# A Cooperative Control Method for Excavation Support Robot with Desired Position/Posture
## 2 ESR System Modeling and Description
### Hydraulic System Model with AHC for ESR
Equation (4) is simplified to:  
\[Q_{L}=(A_{p}-\pi r^{2})\frac{dx_{p}}{dt}+\frac{V_{t}}{4\beta_{e}}\frac{dp_{L}} {dt}+(C_{pi}+\frac{C_{pe}}{2})p_{L} \tag{5}\]  
Further sorting out Equation (5) leads to:  
\[Q_{L}=A_{pe}\frac{dx_{p}}{dt}+\frac{V_{t}}{4\beta_{e}}\frac{dp_{L}}{dt}+C_{pi} p_{L} \tag{6}\]  
where \(A_{pe}\) indicates the equivalent area of the AHC piston, \(A_{pe}=A_{p}-\pi r^{2}\), \(x_{p}\) represents the (displacement) stroke of the AHC piston, and \(C_{pi}\) denotes the total leakage coefficient. (3) The expression of force balance equation with external load is rewritten as  
\[A_{pe}p_{L}=m_{t}\frac{d^{2}x_{p}}{dt}+c_{t}\frac{dx_{p}}{dt}+k_{t}x_{p} \tag{7}\]  
where \(m_{t}\) represents the effective total mass of the ESR (kg), \(c_{t}\) is the viscous damping coefficient of the piston in a transverse AHC with an external load (\(\text{N}\cdot\text{m}/\text{s}\)), and \(k_{t}\) denotes stiffness coefficient with an external load (\(\text{N}/\text{m}\)). Equations (1), (6), and (7) are expanded by the Laplace Transform, leading to:  
\[\left\{ \begin{array}{l}Q_{L}(s)=K_{q}x_{p}(s)-K_{c}p_{L}(s)\\ Q_{L}(s)=A_{pe}\text{s}x_{p}(s)+\frac{V_{t}}{4\beta_{e}}\text{s}p_{L}(s)+C_{ pt}p_{L}(s)\\ p_{L}(s)=\frac{(m_{t}s^{2}+c_{t}s+k_{t})}{A_{pe}}x_{p}(s)\end{array} \right. \tag{8}\]  
Notice that the spool displacement is the input and the piston displacement of the lateral AHC is the output. With this in mind, the transfer function of the system is expressed as:  
\[G(s)=\frac{\frac{m_{t}V_{t}}{4A_{pe}\beta_{e}}s^{3}+\Big{(}\frac{c_{t}V_{t}}{4 A_{pe}\beta_{e}}+\frac{(K_{c}+c_{pi})m_{t}}{A_{pe}}\Big{)}s^{2}+\Big{(}A_{pe}+ \frac{(K_{c}+c_{pi})c_{t}}{A_{pe}}+\frac{k_{t}V_{t}}{4A_{pe}\beta_{e}}\Big{)}s +\frac{(K_{c}+c_{pi})k_{t}}{A_{pe}} \tag{9}\]  
Assuming that \(K_{cpt}=K_{c}+c_{pt}\) holds gives the simplified equation:  
\[G(s)=\frac{K_{p}}{\frac{m_{t}V_{t}}{4A_{pe}\beta_{e}}s^{3}+\Big{(}\frac{c_{t}V_ {t}}{4A_{pe}\beta_{e}}+\frac{K_{cpt}m_{t}}{A_{pe}}\Big{)}s^{2}+\Big{(}A_{pe}+ \frac{K_{cpt}c_{t}}{A_{pe}}+\frac{k_{t}V_{t}}{4A_{pe}\beta_{e}}\Big{)}s+\frac{ K_{cpt}k_{t}}{A_{pe}}} \tag{10}\]Considering that the viscous damping coefficient \(c_{t}\) of the lateral AHC piston under the external load is negligible, then \(\frac{c_{t}K_{opt}}{A_{pe}^{2}}\ll 1\) is satisfied. Simplified Equation (10) reads as  
\[G(s)=\frac{K_{p}/A_{pe}}{\frac{m_{t}V_{t}}{4\beta_{e}A_{pe}^{2}}s^{3}+\left( \frac{c_{t}V_{t}}{4\beta A_{pe}^{2}}+\frac{K_{opt}m_{t}}{A_{pe}^{2}}\right)s^{2 }+\left(1+\frac{k_{t}V_{t}}{4\beta_{e}A_{pe}^{2}}\right)s+\frac{K_{opt}k_{t}}{A _{pe}^{2}}} \tag{11}\]  
Let \(\sqrt{\frac{m_{t}V_{t}}{4\beta_{e}A_{pe}^{2}}}=\frac{1}{\omega_{t}},\frac{K_{ opt}}{A_{pe}}\sqrt{\frac{m_{t}\beta_{e}}{V_{t}}}=\xi_{t},\frac{4A_{pe}^{2} \beta_{e}}{V_{t}}=K_{h}\) hold, and the simplified Formula (11) is expressed as  
\[G(s)=\frac{K_{p}/A_{pe}}{\frac{1}{\omega_{t}^{2}}s^{3}+\frac{2\zeta_{t}}{ \omega_{t}}s^{2}+\left(1+\frac{k_{t}}{K_{h}}\right)s+\frac{K_{opt}k_{t}}{A_{pe }^{2}}} \tag{12}\]  
Set the assumption that \(\frac{K_{opt}\sqrt{m_{t}k_{t}}}{(1+\frac{k_{t}}{K_{h}})A_{pe}^{2}}\ll 1\) holds, further simplifying Equation (12), and is written as:  
\[G(s)=\frac{K_{p}/A_{pe}}{\left[\left(1+\frac{k_{t}}{K_{h}}\right)s+\frac{K_{ opt}k_{t}}{A_{pe}^{2}}\right]\left(\frac{1}{\omega_{tn}^{2}}s^{2}+\frac{2 \zeta_{tn}}{\omega_{tn}}s+1\right)} \tag{13}\]  
where \(\omega_{tn}\) represents the system integrated natural frequency, and \(\omega_{tn}=\omega_{t}\sqrt{1+\frac{k_{t}}{K_{h}}}\) holds, \(\xi_{tn}\) is the system integrated damping ratio, and \(\xi_{tn}=\frac{1}{2\omega_{tn}}\left(\frac{4K_{opt}\beta_{e}}{(1+\frac{k_{t}} {K_{h}})V_{t}}+\frac{c_{t}}{m_{t}}\right)\) holds. As mentioned previously, the poles of the system are expressed as \(s=-\frac{K_{opt}k_{t}K_{h}}{(K_{opt}+k_{t})A_{pe}^{2}}\), \(s=-\omega_{tn}(\xi_{tn}\pm\sqrt{\xi^{2}{}_{tn}-1})\).