# Natural Frequency Prediction Method for 6R Machining Industrial Robot
## 1 Introduction  
Due to the rapid development of industrial automation, the global amount of industrial robot keeps increasing significantly in recent years [1], and new applications attract more attention [2; 3; 4; 5]. Because of the multiple degree of freedom, large workspace, and relative low cost, industrial robot becomes a potential alternative of traditional machine tool in hard material drilling [6], boring [7], and milling [8; 9] processes, which are not maturely and widely applied. For machine tool, in the whole workspace, the stiffness and dynamic behavior nearly remain steady, so machining stability and chatter suppression, which are concerned with multiple factors, are research emphasis [10; 11; 12]. However, for industrial robot, current researches focus on the characteristic of robot itself. Common industrial robot is of open-loop articulated serial structure with six revolute joints (6R); therefore, the stiffness and dynamic performance are much inferior to machine tool, which results in vibration issues and poor surface quality in machining process [13]. Static stiffness issues are studied to avoid the robot deformation in machining, and several stiffness indexes are proposed to evaluate the static stiffness of robot in the workspace [14; 15; 16; 17]. On the other hand, dynamic behavior of robot is studied to evaluate the capacity of dealing with time-variant load in machining process, which may help to suppress vibration [18; 19]. As 6R robot realizes movement through adjusting joint angles, dynamic behavior keeps varying with configurations during motion process. For that reason, research object is gradually expanded from one configuration to the whole workspace. [PERSON] et al. [20] selectedthree configurations from a linear movement path of a 6R robot with interval of 50 cm for dynamic behavior study and vibration analysis. It was found that different natural frequencies led to dynamic performance variation. [PERSON] et al. [21] performed a similar four-point-study and analyzed the change of stability region through modal information. Several configurations may present a linear movement, but the usage of robot should not be such limited. [PERSON] et al. [22] executed modal test with 63 measured points for 23 configurations, through which mode shapes were obtained and approximate distribution of first two step natural frequencies in a plane was depicted. Natural frequency is the vital dynamic parameter in robot machining, but massive tests are inefficient although the accuracy is well ensured. A method was proposed by [PERSON] et al. [23] to predict natural frequency in the whole workspace efficiently, in which sample configurations were selected by design of experiment (DoE) method and prediction model was fitted by least square (LS) method with the natural frequency data of sample configurations, but the prediction deviation reached up to 8.9%, making the method inadequate for practical application. The low order natural frequencies of 6R robot are relatively small and the first order is merely about 10 Hz, which is covered in frequency range of most machining processes, so 6R robot tends to forced vibration during machining processes [24]. To obtain basis for vibration suppression, structure optimization and path planning, low order natural frequencies are particularly concerned among dynamic parameters. It is proposed that low order natural frequencies of robot are decided by the configurations, while the high order natural frequencies are mainly related to the machining system, which do not vary with robot configurations [25], this phenomenon is demonstrated in Figure 1.  
Thus, a method to predict natural frequency of 6R industrial robot efficiently and precisely is proposed in this paper. As shown in Figure 2, sample configurations are selected to present the practical workspace (variable range) for fast prediction, and a regression equation fitted and optimized by partial least square (PLS) method is used as prediction model for higher accuracy.