## 3 Theory/Calculation  
### Physical Fundamentals  
The principle of the action of such an actuator with artificial muscles (AMs) in antagonistic connection can be explained by Figure 4.  
Figure 3: Schematic diagram of the experimental platform. If we have two identical muscles, AM1 and AM2, in antagonistic connection (Figure 2), and these muscles are filled with an identically large amount of compressed air, they shorten to the length \(l_{0}\), and the following applies [33]:  
\[l_{0}=l_{max}\frac{\Delta l_{max}}{2} \tag{1}\]  
where:  
\(l_{max}\)--maximum AM length,  
\(\Delta l\)--change in AM length. AM1 has the tensile force \(F_{m1}\) which is transmitted through a pulley to the AM2 muscle that develops the tensile force \(F_{m2}\). With identical filling pressures in AM1 and AM2, tensile forces become identical at identical values of their contractions and the actuator's arm stabilises in the initial position. In such an initial position, the actuator stiffness is the highest if air pressure in both muscles is at its maximum. At non-identical filling pressures in the muscles (Figure 4), the actuator's arm stabilises in the position corresponding to the equality of tensile forces in both muscles. The angular displacement \(\beta\) for the arm exposed to the external load with the mass \(m\) depends on the pulley radius \(r_{kl}\) for identical changes in the length of pneumatic artificial muscles. The rotation of the arm is subject to the equation:  
\[\beta=\frac{\Delta l}{r_{kl}} \tag{2}\]  
If we designate the derivation of the muscle volume according to the length \(dV/dl\) generally as a non-linear function \(f\), for the two muscles we can write the following:  
\[F_{m1}=-P_{1}f_{1}(l_{10}-\Delta l),F_{m2}=-P_{2}f_{2}(l_{20}+\Delta l) \tag{3}\]  
where:  
\(P_{1}P_{2}\)--pressure in the respective muscle,  
\(l_{10}J_{20}\)--initial lengths of both muscles. The above-mentioned equation (3) indicates that the length of each muscle was changed in the same value \(\Delta l\), while the shortening of one muscle became larger and of the other muscle became smaller (Figure 4). It is interesting that the stiffness of such a mechanism can be changed according to the determined requirement, as the arm's position is proportionate to the difference in pressures in the muscles, whereas the stiffness is proportionate to the sum of pressures in the muscles. Hence, the same pressure difference may be achieved at different values of their sum, i.e., the same arm displacement can be  
Figure 4: Pneumatic artificial muscles in antagonistic connection. achieved at various resulting stiffnesses. The stiffness of the mechanism may be derived on the basis of the expression of the total force affecting the load, as follows:  
\[F=F_{m1}-F_{m2}=-P_{1}\frac{dV_{1}}{dl_{1}}+P_{2}\frac{dV_{2}}{dl_{2}}=P_{1} \frac{dV_{1}}{d(\Delta l)}+P_{2}\frac{dV_{2}}{d(\Delta l)}. \tag{4}\]  
In which case, the following applies to the stiffness:  
\[K=\frac{dF}{d(\Delta l)}=-\frac{dP_{1}}{dV_{1}}\bigg{(}\frac{dV_{1}}{d(\Delta l )}\bigg{)}^{2}-\frac{dP_{2}}{dV_{2}}\bigg{(}\frac{dV_{2}}{d(\Delta l)}\bigg{)} ^{2}-P_{1}\frac{d^{2}V_{1}}{d(\Delta l)^{2}}+P_{2}\frac{d^{2}V_{2}}{d(\Delta l )^{2}}. \tag{5}\]  
According to the new concept of actuator action prepared by the authors, if we change the pressure only in one muscle, we distinguish between active and passive AM. The active muscle is always the muscle with a variable air pressure. The passive muscle serves as a non-linear spring at constant air pressure, thus ensuring the stiffness of the actuator's mechanism and equality of forces for every actuator position. When the air pressure changes (decreases), for example, in the AM2 muscle, the muscle contraction changes (decreases) as well (active muscle). As a result, the mass of the load on the arm attached to the pulley axis makes a rotary motion. Such motion direction is regarded as positive (+) with regard to the initial point. Negative position values (\(-\)) are achieved by the actuator in the same manner.