# Robust Estimation of Vehicle Motion States Utilizing an Extended Set-Membership Filter
## 4 Calculation of Tire Forces
### Lateral Tire Force Estimator  
In order to facilitate the construction of lateral force estimator, some simplifications are made on the dual-track vehicle dynamics model mentioned in Section 2.1. This is conducive to simplifying the calculation formulas for state variables [28] with minor effect on the overall idea of vehicle dynamic behavior. Figure 4 shows the established single-track model.  
According to [PERSON]'s second law and rotational equilibrium, the vehicle's lateral motion and vertical rotation can be expressed by the following equations:  
\[\begin{cases}ma_{y}=F_{yf}\cos\delta+F_{xf}\sin\delta+F_{yr}\\ I_{z}\dot{r}=aF_{yf}\cos\delta+aF_{xf}\sin\delta-bF_{yr}\end{cases} \tag{22}\]  
where \(F_{xf}\) and \(F_{yr}\) represent the longitudinal resultant forces of two front tires and two rear tires, and \(F_{yf}\) and \(F_{yr}\) represent their lateral resultant forces, respectively.  
Based on Equation (22), the lateral forces \(F_{yf}\) and \(F_{yr}\) are rewritten as follows:  
Figure 4: Single-track vehicle dynamic model.  
\[\begin{cases}F_{yf}=\dfrac{bma_{y}+I_{z}\dot{r}-F_{xf}(a+b)\sin\delta}{(a+b)\cos \delta}\\ F_{yr}=\dfrac{ana_{y}-I_{z}\dot{r}}{a+b}\end{cases} \tag{23}\]  
With the purpose of enhancing the estimation accuracy, lateral forces \(F_{yf}\) and \(F_{yr}\) computed from Equation (23) are treated as one part of the measurement values, where they are denoted as \(F_{yf}^{\text{com}}\) and \(F_{yr}^{\text{com}}\). It is feasible because their values can be acquired by utilizing onboard sensors signals, namely, the steering wheel angle, yaw rate and lateral acceleration. As a result, the state vector and measurement vector for the lateral tire force estimator are chosen as follows.  
\[\begin{split}\text{State vector: }\mathbf{x}_{b}=[F_{yf}\ \ F_{yr}\ \ r]^{ \text{T}};\\ \text{Measurement vector: }\mathbf{z}_{b}=[F_{yf}^{\text{com}}\ \ F_{yr}^{ \text{com}}\ \ r\ \ a_{y}]^{\text{T}}.\end{split}\]  
The state space for lateral force estimator is expressed as:  
\[\begin{split}\dot{\mathbf{x}}_{b}&=\mathbf{F}_{b}\mathbf{x}_{b}+ \mathbf{u}_{b1}+\mathbf{w}_{\mathbf{b}}\\ \mathbf{z}_{b}&=\mathbf{H}_{b}\mathbf{x}_{b}+\mathbf{u}_{b2}+\mathbf{v}_{ b}\end{split} \tag{24}\]  
The corresponding state transition matrix \(\mathbf{F}_{b}\), measurement matrix \(\mathbf{H}_{b}\) and input vectors \(\mathbf{u}_{b1}\), \(\mathbf{u}_{b2}\) are as follows:  
\[\begin{split}\mathbf{F}_{b}&=\left[\dfrac{\mathbf{0}_{2 \times 3}}{\dfrac{\cos\delta\cdot a}{I_{z}}\ \ \ -\dfrac{b}{I_{z}}\ \ \ 0}\right],\mathbf{H}_{b}=\left[\dfrac{\mathbf{ \bar{\mathbf{I}}}_{3\times 3}}{\dfrac{\cos\delta}{m}\ \ \ \dfrac{1}{m}\ \ \ 0}\right],\\ \mathbf{u}_{b1}&=\left[\begin{matrix}0\ \ \ \ 0\ \ \ \dfrac{a}{I_{z}}F_{xf}\sin\delta \end{matrix}\right]^{\text{T}},\mathbf{u}_{b2}=\left[\begin{matrix}0\ \ \ \ 0\ \ \ 0\ \ \dfrac{F_{xf}\sin\delta}{m}\end{matrix}\right]^{\text{T}}\end{split}\]  
Furthermore, the aspect of the load transfer between left and right tires should also be taken into account because the lateral force of each tire are required in the robust estimation strategy. According to the proportional relationships between the vertical loads of each tire on the same axis, the lateral resultant forces can be decomposed into different components for each tire.  
\[\begin{split} F_{y1}&=F_{yf}\dfrac{F_{z1}}{(F_{z1}+F_{ z2})}\\ F_{y2}&=F_{yf}\dfrac{F_{z2}}{(F_{z1}+F_{z2})}\\ F_{y3}&=F_{yr}\dfrac{F_{z3}}{(F_{z3}+F_{z4})}\\ F_{y4}&=F_{yr}\dfrac{F_{z4}}{(F_{z3}+F_{z4})}\end{split} \tag{25}\]  
where the expressions of the loads on each tire \(F_{zi}\ \ (i=1,2,3,4)\) are shown in Equation (26), which includes the static load, longitudinally transferred load and laterally transferred load.  
\[F_{z1} =\frac{mgb}{2(a+b)}-\frac{ma_{x}h}{2(a+b)}-\frac{ma_{y}hb}{(a+b)t_{f}} \tag{26}\] \[F_{z2} =\frac{mgb}{2(a+b)}-\frac{ma_{x}h}{2(a+b)}+\frac{ma_{y}hb}{(a+b)t_{f}}\] \[F_{z3} =\frac{mga}{2(a+b)}+\frac{ma_{x}h}{2(a+b)}-\frac{ma_{y}ha}{(a+b)t_{ r}}\] \[F_{z4} =\frac{mga}{2(a+b)}+\frac{ma_{x}h}{2(a+b)}+\frac{ma_{y}ha}{(a+b)t_ {r}}\]