## 3 TSMCSPO for the Hydraulic System  
### Hydraulic Servo System and Dynamics  
The hydraulic manipulator is shown in Figure 3 with its 3 dimensional file. It consists of three arms, two single-rod hydraulic actuated cylinders, and one AC servo motor. In this paper, only a 1 st link hydraulic system is applied to evaluate the control performance by the simulation and experiment. The constitution of the 1 st link is shown in Figure 4. It consists of an asymmetric piston, servo valve, an aluminum link, and an encoder. The rotation of link 1 is derived from the movement of the piston. The rotation of link 1 was derived by the movement of the piston. The relation between the displacement of the piston and rotation angle of link1 are obtained as follows:  
\[\Delta x_{L}\approx R\Delta\theta_{1} \tag{25}\]  
where \(x_{L}\) is the displacement of the piston, \(\theta_{1}\) is the rotation angle of link 1, and \(R\) is the radius of link 1's rotation (Figure 5). Figure 4: Schematic diagram of hydraulic system. Figure 5: Schematic diagram of link 1. The dynamics of a single-rod hydraulic servo system was obtained as a state-space form in the following [8; 31]:  
\[\mathbf{x}=\left[x_{1},x_{2}\right]^{T}\triangleq[x_{L},\dot{x}_{L}]^{T} \tag{26}\]  
\[\dot{x}_{1}=x_{2} \tag{27}\] \[\dot{x}_{2}=\frac{1}{m}(A_{1}P_{1}-A_{2}P_{2}-bx_{2}-F_{d}) \tag{28}\]  
where \(m\) is the mass of the piston, \(P_{1}\) and \(P_{2}\) are the pressures inside the two chambers of the cylinder, \(b\) is the coefficient of viscous friction, \(F_{d}\) contains the payload and the gravitational effects, \(\beta_{e}\) is the bulk modulus of the fluid, \(C_{t}\) is the internal leakage coefficient, \(V_{1}\) and \(V_{2}\) are the total fluid volumes of the two sides of the cylinder, \(A_{1}\) and \(A_{2}\) are the areas of the two sides of the piston, \(k_{q}\) is the flow gain coefficient of the servo valve, \(u\) is the servo valve control input signal, and \(k_{a}\) is the servo amplifier gain [8]. When the leakage is negligible \(P_{1}\) and \(P_{2}\) can be obtained as:  
\[ \begin{array}{l}P_{1}(t_{1})=u\int_{t_{0}}^{t_{1}}\frac{\beta_{e}}{V_{1}}k_ {a}k_{q}\sqrt{\Delta P_{1}}dt-\int_{t_{0}}^{t_{1}}\frac{\beta_{e}}{V_{1}}A_{1} x_{2}dt+P_{1}(t_{0})\\ P_{2}(t_{1})=u\int_{t_{0}}^{t_{1}}\frac{\beta_{e}}{V_{2}}k_{a}k_{q}\sqrt{ \Delta P_{2}}dt-\int_{t_{0}}^{t_{1}}\frac{\beta_{e}}{V_{2}}A_{2}x_{2}dt+P_{2}( t_{0})\end{array}  \tag{29}\]  
where \(t_{0}\) is starting time, \(P_{1}(t_{0})\) and \(P_{2}(t_{0})\) are the initial pressures of each cylinder. Substituting Equation (29) into Equation (27), we get:  
\[ \begin{array}{l}\dot{x}_{2}=\frac{1}{m}(A_{1}P_{1}-A_{2}P_{2}-bx_{2}-F_{d})= u[\frac{A_{1}}{m}\frac{\beta_{e}}{V_{1}}k_{a}k_{q}\int_{t_{0}}^{t_{1}}\sqrt{ \Delta P_{1}}dt-\frac{A_{2}}{m}\int_{t_{0}}^{t_{1}}\frac{\beta_{e}}{V_{2}}k_{ a}k_{q}\sqrt{\Delta P_{2}}dt]\\ \quad\quad-\frac{\beta_{e}}{V_{1}}\frac{A_{1}}{m}A_{1}\int_{t_{0}}^{t_{1}}x_{2} dt+\frac{A_{2}}{m}\frac{\beta_{e}}{V_{2}}A_{2}\int_{t_{0}}^{t_{1}}x_{2}dt-\frac{ b}{m}x_{2}+\frac{A_{1}}{m}P_{1}(t_{0})-\frac{A_{2}}{m}P_{2}(t_{0})-\frac{F_{d}}{m} \end{array}  \tag{30}\]  
**Assumption 3**. : _The two sides cylinders have the same volume \(V_{\text{eq}}\) and the same area \(A_{\text{eq}}\). Equation (30) can be simplified as follows:_  
\[\dot{x}_{2}=u\frac{A_{\text{eq}}}{m}\frac{\beta_{e}}{V_{\text{eq}}}k_{a}k_{q} (P^{\star})-\frac{b}{m}x_{2}+\frac{A_{\text{eq}}}{m}[P_{1}(t_{0})-P_{2}(t_{0}) ]-\frac{F_{d}}{m} \tag{31}\]  
_where \(P^{\star}\) is a constant which is assumed as an average value of \((\int_{t_{0}}^{t_{1}}\sqrt{\Delta P_{1}}dt-\int_{t_{0}}^{t_{1}}\sqrt{\Delta P_ {2}}dt)\). Hence, the dynamics of the entire system can be considered as a SISO (Single-Input and Single-Output) second order system with the control input \(u\) and the rotation output \(\theta_{1}\) of link 1 as follows:_  
\[\ddot{\theta}_{1}=\frac{1}{f_{1}}u-\frac{1}{f_{1}}D_{1}\dot{\theta}_{1}-\Psi. \tag{32}\]  
_where \(\Psi\), \(D_{1}\) and \(f_{1}\) is defined as follows:_  
\[