# Stiffness-Based Cell Setup Optimization for Robotic Deburring with a Rotary Table
## 4 Optimization of Robotic Deburring
### Workpiece Initial Placement
#### 4.1.4 Objective Function Constraints  
Some workpiece offsets accepted by the algorithm may result in an undesirable toolpath placement that makes it impossible to debur the workpiece with the desired accuracy. The individuals that lead to an undesirable toolpath placement are artificially penalized to degrade their fitness and reduce their chances of advancing to the next generation. The traits of the punished individuals will be less common in the individuals that form the future generations.  
The first case of undesirable toolpath placement is when at least one toolpath segment lies outside the cylinder defined by the outer edge of the rotary table, as follows:  
\[\left(x_{t}-x_{0}\right)^{2}+\left(y_{t}-y_{0}\right)^{2}<r^{2},\ t\ =\ 1,\ldots,n \tag{12}\]  
where \(x_{t}\) and \(y_{t}\) are the coordinates of the current machining toolpath segment and \(n\) is the total number of toolpath segments. At such a machining toolpath segment, a workpiece protrusion occurs, which reduces the clamping stiffness of the workpiece. The clamping stiffness of the workpiece is considered negligible without a protrusion.  
Another case of undesirable toolpath placement is when a toolpath segment cannot be reached due to the robot's axis limits. The kinematics of the robot are different from that of a machine tool, which usually has at least three linear axes; therefore, some seemingly easy-to-reach toolpath segments are unreachable by the robot. In addition, the robot is equipped with a deburring head, which further increases the kinematic complexity of the system.  
The limit values for the position axes with respect to the home position of the robot, as defined by the DH-parameter values in Table 1, are as follows:  
\[\left\{\begin{array}{l}\theta_{1}\in(-180^{\circ},180^{\circ})\\ \theta_{2}\in(-55^{\circ},65^{\circ})\\ \theta_{3}\in(-60^{\circ},65^{\circ})\end{array}\right. \tag{13}\]  
where \(\theta_{1},\ \theta_{2},\ \theta_{3}\) are the first three axes of the robot.  
The limit values for the orientation axes are as follows:  
\[\left\{\begin{array}{l}\theta_{4}\in(-200^{\circ},200^{\circ})\\ \theta_{5}\in(-120^{\circ},120^{\circ})\\ \theta_{6}\in(-300^{\circ},300^{\circ})\end{array}\right. \tag{14}\]  
where \(\theta_{4},\ \theta_{5},\ \theta_{6}\) are the last three axes of the robot.  
The last considered case of undesired toolpath placement is when at least one toolpath segment cannot be reached by the robot with the desired accuracy. For position coordinates, the restriction is set above the standard CNC machining accuracy of 3\(\cdot\)10\({}^{-5}\) m. The acceptable deburring accuracy is defined as follows:  
\[\left\{\begin{array}{l}\left|x_{i}-x_{i,des}\right|<3\cdot\)10\({}^{-6}\\ y_{i}-y_{i,des}\right|<3\cdot\)10\({}^{-6}\\ z_{i}-z_{i,des}\left|<3\cdot\)10\({}^{-6}\end{array}\right. \tag{15}\]For orientation, the restriction is set as follows:  
\[\left\{\begin{array}{l}\left|\alpha_{i}-\alpha_{i,des}\right|<3{\cdot}10^{-4}\\ \left|\beta_{i}-\beta_{i,des}\right|<3{\cdot}10^{-4}\\ \left|\gamma_{i}-\gamma_{i,des}\right|<3{\cdot}10^{-4}\\ \end{array}\right. \tag{16}\]