## 5 Identification  
Testing configurations Test 1 and Test 2 made it possible to identify modes of vibration clearly dominated by the stiffness properties of J1 and J2 respectively, and these stiffness properties are the most important from the point of view of robot operations (e.g., machining or assembly). Testing configurations Test 4 and Test 5 made it possible to identify modes of vibration with a large contribution of the stiffness of J4 and J5. Owing to the small moments of end-effector forces about J4 and J5, the stiffness of wrist joints is less critical than the stiffness of arm joints. Testing configuration Test 3 minimized the coupling between J2 and J3, nevertheless the identified modal shape a at 30.8 Hz showed some influence of the rotation about J2 and of structural deformation, since in this configuration the robot has a structural mode ad 34.7 Hz. Therefore, the measured frequency is chiefly influenced by the stiffness of J3, but it includes the contribution of other stiffness properties. For the aforementioned reasons, the stiffness about J1, J2, J4 and J5 was directly calculated from Equation (7), assuming modal coordinate \(\eta_{i}\) coincident with \(q_{i}\) and using a 1 DOF model:  
\[f_{ni}\ =\ \frac{1}{2\pi}\sqrt{\frac{k_{i}}{I_{iz}\big{(}q_{i}\big{)}}} \tag{10}\]  
\[k_{i}\ =\ (2\pi f_{ni})^{2}I_{iz_{i}\big{(}q_{i}\big{)}} \tag{11}\]  
where \(f_{ni}\) is the natural frequency measured in testing configuration \(q_{i}\)  
Figure 11: Test 5, (**a**) direct point frequency response function (FRF) modulus, (**b**) phase, (**c**) complex mode indicator function (CMIF). The damping coefficient of joint \(i\) (\(c_{i}\)) was calculated with the following equation:  
\[c_{i}\ =\ \frac{2\zeta_{i}k_{i}}{2\pi f_{ni}} \tag{12}\]  
where \(\zeta_{i}\) is the identified damping ratio of mode \(i\). The results of these calculations are summarized in Table 2.  
For joint \(\mid\)3, the value given by Equation (11) was considered only a first estimate of the actual joint stiffness (\(k_{3}\)). The identification of this stiffness was then improved carrying out an optimization. The following penalty function was defined:  
\[F_{p}(k_{3})\ =\ \sum_{j\ =\ 1}^{5}\sum_{k\ =\ 1}^{5}\left(f_{njk}-f_{pjk}(k_{3}) \right)^{2} \tag{13}\]  
This is the sum of the squared differences between the measured natural frequencies and the natural frequencies \(f_{pjk}\) predicted by the model with assigned \(k_{3}\). Five modes of vibration (\(j\ =\ 1,\dots,\ 5\)) in the five testing configurations are considered (\(k\ =\ 1,\dots,\ 5\)). Minimization was carried out with the function fmincon of MATLAB. The optimized stiffness \(k_{3}\) = 6485 Nm/rad was rather different from the first attempt value. Finally, damping coefficient was calculated according to Equation (12) and resulted \(c_{3}\) = 1.41 Nms/rad. The identified stiffness and damping properties were the implemented in the mathematical model of the robot (Equation (3)), and the natural frequencies and damping ratios were calculated in some validation configurations that were experimentally tested as well. Joint variables of validation configurations are summarized in Table 3.  
Table 4 shows the comparison between numerical values and experimental values in the validation configurations. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Configuration** & & **Mode 1** & **Mode 2** & **Mode 3** & **Mode 4** & **Mode 5** \\ \hline \multirow{4}{*}{Validation 1} & Calc. frequency [Hz] & 15.4 & 16.1 & 33.0 & 57.2 & 171.9 \\  & Exp. frequency [Hz] & 16.3 & 17.8 & 36.2 & 58.4 & 170.5 \\  & Calc. damping \% & 2.6 & 1.8 & 2.4 & 3.0 & 2.8 \\  & Exp. damping \% & 3.9 & 2.1 & 3.1 & 3.4 & 3.2 \\  & Calc. frequency [Hz] & 12.9 & 15.7 & 46.1 & 57.2 & 172.3 \\  & Exp. frequency [Hz] & 13.3 & 14.5 & 40.2 & 59.1 & 171.8 \\  & Calc. damping \% & 1.33 & 2.8 & 3.4 & 3.0 & 2.8 \\  & Exp. damping \% & 2.7 & 2.4 & 2.5 & 3.7 & 2.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Validation results.