# New Indices for Evaluating Vibration Characteristics of Flexible-Joint Robots
## 4 Numerical Simulation
### Performance Indices for Given Positions  
The proposed evaluation model has been implemented in the MATLAB environment. We first investigate the variation of performance indices \(\eta_{\textit{tmax}}\) and \(\eta_{\textit{tmax}}\) with different frequencies when the robot moves to given positions. As an example, the values of \(\eta_{\textit{tmax}}\) and \(\eta_{\textit{tmax}}\) with frequencies from 0 to 200 Hz at Point A (\(x\) = 0.1 m, \(y\) = 0, and \(z\) = 1 m) and Point B (\(x\) = 1 m, \(y\) = 0, and \(z\) = 1 m) are compared in Figures 4 and 5 respectively. It is seen that when the amplitude of external force/torque is equal to unit force/torque, the maximum amplitude of TCP displacement strongly depends on the excitation frequency and can be much higher than the largest static displacement when a unit static force/torque acts on the end-effector (represented by the values of \(\eta_{\textit{tmax}}\) or \(\eta_{\textit{tmax}}\) when the frequency is equal to zero). This implies that only considering the elastostatic performance is insufficient for dynamic robotic processes. At Point A, it is found that the first two modes of the robot have natural frequencies of 54 rad/s and 103 rad/s respectively, and both mode shapes are related to the deflections of Joint 2 and Joint 3. The third mode of the robot has a natural frequency of 155 rad/s, and its mode shape is related to the deflection of Joint 1. From Figures 4 and 5, we observe two peak values of \(\eta_{\it{1}\it{0}\it{max}}\) and \(\eta_{\it{1}\it{0}\it{max}}\) at frequencies close to the first two natural frequencies. As the moment arm about Joint 1 is small at Point A, we do not observe a third peak value related to the third mode on both curves of \(\eta_{\it{1}\it{0}\it{max}}\) and \(\eta_{\it{1}\it{0}\it{max}}\), which indicates that the TCP vibration in the \(y\) direction is less obvious than those in the \(\it{ozx}\) plane. We further investigate the eigenvectors of matrices \(\mathbf{A_{\rm{p}}}\) and \(\mathbf{A_{\rm{w}}}\) corresponding to \(\eta_{\it{1}\it{0}\it{max}}\) and \(\eta_{\it{1}\it{0}\it{max}}\) at different frequencies. For example, the eigenvector for \(\eta_{\it{0}\it{max}}\) is \(\mathbf{\rm{e}}_{\it{0}\it{max}}=[-0.91-0.38i,\,0,\,0.18]^{T}\) (in N) at 54 rad/s and \(\mathbf{\rm{e}}_{\it{0}\it{max}}\)= \([-0.36-0.02i,\,0,\,0.93]^{T}\) (in N) at 103 rad/s for the two peak values. These vectors represent the amplitudes and phases of the three force components under which the two peak values of \(\eta_{\it{0}\it{1}\it{0}\it{max}}\) occur at these frequencies. The eigenvectors for \(\eta_{\it{1}\it{0}\it{max}}\) are \(\mathbf{\rm{e}}_{\it{0}\it{max}}=[0,\,1,\,0]^{T}\) (in N-m) for both peak values of \(\eta_{\it{1}\it{0}\it{max}}\), therefore, the worst cases of the TCP vibrations due to a harmonic torque at 54 rad/s and 103 rad/s occur when the torque is in the y direction. At Point B, the first and third modes of the robot are related to the deflections of Joint 2 and Joint 3, and their natural frequencies are about 39 rad/s and 132 rad/s respectively. The second mode is related to Joint 1 deflection with a natural frequency of 103 rad/s. Compared with Point A, the moment arm about Joint 1 is large at Point B, thus we observe three peak values of \(\eta_{\it{1}\it{0}\it{max}}\) and \(\eta_{\it{1}\it{0}\it{max}}\) at frequencies close to the natural frequencies. Furthermore, we find that the eigenvectors of matrices \(\mathbf{A_{\rm{p}}}\) and \(\mathbf{A_{\rm{w}}}\) corresponding to the first and third peak values of \(\eta_{\it{1}\it{0}\it{max}}\) are \(\mathbf{\rm{e}}_{\it{0}\it{max}}=[-0.55-0.01i,\,0,\,0.83]^{T}\) (in N) at 35 rad/s and \(\mathbf{\rm{e}}_{\it{0}\it{max}}=[-0.32-0.04i,\,0,\,0.95]^{T}\) (in N) at 132 rad/s, respectively. Again, the eigenvectors for \(\eta_{\it{1}\it{0}\it{max}}\) are \(\mathbf{\rm{e}}_{\it{0}\it{max}}=[0,\,1,\,0]^{T}\) (in N-m) at the first and third peak values.