## 1 Introduction
In [26], experimental modal analysis is used to identify the joint and base stiffness of an industrial robot represented with a four-degrees-of-freedom (four-DOF) planar model. Experimental modal analysis was also used in [6; 16], to identify the stiffness of the first three joints of an industrial robot. The static tests for the identification of joint stiffness have some drawbacks. The equipment for the measurement of end-effector displacements along three directions may be complex and expensive, especially if high accuracy is required and the workspace of the robot is large. The application of vertical forces on the end-effector is rather simple, since weights can be used, but the application of horizontal forces may require specific actuators that have to be connected with the end-effector without introducing additional constraints. Since Cartesian stiffness is configuration dependent, many test configurations are needed to map the robot workspace. The identification of joint stiffness from the measured Cartesian stiffness requires, in most cases, an optimization method. Finally, the measurements are affected by the structural compliance of the links, and it is difficult to separate the contribution of end-effector displacement caused by joint compliance from the one due to link compliance. For the above-mentioned reasons, this research aims to use the methods of impulsive modal analysis to identify the joint stiffness of industrial robot arms. In future, the same methodology could be extended to other kinds of robots e.g., walking robots [27]. Some factors foster the development of testing methods based on impulsive excitation. The basic equipment (which includes an instrumented hammer, an accelerometer, a data acquisition board, and a computer) is nowadays affordable for most laboratories. Hammer excitation can be exerted in three directions, even if the robot workspace is wide. Modal analysis identifies all the modes of vibration, both the ones dominated by joint compliance, and the ones dominated by link compliance (if present), hence, the modal method allows a better understanding of the importance of link compliance than the static method. The modes of vibration of a robot typically involve the rotations of some joints and the natural frequencies are functions of the stiffness of these joints. Therefore, the main issue of the modal methods for the identification of joint stiffness is the possibility of finding robot configurations with modes of vibration dominated by the stiffness of only one joint. In this specific case, modal stiffness coincides with joint stiffness, and the latter can be identified from the measured natural frequency and the calculated value of moment of inertia. Other important issues are the choice of proper excitation directions for the selected configurations and the minimization of test configurations. This paper is organized as follows. In Section 2, a serial six-DOF robot is considered (Omron Adept Viper s650), since it is representative of industrial robots used for a wide range of operations, and a dynamic model of robot vibrations is developed. Section 3 deals with selective modal testing, and some criteria for finding robot configurations with modes of vibration dominated by only one joint, or by a couple of joints, are presented and discussed. Experimental tests carried out in the selected configurations are presented in Section 4, which significantly extends a previous work of the authors [16], in which only the first three joints of the robot were considered. Section 5 deals with the identification of joint stiffness, and an optimization technique is presented to cope with the cases in which the identified mode is dominated by a couple of joints. The use of the robot dynamic model with the identified stiffness values is presented in Section 6, and the variation of the natural frequencies in the robot workspace is analyzed. Finally, in Section 7, conclusions are drawn.