# AOA-Based Three-Dimensional Positioning and Tracking Using the Factor Graph Technique
## 1 Introduction
In contrast, the performances of AOA-based techniques are not so sensitive to measurement errors, while still enjoying the same advantages as TDOA for detecting anonymous targets. Therefore, this paper will focus on AOA. In [18; 19; 16], AOA-based location detection and tracking was studied. However, the previous works only considered the scenarios in 2D. Actually, the use cases in 3D are currently increasing. For example, it can be applied to aviation control for airports, where not only the scheduled aircraft, but also the anonymous invaders can be detected and tracked. Moreover, live TV broadcasts via cameras on drones will become more popular, which requires accurate drone control in 3D by the fusion center. Additionally, future intelligent transportation systems may realize 3D detection capability to assist the navigation of vehicles running in the complex multi-layer overpass roads. Note that the measurement of AOA samples, including azimuth and elevation angles in 3D, may require more complicated soft/hard designs, such as MUSIC and ESPRIT [20; 21], but they are out of the scope of this paper. For location detectors that work in a relatively stable condition, the variance of detection errors will also be stable. In this case, a fixed variance value can be fed to EKF for estimating the performance of the observer. Usually, such a value can be calculated from off-line training. However, this paper considers a dynamic system, which indicates that the measuring abilities of sensors may vary through the tracking process. Therefore, a real-time tuning approach of the observation variance is required. This paper proposes an estimation method by using an approximated Cramer-Rao bound (ACRB). Specifically, the Cramer-Rao bound (CRB) provides a low bound on the detection performance according to the measurement of indirect parameters. For the AOA-based location detection problem, the CRB can be calculated by using the variances of AOA measurements and the real position of the target. However, the latter parameter is unavailable in practice. Instead, it is possible to get a predicted target position with EKF, with which the ACRB can be calculated as the estimation of the observation variance. Such a process can work in dynamic environments and has been shown to achieve a robust tracking performance in this paper. The main contribution of this paper can be summarized as follows. 1. An integrated FG structure is formulated to realize 3D location detection and tracking with AOA measurements. 2. It has been proven that the information from the elevation angle can help to improve the detection of 2D space, from both theoretical and simulation analyses. 3. The proposed technique has exhibited robust performances even with an unstable sensing environment. 4. By utilizing the ACRB as side information (SI) to the detector, the proposed system can greatly reduce the impacts of burst sensing errors. The organization of this paper is as follows. The proposed system model is described in Section 2. Then, the proposed FG algorithms for realizing the location detector and tracking are provided in Sections 3 and 4, respectively. After that, the derivations of CRB and ACRB are given in Section 5. The simulation results that prove the superiority of the proposed technique are provided in Section 6. Finally, the conclusion is given in Section 7 with some remarks.