# Stiffness-Based Cell Setup Optimization for Robotic Deburring with a Rotary Table
## 5 Results and Discussion  
The first solution was obtained as an inverse kinematic problem with the engine block placed in the center of the table, as shown in Figure 1. With the non-optimized setup, the robot manages to debur the outer edges on one side of the workpiece but exceeds the joint limits before reaching the cylinder bores, even though the table rotation is enabled, as explained in Section 4.2. Several feasible solutions were found by optimizing the workpiece placement. The best solution was found with the minimum total inverse of stiffness along the toolpath at 0.339 mm/N. The corresponding optimization variables are \(x_{var}~{}=~{}1.631988\) m, \(y_{var}~{}=~{}1.075345\) m and \(\alpha_{var}~{}=~{}2.6278\) rad. The optimized setup is shown in Figure 8.  
Toolpath tracking was divided into the following two types of motion: cutting motion and non-cutting motion. Based on the type of motion in the current toolpath segment, different sizes of rotary table rotation were allowed. Table 6 shows the maximum rotational difference on the representative toolpath segments when deburring or moving between the deburring features. A comparison between the original (a) and the optimized (b) toolpaths is shown in Figure 9. Both toolpaths are shown relative to the base frame of the robot. The optimized toolpath spans a larger workspace and gives the robot more freedom to adopt a stiff posture. The largest distortions seen in the Cartesian representation of the optimized deburring toolpath occur during non-cutting motions. This can be observed even for short segments such as the path between the deburring of cylinder holes, where the toolpath is compressed into a smaller space compared to the original toolpath. Toolpath distortion is virtually non-existent during cutting motions because the permissible table rotations and the number of toolpath segments are small. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Outer Edges** & **Bore 1 Edges** & **Bore 1–4** & **Traverse** \\ \hline Table rotation (\({}^{\circ}\)) & 0.12 & 0.05 & 5.07 & 147.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Maximum rotational difference on different toolpath segments. Figure 8: Optimized robotic deburring cell setup. When deburring the outer edges on the left side of the engine block, the maximum rotational difference of the table is 0.12\({}^{\circ}\). It should be noted that the toolpath segments created for deburring the outer edges sometimes require long linear motions. To reach the next toolpath segment during a long linear motion, the robot must travel a long distance, which can lead to a reduction in stiffness or even a constraint violation. To increase the performance of the robot and reduce the risk of constraint violation, the table rotation could be activated during the long linear motions. This would require breaking down the long linear motions into smaller toolpath segments, which was not performed in this study. When deburring the edges of the first cylinder bore, the maximum rotational difference of the table is 0.05\({}^{\circ}\). One reason for the small rotational difference is that only every tenth toolpath segment was considered in the optimization. If the complete toolpath was considered, each additional segment would also contribute to the maximum rotation difference. Second, the absolute maximum table rotation was evaluated for a worst-case scenario. If the Euclidean distance, based on which the absolute maximum table rotation was calculated, were to take the current toolpath segment into account, a larger maximum table rotation difference could also be achieved. Between the deburring of each edge and each cylinder bore, non-cutting motions occur. During the non-cutting motions, free table rotation is enabled. The table rotation during deburring of the first and the fourth cylinder bore on the left side of the engine block was 5.07\({}^{\circ}\).