## 4 Results and Discussion
### Creation of the Model of the Dynamic Model Section
The task is to minimise the argument of the function \(V_{N}(\theta,Z^{N})\) by identifying relevant values of coefficients if \(Z^{N}=\{u(1),y(1),\cdots,u(N),y(N)\}\) (i.e., \(N\) input-output pairs), while applying the known procedure. Then, we obtain  
\[V_{N}\Big{(}\theta,Z^{N}\Big{)}=\frac{1}{N}\sum_{t=1}^{N}\left(y(t)-\hat{y}(t |\theta)\right)^{2}=\frac{1}{N}\sum_{t=1}^{N}\left(y(t)-\phi^{T}(t)\theta \right)^{2} \tag{27}\]  
Figure 8: The actuator responses measured at small step changes of the input pressure. \[\hat{\theta}_{N}=\underset{\theta}{\text{argmin}}V_{N}\Big{(}\theta,Z^{N}\Big{)} \tag{28}\]  
If we set the derivation \(V_{N}\big{(}\theta,Z^{N}\big{)}\) as equal to zero and after editing, we obtain an estimation of the parametric vector  
\[\hat{\theta}=\left[\sum_{t=1}^{N}\phi(t)\phi^{T}(t)\right]^{-1}\sum_{t=1}^{N} \phi(t)y(t). \tag{29}\]  
Then, the actuator can be described by the first order transfer function  
\[G_{1}(\text{s})=\frac{1}{0.64595\text{s}+1}. \tag{30}\]  
or by the second order transfer function  
\[G_{2}(\text{s})=\frac{1}{(0.090314\text{s}+1)(0.57873\text{s}+1)}=\frac{1}{0. 052267\text{s}^{2}+0.669\text{s}+1} \tag{31}\]  
Responses of the linear model of the first and second orders after the identification are presented in Figure 9.  
The actuator consisting of pneumatic artificial muscles in an antagonistic arrangement has, in each half of the angular displacement of the pulley shaft, regulated air pressure only in one artificial muscle. The other artificial muscle has a constant pressure and serves as a non-linear and pneumatic spring. Such an actuator is a non-linear system with the end position being a non-linear centrally symmetrical function of the filling air pressure in artificial muscles. With decreasing pressure in the active AM, the angular displacement of the actuator pulley shaft increases along the curve with a decreasing inclination. The static property of such a system proves by its course that actuator strengthening depends on the arm's position which depends on pressures and forces in individual artificial muscles. Actuator configuration, together with a simple control system, facilitates the construction of a rather simple positional servo system with adequate control requirements and execution costs. Dependencies of the actuator end position represent a non-linear function of the filling air pressure in the artificial muscle and their courses are different at different values of actuator load. This follows from non-linear properties of artificial muscles. It is also necessary to respect a non-identical tensile force (or torque) of the actuator, the magnitude  
Figure 9: Responses of the linear model of the first and second orders after the identification. of which is changing with the value of angular displacement of the actuator pulley shaft (or artificial muscle contraction or displacement). This property also causes non-identical mechanism stiffness at different position values, whereas the highest and two-side symmetrical stiffness is reached in the reference point, when the pressures in the AMs are identical and reach the maximum.