## 2 System Dynamics including Actuator  
An n-DOF serial hydraulic manipulator is structured as depicted in Figure 1. The manipulator dynamics is presented by [44]:  
\[\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}\Big{(}\mathbf{q},\dot{ \mathbf{q}}\Big{)}\dot{\mathbf{q}}+\mathbf{G}(\mathbf{q})+\mathbf{\Delta}+ \mathbf{J}^{T}(\mathbf{q})\mathbf{F}_{ext}=\boldsymbol{\tau} \tag{1}\]  
where \(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}}\in R^{n}\) are position, angular velocity, and angular acceleration vectors of each joint, respectively. \(\mathbf{M}(\mathbf{q})\in R^{n\times n}\) is the symmetric and positive definite matrix of nominal inertia. \(\mathbf{C}\Big{(}\mathbf{q},\dot{\mathbf{q}}\Big{)}\in R^{n\times n}\) denotes the nominal Coriolis and Centrifugal term matrix. \(\mathbf{G}(\mathbf{q})\in R^{n}\) is the nominal gravity term. \(\mathbf{\Delta}\) denotes lumped parametric uncertainties, disturbances, and model error. \(\mathbf{F}_{ext}\in R^{n}\) represents external load induced from environment interaction in the end-effector. \(\boldsymbol{\tau}_{ext}\in R^{n}\) is the torque vectors acting on joints. \(\mathbf{J}(\mathbf{q})\in R^{n\times n}\) is the Jacobian matrix to transfer from Cartesian to manipulator joint space as  
\[\mathbf{x} =\mathbf{f}(\mathbf{q}) \tag{2}\] \[\dot{\mathbf{x}} =\frac{\partial\mathbf{f}(\mathbf{q})}{\partial\mathbf{q}}= \mathbf{J}(\mathbf{q})\dot{\mathbf{q}}\]  
**Property 1**. : \(\dot{\mathbf{M}}(\mathbf{q})\ddot{\mathbf{q}}-2\mathbf{C}\Big{(}\mathbf{q}, \dot{\mathbf{q}}\Big{)}\) _is a skew matrix that satisfies the condition \(\mathbf{X}^{T}\Big{(}\dot{\mathbf{M}}(\mathbf{q})-2\mathbf{C}\Big{(}\mathbf{q},\dot{\mathbf{q}}\Big{)}\Big{)}\mathbf{X}=0\) for any non-zero vector \(\mathbf{X}\in R^{n}\)._  
**Property 2**. : _The inertia matrix \(\mathbf{M}(\mathbf{q})\in R^{n\times n}\) is a symmetric and positive definite matrix and is bounded as \(m_{l}||\mathbf{X}||^{2}\leq\mathbf{X}^{T}\mathbf{M}(\mathbf{q})\mathbf{X}\leq M ^{u}||\mathbf{X}||^{2}\), in which \(m_{l}\) and \(M^{u}\) are lower and upper bound positive constants, respectively._  
Movements of each link are driven by an electro-hydraulic actuator. Herein, two type of the electro-hydraulic actuator are utilized as depicted in Figure 2. Due to its complicated characteristics, the actuator dynamics should be derived and included in the whole system dynamics. Define \(\mathbf{x}_{a}=\left( \begin{array}{cccc}x_{a1}&x_{a2}&\ldots&x_{an-1}&q_{an} \end{array} \right)\in R^{n}\) is a vector of actuator positions in actuator-space. The actuator position can be relevant calculated from manipulator joint space as  
\[\mathbf{x}_{a} =f_{a}(\mathbf{q}) \tag{3}\] \[\dot{\mathbf{x}}_{a} =\mathbf{J}_{a}(\mathbf{q})\dot{\mathbf{q}}\]  
Figure 1: Architecture of the n-DOF serial hydraulic manipulator. Without losing generality, let us consider one element of the total robotic system. The actuator force/torque is calculated as  
\[\tau_{i}=\left\{ \begin{array}{ll}A_{r}(P_{ai1}-P_{ai2})&(\text{for rotary actuator})\\ J_{ai}^{T}(q)(A_{ai1}P_{ai1}-A_{ai2}P_{ai2})&(\text{for cylinder})\end{array} \right. \tag{4}\]  
where \(A_{r}\), \(A_{ai1}\), and \(A_{ai2}\) are volumetric area of rotary actuator and areas of piston head part and rod part of cylinders, respectively. \(P_{ai1}\) and \(P_{ai2}\) (\(i=1,\,2,\,\ldots,\,n\)) are pressures of two chambers of each actuator. These parameters are expressed as  
\[\dot{P}_{ai1}=\frac{\beta}{V_{i1_{0}}+A_{ai1}x_{ai}}\bigg{[}+Q_{ai1}-A_{ai1} \frac{\partial x_{ai}}{\partial q_{i}}\dot{q}_{i}-C_{lank\_i}(P_{ai1}-P_{ai2}) \bigg{]} \tag{5}\]  
\[\dot{P}_{ai2}=\frac{\beta}{V_{i2_{0}}+A_{ai2}(L_{ai}-x_{ai})}\bigg{[}-Q_{ai2}+ A_{ai2}\frac{\partial x_{ai}}{\partial q_{i}}\dot{q}_{i}+C_{lank\_i}(P_{ai1}-P_{ai2}) \bigg{]} \tag{6}\]  
where \(\beta\) is the effective bulk modulus, \(V_{ij_{0}}(q)_{(j=1,2)}\) are initial volumes of the two chambers, \(C_{lank\_i}\) is coefficient of internal leakage, and \(Q_{aij}(t)_{(j=1,2)}\) represents two flows into and out of the two chambers. **Assumption 1**.