# Vehicular Localization Enhancement via Consensus
## 3 Proposed Strategy
### Consensus-Based Update  
The first step of the strategy is consensus-based updates. The location estimates of the vehicles are exchanged in a consensus-based and iterative manner. Prior to the initialization of the update, an individual vehicle in the network performs the self-localization. The resulting location estimate is exchanged with each other after the initial measurement is made. These exchanges are not necessarily made in a synchronous manner but only for pairs with pairwise connection available in the network. With the exchanged information at hand, the measurements of distance and AoA with respect to each neighboring vehicle are available. With the collected measurements, new estimates can be calculated by combining the neighboring vehicles' location estimates and the pairs of distance and AoA with respect to the corresponding partner. In this manner, one new location estimate can be obtained for each neighboring vehicle. If each of \(N_{v}\) vehicles can reach to all other neighbors, each vehicle in the network can obtain \(N_{v}-1\) new estimates. With the new estimates obtained, the vehicles updates their location estimates to a weighted average of the new estimates and the previous estimates. After the update, exchanges and updates are repeated. With time instances \(t=\{0,1,2...\}\) denoted as superscripts, the update of the location estimate of vehicle \(v_{i}\) from \(\mathbf{p}_{v_{i}}^{(t)}\) to \(\mathbf{p}_{v_{i}}^{(t+1)}\) can be described as  
\[\mathbf{p}_{v_{i}}^{(t+1)}=\frac{1}{N_{v}}\{\mathbf{p}_{v_{i}}^{(t)}+\sum_{v \in\mathcal{N}(v_{i})}(\mathbf{p}_{v}^{(t)}+\mathbf{d}_{v,v_{i}})\}, \tag{1}\]  
Figure 1: A possible scenario of the vehicle-to-everything (V2X) network. where \(\mathcal{N}(v_{i})\) indicates the set of neighbors of the vehicle \(v_{i}\) excluding itself. Note that the coefficient \(1/N_{v}\) is valid only when the network is strongly connected or complete, that is, all vehicles are able to communicate with all other neighboring vehicles. If the network is sparse and weighted average is applied, the coefficient can be replaced by an appropriate pair of two coefficients. However, the following analysis can be extended in a straightforward way and the corresponding discussion proceeds analogously. Thus, we stick to the complete network case for simple representation. For each update, new location estimate \(\mathbf{p}_{v_{i}}^{(t)}\) is acquired. With some simple algebra, the analytical expression of the localization error of vehicle \(v_{i}\) at an arbitrary time \(t\) is found in a linear combination of four different types of the measurement noises. Through such decomposition, the localization error at time \(t\) can be expressed as weighted sums of four different types of noises, as  
\[\mathbf{p}_{v_{i}}^{(t)}-\mathbf{\tilde{p}}_{v_{i}}=w_{1}\mathbf{n}_{v_{i}}+ \sum_{v\in\mathcal{N}(v_{i})}w_{2}\mathbf{n}_{v}+\sum_{v\in\mathcal{N}(v_{i}) }w_{3}\mathbf{n}_{v,v_{i}}+\sum_{v^{\prime\prime}\in\mathcal{N}(v^{\prime}),v ^{\prime}\in\mathcal{N}(v_{i})}w_{4}\mathbf{n}_{v^{\prime\prime},v^{\prime}}, \tag{2}\]  
where \(w_{1}\), \(w_{2}\), \(w_{3}\), and \(w_{4}\) indicate the weights on the different types of the measurement noises with respect to the vehicle. The first \(w_{1}\) is the weight on the self-localization noise itself. The second \(w_{2}\), on the other hand, is the weight on the sums of the self-localization noises of the neighboring vehicles. The last two \(w_{3}\) and \(w_{4}\) denote the weights of the pair-wise distance and AoA measurement noises, respectively. To be precise, \(w_{3}\) is the weight on the sum of noises that occur upon pair-wise measurements from the neighborhood. \(w_{4}\) is also the weight on the sum of pair-wise measurement noises, but from the neighborhood of a neighbor to the neighbor.