## 2 Dynamical Model for the Robotic Drilling End-Effector
### Motion Equations of the Robotic Drilling End-Effector  
The coordinate system of the designed robotic drilling end-effector is shown as Figure 2. The kinematic model is established using Denavit-Hartenberg (D-H) method. The base coordinate  
Figure 1: Structure and prototype of the designed end-effector: (**a**) the diagram of the end-effector structure; (**b**) the specific of the internal structure of the end-effector; and (**c**) the prototype of the end-effector. system is placed in rotating Joint 1 and concentric exactly with joint coordinate [1]. Because the normal adjustment unit of the end-effector consists of Joints 1 and 2, the world frame \(\{0\}\) is placed at the center of the Joint 1 and is concentric with the frame \(\{1\}\) for the sake of analysis. The transform matrixes in the coordinate system are calculated as:  
\[ \begin{array}{c}{}_{1}^{0}T=\left[\begin{array}{cccc}\cos\theta_{1}&-\sin \theta_{1}&0&0\\ \sin\theta_{1}&\cos\theta_{1}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]\\ \\ {}_{2}^{1}T=\left[\begin{array}{cccc}-\sin\theta_{2}&-\cos\theta_{2}&0&0\\ 0&0&1&0\\ -\cos\theta_{2}&\sin\theta_{2}&0&0\\ 0&0&0&1\end{array}\right]\\ \\ {}_{3}^{2}T=\left[\begin{array}{cccc}0&-1&0&0\\ 0&0&-1&d_{3}\\ 1&0&0&0\\ 0&0&0&1\end{array}\right].\end{array}  \tag{1}\]  
Both spindle feeding unit and pressing unit will not move in the normal adjusting process of the end-effector. Thus, both their acceleration and velocity are 0. A 3 \(\times\) 3-order positive definite inertial matrix is calculated as:  
\[\mathbf{D}(\mathbf{q})=\left[ \begin{array}{cccc}D_{11}&D_{12}&D_{13}\\ D_{21}&D_{22}&D_{23}\\ D_{31}&D_{32}&D_{33}\end{array} \right] \tag{2}\]  
with  
\[\mathbf{q}=\left[ \begin{array}{cccc}\theta_{1}&\theta_{2}&d_{3}\end{array}  \right]^{T} \tag{3}\]  
\[ \begin{array}{c}D_{11}&=I_{zz1}+I_{yy2}+I_{zz3}+I_{a1}+\left(I_{xx2}-I_{yy2}+ I_{yy3}-I_{zz3}-2m_{3}d_{3}c_{z3}+m_{3}d_{3}^{2}\right)\cos^{2}(\theta_{2})\\ &+\left(I_{xy2}+I_{yz3}-m_{3}d_{3}c_{y3}\right)\sin(2\theta_{2})\end{array}  \tag{4}\]  
\[D_{12}=\left(-I_{yz2}+I_{xz3}-m_{3}d_{3}c_{x3}\right)\sin(\theta_{2})+\left(I_ {xz3}-I_{xy3}\right)\cos(\theta_{2}) \tag{5}\]  
\[D_{13}=m_{3}c_{x3}\cos(\theta_{2}) \tag{6}\]  
\[D_{21}=\left(-I_{yz2}+I_{xz3}-m_{3}d_{3}c_{x3}\right)\sin(\theta_{2})+\left(I_ {xz2}-I_{xy3}\right)\cos(\theta_{2}) \tag{7}\]  
\[D_{22}=I_{z2}+I_{xx3}-2m_{3}d_{3}c_{x3}+m_{3}d_{3}^{2}+I_{a2} \tag{8}\]  
\[D_{23}=-m_{3}c_{y3} \tag{9}\]  
Figure 2: Coordination definition of the end-effector. \[D_{31}=m_{3}c_{x3}\cos(\theta_{2}) \tag{12}\]  
\[D_{32}=-m_{3}c_{y3} \tag{13}\]  
\[D_{33}=m_{3}+I_{a3} \tag{14}\]  
where \(\theta_{1}\) and \(\theta_{2}\) denote the rotation angles of the joints 1 and 2, respectively, and \(d_{3}\) denotes the linear displacement of the prismatic Joint 3. \(I_{abi}\) is the product of inertia of Joint \(i\) with respect to a pair of orthogonal axes \(a\) and \(b\), \(a\), \(b\) denote \(x\), \(y\), or \(z\); \(m_{i}\) is the mass of Joint \(i\). \(c_{ai}\) is the centroid coordinate of Joint \(i\) on axis \(a\), \(a\) denote \(x\), \(y\), or \(z\). A 3 \(\times\) 1-order vector of centrifugal and Coriolis force is calculated, regarding that \(\dot{q}_{3}=\dot{d}_{3}=0\), the result is:  
\[\mathbf{H}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}=\left[ \begin{array}{cc }h_{1}&h_{2}&h_{3}\end{array} \right]^{T} \tag{15}\]  
with  
\[