# Analytical Investigation of Vertical Force Control in In-Wheel Motors for Enhanced Ride Comfort
## 2 Dynamics Models and Methods
### Vehicle Dynamics Model
The dynamics model of the system is determined as follows: substitute Equations (1)-(4) and (7)-(10) into Equations (5) and (6), and then the vehicle dynamics of the sprung mass are defined by  
\[\begin{array}{c}\tilde{z}_{b}=-\frac{k_{f}+k_{r}}{m_{s}}z_{b}-\frac{c_{f}+c _{r}}{m_{s}}z_{b}+\frac{bk_{f}-ck_{r}}{m_{s}}\theta+\frac{bc_{f}-cc_{r}}{m_{s} }\ \dot{\theta}\\ \ \ \ \ \ \  
where \(A_{ol}\), \(B_{ol}\), \(C_{ol}\), and \(D_{ol}\) are defined in Appendix A, Equation (A2), and \(x\) and \(u\) are defined as follows:  
\[ \begin{array}{l}x=\begin{bmatrix}z_{b}&\dot{z}_{b}&\theta&\dot{\theta}\end{bmatrix} ^{T},\\ u=\begin{bmatrix}z_{f}&z_{r}&\dot{z}_{f}&\dot{z}_{r}\end{bmatrix}^{T}\end{array}  \tag{14}\]  
The system is a multiple-input single-output (MISO) system. Determining the transfer function of the input or road input to the output or the vertical acceleration of the sprung mass is not easy in this case due to the MISO nature of the system. However, given the characteristics of the system and its linearity, the transfer function of the road input to the vertical acceleration of the sprung mass can be determined. The inputs \(z_{f}\), \(z_{r}\), \(\dot{z}_{f}\), and \(\dot{z}_{r}\) have the relationships shown in Figure 3. The inputs \(\dot{z}_{f}\) and \(\dot{z}_{r}\) are the derivatives of the inputs, \(z_{f}\) and \(z_{r}\) and the input \(z_{r}\) is the input \(z_{f}\) with a delay time. Moreover, the block diagram of the vehicle system based on the MISO system principle, as shown in Figure 4a, can be rewritten using the linear combination principle, as shown in Figure 4b. Thus, the transfer function of the road input to the vertical acceleration of the sprung mass is given in Equations (15)-(18). \[\frac{\dot{Z}_{b,ol}(s)}{Road(s)}=G_{1,ol}(s)_{z_{f}\to\dot{z}_{b}}+G_{2,ol}(s)_ {z_{r}\to\dot{z}_{b}}+G_{3,ol}(s)_{\dot{z}_{f}\to\dot{z}_{b}}+G_{4,ol}(s)_{ \dot{z}_{r}\to\dot{z}_{b}} \tag{15}\]  
\[\frac{\dot{Z}_{b,ol}(s)}{Road(s)}=\frac{\dot{Z}_{b,ol}(s)}{Z_{f}(s)}+\frac{ \dot{Z}_{b,ol}(s)}{Z_{r}(s)}+\frac{\dot{Z}_{b,ol}(s)}{\dot{Z}_{f}(s)}+\frac{ \dot{Z}_{b,ol}(s)}{\dot{Z}_{r}(s)} \tag{16}\]  
\[\frac{\dot{Z}_{b,ol}(s)}{Road(s)}=\frac{\dot{Z}_{b,ol}(s)}{Z_{f}(s)}+\frac{e^{ -\tau_{S}}\dot{Z}_{b,ol}(s)}{Z_{r}(s)}+\frac{s\dot{Z}_{b,ol}(s)}{\dot{Z}_{f}(s )}+\frac{e^{-\tau_{S}}\dot{Z}_{b,ol}(s)}{\dot{Z}_{r}(s)} \tag{17}\]  
\[\frac{\ddot{Z}_{b,ol}(s)}{Road(s)}=\frac{s\dot{Z}_{b,ol}(s)}{Road(s)}=\frac{s \dot{Z}_{b,ol}(s)}{Z_{f}(s)}+\frac{e^{-\tau_{S}}\dot{Z}_{b,ol}(s)}{Z_{r}(s)}+ \frac{s^{2}\dot{Z}_{b,ol}(s)}{\dot{Z}_{f}(s)}+\frac{e^{-\tau_{S}}s^{2}\dot{Z}_{ b,ol}(s)}{\dot{Z}_{r}(s)} \tag{18}\]  
where \(Road(s)\) is the Laplace transform of road input, \(e^{-\tau_{S}}\) is the Laplace transform of exact delay, and \(\tau\) is the delay time. Moreover, each transfer function of Equation (15) can be determined by  
\[\frac{\dot{Z}_{b,ol}(s)}{Z_{f}(s)}=C(sI-A_{ol})^{-1}B_{ol}(;,1)+D(;,1) \tag{19}\]  
\[\frac{\dot{Z}_{b,ol}(s)}{Z_{r}(s)}=C(sI-A_{ol})^{-1}B_{ol}(;,2)+D(;,2) \tag{20}\]  
\[\frac{\dot{Z}_{b,ol}(s)}{\dot{Z}_{f}(s)}=C(sI-A_{ol})^{-1}B_{ol}(;,3)+D(;,3) \tag{21}\]  
\[\frac{\dot{Z}_{b,ol}(s)}{\dot{Z}_{r}(s)}=C(sI-A_{ol})^{-1}B_{ol}(;,4)+D(;,4) \tag{22}\]  
Using the parameters in Table 1, substitute them into Equations (14) and (19)-(22).