# Controller of Pneumatic Muscles Implemented in Active Seat Suspension
## 2 Physical and Mathematical Model of the Suspension System  
The subject of this study is a conventional seat suspension system that has been transformed into an active vibrosolation system through the integration of pneumatic muscle actuators. This enhancement aims to reduce vibrations generated when the working machine moves over uneven terrain, thereby improving operator safety. The research focuses on assessing the effectiveness of this specific seat suspension system without requiring a complex model of the human body seated. Instead, the evaluation uses a simplified representation, modeling the seat and the human body as a concentrated mass system (isolated body m\({}_{1}\)). This approach provides sufficient information to assess the quality and effectiveness of the suspension system in mitigating vibrations. The system (Figure 1) is a hybrid combination of passive elements, such as tension coil springs and end bumpers, with active elements in the form of pneumatic muscles. A pneumatic muscle can be described as a variant of a single-acting pneumatic actuator. It consists of a flexible hose located in a braid. When filled with compressed air, its diameter increases while its length decreases. Thanks to this phenomenon, it generates the pulling force \(F_{\text{ax1}}\) and \(F_{\text{ax2}}\). The motion equation of the system (Figure 1) can be described as an ordinary differential equation:  
\[m_{1}\bar{q}_{1\text{x}}=-2F_{\text{cx1}}\pm F_{\text{bx2}}-F_{\text{dx2}}+F_{ \text{ax1}}-F_{\text{ax2}} \tag{1}\]  
where:  
\(m_{1}\)--mass of the isolated body,  
\(q_{1\text{x}}\)--displacement of the isolated body,  
\(F_{\text{cx1}}\)--reaction force of tension coil springs,  
\(F_{\text{bx2}}\)--reaction forces of end bumpers, limiting the maximum displacement of the suspension system,  
\(F_{\text{dx2}}\)--total friction force of the suspension system,  
\(F_{\text{ax1}}\), \(F_{\text{ax2}}\)--forces generated by pneumatic muscles. Force characteristics of the passive sub-system are presented in Figure 2. The optimized system uses tension springs of the lower forces \(2F_{\text{cx1}}\) and the reduced friction force \(F_{\text{dx2}}\) of a suspension mechanism due to the application of the needle bearings instead of conventional plastic rollers. Such modifications allow for an increase in the intensity of the active vibration control by using the artificial pneumatic muscles. The forces \(\pm F_{\text{bx2}}\) of end-stop buffers are the same as conventional systems that are mounted to limit the maximum displacement of a suspension mechanism. Figure 1: Active Seat Suspension System. The required active force values, which should be generated by the pneumatic muscles to compensate for the vibrations of the isolated body, are described as [19]:  
\[F_{\text{ax1}}=-k_{\text{a1}}\dot{q}_{\text{1x}}-k_{\text{a2}}(q_{\text{1x}}-q_{ \text{ss}})\text{ for }-k_{\text{a1}}\dot{q}_{\text{1x}}-k_{\text{a2}}(q_{\text{1x}}-q_{\text{ss}})>0 \tag{2}\]  
\[F_{\text{ax2}}=-k_{\text{a1}}\dot{q}_{\text{1x}}-k_{\text{a2}}(q_{\text{1x}}-q_{ \text{ss}})\text{ for }-k_{\text{a1}}\dot{q}_{\text{1x}}-k_{\text{a2}}(q_{\text{1x}}-q_{\text{ss}})<0 \tag{3}\]  
where:  
\(q_{\text{ss}}\)--is the displacement describing the forced motion of the suspension system,  
\(k_{\text{a1}}\)--is the gain of the feedback loop from the absolute velocity of the isolated body,  
\(k_{\text{a2}}\)--is the gain of the feedback loop from the relative displacement of the suspension system. The gains \(k_{\text{a1}}\) and \(k_{\text{a2}}\) are the essential control parameters shaping the desired active force that have a significant influence on the suspension dynamics. The value of \(k_{\text{a1}}\) is responsible for vibration reduction of the isolated body, and the higher \(k_{\text{a1}}\) improves the ride comfort of the driver. On the contrary, the value of \(k_{\text{a2}}\) has the main influence on limiting the suspension travel and the higher \(k_{\text{a2}}\) ensures the controllability of a vehicle by the driver.