## 3 Design of the Direct Yaw Moment Control System
### Feed-Forward Control  
Based on the Laplace transformation of governing equations of the linear two-wheel vehicle model, including the additional direct yaw moment from the transverse distribution of the longitudinal tire forces, the transfer function from front steering angle and yaw moment to yaw rate can be obtained:  
\[r(s)=G_{r}^{\delta}\delta_{f}(s)+G_{r}^{M}M_{DYC}(s) \tag{17}\]  
where, each transfer function of vehicle yaw rate can be expressed as follows \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Description** & **Parameter** & **Unit** & **1 Person Unloaded** & **1 Person 40 kg** & **1 Person 80 kg** & **100 kg** \\  & & & **Unloaded** & **Loaded** & **Loaded** & **Loaded** \\ \hline Time to peak of yaw rate response & \(t_{p}\) & sec & 0.328 & 0.396 & 0.477 & 0.527 \\ Absolute steady-state side slip & \(\frac{G_{s}^{2}(0)}{G_{r}^{2}(0)}\) & deg/(m/s\({}^{2}\)) & 0.377 & 0.408 & 0.440 & 0.457 \\ angle w.r.t. lateral acc. & \(TB\) & sec & 0.124 & 0.162 & 0.210 & 0.241 \\ T\& Factor & \(\omega_{n}\) & Hz & 1.048 & 0.198 & 0.812 & 0.764 \\ Natural frequency & \(\omega_{n}\) & \(\xi\) & - & 0.651 & 0.672 & 0.703 & 0.722 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Vehicle dynamics parameters related to handling quality evaluation. Figure 4: Block diagram of the DYC system. [Transfer function with respect to front steering angle input]  
\[G_{r}^{\delta}(s)=\frac{a_{1}^{\delta}s+a_{0}^{\delta}}{s^{2}+b_{1}^{\delta}s+b_ {0}^{\delta}}, \tag{18}\]  
[Transfer function with respect to DYC input]  
\[G_{r}^{M}=\frac{a_{1}^{M}s+a_{0}^{M}}{s^{2}+b_{1}^{M}s+b_{0}^{M}}. \tag{19}\]  
Each coefficient in Equations (17) and (18) can be expressed as follows  
\[ \begin{array}{c}a_{1}^{\delta}=\frac{2l_{f}K_{f}}{l_{z}},\\ a_{0}^{\delta}=\frac{4K_{f}K_{r}}{ml_{z}^{\delta}V},\\ a_{1}^{M}=\frac{1}{2(K_{f}+K_{r})},\\ a_{0}^{M}=\frac{2(K_{f}+K_{r})}{ml_{z}V}+\frac{2(l_{f}^{2}K_{f}+l_{z}^{2}K_{r} )}{l_{z}V},\\ b_{0}^{\delta}=b_{0}^{M}=\frac{4K_{f}K_{r}l^{2}}{ml_{z}V^{2}}-\frac{2(l_{f}K_{ f}-l_{z}K_{r})}{l_{z}}.\end{array} \]  
As stated above, the primary control objective is to compensate for the deteriorated transient yaw response, i.e., small natural frequency in the loaded vehicle condition leaving the steady-state gain unaffected. The desired yaw rate response, \(r_{m}\), to the front steering angle input is determined with a first-order lag yaw rate model:  
\[\frac{r_{m}(s)}{\delta_{f}(s)}=\frac{G_{e}}{\tau_{e}s+1} \tag{20}\]  
where \(G_{e}\) is the desired yaw rate gain and \(\tau_{e}\) is the desired time constant. To reduce the order of the transfer function of the controller, the second order transfer functions Equation (17) of the yaw rate response to front steering angle and the yaw moment are approximated to first order delay systems based on the asymptote of the yaw rate gain in the low and high frequency regions:  
\[G_{r}^{\delta}(s)=\frac{a_{1}^{\delta}s+a_{0}^{\delta}}{s^{2}+b_{1}^{\delta}s+ b_{0}^{\delta}}\approx\frac{G_{r}^{\delta}(0)}{\tau_{r}^{\delta}s+1}, \tag{21}\]  
\[G_{r}^{M}(s)=\frac{a_{1}^{M}s+a_{0}^{M}}{s^{2}+b_{1}^{M}s+b_{0}^{M}}\approx \frac{G_{r}^{M}(0)}{\tau_{r}^{M}s+1}. \tag{22}\]  
The feed-forward compensator to make the vehicle yaw rate follows the desired yaw rate response as indicated in Equation (19), and can be obtained using Equations (16)-(21) as follows  
\[r_{m}(s)=G_{r}^{\delta}(s)\delta_{f}(s)+G_{r}^{M}(s)M_{DYC}(s),\]  
\[M_{FF}(s)=\bigg{(}\frac{G_{e}(\tau_{r}^{M}s+1)}{(\tau_{e}s+1)G_{r}^{M}(0)}- \frac{G_{r}^{\delta}(0)}{G_{r}^{M}(0)}\frac{(\tau_{r}^{M}s+1)}{(\tau_{r}^{ \delta}s+1)}\bigg{)}\delta_{f}(s). \tag{23}\]  
Next, based on the analysis in the previous study [7], the control law in Equation (22) can be further simplified as the time constant of the yaw rate response to yaw moment \(\tau_{r}^{M}\), which is approximately equal to the time constant of the yaw rate response to steering angle in the unloaded condition \(\tau_{r}^{\delta 0}\), i.e., \(\tau_{e}=\tau_{r}^{\delta 0}\approx\tau_{r}^{M}\).