PID controller has the advantages of simple principle, strong robustness, and wide practicality, making it a mature and widely used control system. Therefore, this study will apply it to the optimization of 8BSM dynamic modeling (Hammoodi et al., 2020; Garai et al., 2023). In a PID control system, the output value depends on the deviation between the system set value and the system output value, the linear weighted combination of deviation integration and deviation differentiation (Phu et al., 2020). The diagram of the PID controller is Figure 1.

In Figure 1, r t is the given value. y t represents the actual output value, and e t is the control deviation. The expression between the three is Eq. 1. e t = r t − y t (1)

In addition, u t in Figure 1 represents the output signal of the PID, which is used to control the execution part of the system, as shown in Eq. 2. u t = K p * e t + K i * ∫ 0 , t e τ d τ + K d * d e t / d t (2)

In Eq. 2, K p * e t is the proportional control term. K p is the proportional gain. K i * ∫ 0 , t e τ d τ is the integral control term. K i is the integration coefficient. K d * d e t / d t is the differential control term. K d is the differential coefficient. Due to the fact that traditional PID controllers may not be able to effectively handle nonlinear systems in certain situations, it is necessary to improve traditional PID controllers. RBFNN has excellent learning and adjustment capabilities, and can overcome the adverse effects of uncertainty on system performance through online identification of the system. This solves the problems of poor robustness and limitation by precise mathematical models in traditional PID control. The output formula of RBFNN is Eq. 3 (Chen, 2022). u R B F t = ∑ i = 1 N ω i ⁡ exp − x t − c i 2 2 δ i 2 (3)

In Eq. 3, u R B F t is the output of RBFNN at time t . N is the quantity of hidden layer neurons (HLNs). ω i is the weight between the i -th HLN and the output layer. x t is the input vector. c i is the center vector of the i -th HLN. δ i is the width parameter of the i -th HLN. RBFNN, as a new type of artificial neural network, is often applied in various complex prediction problems because it can effectively find the best training data plane existing in multidimensional space (Elsisi, 2021; Bensafia et al., 2022). The specific structure diagram of RBFNN is Figure 2.

In Figure 2, the three main components of RBFNN are the input, hidden, and output layers. The input layer has several perceptual units. The hidden layer contains several hidden layer nodes, which are generally composed of Gaussian basis functions, and the Gauss function is Eq. 4. α i x = ⁡ exp − X − c i 2 2 σ i 2 , i = 1 , 2 , ⋯ , m (4)

In Eq. 4, α i x represents the output result of the i -th HLN. m represents the number of HLN. c i represents the center vector of the Gosky function corresponding to the i -th hidden layer node. σ i represents the base width of the i -th HLN. And X represents the input sample. X contains x 1 , x 2 , ⋯ x n of different dimensions. The expression of the output layer is Eq. 5. y k = ∑ i = 1 m ω i k α i x , k = 1 , 2 , ⋯ p (5)

In Eq. 5, y k represents the output result of the neural network. ω i k is the weight between the hidden and the output layers. p is the number of nodes in the output layer. In the entire neural network structure, the hidden layer structure mainly performs nonlinear transformation on the data, transforming the input data X into α i x . The output layer structure mainly performs linear transformation on α i x to obtain the final output result y k . By performing two different linear transformations, not only can the overall running speed of the neural network be improved, but also its ability for nonlinear mapping can be enhanced. The overall structure of RBFNN is Eq. 6. y ′ = W T F X (6)

In Eq. 6, y ′ is the output vector of the network. W represents the weight matrix. X is the input sample matrix. To overcome the shortcomings of PID control technology, this study proposes to apply RBFNN to PID parameter optimization. The principle of this way is to utilize the powerful learning and approximation capabilities of RBFNN to identify the DCs of the system online, and adjust the PID control parameters in real time based on the identification results to achieve better control effects (Elsisi, 2020; Ghamari et al., 2022). The process of optimizing PID parameters based on RBFNN is Figure 3.

In Figure 3, the PID parameter optimization based on RBFNN first requires constructing an RBFNN model that includes input layer, hidden layer, and output layer to map the system state and output PID control parameters. Determine the network’s parameters through offline training or online adjustment. Next, design a PID controller based on system requirements and integrate it with RBFNN to build a complete control system. The system operation process utilizes RBFNN to identify the DCs of the system online, and adjusts the PID parameters in real-time to optimize the control effect. Finally, the performance indicators of the evaluation system, such as control accuracy, stability, and response speed, are used to measure the optimization effect. The performance indicator evaluation formula is Eq. 7. J = α ⋅ ∫ 0 + ∞ e t 2 d t + β ⋅ ∫ 0 + ∞ u t 2 d t (7)

In Eq. 7, J is a performance indicator used to measure system performance. α and β are weight coefficients used to balance errors and control the relative importance of inputs. This method can be widely applied to complex systems such as 8BSM and other similar control problems, effectively improving control performance and adaptability.

8BSM is a commonly used stamping equipment in industrial manufacturing, with unique design and excellent performance (Choquette et al., 2021). The core part is composed of eight connecting rods, which are connected together in a specific way, enabling the mechanism to achieve efficient and accurate stamping operations. The specific structure of 8BSM is Figure 4.

The advantage of 8BSM lies in its high flexibility and adjustability, and its impulse pressure formula is Eq. 8. F = k ⋅ l ⋅ ⁡ sin ⁡ θ (8)

In Eq. 8, F is the punching force, which is the force exerted by the punch on the workpiece. k is a constant that is related to the geometric design and material of the linkage mechanism. l is the effective length of the punch connecting rod. θ is the angle between the punch connecting rod and the horizontal line. The expression for stamping speed is Eq. 9. v = ω ′ ⋅ r (9)

In Eq. 9, v is the stamping speed. ω ′ means the angular velocity driven by the motor. r is the radius from the motor driven wheel to the punch connecting rod. In order to better control the eight bar linkage mechanism, it is necessary to conduct dynamic modeling. The dynamic modeling of 8BSM is a complex process that requires consideration of the kinematic relationships, interaction forces, and external constraints of multiple linkages. The specific steps of 8BSM dynamic modeling are shown in Figure 5.

In Figure 5, the dynamic modeling of 8BSM involves multiple steps. Firstly, it is necessary to establish a suitable coordinate system, which should be able to conveniently describe the motion of the mechanism. Next, the relevant parameters of each connecting rod, such as length, angle, mass, and moment of inertia, are defined. These parameters will directly affect the kinematic and DCs of the mechanism. By using the Denavit Hartenberg (D-H) parameter method, the relative position and angle relationship between each connecting rod can be systematically analyzed, and the transformation matrix can be obtained. The formula for the transformation matrix of the connecting rod is Eq. 10. T B A = cos ⁡ θ − ⁡ sin ⁡ θ 0 a sin ⁡ θ cos ⁡ θ 0 b 0 0 1 c 0 0 0 1 (10)

In Eq. 10, θ is the angle between two coordinate systems, and a , b , c is the translation distance between the two coordinate systems. These matrices can reveal the precise position and attitude information of the end effector when applied continuously, as calculated in Eq. 11. P n 0 = T 1 0 ⋅ T 2 1 ⋅ T 3 2 ⋯ T n n − 1 (11)

The establishment of dynamic equations requires considering the effects of multiple forces and constraints. The calculation of kinetic and potential energy of the mechanism is the core part of this process, which involves the mass, moment of inertia, velocity and angular velocity of the connecting rod, as well as the influence of elastic and damping forces. The formula for calculating kinetic energy is Eq. 12. K = 1 2 m v 2 + 1 2 I ω 2 (12)

In Eq. 12, K is the kinetic energy of the connecting rod. m is the mass of the connecting rod. v is the speed of the connecting rod. I is the moment of inertia of the connecting rod, and ω is the angular velocity of the connecting rod. The potential energy calculation formula is Eq. 13. P = 1 2 k ′ x 2 + 1 2 c v 2 (13)

In Eq. 13, P is the potential energy of the mechanism. k ′ is the elastic coefficient. x is the deformation of the connecting rod, and c is the damping coefficient. The Lagrangian equation is an effective tool for establishing dynamic equations. By substituting the calculation results of kinetic energy and potential energy, dynamic equations describing the dynamic behavior of mechanisms can be derived. The basic form of the Lagrange equation is Eq. 14. d d t ∂ T ∂ q j − ∂ T ∂ q j = Q j , j = 1 , 2 , ⋯ , k (14)

In Eq. 14, T is the total kinetic energy of the system. q i is the generalized coordinate, and Q j is the generalized force corresponding to q i ; k is the number of complete constraint equations. The process of using an improved PID controller for dynamic modeling optimization of 8BSM is Figure 6.

In Figure 6, the dynamic model of 8BSM was first established through the application of dynamic principles and methods, and its kinematics, dynamic equations, and impact force were analyzed. Secondly, through in-depth research on dynamic behavior, the changes in its motion trajectory, velocity, acceleration, and impact force under different working conditions were understood. Afterwards, to meet the control requirements, an improved PID controller was designed and a complete control system was constructed by combining the dynamic model. Then, through simulation and experimental verification, the performance of the control system can be evaluated and optimized. Finally, based on actual feedback and application requirements, control parameters will be continuously adjusted, neural network structures will be optimized, and new control strategies will be explored to continuously improve the performance and practicality of the control system. This strategy not only enhances the operational efficiency and product quality of 8BSM, but also provides effective control strategy ideas for similar complex systems.

This study utilized RBFNN to optimize the PID parameters (RBF-PID). To analyze the performance of the optimized PID controller using this method, this study compares its performance with PID controllers based on chaos optimization algorithm (Chaotic PID), GA optimization (GA PID), and Z-N formula method (Z-N PID). The specific experimental environment of the comparison test is mainly divided into hardware environment and software environment, in which the hardware environment is divided into three categories: control system hardware, controlled object and auxiliary equipment. The hardware of the control system mainly includes industrial control computer, data acquisition module and power amplifier. The controlled objects mainly include material testing machine, pressure sensor and electric cylinder; Auxiliary equipment mainly includes power supply, signal conditioner and multi-function tester. The software environment mainly includes MATLAB programming software, LabVIEW data acquisition and processing software and Windows10 operating system. The results of comparing the controller parameters of the PID controller obtained by the four methods are shown in Figure 7.

In Figure 7, the K _p , K _i , and K _d values of the PID obtained by Z-N PID, Chaotic PID, and GA PID are 30.21, 0.388, 0.055, 21.45, 0.578, 0.048, 52.36, 0.273, and 0.083, respectively. The three values obtained by RBF-PID are 22.48, 0.271, and 0.045, respectively. After obtaining the parameters of four PID controllers, to calculate the dynamic characteristic indicators of the unit step response under the action of the four PID controllers. The specific data of the dynamic characteristic indicators under Z-N PID, GA-PID, RBF-PID, and Chaotic PID parameters obtained are displayed in Figure 8.

According to Figure 8, the adjustment time, peak time, and rise time of the RBF-PID controller are 0.513 s, 0.478 s, and 0.443 s, respectively. The Chaotic PID controllers are 0.638 s, 0.629 s, and 0.613 s, respectively. The Z-N PID controller is 1.836 s, 0.721 s, and 0.633 s. The GA-PID controller is 1.391 s, 0.811 s, and 0.724 s. The above data indicates that the control performance of the RBF-PID is superior, and the response speed of the target is faster. To further analyze the effectiveness of each PID controller, the control system pressure was adjusted to 10 MPa and 15 MPa respectively. Performing step experiments on a material testing machine with elastic loads under four different PID controller conditions. The step response curves of the four PID controllers are shown in Figure 9.

In Figure 9A, when the control system pressure is 10 MPa, the peak times of RBF-PID, Chaotic PID, Z-N PID, and GA PID are 0.516 s, 0.615 s, 0.835 s, and 0.838 s, respectively. The adjustment times for RBF-PID, Chaotic PID, Z-N PID, and GA PID are 1.038 s, 1.121 s, 1.356 s, and 1.384 s, respectively. In Figure 9B, when the control system pressure is 15 MPa, the peak times of RBF-PID, Chaotic PID, Z-N PID, and GA PID are 0.618 s, 0.705 s, 0.758 s, and 0.773 s, respectively. The adjustment times for RBF-PID, Chaotic PID, Z-N PID, and GA PID are 1.113 s, 1.152 s, 1.613 s, and 1.664 s, respectively. The above results indicate that under two different control system pressures, the control effect of Chaotic PID is superior to the three improved PID controllers compared. Afterwards, four types of controllers were used for simulation and experimentation, and the results are shown in Figure 10.

In Figure 10, the simulation and experimental curve trends of the four controllers are basically consistent, and the experimental response time is longer than the simulation response time under the same conditions. In Figure 10A, compared to Z-N PID, Chaotic PID reaches stability earlier. In Figure 10B, compared to GA PID, the RBF-PID controller reaches stability earlier. By comparing Figures 10A, B, RBF-PID has the shortest time to reach stability. The above results indicate that the control effect of RBF-PID controller is better and it is effective.

To analyze the actual effect of the proposed dynamic modeling optimization of 8BSM based on improved PID controller, this study will compare the performance of the optimized 8BSM and the pre optimized 8BSM using this method. Table 1 shows the environmental parameters for the comparative experiment.

Compare the performance of two types of 8BSM under the conditions of Table 1, and use dynamic response index and control performance index as comparison indicators. The performance comparison results of the two types of 8BSM are shown in Table 2. The rise time, peak time, overshoot rate, and adjustment time of the optimized 8BSM are 0.5 s, 0.7 s, 10%, and 2.0 s, respectively, all better than before optimization. And the control accuracy, robustness, and stability of the optimized 8BSM are significantly better than before optimization. This result indicates that the proposed optimization scheme has a significant optimization effect on 8BSM.

In addition, to further validate the superiority of the designed improved 8BSM, this study selected 40 relevant experts and divided them into four groups to evaluate the proposed improved 8BSM (Mechanism 1), GA-PID optimized 8BSM (Mechanism 2), and Chaotic PID optimized 8BSM (Mechanism 3). The evaluation scores of the three institutions are shown in Figure 11. In Figure 11A, the average operating efficiency score of mechanism 1 is 9.3 points, significantly better than the 7.7 points of mechanism 2 and the 8.1 points of mechanism 3. In Figure 11B, the average product quality score of Institution 1 is 9.2 points, significantly better than the 7.9 points of Institution 2 and the 8.2 points of Institution 3.