## 4 Design of the End-Effector of Supporting Cutter-Changing Robot
### Parametric Design of End-Effector of Supporting Cutter-Changing Robot
#### 4.2.3 Parametric Design of the Flexible Connecting Mechanism  
The flexible connecting mechanism consists of two universal joints. Let the rod length of universal joint 1 be \(s\) and the angle be \(\varepsilon\). Let the rod length of universal joint 2 be \(t\) and the angle be \(\mu\), as shown in Figure 16.  
When the universal wrench is butted with the fastening bolt, the deviation includes radial deviation \(z\) and angular deviation \(\theta_{1}\). \[Z=R_{1}+L+m \tag{52}\]  
\[\theta_{1}=\max\{\alpha,\beta\}+\theta_{y} \tag{53}\]  
When the universal wrench is docked with the fastening bolt, the fastening bolt enters the guiding mechanism. Due to the radial deviation and the angle deviation, the axes of the two are not collinear and not parallel, and the guiding mechanism will rotate around the universal joint 1 by an angle of \(\theta_{2}\). During the process of the universal wrench jamming the bolt, the flexible connecting mechanism rotates to eliminate this angle deviation. (1) Determination of the rotation angle of the universal joint 1, \(\varepsilon\)  
The connecting process of the guiding mechanism and the fastening bolt is shown in Figure 17, and the geometric relationship of each parameter is shown in Figure 18.  
Figure 16: Parameters of the flexible connecting mechanism. Figure 17: The connecting process of the guiding mechanism and the fastening bolt. Suppose the distance from the small end section of the guiding mechanism to the center of rotation of the universal joint 1 is \(OB=j\). The radius of the small end of the guide cone is \(BC\). In the figure, \(AD\) is the axis of the disc cutter tightening bolt. To ensure reliability, take the safety factor of 1.2. Then, according to the geometry:  
\[ \begin{array}{l}\varepsilon=1.2\times\max\{\theta_{1},\theta_{2}\}\\ =1.2\times\max\Biggl{\{}\begin{array}{l}\max\{\alpha,\beta\},\\ \arcsin\frac{\max\{u,v-\mathrm{m}\}+m}{j}-\arcsin\frac{R_{2}}{j\times\cos 30^{ \circ}}\end{array}\Biggr{\}}\end{array}  \tag{54}\]  
(2)  
Determination of the rotation angle of the universal joint 2, \(\mu\)  
The process of connecting the end-effector of the cutter-changing robot with the bolt is shown in Figure 19. In this process, the universal joint 2 rotates at an angle \(\theta_{6}\), and the universal joint 1 rotates at an angle \(\theta_{5}\) in a reverse direction so that the axis of the tightening bolt is parallel to the axis of the universal wrench. There are two cases of eccentricity and non-eccentricity, and the eccentric distance \(k\) is the radius of the steel bar, that is, \(k=e/2\). Under the condition that there is no eccentricity between the universal wrench and the fastening bolt, the schematic diagram of the universal wrench and the bolt is shown in Figure 20a. When there is an eccentricity \(k\) between the universal wrench and the fastening bolt, the schematic diagram of the universal wrench and the bolt is shown in Figure 20b. Figure 19: The process of connecting the universal wrench with the bolt. Figure 18: The geometric relationship between the guiding mechanism and the fastening bolt. In the figure, \(HI\) is equal to the working depth of the universal wrench steel bar, \(f\). To ensure reliability, take the safety factor of 1.2, then,  
\[\mu=1.2\times\max\left\{\theta_{\theta},\ \theta_{6}^{\prime}\right\} \tag{55}\] \[=1.2\times\max\left\{ \begin{array}{c}\mathrm{arc}\ \sin\frac{(L+s-f)\times\sin(\max\{a,\beta\})+\max\{u,\,v-m\}+m}{t},\\ \frac{(L+s+f)\sin(\max\{a,\beta\})+\max\{u,\,v-m\}+m-\frac{s}{2}\times\cos(\max \{a,\,\beta\})}{t}\end{array} \right\}\]  
The Simulation Analysis of End-Effector of the Cutter-Changing Robot 4.3.1. Function Simulation Analysis of End Effector  
Verify the docking function of the end-effector based on ADAMS software. Add connecting kinematic pairs and forces according to the actual situation, as shown in Figure 21a, and conduct docking simulation without initial error, single radial displacement error, single angular displacement error and comprehensive error, as shown in Figure 21b-e.