# Analytical Investigation of Vertical Force Control in In-Wheel Motors for Enhanced Ride Comfort
## 2 Dynamics Models and Methods
### Vehicle Dynamics Model  
The ride-blending controller leverages the construction constraints of typical suspensions, which have rotational centers for both roll and pitch movements in the longitudinal direction. This rotational movement allows the suspension to transfer some of the tire's driving force into a vertical direction. The vertical component of this driving force can then be utilized to control the dynamics of the vehicle's sprung mass. To examine the impact of controlling the vertical component of the driving force on the vehicle's dynamics, a half-vehicle model is introduced. This model includes longitudinal and vertical dynamics, as illustrated in Figure 1.  
Based on Figure 1, point \(G\) represents the center of gravity of the sprung mass, while points A and B are the positions of the shafts connecting the wheels to the sprung mass. The forces from the wheels can be transmitted to the sprung mass through these points. Additionally, the sprung mass is supported by the suspension system at both the front and rear wheels. For simplicity, the wheels are assumed to be rigid bodies, meaning their positions, \(z_{f}\) and \(z_{r}\), equal the elevation of the road profile. The free body diagram in Figure 2a shows that  
\[A_{x}=-F_{xf} \tag{1}\]  
\[B_{x}=F_{xr} \tag{2}\]  
where \(F_{xf}\) is additional front longitudinal drive force and \(F_{xr}\) is additional rear longitudinal drive force. Figure 1: A half-vehicle dynamics model. Figure 2: Free body diagram of the half-vehicle model: (**a**) free body diagram of front and rear wheels; (**b**) free body diagram of sprung mass. Also, based on Figure 2a, the vertical forces in terms of the tire's driving forces can be presented as follows [1; 3; 31; 32]:  
\[A_{z}=-F_{xf}tan\psi_{f}=-F_{xf}\frac{e_{f}}{d_{f}} \tag{3}\]  
\[B_{z}=F_{xt}tan\psi_{r}=F_{xt}\frac{e_{r}}{d_{r}} \tag{4}\]  
where \(A_{z}\) is front vertical force from the front longitudinal drive force, \(B_{z}\) is rear vertical force from the rear longitudinal drive force, \(e_{f}\) is the vertical distance from the ground to point \(A\), \(e_{r}\) is the vertical distance from the ground to point \(B\), \(d_{f}\) is the horizontal distance from the front wheel to point \(A\), and \(d_{r}\) is the horizontal distance from the rear wheel to point \(B\). Based on Figure 2b, the vehicle dynamics of the sprung mass are  
\[m_{s}\tilde{z}_{b}=F_{sf}+F_{sr}+F_{df}+F_{dr}+A_{z}+B_{z} \tag{5}\]  
\[ \begin{array}{c}I_{z}\ddot{\theta}=\left(-b\left(F_{sf}+F_{df}\right)+c\left( F_{sr}+F_{dr}\right)\right.\\ \left.+\left(h-e_{f}\right)A_{x}-\left(b-d_{f}\right)A_{z}-\left(h-e_{r}\right) B_{x}+\left(c-d_{r}\right)B_{z}\right)\end{array}  \tag{6}\]  
where \(z_{b}\) is the vertical displacement of sprung mass, \(\theta\) is the pitch angel of sprung mass, \(m_{s}\) is the sprung mass, \(I_{z}\) is the pitch inertia of the sprung mass, \(F_{sf}\) and \(F_{sr}\) are front and rear spring forces from the suspension, \(F_{df}\) and \(F_{dr}\) are front and rear damping forces from the suspension, and \(b\), \(c\), \(d_{f}\), \(d_{r}\), and \(h\) are the distance according to Figure 2b. The suspension forces are defined as follows:  
\[F_{sf}=k_{f}\Big{(}z_{f}-z_{sf}\Big{)}=k_{f}\Big{(}z_{f}-\left(z_{b}-b\theta \right)\Big{)} \tag{7}\]  
\[F_{sr}=\ k_{r}(z_{r}-z_{sr})=\ k_{r}(z_{r}-\left(z_{b}+c\theta\right)) \tag{8}\]  
\[F_{df}=\ c_{f}\Big{(}\dot{z}_{f}-\dot{z}_{sf}\Big{)}=\ c_{f}\Big{(}\dot{z}_{f}- \left(\dot{z}_{b}-b\dot{\theta}\right)\Big{)} \tag{9}\]  
\[F_{dr}=\ c_{r}\big{(}\dot{z}_{r}-\dot{z}_{sr}\big{)}=\ c_{r}\Big{(}\dot{z}_{r}- \left(\dot{z}_{b}+c\dot{\theta}\right)\Big{)} \tag{10}\]  
where \(k_{f}\) and \(k_{r}\) are the front and rear spring stiffness, and \(c_{f}\) and \(c_{r}\) are the front and rear damping coefficients. For the case where there is no additional front and rear longitudinal drive force, \(F_{xf}=F_{xt}=0\), there is no vertical component of the driving force; \(A_{z}=B_{z}=0\). This system is an uncontrolled or open-loop system.