## 2 Dynamic Modeling of HDFRMS
### Hydraulic Subsystem Model  
Joint 1 is a rotary joint, which is driven by a rotary hydraulic motor. Joints 2 and 3 are revolute joints, which are driven by a linear asymmetric hydraulic cylinder. The manipulator needs to move the payload along some predefined trajectory in practice. The hydraulic-driven subsystem consists of a hydraulic cylinder and hydraulic motor. Additionally, Table 2 shows the nomenclature of the hydraulic subsystem. The asymmetric four-way valve-controlled hydraulic cylinder is a kind of hydraulic actuator converting hydraulic energy into linear motion mechanical energy as shown in Figure 3. \begin{table}
\begin{tabular}{c c c c} \hline
**Symbol** & **Definition** & **Symbol** & **Definition** \\ \hline \(P_{s}\) & supply pressure & \(x_{v}\) & spool displacement \\ \(P_{r}\) & return pressure & \(y\) & piston displacement \\ \(P_{L}\) & payload pressure & \(L\) & hydraulic cylinder stroke \\ \(\beta_{c}\) & bulk modulus & \(w\) & area gradient of the valve \\ \(C_{d}\) & flow coefficient of the serve-valve port & \(I\) & servo valve current \\ \(V_{t}\) & equivalent volume & \(K_{q}\) & flow gain coefficient \\ \(K_{c}\) & flow/pressure coefficient & \(Q_{L}\) & load flow of hydraulic cylinder/motor \\ \(P_{1}\)\(P_{2}\) & left and right actuator chamber pressure & \(Q_{1}\)\(Q_{2}\) & left and right actuator chamber fluid flow \\ \(D\) & volume displacement of the motor & \(V_{1}\)\(V_{2}\) & left and right actuator chamber volume \\ \(C_{tm}\) & equivalent leakage coefficient & \(A_{1}\)\(A_{2}\) & left and right effective piston area \\ \hline \end{tabular}
\end{table} 
Table 2: Notation, frequently used symbols of hydraulic subsystem. Actually, the time constant of mechanical systems is much greater than the hydraulic serve-valve time constant; therefore, the servo valve spool displacement \(x_{v}\) is proportional to the control input by ignoring the dynamics of the servo valve. We obtain \(K_{i}~{}=~{}K_{v}K_{a}\), where \(K_{v}\) is the proportional valve gain and \(K_{a}\) is the amplifier gain. \[x_{v}~{}=~{}K_{i}i \tag{8}\]  
Considering the law of cosines, depending on the installation location of hydraulic cylinder, we have  
\[ \begin{array}{rcl}y_{2}&=&\sqrt{1_{21}^{2}+L_{11}^{2}-2L_{11}L_{21}\cos(180- \theta_{2})}\\ y_{3}&=&\sqrt{1_{22}^{2}+L_{31}^{2}-2L_{23}\cos(180-\theta_{3})}\\ \end{array}  \tag{9}\]  
\[\dot{y}~{}=~{}j\dot{\theta}\]  
where \(J~{}=~{}diag(J_{1},J_{2})\) is the matrix relating the actuator linear displacement to the link angular displacement. Additionally, the details of this matrix \(J\) are described below. \[ \begin{array}{rcl}J_{2}&=&\frac{\partial y_{2}}{\partial\theta_{2}}~{}=& \frac{-L_{21}L_{11}\sin\theta_{2}}{\sqrt{1_{21}^{2}+L_{11}^{2}+2L_{11}L_{21} \cos\theta_{2}}}\\ J_{3}&=&\frac{\partial y_{3}}{\partial\theta_{3}}~{}=&\frac{-L_{22}L_{31}\sin \theta_{3}}{\sqrt{1_{22}^{2}+L_{31}^{2}+2L_{22}L_{31}\cos\theta_{3}}}\\ \end{array}  \tag{10}\]  
According to the relationships of force balances and the basic theory of the hydraulic system [26], we obtain  
\[\left\{ \begin{array}{l}Q_{L}=K_{q}x_{v}-K_{c}P_{L}\\ Q_{L}=C_{tm}P_{L}+\frac{V_{l}}{4B_{e}}\dot{P}_{L}+A_{p}y\\ x_{v}=K_{i}I\\ \tau=A_{p}Jp_{L}\\ \end{array} \right.A_{p}=\left\{ \begin{array}{l}A_{1}x_{v}\geq 0\\ A_{2}x_{v}<0\\ \tau=A_{p}Jp_{L}\\ \end{array} \right. \tag{11}\]  
Figure 3: Schematic diagram of asymmetric hydraulic cylinder. Applying (9) and (11), we can derive that  
\[\dot{\tau}+[\frac{4\beta_{e}}{V_{t}}(K_{c}+C_{tm})-\frac{\dot{J}}{J}]\tau+\frac{4 \beta_{e}A_{p}^{2}}{V_{t}}J^{2}\dot{\theta}=\frac{4\beta_{e}A_{p}K_{q}K_{i}}{V_{ t}}JI \tag{12}\]  
where \(V_{t}~{}=~{}2 LA_{p}^{4}/(A_{1}^{3}+A_{2}^{3})\). Similarly, the kinetics of the valve-controlled hydraulic motor can be derived as  
\[\left\{ \begin{array}{l}x_{v}=K_{i}I\\ \tau=P_{L}D\\ Q_{L}=K_{q}x_{v}-K_{c}P_{L}\\ Q_{L}=D\dot{\theta}+C_{tm}P_{L}+\frac{V_{t}}{4\beta_{e}}P_{L}\end{array} \right. \tag{13}\]  
where \(K_{q}~{}=~{}C_{d}w\sqrt{(P_{s}-P_{L})/\rho}\).