## 2 Vehicle and Tire Model
### Vehicle Model  
The object of research is an electric vehicle with distributed drive, where the four wheels are independently controllable and it is easy to obtain the torque and speed information on each wheel, so the differential speed control of the wheels should be considered. In order to meet practical research needs and real-time computing, the vehicle model is reasonably simplified to consider movements in the longitudinal, lateral, yaw, and four-wheel rotation directions. The seven-degree-of-freedom vehicle dynamics model developed in this paper is shown in Figure 1 and assumes that:  
1. The vehicle model's centroid coincides with the original point of the vehicle coordinate system;2. Ignore the freedom of the vehicle in pitch, roll, and vertical; 3. Make the suspension a rigid body and the drivetrain a linear system to simplify the system; 4. Neglecting the influence of longitudinal friction resistance on state estimation.  
Equations for 7-DOF vehicle dynamics are presented below:  
\[\dot{v}_{x}=rv_{y}+\frac{1}{m}\Big{(}F_{xfl}cos\delta_{fl}+F_{xfr}cos\delta_{ fr}-F_{yfl}sin\delta_{fl}-F_{yffr}sin\delta_{fr}+F_{xrl}+F_{xrr}\Big{)} \tag{1}\]  
\[\dot{v}_{y}=-rv_{x}+\frac{1}{m}\Big{(}F_{xfl}sin\delta_{fl}+F_{xfr}sin\delta_{ fr}+F_{yfl}cos\delta_{fl}+F_{yffr}cos\delta_{fr}+F_{yrl}+F_{yrr}\Big{)} \tag{2}\]  
\[\dot{r}=\frac{1}{I_{\omega}}\Big{(}T_{dfl}-T_{bfl}-R_{0}F_{xfl}\Big{)} \tag{4}\]  
\[\dot{\omega}_{fr}=\frac{1}{\omega}\Big{(}T_{dfr}-T_{bfr}-R_{0}F_{xrf}\Big{)} \tag{5}\]  
\[\dot{\omega}_{rl}=\frac{1}{\omega}(T_{drl}-T_{brl}-R_{0}F_{xrl}) \tag{6}\]  
\[\dot{\omega}_{rr}=\frac{1}{I_{\omega}}(T_{drr}-T_{brr}-R_{0}F_{xrr}) \tag{7}\]  
where \(v_{x}\) and \(v_{y}\) are respectively the longitudinal speed and lateral speed, \(r\) is the yaw rate, \(\beta\) is the sideslip angle, \(a\) and \(b\) are respectively the front and rear axle distances from the vehicle's mass center, \(B_{f}\) and \(B_{r}\) are the wheelbases of the front and rear wheels of a vehicle, respectively, \(m\) is the vehicle mass, \(I_{z}\) represents the moment of inertia for the vehicle body around the \(Z\)-axis, \(R_{0}\) is the radius of wheel rotation, \(\delta_{ij}\) is the front wheel angle, \(\omega_{ij}\) is the wheel rotation angular velocity, \(I_{\omega}\) indicates the moment of inertia for each wheel, \(F_{xij}\) and \(F_{yij}\) are respectively, longitudinal and lateral forces of tires, \(T_{bij}\) and \(T_{dij}\) respectively,  
Figure 1: 7-DOF vehicle dynamics model.  
represent tire braking torque and driving torque (\(ij=fI,fr,rl\) and \(rr\), which respectively indicate positions of front left, front right, rear left, and rear right).  
To get the observation equations, \(a_{x}\) and \(a_{y}\) are obtained by deforming Equations (1) and (2) according to [PERSON]'s law, respectively, to represent vehicular acceleration in the longitudinal and lateral directions:  
\[\begin{array}{l}a_{x}=\frac{1}{m}\left(F_{xfl}cos\delta_{fl}+F_{xfr}cos\delta _{fr}-F_{yfl}sin\delta_{fl}-F_{yfr}sin\delta_{fr}+F_{xrl}+F_{xrr}\right)\\ a_{y}=\frac{1}{m}\left(F_{xfl}sin\delta_{fl}+F_{xfr}sin\delta_{fr}+F_{yfl}cos \delta_{fl}+F_{yfr}cos\delta_{fr}+F_{yfl}+F_{yrr}\right)\end{array} \tag{8}\]  
From the above equations, the state and measurement equations for the state estimator are expressed as standard state equations, as shown in Equation (9):  
\[\begin{pmatrix}x_{k}=f(x_{k-1},u_{k-1})+q_{k-1}\\ y_{k}=h(x_{k},u_{k})+r_{k}\end{pmatrix} \tag{9}\]  
where the system state vector is \(x=\left[v_{x},v_{y},r,\omega_{fl},\omega_{fr},\omega_{rl},\omega_{rr}\right]^{T}\), the measurement vector is \(y=\left[a_{x},a_{y},r,\omega_{fl},\omega_{fr},\omega_{rl},\omega_{rr}\right]^ {T}\), the input vector is \(u=\left[\delta,\omega_{ij}\right]^{T}\), \(q_{k}eN(0,Q_{k})\) and \(r_{k}eN(0,\tilde{R}_{k})\) are respectively process noise and measurement noise. To conform the signal patterns of model and sensor noises in real environments, it is assumed that the system noise is heavy-tailed non-Gaussian noise that satisfy the mixed Gaussian distribution.