## 3 Theory/Calculation
### Model Formation  
The pneumatic artificial muscle AM2 extends its length while the compressed air is released, and at the same time it exerts the tensile force \(F_{m2}\), the value of which gradually decreases. The relationship between the tensile force \(F_{m}\), the air pressure \(p\), and the length \(l\) (contraction k magnitude) of this artificial muscle is expressed by the properties presented in Figure 5 that are depicted together with the property of the AM1 artificial muscle. In the case of this artificial muscle, the air pressure does not change, so only one of its properties is depicted, the one corresponding to the initial filling pressure \(p_{m}\). The property is drawn so that it describes antagonistic forces of AM1 against AM2. The AM1 muscle serves as a non-linear pneumatic spring. This ensures equality of forces for every position of the positive position value. It also provides stiffness of the actuator's mechanism. Property intersection points in the EC section correspond to the course of increasing concentration of the passive AM of AM1 during a gradual pressure decrease in the active AM of AM2. The above-described procedure facilitates achieving an arbitrary positive value \(l_{d}\) (up to \(l_{thmax}\)). Actuator's negative position values are reached through the same procedure as described in the previous paragraph. Only the roles of individual muscles are switched. AM1 has a variable air pressure; AM2 serves as a non-linear pneumatic spring. In the configuration of the antagonistic actuator, as shown in Figure 5 (position in the reference point E), while identical pneumatic artificial muscles are used, their non-linear functions \(g_{N}\) will be identical too, as well as their tensile forces:  
\[F_{m1}=g_{N}(l_{1})=F_{m2}=g_{N}(l_{2})=F_{0}, \tag{6}\]  
At the same time, the following applies: \(l_{1}=l_{2}=l_{0}\) and \(p_{1}=p_{2}=p_{m}\). In the configuration of the antagonistic actuator, as shown in Figure 5, while identical pneumatic artificial muscles are used, their grids of non-linear functions \(g_{M}\) will be identical, as well as their tensile forces (position in the reference point E):  
\[F_{m1}=g_{M}(l_{1},p_{1})=F_{m2}=g_{M}(l_{2},p_{2}), \tag{7}\]  
while the following applies: \(l_{1}<\!\!>l_{2}\) and \(p_{1}<\!\!>p_{2}\) and \(k_{AM1}+k_{AM2}=k_{max}\) (\(k_{AM1}\) is contraction of the actuator AM1, \(k_{AM2}\) is contraction of the actuator AM2 and \(k_{max}\) is maximum contraction of the AMs antagonistic connection). If in Equation (7), we know the non-linear function \(g_{M}\) (\(l\), \(p\)), then on the basis of the given equality of forces we can express the relationship between the output from the actuator (length \(l\) or \(l_{d}\)) and its input (air pressure \(p\)) in the form of a non-linear function\(f_{N}\):  
\[abs(l_{d})=f_{N}(p), \tag{8}\]  
\[abs(\beta)=\frac{l_{d}}{r_{kl}}=\frac{f_{N}(p)}{r_{kl}}. \tag{9}\]  
Equations (8) and (9) clearly indicate that the values of respective parameters \(\beta\) and \(l_{d}\) will be achieved with a positive and a negative polarity. Therefore, the value of the input pressure \(p\) must also have assigned both polarities, despite the fact that the pressures into individual muscles have positive values (\(p_{AM1}>0\) and \(p_{AM2}>0\)). This procedure is carried out depending on the sign of the required displacement of AM \(l_{dD}\) or the required angular displacement \(\beta_{D}\) as follows:  
if _sign_\(\beta_{D}=+\), then \(p=p_{AM2}\)_sign_\(\beta_{D}>0\) whereas _sign_\(\beta=+\) and \(p_{AM1}=\text{const}>0\),  
if _sign_\(\beta_{D}=-\), then \(p=p_{AM1}\)_sign_\(\beta_{D}<0\) whereas _sign_\(\beta=-\) and \(p_{AM2}=\text{const}>0\).