# Vehicular Localization Enhancement via Consensus
## 2 System Model  
In a local vehicular ad-hoc network, \(N_{v}\) vehicles communicate with each other via utilizing mmWave technology features. By the utilization of a set of mmWave transceivers that work in all angles, the communications is not affected by the respective orientations of vehicles. Furthermore, as the localization takes place with respect to an instance of time, the vehicles' mobility causes no direct impact to the connection between the vehicles. However, some pairs of vehicles may be beyond each other's communication range if the network size is relatively large. The vehicles estimate the locations of themselves by means such as global navigation satellite system (GNSS). Among them, a vehicle that possesses relatively accurate means for self-localization is referred to as an anchor, while other non-anchor vehicles are referred to as agents. To achieve the distributed mode, the vehicles communicate with the available set of neighbors rather than abiding by the pre-configured borders of the network. Hence the network boundaries are not fixed but determined under considerations of communication coverage, signal-to-noise ratio (SNR), and propagation delay. Based on this strategy, the anchor is selected on a consensus basis, where each vehicle possesses the knowledge of the noise level of its sensory devices. With this knowledge, each vehicle can determine if it can serve as the anchor to the neighbors. If present, road-side infrastructures with measurement and communication capabilities can also serve as anchors. Such infrastructures can serve as high-quality anchors because of their deterministic position and possible availability of additional computation and communication resources. For the most intended systems, LTE device-to-device (D2D) protocol of LTE Sidelink [21] is deployed for the message delivery. Under this deployment, vehicles broadcast the messages with source information, while an asynchronous collision avoidance scheduling is done if an infrastructure is present. The ground truth position of vehicle \(v_{i}\), where \(i=\{1,2,...,N_{v}\}\), is represented in two-dimensional coordinates denoted by \(\mathbf{\hat{p}}_{v_{i}}=(\hat{p}_{x_{v_{i}}},\hat{p}_{y_{v_{i}}})\). The measurements of \(\mathbf{\hat{p}}_{v_{i}}\) are subject to additive measurement noises of self-localization with the same dimensions denoted by \(\mathbf{n}_{v_{i}}=(n_{x_{v_{i}}},n_{y_{v_{i}}})\). Vehicle \(v_{i}\) thus obtains an erroneous measurement \(\mathbf{p}_{v_{i}}=\mathbf{\hat{p}}_{v_{i}}+\mathbf{n}_{v_{i}}\). Upon the measurement, vehicle \(v_{i}\) transmits a message pertaining to \(\mathbf{p}_{v_{i}}\) to its neighbors via mmWave communication. By the transmission of mmWave, the inter-vehicular distance and AoA with respect to each neighboring vehicle is measured [22; 23]. In particular, the distance from vehicle \(v_{i}\) to vehicle \(v_{j}\) is defined as \(\mathbf{\hat{d}}_{v_{i}v_{j}}\equiv\mathbf{\hat{p}}_{v_{j}}-\mathbf{\hat{p}}_ {v_{i}}\). In a similar manner to observing \(\mathbf{p}_{v}\) instead of \(\mathbf{\hat{p}}_{v}\), additive noise \(\mathbf{n}_{v_{i}v_{j}}=(n_{x_{v_{i}v_{j}}},n_{y_{v_{i}v_{j}}})\) occur. The resulting erroneous measurement is therefore \(\mathbf{d}_{v_{i}v_{j}}=\mathbf{\hat{d}}_{v_{i}v_{j}}+\mathbf{n}_{v_{i}v_{j}}\). Figure 1 illustrates an example configuration of the system model. Three vehicles \(v_{1}\), \(v_{2}\) and \(v_{3}\) can respectively observe \(\mathbf{p}_{v_{1}}\), \(\mathbf{p}_{v_{2}}\) and \(\mathbf{p}_{v_{3}}\) by self-localization. The vehicles send their location estimates to each other through mmWave communication. The measurement of the distance and angle of arrival (AoA) to each other is also made. For example, in communication between \(v_{1}\) and \(v_{2}\), pairs of distance and AoA \(\mathbf{e}_{v_{1},v_{2}}\) and \(\mathbf{d}_{v_{2},v_{1}}\) are measured. The figure collectively denotes such pair of distance and AoA measurements as \(D_{v_{1},v_{2}}\).