## 2 Robot Manipulator Dynamics
### Electro-Hydraulic Dynamics  
Let \(x_{a}=\left[\begin{array}{cccc}x_{1a}&x_{2a}&x_{3a}\end{array}\right]^{T}=\left[ \begin{array}{cccc}\theta_{1a}&\theta_{2a}&d_{3}\end{array}\right]^{T}\in R^{3 \times 1}\) represents the actuator space that is related to the robot joint-space as (Chapters 6-29-29)  
\[\begin{array}{l}x_{a}=h(\theta)\\ \dot{x}_{a}=J(\theta)\dot{\theta}\end{array} \tag{3}\]  
where \(h(\theta)\in R^{3\times 1}\) denotes the forward kinematics of the actuator and \(J(\theta)=\frac{\partial h}{\partial\theta}\in R^{3\times 3}\) represents the differentiable actuator Jacobian matrix as shown in Appendix B.  
The torque vector is calculated as follows (30):  
\[\tau=J(\theta)F_{a}=J(\theta)(A_{1}P_{1}-A_{2}P_{2}) \tag{4}\]  
where \(A_{i(i=1,2)}=diag(A_{i1},\ldots,A_{i3})\in R^{3\times 3}\) are an area matrix of piston head part and an area matrix of rod part. \(P_{i(i=1,2)}=\left[\begin{array}{cccc}P_{i1}&\cdots&P_{i3}\end{array}\right]^ {T}\) are the pressure vector of two chambers of each actuator.  
The hydraulic actuator pressure dynamics can be presented as follows (31):  
\[\dot{P}_{1}=\beta V_{1}^{-1}(x_{a})\left[-A_{1}\frac{\partial x_{a}}{\partial \theta}\dot{\theta}+\chi_{1}[P_{1},U[u]]U[u]\right]-\zeta_{1}(t) \tag{5}\]  
\[\dot{P}_{2}=\beta V_{2}^{-1}(x_{a})\left[A_{2}\frac{\partial x_{a}}{\partial \theta}\dot{\theta}-\chi_{2}[P_{2},U[u]]U[u]\right]+\zeta_{2}(t) \tag{6}\]  
where \(\beta\) is the effective bulk modulus, \(V_{i}(\theta)_{(i=1,2)}=diag(V_{i1}(\theta),\ldots,V_{i3}(\theta))\in R^{3 \times 3}\) are volume matrix of two chambers, \(V_{ij}=V_{ij0}+(-1)^{i+1}x_{ju}A_{ij}\), \((i=1,2;j=1,2,3)\), \(V_{ij0}(i=1,2;j=1,2,3)\) is initial volumes of two chambers, \(\zeta_{i}(t)_{(i=1,2)}\in R^{3\times 1}\) are the lumped disturbances of two chambers (internal/external leakage, modelling error), \(U=\left[\begin{array}{cccc}u_{1}&u_{2}&u_{3}\end{array}\right]^{T}\) is a control voltage vector, \(\chi_{i}(P_{i},U)_{(i=1,2)}=K_{qi}diag(\xi_{i1},\ldots,\xi_{3})\in R^{3\times 3}\), and \(K_{qi}=diag\big{(}k_{qi1},\ldots,k_{qi3})\in R^{3\times 3}\) are flow gain coefficients matrices in orifice equations of the actuators.  
\[\left\{\begin{array}{l}\xi_{1i[i=1,\ldots,3}=\sqrt{P_{s}-P_{1i}},u_{i}\geq 0 \\ \xi_{1i[i=1,\ldots,3}=\sqrt{P_{1i}-P_{r}},u_{i}<0\\ \xi_{2i[i=1,\ldots,3}=\sqrt{P_{2i}-P_{r}},u_{i}\geq 0\\ \xi_{2i[i=1,\ldots,3}  
The Equation (8) can be rewritten as follows  
\[\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=M_{o}{}^{-1}(-C_{o}(x_{1},x_{2})x_{2}-G_{o}(x_{1})-\Delta_{1}(t)+J(x )x_{3})\\ \dot{x}_{3}=F_{1}x_{2}+F_{2}U+\Delta_{2}(t)\end{array} \tag{9}\]  
where \(F_{1}=-[\kappa_{1}A_{1}+\kappa_{2}A_{2}]\frac{\partial x_{2}}{\partial x_{1}} \in R^{3\times 3}\), and \(F_{2}=\kappa_{1}\xi_{1}(P_{1},U(u))+\kappa_{2}\xi_{2}(P_{2},U(u))\in R^{3 \times 3}\).  
**Remark 1**.: _In practice, \(\det(\xi_{1}(P_{1},U(u)))\) and \(\det(\xi_{2}(P_{2},U(u)))\) are both seldom zero when the system is operating smoothly, since \(P_{1}\) and \(P_{2}\) are rarely close to \(P_{s}\) and \(P_{r}\). In the seldom case that \(\det(\xi_{1}(P_{1},U(u)))\) and \(\det(\xi_{2}(P_{2},U(u)))\) equal to zero (e.g., due to the noise in \(P_{1}\) and \(P_{2}\)) it is set to a small positive number to avoid the problem of dividing zero._