# Vehicular Localization Enhancement via Consensus
## 3 Proposed Strategy
### Consensus-Based Update
To observe particular values of weights \(w_{1}\), \(w_{2}\), \(w_{3}\), and \(w_{4}\) upon the iteration, further algebraic manipulations obtain closed-form expressions of the weights at time \(t\), as  
\[w_{1}^{(t)}= w_{2}^{(t)}=\frac{1}{N_{v}}, \tag{8}\] \[w_{3}^{(t)}= \frac{1}{N_{v}}+\frac{t-1}{N_{v}^{2}},\] (9) \[w_{4}^{(t)}= \frac{t-1}{N_{v}^{2}}. \tag{10}\]  
The above expressions show that the update rules on self-localization noises \(w_{1}\) and \(w_{2}\) immediately converge to \(N_{v}^{-1}\). On the other hand, the weights on the pair-wise measurement noises, \(w_{3}\) and \(w_{4}\) increase linearly. Such a contrast between the pairs of \(\{w_{1},w_{2}\}\) and \(\{w_{3},w_{4}\}\) indicates that the localization accuracy during the consensus-based updates is dependent on the qualities of the pair-wise measurements. In particular, the update of weight \(w_{3}\) involves a constant addend \(N_{v}^{-1}\) from (4), which then is again added to \(w_{4}\) during the subsequent update. As a result, \(w_{3}\) and \(w_{4}\) are expected to increase by a constant amount of \(N_{v}^{-2}\). Figure 2 illustrates the characteristics of the changes in the weights upon 10 iterations of the updates, where \(N_{v}=10\) vehicles are available in a network. At the initial measurement of a vehicle's own location at step 0, only the noise corresponding to \(w_{1}\) exists since neither any exchange of the location estimates nor pair-wise measurement is made. Thus, with no consensus-based update, the value of \(w_{1}\) equals unity, while those of \(w_{2}\), \(w_{3}\), and \(w_{4}\) equal zero. Afterwards, as the consensus-based updates are executed, \(w_{1}\) is diminished and \(w_{2}\) is raised because the initially occurred self-localization error is exchanged and averaged with the neighboring vehicles. On the other hand, \(w_{3}\) and \(w_{4}\) begin to increase after the exchanges are made. Specifically, the figure shows that \(w_{1}\) and \(w_{2}\) converge to 0.1 immediately after the first update. This coherently corresponds to the analysis that \(w_{1}\) and \(w_{2}\) converge to \(N_{v}^{-1}\). On the other hand, \(w_{3}\) and \(w_{4}\) steadily increase by 0.01 as the iteration continues. This trend also appropriately corresponds to the analysis that \(w_{3}\) and \(w_{4}\) increase by \(N_{v}^{-2}\). In general cases where the self-localization and pair-wise measurement noises occur in random directions, the increase of \(w_{3}\) and \(w_{4}\) upon the repeated updates lead to the increase in the total amount of localization errors. However, the consensus-based updates are not designed for the immediate reduction of the localization error; the effective reduction of the localization error is achieved by the compensation step described in the following subsection. Instead, the primary aim of the consensus-based updates is to average out the estimate errors of the neighboring vehicles. A proper stopping point for the consensus-based updates is therefore needed since the continued addition of the noises pertaining to \(w_{3}\) and \(w_{4}\) is rather aimless. The condition to stop the iteration can be found by the collective trend in the change of the location estimates upon a sufficient number of the iteration. The convergence of \(\{w_{1},w_{2}\}\) and the linear increase of \(\{w_{3},w_{4}\}\) cause the location estimates to be updated by similar amounts and directions for all vehicles at each iteration. The repeated addition of similar magnitudes can thus serve as an indicator to halt the iteration. Such a halt condition regarding the amount of the changes of the location estimates can be described in terms of threshold \(C_{h}\) as  
\[\|\mathbf{p}_{v}^{(t)}-\mathbf{p}_{v}^{(t-1)}\|-\|\mathbf{p}_{v}^{(t-1)}- \mathbf{p}_{v}^{(t-2)}\|\leq C_{h}, \tag{11}\]  
where a specific value of \(C_{h}\) is determined according to conditions of the network. Upon the achievement of (11), the compensation takes place to reduce the localization error effectively.