Localization of a Moving Object with Sensors in Motion by Time Delays and Doppler Shifts

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.