## 2 Estimation of Sideslip Angle
### The 7-DOF Nonlinear Vehicle Dynamic Model  
To fully reflect the state information of a driving vehicle, we built a 7-DOF nonlinear vehicle dynamic model involving the vehicle longitudinal movement, lateral movement, and yawing in the tire model (Figure 2). It is supposed that the center-of-mass of the vehicle is superposed with the origin of its driving coordinates, and the action of the steering system is ignored. Let the direction of its longitudinal motion be X-axis, the forward direction be  
Figure 1: The overall structure of the vehicle dynamics stability controller.  
positive; let the Y-axis pass the center-of-mass, the leftward direction be positive; let the torque and rolling angle on the horizontal plane pointing to the anticlockwise direction as positive.  
The longitudinal movement, lateral movement and yawing of the vehicle are shown below.  
\[\left\{\begin{array}{l}m(\overset{\bullet}{u}-\omega_{r}\cdot v)=\sum\limits_{ i=1}^{2}F_{xi}\cos\delta-\sum\limits_{i=1}^{2}F_{yi}\sin\delta+F_{x3}+F_{x4}\\ m(\overset{\bullet}{v}+\omega_{r}\cdot u)=\sum\limits_{i=1}^{2}F_{xi}\sin \delta+\sum\limits_{i=1}^{2}F_{yi}\cos\delta+F_{y3}+F_{y4}\\ l_{z}\cdot\overset{\bullet}{\omega_{r}}&=\left[\sum\limits_{i=1}^{2}F_{xi}\sin \delta+\sum\limits_{i=1}^{2}F_{yi}\cos\delta\right]a+(F_{x4}-F_{x3})t_{r}\\ &+\left[\sum\limits_{i=1}^{2}(-1)^{i}F_{xi}\cos\delta+\sum\limits_{i=1}^{2}(- 1)^{i}F_{yi}\sin\delta\right]t_{f}-(F_{y3}+F_{y4})b\end{array}\right.. \tag{1}\]  
The accelerations in the body coordinate system are converted to the accelerations and sideslip angle in the inertia coordinate system:  
\[\left\{\begin{array}{l}a_{x}=\overset{\bullet}{u}-v\omega_{r}\\ a_{y}=\overset{\bullet}{v}+u\omega_{r}\\ \beta=\arctan(\frac{v}{u})\end{array}\right., \tag{2}\]  
The sideslip angles and vertical loads of the four tires are shown below:  
\[\left\{\begin{array}{l}\alpha_{1}=\arctan(\frac{v+u\omega_{r}}{u-f\omega_{ r}})-\delta\\ \alpha_{2}=\arctan(\frac{v+u\omega_{r}}{u+f\omega_{r}})-\delta\\ \alpha_{3}=\arctan(\frac{v-f\omega_{r}}{u-f\omega_{r}})\\ \alpha_{4}=\arctan(\frac{v-f\omega_{r}}{u+f\omega_{r}})\end{array}\right., \tag{3}\]  
\[\left\{\begin{array}{l}F_{z1}=mg\frac{b}{2L}-m\overset{\bullet}{u}\frac{h_ {x}}{L}-m\overset{\bullet}{V}\frac{2h_{x}}{l_{r}}\cdot\frac{b}{L}\\ F_{z2}=mg\frac{b}{2L}-m\overset{\bullet}{u}\frac{h_{x}}{L}+m\overset{\bullet }{V}\frac{2h_{x}}{l_{r}}\cdot\frac{b}{L}\\ F_{z3}=mg\frac{a}{2L}+m\overset{\bullet}{u}\frac{h_{x}}{L}-m\overset{\bullet }{V}\frac{2h_{x}}{l_{r}}\cdot\frac{b}{L}\\ F_{z4}=mg\frac{a}{2L}+m\overset{\bullet}{u}\frac{h_{x}}{L}+m\overset{\bullet }{V}\frac{2h_{x}}{l_{r}}\cdot\frac{a}{L}\end{array}\right., \tag{4}\]  
where \(m\) is the entire vertical mass; \(a\) is the distance from the center-of-mass to the front axle; \(b\) is the distance from the center-of-mass to the rear axle; \(L\) is the between-axle distance; \(t_{f}\) and \(t_{r}\) are half of the front-wheel distance and the rear wheel distance; \(h_{g}\) is the above  
Figure 2: Seven-degrees-of-freedom vehicle dynamic model.  
ground height of the center-of-mass; \(u\) and \(v\) are the longitudinal and lateral velocity; \(a_{x}\) and \(a_{y}\) are the longitudinal and lateral accelerations in the inertia coordinate system; \(l_{z}\) is the rotary inertia around the z-axis; \(\omega_{r}\) is the yaw rate; \(\delta\) is the front-wheel steering angle; \(\beta\) is the sideslip angle; \(F_{xi}\) and \(F_{yi}\) are the longitudinal and lateral forces of tires; \(\alpha_{i}\) is the sideslip angle of tires; \(F_{zi}\) is the vertical loading of tires (where \(i=[1,2,3,4]\), \(1\), \(2\), \(3\), and \(4\) represent the left front wheel; right front wheel, left rear wheel, and right rear wheel respectively).