## 2 Modeling of Vehicle Dynamics for Dynamical Driving Conditions
### Validation and Analysis of CDM  
To verify the accuracy of CDM, it is compared with the kinematic model and the single-track model in the natural driving scenarios. The kinematic model is purely based on geometric relationships that works well at low speed. The single-track model considered the tire force with a small angle assumption. Both the kinematic and single-track models have been validated clearly, which can effectively describe vehicle dynamics in them owe application range and the equations are in the Appendix A.  
Figure 3: Vehicle dynamics model considering force coupling. According to the characteristics of the kinematic and single-track models [15,22], and the natural driving conditions analyzed in Section 2.1, the following typical ones are selected from the natural driving data and the statistical results of model errors are shown in Figure 4:  
* Condition 1: The vehicle moved at low speed (\(v\leq 5\) m/s) and the range of the steering angle is \(\left|\delta_{f}\right|\geq 35^{\lx@math@degree}\), which is comparatively larger. * Condition 2: The vehicle moved in the speed range from 5 m/s to 20 m/s and the steering angle is bigger than \(15^{\lx@math@degree}\). According to the analysis in Section 2.1, both the kinematic and single-track models are inapplicable to such conditions. * Condition 3: The vehicle moved at high speed (\(v\geq 20\) m/s) and the steering angle satisfies \(\left|\delta_{f}\right|\leq 3^{\lx@math@degree}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Value** \\ \hline \(m\) & Vehicle mass & 1460 kg \\ \(I_{z}\) & Yaw moment of inertia & 1943 kg\(\cdot\)m\({}^{2}\) \\ \(C_{yf},C_{yr}\) & Cornering stiffness of front/rear tire & 54,600 N/rad \\ \(I_{f},I\) & Distance from central of gravity to front/rear axle & 1.17 m/1.77 m \\ \(\alpha_{f},\alpha_{r}\) & Sideslip angle at the of front/rear tire & / \\ \(F_{yf},F_{yr}\) & Lateral forces at the of front/rear tire & / \\ \(v_{x},v_{y}\) & Longitudinal/lateral velocity in vehicle frame \((x,y)\) & / \\ \(v,v_{f},v_{r}\) & Vehicle velocity at central/front/rear & / \\ \(F_{T}\) & Longitudinal driving force & / \\ \(\omega\) & Yaw rate & / \\ \(\beta\) & Vehicle slip angle & / \\ \(\varphi\) & Heading angle & / \\ \(\delta_{f}\) & Front wheel steering angle & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of parameters and variables. Figure 4: Statistical results of model errors: (**a**) longitudinal speed; (**b**) lateral speed; (**c**) yaw rate. It is known from Figure 4a that the longitudinal speed error of the single-track model is bigger than another two vehicle models in Condition 1, because the projection of the lateral force of the tyre on the longitudinal driving direction has a significant influence on the longitudinal vehicle dynamics when the steering angle is big enough. Moreover, it is found from Figure 4b,c that both lateral speed error and yaw rate error of the kinematic model increase with the velocity, and the maximum errors of the lateral speed and yaw rate reach 0.12 m/s and 1.92 deg/s respectively in Condition 3. Because the kinematic model ignores the cornering characteristic of tire, which becomes very important to the vehicle lateral dynamics when the lateral force of tyre is large. Summarizing the above analysis, the single-track model and the kinematic model are only applicable to some of the natural driving conditions. Comparatively, CDM achieves a fine accuracy under all the evaluation conditions. Though CDM is more accurate than the kinematic model and the single-track model by considering the coupling effect of tyre forces in longitudinal and lateral directions, its nonlinearity becomes stronger because of the projection of forces. This requires much more discrete points to fit the nonlinear vehicle dynamics when designing the motion planer by such methods as NMPC. The consumption of computation resources for numerical optimization increases sharply with the dimension of optimization variables [30].