## 4 Yaw Stability Control Method for the Liquid Tank Semi-Trailer and Its Implementation
### Wheel Slip Rate Control in Yaw Stability Control Method for the Liquid Tank Semi-Trailer  
The calculation formula of the wheel slip ratio is  
\[\lambda=\frac{u-r_{w}w}{u} \tag{28}\]  
where \(u\) is the longitudinal velocity of the tractor or semi-trailer; \(r_{w}\) is the rolling radius of the wheel; and \(w\) is the angular velocity of the wheel. Figure 13 shows that when the slip ratio is 0.15-0.20, the longitudinal adhesion coefficient and lateral adhesion coefficient are both large, and the longitudinal force and lateral force on the tire are also sufficient, so the vehicle can maintain good dynamic performance and stability. To control the vehicle tire slip rate within 0.15-0.20, when the wheel slip rate is greater than 0.20, the braking torque shall be reduced to reduce the wheel slip rate; when the wheel slip ratio is between 0.15 and 0.20, the braking torque should remain unchanged; when the wheel slip rate is lower than 0.15, the wheel braking torque shall be increased to increase the wheel slip rate. It should be noted that in the process of increasing or decreasing the braking torque, the appropriate growth rate and decline rate should be selected. When the growth rate and decline rate are too large, the slip rate will overshoot and even cause system vibration; when the values of the growth rate and the decline rate are too small, the slip rate changes slowly, and the system response is not sensitive. According to the characteristics of the wheel, the appropriate growth rate and decline rate are selected, and the corresponding braking torque can be obtained  
\[\left\{ \begin{array}{ll}T_{t+1}=T_{t}+k_{t}T_{s},&S<0.15\\ T_{t+1}=T_{t},&0.15\leq S\leq 0.2\\ T_{t+1}=T_{t}-k_{d}T_{s},&S>0.2\end{array} \right. \tag{29}\]  
where \(T_{t+1}\) is the output value of braking torque at time \(t+1\); \(T_{t}\) is the output value of braking torque at time \(t\); \(T_{s}\) is the calculation time interval, 0.005 s; \(k_{i}\) is the growth rate, which is 6000 N\(\cdot\)m/s; and \(k_{d}\) is decline rate, 6000 N\(\cdot\)m/s. Verification of the Effectiveness and Robustness of the Yaw Stability Control Method for the Liquid Tank Semi-Trailer  
To verify the effectiveness of the proposed control method, based on the liquid tank semi-trailer co-simulation model, the different vehicle state responses are compared while  
Figure 13: Relationship between longitudinal and lateral adhesion coefficient and slip ratio. taking the multi-object PID differential braking-control (multi-object), tractor yaw rate + semi-trailer yaw rate differential braking-control (\(\omega_{1}+\omega_{2}\)), tractor yaw rate + articulation angle differential braking-control (\(\omega_{1}+\theta\)), and no control methods, respectively. For the verification of the proposed control method, the double-lane change and the step-steering-angle input's working conditions are selected, the initial vehicle speed is set as 80 km/h, the road adhesion coefficient is set as 0.3, and the basic vehicle parameters can be referred to in Table 2.  
To verify the robustness of the proposed method, the robustness of the proposed control method is analyzed and verified when the multi-object PID control parameters are changed. Verification of the Effectiveness of the Yaw Stability Control Method under the Double-Lane Change Working Condition  
In this section, the double-lane change working condition simulation test is performed when the liquid tank semi-trailer is running on a low-adhesion coefficient road. The comparison diagrams of the yaw rate, the vehicle track, and the articulation angle of the tractor and semi-trailer are shown in Figures 14-16, respectively. Figure 14: Comparison of vehicle yaw rate under the double-lane change working condition: (**a**) comparison of tractor yaw rate and (**b**) comparison of semi-trailer yaw rate. Figure 15: Comparison of vehicle track under the double-lane change working condition: (**a**) comparison of tractor track and (**b**) comparison of semi-trailer track.