## Manipulator Analysis
### Dynamics Manipulator  
The dynamics of the robot with a degree of freedom can be expressed as:  
\[M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)+F_{d}\ddot{q}+F_{s}\big{(}\dot{q}\big{)}= \tau+\tau_{load} \tag{1}\]  
where \(q,\dot{q},\ddot{q}\in R^{3}\) are position, angular velocity, and angular acceleration vectors of each joint, respectively, \(M(q)\in R^{3\times 3}\) is the symmetric and positive definite matrix of inertia, \(C(q,\dot{q})\in R^{3\times 3}\) denotes the Coriolis and Centrifugal term matrix, \(G(q)\in R^{3}\) is the gravity term, \(\tau\) is torque acting on joints, \(F_{d}\in R^{3\times 3}\) is a diagonal matrix of viscous and dynamic friction coefficients, \(F_{s}\big{(}\dot{q}\big{)}\in R^{n}\) is the vector of unstructured friction effects such as static friction terms, \(\tau_{load}\) denotes the torque load induced by an external payload. However, the dynamics of the manipulator is difficult to determine clearly because of the mass distribution, and oscillation in the process of operation. Then the Equation (1) is re-written as:  
\[\hat{M}(q)\ddot{q}+\hat{C}(q,\dot{q})\dot{q}+\hat{G}(q)+d=\tau+\tau_{load} \tag{2}\]  
where \(\hat{M}(q)\ddot{q}+\hat{C}(q,\dot{q})\dot{q}+\hat{G}(q)\) represent the estimated terms and can be assumed to be known (Appendix A), \(d=\Delta M(q)\ddot{q}+\Delta C(q,\dot{q})\dot{q}+\Delta G(q)+F_{s}\big{(}\dot{ q}\big{)}+\Delta F_{d}\dot{q}\in R^{3}\) is the vector of any generalized input because of uncertainties, disturbances, or un-modeled dynamic. The detailed structure of the \(\hat{M}(q),\hat{C}(q,\dot{q}),\hat{G}(q)\) is discussed in Appendix A.  
**Assumption 1**.: _All \(\Delta(.)\) are bounded, i.e., \(\|\Delta(.)\|_{\infty}\leq\xi(.)\)._  
Unlike conventional manipulator, the last link of our proposed hydraulic manipulator employs cylinder to load heavier object. The torque acting on the joint is essentially induced by the force generated by the cylinder then the relationship between the rotational motion of the joint and the  
Figure 1: The overview of the 3-DOF hydraulic manipulator.  
translational motion of the actuator is investigated. The architecture of the last link is illustrated in Figure 2.  
The correlation between the motion of the actuator in the actuator space and the motion in joint space can be derived by:  
\[d_{3}+d_{0}=\sqrt{{d_{1}}^{2}+{d_{2}}^{2}-2d_{1}d_{2}\cos(\pi-q_{3})} \tag{3}\]  
where \(d_{0}\) is the initial length in case of maximum retracting; the positive direction in the actuator space is chosen for extracting the piston. \(d_{3}\) denotes the position of the end-effector under the motion of the cylinder, \(d_{1}\) and \(d_{2}\) are fixed length of the hinge joints.  
To obtain the correlational velocity between the actuator and the joint motion, taking the derivation respect to time, the Equation (3) results into:  
\[\dot{d}_{3}=\frac{\partial d_{3}}{\partial q_{3}}\dot{q}_{3}=\frac{d_{1}d_{2} \sin(\pi-q_{3})}{\sqrt{{d_{1}}^{2}+{d_{2}}^{2}-2d_{1}d_{2}\cos(\pi-q_{3})}} \dot{q}_{3}=J_{a3}(q_{3})\dot{q}_{3} \tag{4}\]  
where \(J_{a3}\) (\(q_{3}\)) is a Jacobian matrix of the 3 rd joint.  
Thus, a torque acting on this joint can be derived as:  
\[\tau_{3}=J_{a3}^{T}(q_{3})F_{3} \tag{5}\]  
The completely correlational motion between the actuator and the joint motion is described by:  
\[\left(\begin{array}{c}\dot{q}_{1a}\\ \dot{q}_{2a}\\ \dot{d}_{3}\end{array}\right)=\left(\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&J_{a3}(q_{3})\end{array}\right)\left(\begin{array}{c}\dot{q}_{1}\\ \dot{q}_{2}\\ \dot{q}_{3}\end{array}\right) \tag{6}\]  
where \((q_{1a}\)\(q_{2a}\)\(d_{3})^{T}\in R^{3}\) denotes position vector of the actuator in actuator space.