1. Introduction
Computer numerical control (CNC) machining is a manufacturing process automatically controlling the machine tools for producing high-precision workpieces. To achieve high-speed and high-precision machining, on-site operators typically adjust the feed rate in computer-aided manufacturing (CAM) software based on their experiences derived from previous cutting experiments. Previous studies presented many advanced control methods such as optimal control and adaptive control of cutting conditions in order to improve machining accuracy [
1,
2,
3]. Among them, dynamic feed rate scheduling uses two indicators for evaluating the feed rate: constant cutting force-based and processing path-based dynamic feed rate scheduling.
First, in constant cutting force-based dynamic feed rate scheduling, the cutting force model is usually established depending on the cutting tool and the direction of the cutting velocity vector. In other words, the model provides a suitable feed rate instantly under a constant cutting force and spindle speed. Wang et al. [
4] presented a feed rate optimization method for constant peak cutting force in the five-axis flank milling process to resolve the unstable machining of parts with ruled surfaces in the aviation industry. The optimization method used least squares theory with the cutter entry angle and feed rate as variables for establishing the peak cutting force for each cutting point. Moreover, a feed rate scheduling method was also designed to quickly solve the appropriate feed rate under constant peak cutting force. Kim et al. [
5] proposed a mechanistic cutting force model to perform effective feed rate scheduling for indexable end milling in process planning. The developed cutting force model applied cutting-condition-independent cutting force coefficients which took run out, cutter deflection, geometry variation and size effect as consideration for accurate cutting force prediction. Lee and Cho [
6] developed a reference cutting force model based on considering the transverse rupture strength of the tool material and the area of the rupture surface. The experimental results revealed that the model provides an effective criterion for a feed rate scheduling system that regulates cutting force at a given criterion. All these studies have implemented a cutting force model to adjust the feed rate; however, feed rate scheduling is still associated with many difficulties, such as incompatibility of the cutting force model, sensors installation, and cost much time for collecting cutting experimental data.
Secondly, the poor processing quality such as large surface roughness and insufficient contour precision is mainly attributed to an incorrect processing feed rate or large jerk produced by excessive acceleration and deceleration. These situations likely occur when machining with a larger curvature therefore producing the chord errors [
7,
8]. Luan et al. [
9] noted that the chord error was affected by the interpolation algorithm, curvature, and feed rate. In a condition of the same feed rate, an area with larger curvature produces a larger chord error, because the interpolation algorithm of the conventional controller is constant, the feed rate must be reduced to maintain the chord error within the requirements; therefore, an offline dynamic feed rate scheduling method was proposed to alleviate the abovementioned problem. Yeh and Hsu [
10] found that due to the non-uniform map between curves and parameters, it is very difficult to maintain a constant feed rate and chord accuracy between two interpolated points along parametric curves; therefore, an adaptive feed rate calculation using speed-controlled interpolation algorithm was proposed. Both simulation and experimental results for non-uniform rational B-spline (NURBS) examples were verified for the feasibility and precision of the proposed interpolation algorithm. Giannelli et al. [
11] proposed a configurable trajectory planning strategy which can be applied to any planar path with a piecewise sufficiently smooth parametric representation. The processing path-based dynamic feed rate scheduling does not need to install an extra sensor. The establishment of a cutting force model does not require cutting experiments; therefore, employing the processing path-based dynamic feed rate scheduling is better than the other method in terms of practicality and versatility. Thus, the present study primary focuses on processing path-based dynamic feed rate scheduling method. An extended cutting experiment is needed to establish and verify the cutting force model. Therefore, this study uses fuzzy theory to control the dynamic cutting feed rate of a machine tool under the same processing software, cutting tool, and processing material to replace manual feed rate adjustment and improve the machining accuracy while shortening the processing time.
Fuzzy control was first proposed by L. A. Zadeh in 1964 [
12] to develop a versatile control system and to avoid some of the difficulties associated with force-based controllers, it has been widely used in various fields, such as signal processing, servo control, and image processing. Ratava et al. [
13] developed an adaptive optimizing fuzzy controller for controlling feed rate. The system designed based on the concept of the cutting state and collected expert rules. The experiment has shown that the system performed adequate results. Huang et al. [
14] used fuzzy control to realize high cutting rate and high surface quality requirement. The control system is based on controlling the cutting force being milled by digital adaptation of cutting parameters. The experimental results show that the milling system with the designed controller has high robustness and the machining efficiency of the milling system with the adaptive controller is much higher than for the conventional CNC milling system. Chen et al. [
15] designed an intelligent fuzzy proportional–integral–derivative (PID) controller to increase the processing efficiency of the CNC machine tool. The main feature of the proposed fuzzy PID controller was to change parameters in different degrees according to time-varying working conditions, thereby improving the adaptability and reliability of the controller. The advantages of fuzzy controller are that it is suitable for nonlinear systems and that it is described by control laws made by experts in their field. Miao and Li [
16] developed a fuzzy control system based on the assumed feed rate, radial and axial depths of cut for CNC profile milling feed rate determination to predict the cutting force. Liang et al. [
17] established a fuzzy logic-based torque control system for optimizing the material removal rate in high-speed milling processes. The aforementioned studies have all used fuzzy controllers to achieve good results. The reason is that fuzzy controller has a human-like reasoning mechanism that can solve complex problems with unclear inputs and make decisions accordingly. Therefore, this study uses fuzzy control in processing path-based dynamic feed rate scheduling method.
In this study, a fuzzy control for feed rate scheduling based on curvature and curvature variation is proposed. The proposed system is implemented in actual cutting, and to verify the data an optical three-dimensional scanner is used to measure the cutting trajectory of the workpiece. The major contributions of this study are presented as follows: (1) A fuzzy control method is used to effectively schedule a reasonable feed rate; (2) an average filter is developed to reduce jerks to avoid excessive impact on the machine, and (3) the proposed system increases the cutting accuracy under the same cutting time; moreover, it decreases the cutting time under the same cutting accuracy.
The remainder of this paper is organized as follows.
Section 2 describes materials and methods which includes fuzzy control for feed rate scheduling.
Section 3 presents the experimental results. Finally,
Section 4 presents the conclusions of this study, and future research directions are recommended.
3. Experimental Results
The ∞ shape, trident shape, and butterfly shape graphics [
18,
20], represented as NURBS graphs in
Figure 7a–c, respectively, are used in the cutting experiment. The sizes of those shapes were shrunk to fit into the aluminum workpiece for cutting experiment in addition 2000-point coordinates are generated by a resolution of 0.0005, and the curvature and curvature variation are extracted through the machining path.
Figure 8,
Figure 9 and
Figure 10 show the curvature and curvature variation of the ∞ shape, trident shape, and butterfly shape, respectively.
To ensure all the shapes are engraved with the same depth, each aluminum 6061 alloy workpiece will be face-processed before cutting. Furthermore, to prevent the tool from bending, the depth of cut is set to 2 mm.
Figure 11a shows the cutting process of an aluminum workpiece, and
Figure 11b shows the aluminum workpiece after cutting.
3.1. Optical 3D Scanner Measurement
Figure 12a shows a flowchart of optical three-dimensional measurement. First, the surface of the aluminum workpiece can easily refract light; therefore, a developer for measurement must be sprayed before optical measurement, as shown in
Figure 12b.
Figure 12c shows the scanning with the optical 3D scanner to obtain the geometric features of the workpiece.
Figure 13 shows the software output as a binary 3D graphics file. After obtaining the 3D graphics, the deepest coordinate point of the cutting path groove is acquired as the tool tip cutting path. This coordinate is matched to the path of the original machining program, and the shortest distance is calculated as the path error, as shown in
Figure 13.
This study performs three experiments. At the same time, three methods proposed by Luan et al. [
9], Yeh and Hsu [
10], and Giannelli et al. [
11] are also adopted to test and compare the results of the proposed method. The lower the tracking error the higher the accuracy and the less machining time the better the machining efficiency is. To performing suitable comparisons, the maximum feed rates, 300, 150, 185, and 58 mm/min, were calculated first according to the methods proposed by scholars before real cutting experiments. Additionally, the properties of the 5-axis CNC processing machine were also taken into consideration during the calculation.
3.2. Experiment 1: Cutting ∞ Shape
Figure 14,
Figure 15,
Figure 16 and
Figure 17 show the feed rate, acceleration, and jerk of cutting the ∞ shape using the three previous methods and our method, respectively.
Table 7 lists the processing time, maximum tracking error, minimum tracking error, average tracking error, and standard deviation. Cutting time indicates that CNC finishes executing the G-code of the machining.
3.3. Experiment 2: Cutting Trident Shape
Figure 18,
Figure 19 and
Figure 20 show the feed rate, acceleration, and jerk when cutting the trident shape using the three previous methods.
Figure 21 and
Figure 22 show the feed rate, acceleration, and jerk when cutting the trident shapes using our method at maximum feed rates of 150 and 300 mm/min, respectively.
Figure 23 shows a comparison of the jerk using the three previous methods and our method.
Table 8 lists the processing time, maximum tracking error, minimum tracking error, and average tracking error.
3.4. Experiment 3: Cutting Butterfly Shape
Figure 24,
Figure 25 and
Figure 26 show the feed rate, acceleration, and jerk when cutting the butterfly shape using the three previous methods.
Figure 27,
Figure 28 and
Figure 29 show the feed rate, acceleration, and jerk when cutting the butterfly shape using our method at maximum feed rates of 58, 185, and 300 mm/min, respectively.
Table 9 lists the processing time, maximum tracking error, minimum tracking error, and average tracking error.
4. Discussion
The ∞ shape experimental data indicate that although the method by Giannelli et al. [
11] has the largest jerk, the cutting speed is mostly as low as 20 mm/min; therefore, it shows the best tracking error while sacrificing the processing time. The proposed method has the same processing time as the method by Luan et al. [
9]. The maximum tracking error is better than 0.041 mm, minimum tracking error is better than 0.006 mm, and average tracking error is better than 0.039 mm. Compared with the method by Yeh and Hsu [
10], the maximum tracking error is less than 0.004 mm, minimum tracking error is better than 0.001 mm, and average tracking error is less than 0.005 mm; however, our method requires only half the processing time. In experiment 1, the proposed system increases the cutting accuracy by 41% under the same cutting time; moreover, it decreases the cutting time by 50% under approximative same cutting accuracy (0.023 and 0.028).
The trident shape experimental data prove that the method by Giannelli et al. [
11] has the best path error performance, although it somewhat sacrifices the processing time. The proposed method uses a maximum cutting speed of 300 mm/min, in which case its processing time is less than 1 min as with the method by Luan et al. [
9], and its maximum tracking error is better than 0.029 mm and average tracking error is better than 0.008 mm. Compared with the method by Yeh and Hsu [
10], the average tracking error is poorer than 0.004 mm; yet the processing time is better than 82 s. In experiment 2, the proposed system increases the cutting accuracy by 70% under the approximative same cutting time (46 and 45). Even though our method shows higher average tracking errors in comparison with [
10] and [
11], the machining time can be saved around 2.8 and 6.5 times, respectively.
The butterfly shape experimental data illustrate that the maximum tracking error of 0.004 mm is better than that of the method by Giannelli et al. [
11], average tracking error is better than 0.002 mm, and processing time is half for the same maximum speed. Under the same maximum speed as in the method by Yeh and Hsu [
10], the maximum tracking error is better than 0.019 mm, average tracking error is better than 0.003 mm, and processing time is less than 11 s. Our method uses the same maximum speed as the method by Luan et al. [
9], and the maximum tracking error is better than 0.029 mm, average tracking error is better than 0.009 mm, and processing time is less than 6 s. In experiment 3, the average tracking error and machining time are all decreasing using the proposed method overall.
5. Conclusions
In this study, fuzzy control is used to productively schedule a reasonable feed rate, and an average filter is used to effectively reduce jerks to avoid excessive impact on the machine. The experimental results prove that our proposed method can effectively shorten the processing time, improve the cutting precision, as well as providing a stable machining which lower standard deviation of tracking error compared with those of three previously proposed methods.
In the future, dynamic feed rate scheduling could be applied to noncontact processing such as laser cutting and 3D printing. The current study only considers the XY plane, so the Z direction can be considered in future studies for dynamic feed rate scheduling to reach an optimal machining. For fuzzy control system, a fuzzy neural network could be implemented to learn the machining parameters of the machine, thus improving the performance of this method.