# Vehicular Localization Enhancement via Consensus
## 3 Proposed Strategy
### Compensation  
As discussed in the preceding subsection, the noises pertaining to the pairwise measurements steadily increase as the updates are repeated. While this causes the overall magnitude of the localization error to increase, similarities between the remaining localization errors are made. The compensation is designed to counteract the accumulated error by taking advantage of such similarities and the anchor's superior localization performance.  
If the measurement noise has zero mean, the magnitude of the average of the noises is less significant compared to many of the vehicles' innate noises since noises in the different directions cancel out upon the addition. Since the agents have relatively low measurement accuracy, the agents' localization estimates become reliable after the consensus-based updates. On the contrary, the anchor initially possesses more accurate location estimate than the agents. During the updates, the anchor receives relatively inaccurate measurements from its neighbors, thus resulting with the aggravated  
Figure 2: Weight changes upon consensus-based updates.  
estimate after the updates. These characteristics can be manipulated by undoing the updates of all vehicles with respect to the anchor. In other words, the vehicles' location estimates can be compensated by \(\mathbf{p}_{v_{*}}^{(0)}-\mathbf{p}_{v_{*}}^{(t_{h})}\), where \(v_{*}\) and \(t_{h}\) denote the anchor and the time of the iteration halt, respectively. Apparent is the restoration of the anchor's localization to the initial and superior state. Since all vehicles' estimates contain similar magnitude and orientation of error, the compensation brings not only the anchor's but also all agents' location estimates closer to the ground truth locations. The compensation step can thus simultaneously bring down the localization errors within the network.  
Continuing from the analysis of the previous subsection, the compensated estimate of vehicle \(v_{i}\), denoted by \(\mathbf{p}_{v_{i}}^{(comp)}\), is expressed as  
\[\mathbf{p}_{v_{i}}^{(comp)}=\mathbf{\tilde{p}}_{v_{i}}+\frac{1}{N_{\varrho}} \sum_{v_{\mathsf{D}}\in\mathcal{N}(v_{\mathsf{I}})}\mathbf{n}_{v_{\mathsf{D}}v _{i}}-\frac{1}{N_{\varrho}}\sum_{v_{\mathsf{D}}\in\mathcal{N}(v_{*})}\mathbf{n }_{v_{\mathsf{D}}v_{*}}+\mathbf{n}_{v_{*}}. \tag{12}\]  
With the above expression of the estimate, the CRLB of the estimator can be found as  
\[\mathrm{E}[\|\mathbf{p}_{v_{i}}^{(comp)}-\mathbf{\tilde{p}}_{v_{i}}\|^{2}] \geq\frac{1}{\mathcal{I}(\sigma_{v_{*}})}=2\sigma_{v_{*}}^{2}. \tag{13}\]  
The corresponding Fischer information \(\mathcal{I}(\sigma_{v_{*}})\) is given by  
\[\mathcal{I}(\sigma_{v_{*}})=-\mathrm{E}\left[\frac{\partial^{2}}{\partial \mathbf{\tilde{p}}_{v_{i}}^{2}}\ln\mathcal{L}(\mathbf{\tilde{p}}_{v_{i}}\mid \mathbf{p}_{v_{i}}^{(comp)},\sigma_{v_{*}})\right]=[\frac{1}{\sigma_{v_{*}}^{ 2}},\frac{1}{\sigma_{v_{*}}^{2}}], \tag{14}\]  
where \(\mathcal{L}(\mathbf{\tilde{p}}_{v_{i}}\mid\mathbf{p}_{v_{i}}^{(comp)},\sigma_ {v_{*}})\) denotes the likelihood of the estimator given by  
\[\mathcal{L}(\mathbf{\tilde{p}}_{v_{i}}\mid\mathbf{p}_{v_{i}}^{(comp)},\sigma_ {v_{*}})=\frac{1}{2\pi\sigma_{v_{*}}^{2}}\exp\left(-\frac{1}{2\sigma_{v_{*}}^ {2}}(p_{v_{i},x}^{(comp)}-\tilde{p}_{v_{i},x})^{2}-\frac{1}{2\sigma_{v_{*}}^ {2}}(p_{v_{i},y}^{(comp)}-\tilde{p}_{v_{i},y})^{2}\right). \tag{15}\]  
The resulting CRLB in (13) indicates that the accuracy of the compensated estimate largely depends upon the accuracy of the anchor.