## 2 Statistical Analysis of Operating Range  
When modelling vehicle dynamics for motion planning, the following are three typical issues [28; 29]:  
1. Using linear or nonlinear tire models;
2. Whether the transfer of the vertical load is negligible;
3. Whether the coupling between the longitudinal and lateral dynamics can be considered.  
Methods of dealing with these problems are directly related to the operation conditions. Therefore, the naturalistic field operational test (FOT) was adopted to study the working conditions of autonomous vehicles [30]. The natural driving data was collected in Chongqing, China.  
As shown in Figure 1, the trajectory was about 30 km, including highways, urban expressways, urban common roads, and several roundabouts. Ten experienced Chinese drivers (six males and four females) participated in the experiment. The participants were required to drive normally according to their habits. Each participant drove during working hours and off hours respectively. A total of 20 sections of driving data were recorded.  
Normally, the motion planner uses the vehicle state, i.e., longitudinal speed, yaw rate, and the lateral acceleration as its inputs, and controls the vehicle using longitudinal acceleration and the steering wheel angle. Therefore, these signals were selected to statistically analyze the operating range, and the results are shown in Figure 2.  
As shown in Figure 2, the 3\(\sigma\) boundary of the vehicle longitudinal acceleration was between \([-2.9,\ 2.7]\) m/s\({}^{2}\) and decreased gradually with increases of the vehicle speed. The 3\(\sigma\) boundary of the lateral acceleration was between \([-3.9,\ 3.7]\) m/s\({}^{2}\) and also decreased with the increase of the vehicle speed. When the vehicle speed was more than 10 m/s, the longitudinal and lateral acceleration was less than 2.2 m/s\({}^{2}\). The absolute steering angle of the front tire was smaller than 31 degree, and the absolute yaw rate was smaller than 46 degree/s. To ensure the stability and comfort of driving, the steering angle of the front tire decreased with the increase of the vehicle speed, as did the lateral acceleration and yaw rate.  
Figure 1: Trajectory for collection of natural driving data.  
Based on the statistical results above, the following conclusions were drawn for modelling of vehicle dynamics under normal driving:  
1. Because the absolute value of the lateral acceleration was less than 4 m/s\({}^{2}\), which lies in the linear region of the tires, the vertical load transfer along the lateral direction can be ignored [28].
2. The longitudinal acceleration was less than 2.2 m/s\({}^{2}\) when the vehicle speed was more than 10 m/s, so the vertical load transfer along the longitudinal direction can be ignored [29].  
The steering angle of the front tire varied widely, especially when the vehicle speed was less than 10 m/s. When the steering angle was large, the lateral force of the front tire produced a large component in the longitudinal direction and the longitudinal driving force also generated a component in the lateral direction of the vehicle. This resulted in coupling between the longitudinal and lateral dynamics.