## 1 Introduction  
In the milling process, the finished product requires a fine surface texture, high-precision contour, and short machining time (MT). To achieve high-precision machining, machine errors, such as static error, dynamic error, and process error, should be minimized [1]. Generally, dynamic errors are caused by interpolation error, servo lag [2, 3], and vibrations [3, 4]. Surface quality is typically determined through considerations of surface roughness [5, 6] and surface finish. Among the causes of machine errors, vibrations not only affect dynamic errors and surface texture [7] but can also shorten the tool life [8]. Vibrations may be passively suppressed by machine tools with high damping capability. However, for existing machine tools, path planning may play a critical role in reducing contour errors and improving surface quality. Computer numerical control (CNC) controller interpolation that considers the acceleration/deceleration (Acc/Dec) planning of the machining trajectory has considerable effects on the finished product. Therefore, adjusting Acc/Dec planning [9] not only improves contouring accuracy and suppresses machine vibration, improving surface quality, but may also reduce the overall MT. A specific instrument was developed to measure the relative dynamic displacement between the tool and workpiece at the tool center point (TCP) such that the limits for the axis acceleration and jerk could be adjusted to improve the contour accuracy [10]. In [11], the findings revealed that vibrations are mainly caused by the acceleration and jerk of each axis, especially in the corners. An integrated dynamic Acc/Dec method [12] was proposed to determine the corner velocity according to the maximum dynamic contour error; the method entails considering the command and servo dynamic errors. Related studies on predicting contour errors have typically employed an analytical formulation that is generally based on the rigid body assumption. Another approach to minimizing tracking errors and unwanted vibrations, an interpolation using a linear combination of B-spline basis functions that employ the forward filter was proposed [9]. Other methods have involved the use of a virtual cyber physical system (CPS) to design the Acc/Dec parameters to suppress the unwanted vibrations. However, extensive experiments are required to determine the parameters of a CPS system [13], making it difficult to implement. Other methods for developing CPS models are data-driven approaches that capture data through experiments that employ various CNC controller parameters. An artificial neural network (ANN) [14] and an ANN integrated with a genetic algorithm were applied to compute performance indices, such as the minimum surface roughness [6] and tracking and contour errors [15]. In multi-objective optimization methods, weighted sum approaches have been adopted to determine the optimal parameters. Search methodologies such as a genetic algorithm [16] or particle swarm optimization [15] have been applied. The adaptive neuro-fuzzy inference system was applied to predict the contour error and tracking error [15]; however, vibration data at the TCP were not included in the data-driven approach. To reduce the data capture time, various experimental designs, such as the Taguchi experiment design (TED) [6; 17] method and full factorial design (FFD) [18], have been proposed. The designers claimed that a FFD could deliver a more accurate model with fewer data. However, this paper reveals that many experiments must be conducted to establish the ANN model for the CNC dynamic model. Thus, the data acquisition process is not only time consuming but also labor intensive. Currently, no paper has developed a CPS model that includes vibration effects using a data-driven method. To predict the chatter vibration, the single frequency model (SFM) and the modern collocation method Chebyshev polynomials (CCM) were applied, and the mounting positions of accelerometers were indicated, in [19].