# AOA-Based Three-Dimensional Positioning and Tracking Using the Factor Graph Technique
## 6 Results
### Robustness Analysis  
In this subsection, the robustness of tracking performance is evaluated, which is also regarded as the main advantage of the proposed technique. First of all, the effect of sudden sensing errors is considered, such as the false alarm, which may cause unreasonable measurement AOAs. In contrast to the previous simulations, four sensors are used with their positions at \([-100,-10,0]^{T}\), \([-10,-10,0]^{T}\), \([-50,-100,0]^{T}\), and \([-50,-5,10]^{T}\). The reason why more sensors are utilized is to avoid the case in which not enough sensors can work normally when some suffer from significant sensing errors. The FG iteration number is fixed at 15, while 100 snapshots are assumed for measurement at each time index. The real path of the target is set the same as in the previous simulations.  
Specifically, at each time index, one of the four sensors is assumed to encounter sudden errors with a probability equal to 0.2, besides the true AOA measurement. Without the help of SI, the fusion center can simply discard the data of a sensor if AOAs from more than one origin are detected. However, the system performance will also decrease when fewer sensors are used. In the proposed system, a predicted AOA will be calculated for each sensor by the fusion center, based on the predicted sensor location by EKF. If the measured AOA is close to the predicted one within a predefined threshold, set by \(10^{\circ}\) in the simulation, it will be used for location detection at the current time index. Otherwise, the measured data will be discarded. By doing this, sudden errors such as false alarms can be eliminated while still keeping the desired measurement, in order to achieve a better tracking performance.  
The performance improvement with SI is shown in Figure 7. Actually, SI is regarded as the a priori information generated from ACRB, which helps the fusion center make a pre-decision to eliminate the impacts of burst sensing errors. In other words, ACRB enables the prediction of the expected AOA measurements. For example, in this simulation of false alarms, if the measured mean AOAs are larger than the predicted values beyond a threshold, i.e., \(5^{\circ}\), they will be directly eliminated. It can be seen that the RMSEs achieved with SI are always smaller than the cases without SI. The improvement is found to be more significant when the standard deviation of the AOAs becomes larger.  
Figure 6: RMSE performances versus FG iteration number.  
Moreover, simulations are conducted to evaluate the robustness of the system against the dynamic measurement variances. In this case, \(\sigma_{\phi,\theta}\) is assumed to be randomly chosen from a set of \(\{5^{\circ},10^{\circ},15^{\circ},20^{\circ}\}\), for each time index. After location detection, the observation variance is calculated based on the proposed ACRB, which is dynamically fed to the EKF. As a comparative study, another scheme assuming a fixed \(\sigma_{\phi,\theta}\) equal to two is also simulated. According to Figure 8, the RMSE achieved by the proposed technique is 4.3, which is smaller than the comparative scheme with the RMSE being 5.4. Obviously, the two local maximum RMSEs appeared for the comparative scheme happen when \(\sigma_{\phi,\theta}=20^{\circ}\) was chosen. However, such errors are much reduced with the proposed technique. Therefore, the robustness of the proposed system is proven.