This paper investigates the problem of active localization of a moving object in its initial position and velocity, using time delay only or with Doppler shift measurements acquired by a number of monostatic sensors. Each sensor has non-negligible motion during the observation period, causing it at different positions when it sends and receives the signal, with the separation proportional to the signal travel time in reaching the object and returning back. The object is not at the same position when it reflects the signals from various sensors due to its motion. Both motion effects lead to recursive model equations for time delay and Doppler shift, making the localization problem interesting and challenging. We shall derive the measurement model equations under this scenario, evaluate the Cramer-Rao lower bound (CRLB) of the estimation problem and analyze the proposed models by contrasting with the performance loss when ignoring the object and sensor motion effects. The Maximum Likelihood Estimators (MLEs) are next developed, using the Gauss-Newton or Quasi-Newton iterations. Algebraic solution for the special case of moving object non-moving sensors is derived and analyzed, and it can serve as an effective initialization of the iterative MLEs if sensor motion is present. Both the theoretical analysis and simulation studies corroborate the importance of taking the object and sensor motions into consideration during the observation period, when the relative velocity between the object and sensor is significant compared to the signal propagation speed, such as in an acoustic or underwater environment.

This paper investigates active localization of a stationary object using time delay alone or together with Doppler shift measurements observed by a number of dynamic monostatic sensors having collocated transmitters and receivers, where the sensor motion during the interrogation period is not negligible. Non-negligible motion effect appears when the sensor speed relative to the signal propagation speed is significant, such as in an acoustic or an underwater environment. The motion effect causes the signal sent and received locations different, with their separation proportional to the two-way propagation time delay that is dependent on the object location. We first provide new measurement models for time delay and Doppler shift in the presence of sensor motion. They come as recursive equations and their non-recursive forms are derived. Analysis of the motion effect based on the Cramér-Rao lower bound (CRLB) is conducted, and the performance loss in terms of the location bias and variance when ignoring the sensor motion effect is evaluated. Two kinds of estimators for the object location are next proposed by exploiting the non-recursive forms of the model equations. One is a closed-form solution by algebraic evaluation and the other is a semi-definite relaxation solution by convex optimization. The first is computationally efficient and is shown to achieve the CRLB performance over the small error region under Gaussian noise, while the second can handle the positioning problem better when the noise level is high. Simulations validate the theoretical analysis and the performance of the proposed estimators.

The motion of sensors during the measurement period, if not accounted for, can degrade significantly the localization accuracy. This paper investigates the sensor motion effect for the positioning of an object, using angles of arrival only or together with time of arrival measurements. The biases from the AOA and TOA data models when ignoring the motion effect are examined. Positioning algorithms for AOA localization and mixed AOA-TOA localization that account for the motion effect are developed by the pseudo-linear formulation. The algorithms derived include the computationally attractive closed-form estimators and the noise resilient semidefinite programming solutions. The bias coming from the pseudo-linear formulation is analyzed in detail, and it can be subtracted from the closed-form solution to obtain a bias-suppressed estimate. Simulation validates the effectiveness of the proposed solutions in achieving the CRLB performance under Gaussian noise before the thresholding effect occurs.

This article considers the localization of a stationary target using time-delay and Doppler shift measurements obtained by moving monostatic sensors that takesthe sensor motion during the localization period into consideration. Starting from the model equation of a previous work, we propose a new formulation of optimization through constrained weighted least-squares to solve for the target location by developing two new localization solutions, one in closed form and the other by semi-definite relaxation (SDR). The new formulation enables the reduction of the minimum number of auxiliary variables in the closed-form solution and allows additional constraints in the SDR solution, resulting in fewer sensors needed in the former and less relaxation error and higher noise tolerance in the latter. Simulations validate the better performance of the proposed solutions than the previous ones.

The sensor position uncertainties and synchronization offsets can cause substantial performance degradation if the sensors are not properly calibrated. This paper investigates the localization of a constant velocity moving target and the self-calibration of sensors using a sequence of range and azimuth measurements observed at successive instants. A theoretical study by the Cramer-Rao Lower Bound (CRLB) reveals that the sensor positions can only be self-calibrated when there are at least two sensors and synchronization offsets can be handled by joint estimation. A low complexity sequential closed-form solution is proposed to estimate the target position and velocity first, and the coordinates of each sensor and synchronization offset afterward. While less intuitive, the analysis shows that the closed-form solutions for both the target and sensor parameters can reach the CRLB accuracy under small Gaussian noise. We also develop a semidefinite programming (SDP) solution by semidefinite relaxation (SDR) for joint localization and calibration from the Maximum Likelihood formulation, which exhibits higher noise tolerance than the closed-form solution. Simulations validate the analysis and the performance of the proposed methods.