## 4 Experimental Tests  
The equipment needed to carry out the modal analysis of a robot with the impulsive method is rather simple and cheap. It includes a hammer for modal testing, an accelerometer, and a data acquisition board. In the framework of this research, a PCB 086C01 hammer (with load cell sensitivity 0.2549 mV/N), a PCB 356A17 tri-axial accelerometer (sensitivity 50.5 mV/(m/s\({}^{2}\))), and a NI9234 data acquisition board were used. The hammer and the accelerometer were made by PCB Piezotronics Inc., Depew, NY, USA, whereas the board was made by National Instruments Inc., Austin, TX, USA. Measured signals were analyzed in the frequency domain by means of ModaVIEW R2 (2017), which is a specific software for modal analysis developed by ABSignal-National Instruments Inc., Austin, TX, USA. In order to characterize the vibrations of the robot 10 testing points were defined. Figure 6 shows that testing point 1 is located on the flange of the end-effector, testing point 10 is located on the base, before joint 1, and the other testing points are evenly distributed on the links. The number of testing points is a trade-off between the need for a large number of measurement locations, to accurately describe robot vibrations, and the need for a quick testing procedure. Modal analysis was carried out with the rowing response approach [34] Excitation was always exerted on the end-effector,  
Figure 5: Effect of joint variable \(q_{3}\) on inertial cross coupling between J2 and J3. whereas the tri-axial accelerometer was sequentially moved to the 10 testing points. The three axes of the accelerometer were always parallel or perpendicular to the robot surface. Since motions of the base of the robot may have a negative effect on the quality of measurements, the robot was rigidly fastened to a large steel base. After some preliminary tests, data acquisition was carried out with a sampling frequency of 1024 Hz and 2048 samples. Figure 7 deals with the experimental results obtained in configuration Test 1, with hammer excitation in the lateral direction (y). Plots (a) and (b) show the modulus and phase of the direct point frequency response function (FRF), which is the FRF measured in the excitation point between the acceleration in the direction of the hammer force and the hammer force. Plot (c) shows the complex mode indicator function (CMIF) calculated over the 30 FRFs [30]. Both the modulus of the FRF and the CMIF show a high and isolated peak at about 13 Hz. The phase of the direct point FRF shows a large phase change at the same frequency. These properties are hints of the presence of a well-defined mode of vibration at about 13 Hz. ModalVIEW made it possible to identify a mode of vibration at 12.8 Hz with a damping ratio of 2.3% [30]. This mode, which is represented at the bottom of Figure 7, is dominated by the rotation of the whole robot arm about J1. Figure 6: Measurement points for modal testing. Figure 7: Test 1 configuration, (**a**) direct point frequency response function (FRF) modulus, (**b**) phase, (**c**) complex mode indicator function (CMIF). Figure 8 shows the experimental results obtained in configuration Test 2 with hammer excitation in the vertical direction (z). In this case, the direct point FRF and the CMIF show a high and isolated peak at about 17 Hz. ModalVIEW made it possible to identify at 17.5 Hz a mode of vibration dominated by the rotation of the whole robot arm about J2, which is represented in Figure 8. The corresponding damping ratio is 1.9%, this value is congruent with the damping ratio of J1. The experimental results obtained in configuration Test 3 are summarized in Figure 9. The direct point FRF shows a high peak at about 30 Hz with a large phase change, but in this case the main peak is not alone, since there is another peak at about 35 Hz. The CMIF plot confirms this scenario. ModalVIEW made it possible to identify two modes. The first mode is at 30.8 Hz with a damping ratio of 2.1%.