# New Indices for Evaluating Vibration Characteristics of Flexible-Joint Robots
## 3 Theoretical Development
### Frequency Response Ratios  
Now we investigate the forced vibration characteristics of the robot under a harmonic force. From (7), we have:  
\[\Delta\mathbf{E}_{\
u}^{H}\Delta\mathbf{E}_{\
u}=\overline{\mathbf{F}}^{H} \mathbf{H}_{\
u}^{H}(\omega)\mathbf{H}_{\
u}(\omega)\overline{\mathbf{F}} \tag{14}\]  
where the superscript \(H\) represents the conjugate transpose of a complex matrix. Then we define the displacement-force frequency response ratio as:  
\[\eta_{\
u}=\sqrt{\frac{\Delta\mathbf{E}_{\
u}^{H}\Delta\mathbf{E}_{\
u}}{ \overline{\mathbf{F}}^{H}\overline{\mathbf{F}}}}=\sqrt{\frac{\overline{ \mathbf{F}}^{H}\mathbf{A}_{w}(\omega)\overline{\mathbf{F}}}{\overline{ \mathbf{F}}^{H}\overline{\mathbf{F}}}} \tag{15}\]  
where \(\mathbf{A}_{\
u}(\omega)=\mathbf{H}_{\
u}^{H}(\omega)\mathbf{H}_{\
u}(\omega)\) is a \(3\times 3\) Hermitian matrix. The ratio \(\eta_{\
u}\) represents the mapping from the amplitude of external force to the amplitude of tool displacement due to structural vibration, and \(\eta_{\
u}\) depends on the robot configuration and the force frequency. Furthermore, the ratio is related to the amplitudes and phases of the three force components in the complex vector \(\overline{\mathbf{F}}\). When the robot configuration and force frequency are given, the upper and lower bounds of \(\eta_{\
u}\) with different \(\overline{\mathbf{F}}\) can be computed as [35]:  
\[\eta_{\
u\text{min}}=\sqrt{\lambda_{\
u\text{min}}},\ \eta_{\
u\text{max}}=\sqrt{ \lambda_{\
u\text{max}}} \tag{16}\]where \(\lambda_{\rm{vmin}}\) and \(\lambda_{\rm{vmax}}\) are the minimum and maximum eigenvalues of the Hermitian matrix \(\mathbf{A}_{\rm{v}}\), respectively. The force vector \(\overline{\rm{F}}\) corresponding to the upper and lower bounds of \(\eta_{\rm{v}}\) are:  
\[\overline{\mathbf{F}}_{\rm{min}}=\mathbf{e}_{\rm{vmin}},\ \ \ \overline{\mathbf{F}}_{\rm{ max}}=\mathbf{e}_{\rm{vmax}} \tag{17}\]  
where \(\mathbf{e}_{\rm{vmin}}\) and \(\mathbf{e}_{\rm{vmax}}\) are the eigenvectors of \(\mathbf{A}_{\rm{v}}\) corresponding to \(\lambda_{\rm{vmin}}\) and \(\lambda_{\rm{vmax}}\) respectively. Equations (16) and (17) give the range of the displacement-force frequency response ratio \(\eta_{\rm{v}}\) and the amplitudes and phases of the force components for the boundary values of \(\eta_{\rm{v}}\). Similar analysis can be carried out on the forced vibration characteristics of the robot under a simple harmonic torque. We define the displacement-torque frequency response ratio as:  
\[\eta_{\rm{w}}=\sqrt{\frac{\Delta\mathbf{E}_{\rm{w}}^{H}\Delta\mathbf{E}_{\rm{ w}}}{\overline{\mathbf{\tau}}^{H}\overline{\mathbf{\tau}}}}=\sqrt{\frac{ \overline{\mathbf{\tau}}^{H}\mathbf{A}_{\rm{w}}(\omega)\overline{\mathbf{\tau }}}{\overline{\mathbf{\tau}}^{H}\overline{\mathbf{\tau}}}} \tag{18}\]  
where \(\mathbf{A}_{\rm{w}}(\omega)=\mathrm{H}_{\rm{w}}^{\rm{H}}(\omega)\mathrm{H}_{ \rm{w}}(\omega)\) is also a \(3\times 3\) Hermitian matrix. The ratio \(\eta_{\rm{w}}\) represents the mapping from the amplitude of external torque to the amplitude of tool displacement due to structural vibration. When the robot configuration and torque frequency are given, the upper and lower bounds of \(\eta_{\rm{w}}\) with different \(\overline{\mathbf{\tau}}\) can be obtained as [34]:  
\[\eta_{\rm{wmin}}=\sqrt{\lambda_{\rm{wmin}}},\ \ \ \eta_{\rm{wmax}}=\sqrt{\lambda_{\rm{ wmax}}} \tag{19}\]  
where \(\lambda_{\rm{wmin}}\) and \(\lambda_{\rm{wmax}}\) are the minimum and maximum eigenvalues of the Hermitian matrix \(\mathbf{A}_{\rm{w}}\) respectively, and the corresponding torque vector are given as:  
\[\overline{\mathbf{\tau}}_{\rm{min}}=\mathbf{e}_{\rm{wmin}},\ \ \ \overline{\mathbf{\tau}}_{\rm{max}}= \mathbf{e}_{\rm{wmax}} \tag{20}\]  
where \(\mathbf{e}_{\rm{wmin}}\) and \(\mathbf{e}_{\rm{wmax}}\) are the eigenvectors of \(\mathbf{A}_{\rm{w}}\) corresponding to \(\lambda_{\rm{wmin}}\) and \(\lambda_{\rm{wmax}}\) respectively.