# Design and Control of a Pneumatic Muscle Servo Drive Applied to a 6-DoF Parallel Manipulator
## 3 Results  
### Functional Model of the Pneumatic Muscle  
Analyzing the static characteristics of the muscle, it was observed that the static relationship between the muscle position \(x\) and the control signal \(u\) is highly nonlinear. However, the relationship between the force generated by the muscle \(F\) and the control signal \(u\) can be approximated as linear within the full range of muscle positions \(x\). Analyzing the dynamic characteristics determined in Section 2.3, it was noticed that the behavior of the object around a given operating point can be approximately described by a second-order differential equation with constant coefficients and with initial conditions as presented in Equation (1). \[ \begin{split}&\ddot{x}+b\dot{x}+cx=du\\ & x(0)=x_{a,0},\quad\dot{x}(0)=0,\quad u(0)=u_{a,0}(F_{a},x_{a,0} )\end{split}  \tag{1}\]  
where \(x\) represents the muscle position expressed in coordinates relative to the operating point, \(u\) denotes the control signal expressed in coordinates relative to the operating point, and \(b\), \(c\), and \(d\) are the constant coefficients of the equation. The operating point, on the other hand, was determined in absolute coordinates by the relationship \(u_{a,0}=g(F_{a},x_{a,0})\) by using parameters such as the load force on the muscle \(F_{a}\), which was maintained at a constant level throughout the study, and the initial muscle position \(x_{a,0}\), which corresponds directly to the initial control signal value \(u_{a,0}\). Then, utilizing Wolfram Mathematica software version 13.3.1, the responses to step inputs obtained in Section 2.3 for individual operating points were fitted to the solution of Equation (1) and are presented as Equation (2). \[ \begin{split} x(t)&=\frac{1}{2\sqrt{b^{2}-4c}c}e^{- \frac{1}{2}(b+\sqrt{b^{2}-4c})t}(-b(-1+e^{\sqrt{b^{2}-4c}t})(du+u_{a,0}(F_{a},x _{a,0})-cx_{a,0})\\ &+\sqrt{b^{2}-4c}(d(-1+2e^{\frac{1}{2}(b+\sqrt{b^{2}-4c})t}-e^{ \sqrt{b^{2}-4c}t})u)\\ &+(-1+2e^{\frac{1}{2}(b+\sqrt{b^{2}-4c})t}-e^{\sqrt{b^{2}-4c}t}) u_{a,0}(F_{a},x_{a,0})+c(1+e^{\sqrt{b^{2}-4c}t})x_{a,0})\end{split}  \tag{2}\]  
The results of this fitting for several selected step responses are presented below. In Figures 11 and 12, the real object responses to step inputs are shown with solid lines on the left side, while the responses of the fitted model are represented with dashed lines. In Figures 11 and 12 on the right side, the absolute fitting error for each response is depicted. Figure 11: Results of model fitting at selected operating points of the object (on the left) and fitting error (on the right). Figure 14: Results of the coefficient \(c\) approximation: actual surface (**top left**), approximation surface (**top right**), and approximation error (**bottom**). Figure 13: Results of the coefficient \(b\) approximation: actual surface (**top left**), approximation surface (**top right**), and approximation error (**bottom**). Next, the obtained polynomials were inserted into Equations (1) and (2), resulting in a model describing the behavior of the object in the form of a second-order differential  
Figure 16: Results of the coefficient \(u_{a,0}\) approximation: actual surface (**top left**), approximation surface (**top right**), and approximation error (**bottom**). Figure 15: Results of the coefficient \(d\) approximation: actual surface (**top left**), approximation surface (**top right**), and approximation error (**bottom**). which is then applied to the control valve. Similarly, the static inverse model requires the current load force \(F_{a}\) and the desired position value for the muscle to achieve \(x_{a,zad}\). The result of the static inverse model's operation is also the generation of the control voltage applied to the valve. In Figures 18 and 19, the results of the verification of both developed inverse models are presented.