## 1 Introduction  
Due to their flexibility, efficiency, and reliability, industrial robots have played an important role in production automation for a wide spectrum of applications including welding, painting, assembly, packaging, and material handling [1]. Recently, these machines have also been proposed for more complex tasks with a large dynamic loading such as machining [2], and these robotic applications require a high positioning accuracy and involve large time-varying loading during the processes. A few examples are the European projects HEPESTOS and COMET that were launched to develop innovative robotic machining systems for small-batch production [3; 4] and Boeing's initiative that aimed at using robots to automate drilling and percussive riveting processes in aircraft assembly [5]. On the other hand, the challenges for these robotic applications have also been highlighted in literature due to the inherent disadvantages of industrial robots compared with machine tools, such as having a low positioning accuracy and being prone to vibrations [2]. With the development of modern metrology techniques such as laser tracking [6; 7] and photogrammetry-based measurement [8; 9], the robotic positioning accuracy can be improved by kinematic parameter calibration [10; 11; 12] and positioning error compensation [7; 8; 13; 14; 15; 16]. For example, [PERSON] et al. [12] developed a calibration method to estimate the external joint configuration parameters using a laser displacement sensor. [PERSON] et al. [16] developed a kinematic model for a robot arm through a coordinate measuring machine (CMM), and experimental measurements at some robot poses were taken using the CMM. However, industrial robots still face issues due to structural vibrations for these applications, and the resulting influence is difficult to eliminate by metrology techniques. Robotic machining involves dynamic loading in a way that the magnitudes and directions of the force and torque acting on the robot dramatically change with time during machining. The resulting forced vibrations of the robot structure lead to much more complicated dynamics behaviors than many other robotic tasks such as pick-and-place and welding where static or very low contact forces act on the robot. First, the forced vibrations of the robot can seriously affect the process, for example, by decreasing the efficiency [17], damaging the surface quality [18], causing tool misalignment [19], and reducing the fatigue lives of the tool and the robot to a great extent [20; 21]. Second, not only static but also inertial forces are important for these applications. Furthermore, the interaction between the machining forces and the robot vibrations can even cause instability to the robotic machining system [22; 23] if some machining parameters such as spindle speed and feed rate are not selected properly [24; 25]. All these works have shed some light on the complexity of structural dynamics of robotic machining. Several approaches have been proposed to improve the robotic machining quality for a specific task considering the forced vibrations caused by machining. [PERSON] et al. [26] used two genetic algorithms to find the region with the best manipulability and the lowest joint torque in the robot workspace. [PERSON] et al. [27] developed a parametric modeling procedure that can correctly simulate the robot dynamics behavior. [PERSON] et al. [28] modeled the interaction between the milling tool and the workpiece and proposed a method for compensating the compliance errors based on a nonlinear stiffness model. [PERSON] et al. [29; 30] adjusted the tool path to counteract the tool deflections during the milling operations after estimating the tool force and robot joint stiffness. [PERSON] et al. [31] proposed an online method for compensating the backlash errors induced by the drive reversion. [PERSON] et al. [32] proposed a new joint space model that combined the joint stiffness and the reversal error and then used the model to compensate the milling path deviation. [PERSON] et al.