## 2 Manipulator Dynamics  
The structure of the 3-DOF manipulator is described in Figure 1.  
The dynamics of the 3-DOF can be expressed as:  
\[M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)+F_{d}\dot{q}+F_{s}\Big{(}\dot{q}\Big{)}=\tau \tag{1}\]  
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, and \(F_{s}\in R^{3}\) is the vector of unstructured friction effects such as static friction terms.  
In practice, the dynamics of the manipulator is difficult to determine clearly due to the mass distribution, and oscillation in the process of operation. Then the Equation (1) is represented as follows:  
\[\hat{M}(q)\ddot{q}+\hat{C}(q,\dot{q})\dot{q}+\hat{G}(q)+\tau_{d}=\tau \tag{2}\]  
where \(\hat{M}(q)\ddot{q}+\hat{C}(q,\dot{q})\dot{q}+\hat{G}(q)\) represents the estimated terms of \(M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)\), and can be assumed to be known, and \(\tau_{d}=\Delta M(q)\ddot{q}+\Delta C(q,\dot{q})\dot{q}+\Delta G(q)+F_{s} \Big{(}\dot{q}\Big{)}+F_{d}\dot{q}\in R^{3}\) is the vector of any generalized input due to uncertainties, disturbances or un-modeled terms.  
**Assumption.**_All \(\Delta(.)\) are supposed to be bounded, i.e., \(\left\|\Delta(.)\right\|_{\infty}\leq\xi(.)\)._  
Unlike a conventional manipulator, the last link of our proposed hydraulic manipulator employs a cylinder to load heavier objects. The torque acting on the joint is essentially induced by force generated by the cylinder; then the relationship between the rotational motion of the joint and the translational motion of the actuator will be investigated. The architecture of the last link is illustrated in Figure 2.  
Figure 1: An overview of the 3-degrees of freedom (DOF) hydraulic manipulator: (**a**) 3D drawing; (**b**) principle drawing.  
The correlation between the motion of the actuator in the actuator space and the motion in joint space can be derived by:  
\[d_{0}+d_{3}=\sqrt{{d_{1}}^{2}+{d_{2}}^{2}-2d_{1}d_{2}\cos(\pi-q_{3})} \tag{3}\]  
where \(d_{0}\) is an 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) is:  
\[\dot{d}_{3}=\frac{\partial d_{3}}{\partial q_{3}}\dot{q}_{3}=\frac{d_{1}d_{2} \sin(\pi-q_{3})}{\sqrt{{d_{1}}^{2}+{d_{2}}^{2}-2d_{1}d_{2}\cos(\pi-q_{3})}} \dot{q}_{3}=J_{a3}(q_{3})\dot{q}_{3} \tag{4}\]  
where J\({}_{a3}\) (q3) is a Jacobian matrix of the 3 rd joint.  
Thus, a torque acting on this joint can be derived as:  
\[\tau_{3}=\dot{J}_{a3}^{T}(q_{3})F_{3}. \tag{5}\]  
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 \(\left(\begin{array}{ccc}q_{1a}&q_{2a}&d_{3}\end{array}\right)^{T}\in R^{3}\) denotes a position vector of the actuator in actuator space.