# Characterization and Joint Control Study of Pneumatic Artificial Muscles
## 2 Pneumatic Artificial Muscle Modeling and Dynamic Simulation
### Simulation Analysis of Dynamic Characteristics of Pneumatic Artificial Muscles  
The single joints connected by pneumatic artificial muscles are all rotating joints; thus, they play a big role in securing the parts and rotating them. The desired angle of each joint is achieved through the actuator's output, thus completing the motion control of the entire joint. However, due to the weave's texture and working principle, it is impossible to output thrust, just like the antagonism of human muscles against pulling, so it can be approximated as a single-acting adjustable pneumatic component. Therefore, to achieve the output of the desired angle and thus control the joint, the process of converting the linear motion of the pneumatic tendon into rotation must be completed using a pair of pulls. The following two methods are available, as shown in Figure 2.  
The first way is achieved by using a pneumatic artificial muscle and a spring composed of a pulling mechanism. The second uses the antagonistic arrangement of a pair of antagonistic connections of the pulling pneumatic muscle. In comparison, the price of the first pneumatic muscle-spring combination is better than the second; the disadvantage is that this mechanism constitutes the antagonistic joint, and to achieve the same expected angle, the first requires a more considerable pneumatic muscle output tension than the second, which often reduces the service life of the pneumatic muscle. Moreover, in the control of the spring compared to the controllability of the pneumatic artificial muscle, jitter and other problems often arise with use. Furthermore, the long-time use of the spring can easily produce fatigue, resulting in control accuracy as well as elasticity decline, but it is suitable for the study of pneumatic artificial muscle's own physical characteristics of the law. Figure 2: Joint-to-pull mode. Therefore we use the structure of the PAM-PAM series pull-pair for dynamic characteristics simulation, control simulation, and control experiments. As the dynamic characteristics experiment is used to discuss and verify the physical properties of pneumatic artificial muscles, the use of PAM + spring also satisfies the requirements of studying the laws of PAM's physical properties. From an economic viewpoint, the experiments on the dynamic characteristics of PAM in the thesis are conducted according to the PAM-spring structure. The simulation model is established in the Matlab Simulink software according to Equations (1) and (2), (8)-(10), Figure 3 shows the pneumatic artificial muscle-driven joint model, and the parameters used in the simulation are shown in Table 1. Gas source temperature is given in degrees Kelvin, pneumatic muscle initial length and single fiber length in meters, valve port circulation effective cross-sectional area in square meters, upstream inflation pressure in MPa, gas constant in N\(\cdot\)m\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\), connecting rod length in meters, load mass in kg, joint radius in meters, and damping factor in N\(\cdot\)s\(\cdot\)m\({}^{-1}\). Figures 4-9 show the curves of each variable of the rotating joint model under the condition that the input signal is a step signal. The maximum joint rotation angle in Figures 4 and 5 is about 68\({}^{\circ}\), and the steady state time is 2 s. It can be seen from Figures 4 and 5 that the single joint impact with pneumatic muscles is flexible and does not produce sudden changes in velocity acceleration. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **Takes Values** & **Parameters** & **Takes Values** \\ \hline \(K\) & 1.4 & \(\gamma\) & 20\({}^{\circ}\) \\ \(R_{1}\) & 0.528 & \(S\) & \(6\times 10^{-7}\) \\ \(n\) & 7.50 & \(m\) & 15 \\ \(T\) & 300 & \(C\) & 287.1 \\ \(E_{0}\) & 0.300 & \(e_{n}\) & 0.4 \\ \(e\) & 0.585 & \(\zeta\) & 200 \\ \(r\) & 0.04 & \(p_{1}\) & 0.6 \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation parameter setting table. Figure 3: Pneumatic artificial muscle-driven joint model. Figure 4: Angle and speed profile.