## 6 Numerical Simulations  
Recent studies [2; 6; 35; 36] have highlighted that, in some robot operations (e.g., milling tasks), the dynamic properties of the robot arm are important, and that a simple optimization of robot configurations based on static stiffness does not lead to the best performance [2]. The modes of vibration of a robot are configuration-dependent, and the variation of natural frequency and modal damping in the workspace is an important feature of robot dynamics. It gives some hints useful to improve robot performance selecting robot configurations with highly damped modes having natural frequencies far from the excitation frequencies. Experimental modal analysis is time consuming, thus, the mathematical model based on modal analysis in a small number of selected configurations is a very useful tool to predict the variation in the dynamic properties of the robot over the whole workspace. The first numerical analysis aimed to understand the effect of robot configuration on the natural frequency and damping of the modes of vibration. A total of 10,000 random configurations of the robot corresponding to 10,000 random combinations of joint variables J2, J3, J4, and J5 (within joint ranges) were defined, and the modal properties were calculated. Since the robot model is axial-symmetric about J1, the variation of J1 was not taken into account, and this joint variable was set to zero. The results of these calculations are collected in Figures 12 and 13, which depict the mean values and standard deviations of natural frequencies and modal dampings, respectively. Figure 12: Mean values and standard deviations of natural frequencies in the workspace, (**a**) random variations of J2, J3, J4, J5, (**b**) random variations of J2 and J3. The natural frequencies of the second and third mode, which are usually characterized by large displacements in the meridian plane, show the largest standard deviations. Therefore, these modes are influenced by the robot configuration. The first natural frequency, which usually corresponds to a mode of vibration with large displacements in the horizontal plane, shows a mild standard deviation, since the robot configuration affects the moment of inertia about J1. The natural frequencies of the last two modes, which are chiefly dominated by J4 and J5, show very small standard deviations. This happens because the robot configuration has a small effect on the direct inertia term of J4 and no effect on the direct inertia term of J5, but it only changes the inertial cross-coupling terms (see Figure 3). The statistical analysis of modal damping variation with robot configurations is depicted in Figure 13. This figure shows that robot configuration affects the modal damping of the first three modes, since standard deviations are comparable with the mean values. Conversely, robot configuration has a very small effect on the damping of the last modes, which chiefly involve wrist joints, because standard deviations are very small. Figures 12b and 13b show the result of a similar calculation, in which only random variations in J2 and J3 were performed. The standard deviations of the natural frequencies and modal dampings are very similar to the ones of Figures 12a and 13a, therefore, this result points out that the variations in J2 and J3 have the largest effect on the modal properties of the robot. Figures 12 and 13 highlight another important feature of the robot. Variations in joint variables are able to modify the natural frequencies, but they do not change the typical frequency bands of the modes. In other words, the frequencies of the modes associated to the wrist joints are always higher than the frequencies associated to the arm joints. The first two modes belong to the same frequency band (\(15\div 30\) Hz), and this band has a negligible overlap with the bands containing the other modes. The statistical analysis showed mild variations in the properties of the first modes of vibrations of the robot, due to variations in J2 and J3.