## 2 Mathematical Model of the Pressure Control System for the Tractor Electronic Hydraulic Hitch
### Analysis of Kinematics of the Tractor Electronic Hydraulic Hitch  
The structure of the tractor electronic hydraulic hitch is shown in Figure 1. The triangle ANC structure is composed of hydraulic cylinder AC and lifting arm ND. The angle of lifting arm ND is changed by the telescoping hydraulic cylinder. The quadrilateral structure NDEB is composed of lower pull rod BV, lifting rod de and lifting arm ND. The angle of lifting arm ND is transmitted to the pull-down rod BV to drive the lower pull rod BV to rotate. Another quadrilateral mechanism ABCD is composed of the upper pull rod mg, pull-down rod BV and implemented so as to lift and lower the plough. The vertical line passing through point B as the horizon is recorded as the y-axis, positive upward; the horizon is recorded as the x-axis, positive left; the intersection point O is recorded as the coordinate origin. In the triangular ANC, \(l\) represents the length of two points, and the coordinates of each point are as follows:  
\[\left\{ \begin{array}{ll}x_{\rm N}-l_{\rm NC}\cos(\alpha_{\rm C})&=x_{\rm A}-l _{\rm AC}\cos(\alpha_{\rm AC})\\ y_{\rm N}+l_{\rm NC}\sin(\alpha_{\rm C})&=y_{\rm A}+l_{\rm AC}\sin(\alpha_{\rm AC })\end{array} \right. \tag{1}\]  
In the quadrilateral NDEB, the coordinates of each point are as follows:  
\[\left\{ \begin{array}{ll}x_{\rm D}-l_{\rm DE}\cos(\beta_{\rm E})&=x_{\rm B}-l _{\rm BE}\cos(\alpha_{\rm V})\\ y_{\rm D}+l_{\rm DE}\sin(\beta_{\rm E})&=y_{\rm B}+l_{\rm BE}\sin(\alpha_{\rm V })\end{array} \right. \tag{2}\]  
Figure 1: The structure of the tractor electronic hydraulic hitch: (**a**) whole part; (**b**) magnified part. Similarly, in the quadrilateral MGVB, the coordinates of each point are as follows:  
\[\left\{ \begin{array}{l}x_{\text{M}}-l_{\text{MG}}\cos(\alpha_{\text{G}})-l_{ \text{GV}}\cos(\beta_{\text{V}})=x_{\text{B}}-l_{\text{BV}}\cos(\alpha_{\text{V}} )\\ y_{\text{M}}-l_{\text{MG}}\sin(\alpha_{\text{G}})-l_{\text{GV}}\sin(\beta_{\text{V} })=y_{\text{B}}+l_{\text{BV}}\sin(\alpha_{\text{V}})\end{array} \right. \tag{3}\]  
The coordinates (\(x_{\text{W}},y_{\text{W}}\)) of the plough center of mass point W, the coordinates (\(x_{\text{U}}\), \(y_{\text{U}}\)) of the axle center of land wheel point U, and the coordinates (\(x_{\text{P}}\), \(y_{\text{P}}\)) of the tip of the plough in the middle point P in the x-o-y coordinate system were defined as follows, respectively:  
\[\left\{ \begin{array}{l}x_{\text{W}}=x_{\text{G}}-l_{\text{GW}}\cos(\beta_{ \text{V}}-\beta_{\text{W}})\\ y_{\text{W}}=y_{\text{G}}-l_{\text{GW}}\sin(\beta_{\text{V}}-\beta_{\text{W}}) \\ x_{\text{U}}=x_{\text{G}}-l_{\text{GU}}\cos(\beta_{\text{V}}-\beta_{\text{U}}) \\ y_{\text{U}}=y_{\text{G}}-l_{\text{GU}}\sin(\beta_{\text{V}}-\beta_{\text{U}}) \\ x_{\text{P}}=x_{\text{G}}-l_{\text{GP}}\cos(\beta_{\text{V}}-\beta_{\text{P}}) \\ y_{\text{P}}=y_{\text{G}}-l_{\text{GP}}\sin(\beta_{\text{V}}-\beta_{\text{P}}) \end{array} \right. \tag{4}\]  
In accordance with the geometric relationship shown in Figure 1, the coordinates (\(x_{\text{Q}}\), \(y_{\text{Q}}\)) of the velocity instantaneous center point Q in the x-o-y coordinate system were defined as follows:  
\[\left\{ \begin{array}{l}x_{\text{Q}}=\frac{y_{\text{V}}-y_{\text{G}}+k_{1}x_ {\text{G}}-k_{2}x_{\text{V}}}{k_{1}-k_{2}}\\ y_{\text{Q}}=y_{\text{V}}+k_{2}(x_{\text{Q}}-x_{\text{V}})\end{array} \right. \tag{5}\]  
where \(k_{1}=\frac{y_{\text{M}}-y_{\text{G}}}{x_{\text{M}}-x_{\text{G}}},k_{2}=\frac {y_{\text{B}}-y_{\text{V}}}{x_{\text{B}}-x_{\text{V}}}\). The turning speed of the plough's center of mass could be calculated as follows:  
\[\dot{\alpha}_{\text{W}}=\frac{l_{\text{MC}}\dot{\alpha}_{\text{G}}}{l_{\text{ CQ}}} \tag{6}\]  
where \(\alpha_{\text{w}}\) is the turning angle of the plough's center of mass. When tractor ploughing operation is approximately stable, the road roughness can be recorded as Zg (t) [32], and it can be obtained from Figure 1 that the land wheel settlement \(Z_{\text{R}}\) is as follows:  
\[\left\{