## 1 Introduction  
Chatter in the machining process may cause problems such as poor workpiece quality, upset noise, increased tool wear, and continuous damage to the cutting spindle. Thus, predicting and avoiding the onset of chatter are important and desirable in practice. In the past, a large number of scholars carried out the research associated with cutting parameter optimization in machining to ensure the maximum material removal rate without chatter. It is generally believed that one of the greatest sources of cutting chatter is the regeneration effect, where the displacement of the current and previous teeth of a cutter will change dynamically, resulting in dynamic cutting thickness and dynamic milling force with phase difference. [PERSON] and [PERSON] [1] and [PERSON] and [PERSON] [2] were the pioneers in researching chatter mechanisms and they determined the presence of the regeneration mechanism firstly in the 1960s. The main method to predict and control regeneration chatter is to obtain a stability lobe diagram (SLD), from which one can choose the process parameters corresponding to chatter-free cutting. Consequently, following the regeneration mechanism, great deal of investigations have been done on the stability prediction methods for cutting process, trying to obtain SLD accurately and efficiently. Some typical methods are: zeroth-order frequency-domain method proposed by [PERSON] and [PERSON] [3], temporal finite element method by [PERSON] et al. [4], multi-frequencysolution by [PERSON] and [PERSON] [5], Chebyshev polynomial approximation by [PERSON] and [PERSON] [6], semi-discretization method (SDM) by [PERSON] and [PERSON] [7; 8; 9], Galerkin projection by [PERSON] et al. [10], Lambert W function based method by [PERSON] and [PERSON] [11], Homotopy perturbation method(HPM) by [PERSON] [12], Chebyshev collocation method (CCM) by [PERSON] et al. [13], full-discretization method (FDM) by [PERSON] et al. [14] and [PERSON] et al. [15], implicit subspace iteration by [PERSON] et al. [16], spectral element method by [PERSON] et al. [17] and [PERSON] et al. [18], etc. [PERSON] et al. [19] and [PERSON] et al. [20] conducted the stability analysis of a multivariable milling tool by enhanced multistage HPM and high-order HPM, respectively. [PERSON] et al. use CCM to deal with the turning stability prediction considering the cases with regard to the flexible workpiece [21], the slender tools [22], the large horizontal lathes [23], or the low rotational speeds [24]. [PERSON] et al. [25; 26] made a comparison of SDM, FDM, CCM, and HPM in terms of accuracy and computational time and pointed out that CCM and HPM are the most efficient methods. [PERSON] et al. [27] experimentally investigated the influence of the relative position of the face milling tool towards the workpiece on milling dynamics. Recently, [PERSON] et al. [28] reviewed the current state of research with respect to the prediction methods and experimental techniques for chatter avoidance in turning in detail. Generally, the dynamics of tool-workpiece system in milling process, considering regeneration effect, can be modeled by linear delay-differential equations (DDEs) theoretically. With the change of milling cutter style, milling strategy, actual milling conditions, and other factors in the milling process or the consideration of modeling complexity, the time-delay term in system equation changes from single one to multiple ones, to single variable one, to multiple variable ones, to distributed ones or to other forms. For instance, to deal with the problems of stability prediction for the milling with variable pitch cutter (VPC)/cutter runout [29] where the difference of the passing time between two continuous teeth leads to the existence of multiple time delays, different versions of SDM methods [30; 31; 32; 33; 34] have been put forward and their effectiveness is verified too. For predicting/optimizing the stability for milling processes with variable helix cutter (VHC) or serrated cutter characterized by the distributed delays, [PERSON] et al.