## 4 Simulation Experiment and Mathematical Model Validation
### Simulation of Driving Under Pressure Dynamic Model  
To analyze the variation in the pushing and pulling forces of the moving cylinder, as well as the supporting force of the top beam of support 1 and support 2 on the roof, and to verify the accuracy of the dynamic model during the driving under pressure motion of  
Figure 14: Critical values of motion for support 1 and support 2 at \(P_{22d}\) = 0.1 MPa. Figure 15: Critical values of motion for support 1 and support 2 at \(P_{22d}\) = 0.19 MPa. support 1 and support 2, the driving under pressure process and its dynamic model of the ISR are simulated through the Adams software based on the simulation model of the boundary condition simulation experiment of the driving under pressure dynamic model taking \(P_{12s}=P_{22s}=0.2\) MPa. The effective area of the piston rod in the piston chamber of the push hydraulic cylinder compared to the effective area of the piston rod in the rod chamber typically ranges from 1.2 to 2. Therefore, the ratio of the thrust to the pull force provided by the push hydraulic cylinder is also between 1.2 and 2. Taking the effective area ratio of the piston chamber piston rod to the rod chamber piston rod as 1.3, during pressurized operation of the temporary support robot, in order to achieve a more rational distribution of thrust and pull force of the push hydraulic cylinder, \(P_{12d}\) is set to 0.1 (MPa). Based on the parameters of the virtual prototype model of the TSR, and assuming that support 1 and support 2 are in uniform motion, the maximum thrust required for support 1 is calculated to be approximately 427 (KN) using the pressure formula, Equations (2) and (9). Consequently, the maximum pull force for support II during pressurized operation is about \(427/1.3\approx 328\) (KN). Further calculations using the pressure formula, Equations (5) and (10), indicate that \(P_{22d}\) is approximately 0.07 (MPa), with both \(P_{12d}\) and \(P_{22d}\) remaining within their respective value ranges. And, the critical Stribeck velocity \(v_{st}=0.01\) m/s, a step function of the displacement of the moving cylinder piston rod relative to the cylinder, is constructed in the Adams software as a driver for simulation. The displacement increases from 0 m to 0.8 m within 0 to 60 s and decreases from 0.8 m to 0 m within 70 s to 130 s. The simulation time is 140 s. The displacement curve of the moving cylinder piston rod relative to the cylinder and the displacement variation in support 1 and support 2 obtained by simulation are shown in Figure 16. Using the MATLAB R2023a software, the displacement data of support 1 and support 2 are substituted into their respective dynamic models to simulate two cases: considering the Stribeck friction theory and not considering the Stribeck friction theory (when the Stribeck friction theory is not considered, take \(f_{1}=f_{2}=0.28\)). The theoretical values of the pushing and pulling forces of the moving cylinder are obtained. The force value pushing support 1 is positive, and the force value pulling support 2 is negative. The pushing and pulling forces of the moving cylinder obtained by the Adams simulation are compared with them, and the results are shown in Figure 17.  
Figure 16: Displacement and supporting force curves of support 1 and support 2. As shown in Figure 16, the stage from 0 s to 60 s is the driving under pressure stage of support 1. In this stage, driven by the moving cylinder, the speed of support 1 gradually increases from zero within 0 s to 16 s, remains constant within 16 s to 44 s, and gradually decreases to zero within 44 s to 60 s, with a displacement of 0.8 m during the whole process. The stage from 60 s to 70 s is the transition stage. The stage from 70 s to 130 s is the driving under pressure stage of support 2. In this stage, pulled by the moving cylinder, the speed of support 2 gradually increases from zero between 70 s and 86 s, remains constant between 86 s and 114 s, and gradually decreases to zero between 114 s and 130 s, with a displacement of 0.8 m during the whole process.