## Reference Vector Field of Vehicle Following
### Sufficient Conditions for Following Errors' Convergence  
In addition to the influence of the friction circle on the path tracking and the spacing control, the parameter \(k\) in Equation (3) also affects the perspective of subjective guessing. To explore the \(k\)'s influence on the error convergence of the trajectory and spacing, the same numerical simulation scenario as in 2.2 is used, and different control gain values are applied. The result is shown in Figure 5. It can be found from Figure 5 that, in the step input, the overshoot will increase when the error is enhanced with the increase of \(k\). When \(k\leq 0.2\), the overshoot is eliminated; Figure 5 b shows the longitudinal spacing error. The convergence speed becomes faster as \(k\) increases, and when \(k=0\), the norm of the flow vector \(w\) is directly related to the speed of the leading vehicle. At this time, the current error can only be prevented from further enlarging, but the error will not converge. Therefore, the selection of \(k\) is a contradictory parameter for the convergence of trajectory tracking error and the longitudinal following error. Two tracking targets should be coordinated in the application. To explore the conditions that enable the following vehicle to converge to the balanced point when the leading vehicle drives in a stable driving state (longitudinal uniform speed), this paper constructs a smooth positive function related to the distance between the vehicle and the balanced point, as shown in Figure 6. The expressions of \(R(x)\) is as shown in (16). \(x_{I}\) denotes the position vector of balanced point and \(x_{g}\) denotes the position vector of the following vehicle. \[R\left(x_{g}\right)=\left(x_{I}\text{-}x_{g}\right)^{2} \tag{16}\]  
Figure 5: Particle motion under the different values of \(k\): (**a**) The path of particle model under the different values of \(k\); and, (**b**) The following error of particle model under the different values of \(k\). Figure 6: Effect of the tracking error convergence. To make \(R\big{(}x_{g}(t)\big{)}\to 0\) as \(t\to\infty\), \(R(x_{g})\) is required to monotonically decrease. When \(t_{1}\geq t_{0}\), there is \(R\big{(}x_{g}(t_{1})\big{)}\leq R\big{(}x_{g}(t_{0})\big{)}\). Assuming that \(R\big{(}x_{g1}\big{)}=\big{(}x_{11}-x_{g1}\big{)}^{2}=S_{0}^{2}+y_{0}^{2}\), and the leading vehicle is in a stable and uniform straight-line driving state, the component of the speed in the path direction is zero. Suppose that the position of the vehicle at time \(t_{1}\) is \(x_{g1}\) and the position at time \(t_{2}\) is \(x_{g2}\), then \(R(x_{g})\) is asked to satisfy Equation (17). \[ \begin{split} R\big{(}x_{g2}\big{)}&=\big{(}x_{ \Omega}-x_{g2}\big{)}^{2}=\big{(}x_{\Omega}+v_{\Omega}\Delta t-x_{g1}-v_{xg1} \Delta t\big{)}^{2}+\big{(}y_{\Omega}+y_{\Omega}\Delta t-y_{g1}-v_{yg1}\Delta t \big{)}^{2}\\ &=\big{(}S_{0}+v_{\Omega}\Delta t-(v_{\Omega}+kS_{0})\Delta t \big{)}^{2}+\Big{(}y_{0}+0-\frac{(v_{\Omega}+kS_{0})R_{0}}{y_{0}}\Delta t \Big{)}^{2}\\ &=\big{(}L_{0}-kS_{0}\Delta t\big{)}^{2}+\Big{(}y_{0}-\frac{(v_{ \Omega}+kS_{0})R_{0}}{y_{0}}\Delta t\Big{)}^{2}\leq L_{0}^{2}+y_{0}^{2}\leq R \big{(}x_{g1}\big{)}\end{split}  \tag{17}\]  
where \(x_{\Omega}=\left( \begin{array}{c}x_{\Omega}\\ y_{\Omega}\end{array} \right)\), \(x_{\Omega}=\left( \begin{array}{c}x_{\Omega}\\ y_{\Omega}\end{array} \right)\), \(x_{g1}=\left( \begin{array}{c}x_{g1}\\ y_{g1}\end{array} \right)\), \(x_{g2}=\left( \begin{array}{c}x_{g2}\\ y_{g2}\end{array} \right)\), \(\Delta t=t_{2}-t_{1}\). According to the previous analysis, the conditions that the error function trends to zero needs to satisfy:  
(1) the flow vector converges to the balanced point in the reference trajectory;  
(2) the error between the vehicle speed and the flow vector is bounded;  
(3) the controlled variable-the acceleration is limited by the friction circle; and,  
(4) the control gain k is greater than zero. The above conditions need to be satisfied at the same time to ensure that the following vehicle gradually approaches the balanced point.