Bias-Reduced Closed-Form Method for 3-D Moving Object Localization by AOA Using Sensors at Known and Unknown Positions

Considering all or part of the sensors are at known positions, this paper addresses the problem of locating a 3-D moving object with a linear constant velocity trajectory using angle-of-arrival (AOA) observations. We start by analyzing the minimum number of sensors and show that at least two sensors with known positions are needed for this problem. For the case of all sensors at known positions, we formulate a weighted least squares (WLS) problem with the capability of bias reduction, called the bias-reduced WLS (BR-WLS) problem, to limit the bias caused by the transformation of the measurement models. The BR-WLS problem has a closed-form solution, and thus, solving it is very computationally efficient. For the case where only part of the sensors are at known positions, we first theoretically analyze the performance gain by including the sensors at unknown positions for localization, and then extend the proposed BR-WLS method to jointly estimate the object motion parameters and the unknown sensor positions. For both cases, we show that the localization mean square error (MSE) can reach the Cramér-Rao lower bound (CRLB) when the noise is small and Gaussian distributed. Moreover, we also derive the theoretical expressions of the residual biases for the object position and velocity estimation by the proposed solutions. Simulation results confirm the good performance of the proposed solutions and also validate the theoretical results.