## 3 Selective Modal Testing  
Vibration signals contain information about mechanical systems [29], and this paper aims to use impulsive modal analysis to identify the stiffness properties of an industrial robot, and to develop a compliant robot model. Modal analysis is a mature technology for the study of vibrations [30], which has been successfully used in many fields of aerospace [31], vehicle, and automation engineering [16; 32; 33]. In the field of industrial robotics, it is well recognized that joint stiffness has a larger effect on robot performance than link stiffness, therefore the main task of impulsive modal analysis is the identification of the stiffness of the joints. A robot with assigned joint values (\(\mathbf{q}\)) is considered. When an impulse force is exerted on a point of the robot by means of a hammer for modal testing and \(N\) acceleration components are measured by means of accelerometers, a set of \(N\) modes of vibrations can be identified. These modes of vibration  
Figure 2: Robot reference frames. transform the equations of motions (3) into a set of \(N\) independent equations in the modal coordinates \(\eta_{k}\;k\;=\;1,\ldots,N\):  
\[\left[ \begin{array}{cc}M\star_{11}(\mathbf{q})&0&0\\ 0&M\star_{kk}(\mathbf{q})&0\\ 0&0&M\star_{NN}(\mathbf{q})\end{array} \right]\left\{ \begin{array}{cc}\ddot{\eta}_ {1}\\ \ddot{\eta}_{k}\\ \ddot{\eta}_{N}\end{array} \right\}+\left[ \begin{array}{cc}C\star_{11}(\mathbf{q} )&0&0\\ 0&C\star_{ik}(\mathbf{q})&0\\ 0&0&C\star_{NN}(\mathbf{q})\end{array} \right]\left\{ \begin{array}{cc}\dot{\eta}_ {1}\\ \dot{\eta}_{k}\\ \dot{\eta}_{N}\end{array} \right\} \tag{6}\] \[+\left[ \begin{array}{cc}K\star_{11}(\mathbf{q})&0&0\\ 0&K\star_{kk}(\mathbf{q})&0\\ 0&0&K\star_{NN}(\mathbf{q})\end{array} \right]\left\{ \begin{array}{cc}\eta_{1} \\ \eta_{k}\\ \eta_{N}\end{array} \right\}\;=\;\left\{ \begin{array}{cc}0\\ 0\\ 0\end{array} \right\}\]  
where the mass, damping and stiffness matrices are diagonal, and elements \(M\star_{kk}(\mathbf{q}),C\star_{kk}(\mathbf{q})\) and \(K\star_{kk}(\mathbf{q})\) represent modal mass, damping and stiffness associated to the \(k\)th mode. When the robot is tested in a generic configuration the modes of vibration typically involve rotations about many joints of the robot. Therefore, modal coordinate \(\eta_{k}\) is a combination of joint rotations, which is defined by the corresponding mode of vibration. Modal mass, damping and stiffness do not have a direct physical meaning, and are not very useful to identify the stiffness and damping characteristics of the robot joints. Conversely, when the same robot is modally tested in a specific configuration in which the mass matrix shows that a joint variable \(i\) has negligible inertial cross coupling terms, one of the modes of vibration is dominated by the rotation about joint \(i\), and the corresponding modal equation becomes:  
\[I_{i_{z}}(\mathbf{q})\ddot{q}_{i}+c_{i}\dot{q}_{i}+k_{i}q_{i}\;=\;F_{i}(t) \tag{7}\]  
The modal coordinate \(\eta_{i}\) coincides with joint variable \(q_{i}\), modal mass coincides with inertia about joint \(i\) (\(I_{i_{z}}(\mathbf{q})\)), modal stiffness and damping coincide with joint stiffness (\(k_{i}\)) and damping (\(c_{i}\)), respectively. Mode \(i\) is very useful to identifying the stiffness of joint \(i\). Inertia about joint \(i\) (\(I_{i_{z}}(\mathbf{q})\)) depends on the inertias of the following links and can be calculated by using the Huygens-Steiner theorem:  
\[I_{i_{zz}}(\mathbf{q})\;=\;\sum_{j\;=\;i}^{N}\bigl{(}\mathbf{R}_{ji}\mathbf{I}_{i}\mathbf{R}_{ ji}^{T}+m_{j}\mathbf{P}_{ji}\bigr{)} \tag{8}\]  
\[\mathbf{P}_{ji}\;=\;\left[ \begin{array}{ccc}y_{ji}^{2}+z_{ji}^{2}&-x_{ji}y_{ji}& -x_{ji}z_{ji}\\ -y_{ji}x_{ji}&x_{ji}^{2}+z_{ji}^{2}&-y_{ji}z_{ji}\\ -z_{ji}x_{ji}&-z_{ji}y_{ji}&x_{ji}^{2}+y_{ji}^{2}\end{array}