## 4 Yaw Stability Control Method for the Liquid Tank Semi-Trailer and Its Implementation
### General Architecture of Yaw Stability Control Method for the Liquid Tank Semi-Trailer
\begin{array}{l}\Delta M_{1}=F_{x11}.\frac{t_{y1}}{2}\\ \Delta M_{2}=(F_{x14}+F_{x15}+F_{x16}).\frac{t_{y3}}{2}\end{array} \right. \tag{23}\]  
where \(F_{x11}\) is the braking force applied to the left wheel of the first axle of the tractor; \(F_{x14}\), \(F_{x15}\), \(F_{x16}\) are the braking forces applied to the left wheels of the fourth, fifth, and sixth axles of the semi-trailer, respectively; and \(t_{w1}\), \(t_{w3}\) are the first axle base of the tractor and the fourth, fifth, and sixth axle bases of the semi-trailer, respectively. The cornering stiffness of the tire will change due to the change in the vertical load of the tire, and the change in the cornering stiffness of the tire will affect the yaw stability of the vehicle. Assuming that the moment of the braking wheel when the tire is not locked is approximately proportional to its vertical load, the braking force applied by the left wheel of the first axle of the tractor and the braking force distribution formula of the left wheel of the fourth, fifth, and sixth axles of the semi-trailer can be expressed, respectively, as follows:  
\[F_{xfl}=\frac{2\Delta M_{1}}{t_{w1}} \tag{24}\]  
\[\left\{ \begin{array}{l}F_{x14}=\frac{F_{x14}}{F_{x14}+F_{x15}+F_{x16}}.\frac {2\Delta M_{2}}{t_{w3}}\\ F_{x15}=\frac{F_{x15}}{F_{x14}+F_{x15}+F_{x16}}.\frac{2\Delta M_{2}}{t_{w3}}\\ F_{x16}=\frac{F_{x16}}{F_{x14}+F_{x15}+F_{x16}}.\frac{2\Delta M_{2}}{t_{w3}} \end{array} \right. \tag{25}\]  
where \(F_{x14}\), \(F_{x15}\), \(F_{x16}\) are the vertical load of the left wheel of the fourth, fifth, and sixth axles of the semi-trailer, respectively. The braking moment of the left wheel on the first axle of the tractor and the left wheel on the fourth, fifth, and sixth axles of the semi-trailer can be expressed, respectively, as follows:  
\[T_{l1}=F_{x11}R_{1}=\frac{2\Delta M_{1}}{t_{w1}}.R_{1} \tag{26}\]  
Figure 12: Braking characteristics of different brake wheels: (**a**) relationship between yaw moment generated by braking of different wheels and longitudinal braking force and (**b**) influence of single wheel braking on vehicle steering. \[ \begin{cases}T_{14}=F_{x14}R_{3}=\frac{F_{14}}{F_{x14}+F_{x15}+F_{x16}},\frac{2\Delta M _{1}}{t_{03}}.R_{3}\\ T_{15}=F_{x15}R_{3}=\frac{F_{x15}}{F_{x14}+F_{x16}+F_{x16}}.\frac{2\Delta M_{2}}{t _{03}}.R_{3}\\ T_{16}=F_{x16}R_{3}=\frac{F_{x16}}{F_{x24}+F_{x15}+F_{x16}}.\frac{2\Delta M_{2}}{t _{03}}.R_{3}\end{cases}  \tag{27}\]  
where \(R_{1}\) is the wheel rolling radius of the first axle of the tractor; and \(R_{3}\) is the wheel rolling radius of the fourth, fifth, and sixth axles of the semi-trailer. From the previous analysis, the additional yaw moment required by the tractor and semi-trailer can be converted into the braking moment of the wheel. After that, the target braking wheel needs to be determined to generate the additional yaw moment efficiently. As shown in Figure 11(a), when the outer front wheels of the vehicle brake, the generated yaw moment gradually increases outwards; when the outer rear wheels of the vehicle brake, the yaw moment produced gradually increases and then decreases outwards, and finally gradually increases inwards; when the inner front wheels of the vehicle brake, the yaw moment generated gradually increases, then decreases inwards, and finally increases outwards; when the inner rear wheels of the vehicle brake, the yaw moment generated gradually increases inwards. As shown in Figure 11(b), when the liquid tank semi-trailer runs in a steady circular motion, the brake of the vehicle's outer front wheel will cause the vehicle to understeer, and the trend is large; the brake of the vehicle's outer rear wheel will cause the vehicle to understeer, and the trend is relatively small; the front wheel braking in the vehicle will cause the vehicle to oversteer, and the trend is relatively small; and the rear wheel braking in the vehicle will cause the vehicle to oversteer, and the trend is large.