## 1 Introduction  
In robotic processes, the compliance of the robot arm plays a very important role. In some conditions, for example, in robotic assembly, robot arm compliance can compensate for small position and orientation errors of the end-effector. In other processes, like machining, robot compliance may generate chatter vibrations, with an impairment in the quality of the machined surface. Indeed, still very few robots have been applied in this economically important application area [1], mainly due to their low stiffness [2]. In industrial robots, the compliance of the end-effector is mainly due to joint compliances [3; 4; 5], even if there are some examples of robots [6] having structural modes (dominated by link or bearing compliance) in the band of frequency that contains the modes dominated by joint compliance. The compliance properties of robots are very important in emerging fields as well, such as flexible assembly systems [7; 8] and collaborative robotics [9]. Once robot joint stiffness has been identified, the kinematic redundancy of the arm with respect to the robotic task can be exploited to minimize (or maximize) the Cartesian compliance of the end-effector through optimization methods [10; 11; 12; 13], as already proposed for the minimization of base reactions and kinetic energy. In material removal processes, low stiffness causes imprecise products, due to the robot deflections during the robotic task. From a dynamic point of view, low frequency chatter vibrations [3; 14; 15] can be induced when low-stiffness robots are used, with an impairment in the quality of the machined surface. Moreover, vibrations cause a reduction of tool life and can lead to breakages in robot joint transmission. In particular, a low joint stiffness of the robot is one of the main issues in using robotic machining instead of computer numerical control (CNC) machining. In robotic processes, the directions along which the robot arm shows large compliance are important. The stiffness matrix in the joint space does not directly give this piece of information; moreover, for a serial robot, the stiffness matrix in the Cartesian space is not diagonal and it is configuration-dependent. This means that the force and deformation in the Cartesian space are coupled and this can generate some counterintuitive results. A geometric and intuitive interpretation of the end-effector displacements in different robot operative conditions has been presented in [16]. Static tests are widely used to obtain the joint stiffness of industrial robots [17]. In the static tests, a set of forces is applied to the robot end-effector in different robot configurations, while the displacement sensors (laser sensors, vision systems, coordinate measuring machines) measure the static deformation of the end-effector. Therefore, the Cartesian stiffness of the robot can be calculated. Then, the stiffness of the joints is obtained through the analytical relation between the joint and Cartesian stiffness based on the kinematic model of robot. Various identification methods (least squares or genetic algorithms) are employed to numerically solve the problem. Most of the researches (e.g., see [4; 18; 19; 20; 21]) neglect link flexibility and identify the joints' rotational stiffness (modeled with linear rotational springs) using the above-mentioned method. In [22], the robot arm is modeled considering rigid links and three lumped rotational springs for each joint to take into account joint compliance, bearings compliance, and link deflections. In [23], an analysis method in which both joint and link stiffness are considered is presented. In [24; 25], two types of robot dynamic models are presented by considering either only joint compliance or its combination with link compliance, in order to predict the robot dynamic behavior in different postures. In [6] a hybrid approach is adopted, in which a critical structural compliance in a direction perpendicular to a joint axis is taken into account, together with joint compliance.