## 1 Introduction
### Simplified Six Degrees of Freedom Model of the Liquid Tank Semi-Trailer for Yaw Stability Control  
In this paper, a six-axle liquid tank semi-trailer is selected for research. The schematic diagram of the single-track model of the vehicle is shown in Figure 5. Figure 5a shows the XOY plane layout of the selected vehicle, Figure 5b,c explain the roll motion between the sprung mass and the chassis when the vehicle is turning on a road. The origin point of the tractor and semi-trailer are, respectively, the intersection of the centroid cross section and its roll axis. The X-axis points to the front of the vehicle, the Y-axis points to the left of the driver, and the Z-axis points to the top of the vehicle. According to the definition in Figure 5, considering the lateral, yaw, and roll motions of the tractor and semi-trailer, the differential motion equations of the two are listed as follows. For the lateral motion of the tractor:  
\[m_{1}u_{1}\left(\dot{\beta}_{1}+\dot{\psi}_{1}\right)-m_{1s}h_{1}\dot{\psi}_{1} =F_{y1}\cos\delta+F_{y2}+F_{yH} \tag{6}\]  
where \(m_{1}\) is the mass of the tractor, kg; \(m_{1s}\) is the sprung mass of the tractor, kg; \(u_{1}\) is the longitudinal velocity of the tractor, \(\mathrm{m}/\mathrm{s}\); \(\dot{\beta}_{1}\) is the sideslip angle of the tractor's center of mass, rad; \(\dot{\psi}_{1}\) is the yaw rate of the tractor, rad\(/\mathrm{s}\); \(\dot{\phi}_{1}\) is the tractor roll angular acceleration,  
Figure 4: TruckSim 1/O setting module: (a) input module and (b) output module. Figure 3: Schematic diagram of co-simulation model for liquid tank semi-trailer. \(\mathrm{rad}/\mathrm{s}^{2}\); \(F_{y\!i}\) is the lateral force of wheel \(i(i=1,\,2,\,3)\), N; \(F_{y\!H}\) is the lateral force at the articulated point, N; \(\delta\) is the steering wheel angle, rad; and \(h_{1}\) is the distance from the tractor centroid to its roll axis, m.  
For the yaw motion of the tractor:  
\[I_{1 zz}\ddot{\psi}_{1}-I_{1 xz}\ddot{\phi}_{1}=F_{y1}a\cos\delta-F_{y2}-F_{y\!H}c \tag{7}\]  
where \(I_{1 zz}\) is the moment of inertia of the tractor around the \(Z\) axis, kg:m\({}^{2}\); \(I_{1 xz}\) is the centroid yaw-roll inertia product of the tractor, kg:m\({}^{2}\); \(a\) is the distance from the front axle of the tractor to its centroid, m; and \(c\) is the distance from the tractor centroid to the articulated point, m.  
For the roll motion of the tractor:  
\[ \begin{array}{c}\big{(}I_{1 xx}+m_{1s}h_{1}^{2}\big{)}\ddot{\phi}_{1}-I_{1 xx }\ddot{\psi}_{1}=m_{1s}gh_{1}\sin\phi_{1}m_{1s}\\ +u_{1}\Big{(}\Big{(}\dot{\beta}_{1}+\dot{\psi}_{1}\Big{)}-h_{1}\ddot{\phi}_{1} \Big{)}h_{1}-k_{r1}\phi_{1}-c_{1}\dot{\phi}_{1}+k_{12}(\phi_{2}-\phi_{1})-F_{ y\!H}h_{1c}\end{array}  \tag{8}\]  
where \(I_{1 xx}\) is the moment of inertia of the tractor around the X-axis, kg:m\({}^{2}\); \(h_{1c}\) is the distance from the articulation point to the tractor roll axis, m; \(k_{r1}\) is the roll stiffness of the tractor, N:m/rad; \(c_{1}\) is the roll damping of the tractor, N:m-s/rad; \(k_{12}\) is the coupling roll stiffness of the fifth wheel, N:m/rad. For the lateral motion of the semi-trailer:  
\[m_{2}u_{2}\Big{(}\dot{\beta}_{2}+\dot{\psi}_{2}\Big{)}-m_{2s}h_{2}\dot{\phi}_{ 2}=-F_{yT}+F_{y3} \tag{9}\]  
Figure 5: Schematic diagram of the liquid tank semi-trailer: (**a**) XOY plane diagram, (**b**) tractor roll model, and (**c**) semi-trailer roll model.