# Application and Assessment of Cooperative Localization in Three-Dimensional Vehicle Networks
## 8 Simulation Results
### Effect of Absolute Position Measurements in CL  
It has previously been reported that measuring the absolute position of one vehicle can reduce the localization errors of other vehicles in CL [14]. Therefore, CL with only relative position measurements may still be useful if a vehicle is able to attain the absolute position measurement via GPS, SLAM, etc. The three-vehicle system of Scenario 2 was utilized to illustrate the effect of absolute measurements. We consider two cases. In one case, the relative positions of vehicles 2 and 3 are measured by vehicles 1 and 2, respectively, for the first 100 s (CL). Afterwards, vehicle 1 measures its own absolute position for a short period of 1 s in addition to the relative measurements. Afterwards, there are no subsequent measurements by any of the vehicles. In the other case, there are no measurements for the first 100 s (no CL), after which vehicle 1 measures its own absolute position. There are no other measurements involved.  
The mean error norm \(\Delta r\) across all vehicles and simulations was calculated to be \(0.85\pm 0.04\) for the CL case and \(1.05\pm 0.01\) for the no CL case. This clearly indicates that the absolute measurement reduces the overall localization errors much more significantly when there has been CL with prior relative position measurements.  
Additional details can be found in Figure 9 where the comparisons between the time histories of the two cases for the three vehicles are plotted. It is clear that CL does not make a significant contribution during the first 100 s. After the first 100 s vehicle 1 measures its own position and its localization error drops to near zero in both cases. However, despite the lack of relative position measurements from 100 s to 200 s, the localization errors for the \"CL\" case are significantly lower than the \"no CL\" case for vehicles 2 and 3. This is due to the cross-covariance evolution in the CL algorithm during the first 100 s of the CL case. In conclusion, we propose that when dependable absolute orientation measurements are accessible, CL with relative position measurement does not improve localization errors using the kinematic model. Nevertheless, if any vehicle can measure its absolute position at some point during the experiment, CL can still greatly reduce the position errors of all vehicles.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{e_{r_{mean}}}\) & \(\mathbf{e_{r_{std}}}\) & \(\mathbf{e_{r_{min}}}\) & \(\mathbf{e_{r_{max}}}\) \\ \hline Propagate & 1.35 & 0.01 & 1.32 & 1.39 \\ Case 1 & 1.34 & 0.01 & 1.31 & 1.38 \\ Case 2 & 1.30 & 0.10 & 1.25 & 2.28 \\ Case 3 & 1.28 & 0.02 & 1.24 & 1.32 \\ Case 4 & 1.28 & 0.02 & 1.25 & 1.32 \\ Case 5 & 1.28 & 0.01 & 1.25 & 1.32 \\ Case 6 & 1.27 & 0.01 & 1.25 & 1.31 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Mean position error (m) statistics across 100 simulations of _all vehicles_ for Scenario 2 with only relative position measurements.  
Figure 9: Comparison of error norms when vehicle 1 measures its own position for 1 s at the 100 s mark. Dashed lines indicate CL with relative position measurements for the first 100 s, and solid lines indicate no CL.