## Reference Vector Field of Vehicle Following
### Particle Dynamic Model and Reference Vector Field
\[\dot{\mathbf{e}}(t)=\mathbf{u}-(\mathbf{\omega}\cdot\
abla)\mathbf{\omega}=0\rightarrow\mathbf{u} _{1}(\mathbf{x}(t))\equiv(\mathbf{\omega}\cdot\
abla)\mathbf{\omega} \tag{8}\]  
(2) Assuming that the initial error vector e is not 0, then  
\[\dot{\mathbf{e}}(\mathbf{t})=\mathbf{u}-(\mathbf{\omega}\cdot\mathbf{\
abla})\mathbf{\omega}-(\mathbf{e} \cdot\mathbf{\
abla})\mathbf{\omega}\overset{\mathbf{u}=\mathbf{u}_{1}\mathbf{\tilde{u}}}{\Leftrightarrow} \dot{\mathbf{e}}(\mathbf{t})=\tilde{\mathbf{u}}-(\mathbf{e}\cdot\mathbf{\
abla})\mathbf{\omega} \tag{9}\]  
where the first item of control \(\mathbf{u}\) is based on the position feedback information of the vehicle, and the second item \(\tilde{\mathbf{u}}\) is used to eliminate the initial error. The usage of the friction circle is as shown in Figure 3.  
To prevent the control variable from exceeding the limitation of the friction circle, there is a restriction, as shown in (10). \[\frac{dy}{dx}=\frac{y}{S} \tag{4}\]0  
Based on the previous analysis and requirements, the control law for eliminating the error between the vehicle speed and the flow vector is constructed:  
\[\tilde{\mathbf{u}}=\left\{ \begin{array}{ll}-\varepsilon\mathbf{e}/\left|\mathbf{e} \right|&\text{If }\left|e\right|\geq v_{0}\\ -\mathbf{e}&else\end{array} \right. \tag{11}\]  
When the error is small enough, the use of the tire force can be reduced to make the vehicle converge to the path stably, so \(v_{0}\) is introduced as a judgment condition. It can be seen from the analysis that the convergence time of the error is related to the use of the tire force. This paper numerically simulates different values of \(\varepsilon\) (the value of \(\varepsilon\) is defined between 0-\(\mu g\), where \(\mu\) is considered as 1.0 and g denotes the gravity acceleration, which is selected as 9.8 m/s\({}^{2}\)). The leading vehicle is driving along with a straight line--y = 0, while the initial position of the following vehicle is (0, \(-\)3). The results of numerical simulations are as shown in Figure 4.  
Figure 3: Control variables in friction circle. As shown in Figure 4a, when the particle model is tracking the flow vector, the value of control law is used to eliminate the error between the actual vehicle speed and the streamline vector. If the tire force is insufficient, the error convergence speed will be small. As a result, the vehicle will oscillate around the reference path. As the \(\epsilon\) increases, the oscillation amplitude and the error convergence speed increase gradually, and the overshoot gradually decreases. As can be seen in Figure 4b, the value of \(\epsilon\) has a particular influence on the longitudinal spacing control. Moreover, as the utilization of the friction circle increases, the longitudinal spacing converges more rapidly and stably. For theoretical analysis about the influence of \(\epsilon\) on the convergence speed of the error between the velocity vector and the reference vector, Equation (9) is differentiated to obtain the Jacobian matrix \(\mathrm{H}\), and its eigenvectors are \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\). \[H=\left[ \begin{array}{cc}\frac{\partial\omega_{x}}{\partial x}&\frac{ \partial\omega_{y}}{\partial x}\\ \frac{\partial\omega_{x}}{\partial y}&\frac{\partial\omega_{y}}{\partial y} \end{array} \right]\]  
Let \(e\) be the norm of the error vector \(\mathbf{e}\), then (12) is obtained from the (7). \[e\dot{e}=\mathbf{e}\dot{\mathbf{e}}=\mathbf{e}\mathbf{\ddot{u}}-\mathbf{e} \cdot\left[(\mathbf{e}\cdot\
abla)\boldsymbol{\omega}\right] \tag{12}\]  
There is \(m_{1}+m_{2}=\frac{\partial\omega_{x}}{\partial x}+\frac{\partial\omega_{y}}{ \partial y}=\
abla\cdot\boldsymbol{\omega}\), (13) could be obtained with the combination of (7) and (12). \[ \begin{array}{rl}e\dot{e}&\leq-e\epsilon-\mathbf{e}^{\mathbf{T}}(m_{1}+m_{2} )\mathbf{e}=-e\epsilon-(m_{1}+m_{2})e^{2}\\ &\Rightarrow\dot{e}\leq-\epsilon-(m_{1}+m_{2})e\end{array}