## 4 Robotic Deburring Process Parameter Control Method based on Fuzzy Control
In this section, a fuzzy controller with the above advantages of the fuzzy control is designed for a proposed robotic deburring process parameter control method to systematically accommodate robotic deburring. The control objectives of the designed fuzzy control system in this section are the robot manipulator deburring process parameters, i.e., robotic spindle speed and robotic feed. The robotic spindle speed and the robotic spindle load of the robot manipulator deburring system can be controlled by adjusting the control voltage of the robotic spindle and the robotic feed, in order to realize the robotic deburring with a constant cutting speed and a constant cutting force. Hence, the designed fuzzy controller in this section is a two-input and two-output control system (shown in Figure 7). Here, input variables are robotic spindle speed and robotic spindle load, and output variables are control voltage of the robotic spindle and robotic feed. In this article, a parallel structure adopted is used by the fuzzy controller to independently control the control voltage of the robotic spindle and the robotic feed. There are two parts of representing fuzzy variables in the designed fuzzy controller. First, error of robotic spindle speed \(E_{1}\) and change in error of robotic spindle speed \(EC_{1}\), and error of robotic spindle load \(E_{2}\) and change in error of robotic spindle load \(EC_{2}\), are represented as input fuzzy variables for controlling the control voltage of the robotic spindle and the robot feed, respectively. Second, change in control voltage of the robotic spindle \(u_{1}\) and change in robotic feed \(u_{2}\) are represented as output fuzzy variables. To achieve discrete membership functions, a discrete set with thirteen elements and a discrete set with seven elements are defined in the designed fuzzy controller for \(E_{1}\), \(EC_{1}\), and \(u_{1}\), and for \(E_{2}\), \(EC_{2}\), and \(u_{2}\), respectively, and corresponding sets are \(U_{1}=\left\{ \begin{array}{cccccccc}-6&-5&\cdots&-1&0&+1&+2&\cdots&6\\ \end{array} \right.\) and \(U_{2}=\left\{ \begin{array}{cccccccc}-3&-2&-1&0&+1&+2&+3\\ \end{array} \right\}\), respectively. Assume that variation ranges of \(E_{1}\), \(EC_{1}\), \(E_{2}\) and \(EC_{2}\) are \([-S_{e},S_{e}]\), \([-S_{ec},S_{ec}]\), \([-L_{e},L_{e}]\) and \([-L_{ec},L_{ec}]\), respectively. Combining these variation ranges and corresponding discrete sets, several quantizers corresponding to \(E_{1}\), \(EC_{1}\), \(E_{2}\) and \(EC_{2}\) are defined respectively as follows:  
\[K_{Se}=6/S_{e}, \tag{3}\]  
\[K_{Sec}=6/S_{ec}, \tag{4}\]  
\[K_{Le}=3/L_{e}, \tag{5}\]  
\[K_{Lec}=3/L_{ec}. \tag{6}\]  
Here, each quantizer maps an inbound measurement within the variation range to a nearest integer element in the corresponding discrete set. Similarly, scaling factors for the output variables (i.e., control variables) of \(u_{1}\) and \(u_{2}\), where corresponding variation ranges are \([-V_{u},V_{u}]\) and \([-F_{u},F_{u}]\), respectively, are defined as follows:  
\[K_{Vu}=V_{u}/6, \tag{7}\]  
\[K_{Fu}=F_{u}/3. \tag{8}\]  
Assuming exact values obtained by the fuzzy reasoning for change in control voltage of the robotic spindle \(u_{1}\) and change in robotic feed \(u_{2}\) are \(u_{1i}\) and \(u_{2i}\), respectively, then corresponding exact values of variation ranges \(V_{ui}\) and \(F_{ui}\) can be obtained by using above defined scaling factors. The detailed representations are given by  
\[V_{ui}=K_{Vu}\cdot u_{1i}, \tag{9}\]  
\[F_{ui}=K_{Fu}\cdot u_{2i}. \tag{10}\]  
Figure 7: Schematic diagram for deburring process parameter control. The fuzzy rule with automatic adjusting factors in the whole set using analytical expressions for fuzzy controller are usually designed as follows. Suppose that sets for error \(E\), change in error \(EC\) and control variable \(u\) select as \(E=EC=u=\{-N,\cdots,-2,-1,0,1,2,\cdots,N\}\), the fuzzy rule can be expressed as  
\[\left\{