# Application and Assessment of Cooperative Localization in Three-Dimensional Vehicle Networks
## 8 Simulation Results
### Simultaneous Relative Pose Measurement Assessment
#### 8.2.1 Scenario 1--Effect of Other Vehicle Measurements on Single-Vehicle Localization  
The idea for Scenario 1 is to see how other vehicles taking additional simultaneous relative measurements affects the localization of a single vehicle and, consequently, all vehicles in the network. Table 3 shows five cases that were simulated, concentrating on the localization of vehicle 1. In case 1, relative measurements of vehicle 2 are taken by vehicle 1, while in case 2, vehicle 2 simultaneously takes relative measurements of vehicle 3 and so on. In each case, errors are averaged across the 100 simulations. The mean of the position error norm (\(\Delta r\)) and the orientation error norm (\(\Delta q\)) for vehicle 1 in all 100 simulations are plotted in Figures 7 and 8, respectively, for propagation without measurements and the five CL cases. Both \(\Delta r\) and \(\Delta q\) show that the performance increases as more relative measurements are added, and all CL cases show significant improvement over the propagation. An interesting observation is the significant error reduction in both \(\Delta r\) and \(\Delta q\) of vehicle 1 from Case 1 to Case 2 when vehicle 2 measures the relative pose of vehicle 3. Hence, the overall error for vehicle 1 is reduced even though it is  
Figure 6: Time history of vehicle orientation error norm in the test case. not involved in the additional measurement. When subsequent measurements are added in cases 3 through 5, the localization errors continue to decrease but are relatively marginal. This demonstrates that the beneficial effects of CL may not outweigh its additional computational cost as the number of simultaneous measurements is increased. It should also be noted that the temporary dip in quaternion propagation error in Figure 8 is due to the complete rotation of the vehicles, bringing some components of the quaternion closer to their actual values. Table 4 lists the error norm statistics across the 100 simulations for vehicle 1, which includes statistical terms such as mean and standard deviation for position \(r\) and orientation \(q\). Figure 8: Scenario 1—vehicle 1 mean \(\Delta q\) norm. \begin{table}
\begin{tabular}{c c} \hline \hline  & **Simultaneous Measurements (\(b\to t\))** \\ \hline Case 1 & \(1\to 2\) \\ Case 2 & \(1\to 2,2\to 3\) \\ Case 3 & \(1\to 2,2\to 3,3\to 4\) \\ Case 4 & \(1\to 2,2\to 3,3\to 4,4\to 5\) \\ Case 5 & \(1\to 2,2\to 3,3\to 4,4\to 5,5\to 6\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simultaneous measurement descriptions for Scenario 1. Figure 7: Scenario 1—vehicle 1 mean \(\Delta r\) norm. To calculate these terms, we determine the norms of \(\Delta r\) and \(\Delta q\) at each time step of each simulation, and their root means squared are derived over time. Finally, the mean, standard deviation, min, and max are derived across all simulations. In CL, the cross-covariances contribute to lower overall error for vehicles when additional measurements are performed. The statistics again illustrate that the error reduction beyond Case 2 is relatively marginal. However, it also demonstrates that the variability is significantly increased in CL (see the standard deviation of CL cases compared to propagation). We also compared the same statistics across all vehicles. In this case, \(\Delta r\) and \(\Delta q\) are again calculated at each time step of each simulation, and the root mean squared of the means is derived across all time steps and then averaged for all vehicles. Finally, the mean, standard deviation, min, and max are derived across all simulations. The results are listed in Table 5. The data show a similar trend to Table 4, where there is a reduction in the error estimates of both \(\Delta r\) and \(\Delta q\) as simultaneous measurements are added. However, as expected, the reduction in overall localization errors continues to be significant as the number of vehicles that rely solely on propagation decreases. The same increase in variability is also observed in the location of all vehicles.