Consistent TOA Localization with Sensor Position Uncertainties

‍ ‍Target localization refers to estimating the spatial position of a target in a specific coordinate system using sensing information. It has widespread applications， ranging from advertisement recommendations， global navigation systems， and disaster rescue to battlefield monitoring. Time of arrival （TOA）-based or equivalent range-based localization methods have been widely studied due to their high-precision characteristics. The localization principle is to find the intersection of the circles （for a 2D case） or spheres （for a 3D case） induced from range measurements. However， due to the existence of measurement noises， these circles and spheres generally do not intersect at one point， and thus， optimization-based methods are required to estimate the target position. Existing TOA localization methods typically assume precise sensor positions and only consider range measurement noises， whereas previous studies that have taken sensor position uncertainties into account often lack statistical optimization and analysis， making it difficult to obtain consistent estimates. In this study， we considered both range measurement noises and sensor position uncertainties， and constructed a maximum likelihood problem by treating both the target position and sensor coordinates as unknown variables. The maximum likelihood problem turned out to be a weighted nonlinear least-squares problem. This paper first presented the assumptions on measurement noises and sensor spatial distribution； that is， measurement noises are independent and identically distributed Gaussian noises， and sensor positions have an asymptotic distribution whose probability measure does not concentrate on a line （for a 2D case） or a plane （for a 3D case）. Based on these assumptions， the asymptotic identifiability of the target position is guaranteed. Of note， the maximum likelihood estimate is not necessarily consistent， which is quite different from the typical case without sensor position uncertainties. The original measurement equation was further transformed by squaring both sides to construct an optimization problem that can be optimally solved. Specifically， for situations where the variance of range measurement noise is known， a generalized trust region problem with a quadratic objective function and a quadratic equality constraint was formulated， and a corresponding algorithm that calculates its global minimizer was given. For situations where the variance of range measurement noise is unknown， an ordinary linear least-squares problem was constructed to simultaneously estimate the target position and range measurement noise variance. For both situations， bias elimination methods based on probability theory and statistics were proposed to eliminate asymptotic biases and achieve consistent estimation； that is， as the number of range measurements increases， the estimate can converge to the true target position. The consistency feature enables the proposed algorithm to achieve ultra-high precision localization in scenarios with large sample measurements， such as satellite-based positioning systems and high-frequency indoor Internet of Things localization systems. In addition， the Gauss-Newton iterative algorithm that simultaneously refines target and sensor position estimates was derived， which can improve the algorithm’s localization accuracy when the measurement number and the uncertainties of the sensor position are relatively small. The simulation results show that the proposed bias-eliminated estimators are asymptotically unbiased and consistent， which verifies the correctness of the derived theoretical results. In addition， the simulation also demonstrates the superiority of the proposed algorithm in the large sample case and gives guidelines for selecting estimators in different scenarios.