## 3 Dynamic Force-Thermal Follow-Up Loading Method for Multi-Stage Telescopic Wings
### Force-Thermal Follow-Up Loading System Modeling and Simulation Analysis  
To ensure that each force loading unit can apply the preset force-load curve, a force feedback control system is designed for real-time feedback control of the loading unit's force. The loading units are organized into a modular system, so an equivalent system model is established for a single loading unit. Figure 28 shows the equivalent impedance control model of the loading unit. The servo cylinder in the follow-up loading unit is simplified and equivalent to a moving slider with the mass m\({}_{d}\), with the driving input F(t) and displacement x\({}_{1}\)(t). The loading unit and the deformed wing surface are considered to be in constant contact during the loading process. k\({}_{d}\) is the equivalent stiffness of the disc spring in the loading unit, and c\({}_{d}\) is the system's equivalent damping. The theoretical model for the follow-up loading unit is established as follows:  
\[\mathrm{m_{d}\ddot{x}_{1}=F_{d}(t)-k_{d}x_{1}-c_{d}\dot{x}_{1}} \tag{17}\]  
Taking the Laplace transformation of Equation (13), the expression is obtained as follows:  
\[\mathrm{m_{d}s^{2}x_{1}(s)=F_{d}(s)-k_{d}x_{1}(s)-c_{d}sx_{1}(s)} \tag{18}\]  
Figure 27: Comparison of fitting errors. (**a**) RMSE; (**b**) R\({}^{2}\). Equation (14) represents the transfer function of the moving slider displacement x1 corresponding to the driving input force F in the deformable wing tracking loading control system, as follows:  
\[\mathrm{x_{1}(s)=\frac{F_{d}(s)}{m_{d}s^{2}+c_{d}s+k_{d}}} \tag{19}\]  
Based on the transfer function from Equation (19), the control program block diagram of the tracking loading system is provided, as shown in Figure 29. The input force \(F_{d}\) serves as the system input, and the equivalent spring force F during the tracking process is the output, which is fed back to the PID controller. The PID controller adjusts the tracking loading unit to ensure the output force follows the desired force. The output of the PID algorithm is shown in Equation (16):  
\[F(t)=K_{p}\cdot e(t)+K_{i}\int_{0}^{t}e(t)dt+K_{d}\frac{de(t)}{dt} \tag{20}\]  
where F(t) is the output of the controller; \(e(t)\) is the response error; and \(K_{p}\), \(K_{i}\), \(K_{d}\) are the coefficients for the proportional, integral, and derivative terms, respectively. According to the design scheme of the force-thermal follow-up loading platform, a mechanical system model was established and imported into ADAMS2020. The material properties of each component were defined, appropriate motion pairs and contacts were added at the joints, and the corresponding driving conditions were applied to construct the loading platform mechanical system, as shown in Figure 30. Combining the established control system model, a control system simulation program block diagram was built. The integrated simulation of the follow-up loading platform with force feedback control was conducted to analyze the loading force error and the stability of the control system. Figure 28: Impedance control model. Figure 29: Block diagram of the control system. Figure 31 shows the entire process of dynamic loading. The twelve force loading units of the dynamic loading platform are divided into four rows, with three loading units in each row. The loading units in the same row act on the same section of the wing surface. For the analysis of the loading process, the four rows of loading units are taken as an example from the side view. The dynamic loading process is as follows: In the initial state, the four rows of loading units are positioned away from the external surface of the telescoping wing. During the loading process, the first two rows of loading units apply force to the telescoping wing. As the wing extends, the resultant force from the loading units acting on the wing increases. When the wing extends to the position of the third row of loading units, the third row of loading units rises and makes contact with the external surface of the wing, applying force.