# Natural Frequency Prediction Method for 6R Machining Industrial Robot
## 3 Results  
### Prediction Model Verification and Analysis  
#### 3.1.1 Model Verification with Random Configurations  
Sixteen groups of \(q_{2}\)-\(q_{5}\) values are picked randomly from the variable range defined in Section 2.1.1, through which 16 corresponding robot configurations are defined for model verification. The simulation images of the 16 configurations are reranked by joint angles \(q_{2}\) and \(q_{3}\) as shown in Figure 7, and the measured natural frequency is listed in the bottom left corner of each image. The configurations are nearly uniformly distributed in the variable range, which ensures the comprehensiveness of verification objects and rationality of verification result.  
\begin{table}
\begin{tabular}{c c c} \hline
**Items** & **Coefficient/M** & **Coefficient/M\({}_{2}\)** \\ \hline Constant & \(1.138\times 10^{1}\) & \(1.086\times 10^{1}\) \\ \(q_{2}\) & \(8.746\times 10^{-3}\) & \(1.214\times 10^{-2}\) \\ \(q_{3}\) & \(5.987\times 10^{-2}\) & \(6.611\times 10^{-2}\) \\ \(q_{4}\) & \(-7.532\times 10^{-5}\) & \(7.873\times 10^{-4}\) \\ \(q_{5}\) & \(1.374\times 10^{-4}\) & \(-3.785\times 10^{-3}\) \\ \(q_{2}^{2}\) & \(8.821\times 10^{-4}\) & \(1.179\times 10^{-3}\) \\ \(q_{3}^{2}\) & \(5.322\times 10^{-4}\) & \(5.894\times 10^{-4}\) \\ \(q_{4}^{2}\) & \(-2.464\times 10^{-5}\) & \(1.242\times 10^{-5}\) \\ \(q_{5}\) & \(-1.018\times 10^{-4}\) & \(7.342\times 10^{-5}\) \\ \(q_{2}q_{3}\) & \(1.171\times 10^{-4}\) & \(1.253\times 10^{-4}\) \\ \(q_{2}q_{4}\) & \(4.323\times 10^{-6}\) & - \\ \(q_{2}q_{5}\) & \(4.224\times 10^{-5}\) & - \\ \(q_{3}q_{4}\) & \(-7.206\times 10^{-6}\) & - \\ \(q_{3}q_{5}\) & \(-4.939\times 10^{-5}\) & \(-5.285\times 10^{-5}\) \\ \(q_{4}q_{5}\) & \(1.099\times 10^{-5}\) & - \\ \hline \end{tabular}
\end{table}
Table 4: Coefficients of items in M and M\({}_{2}\).  
\begin{table}
\begin{tabular}{c c c} \hline
**Model** & **h** & \(\mathbf{R_{Y}^{2}(cum)}\) & \(\mathbf{Q_{B}^{2}(cum)}\) \\ \hline M\({}_{1}\) & 2 & 0.958 & 0.776 \\ \hline M\({}_{2}\) & 2 & 0.960 & 0.778 \\ \hline M\({}_{3}\) & 2 & 0.958 & 0.764 \\ \hline M\({}_{4}\) & 2 & 0.958 & 0.762 \\ \hline M\({}_{5}\) & 2 & 0.958 & 0.723 \\ \hline M\({}_{6}\) & 2 & 0.958 & 0.717 \\ \hline \end{tabular}
\end{table}
Table 3: Validation results of M\({}_{1}\)–M\({}_{6}\).  
Figure 7: Random configurations for model verification.  
The first order natural frequencies of the 16 configurations are obtained by EMT. Then, original model M and optimized model M\({}_{2}\) are utilized to predict the first order natural frequencies. The deviation between prediction value and measured value is defined as prediction error, the details are listed in Table 5; Table 6.  
The average error and average relative error of optimized model M\({}_{2}\) are both much lower than those of original model M. The average relative error of M\({}_{2}\) is less than 5% which brings more reliability. On the other hand, seven high relative errors (more than 5%) which are underlined show up in the prediction process of M, and the maximum error reaches up to 1.78 Hz. While M\({}_{2}\) has no predictions with over 5% relative error, and the maximum error is 0.51 Hz which is acceptable. Obviously, optimized model M\({}_{2}\) has better prediction ability.  
A model (M\({}_{\text{LS}}\)) fitted by least square (LS) method with the same sample data is taken as the contrast. Moreover, the prediction errors are displayed in Table 7. Average error and average relative error of M\({}_{\text{LS}}\) are both higher than those of M and M\({}_{2}\), that is, PLS method have better performance in construct prediction model of natural frequency than LS method. In conclusion, model fitted and optimized by PLS method has the best prediction ability.