## Manipulator Analysis
### The Dynamic of the Electro-Hydraulic Actuator  
The schematic of the EHA is described in Figure 3. The dynamic is presented by the following equation:  
\[J_{i}\ddot{q}_{ia}=\tau_{hydraulic\_i}-b_{act\_i}\dot{q}_{ia}-d_{act\_i}(q_{ ia},\dot{q}_{ia},t)-\tau_{load\_i},(i=1,2,3) \tag{7}\]  
where \(J_{i}\) is an inertia mass of the link \(i\)th, \(q_{ia},\dot{q}_{ia},\ddot{q}_{ia}\) are the angular position, velocity, and acceleration of the actuator which have a same movement as each link of the manipulator, \(b_{act\_i}\) is a coefficient of the viscosity friction, \(d_{act\_i}(q_{ia},\dot{q}_{ia},t)\) represents uncertainty parameters such as friction or disturbance, \(\tau_{load\_i}\) is an arbitrary load torque on the actuator \(i\)th, and \(\tau_{hydraulic\_i}\) is torque applied on the link \(i\)th of the manipulator, same as Equation (2). \(\tau_{hydraulic\_i}\) is produced by:  
Figure 2: The architecture of the last link. \[\tau_{hydraulic\_,i}=\left\{ \begin{array}{ll}A_{q_{ia}}(P_{i1}-P_{2})=A_{q_{ia}} P_{iL}&(\text{for rotary actuator})\\ J_{a3}^{T}(q)F_{a}=J(q)(A_{1}P_{1}-A_{2}P_{2})&(\text{for cylinder})\end{array} \right. \tag{8}\]  
where \(A_{q_{ia}}\) is the radian displacement of the rotary actuator, \(A_{i(i=1,2)}\) are an area of the bore side and rod side of the cylinder, respectively. \(P_{i(i=1,2)}=\left[P_{i1}\ \ \cdots\ \ P_{i3}\right]^{T}\) are pressure in chamber 1 and chamber 2 of each actuator, respectively. The pressure in each chamber can be computed by [44]:  
\[\frac{dP_{i1}}{dt}=\frac{\beta}{V_{i1}}\bigg{(}Q_{i1}-A\frac{\partial q_{a}}{ \partial q}q\bigg{)}-\zeta_{i1}(t) \tag{9}\]  
\[\frac{dP_{i2}}{dt}=\frac{\beta}{V_{i2}}\bigg{(}-Q_{i2}+A\frac{\partial q_{a}}{ \partial q}q\bigg{)}+\zeta_{i2}(t) \tag{10}\]  
where \(V_{i1}=V_{i01}+A_{i}q\) is the volume of chamber 1, \(V_{i2}=V_{i02}-A_{i}q\) is volume of chamber 2, \(V_{i01}\) and \(V_{i02}\) are the initial volumes of the two chambers, \(\beta\) is the Bulk modulus, \(q_{link}\) is the leakage volume between the two chambers, \(Q_{i1}\) is the flow rate supplied by the hydraulic source to the forward chamber, and \(Q_{i2}\) is the returned flow rate from the backward chamber. Both \(Q_{i1}\) and \(Q_{i2}\) are driven by position of the spool inside the servo valve. The equation expressed in relation between \(Q_{i1}\) and \(Q_{i2}\) and displacement of the spool are presented in [45; 46; 47]:  
\[Q_{i1}=k_{q}\sqrt{\frac{2}{\rho}}[sign(x_{v})\sqrt{P_{S}-P_{i1}}+sign(-x_{v}) \sqrt{P_{1}-P_{ir}}] \tag{11}\]  
\[Q_{i2}=k_{q}\sqrt{\frac{2}{\rho}}[sign(x_{v})\sqrt{P_{i2}-P_{r}}+sign(-x_{v}) \sqrt{P_{S}-P_{i2}}] \tag{12}\]  
where \(x_{v}\) is the valve displacement, \(\omega\) is the spool valve area gradient, \(\rho\) is the density of the fluid, \(C_{d}\) is the discharge coefficient, \(P_{s}\) and \(P_{r}\) are the supply and return pressure, respectively. The dynamic of the valve was presented in [48; 49]:  
\[m_{v}\ddot{x}_{v}+b_{v}\dot{x}_{v}+k_{v}x_{v}=F_{flow}+f(u) \tag{13}\]  
Since \(x_{v}\) relates to the input voltage, and if neglecting servo-valve dynamics [50], we can assume approximately that an input signal \(u\) is a linear function and directly proportional to the spool position [42]\(x_{v}\), i.e.,  
\[x_{v}=k_{v}u \tag{14}\]  
where \(k_{v}\) is a positive constant and \(u\) is the control input voltage. Figure 3: The electro-hydraulic actuator schematic. For convenient controller design, the dynamic equation of the system should be performed in state space. The variables \((x_{v},q,P_{L})\) should be considered because of the state of the system.