , 8 ], wireless communication [ 2 , 3 ], target tracking [ 9 , 10 ], navigation [ 11 ], underwater [ 12 ], tactile interfacing for human–computer interaction [ 13 ], and seismic exploration [ 14 ]. The determination of a unique target node’s location using TDoA measurements in closed form requires at least four anchor nodes, i.e., three TDoA measurements, in the two-dimensional (2D) space and five anchor nodes, i.e., four TDoA measurements, in the three-dimensional (3D) space [ 15 ].…”