## 3 Selective Modal Testing
\right] \tag{9}\]  
where \(\mathbf{R}_{ji}\) is the rotation matrix that transforms the reference frame of link \(j\) to the reference frame of link \(i\), and \(x_{ij}\), \(y_{ij}\) and \(z_{lj}\) are the coordinates of the center of mass of link \(j\) defined in the reference frame of link \(i\). It is worth noticing that usually mode \(i\) (dominated by joint \(i\)) includes small contributions of the other joints, and sometimes of structural compliance of the links, therefore assuming \(M\star_{ii}(\mathbf{q})\;=\;I_{i_{z}}(\mathbf{q})\), \(K\star_{ii}(\mathbf{q})\;=\;k_{i}\) and \(C\star_{ii}(\mathbf{q})\;=\;c_{i}\) is only an approximation. For the above-mentioned reasons, a specific modal testing method is proposed, which is based on robot configurations and excitation directions that make possible the excitation of modes of vibrations dominated by only one joint. In order to have an insight into the properties of the mass matrix of the tested robot, the elements of the mass matrix were evaluated in random configurations. In particular 1,000,000 random combinations of \(q_{1}\ldots q_{5}\) with values within the actual joint ranges were considered. Since the inertial cross-coupling terms may be positive or negative, depending on the configuration, the mean values and standard deviations of the moduli of the elements of the mass matrix were calculated to avoid cancellations. The results of this analysis are summarized in Figure 3, and show the following properties. 1. The direct inertia terms of the first three joints are by far larger than the ones of the wrist joints--notice that a logarithmic scale is adopted in Figure 3. Joint 2 (J2) has the largest mean value followed by Joint 1 (J1). 2. The direct inertia term of joint \(i\) (\(M_{ii}\)) depends on the joint variables of the following links, hence \(M_{11}\) depends on \(q_{2}\ldots q_{N}\), whereas \(M_{NN}\) is constant. Therefore, the standard deviation decreases from J1 to J5. 3. The inertial cross-coupling terms between the first three joints are much smaller than the direct inertia terms with the exception of \(M_{23}\), which is comparable with \(M_{33}\). 4. \(M_{23}\) has a large standard deviation, and this means that there are robot configurations with a minimum coupling between J2 and J3. 5. The mean values of the inertial cross-coupling terms of the first three joints with the wrist joints are very small in comparison with the direct inertia terms of the first three joints. They are more important if compared with the direct inertia terms of the wrist joints
6. There is a negligible coupling between wrist joints (\(M_{45}\)). In a specific configuration, the inertial cross-coupling terms of joint \(i\) (\(M_{ij},\ j\
eq i\)) are small or negligible if the velocity of joint \(i\) generates a distribution of robot arm velocities that cannot be generated by the velocities of the other joints. Even if the previous condition is satisfied, cross-coupling terms may be present, if the axes of the joints are not principal axes. According to the results of Figure 3, the configurations for selective modal testing summarized in Figure 4 and Table 1 were chosen. Figure 3: Mean values and standard deviations of mass matrix terms (\(10^{6}\) random configurations), direct inertia terms (red bars) and inertial cross-coupling terms (blue bars). J1 is always perpendicular to joints 2 and 3 (J2 and J3). If joint variables \(q_{2}\), \(q_{3}\) are set to \(0^{\circ}\) and \(90^{\circ}\), respectively, the robot arm is horizontal, and if \(q_{4}\)\(q_{5}\) are set to \(0^{\circ}\), the approach axis is horizontal as well. In this configuration (Test 1) only J1 is able to generate relevant velocities in the horizontal plane. This configuration minimizes the inertial coupling between J1 and the other joints (which is usually small) and enhances the excitation of J1, when a lateral force (in \(y\) direction) is exerted on the end-effector.