## 2 Mathematical Model of the Pressure Control System for the Tractor Electronic Hydraulic Hitch
### The Dynamic Model of the Tractor Electronic Hydraulic Hitch  
To simplify the analysis, the soil resistance of plough is equivalent to that of the middle plough. As shown in Figure 2, the force on the plough body could be divided into horizontal resistance and vertical resistance. The vertical resistance included the steady part of the plough body in the vertical direction and the damping force in the process of the plough body movement [33]. The total steady resistance of the plough could be calculated as follows:  
\[R=n_{\text{P}}b_{\text{P}}k_{\text{P}}n_{\text{P}} \tag{8}\]  
where \(n_{\text{P}}\) is the number of ploughs, \(k_{\text{P}}\) is the resistance coefficient; \(b_{\text{P}}\) is the width of the single plough; \(h_{\text{P}}\) is the depth of plough. The horizontal resistance was calculated as follows:  
\[R_{\text{H}}=R\cos(\varphi) \tag{9}\]where \(\varphi\) is the angle between the total steady resistance of the plough body and the horizontal line. It was assumed that the acting line of the vertical transient force of the plough body is approximately coincident with the acting line of the vertical component of the steady working resistance of the plough body. The vertical resistance of the plough body is as follows:  
\[R_{\rm V}=R\sin(\varphi)+R_{\rm H}\frac{\dot{y}_{\rm RV}}{v} \tag{10}\]  
The location of the force bearing point of the plough body is as follows:  
\[\left\{ \begin{array}{l}S_{\rm RH}\approx 0.36h_{\rm p}\\ S_{\rm RV}\approx 0.54h_{\rm p}\end{array} \right. \tag{11}\]  
As shown in Figure 3, before establishing the dynamic model of the tractor electronic hydraulic hitch, the following assumptions were made:  
(1) Ignore the shock of the wheel on the uneven soil, due to the low speed of ploughing. (2) Ignore the mass of each member bar. (3) The deformation stiffness of soil is far less than that of the implement members. Accordingly, the deformation of the plough and members is ignored. As shown in Figure 3c, the forces applied to the plough include the soil resistance of the plough body, the vertical reaction force \(\rm F_{\rm Ry}\) and the horizontal rolling resistance \(\rm F_{\rm Rx}\) of the soil-applied to the wheel, the plough gravity \(\rm m_{\rm Wg}\), the force \(\rm F_{\rm G}\) at the upper pull rod, and the horizontal \(\rm F_{\rm Vx}\) and vertical force \(\rm F_{\rm Vy}\) at the pull rod. Figure 3: Simplified force analysis for tractor electronic hydraulic hitch:(**a**) force analysis of lifting arm; (**b**) force analysis of lower pull rod; (**c**) force analysis of plough gear. Figure 2: Force analysis of plough. We are taking though as the research object. The equation of longitudinal and lateral dynamics of the plough and the equation of using D'Alembert's principle for the sum of torques around V can be written as follows:  
\[\left\{ \begin{array}{l}R_{\rm H}+F_{\rm G}\cos\alpha_{\rm G}-F_{\rm Vx}+F_{\rm Rx }+m_{\rm W}a_{\rm Wx}=0\\ F_{\rm G}\sin\alpha_{\rm G}+F_{\rm Vy}-R_{\rm V}+F_{\rm Ry}-m_{\rm W}\Big{(}g+a_ {\rm Wy}\Big{)}=0\\ R_{\rm H}(y_{\rm V}-y_{\rm P}-s_{\rm Ry_{\rm L}})-R_{\rm V}(x_{\rm V}-x_{\rm P}+s _{\rm Ry_{\rm V}})+F_{\rm Ry}(x_{\rm V}-x_{\rm U})+F_{\rm Rx}(y_{\rm V}-y_{\rm U} )\\ -m_{\rm W}\Big{(}g+a_{\rm Wy}\Big{)}(x_{\rm V}-x_{\rm W})+m_{\rm W}a_{\rm Wx}(y_ {\rm V}-y_{\rm W})-F_{\rm G}\cos(\alpha_{\rm G})(y_{\rm G}-y_{\rm V})\\ -F_{\rm G}\sin(\alpha_{\rm G})(y_{\rm G}-y_{\rm V})-(J_{\rm W}+m_{\rm W}((x_{ \rm V}-x_{\rm W})^{2}+(y_{\rm V}-y_{\rm W})^{2}))\tilde{\alpha}_{\rm W}=0\\ \end{array} \right. \tag{12}\]  
The vertical force of the land wheel could be calculated by the following formula [34; 35]:  
\[F_{\rm Ry}=\left\{ \begin{array}{l}\frac{1}{3}(3-n)(k_{\rm c}+bk_{\rm p})\sqrt {2r_{\rm R}}z_{R}^{\frac{2n+1}{2}}\ \ \ \ \ z_{R}>0\\ 0\ \ \ z_{R}\leq 0\\ \end{array} \right. \tag{13}\]  
The rolling resistance of the land wheel is approximately taken as follows:  
\[F_{\rm Rx}\approx F_{\rm Ry}f \tag{14}\]  
where \(f\) is rolling resistance coefficient. As shown in Figure 3b, the pull-down rod BV is taken as the research object.