# New Indices for Evaluating Vibration Characteristics of Flexible-Joint Robots
## 3 Theoretical Development
### Frequency Response Functions  
We first analyze the frequency response of the displacement of the tool tip due to an external force. To do this, we assume that the end-effector is subject to a simple harmonic force with a frequency \(\omega\) which can be written in vector form as:  
\[\mathbf{F}=\mathbf{\overline{F}}e^{j\omega t} \tag{4}\]  
where \(\mathbf{\overline{F}}\) is a \(3\times 1\) complex vector whose norm and phase angle of each component represent the amplitude and phase of the force in the corresponding direction. Then the steady-state response of the joint deflection \(\Delta\mathbf{q}\) can be expressed as:  
\[\Delta\mathbf{q}=\Delta\mathbf{q}_{v}e^{j\omega t} \tag{5}\]  
where \(\Delta\mathbf{q}_{v}\) is an \(n\times 1\) complex vector which contains the amplitude and phase of the steady-state response of the joint deflections. Substituting (4) and (5) into (1) leads to:  
\[\Delta\mathbf{q}_{v}=\left(-\omega^{2}\mathbf{M}+j\omega\mathbf{C}+\mathbf{K }\right)^{-1}\mathbf{J}_{v}^{T}\mathbf{\overline{F}} \tag{6}\]  
The steady-state response of the tool displacement due to the harmonic force \(\mathbf{F}\) is then computed from (3) as below:  
\[\Delta\mathbf{E}_{v}=\mathbf{H}_{v}(\omega)\mathbf{\overline{F}} \tag{7}\]where \(\mathbf{H}_{\
u}(\omega)\) is the \(3\times 3\) displacement-force frequency response function as follows:  
\[\mathbf{H}_{\
u}(\omega)=\mathbf{J}_{\
u}\Big{(}-\omega^{2}\mathbf{M}+j\omega \mathbf{C}+\mathbf{K}\Big{)}^{-1}\mathbf{J}_{\
u}^{T} \tag{8}\]  
Similarly, when the end-effector is subject to a simple harmonic torque with a frequency \(\omega\) as:  
\[\boldsymbol{\tau}=\boldsymbol{\tau}e^{j\omega t} \tag{9}\]  
where \(\boldsymbol{\tau}\) is a \(3\times 1\) complex vector which contains the amplitudes and phases of the three torque components, then the steady-state response of joint deflection can be written as:  
\[\Delta\mathbf{q}_{w}=\Big{(}-\omega^{2}\mathbf{M}+j\omega\mathbf{C}+\mathbf{ K}\Big{)}^{-1}\mathbf{J}_{w}^{T}\boldsymbol{\tau} \tag{10}\]  
Now, the steady-state response of the tool displacement due to the harmonic torque \(\boldsymbol{\tau}\) can be computed as:  
\[\Delta\mathbf{E}_{w}=\mathbf{H}_{w}(\omega)\boldsymbol{\overline{\tau}} \tag{11}\]  
where \(\mathbf{H}_{w}(\omega)\) is the \(3\times 3\) displacement-torque frequency response function as follows:  
\[\mathbf{H}_{w}(\omega)=\mathbf{J}_{\
u}\Big{(}-\omega^{2}\mathbf{M}+j\omega \mathbf{C}+\mathbf{K}\Big{)}^{-1}\mathbf{J}_{w}^{T} \tag{12}\]  
The frequency response functions \(\mathbf{H}_{\
u}(\omega)\) and \(\mathbf{H}_{w}(\omega)\) represent the elastodynamic compliance of the robot under simple harmonic force and torque, respectively. They not only depend on the robot configuration, but also depend on the excitation frequency. When \(\omega\) is equal to zero, i.e., the robot is subject to a static force or torque, then (8) and (12) are reduced to:  
\[\mathbf{H}_{\
u}=\mathbf{J}_{\
u}\mathbf{K}^{-1}\mathbf{J}_{\
u}^{T}\ \,\ \ \ \mathbf{H}_{w}=\mathbf{J}_{\
u}\mathbf{K}^{-1}\mathbf{J}_{w}^{T} \tag{13}\]  
which represent the elastostatic compliance of the robot under static force and torque respectively.