# Integrated Adaptive Steering Stability Control for Ground Vehicle with Actuator Saturations
## 2 Problem Formulation  
During a turning manoeuvre by a vehicle, both yaw movement and roll movement are brought about; a 6-DOF turning dynamics model considering heave and roll movement of the sprung mass, left and right hop movement of the unsprung mass, and lateral and yaw movement of the whole vehicle is established in Figure 1, where \(z_{s}\) and \(\theta\) are the vertical displacement and roll angle, respectively, of the sprung mass, \(z_{u\!rl}\) and \(z_{u\!rr}\) are the respective hop of the left and right unsprung mass, \(z_{r\!rl}\) and \(z_{r\!r}\) are the respective vertical inputs of the left and right road, \(\beta\) and \(\gamma\) are the respective slip angle and yaw rate of the whole vehicle, \(m_{s}\) is the sprung mass, \(v\) is the vehicle velocity, \(F_{y}\) is the total lateral force of the four tires, \(h_{g}\) is the height of CG, \(k_{s}\) and \(c_{s}\) are the stiffness and damping coefficient of the unilateral (left or right, hereinafter) suspension springs and dampers, respectively, \(k_{w}\) and \(c_{w}\) are the respective stiffness and damping coefficient of the unilateral tire springs and damppers, \(m_{w}\) is the unilateral mass of the unsprung mass, \(F_{yl}\) and \(F_{yr}\) are the respective lateral force of left and right tires, \(d\) is the half-tread, \(\alpha_{f}\) and \(\alpha_{r}\) are the respective slip angles of the front and rear wheels, \(\delta_{f}\) is the steering angle of front wheel, \(a\) and \(b\) are the distance from the CG of the sprung mass to the front axle and rear axle, respectively, \(u_{z}\) and \(u_{\theta}\) are respectively the equivalent vehicle body vertical force and roll moment, which are essentially determined by \(u_{l}\) and \(u_{r}\), and \(u_{l}\) and \(u_{r}\) are the respective input forces of the left and right active suspensions. Conventionally, the control input forces \(u_{l}\) and \(u_{r}\) can be generated by linear electric motors. The corresponding dynamics can be formulated as follows:  
\[m_{s}z_{s}+F_{sdl}+F_{sdr}+F_{ssl}+F_{ssr}=u_{z} \tag{1}\]  
\[I_{x}\ddot{\theta}+d(F_{sdl}+F_{ssl})-d(F_{sdr}+F_{ssr})=u_{\theta}+F_{y}h_{g} \tag{2}\]  
\[m_{w}\underline{u}_{val}-F_{sdl}-F_{ssl}+F_{uml}+F_{usl}=-u_{l} \tag{3}\]  
\[m_{w}\underline{u}_{uv}-F_{sdr}-F_{ssr}+F_{uml}+F_{uwr}=-u_{r} \tag{4}\]  
\[\dot{\beta}=-\frac{2(c_{f}+c_{r})}{mv}\beta+\left(\frac{2(bc_{r}-ac_{f})}{mv^{ 2}}-1\right)\gamma+\frac{2c_{f}}{mv}\delta_{f} \tag{5}\]  
\[\dot{\gamma}=\frac{2(bc_{r}-ac_{f})}{l_{z}}\beta-\frac{2(a^{2}c_{f}+b^{2}c_{r })}{l_{z}v}\gamma+\frac{2 ac_{f}}{l_{z}}\delta_{f}+\frac{1}{l_{z}}M \tag{6}\]  
where  
\[\left\{ \begin{array}{l}F_{uml}=c_{w}(\dot{z}_{sml}-\dot{z}_{rl})\\ F_{udr}=c_{w}(\dot{z}_{smr}-\dot{z}_{rr})\\ F_{usl}=k_{w}(z_{sml}-z_{rl})\\ F_{usr}=k_{w}(z_{smr}-z_{rl})\end{array} \right.,\quad\left\{ \begin{array}{l }F_{sdl}=c_{s}\Delta\dot{y}_{l}\\ F_{sdr}=c_{s}\Delta\dot{y}_{l}\\ F_{ssl}=k_{s}\Delta y_{l}\\ F_{ssr}=k_{s}\Delta y_{r}\end{array} \right.,\quad\left\{ \begin{array}{l} \Delta y_{l}=z_{s}+d\sin\theta-z_{wl}\\ \Delta y_{r}=z_{s}-d\sin\theta-z_{wr}\\ \Delta y_{l}=\dot{z}_{s}+d\theta\cos\theta-\dot{z}_{wl}\\ \Delta y_{r}=\dot{z}_{s}-d\theta\cos\theta-\dot{z}_{wr}\end{array} \right.,\]  
\[F_{y}=F_{yr}\cos\delta_{f}+F_{yr}=2c_{f}\alpha_{f}\cos\delta_{f}+2c_{r}\alpha _{r},\]  
\[K=\frac{m}{2L^{2}}(\frac{b}{c_{f}}-\frac{a}{c_{r}}),\quad L=a+b,\quad m=m_{s}+ 2m_{w},\]  
\[\left\{ \begin{array}{l}u_{l}+u_{r}=u_{z}\\ u_{l}d-u_{r}d=u_{\theta}\end{array} \right.,\quad\left\{ \begin{array}{l}\alpha _{f}=\delta_{f}-\frac{\alpha_{f}}{v}-\beta\\ \alpha_{r}=\frac{\alpha_{r}}{v}-\beta\end{array} \right.,\]  
where \(F_{uml}\) and \(F_{udr}\) are the respective damper forces of the left tire and right tire, \(F_{usl}\) and \(F_{usr}\) are the respective spring forces of the left tire and right tire, \(F_{sdl}\) and \(F_{sdr}\) are the  
Figure 1: Vehicle model.