## 2 System Description
### System Modeling  
Based on the geometry of the typical antagonistic configuration, which is illustrated in Figure 1a, the length of each PAMs can be obtained from the measured joint angle, as in the following equations:  
\[y_{A}=y_{AN}+R\theta \tag{1a}\] \[y_{P}=y_{PN}-R\theta, \tag{1b}\]  
where \(y_{AN}\) and \(y_{PN}\) are the nominal length of the anterior and posterior PAMs when the joint angle \(\theta=0\), and \(R\) is the rotation radius of the actuator. Because two similar PAMs are used in the system, we can consider that \(y_{AN}=y_{PN}=y_{N}\). Following that, the relationship between contraction of PAMs and the measured angle can be expressed as  
\[\varepsilon_{A}= \frac{y_{0}-y_{A}}{y_{0}}\times 100\%=\frac{y_{0}-(y_{N}+R\theta)}{ y_{0}}\times 100\% \tag{2a}\] \[\varepsilon_{P}= \frac{y_{0}-y_{P}}{y_{0}}\times 100\%=\frac{y_{0}-(y_{N}-R\theta)}{ y_{0}}\times 100\% \tag{2b}\]  
Figure 1: (**a**) Typical antagonistic configuration of two pneumatic artificial muscles (PAMs) and (**b**) experiment platform of an antagonistic actuator powered by PAMs. where \(y_{0}\) is the length of PAMs in the complete deflation state. In (2), \(y_{0}\) and \(y_{N}\) are fixed by the deflation and nominal lengths of PAMs. Therefore, the contraction of PAMs can be expressed as the function of the measured joint angle \(\theta\). As a result, the dynamic behaviour of an antagonistic muscle can be described by a single input single output (SISO) system, in which the input is the difference pressure of two PAMs (\(\Delta P\)), and the output is the measured angle \(\theta\). The input pressure inside the anterior and posterior PAMs can be expressed as  
\[P_{P} =P_{0}+\Delta P \tag{3a}\] \[P_{A} =P_{0}-\Delta P, \tag{3b}\]  
where \(P_{0}\) is the nominal pressure which determines the initial position of antagonistic actuator. The nominal pressure can be chosen so that the joint has the desired compliance for a specific application. It was fixed, so \(\Delta P\) was chosen as a control variable of trajectory-tracking controller. All the system parameters \(P_{0}\), \(y_{0}\), and \(y_{N}\) are provided as in Table 1. In this research, the following discrete-time SISO system was chosen to describe the model of antagonistic actuator:  
\[y_{k+1}=-\sum_{i=1}^{n}a_{i}y_{k-i+1}+\sum_{j=1}^{m}b_{j}u_{k-j-d+1}+p_{k}, \tag{4}\]  
where \(u_{k}\) represents the control pressure \(\Delta P\), \(y_{k}\) is the joint angle, \(d\) is a positive integer representing the dead time of the system (as a number of the sampling time), \(p_{k}\) is the unknown disturbance of the system, \(a_{i}\) and \(b_{j}\) are the model parameters with \(b_{1}\
eq 0\), \(n\) and \(m\) are integers which satisfy \(n\leq m\). The model parameters of the system are obtained by the identification experiment. To verify the mathematical model of PAM, the following experiment procedure was carried out. * The initial position of the actuator was set at \(0^{\circ}\) by supplying nominal pressure \(P_{0}\) to each PAMs of the actuator. * The actuator angle can be changed by sending different types of control signal to the electrical control valves. Three types of control signals were used in this experiment:
* Step response: the control signal was a step wave with the final values 0.2, 0.4, 0.5, and 0.8 MPa. * Sinusoidal signal: The control signal is the 0.2 MPa amplitude sinusoidal signal, where frequency varies from 0.2 to 1.0 Hz. * A sine wave signal with time-varying amplitude and frequency, as in the following equation: \[u(t)=Asin(2\pi ft)+0.8 Asin(2\pi 0.2 ft)+0.5 Asin(2\pi 1.5 ft)+0.2 Asin(2\pi 3 ft),\] (5) where \(A=0.05\) MPa and \(f=0.5\) Hz are the basis amplitude and frequency of the control signal, respectively. All the data, including control signals and measured angles of actuator, were recorded with sampling time \(T_{s}=5\) ms for further analysis. * The discrete-time SOPDT, in which \(m=n=2\), was chosen as the mathematical model of the actuator with good accuracy.