# A Cooperative Control Method for Excavation Support Robot with Desired Position/Posture
## 2 ESR System Modeling and Description
### Kinematic Model with AHC for ESR  
The ESR proposed in this study is based on the extension and retraction of the lateral AHC to adjust position/posture. The coupled motion model of the ESR and the support body is illustrated in Figure 7, where the scroll angles of ESR are set as \(\gamma_{L}\) and \(\gamma_{R}\), and the declination angles are set as \(\alpha_{L}\) and \(\alpha_{R}\). The distances from the position key points to the defined edges are, respectively, identified as \(d_{1}\) to \(d_{8}\). The specified space width is \(B\) and the height is \(H\). The expected distances between the ESR and the two edge sidewalls are identified as \(d_{Le}\) and \(d_{Re}\). The overall longitudinal length of the ESR is \(L\), and the lateral width is \(b\). The distances from the support point of the core position to its center point are respectively calibrated \(m\) and \(n\). Taking the geometric center points \((x_{L0},y_{L0},Z_{L0})\) and \((x_{R0},y_{R0},Z_{R0})\) of the unilateral ESR as the origin of the body coordinate system, the ESR body coordinate systems \((O_{L}-X_{L}Y_{L}Z_{L})\) and \((O_{R}-X_{R}Y_{R}Z_{R})\) are both feasible in the kinematic model. The geometric center of the roadway section is set as the origin, the  
Figure 5: A model of the hydraulic system with AHC for ESR. Figure 6: The hydraulic system with the lateral AHC for ESR. direction of the roadway is the \(X_{0}\) axis direction of the coordinate system, and the direction perpendicular to the side of the roadway is the axis direction of the coordinate system. The right-hand rule determines the direction of the \(Z_{0}\) axis coordinate system, which follows the roadway absolute coordinate system (\(O_{0}-X_{0}Y_{0}Z_{0}\)). In the ESR ontology coordinate system, the left support is taken as an example to illustrate, and its position key point coordinates are marked \((m,B/2,H/2),(-n,B/2,H/2),(m,B/2,-H/2)\), and \((-n,B/2,-H/2)\). The numerical algorithms for roll and declination angles are written as  
\[\left\{ \begin{array}{l}\alpha_{L}=\arccos\frac{(m+n)}{\sqrt{(m+n)^{2}+(d_{i +1}-d_{i})^{2}}}(i=1,3)\\ \alpha_{R}=\arccos\frac{(m+n)}{\sqrt{(m+n)^{2}+(d_{i+1}-d_{i})^{2}}}(i=5,7) \end{array} \right. \tag{17}\]  
\[\left\{ \begin{array}{l}\gamma_{L}=\arccos\frac{H\cos\gamma_{L}}{\sqrt{H^{2} \cos^{2}\gamma_{L}+(d_{3}-d_{1})^{2}\cos^{2}\gamma_{L}}}=\arccos\frac{H}{ \sqrt{H^{2}+(d_{i+2}-d_{i})^{2}}}(i=1,2)\\ \gamma_{R}=\arccos\frac{H\cos\gamma_{L}}{\sqrt{H^{2}\cos^{2}\gamma_{L}+(d_{3} -d_{1})^{2}\cos^{2}\gamma_{L}}}=\arccos\frac{H}{\sqrt{H^{2}+(d_{i+2}-d_{i})^{2 }}}(i=5,6)\end{array} \right. \tag{18}\]  
The main view based on the kinematics model can be derived, which is expressed by the distance \(x_{L}\) from the coordinate origin \(O_{L}\) to the left edge of the roadway side  
\[x_{L}=\left(\frac{m}{(m+n)}(d_{1}+d_{3})+\frac{b}{2}\right)\cos\alpha_{L} \tag{19}\]  
Figure 7: The kinematic model of ESR. (**a**) Main view of the model. (**b**) Top view of the model. Then, the distance \(x_{R}\) from the coordinate origin \(O_{R}\) to the right edge of the roadway is expressed as  
\[x_{R}=\left(\frac{m}{(m+n)}(d_{3}+d_{4})+\frac{b}{2}\right)\cos\alpha_{R} \tag{20}\]  
The distance from the core position key point to the space edge is written as  
\[x_{i}=(b+d_{i})\cos\alpha\ (i=1-8) \tag{21}\]  
When solving for the distance to the left of the edge, set \(\alpha=\alpha_{L}\) to hold. When solving the distance to the right side of the edge, set \(\alpha=\alpha_{R}\) to hold.