# Vehicular Localization Enhancement via Consensus
## 3 Proposed Strategy
### Consensus-Based Update
Since the initial measurement noises are exchanged throughout the consensus-based updates, the change in \(\mathbf{p}_{v_{i}}^{(t)}\) upon the iterative updates is described by the change of the weights \(w_{1}\), \(w_{2}\), \(w_{3}\), and \(w_{4}\) over time. Assume that the self-localization noises that correspond to \(w_{1}\) and \(w_{2}\) are of zero mean and variance \(\sigma_{self}^{2}\), while the means and variances of the pair-wise distance and AoA measurement noises that correspond to \(w_{3}\) and \(w_{4}\) are also set to zero and \(\sigma_{pair}^{2}\), respectively. Although specific values of \(\sigma_{self}^{2}\) and \(\sigma_{pair}^{2}\) differ by individual vehicles, the expectation of the squared localization error can be collectively expressed as a combination of \(\sigma_{self}^{2}\) and \(\sigma_{pair}^{2}\) as  
\[ \begin{split}\mathbb{E}[\|\mathbf{p}_{v_{i}}-\mathbf{\tilde{p}}_{ v_{i}}\|^{2}]&=\sigma_{self}^{2}\left(w_{1}^{2}+w_{2}^{2}\cdot(N_{v}-1)^ {2}+2w_{1}w_{2}\cdot(N_{v}-1)\right)\\ &+\sigma_{pair}^{2}\left(w_{3}^{2}\cdot(N_{v}-1)^{2}+w_{4}^{2} \cdot(N_{v}-1)^{4}+2w_{3}w_{4}\cdot(N_{v}-1)^{3}\right).\end{split}  \tag{3}\]  
Since the sensory noise levels \(\sigma_{self}^{2}\) and \(\sigma_{pair}^{2}\) are determined by the qualities of the devices and the environment, the reduction of the localization error by the proposed scheme primarily relies upon the reduction of weights \(w_{1}\), \(w_{2}\), \(w_{3}\), and \(w_{4}\). To examine the changes of the weights, update rule of their values upon an iteration can be described as a linear system of equations as  
\[\mathbf{w}^{(t+1)}=\mathbf{A}\mathbf{w}^{(t)}+\mathbf{b}, \tag{4}\]  
where \(\mathbf{w}=[w_{1},w_{2},w_{3},w_{4}]^{T}\) and \(\mathbf{b}=[0,0,N_{v}^{-1},0]^{T}\). \(\mathbf{A}\) is partitioned as  
\[\mathbf{A}= \begin{bmatrix}\tilde{\mathbf{A}}&\mathbf{0}_{2,2}\\ \mathbf{0}_{2,2}&\tilde{\mathbf{A}}\end{bmatrix} , \tag{5}\]  
where \(\mathbf{0}_{m,n}\) indicates a \(m\)-by-\(n\) zero matrix. In a fully connected network, \(\tilde{\mathbf{A}}\) is given by  
\[\tilde{\mathbf{A}}= \begin{bmatrix}\frac{1}{N_{v}}&\frac{N_{v}-1}{N_{v}}\\ \frac{1}{N_{v}}&\frac{N_{v}-1}{N_{v}}\end{bmatrix} . \tag{6}\]The resulting linear system described by (4)-(6) suggests that the weight update rules of pair \(\{w_{1},w_{2}\}\) and pair \(\{w_{3},w_{4}\}\) are independent but similar to each other, with a slight difference of adding \(N_{v}^{-1}\) to \(w_{3}\). To observe the changes of the weights upon multiple repetitions of the updates, the \(n\)th power of \(\bar{\mathbf{A}}\) can be found by the diagonalization of \(\bar{\mathbf{A}}\). Two eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) of \(\bar{\mathbf{A}}\) are found as 1 and 0, while the corresponding eigenvectors \(v_{1}\) and \(v_{2}\) are found as \([1,1]^{T}\) and \([1,-(N_{v}-1)^{-1}]^{T}\), respectively. Thus, \(\bar{\mathbf{A}}\) can be diagonalized as \(\mathbf{PDP}^{-1}\) where \(\mathbf{P}=[v_{1}\;v_{2}]\) and the diagonal matrix \(\mathbf{D}\) is given by  
\[\mathbf{D}=\mathbf{P}^{-1}\bar{\mathbf{A}}\mathbf{P}= \begin{bmatrix}1&0\\ 0&0\end{bmatrix} . \tag{7}\]  
Noting that \(\bar{\mathbf{A}}^{n}=\mathbf{PD}^{n}\mathbf{P}^{-1}\), the \(n\)th power of \(\bar{\mathbf{A}}\) equals \(\bar{\mathbf{A}}\) itself. The addition of \(\mathbf{b}\) in Figure 2 upon the consensus-based update is therefore the only cause of the change of the weights. This indicates that if all vehicles are within each other's communication range, a single attempt of the update suffices. In other words, the purpose of repeating the updates is in propagating the location estimates throughout the network, not in increasing the localization accuracy.