## Reference Vector Field of Vehicle Following
### Particle Dynamic Model and Reference Vector Field  
Vehicle motion is simplified in a 2-D plane, and the vehicle is considerate as a single particle. At this point, the particle has two degrees of freedom in the plane, including the longitudinal direction and lateral direction. Since the acceleration of the vehicle is limited by road adhesion, the control  
Figure 1: Platoon path following strategy. Vehicle 0 denotes the leading vehicle. Other vehicles denote the following vehicles in platoon. variable \(\mathbf{u(t)}\) of the vehicle particle dynamics model is selected as acceleration, and the particle dynamics model can be simplified, as shown in formula (1), where \(\mathbf{x_{G}}\) and \(\mathbf{v_{G}}\) are 2-D vectors. \[\dot{\mathbf{x}_{G}}=\mathbf{v_{G}} \tag{1}\] \[\dot{\mathbf{v}_{G}}=\mathbf{u}(t)\]  
As is shown in Figure 2, the following flow vector \(w\) was built by following vehicle's position and the balanced point. The integration of the vector forms a streamline that converges on the reference path. The tangent direction of each point on the streamline is the same as the vector \(\mathbf{w}\). R denotes the distance between the current position of the following vehicle and its balanced points. There is a flow vector that is associated with the current position of the following vehicle, regardless of whether the following vehicle is on either side of the path, and the vector \(\mathbf{w}\) could be described as formula (2). \[\mathbf{\omega}(x,y)=\frac{v}{R}\left( \begin{array}{c}S\\ y\end{array} \right) \tag{2}\]  
Apparently, when the leading vehicle is traveling in a straight line, there is \(R=\sqrt{S^{2}+y^{2}}\). The value of \(\mathbf{v}\) is the norm of the vehicle speed vector, which can be selected as a fixed value when the path tracking control is performed separately; in the coordinated control of the platoon, \(\mathbf{v}\) should be consistent with the speed of other intelligent vehicles in the platoon. To ensure that the distance between vehicles meets the requirements of the spacing control strategy, the value of \(\mathbf{v}\) should be chosen carefully. The calculation method used in this paper is as shown in (3). \[v=|\mathbf{V_{0}}|+k\big{(}x_{0}-x_{g}-p_{d}\big{)}=|\mathbf{V_{0}}|+kS \tag{3}\]  
where \(\mathbf{V_{0}}\) is the speed vector of the leading vehicle. According to (2), the curve that is formed by the integration of the flow vector has the relationship as is shown in (4). \[\frac{dy}{dx}=\frac{y}{S} \tag{4}\]  
To obtain a general solution to the differential Equation (4), the expression of the curve can be obtained as is shown in (5). \[y=y_{0}e^{\frac{(x-x_{0})}{S}} \tag{5}\]  
There is an initial error between the desired velocity vector determined by the flow vector and the actual velocity vector of the particle at the current location, as shown in (6). \[\mathbf{e}(t)=\mathbf{v_{g}}(t)-\mathbf{\omega}\big{(}\mathbf{x_{g}}\big{)} \tag{6}\]  
Figure 2: The diagram of reference vector. To make the following vehicle's path converge to the reference path of the leading vehicle, the following vehicle's driving direction tends to the course of the streamline. Differentiate the (6) and (7) can be obtained. \[\dot{\mathbf{e}}(t) =\dot{\mathbf{v}}_{\mathbf{G}}(t)-\dot{\mathbf{\omega}}(\mathbf{x}_{\mathbf{G}}(t)) \tag{7}\] \[=\mathbf{u}-\mathbf{v}_{\mathbf{G}}\cdot\
abla\mathbf{\omega}\]  
where \(\
abla=\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)^ {T}\), \(\
abla\mathbf{\omega}=\left(\frac{\partial\omega_{x}}{\partial x}+\frac{\partial \omega_{y}}{\partial y}\right)\). (1) Assuming that the initial error vector \(\mathbf{e}\) is 0, the error can be kept at 0 by controlling value \(u_{1}\).