Joint Localization of Passive Object and Rigid Body Receiver Using Time Delay: Analysis and Solution

This paper addresses the multistatic localization problem with the receiving sensors distributed on a rigid body (RB) whose position and orientation are not known. We propose to jointly estimate the unknown object position together with the RB position and orientation, simply called joint object and rigid body localization (JORBL), using time delay (TD) measurements from both the direct-propagation and indirect-reflection paths between the transmitters and sensors. Two scenarios are considered, one with synchronization between the sensors and transmitters and the other without. We begin by investigating the benefits of having the direct-path TD measurements compared to using the indirect-path TDs only for JORBL, in terms of estimation accuracy and minimum number of transmitters and sensors needed, through analyzing the Cramér-Rao lower bound (CRLB). Afterward, in the presence of synchronization, we propose a JORBL method by formulating a non-convex constrained weighted least squares problem and applying semidefinite relaxation to obtain the solution by semidefinite programming. The proposed method is then extended to the case without synchronization. The mean squared error analysis demonstrates that the proposed methods can achieve the CRLB accuracy for small Gaussian noise, which is further confirmed by numerical simulations.