# Estimation of Intelligent Commercial Vehicle Sideslip Angle Based on Steering Torque
## 2 Materials and Methods
### Transfer Function Analysis of Vehicle Dynamics Model  
The transfer function is an important tool for modeling, analysis, and design of control systems. By establishing the transfer function between the steering wheel angle, torque, and 2-DOF vehicle model, the characteristics of the center of sideslip angle during vehicle steering can be further analyzed. The Laplace transform of Formula (18) is as follows:  
\[\left\{ \begin{array}{c}mu[s\cdot\beta(s)+\gamma(s)]=(k_{1}+k_{2})\beta(s)+ \frac{l_{f}k_{1}-l_{r}k_{1}}{u}\gamma(s)-k_{1}\delta(s)\\ I_{z}\gamma(s)=(k_{1}+k_{2})\beta(s)+\frac{l_{f}^{2}k_{1}+l_{r}^{2}k_{1}}{u} \gamma(s)-l_{f}k_{1}\delta(s)\end{array} \right. \tag{19}\]  
After simplification, the transfer function between the sideslip angle and the front wheel angle can be obtained, where the input signal is the front wheel angle and the output signal is the sideslip angle, as shown in Equation (18). \[G_{\beta\delta}(s)=\frac{\beta(s)}{\delta(s)}=\frac{Y(s)}{X(s)}=\frac{5.64 \times 10^{9}s-2.74\times 10^{10}}{6.20\times 10^{9}s^{2}+1.80\times 10^{10}s+5.2 \times 10^{10}} \tag{20}\]where  
\[\left\{ \begin{array}{l}X(s)=\ muI_{z}s^{2}-[(k_{1}+k_{2})\ I_{z}+muM]s+(k_{1}+k_{2 })M+\Big{(}l_{f}k_{1}-l_{r}k_{2}\Big{)}(mu-N)\\ Y(s)=-k_{1}\ I_{z}s+k_{1}M+l_{f}k_{1}(mu-N)\\ M=\frac{l_{f}k_{1}+l_{r}k_{2}}{u}\\ N=\frac{l_{f}k_{1}-l_{r}k_{2}}{u}\end{array} \right. \tag{21}\]  
By solving Equations (12), (13) and (20), we can obtain the transfer function between steering wheel angle and sideslip angle, as well as the transfer function relationship between steering wheel torque and sideslip angle. \[G_{1}(s)=\frac{\beta(s)}{\theta_{s}(s)}=\frac{\beta(s)}{\delta(s)}\frac{\delta (s)}{\theta_{s}(s)}=\frac{3.96\times 10^{13}s-1.92\times 10^{14}}{2.50\times 10^{9 }s^{4}+1.10\times 10^{13}s^{3}+2.80\times 10^{14}s^{2}+8.10\times 10^{14}s+2.09 \times 10^{15}} \tag{22}\]  
\[G_{2}(s)=\frac{\beta(s)}{T_{i}(s)}=\frac{\beta(s)}{T_{i}(s)}=\frac{\beta(s)}{ T_{i}(s)}\frac{T_{i}(s)}{\theta_{s}(s)}=\frac{3.96\times 10^{13}s-1.92\times 10^{14}}{ \Big{(}6.95\times 10^{7}s^{6}+2.84\times 10^{11}s^{5}+1.57\times 10^{13}s^{4}+1.80 \times 10^{15}s^{3}} \tag{23}\]  
\[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-1.39 \times 10^{18}s^{2}-4.02\times 10^{18}s-1.18\times 10^{19}\Big{)}\]  
Through the previous derivation, we obtained the transfer function between the steering wheel angle, steering wheel torque, and the sideslip. The frequency analysis of the transfer function can obtain the corresponding amplitude and frequency characteristics in Figure 4.  
In Figure 4, it can be seen that when the frequency of the steering system is in the range of \(10^{-2}\) to \(10^{-1}\) Hz, the phase in Figure 4a has a phase advance of \(180^{\circ}\), while there is a small phase lag in Figure 4b, which ranges from \(0\) to \(-19^{\circ}\). In the frequency range of \(10^{-1}\)-\(10^{0}\) Hz, there is a significant decrease in the phase of both plots, from \(158^{\circ}\) advance to \(38^{\circ}\) lag (\(158\)\(\leftarrow-38^{\circ}\)) in Figure 4a, while from \(19^{\circ}\) lag to \(200^{\circ}\) lag (\(-19\)\(\leftarrow-200^{\circ}\)) in Figure 4b. It can be seen that the phase difference in Figure 4a is larger than in Figure 4b, which is \(196^{\circ}\) and \(181^{\circ}\), respectively. Moreover, it can be seen from the amplitude characteristics that the amplitude of Figure 4a is greater than that of Figure 4b, indicating that the amplitude ratio of Figure 4a is greater than that of Figure 4b, which is \(-20.4\) dB and \(-95.4\) dB, respectively. According to Equations (22) and (23), the transfer function of the sideslip angle includes the steering angle, steering torque, and front-wheel angle parameters. In order to further explore its characteristics, the transfer function is decomposed into transfer functions based on the EHPS model and 2-DOF model for analysis.