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.

中文翻译：

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使用 AOA 进行 3-D 刚体定位的偏置减少半定松弛方法

本文解决了基于到达角 (AOA) 的刚体定位问题，其中估计了刚体的位置和方向。在高噪声环境中操作时，提高鲁棒性和减少估计偏差对于 AOA 定位至关重要。我们为这个定位问题开发了一种新的半定松弛 (SDR) 方法，具有减少偏差的额外新颖性。我们首先转换 AOA 测量模型，然后使用它来制定约束加权最小二乘 (CWLS) 最小化问题，其中旋转矩阵和位置向量作为优化变量。通过引入辅助变量并施加一个二次约束来减少偏差。旋转矩阵的约束使得 CWLS 问题非凸且难以处理。我们通过执行 SDR 将 CWLS 问题放松为凸半定规划 (SDP)，并添加二阶锥约束以收紧放松的 SDP 问题。由此产生的收紧 SDP 能够以相当小的偏差达到 1 级解决方案。此外，我们提供了一种进行性能分析的新技术。我们通过均方误差 (MSE) 分析表明，解决方案的性能能够接近 Cramer-Rao 下限 (CRLB)，并推导出估计偏差的理论表达式。单独处理单锚情况，并导出不同的 SDP 问题以获得良好的性能。仿真结果验证了所提出的估计器具有比现有解决方案低得多的偏差，并保持 MSE 在更高的噪声水平上接近 CRLB。