# Hybrid Model-Based Analysis of Underground Articulated Vehicles Steering Characteristics
## 2 Dynamic Model of UAVs
### Model of Hydraulic Steering System  
The hydraulic steering system of the UAV is shown in Figure 2. The rodless cavity of the left cylinder is connected with the rod cavity of the right cylinder, and the main directional valve, which is from the M4 series of Bosch Rexroth, determines the direction of oil under pressure, which drives the extension and retraction of the cylinder to steer. Figure 2: Hydraulic steering system of UAV. It should be emphasized that the steering valve has a pressure compensation in front of the spool, which can ensure a constant pressure difference between the inlet pressure and load pressure. Therefore, the input flow of the system is only related to the open area of the spool, which is determined by the control current of spool. Then the input and output flow can be described as [22]:  
\[\left\{ \begin{array}{l}Q_{in}=f(I_{in})\\ Q_{out}=C_{d}A_{0}\sqrt{2(P_{2}-P_{0})/\rho}\end{array} \right. \tag{9}\]  
where \(I_{in}\) denotes the control current, \(C_{d}\) denotes the flow coefficient, and \(A_{0}\) is the area of the orifice, which is related to the control current. \(P_{2}\), and \(P_{0}\) are the pressure of the return oil and atmosphere, respectively; \(\rho\) is the density of hydraulic oil; and \(Q_{in}\), \(Q_{out}\) are the input and output flow of the system. The relationship of the control current and the input flow is shown in Figure 3, which is obtained from experiment data of the steering valve company, i.e., Bosch Rexroth. As shown in Figure 3, the positive current corresponds to the positive flow, which means the flow is led into the rodless cavity of the left cylinder to steer towards the right; conversely, negative current and flow means the flow is led into the right cylinder to steer towards the left. In addition, there are dead zones at \(-0.43\) to \(0.43\) A, \(-0.77\) to \(-0.80\) A, and \(0.77\) to \(0.80\) A, where the flow shows obvious nonlinear characteristics, which originate from the friction, dead zone, etc., between the valve spool and sleeve. There is a linear relationship during other intervals, and the maximum flow is \(68\) L/min. According to Figure 3, the input and output flows of the system contain two parts, respectively, and the relationship can be described as:  
\[\left\{\begin{array}{l}Q_{in}=Q_{L1}+Q_{R2},Q_{out}=Q_{L2}+Q_{R1}\ \ I_{in}>0.43\\ Q_{in}=0,Q_{out}=0\  
and C\({}_{ip}\) denotes the coefficient of internal leakage. \(P_{ij}\) (\(i\) = L, R; \(j\) = 1, 2) denotes the pressure of the rodless and rod cavities of the left and right cylinders, respectively, and can be described as:  
\[\left\{ \begin{array}{l}P_{1}=P_{L1}=P_{R2},P_{2}=P_{R1}=P_{L2},I_{in}>0.45\\ P_{1}=P_{R1}=P_{L2},P_{2}=P_{R2}=P_{L1},I_{in}<-0.45\end{array} \right. \tag{12}\]  
The force balance equation of steering cylinders can be expressed as [22]:  
\[\left\{ \begin{array}{l}F_{L}=A_{1}P_{L1}-A_{2}P_{L2}-f_{s}\dot{X}_{L}\\ F_{R}=A_{2}P_{R2}-A_{1}P_{R1}-f_{s}\dot{X}_{R}\end{array} \right. \tag{13}\]  
where \(f_{s}\) is the coefficient of dynamic friction. In addition to the above relationship of flow and force, there is also a kinematic relationship between the frames and steering cylinders [10], as illustrated in Figure 4. The angles between two cylinders and frames are represented by \(\gamma\) and \(\beta\), respectively. According to the relationship in Figure 4, the displacement of the two cylinders can be expressed as:  
\[\left\{ \begin{array}{l}X_{L}=\sqrt{OP_{1}^{2}+OR_{1}^{2}-2 OP_{1}OR_{1}\cos (\varphi_{0}+\varphi)}\\ X_{R}=\sqrt{OP_{2}^{2}+OR_{2}^{2}-2 OP_{2}OR_{2}\cos(\varphi_{0}-\varphi)} \end{array} \right. \tag{14}\]  
In addition, the angles of cylinders and frames are related to the articulation angle, which can be described as:  
\[\left\{ \begin{array}{l}\gamma_{1}+\varphi=\gamma_{2}\\ \beta_{1}-\varphi=\beta_{2}\end{array} \right. \tag{15}\]  
Based on the sine theorem, the angle of cylinders and front frame can be described as:  
\[\left\{