# Radar-Assisted Multiple Base Station Cooperative mmWave Beam Tracking
## 2 System Model
### Vehicle Mobility Evolution Model  
In order to use the EKF to accurately predict the angles needed for beamforming, a precise model of vehicle kinematic evolution is particularly necessary. Many existing methods focus on linear kinematics, as depicted in Figure 2a, while we propose a novel vehicle curvilinear kinematics model based on previous research, as shown in Figure 2b. For the linear mobility scenario, at the \(n\)-th slot, \(d_{n}\) denotes the distance between RSU and the vehicle, \(\theta_{n}\) denotes the angle of the vehicle relative to the RSU, \(v_{n}\) denotes the speed of the vehicle, \(\Delta d\) denotes the distance moved by the vehicle in a straight line at adjacent moments, \(\Delta\theta\) denotes the variation of the angle at adjacent moments, \(\beta_{n}\) denotes  
Figure 1: Radar-assisted V2I system with multi-RSU serving multiple vehicles. Figure 2: The kinematic model of vehicles in radar-assisted V2I system. (**a**) denotes linear kinematics. (**b**) denotes curvilinear kinematics. the reflection coefficient which contains both the radar cross-section (RCS) of the vehicle and the path-loss of the signal, specifically, \(\beta_{n}=\frac{\varepsilon_{n}}{2d_{n}}\), where \(\varepsilon_{n}\) denotes fading coefficient based on the radar cross-section. Both transmit and receive beams are formulated based on the angles predicted by step I and step II as mentioned previously. According to [16], the linear mobility evolution model is as follows,  
\[\left\{ \begin{array}{c}\theta_{n}=\theta_{n-1}+d_{n-1}^{-1}v_{n-1}\Delta Tsin \theta_{n-1}+q_{\theta}\\ d_{n}=d_{n-1}-v_{n-1}\Delta Tcos\theta_{n-1}+q_{d}\\ v_{n}=v_{n-1}+q_{\
u}\\ \beta_{n}=\beta_{n-1}+\left(1+d_{n-1}^{-1}v_{n-1}\Delta Tcos\theta_{n-1} \right)+q_{\beta}\end{array} \right. \tag{1}\]  
where \(\Delta T\) denotes the time interval between adjacent slots, \(q_{\theta}\), \(q_{d}\), \(q_{\varphi}\), \(q_{\beta}\) denote Gaussian noise which has a mean of zero and a variance of \(\sigma_{\theta}^{2}\), \(\sigma_{d}^{2}\), \(\sigma_{\varphi}^{2}\), \(\sigma_{\beta}^{2}\), respectively. For the curvilinear mobility scenario, we consider the curvilinear mobility of the vehicle as the composition of two uniformly accelerated linear moving objects. It is worth noting that, to guarantee that the trajectory of the vehicle is curvilinear, the direction of the combined velocity and the direction of the combined acceleration at the initial moment could not be the same. Therefore, based on kinematic knowledge and the illustration in Figure 2b, we can obtain the following geometric and kinematic equations. \[\left\{ \begin{array}{c}\Delta x=v_{x_{n-1}}\Delta T+\frac{1}{2}a_{x}\Delta T ^{2}\\ \Delta y=v_{y_{n-1}}\Delta T+\frac{1}{2}a_{y}\Delta T^{2}\\ \Delta d=\sqrt{\Delta x^{2}+\Delta y^{2}}\\ \varphi=arctan\left(\frac{\Delta y}{\Delta x}\right)\\ d_{n}{}^{2}=d_{n-1}{}^{2}+\Delta d^{2}-2d_{n-1}\Delta dcos(\theta_{n-1}- \varphi)\\ \frac{\Delta d}{\sin\Delta\theta}=\frac{d_{n}}{\sin(\theta_{n-1}-\varphi)} \end{array} \right. \tag{2}\]  
where \(\Delta x\) and \(\Delta y\) denote the distance that the vehicle moves in the horizontal (X-axis) and vertical direction (Y-axis), respectively. \(v_{x_{n-1}}\) and \(v_{y_{n-1}}\) denote the velocities of the vehicle in the X-axis and the Y-axis at the \((n-1)\)-th slot, respectively. \(a_{x}\) and \(a_{y}\) denote the acceleration of the vehicle in the X-axis and Y-axis, respectively, which are constant during the vehicle's mobility. \(\varphi\) denotes an angle used for calculation. Similar to the linear mobility scenario in [16], the curvilinear mobility evolution model is as below,  
\[\left\{ \begin{array}{c}\theta_{n}=\theta_{n-1}+\Delta\theta+q_{\theta}\\ d_{n}=d_{n-1}-\Delta d\cos(\theta_{n-1}-\varphi)+q_{d}\\ y_{y_{n}}=v_{y_{n-1}}+q_{y}\Delta T+q_{y_{y}}\\ v_{x_{n}}=v_{x_{n-1}}+a_{x}\Delta T+q_{y_{x}}\\ \beta_{n}=\beta_{n-1}\Big{(}1+\frac{\Delta dcos(\theta_{n-1}-\varphi)}{d_{n-1 }}\Big{)}+q_{\beta}\end{array} \right.