## 2 Intelligent Vehicle Dynamics Model  
### Vehicle Dynamics Model  
#### 2.1.1 Three-Degrees-of-Freedom Vehicle Model  
This article involves the design of observers for the driving state variables of smart electric vehicles: the yaw rate, the side slip angle of the center of mass, and the longitudinal vehicle speed. The design of the driving-state observer first needs to establish a nominal model. Therefore, this section establishes a three-degrees-of-freedom vehicle model as the nominal model of the smart electric vehicle driving-state observer, as shown in Figure 1. For modeling convenience, first make the following assumptions: (1) ignore the influence of air resistance, tire rolling resistance, etc. on the vehicle system; (2) the sideslip angles of the left and right tires of the front and rear axles are respectively equal; (3) the tire works in the linear region, that is, the tire force is proportional to the tire corner angle, and the dynamic characteristics of each tire are completely consistent; (4) ignore the role of the steering system, and the front wheel angle is an input to the vehicle system, ignore the role of the suspension system, ignore the wheel beating, and ignore the vertical movement of the vehicle body, etc.; the vehicle system moves parallel to the ground.  
The vehicle dynamics coordinate axis stipulates that the center of mass of the vehicle coincides with the origin of the dynamic coordinate system fixed on the vehicle, the \(x\)-axis coincides with the axis of symmetry of the vehicle, and the forward direction is specified; the \(y\)-axis passes through the center of mass \(o\) of the vehicle, and the left direction is specified as positive. All of the angles and moments in the coordinate system shown are positive counterclockwise, and all of the vector components are positive with the positive direction of the coordinate axis.  
In order to simplify the calculation, another assumption is made: the front wheel steering angle is small, so take it that the center of mass slip angle is small, \(\beta=\frac{v_{y}}{v_{x}}\).  
The three-degrees-of-freedom vehicle model includes the longitudinal, lateral and yaw motions of the vehicle. According to [PERSON]'s second law, the dynamic equations of the respective degrees of motion can be obtained as follows:  
Longitudinal movement:  
\[\dot{v}_{x}=v_{y}\gamma+a_{x} \tag{1}\]  
Lateral movement:  
\[m(\dot{v}_{y}+v_{x}\gamma)=F_{yf}+F_{yr} \tag{2}\]  
Yaw movement:  
\[I_{z}\dot{\gamma}=aF_{yf}-bF_{yr} \tag{3}\]  
Side slip angle of the front and rear axles:  
\[\left\{\begin{array}{c}\alpha_{f}=\beta+\frac{a_{T}}{v_{x}}-\delta\\ \alpha_{r}=\beta-\frac{b_{T}}{v_{x}}\end{array}\right. \tag{4}\]  
Front and rear axle lateral tire force:  
\[\left\{\begin{array}{c}F_{yf}=k_{1}\alpha_{f}\\ F_{yr}=k_{2}\alpha_{r}\end{array}\right. \tag{5}\]  
Figure 1: Three-degrees-of-freedom vehicle model.  
According to Formulas (1)-(5), the three-degrees-of-freedom vehicle model is obtained as follows:  
\[\left\{\begin{array}{c}\dot{\gamma}=\frac{a^{2}k_{1}+b^{2}k_{2}}{12\
u v}\gamma+ \frac{ak_{1}-k_{2}}{k_{1}}\beta-\frac{ak_{1}}{k_{2}}\delta\\ \dot{\beta}=(\frac{ak_{1}-k_{2}}{m\
u_{x}^{2}}-1)\gamma+\frac{k_{1}+k_{2}}{mv_{ x}}\beta-\frac{k_{1}}{mv_{x}}\delta\\ \dot{v}_{x}=\gamma\beta v_{x}+a_{x}\end{array}\right. \tag{6}\]  
\[a_{y}=\frac{ak_{1}-bk_{2}}{mv_{x}}\gamma+\frac{k_{1}+k_{2}}{m}\beta-\frac{k_{1} }{m}\delta \tag{7}\]  
In Formulas (1)-(7), \(\gamma\) is the yaw rate, \(\beta\) is the side slip angle of the center of mass, \(v_{x}\) is the longitudinal vehicle speed, \(v_{y}\) is the lateral vehicle speed, \(\delta\) is the front wheel angle (\(\delta=\delta_{sw}/i\), \(\delta_{sw}\) is the steering wheel angle), \(a_{x}\) is the longitudinal acceleration, \(a_{y}\) is the lateral acceleration, and the vehicle model parameters are shown in Table 1.