Bias Reduced Semidefinite Relaxation Method for 3-D Rigid Body Localization Using AOA

This paper addresses the angle-of-arrival (AOA) based rigid body localization problem, where the position and orientation of the rigid body are estimated. Improving the robustness and reducing the estimation bias are crucial in AOA localization when operating in high noise environments. We develop a new semidefinite relaxation (SDR) method for this localization problem, with the additional novelty of having bias reduction. We begin by transforming the AOA measurement model, and then use it to formulate a constrained weighted least squares (CWLS) minimization problem with the rotation matrix and position vector as the optimization variables. Bias reduction is accomplished by introducing an auxiliary variable and imposing one quadratic constraint. The constraints on rotation matrix make the CWLS problem non-convex and difficult to handle. We relax the CWLS problem as a convex semidefinite program (SDP) by performing SDR, with second-order cone constraints added to tighten the relaxed SDP problem. The resulting tightened SDP has the ability of reaching a rank-1 solution with considerable small bias. Moreover, we provide a new technique in conducting the performance analysis. We show by mean square error (MSE) analysis that the solution performance is able to approach the Cramer-Rao lower bound (CRLB), and also derive the theoretical expression of the estimation bias. The one-anchor case is treated separately and a different SDP problem is derived for achieving good performance. Simulation results validate that the proposed estimator has a much lower bias than the existing solution and maintains the MSE approaching the CRLB at higher noise level.