# Stiffness-Based Cell Setup Optimization for Robotic Deburring with a Rotary Table
## 3 Stiffness Model  
An approach based on structural stiffness measurements and FEM model calibration was used to build a stiffness model of the ACMA XR701 robot. The full parameter identification procedure is explained in our previous work [43]. The experimental matrix can be found in Table S1.  
ANN \(stif\_net\) was used to correlate joint rotations \(\theta_{2},\ \theta_{3},\ \theta_{4},\theta_{5}\) and the load with structural stiffness. To train the ANN \(stif\_net\), the Levenberg-Marquardt backpropagation algorithm was used with two hidden layers, the first layer with ten neurons and the second with five neurons. The best architecture of the network was found by a random search. To build the structural stiffness model, 70% of the data were used for training, 15% for validation and 15% for testing. Based on the available data, the ANN \(stif\_net\) model errors were found to be between \(-\)54.5793 and 59.5260 N/mm, corresponding to an absolute deformation error of \(-\)0.01929 and 0.046024 mm, respectively.  
The structural stiffness was determined to be load dependent, and the dependence is nonlinear. In the present study, only the output of the structural stiffness model obtained for the end-effector load of 150 N is used. The low load case is considered because the contact forces during deburring are small and mainly depend on the workpiece material and the current burr size. The same load was also used as a limit value for robotic deburring with the support of a variable stiffness mechanism [19]. However, the stiffness modeling approach presented in this work allows lower, even adjustable, stiffness values. With online force measurements, the model can also adapt to the current load. A drawback of the presented approach is the inability of the model to predict stiffness in more than one direction. To be able to align the stiffness ellipsoid of the robot along the direction of the forces applied by the machining process, additional stiffness measurements should be performed on the robot.  
Figure 7 shows the structural stiffness \(T\) in a section of the theoretical workspace of the robot for the end-effector load of 150 N. The figures were obtained by discretizing the workspace of the robot. Each point represents the end-effector position of the robot without the tool. The color of the point represents the structural stiffness of the robot. The points were generated with rotations of each robot axis as shown in Table 3. In the workspace of the robot, the distribution of the structural stiffness shows two divided areas of low stiffness, with the highest stiffness in the middle of the workspace. This is due to the topology of the robot and is characteristic of quasi-serial robots [44; 45].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\theta_{1}\) (\({}^{\circ}\)) & \(\theta_{2}\) (\({}^{\circ}\)) & \(\theta_{3}\) (\({}^{\circ}\)) & \(\theta_{4}\) (\({}^{\circ}\)) & \(\theta_{5}\) (\({}^{\circ}\)) & \(\theta_{6}\) (\({}^{\circ}\)) \\ \hline min. & 0 & \(-\)50 & 30 & \(-\)200 & \(-\)120 & 0 \\ max. & 0 & 65 & 155 & 200 & 120 & 0 \\ step & 0 & 5 & 5 & 50 & 30 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Limit values and step sizes for discrete rotations of the robot axis.  
Figure 6: Cutting and non-cutting motions on the deburring toolpath.