## 3 Selective Modal Testing
Joints J2 and J3 are always parallel, and generate velocity components in the same plane, Figure 3 shows that the inertial cross coupling may be large. The mass matrix of a robot with two parallel joints [28] shows an inertial cross-coupling term depending on joint variable \(q_{3}\) (the angle between the links). A numerical calculation of the ratios between the inertial cross-coupling term \(M_{23}\) and the direct inertias of J2 and J3 (\(M_{22}\) and \(M_{33}\) respectively) is depicted in Figure 5. They show that the inertial cross-coupling term tends to decrease when the two links tend to overlap, notice that the two links are aligned when \(q_{3}\ =\ 90^{\circ}\). Minimum coupling configurations can be found for \(q_{3}\ =\ -38^{\circ}\) and \(q_{3}\ =\ 293^{\circ}\). These theoretical values have to be compared with the actual range of the joint (\(-29^{\circ}\leq q_{3}\leq 256^{\circ}\)), and, for this reason, a configuration with \(q_{3}\ =\ 243^{\circ}\) (Test 3) is suited to identifying the stiffness both of J2 and J3. It is worth noticing that, in this configuration, a vertical force generates a large moment about J3, thus, it is able to excite the mode dominated by this joint. Conversely, in this configuration a vertical force produces a small moment about J2, thus, the excitation of the mode dominated by this joint is poor. Figures 3 and 5 show that the inertial cross-coupling term is less important for J2 than for J3, since the direct inertia \(M_{22}\) term is much larger than \(M_{23}\). For this reason, configuration Test 2 of Table 1, in which links 2 and 3 are almost perpendicular, gives a good compromise between the reduction of inertial cross coupling and the generation of a large moment about J2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Configuration** & **Joint 1** [\({}^{\circ}\)] & **Joint 2** [\({}^{\circ}\)] & **Joint 3** [\({}^{\circ}\)] & **Joint 4** [\({}^{\circ}\)] & **Joint 5** [\({}^{\circ}\)] & **Joint 6** [\({}^{\circ}\)] \\ \hline Test 1 & 0 & 0 & 90 & 0 & 0 & 0 \\ Test 2 & 0 & 0 & 17 & 0 & \(-17\) & 0 \\ Test 3 & 0 & \(-189\) & 243 & 0 & \(-54\) & 0 \\ Test 4 & 0 & 0 & 90 & 0 & 90 & 0 \\ Test 5 & 0 & \(-90\) & 180 & 0 & 90 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Robot testing configurations. Figure 4: Robot configurations. Joint J4 is able to generate velocity components perpendicular to the vertical plane that contains the robot arm (meridian plane). J2, J3 generate velocity components in meridian plane, and the inertial cross coupling with J4 is small. Conversely, J1 is able to generate velocity components in the same direction as J4, and there is a significant cross coupling with J4 (see Figure 3). The inertial cross coupling between J4 and J5 is always negligible (see Figure 3). In order to enhance the moment that a lateral hit can generate about J4, \(q_{5}\) was set to \(\pm 90^{\circ}\) and configuration Test 4 was defined. Finally, when \(q_{4}\) is set to zero, Joint 5 (J5) is able to generate velocity components of the last links in the meridian plane of the robot arm, but these components can be generated by J2 and J3 as well. When \(q_{4}\) is set to \(90^{\circ}\), J5 is able to generate velocity components out of the meridian plane of the robot arm, but these components can be generated by J1 as well. Therefore, Figure 3 shows that there is always a significant coupling between J5 and the arm joints. The test configuration can be chosen only with the aim of increasing the moment about J5 generated by the hammer hit. Test configuration Test 5 satisfies this condition for a hammer hit in the longitudinal direction. The relevance of gravity restoring (or unbalancing) torques was evaluated in the selected configurations. The calculated values resulted much smaller than reference values of joint torques caused by joint stiffness [16].