## 3 The Proposed Model and Methods  
### PAM for a Nonlinear System Model  
Modeling and designing the control algorithm for a nonlinear system is not always a straightforward operation, and the complexity further intensifies when dealing with multivariable systems. PAM has a nonlinear displacement, force, and damping characteristic, with the stimulation pressure used for contraction considered as the independent variable. Although the applied pressure is considered as input with a linear and proportional increase, the output parameters do not respond in the same way. Considering this challenge, we have used the inverse problem of a modular assembly system that includes a single-acting actuator known as the McKibben muscle. Inverse Modeling of a PAM-Based Pressing Unit for a Nonlinear Case  
To achieve the desired results and maintain control over a system, it is essential to accurately model the impact of the input signal from the control device on the actuator's output. This requires a precise and deterministic approach. In the modular pressing and assembly station selected as a case study, PAM is used as the actuator. For the PAM's force-displacement characteristic, the static system models in [29], as depicted in (4), and the derived total displacement models in (5) have been proposed, but the experimentally obtained curves do not exactly fit the various system models [12]. The expressions \(r_{\sigma}\), \(L_{0}\)and \(\mathcal{O}_{0}\) in the equation represent, respectively, the initial values of radius, nominal length, and braid angle, which are the structural variables of the PAM. In addition to the structural variables of the system, \(F\) is the resultant force, \(P\) is the applied inlet pressure, \(\varepsilon\) is the strain, and \(L_{final}\) is the resultant length after the elongation. Accurate and precise measurements or knowledge of all the structural variables of the actuator used in the system is needed for the use of the proposed models. \[F=\pi r_{o}^{2}P\Big{[}a(1-\varepsilon)^{2}-b\Big{]} \tag{4}\]  
where \(a=\frac{3}{tan^{2}\mathcal{O}_{0}}\), \(b=\frac{1}{sin^{2}\mathcal{O}_{0}}\), \(\varepsilon=\frac{L_{final}-L_{0}}{L_{0}}\)  
\[L_{\text{final}}\ -L_{0}=L_{0}\Bigg{(}1-\sqrt{\frac{\frac{F}{\pi r_{o}^{2}P}\ +\ b}{a}}\Bigg{)} \tag{5}\]  
Furthermore, while Festo provides a force-contraction plot for nonlinear PAM characteristics, as shown in Figure 3, it is essential to re-test its accuracy in applications with different components to ensure precise position control. PAM is a flexible actuator that can be used as a spring with variable values, showing different characteristics for different pressure inputs. This feature has practical benefits in terms of use in field applications but presents a mathematical modeling challenge. Different pressure values applied to the PAM input affect the displacement and force characteristics of the muscle, leading to different nonlinear characteristics for each pressure condition. Additionally, mathematical models proposed in the literature generate errors ranging from 4% to 46% for the static force of the muscle [11]. When the system includes any damping or stiffness elements other than the pneumatic muscle actuator and static mass, their displacement and force characteristic curves under different inlet pressures will cause the system to deviate from linearity. In the scope of this study, a modular pressing unit consisting of two pre-tensioning springs and a PAM (Festo DMSP-20) was used as a case study, as shown in Figure 4. In the system model shown in Figure 4, the force is generated by the combination of the contraction caused by the pressure applied to the PAM and the mass of the aluminum body holding the assembled components. Depending on the pressure applied to the PAM, the displacement during assembly and the resulting static force will vary. The PAM is modeled with a variable capacity damping element [29] and spring combination and parallel prestressing springs are also considered.