Rigid Body Localization in Unsynchronized Sensor Networks: Analysis and Solution

This paper addresses the rigid body localization (RBL) problem using an unsynchronized sensor network, where clock offsets between sensors and anchors are present. We first investigate RBL using unsynchronized time-of-arrival (TOA) measurements to determine the necessary and sufficient conditions for this problem by analyzing the rank of the Fisher Information Matrix. When the condition is fulfilled, a constrained weighted least squares (CWLS) problem is formulated to jointly estimate the position and rotation angles of the rigid body together with the clock offsets. Besides the non-convex nature, the CWLS problem contains nonlinear constraints resulting from the orthogonality property of the rotation matrix. To solve this difficult problem, we propose to apply semidefinite relaxation (SDR) to relax it as a convex semidefinite program. Furthermore, we investigate the RBL problem using hybrid TOA and angle-of-arrival (AOA) measurements. The Cramer-Rao Lower Bound (CRLB) analysis shows better performance of RBL by hybrid TOA-AOA than by AOA only when more than one anchors are used. The CWLS problem is extended to include the AOAs and the associated SDR method is derived. Finally, mean square error analysis shows that the proposed methods are able to achieve the CRLB performance under small Gaussian noise. Numerical results confirm the theoretical analyses and validate the performance of the proposed methods.