## 3 Dynamical Model for Motion Planning
### Simplification of Dynamical Model by Coupling Analysis  
In this section, (1) is simplified by numerical analysis of the coupling between lateral and longitudinal dynamics. Let \(\xi=\left[v_{x}\ v_{y}\ \omega\right]^{\mathrm{T}}\) and \(\mathbf{u}=\left[F_{T}\ \delta_{f}\right]^{\mathrm{T}}\), (1) is written as  
\[\dot{\xi}=g(\xi,\mathbf{u}). \tag{2}\]  
To reveal the coupling characteristics and simplify the model, it is expanded by the first-order Taylor equation at the equilibrium point, \((\xi_{0},\mathbf{u}_{0  
\([-4000,\,4000]\) N. They are normalized firstly to compare the effects on the vehicle dynamics at the same scale. The normalized parameters are selected according to the statistical results in Figure 2 as  
\[\overline{v}_{x}=30\text{ m/s},\,\,\overline{F}_{T}=m\cdot a(v_{x}),\,\,\overline{ \delta}_{f}=\delta(v_{x}),\,\,\overline{\omega}=46\text{ deg/s}, \tag{5}\]  
where \(a(v_{x})\) and \(\delta(v_{x})\) are both look-up table functions with \(v_{x}\) as input. The table data of \(a(v_{x})\) is obtained from the 3 \(\sigma\) boundary in Figure 2a and that of \(\delta(v_{x})\) is from Figure 2d. The normalized transfer function is  
\[\mathbf{G}(s)^{\prime}=\left[ \begin{array}{cc}\frac{\Delta v_{x}(s)}{\Delta F_{ T}(s)}*\frac{\overline{F}_{T}}{\overline{v}_{x}}&\frac{\Delta v_{x}(s)}{ \Delta\delta_{f}(s)}*\frac{\overline{\delta}_{f}}{\overline{v}_{x}}\\ \frac{\Delta\omega(s)}{\Delta F_{T}(s)}*\frac{\overline{\omega}}{\Delta\delta _{f}(s)}*\frac{\overline{\delta}_{f}}{\overline{\omega}}\end{array} \right], \tag{6}\]  
where \(s\) here represents the Laplacian operator. In order to analyze the transfer characteristics from \(\delta_{f}\) to \(v_{x}\) over the entire operating range, we have calculated the amplitude-frequency characteristics of a total of 22,500 equilibrium points in the range of \(v_{x}\in[0,\,30]\) m/s and \(\delta_{f}\in[0,\,30]\) degree with the step of 0.2. Since the closed-loop system composed of human and vehicle is a low-pass one with a cut-off frequency of less than 1 Hz [31], only the data in the low frequency band was used, for simplicity. The typical amplitude-frequency characteristics are shown in Figure 4a, which are all close to the first-order inertial system. Their open-loop gains and cut-off frequencies at different linearization points are shown in Figure 4c,d, respectively. Both the open-loop gain and cut-off frequency decrease with \(v_{x}\), while increasing with \(\delta_{f}\). The minimum open-loop gain is greater than 1 dB, which implies that the lateral input has an obvious effect on the longitudinal dynamics. To compare the influence of \(F_{T}\) with respect to \(\delta_{f}\) on \(v_{x}\), the amplitude-frequency characteristics of \(F_{T}\) to \(v_{x}\) are illustrated in Figure 4b, where \(\delta_{f}=5,\,18\) degree. The solid lines are the amplitude-frequency characteristics of \(F_{T}\) to \(v_{x}\), and the dashed lines are that of \(\delta_{f}\) to \(v_{x}\). As shown in Figure 4b, \(\delta_{f}\) has the same order of the magnitude of the open-loop gain and cut-off frequency as that of \(F_{T}\) on \(v_{x}\), so \(\delta_{f}\) has a significant impact on \(v_{x}\). The same method was also used to analyze the transfer characteristics from \(F_{T}\) to \(\omega\), which are shown in Figure 5a when \(\delta_{f}=2,\ 18\) degree. Their amplitude-frequency characteristics were also close to the first-order inertial system. The open-loop gains are given in Figure 5c, fin which it can be seen that the open-loop gains are negative in most cases and decrease with the increase of \(\delta_{f}\) and \(v_{x}\). This implies that the coupling effect of longitudinal input on the lateral dynamics decreases with the increase of \(\delta_{f}\) and \(v_{x}\). To compare the influence of \(F_{T}\) with respect to \(\delta_{f}\) on \(\omega\), the amplitude-frequency characteristics are shown in Figure 5b when \(v_{x}=6,\ 30\ \text{m/s}\). The solid lines are the amplitude-frequency characteristics and the dashed lines are that from \(F_{T}\) to \(\omega\).