# New Indices for Evaluating Vibration Characteristics of Flexible-Joint Robots
## 3 Theoretical Development
### Frequency Response Ratios
Equations (19) and (20) give the range of the displacement-torque frequency response ratio \(\eta_{\rm{w}}\) and the amplitudes and phases of the torque components for the boundary values of \(\eta_{\rm{w}}\). The maximum frequency response ratios \(\eta_{\rm{vmax}}\) and \(\eta_{\rm{wmax}}\) provide a measure of the maximum amplitudes of tool displacement due to harmonic loading with an amplitude of unit force and unit torque respectively, and the corresponding eigenvectors represent the amplitudes and phases of the force and torque components under which the worst vibration situation occurs. The ratios \(\eta_{\rm{vmax}}\) and \(\eta_{\rm{wmax}}\) are inherent characteristics of the robot and independent of the application processes, therefore they can be used as indices to evaluate the vibration characteristics of the robot under harmonic excitation. More importantly, the eigenvectors \(\mathbf{e}_{\rm{cmax}}\) and \(\mathbf{e}_{\rm{wmax}}\) represents the amplitudes and phases of the force and torque components corresponding to \(\eta_{\rm{vmax}}\) and \(\eta_{\rm{wmax}}\). The two indices can be used to predict the worst situation of forced vibrations of the robot for given processes. If the rotation matrix from the work frame (see Figure 2) to the inertial frame is written as \(\mathbf{R}\), then the force and torque acting on the tool can be written as:  
\[\mathbf{F}=\mathbf{R}\mathbf{F}^{\prime},\ \ \ \mathbf{\tau}=\mathbf{R} \mathbf{\tau}^{\prime} \tag{21}\]  
where \(\mathbf{F}^{\prime}\) and \(\mathbf{\tau}^{\prime}\) are the process-dependent force and torque written in the work frame. In the case of machining, when the effect of robot deformation on \(\mathbf{F}^{\prime}\) and \(\mathbf{\tau}^{\prime}\) can be neglected, these force and torque can be predicted via machining force modeling or via machining tests on Computer Numerical Control (CNC) machines. Usually, the process force and torque are periodic functions about time and can be approximated as an expansion of \(2N+1\) Fourier series as follows:  
\[\mathbf{F}^{\prime}\approx\sum_{k=-N}^{N}\mathbf{F}_{k}e^{jk\omega_{k}t},\ \mathbf{\tau}^{ \prime}\approx\sum_{k=-N}^{N}\mathbf{\tau}_{k}e^{jk\omega_{k}t} \tag{22}\]where \(\omega_{v}\) and \(\omega_{w}\) are the force and torque frequencies, \(\mathbf{F}_{k}\) and \(\mathbf{\tau}_{k}\) are complex vectors containing the Fourier coefficients and:  
\[\mathbf{F}_{-k}=\mathbf{F}_{k^{\prime}}^{\star},\ \mathbf{\tau}_{-k}=\mathbf{\tau}_{k}^{\star} \tag{23}\]  
where \(\mathbf{F}_{k}^{\star}\) and \(\mathbf{\tau}_{k}^{\star}\) are the conjugates of \(\mathbf{F}_{k}\) and \(\mathbf{\tau}_{k}\) respectively. Then, the maximum amplitude of tool displacement due to the \(k\)th Fourier series in \(\mathbf{F}^{\prime}\) and \(\mathbf{\tau}^{\prime}\) are respectively computed as:  
\[\eta_{\
u}=\sqrt{\frac{\Delta\mathbf{E}_{\
u}^{H}\Delta\mathbf{E}_{\
u}}{ \overline{\mathbf{F}}^{H}\overline{\mathbf{F}}}}=\sqrt{\frac{\overline{ \mathbf{F}}^{H}\mathbf{A}_{w}(\omega)\overline{\mathbf{F}}}{\overline{ \mathbf{F}}^{H}\overline{\mathbf{F}}}} \tag{15}\]0  
In (24), the Fourier coefficients \(\mathbf{F}_{k}\) and \(\mathbf{\tau}_{k}\) are process dependent and the indices \(\eta_{\text{rmax}}\) and \(\eta_{\text{rmax}}\) are robot dependent. Then, once \(\eta_{\text{rmax}}\) and \(\eta_{\text{rmax}}\) had been provided, for example, by robot manufacturers and \(\mathbf{F}_{k}\) and \(\mathbf{\tau}_{k}\) are modeled or measured by robot users, the largest tool misalignment due to each Fourier series can be easily evaluated using (24).