Optimization of Computer Numerical Control Interpolation Parameters

applied sciences

Article Optimization of Computer Numerical Control Interpolation Parameters Using a Backpropagation Neural Network and Genetic Algorithm with Consideration of Corner Vibrations

Hsiang-Chun Tseng 1, Meng-Shiun Tsai 2,*, Chih-Chun Cheng 1 and Chen-Jung Li 3

1 Department of Mechanical Engineering, National Chung Cheng University, Chiayi 62102, Taiwan; [email protected] (H.-C.T.); [email protected] (C.-C.C.) 2 Department of Mechanical Engineering, National Taiwan University, Taipei 106319, Taiwan 3 Department of Mechatronics Engineering, National Kaohsiung University of Science and Technology, Kaohsiung 824005, Taiwan; [email protected] * Correspondence: [email protected]

Abstract: This paper presents an optimization algorithm for tuning the interpolation parameters of computer numerical control (CNC) controllers; it operates by considering multiple objective functions, namely, contour errors, the machining time (MT), and vibrations. The position commands, position errors, and vibration signals from 1024 experiments were considered in the designed trajectory. The experimental data—the maximum contour error (MCoE), MT, and corner vibration (CVib)—were analyzed to compute the performance index. A backpropagation neural network (BPNN) with 20 hidden layers was applied to predict the performance index. The correlation coefﬁcients for the Citation: Tseng, H.-C.; Tsai, M.-S.; predicted values and experimental results for the MCoE, MT, and CVib based on the validation data Cheng, C.-C.; Li, C.-J. Optimization of were 0.9984, 0.9998, and 0.9354, respectively. The high correlation coefﬁcients highlight the accuracy Computer Numerical Control of the model for designing the interpolation parameter. After the BPNN model was developed, Interpolation Parameters Using a a genetic algorithm (GA) was adopted to determine the optimized parameters of the interpolation Backpropagation Neural Network under different weighting of the performance index. A weighted sum approach involving the and Genetic Algorithm with objective function was employed to determine the optimized interpolation parameters in the GA. Consideration of Corner Vibrations. Thus, operators can judge the feasibility of the interpolation parameter for various weighting settings. Appl. Sci. 2021, 11, 1665. https:// Finally, a mixed path was selected to verify the proposed algorithm. doi.org/10.3390/app11041665

Academic Editor: Luis Norberto Keywords: interpolator; data driven; optimization; vibration; contour error; machine performance López De Lacalle

Received: 4 January 2021 Accepted: 8 February 2021 1. Introduction Published: 12 February 2021 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, Publisher’s Note: MDPI stays neutral machine errors, such as static error, dynamic error, and process error, should be mini- with regard to jurisdictional claims in mized [1]. Generally, dynamic errors are caused by interpolation error, servo lag [2,3], and published maps and institutional affil- vibrations [3,4]. Surface quality is typically determined through considerations of surface iations. 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 Copyright: © 2021 by the authors. contour errors and improving surface quality. Licensee MDPI, Basel, Switzerland. Computer numerical control (CNC) controller interpolation that considers the acceler- This article is an open access article ation/deceleration (Acc/Dec) planning of the machining trajectory has considerable effects distributed under the terms and on the finished product. Therefore, adjusting Acc/Dec planning [9] not only improves conditions of the Creative Commons contouring accuracy and suppresses machine vibration, improving surface quality, but Attribution (CC BY) license (https:// may also reduce the overall MT. A specific instrument was developed to measure the creativecommons.org/licenses/by/ relative dynamic displacement between the tool and workpiece at the tool center point 4.0/).

Appl. Sci. 2021, 11, 1665. https://doi.org/10.3390/app11041665 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 1665 2 of 15

(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]. The finite element approach [20,21] is generally applied to predict vibration behavior. However, because of highly uncertain structural characteristics (e.g., stiffness), the damping ratio cannot be identified accurately. Thus, the finite element approach is not feasible for predicting the vibrations of the TCP accurately under various Acc/Dec conditions. A more suitable approach for constructing a CPS model that can predict the contour error, MT, and TCP vibrations simultaneously is using an ANN with a well-designed experimental process. The design process in this paper involved the use of a data acquisition system that could capture the data automatically without any operator. The ranges of the Acc/Dec parameters were selected in accordance with the coarse machining to fine machining process. The data were applied to construct an ANN model. Subsequently, the weighted sum and GA approaches were adopted to determine the optimized Acc/Dec parameters when subject to various cost functions of the contour error, MT, and vibration. This paper is divided into five sections. Section2 introduces the interpolation design. Section3 reveals the experimental process and the data analysis procedures for converting the machine performance index. In Sections 4.1 and 4.2, the Z-score and a backpropagation neural network (BPNN) for prediction are described. In Section 4.3, how a weighted sum approach was employed to identify design optimization concerns to determine the optimized parameters is described. Finally, a mixed path was selected to verify this paper’s proposed algorithm. Appl. Sci. 2021, 11, 1665 3 of 15

2. Introduction of Interpolator An interpolator is used to generate the velocity profile and position commands by interpreting a part program for each axis. For a linear interpolation (G01), Figure1 reveals the S-shaped velocity and Acc/Dec profiles [22,23] for a single block that consists of constant speed periods (A), a constant slope for the Acc/Dec periods (B), and a constant Acc/Dec period (C). Based on the S-shaped Acc/Dec control, the velocity profile can be calculated according to the maximum velocity fmax, the maximum acceleration Amax, the time constant of the S-shaped Acc/Dec T2, and the maximum jerk Jmax, as illustrated in Figure1.

Figure 1. S-shaped acceleration/deceleration (Acc/Dec) control for a signal block.

Generally, each axis has an allowable acceleration value that is determined according to the driving capability of its servo motor. Compared with a single block, multiple blocks of numerical control (NC) codes require one more parameter, which is given as the corner velocity Vc [24] (Figure2b). However, velocity discontinuity may occur when the velocity proﬁle is distributed to each axis, as indicated in Figure2c,d. Clearly, a higher corner velocity reduces the MT, but discontinuity can cause high acceleration and jerk for each axis as well as a large vibration in the corner.

Figure 2. Corner velocity in velocity profile: (a) a sample NC code and multiple blocks (b) tangent velocity profile (c) x-axis velocity profile (d) y-axis velocity profile. Appl. Sci. 2021, 11, 1665 4 of 16

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Appl. Sci. 2021, 11, 1665 To reduce discontinuity, the Acc/Dec after interpolation4 of 15 (ADAI) [12] for each axis is applied. The ADAI is a finite impulse response 25 filter with a time constant equal to Ta To reduce discontinuity,that smooths the Acc/Dec the acceleration after interpolation and jerk (ADAI)profile, [12]as displayed for each axisin Figure is 3. The application of applied. The ADAI is a ADAIfinite impulsesacrifices response contour 25accuracy filter with and aincrea timeses constant the MT. equal Table to 1T revealsa how the machine To reduce discontinuity, the Acc/Dec after interpolation (ADAI) [12] for each axis is that smooths the accelerationperformance and jerk varies profile, as asthe displayed interpolation in Figure para 3.meters The application change. The of tradeoff among various applied. The ADAI is a finite impulse response 25 filter with a time constant equal to T parameter settings for the contour errors, MT, and corner vibrationa (CVib) motivated the that smoothsADAI sacrifices the acceleration contour and accuracy jerk profile, and increa as displayedses the MT. in Figure Table 31. reveals The application how the ofmachine current researchers to propose an optimization method based on the constructed ANN ADAIperformance sacrifices contour varies accuracy as the interpolation and increases para themeters MT. Table change.1 reveals The howtradeoff the among machine various model. performanceparameter varies settings as the for interpolation the contour parameterserrors, MT, change.and corner The vibration tradeoff (CVib) among motivated various the parametercurrent settings researchers for the to contour propose errors, an optimiza MT, andtion corner method vibration based (CVib) on the motivated constructed the ANN currentmodel. researchers to propose an optimization method based on the constructed ANN model.

Figure 3. (a) Velocity profiles in the Acc/Dec after interpolation (ADAI). (b) Velocity profiles out of the ADAI [25]. Figure 3. (a) Velocity profiles in the Acc/Dec after interpolation (ADAI). (b) Velocity profiles out of Figurethe 3. (ADAIa) Velocity [25]. proﬁles in theTable Acc/Dec 1. Trends after in interpolation machine performance (ADAI). (b as) Velocitythe interpolation proﬁles out parameters of change. the ADAI [25]. Corner Para. Contour Errors Machining Time Table 1. Trends in machine performance as the interpolation parameters change. Vibrations Table 1. Trends in machine performance as the interpolation parameters change. ↗ Increase ↘ DecreaseCorner ↗ Increase Para. Contour Errors Machining Time Para. Contour Errors Machining↗ Increase Time Corner Vibrations↗Vibrations Increase ↘ Decrease

Amax ↗Increase Increase2 ↘Decrease↘ Decrease Decrease Increase↗↗ Increase Increase ↘ Decrease

Ta ↗Increase Increase ↗Increase Increase↗ Increase Decrease↘↘ Decrease Decrease ↗ Increase T2 2 ↘Decrease Decrease ↗Increase Increase↗ Increase Decrease↘↘Decrease Decrease ↗ Increase

Vc ↗Increase Increase Decrease↘ Decrease Increase↗ Increase Vmax ↗Increase Increase3. Experimental ProcedureDecrease↘Decrease and Analysis ProcessIncrease↗ Increase 3.1. Experimental Designs 3. Experimental3. Experimental Procedure Procedure and Analysis andTo Analysisestablish Process Processthe data-driven model, an automated acquisition system (AAS) was 3.1. Experimental3.1. Experimental Designs Designsdeveloped. The AAS uses the Fuji Automatic Numerical Control (FANUC) Open CNC To establishTo establish the data-driven the data-drivenAPI model,Specifications model, an automated anlibrary automated acquisition(FOCAS) acquisition to system capture system (AAS) the (AAS)position was was command and position developed.developed. The AAS The uses AAS the uses Fujifeedback. the Automatic Fuji The Automa Numericaltrackingtic Numerical error Control and (FANUC)Control contour (FANUC) Open error CNC can Open APIbe CNCcomputed accordingly. The SpecificationsAPI Specifications library (FOCAS) libraryvibration to (FOCAS) capture signal the to measured positioncapture command theby aposition three-axis and command position accelerometer feedback. and (PCBposition 356A32 was made by PCB The trackingfeedback. error The and tracking contourPiezotronics, error error and can becontour Depew, computed errorNY, accordingly.USA) can isbe captured computed The vibration using accordingly. National signal Instrument The (NI) devices (NI- measuredvibration by a signal three-axis measured accelerometer9234). by aTo three-axis ensure (PCB the 356A32accelerometer AAS could was made acquire(PCB by 356A32 data PCB without Piezotronics,was made operators, by PCB the library provided by Depew,Piezotronics, NY, USA) is Depew, captured NY, usingNI USA)ANSI National is C captured support Instrument usingwas used (NI)National devicestogether Instrument (NI-9234). with the (NI)To FOCAS ensuredevices library. (NI- The human–machine the AAS9234). could To acquireensure datathe AAS withoutinterface could operators, acquire(HMI) theprogramdata library without is provided presented operators, by in NI the Figure ANSI library C4. support providedThe program by includes functions to was usedNI ANSI together C support with the was FOCASconnect used library. togetherthe CNC The with human–machinecontroller the FOCAS and NI interfacelibrary. device The(HMI)and humanto program execute–machine experiments sequentially by is presentedinterface in (HMI) Figure 4program. The programusing is presented a designed includes in script functionsFigure file 4. that toThe connect can program change the CNCthincludese values controller functions of five interp to olator parameters and and NI device and to execute experiments sequentially by using a designed script file that connect the CNC controllerthe total and acquisition NI device time.and to Here, execute the samplingexperiments rate sequentially for the CNC bywas 1000 Hz, and that for can change the values of five interpolator parameters and the total acquisition time. Here, using a designed script filethe thatNI-9234 can changewas 2048 th eHz. values of five interpolator parameters and the sampling rate for the CNC was 1000 Hz, and that for the NI-9234 was 2048 Hz. the total acquisition time. Here, the sampling rate for the CNC was 1000 Hz, and that for the NI-9234 was 2048 Hz. Appl. Sci. 2021, 11, 1665 5 of 15

Figure 4. Human–machine interface (HMI) of the automated acquisition system.

The experiments were conducted on a three-axis horizontal milling machine (Tongtai HA-500II was manufactured by Tongtai Machine and Tool Co., LTD., R.O.C., Kaohsiung City, Taiwan), with the three-axis accelerometer (PCB 356A32) installed on the spindle housing, as shown in Figure5.

Figure 5. Experimental machine and accelerometer setup.

To investigate the dynamic responses for setting various interpolation parameters, a FFD was adopted in this study. The design of the interpolation parameters is presented in Table2; the values of the parameters were selected according to the various machining conditions. Each parameter, namely, Amax, Ta, T2, Vc, and fmax, was set to four different levels. In total, 1024 experiments were conducted for the FFD, and the designed parameter ranges were applied to various machining processes. Appl. Sci. 2021, 11, 1665 6 of 15

Appl. Sci. 2021, 11, 1665 6 of 16

Table 2. Levels of each interpolation parameter. Compared with the TED, the FFD obtained an enhanced ANN model; this finding is Parameter Unit Level 1 Level 2 Level 3 Level 4 described in Section 4.2. 2 Amax mm/s 900 1500 2100 2700 Ta ms 35 42 49 56 Table 2.T Levels2 of each inte ms rpolation parameter.0 8 16 24 ParameterVc mm/minUnit 220Level 1 340Level 2 Level 460 3 Level 580 4 fmax mm/min 300 500 1000 1500 mm/s 900 1500 2100 2700 ms 35 42 49 56 The 1024 experimentsms for the given0 trajectory discussed8 in Section16 3 required24 9 h

to complete. However,mm/min the AAS could 220 command the340 machine to460 conduct the designed580 trajectory and capturemm/min the data automatically300 without500 pressing1000 the cycle-start1500 button. Compared with the TED, the FFD obtained an enhanced ANN model; this ﬁnding is 3.2.described Preprocessing in Section of 4.2the. Data into Performance Index

3.2. PreprocessingIn this study, of the the minimized Data into Performance cost functi Indexon was selected to be the weighted sum of different performance indices. The three performance indices were the maximum contour In this study, the minimized cost function was selected to be the weighted sum of error (MCoE), MT, and corner vibration (CVib). The captured data had to be converted different performance indices. The three performance indices were the maximum contour into performance indices before the training process commenced. error (MCoE), MT, and corner vibration (CVib). The captured data had to be converted into 3.2.1.performance Analysis indices of Contour before Error the trainingand MT process commenced. 3.2.1.The Analysis contour of error Contour is defined Error and as the MT short distance between the desired contour and machiningThe contour contour, error as shown is deﬁned in Figure as the 6, short where distance the desired between contour the desiredis obtained contour from andthe G-codemachining and contour, the machining as shown incontour Figure 6is, whereobtained the desiredfrom the contour position is obtained feedback. from The the rectangularG-code and thetrajectory, machining as contourdisplayed is obtained in Figu fromre 7, the can position be tested feedback. by tuning The rectangular various interpolatortrajectory, as parameters. displayed in The Figure ma7ximum, can be acceleration, tested by tuning ADAI, various time constant interpolator of the parameters. S-shaped Acc/DecThe maximum corner acceleration,velocity, and ADAI, maximum time constantvelocity all of themay S-shaped have dramatic Acc/Dec effects corner on velocity, corner errors,and maximum vibration, velocity and MT. all mayThus, have the dramatic testing trajectory effects on in corner Figure errors, 7 was vibration, applied and as MT.the machiningThus, the testingtrajectory trajectory under various in Figure interpolator7 was applied parameters. as the machining For the trajectory trajectory consisting under ofvarious a straight-line interpolator segment, parameters. the contour For the erro trajectoryr could be consisting computed of by a straight-line simply applying segment, the point-to-linethe contour errorformulation. could be computed by simply applying the point-to-line formulation.

Figure 6. Deﬁnition of contour error. Figure 6. Definition of contour error. Appl.Appl. Sci. Sci. 2021 2021, ,11 11, ,1665 1665 7 7of of 16 16 Appl. Sci. 2021, 11, 1665 7 of 15

Figure 7. Testing trajectory. (The corner 1, 2 and 3 were described in Section 3.2.1 and the period FigureA,Figure B, C 7. and Testing 7. TestingD were trajectory. trajectory.described (The in (The Section corner corner 3.2.2.).1, 1,2 and 2 and 3 we 3 werere described described in inSection Section 3.2.1 3.2.1 and and the the period period A, A, B,B, CC andand DD werewere describeddescribed inin SectionSection 3.2.2 3.2.2.)..). Figure 8 reveals the contour errors of the experiment, with the , , T2, , and FigureFigure 8 reveals8 reveals the the contour contour errors errors of the of experiment, the experiment, with withthe the , Amax, T,2,T a, ,T and2, V c, being given as 2700 mm/s , 35 ms , 0 ms , 220 mm/min , and 1000 mm/min , 2 respectively.and beingfmax beinggiven The maximum givenas 2700 as 2700mm/scontourmm/s, in35 corners ,ms 35, ms0 1, ,ms 2, 0 and,ms 220, 3 220 wasmm/minmm/min approximately, and, and 1000 100060 mm/minμmmm/min. The , , respectively.otherrespectively. contour The errors The maximum maximum away from contour contour the incorner incorners corners were 1, 1,2,much 2,and and 3less was 3 wasthan approximately approximately that of the maximum60 60μmµm. The. The othercontour.other contour contour On the errors basis errors awayof awaythis from observation, from the the corner corner the wereMCoE were much was much selected less less than than as thatthe that first of of theperformance the maximum maximum contour.index.contour. The On OnMT the the isbasis also basis of a ofthiscritical this observation, observation, factor and the thus the MCoE MCoEwas wasselected was selected selected as the as assecondthe the first ﬁrst performance performance performance index.index.index. TheIn Thethis MT MTcase, is isalso the also MTa critical a equaled critical factor factor 7.361 and s. and thus thus was was selected selected as as the the second second performance performance index.index. In Inthis this case, case, the the MT MT equaled equaled 7.361 7.361 s. s.

Figure 8. Maximum contour error (MCoE) and machining time (MT) for one experiment. (The period Figure 8. Maximum contour error (MCoE) and machining time (MT) for one experiment. (The 1, 2 and 3 were marked as the corner 1, 2 and 3 in Figure7, respectively.) period 1, 2 and 3 were marked as the corner 1, 2 and 3 in Figure 7, respectively.) Figure 8. Maximum contour error (MCoE) and machining time (MT) for one experiment. (The period3.2.2. 1, Analysis2 and 3 were of CVibsmarked as the corner 1, 2 and 3 in Figure 7, respectively.) 3.2.2. AnalysisAs shown of inCVibs Figure 2, the discontinuity of the velocity proﬁle in the corner can cause a 3.2.2.large AsAnalysis acceleration shown of in CVibsFigure and jerk.2, the The discontinuity vibrations inof cornerthe velocity 3 in the profile x direction in the corner in Figure can9 causevalidate a thelargeAs observation. shownacceleration in Figure Theand parameters2,jerk. the Thediscontinuity vibrations of Amax of, inT thea ,cornerT velocity2, Vc ,3 andin profile thefmax x indirectionused the corner in thein Figure experimentcan cause 9 validate the observation.2 The parameters of , , T2, , and used in the a largewere 2700accelerationmm/s ,and 35 ms jerk., 0 ms The, 580 vibrationsmm/min in, and corner 1500 3mm/min in the x ,direction respectively. in Figure To further 9 analyze the vibration behavior in the corner, four periods were marked as A, B, C, and validate the observation. The parameters of , , T2, , and used in the D. As indicated in Figure2, period A corresponds to the motion of the x axis moving in a Appl. Sci. 2021, 11, 1665 8 of 16

Appl. Sci. 2021, 11, 1665 experiment were 2700 mm/s , 35 ms , 0 ms , 580 mm/min , and 1500 mm/min , 8 of 15 respectively. To further analyze the vibration behavior in the corner, four periods were marked as A, B, C, and D. As indicated in Figure 2, period A corresponds to the motion of the x axis moving in a state of constant velocity. Period C is the state right after the statecorner of as constant shown in velocity. Figure 7. Period Due to Cthe is disc theontinuity state right velocity after theprofile corner as shown as shown in Figure in Figure 7. Due2, excessive to the discontinuityvibration might velocity occur as proﬁle shown as in shown Figure in 9. Figure Period2 ,D excessive corresponds vibration to the might occurmotion as for shown the y inaxis Figure moving9. Period at a constant D corresponds velocity. toBecause the motion the vibration for the y of axis period moving B at a constantcorresponds velocity. to the Because designed the Acc/Dec, vibration the of re periodsidual Bvibration corresponds at period to the C designedcould cause Acc/Dec, themachining residual marks vibration in the at periodcorner. CThe could vibration cause machiningin period C marks was inselected the corner. as the The third vibration inperformance period C wasindex. selected as the third performance index.

Appl. Sci. 2021, 11, 1665 9 of 16

Figure 9. 9. RawRaw vibrations vibrations of ofthe the high high Acc/Dec Acc/Dec experiment experiment.. (The period (The period A, B, CA, and B, D C were and Dmarked were marked as in Figure 7, respectively.) as in Figure7, respectively.) To further observe the vibration behavior around the corner, the fast Fourier

transform60 (FFT) for the periods A, C, and D was calculated (Figure 10). Four major peaks were observed at 30, 40, 116, and 426 Hz. The first natural frequencyA part was revealed to be 30 Hz, which affects the machining process more dramaticallyC part compared with the other D part modes.50 Furthermore, the magnitude of the 30 Hz mode was highly dependent upon the

interpolator2 parameters. The analysis of 1024 experiments revealed that the 30 Hz vibration mode can be effectively reduced by tuning the interpolation parameters, 40 whereas the other frequency modes were less susceptible to change. Therefore, the vibration magnitude of 30 Hz was selected as the third performance index. 30

20 Acceleration, mm/s

0 100 200 300 400 500 Frequency (Hz) FigureFigure 10. 10.FastFast Fourier Fourier transform transform (FFT) (FFT) for vibrations for vibrations of periods of periods A, C, and A, D C, in and Figure D in 9. Figure9.

4. ModelingTo further and Optimization observe the vibration Concerns behavior around the corner, the fast Fourier transform (FFT)The formachine the periods performance A, C, inde andx D is waswell calculatedknown to be (Figure highly 10 affected). Four by major Acc/Dec peaks were planning.observed A atdata 30,-driven 40, 116, model and 426is presented Hz. The ﬁrstin this natural section frequency for predicting was revealedthe machine to be 30 Hz, performancewhich affects index. the The machining MCoE and process MT were more computed dramatically from the compared command with and theposition other modes. feedback,Furthermore, and the the CVib magnitude was computed of the30 from Hz t modehe accelerometer. was highly dependentThe Z-scoreupon was applied the interpolator toparameters. normalize the The performance analysis of index, 1024 experimentsand the BPNN revealed was applied that theto establish 30 Hz vibration the model mode can andbe used effectively to evaluate reduced the testing by tuning trajectory. the interpolation parameters, whereas the other frequency modes were less susceptible to change. Therefore, the vibration magnitude of 30 Hz was 4.1.selected Z-Score as (Standard the third Score performance) index. Because each dimension of the performance index is different, the accuracy of the trained model might be biased. To address this problem, a normalization and standardization process is generally used in machine learning. The Z-score is used to convert the raw data according to = ( )/ , where μ and are the mean and standard deviation of the population, respectively. For the testing and training data, the μ and of the MCoE were 46.08 and 𝑧𝑧21.17𝑥𝑥 −m𝜇𝜇, respectively.𝜎𝜎 The μ and𝜎𝜎 of the MT were 12.69 and 7.45 s, respectively. The μ and of the CVib were 7.62 and 6.16 mm/s , respectively.𝜎𝜎 µ 𝜎𝜎 2 𝜎𝜎 4.2. BPNN Algorithm A conventional BPNN algorithm [26], displayed in Figure 11, was applied to predict the machine performance indices. Several methodologies, such as [16,18], were tested, but the BPNN was the most accurate for predicting the information index and could be applied later in the optimization process with less computational burden. The architecture contained five inputs, three output elements, 20 hidden layers, and one output layer. Appl. Sci. 2021, 11, 1665 9 of 15

4. Modeling and Optimization Concerns The machine performance index is well known to be highly affected by Acc/Dec planning. A data-driven model is presented in this section for predicting the machine performance index. The MCoE and MT were computed from the command and position feedback, and the CVib was computed from the accelerometer. The Z-score was applied to normalize the performance index, and the BPNN was applied to establish the model and used to evaluate the testing trajectory.

4.1. Z-Score (Standard Score) Because each dimension of the performance index is different, the accuracy of the trained model might be biased. To address this problem, a normalization and standardization process is generally used in machine learning. The Z-score is used to convert the raw data according to z = (x − µ)/σ, where µ and σ are the mean and standard deviation of the population, respectively. For the testing and training data, the µ and σ of the MCoE were 46.08 and 21.17 µm, respectively. The µ and σ of the MT were 12.69 and 7.45 s, respectively. The µ and σ of the CVib were 7.62 and 6.16 mm/s2, respectively.

4.2. BPNN Algorithm A conventional BPNN algorithm [26], displayed in Figure 11, was applied to predict the machine performance indices. Several methodologies, such as [16,18], were tested, but the BPNN was the most accurate for predicting the information index and could be Appl. Sci. 2021, 11, 1665 10 of 16 applied later in the optimization process with less computational burden. The architecture contained ﬁve inputs, three output elements, 20 hidden layers, and one output layer.

Figure 11. Architecture of the backpropagation neural network (BPNN). Figure 11. Architecture of the backpropagation neural network (BPNN). The inputs were the interpolation parameters listed in Table2, and the outputs were theThe performance inputs were indices, the interpolation namely, the MCoE, parameters MT, and listed CVib. in Table The training2, and the dataset outputs was were from thethe performance 1024 experiments indices, described namely, inthe Section MCoE,3 ,MT, and and the CVib. validation The training dataset compriseddataset was 64 from ran- thedomly 1024 selected experiments experiments described and thein Section parameter 3, and ranges the in validation Table2. Besides, dataset 1024 comprised experiments 64 randomlywere conducted selected for experiments the training and dataset; the anotherparameter 64 experimentsranges in Table were conducted2. Besides, for1024 the experimentsvalidation of were the training conducted model. for The the other training four experimentsdataset; another were executed64 experiments for testing were the conductedoptimization for process.the validation A total of 1092the training experiments model. were The conducted other four in theexperiments study. In Table were3 , executedthe training for testing and validation the optimization results for process. the root A total mean of square 1092 experiments error (RMSE) were and conducted maximum inerror the study. (ME)are In Table listed 3, to the demonstrate training and the vali accuracydation of results the trained for the model. root mean square error (RMSE)In and the validationmaximum dataset, error (ME) the RMSEare listed and to ME demonstrate of the MCoE the were accuracy 1.0976 of and the 5.5356 trainedµm , model.respectively. The RMSE and ME of the MT were 0.1305 and 0.5847 s, respectively. The RMSE error was approximately 1% when a mean value of MT µ equal to 12.69 s was Tableapplied. 3. Training Finally, and the validati RMSEon andresults ME for of the the artificial CVib were neural 4.48 network and 8.28(ANN)mm/s model:2, respectively. the maximumThe correlation contour coefficients error (MCoE) of and the predictedmachining valuestime (MT) and and experimental corner vibration results (CVib). were calculated using the Excel CORREL function. TheMCoE equation for theMT correlation coefficientCVib for a two random variable array of X and Y is defined0.0587 as 0.0006 0.38 RMSE of training (μm) n (s) (mm/s ) ∑i=1(x − x)(y − y) Correl(X, Y) = q1.0976 0.1305 3.14 (1) RMSE of validation n 2 n 2 (∑μm)i=1 (x − x) ∑i=(s)1( y − y) (mm/s ) 4.9264 0.2883 4.48 ME of training (μm) (s) mm/s) 5.5356 0.5847 8.28 ME of validation (μm) (s) (mm/s)

In the validation dataset, the RMSE and ME of the MCoE were 1.0976 and 5.5356 μm, respectively. The RMSE and ME of the MT were 0.1305 and 0.5847 s, respectively. The RMSE error was approximately 1% when a mean value of MT μ equal to 12.69 s was applied. Finally, the RMSE and ME of the CVib were 4.48 and 8.28 mm/s, respectively. The correlation coefficients of the predicted values and experimental results were calculated using the Excel CORREL function. The equation for the correlation coefficient for a two random variable array of X and Y is defined as ∑ ( − ̅)( − ) Correl(X, Y) = (1) ∑( −̅) ∑( −)

where ̅ and are the means of the data of x and y, respectively. The correlation coefficients of the predicted values and experimental results for the MCoE, MT, and CVib based on the validation data were 0.9984, 0.9998, and 0.9354, respectively. As indicated in the introduction, the vibration of a machine tool is extremely difficult to predict using the finite element method. The high correlation coefficients suggest the developed model is accurate for designing the interpolation parameters. Experimental design is a crucial concern in developing data-driven models. The TED has the advantage of requiring a relatively low number of experiments, which reduces the Appl. Sci. 2021, 11, 1665 10 of 15

where x and y are the means of the data of x and y, respectively. The correlation coefficients of the predicted values and experimental results for the MCoE, MT, and CVib based on the validation data were 0.9984, 0.9998, and 0.9354, respectively. As indicated in the introduction, the vibration of a machine tool is extremely difficult to predict using the finite element method. The high correlation coefficients suggest the developed model is accurate for designing the interpolation parameters.

Table 3. Training and validation results for the artiﬁcial neural network (ANN) model: the maximum contour error (MCoE) and machining time (MT) and corner vibration (CVib).

MCoE MT CVib 0.0587 0.0006 0.38 RMSE of training (µm) (s) (mm/s2) 1.0976 0.1305 3.14 RMSE of validation (µm) (s) (mm/s2) 4.9264 0.2883 4.48 ME of training (µm) (s) mm/s2) 5.5356 0.5847 8.28 ME of validation (µm) (s) (mm/s2)

Experimental design is a crucial concern in developing data-driven models. The TED has the advantage of requiring a relatively low number of experiments, which reduces the data acquisition time. However, the results might not be as accurate as those gener- ated through a FFD [27]. To compare the performance of the two experimental design approaches, L16(45) arrays—including ﬁve factors and four levels—were used in the TED in this study. Only 16 experiments were conducted with the training dataset. The RMSE and ME of the MCoE were 19.33 and 40.47 µm, respectively. The RMSE and ME of the MT were 2.1144 and 7.9993 s, respectively. The CVib was 13.52 and 26.12 mm/s2, respectively. Thus, the accuracy of the TED-based model was much lower than that of the FFD. Although the number of experiments needed in the FFD was much higher than that in the TED, the AAS provided a convenient platform for conducting more experiments with less effort.

4.3. Design Optimization Concerns and Components of the Multi-Objective Function After the BPNN model was developed, the next step was to deﬁne the cost function for optimization. Here, the genetic algorithm (GA) was applied to search for the optimum solution based on unconstrained functions. The GA, relying on biologically inspired operators such as selection, crossover, and mutation to generate high-quality solutions, is commonly used in nonlinear optimization problems. The GA provided the multi-objective optimization solution with the given constraints on the interpolation parameters. Here, a weighted sum approach 16 was adopted, and the objective function was given as

L = w1 × f 1 + w2 × f 2 + w3 × f 3 (2)

where f1, f2, and f3 represent the BPNN outputs, and w1, w2, and w3 are the weighting functions, respectively. The objective function with constraints is listed in Table4. Depend- ing on the machining process, different weightings were selected; the results are listed in Table5. For example, if w1, w2, and w3 were set to (1, 0, 0), accuracy was the major priority. The GA algorithm determined the optimization interpolation parameters with the lowest acceleration Amax, feedrate fmax, and corner velocity Vc, and the largest time constants Ta and T2 within the given constraints. For the weighting selected as (0, 1, 0), the major priority was MT, and the corresponding results are given in Table6. The machine time was approximately 5 s, which was the shortest among the four cases. Generating noteworthy results that had not been reported in the literature was the priority; thus, the vibration for Case 3 was selected. In Case 3, the CVib was the lowest among the four cases, but the MT was longer than that of Case 2. Neither the MCoE or MT, nor the CVib of Case 4 Appl. Sci. 2021, 11, 1665 11 of 15

achieved the highest performance among the four cases. Without the model, determining the optimization parameters was impossible.

Table 4. Multi-objective function and constraints following the weighted sum approach.

Multi-Objective Function Constraints 900 ≤ Amax ≤ 2700 L = w1 × f 1 + w2 × f 2 + w3 × f 3 35 ≤ Ta ≤ 56 0 ≤ T2 ≤ 24 Minimum L 220 ≤ Vc ≤ 580 300 ≤ Vmax ≤ 1500

Table 5. Results for optimized interpolation parameters according to the optimized mode.

Case 1 2 3 4 (w1, w2, w3) (1, 0, 0) (0, 1, 0) (0, 0, 1) (0.5, 0.2, 0.5) A max 900 2079 1575 900 (mm/s2) T a 35 42 55 35 (ms) T2 23 0 6 23 (ms) V c 220 579 220 220 (mm/min) f max 300 1499 300 884 (mm/min)

Table 6. Artiﬁcial neural network (ANN)-predicted and experimental results for MCoE, MT, and CVib.

MCoE MT CVib Case (µm) (s) (mm/s2) 1 Pred. 15.9 24.012 10.4 Exp. 17.1 24.085 14.1 Error * −1.2 −0.073 −3.7 2 Pred. 100.1 5.009 63.8 Exp. 99.7 4.867 69.7 Error * 0.4 0.142 −5.9 3 Pred. 24.5 24.029 5.0 Exp. 25.0 24.075 4.6 Error * −0.5 −0.046 0.4 4 Pred. 23.4 8.060 5.4 Exp. 23.8 8.299 9.0 Error * −0.4 −0.239 −3.6 Note*: The error was calculated between the predicted value of the experimental result.

The predicted and experimental results are listed in Table6; these data were used to validate the accuracy of the BPNN model. Notably, the MT accuracy for all the cases was less than 3%, which is challenging to achieve without knowing the CNC kernel for the FANUC controller.

4.4. Veriﬁcation by Employing a Complex Path To apply the developed model in real machining, a complex trajectory, illustrated in Figure 12a, was tested. The trajectory, including the G01, G02, and corner errors for the conjunction of each segment, is displayed in Figure 12b–e. The interpolation parameters listed in Table5 were used in the experiments. Clearly, contour errors generally occurred in sharp corners. The contour errors for the smooth conjunctions between G01 and G02 and from G02 to G03 in the period K5–K7 were extremely small compared with those in the period K2–K4. As expected, the MCoE for Case 1 at the peak of the period K2–K4 was Appl. Sci. 2021, 11, 1665 12 of 15

the smallest, and the MCoE of Case 2 was the largest. The vibrations for Case 1 to Case 4 in the period K1 are revealed in Figure 13. Case 3 had the lowest vibration among the four cases. To further illustrate the spectrum of the vibration, FFT was conducted on the Appl. Sci. 2021, 11, 1665 13 of 16 vibration signal when passing the corner; the results are presented in Figure 13b. Clearly, the CVib of Case 2 was the largest among the four cases, and that of Case 3 achieved the least vibration at the ﬁrst resonance.

(a) (b)

(e) (f)

Figure 12. Cont. Appl. Sci. 2021, 11, 1665 14 of 16

Appl. Sci.Appl.2021 Sci., 11 2021, 1665, 11, 1665 14 of 1316 of 15

(g) (g)

FigureFigureFigure 12. Experimental 12. 12. Experimental Experimental contour contourcontour for Cases for for Cases Cases 1 to 1 4: to1 ( ato4:) ( overview,4:a) (overview,a) overview, (b )(b period) period(b) period K2, K2, (c () cK2,) period period (c) period K3, K3, ((dd) )K3, periodperiod (d) K4,period K4, ( (ee)) period periodK4, (e K5,) K5,period (f) K5, (f) period K6, and (g) period K7. period(f) K6,period and K6, (g) periodand (g) K7. period K7.

50 50 Case 1 Case 1 Case 2 Case 3 Case 2 40 Case 3 40 Case 4 Case 4 2 30 30

20 20

10 Acceleration, mm/s

0 0 100 200 300 400 500 0 Frequency (Hz) (a) 0 100 (b)200 300 400 500 Frequency (Hz) (a) Figure 13. (a) CVib of Case 1 to Case 4. (b) FFT result. (b)

5. FigureConclusionsFigure 13. (a 13.) CVib (a) CVib of Case of Case 1 to Case1 to Case 4. (b )4. FFT (b) result.FFT result. This study proposed an optimization algorithm for tuning the CNC interpolation 5. Conclusions 5. Conclusionsparameters by considering multiple objective functions such as contour errors, the machiningThis time,study and proposed corner vibrations. an optimization It is known algorithm that the forfive tun differenting the parameters CNC interpolation of This study proposed an optimization algorithm for tuning the CNC interpolation pa- , , 2, and of the interpolator can influence the objective functions rametersparameters by considering by considering multiple objectivemultiple functionsobjective suchfunctions as contour such errors, as contour the machining errors, the significantly.machining time,For example, and corner a higher vibrations. level of accelerationIt is known will that reduce the five the differentmachining paramete time rs of time, and corner vibrations. It is known that the five different parameters of Amax, Ta, and increase, , the2 ,contour and error andof cornerthe interpolator vibrations. However,can influence no systematic the objective approach functions T2, hasVc andbeen fadoptedmax of the to determine interpolator the optimal can influence solutions the under objective different functions machining significantly. processes. For significantly. For example, a higher level of acceleration will reduce the machining time example,In𝐴𝐴 𝑚𝑚𝑚𝑚𝑚𝑚this astudy, higher𝑇𝑇𝑎𝑎 𝑇𝑇1024 level𝑉𝑉 𝑐𝑐experiments of acceleration𝑓𝑓𝑚𝑚𝑚𝑚𝑚𝑚 were conducted, will reduce and the the machiningexperimental time data and—the increase MCoE, the contourMT,and and errorincrease CVib and— the cornerwere contour captured vibrations. error automatically and However, corner and novibrations analyzed systematic. However,to compute approach nothe has systematicperformance been adopted approach to determineindex.has been A BPNN adopted the optimal with to 20 determine hidden solutions layers the under optimal was differentapp solutionslied to machiningestablish under the different processes. relationship machining In between this study,processes. 1024theIn experiments thisinterpolation study, were1024 parameter conducted,experiments and and wereperforma the conducted experimentalnce index., and data—theThe the correlations experimental MCoE, between MT, data and— the CVib—the MCoE, werepredictedMT, captured and valuesCVib automatically— andwe reexperimental captured and analyzed automatically results tofor compute the and MCoE, anal the MT,yzed performance and to computeCVib index.were the 0.9984, Aperformance BPNN with0.9998,index. 20 hidden andA BPNN 0.9354, layers withrespectively. was 20 applied hidden Then, to layers establish the GA was was the applied adopted relationship to based establish between on the the BPNN relation the interpolation modelship to between parameterdeterminethe interpolation and the performance optimized parameter interp index.olation and The parameter.performance correlations The betweenindex.experimental The the correlationresults predicted validate valuess between the and the experimentalaccuracypredicted and resultsvalues efficiency for and the ofexperimental MCoE, the proposed MT, andresults algorithm. CVib for were theWith 0.9984,MCoE, the developed 0.9998, MT, and and optimization CVib 0.9354, were respec- 0.9984, methodology, operators can set up the parameters systematically by selecting different tively.0.9998, Then, and the 0.9354, GA was respectively. adopted based Then on, the the GA BPNN was modeladopted to based determine on the the BPNN optimized model to weighting without a trial-and-error process. The future work will be adding the surface interpolationdetermine parameter. the optimized The experimentalinterpolation resultsparameter. validate The theexperimental accuracy and results efficiency validate of the the proposedaccuracy algorithm.and efficiency With of the the developed proposed optimization algorithm. methodology,With the developed operators optimization can set up themethodology, parameters systematicallyoperators can byset selecting up the parameters different weighting systematically without by a trial-and-errorselecting different process.weighting The future without work a willtrial be-and adding-error the process. surface The roughness future work into the will performance be adding indicesthe surface Appl. Sci. 2021, 11, 1665 14 of 15

such that one will be able to determine the surface quality, machining time, and contour errors simultaneously.

Author Contributions: H.-C.T., M.-S.T., C.-C.C. and C.-J.L. initiated and developed the concepts related to this research work. H.-C.T. performed the collected data for the designed experiment and analyzed the data. Both of them discussed the experimental results and developed the tuning methodology. H.-C.T. wrote the paper draft under M.-S.T. and C.-C.C.’s guidance. All authors discussed the results and commented on the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Ministry of Science and Technology (MOST), R.O.C., under the contracts MOST 108-2634-F-194-001, MOST 108-2221-E-002-156-MY3, and MOST 108-2218- E-002-071, and the Tongtai Machine & Tool Co., LTD., R.O.C. Data Availability Statement: Not applicable. Acknowledgments: The authors would like to thank the reviewers and editors for their suggestions. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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