## 2 Dynamic Model  
In the framework of this research an Adept Viper s650 robot manufactured by Omron Adept Inc. (Pleasanton, CA, USA) is considered. It is a medium size six-DOF industrial robot, its workable space is roughly a hemisphere with a radius of about 600 mm. Figure 1 shows that this robot has a typical anthropomorphic structure. There are three main joints that define the position of the wrist and three minor joints that define the orientation of the end-effector. All joints are driven by alternate current (AC) servomotors. The vertical plane passing through the axis of joint 1, perpendicular to the axes of joint 2 and 3, is named the meridian plane of the robot arm. Figure 1: The robot Adept Viper s650 with the tested axes. This paper aims at developing a dynamic model for the study of vibrations around a working configuration of the robot defined by vector \(\mathbf{q}_{0}\) of joint variables, therefore joint variables, velocities, and accelerations are defined by the following equations:  
\[ \begin{array}{rcl}\mathbf{q}&=&\mathbf{q_{0}}+\Delta\mathbf{q}\\ \dot{\mathbf{q}}&=&\Delta\dot{\mathbf{q}}\\ \ddot{\mathbf{q}}&=&\Delta\ddot{\mathbf{q}}\end{array}  \tag{1}\]  
where vector \(\Delta\mathbf{q}\) contains the small variations in joint variables around the selected configuration. Joint compliance around the rotation axis is considered, whereas link and bearing compliance is neglected. The axis of joint 6 coincides with the approach axis of the end-effector. Since, in most cases, the external force exerted on the end-effector is aligned with or intersects the approach axis, compliance about joint 6 is neglected. If the non-linear Coriolis and centrifugal terms [28] are neglected, according to the assumption of small oscillations, the equations of free vibrations take the form [6]:  
\[M\big{(}\mathbf{q_{0}}\big{)}\Delta\ddot{\mathbf{q}}+\mathbf{C_{q}}\Delta\dot{\mathbf{q}}+\mathbf{ K}\Delta\mathbf{q}\ =\ 0 \tag{2}\]  
where \(M\big{(}\mathbf{q_{0}}\big{)}\) is the mass matrix that depends on \(\mathbf{q_{0}}\). The elements on the diagonal of \(M\big{(}\mathbf{q_{0}}\big{)}\) are the direct inertia terms, the off-diagonal terms are the inertial cross-coupling terms. \(\mathbf{C_{q}}\) is a diagonal matrix that accounts for joint damping, \(\mathbf{K}\) is a stiffness matrix that accounts for joint compliance and configuration-depending gravity torques. Gravity can generate large restoring or unbalancing torques about joint axes only when the robot arm is close to a vertical configuration (upwards or downwards). If configuration-depending gravity torques are negligible the equations of free vibrations become:  
\[M\big{(}\mathbf{q_{0}}\big{)}\Delta\ddot{\mathbf{q}}+\mathbf{C_{q}}\Delta\dot{\mathbf{q}}+\mathbf{ K_{q}}\Delta\mathbf{q}\ =\ 0 \tag{3}\]  
where \(\mathbf{K_{q}}\) is a diagonal matrix that accounts for joint compliance. Specific calculations have been be carried out to estimate the effect of gravity torques in the selected configurations. The identification of matrices \(\mathbf{K_{q}}\) and \(\mathbf{C_{q}}\) is the main task of this research. The mass matrix \(M\big{(}\mathbf{q_{0}}\big{)}\) can be calculated using the inertial data provided by the computer aided design (CAD) models of the robot. These CAD files have been retrieved from the website of the manufacturer and the links are assumed to have uniform density and no free spaces within the external shell. This is due to the uncertainty of the mass distribution within the links of the robot that could not be eliminated without a complete disassembly of the robot.