Development of a Novel Tuning Approach of the Notch Filter of the Servo Feed Drive System

Journal of Manufacturing and Materials Processing

Article Development of a Novel Tuning Approach of the Notch Filter of the Servo Feed Drive System

Chung-Ching Liu 1,* , Meng-Shiun Tsai 2, Mao-Qi Hong 2 and Pu-Yang Tang 1

1 Department of Mechanical Engineering, National Chung Cheng University, Chiayi 62102, Taiwan; [email protected] 2 Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwan; [email protected] (M.-S.T.); [email protected] (M.-Q.H.) * Correspondence: [email protected]

Received: 23 January 2020; Accepted: 2 March 2020; Published: 5 March 2020

Abstract: To alleviate the vibration effect of the feed drive system, a traditional approach is to apply notch filters to suppress vibrations in the velocity loop. The current approach is to adjust the notch filter parameters such as the center frequency, damping and depth based on observing the frequency response diagram of servo velocity closed loop. In addition, the notch filters are generally provided in the velocity loop control for the commercialized controllers such as FANUC, Siemens, etc. However, the resonance of the transmission system also appears in the position loop when the linear scale is used as the position feedback. The notch filter design without consideration of the resonance behavior of the position loop might cause degradation of performance. To overcome this problem, the paper proposes an innovative method which could automatically determine the optimal parameters of the notch filter under the consideration of resonance behavior of position loop. With the optimal parameters, it is found that both the gains of position and velocity loop controller could be increased such that the bandwidths of the position/velocity loops are higher. Based on the simulation results, the rising time is improved by 33% and the time for reaching the steady state is improved by 72% as comparing the cases of using the optimal approach and traditional approach.

Keywords: notch ﬁlter; auto-tuning; feed drive system

1. Introduction High-speed machine tools are widely used in manufacturing the parts for aviation, automotive, mold, and other industrial fields. The design objectives of machine tools are to obtain high efficiency, high precision and high surface quality. However, vibrations of machine structure may be excited when the machine is under high speed and acceleration condition, and the surface quality would also be affected. To avoid the vibration effects, many factors should be considered. One of the most critical factors is how to adjust the servo control parameters. The vibrations could deteriorate the manufacturing qualities and performances [1–3]. To suppress the structural vibration of the feed drive system, many scholars have proposed different approaches. For example, motion commands are filtered before entering the servo controller [4–6]. Some scholars have also proposed strategies that adopted active damping of structural vibrations inside the control loop [7–9]. In the modern time, there are many various ways to control vibration behavior, such as neural networks [10,11], fuzzy control [12,13], and sliding mode control [14,15]. These methods can achieve higher control bandwidth by applying notch filters in the control loop. However, the design of such controllers could be difficult because higher-order plant models need to be considered in order to include the vibration modes into the dynamic model. For commercialized controllers, the control loops are designed by three main loops: current, velocity and position loops.

J. Manuf. Mater. Process. 2020, 4, 21; doi:10.3390/jmmp4010021 www.mdpi.com/journal/jmmp J. Manuf. Mater. Process. 2020, 4, x FOR PEER REVIEW 2 of 11

commercialized controllers, the control loops are designed by three main loops: current, velocity and J. Manuf. Mater. Process. 2020, 4, 21 2 of 11 position loops. The commercial controllers generally apply the notch filters in velocity loop to alleviate the vibration signal near the resonance frequency [16] such that the bandwidth of the control Theloop commercial could be increased. controllers However, generally the apply traditional the notch method filters of in adjusting velocity loopthe notch to alleviate filter only the vibrationconsiders signalthe resonance near the resonancebehavior frequencyof the velocity [16] such loop. that The the resonance bandwidth behaviors of the control of the loop components could be increased. located However,outside the the velocity traditional loop methodhave not of been adjusting considered. the notch The filterresearch only of considers how to design the resonance the notch behavior filter by ofconsidering the velocity both loop. the Thevelocity resonance and position behaviors loop of dynamics the components is still an located open issue. outside In this the paper, velocity a novel loop haveapproach not been which considered. could determine The research the optimal of how parameters to design the of notchthe notch filter filter by considering is proposed. both In thethe velocityapproach, and the position vibration loop effects dynamics in velocity is still anand open position issue. loops In this are paper, considered a novel, approachand the parameters which could of determinenotch filters the could optimal automatically parameters ofbe the adjusted. notch filter The is paper proposed. is arranged In the approach, as follows. the In vibration Section e2,ffects the indynamics velocity andof the position feed system loops areand considered, the traditional and theapproach parameters are introduced. of notch filters Then, could the automaticallynovel optimal beapproach adjusted. is proposed The paper in isSection arranged 3. The as Bode follows. plots In velocity Section and2, the position dynamics loop of of the using feed traditional system and and theoptimal traditional approaches approach are arecompared introduced. in Section Then, 4. the The novel servo optimal performances approach using is proposed the traditional in Section and3. Theoptimal Bode approaches plots velocity for and the position step input loop and of usingthe trajectory traditional are and compared. optimal approaches Finally, Section are compared 5 gives the in Sectionconclusion.4. The servo performances using the traditional and optimal approaches for the step input and the trajectory are compared. Finally, Section5 gives the conclusion. 2. The Dynamic Model of the Feed Drive System 2. The Dynamic Model of the Feed Drive System In this section, the dynamic model of the feed drive system is first derived. Then, the dynamic modelIn thisis simplified section, theto explain dynamic the model parameter of the tuning feed drive process. system is first derived. Then, the dynamic model is simplified to explain the parameter tuning process. 2.1. The Derivation of the Feed Drive System 2.1. The Derivation of the Feed Drive System To describe the dynamic behavior of feed drive system, this paper utilizes the model proposed by theTo paper describe [4]. the The dynamic model includes behavior the of feeddynamics drive effects system, of this couplings, paper utilizes ball-screws, the model structure proposed base, byand the servo paper loop [4]. controls. The model The includes schematic the diagram dynamics of eaff feedects ofdrive couplings, system ball-screws,is shown in structureFigure 1a. base, The androtary servo motion loop controls.of the motor The drives schematic the platform diagram ofthrough a feed drivethe coupling system isand shown the ball in Figure-screw.1a. The The rotation rotary motionof the motor of the motorcan be drives measured the platform by the throughencoder themounted coupling on andthe themotor. ball-screw. The linear The rotationmotion of thethe motorplatform can can be measuredbe measured by theby encoderthe optical mounted linear scale on the on motor. the platform. The linear The motion feed drive of the system platform can can be besimplified measured as bya mass the optical-damper linear-spring scale (MCK) on the system platform. as shown The feed in Figure drive system1b. can be simplified as a mass-damper-spring (MCK) system as shown in Figure1b.

(a)

(b)

FigureFigure 1.1. TheThe feedfeed drive drive system. system. ( a()a) The The schematic schematic of of the the feed feed drive drive system. system. (b )(b The) The 2 DOF 2 DOF diagram diagram of theof the feed feed drive drive system. system.

InIn Figure1 1,, thethe parameters parameters could could be be defined defined as as the the following. following. isX thet is displacement the displacement of the of table, the table, is theXl isdisplacement the displacement of the ofball the-screw ball-screw,, and and is theXb displacementis the displacement of the base. of the base. and θl and are θthem 𝑋𝑋𝑡𝑡 are the ball-screw and motor rotation angle. Jl and Jm are the inertia of the ball-screw and motor. 𝑋𝑋𝑙𝑙 𝑋𝑋𝑏𝑏 𝜃𝜃𝑙𝑙 𝜃𝜃𝑚𝑚 J.J. Manuf. Manuf. Mater. Mater. Process. Process. 20202020, ,44, ,x 21 FOR PEER REVIEW 33 of of 11 11 ball-screw and motor rotation angle. and are the inertia of the ball-screw and motor. and Mt arande theM bmassare theof table mass and of tablebase. and , base. andC t, C arel and theC dampingb are the of damping guideway, of guideway, ball-screw ball-screwand base, 𝐽𝐽𝑙𝑙 𝐽𝐽𝑚𝑚 𝑀𝑀𝑡𝑡 respectively.and base, respectively. , andK b, Kareg and the Kstiffnesst are the of sti base,ffness coupling of base, stiffness coupling and sti longitudinalffness and longitudinal stiffness of 𝑀𝑀𝑏𝑏 𝐶𝐶𝑡𝑡 𝐶𝐶𝑙𝑙 𝐶𝐶𝑏𝑏 ballstiff-nessscrew, of respectively. ball-screw, respectively. The rotary Themotion rotary of motionthe ball of-screw, the ball-screw, , is transmittedθl, is transmitted to the linear to the 𝐾𝐾𝑏𝑏 𝐾𝐾𝑔𝑔 𝐾𝐾𝑡𝑡 displacement,linear displacement, , throughXl, through the converting the converting ratio, ratio, . Risl. givenRl is given as / as2 P , /where2π, where is Ptheis pitch the pitch of the of 𝜃𝜃𝑙𝑙 ballthe- ball-screw.screw. 𝑋𝑋𝑙𝑙 𝑅𝑅𝑙𝑙 𝑅𝑅𝑙𝑙 𝑃𝑃 𝜋𝜋 𝑃𝑃 AfterAfter the the integration integration of of the the feed feed drive drive syste systemm and and servo servo loop loop control control system, system, the the complete complete system system cancan be be expressed expressed as asthe the block block diagram diagram shown shown in Figure in Figure 2 and2 isand named is named as the asservo the feed servo drive feed system drive. Thesystem. figure The can figure be divided can be dividedinto four into subsystems four subsystems such as such motor as motormodel, model, the transmission the transmission model, model, the velocitythe velocity loop loopand andthe position the position loop. loop. The four The fourparts parts can be can introduced be introduced as follows. as follows.

Position Loop

Velocity Loop Transmission Model

Motor Model

+ + Notch + 𝑚𝑚 1 1 + 𝜏𝜏 + c Filters 𝑐𝑐 + 𝑚𝑚 𝑐𝑐 - 𝜔𝜔 - vi 𝜏𝜏 𝜔𝜔 𝜃𝜃𝑚𝑚- 𝜃𝜃 pp vp 𝐾𝐾 𝑔𝑔 𝐾𝐾 𝐾𝐾 𝑚𝑚 𝐾𝐾 𝑠𝑠 𝐽𝐽 𝑠𝑠 𝑏𝑏 𝑠𝑠 𝜃𝜃𝑠𝑠

- + 1 + 1 + 1

- + 𝑙𝑙 𝑙𝑙 𝑡𝑡 + 𝑡𝑡 - 𝑠𝑠 𝑠𝑠 𝜃𝜃 𝑙𝑙 𝑋𝑋 𝑡𝑡 𝐹𝐹 𝑋𝑋 𝑋𝑋 𝑠𝑠 𝜔𝜔 2 2 𝑙𝑙 𝑙𝑙 𝑅𝑅 𝐾𝐾 𝑡𝑡 𝑡𝑡 𝑙𝑙 𝐽𝐽 𝑠𝑠 𝐶𝐶 𝑠𝑠 𝑀𝑀 𝑠𝑠1 𝐶𝐶 𝑠𝑠 𝑅𝑅 𝑠𝑠

+ + 𝑏𝑏 𝑡𝑡 − 𝑋𝑋 𝜏𝜏 𝑙𝑙 2 𝑅𝑅 𝑀𝑀𝑏𝑏𝑠𝑠 𝐶𝐶𝑏𝑏𝑠𝑠 𝐾𝐾𝑏𝑏

FigureFigure 2. 2. TheThe block block diagram diagram of of the the servo servo feed feed drive drive system system..

InIn the the motor motor model, model, τ c isis the the motor motor torque, torque, b isis the the friction friction torque torque of of the the motor, motor, ωm isis the the motormotor angular angular velocity, velocity, and and τm isis the the torque torque which which is is generated generated by by the the resonance resonance behavior behavior of of the the 𝜏𝜏𝑐𝑐 𝑏𝑏 𝜔𝜔𝑚𝑚 transmissiontransmission model. model. 𝜏𝜏𝑚𝑚 TheThe velocity velocity loop loop consists consists of ofthe the velocity velocity controller, controller, the thenotch notch filters, filters, and andthe motor the motor model. model. The velocityThe velocity controller controller includes includes proportional proportional gain gain K andvp and integral integral gai gainsns K vi; the; the controller controller is is designed designed toto control control the the motor motor model model under under different different effects. effects. The The notch notch filters filters are are used used for for suppress suppressinging the the 𝐾𝐾𝑣𝑣𝑣𝑣 𝐾𝐾𝑣𝑣𝑣𝑣 resonanceresonance behavior behavior and and it it can can be represented as EquationEquation (1).(1). InIn thethe Equation,Equation, ωn, ζ, n and andκn are are the thecenter center frequency, frequency, damping damping and and depth depth of the of notchthe notch filter, filter, respectively. respectively. The BodeThe Bode plot plot of the of notch the notch filter 𝜔𝜔𝑛𝑛 𝜁𝜁𝑛𝑛 𝜅𝜅𝑛𝑛 filteris shown is shown in Figure in Figure3. 3. s2 + 2ζ ω s + ω 2 N(s) = + 2n n + n (1) 2 2 ( ) =s + 2ζn κn ωn s + ωn (1) 2+ 2 + 2 𝑠𝑠 𝜁𝜁n𝜔𝜔n𝑠𝑠 𝜔𝜔n In the transmission model, the input of2 the model is the motor2 angular velocity ωm, and the In the transmission model, the input𝑁𝑁 𝑠𝑠 of the modeln 𝑛𝑛 isn the motorn angular velocity , and the output is the velocity ωs of the moving platform,𝑠𝑠 which𝜁𝜁 𝜅𝜅 𝜔𝜔 is𝑠𝑠 related𝜔𝜔 to the linear displacement Xs by output= is the velocity of the moving platform, which is related to the linear displacement by ωs sXs/Rl. The transmission model is described by the transfer function between ωs and𝜔𝜔𝑚𝑚 ωm. = / . The transmission model is described by the transfer function between and . 𝜔𝜔𝑠𝑠 𝑋𝑋𝑠𝑠 𝜔𝜔𝑠𝑠 𝑠𝑠𝑋𝑋𝑠𝑠 𝑅𝑅𝑙𝑙 𝜔𝜔𝑠𝑠 𝜔𝜔𝑚𝑚

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-5 -10 -15 n -20 𝜅𝜅 magnitude (dB) 𝜁𝜁n -25

10 0 𝜔𝜔10n 1 10 2 10 3 frequency (rad/s)

-20 phase (deg)

-40

-60 10 0 10 1 10 2 10 3 frequency (rad/s) The Bode plot of the notch filter as = 100 (rad/s), = 0.05 , and = 20 (dB). Figure 3. The Bode plot of the notch ﬁlter as ωn = 100 (rad/s), ζn = 0.05, and κn = 20 (dB). 𝜔𝜔n 𝜁𝜁n 𝜅𝜅𝑛𝑛 TheThe outer outer loop loop is is the the position position loop, loop, which which contains contains the the velocity velocity loop, loop, the the transmission transmission model model andand the the position position controller. controller. TheThe positionposition controllercontroller isis aa proportional gain and is represented as as Kpp. . TheThe parameters parameters of of the the servo servo feed feed drive drive system system are are identiﬁed identified from from the the experiment experiment [ 4[4]] as as shown shown in in 𝐾𝐾𝑝𝑝𝑝𝑝 TableTable1. 1.

TableTable 1. 1.Parameters Parameters of of the the servo servo feed feed drive drive system. system.

Parameters Value Value (Unit)(Unit) Parameters Parameters ValueValue (Unit) (Unit)

Cb 800800 (Ns(Ns/m)/m) K g 85208520 (Nm (Nm/rad)/rad) 8 Cl 1.951.95 (Ns(Ns/m)/m) K t 1.951.95 ×10 10(N /(N/m)m) 𝐶𝐶𝑏𝑏 𝐾𝐾𝑔𝑔 × Ct 500500 (Ns(Ns/m)/m) M t 138138 (kg) (kg)8 𝑙𝑙 2) 𝑡𝑡 J𝐶𝐶l 0.00823 (kgm M 𝐾𝐾b 570 (kg) 𝑡𝑡 0.00823 (kgm2 ) 𝑡𝑡 570 (kg) J𝐶𝐶m 0.04 (kgm ) R𝑀𝑀l 0.0032 (m/rad) kgm )2 K 𝑙𝑙 1.830.0410 ( 6 (N/m) b 𝑏𝑏 0.0033420.0032 (Nms (m/rad)/rad) 𝐽𝐽b × 𝑀𝑀 K 1.831.38 × 10 (Nm /(N/m)A2 ) 0.003342 (Nms/rad) 𝐽𝐽𝑚𝑚f 𝑅𝑅𝑙𝑙 1.38 (Nm6 /A ) 𝐾𝐾𝑏𝑏 𝑏𝑏 In order to explain𝐾𝐾𝑓𝑓 the influence of the feed drive system on the design of the servo loop, In order to explain the influence of the feed drive system on the design of the servo loop, the the model is divided into different modules for simplification, which would be explained in the model is divided into different modules for simplification, which would be explained in the following following paragraphs. paragraphs. 2.2. Transformed Representation of the Servo Feed Drive System 2.2. Transformed Representation of the Servo Feed Drive System In this section, in order to discuss the impact caused by motor and transmission models, the feed driveIn system this section, shown in in order Figure to2 discuss is transformed the impact into caused di fferent by motor representations and transmission as shown models, in Figure the feed4 wheredrive thesystem motor shown and transmissionin Figure 2 is models transformed are obtained into different by Mason representation rules and thes as respective shown in transferFigure 4 where the motor and transmission models are obtained by Mason rules and the respective transfer functions Gm and Gp are given as follows. functions and are given as follows. 2 𝑚𝑚 𝑝𝑝 G1s(KgG2+KtG3 KtG4+KgKtG2G3 KgKtG2G4 R1 KtG2+1) 𝐺𝐺Gm = 𝐺𝐺 − − − s(KgG +KtG KtG +KgKtG G KgKtG G R KtG G +1)+KgG +KgKtG G KgKtG G ( 2 +3− 4 2 3+− 2 4− l 2 4 1 1 +3−1) 1 4 𝑚𝑚 KgRlKtsG2(G3 G4) (2) 𝐺𝐺= Gp = − 2 (2) ( + 1 𝑔𝑔+2 Rls(𝑡𝑡Kg3G2+Kt𝑡𝑡G34 KtG4𝑔𝑔+K𝑡𝑡gKt2G23G3 Kg𝑔𝑔KtG𝑡𝑡+2G214)4R+lKtG2𝑙𝑙G4++𝑡𝑡1)2 𝐺𝐺 𝑠𝑠 𝐾𝐾 𝐺𝐺 1 𝐾𝐾 𝐺𝐺 − 𝐾𝐾 𝐺𝐺1− 𝐾𝐾 𝐾𝐾 𝐺𝐺 𝐺𝐺 −−1 𝐾𝐾 𝐾𝐾 𝐺𝐺 𝐺𝐺− − 𝑅𝑅 𝐾𝐾1 𝐺𝐺 G1 = + , G2 = 2 , G3 = 2 , G4 = 2 − 𝑠𝑠 𝐾𝐾𝑔𝑔𝐺𝐺2 𝐾𝐾𝑡𝑡𝐺𝐺3 − 𝐾𝐾𝑡𝑡𝐺𝐺4 Jm𝐾𝐾s𝑔𝑔𝐾𝐾b𝑡𝑡𝐺𝐺2𝐺𝐺3 −Jl𝐾𝐾s 𝑔𝑔+𝐾𝐾C𝑡𝑡ls𝐺𝐺2𝐺𝐺4 − 𝑅𝑅M𝑙𝑙𝐾𝐾ts𝑡𝑡𝐺𝐺+2C𝐺𝐺ts4 M𝐾𝐾b𝑔𝑔s𝐺𝐺+1Cbs+𝐾𝐾K𝑔𝑔b𝐾𝐾𝑡𝑡𝐺𝐺1𝐺𝐺3 − 𝐾𝐾𝑔𝑔𝐾𝐾𝑡𝑡𝐺𝐺1𝐺𝐺4

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( ) = ( + 𝑔𝑔+ 𝑙𝑙 𝑡𝑡 2 3 4 + 1) 𝑝𝑝 𝐾𝐾 𝑅𝑅 𝐾𝐾 𝑠𝑠𝐺𝐺 𝐺𝐺 − 𝐺𝐺 𝐺𝐺 1 𝑙𝑙 𝑔𝑔 2 𝑡𝑡 13 𝑡𝑡 4 𝑔𝑔 𝑡𝑡12 3 𝑔𝑔 𝑡𝑡 2 4 1𝑙𝑙 𝑡𝑡 2 4 = 𝑅𝑅 𝑠𝑠 𝐾𝐾, 𝐺𝐺= 𝐾𝐾 𝐺𝐺 − 𝐾𝐾, 𝐺𝐺 = 𝐾𝐾 𝐾𝐾 𝐺𝐺 𝐺𝐺 −, 𝐾𝐾 𝐾𝐾= 𝐺𝐺 𝐺𝐺 − 𝑅𝑅 𝐾𝐾 𝐺𝐺 𝐺𝐺 J. Manuf. Mater. Process. 2020, 4+, 21 + + + + 5 of 11 − 𝐺𝐺1 𝐺𝐺2 2 𝐺𝐺3 2 𝐺𝐺4 2 𝐽𝐽𝑚𝑚𝑠𝑠 𝑏𝑏 𝐽𝐽𝑙𝑙𝑠𝑠 𝐶𝐶𝑙𝑙𝑠𝑠 𝑀𝑀𝑡𝑡𝑠𝑠 𝐶𝐶𝑡𝑡𝑠𝑠 𝑀𝑀𝑏𝑏𝑠𝑠 𝐶𝐶𝑏𝑏𝑠𝑠 𝐾𝐾𝑏𝑏

FigureFigure 4. 4.The The transformedtransformed blockblock diagramdiagram ofof the servo feed drive system.

TheThe velocity velocity looploop transfer function function between between ωc andand ωm isis defined defined as as Gv . Gvp is given given as as GvGp, , whichwhich is is the the cascaded cascaded functionfunction ofof velocityvelocity loop G andand thethe transmission model model G .. The transformed transformed v𝜔𝜔𝑐𝑐 𝜔𝜔𝑚𝑚 𝐺𝐺𝑣𝑣 𝐺𝐺𝑣𝑣𝑣𝑣p 𝐺𝐺𝑣𝑣𝐺𝐺𝑝𝑝 representationrepresentation could could provide provide a more a more systematic systematic approach approach to evaluate to eval theuate dynamics the dynamics within thewithin velocity the 𝐺𝐺𝑣𝑣 𝐺𝐺𝑝𝑝 loopvelocityGv and loop the dynamics and the dynamics outside of outside the velocity of the velocity loop Gp .loop Figure 4. Figuremotivates 4 motivates the objective the objective of this paperof this for paper designing for designing the notch the filter notch with filter or with without or without consideration consideration of G . of . 𝐺𝐺𝑣𝑣 𝐺𝐺𝑝𝑝 p 𝑝𝑝 2.3.2.3. Traditional Traditional Tuning Tuning Method Method of of Notch Notch Filter Filter 𝐺𝐺 TheThe process process of of the the traditional traditional tuningtuning methodmethod ofof thethe controllercontroller is to tune the parameter of of the the velocityvelocity loop loop to to a a higher higher bandwidth. bandwidth. Then,Then, thethe positionposition loop parameter is then tuned tuned based based on on either either BodeBode plot plot or or step step response. response. However,However, whenwhen tuningtuning thethe controllercontroller in the velocity loop, the the resonance resonance behaviorbehavior outside outside thethe velocity loop, loop, such such as as the the resonance resonance behavior behavior of transmission of transmission model model,, would would not notbe beconsidered. considered. To Toavoid avoid vibration, vibration, the the notch notch filter filter is is applied applied in in the the velocity velocity loop. loop. The The traditional traditional approachapproach to to tune tune the the parameters parameters ofof notchnotch filterfilter onlyonly considerconsider the dynamics of velocity loop loop based based on on thethe frequency frequency response response of of the the velocity velocity closedclosed loop.loop. TheThe resonanceresonance could be identified identified and and the the three three parametersparameters in in Equation Equation (1) (1) could could be befound found consequently.consequently. AfterAfter tuning, the dynamics response response of of the the velocityvelocity loop loop is generallyis generally smooth. smooth. However, However, the resonancethe resonance behavior behavior occured occur in aed position in a position loop because loop bothbecause velocity both loop velocity and positionloop and loop position contain loop the contain resonance the resonance behavior asbehavior shown as in Figureshown4 in. Generally, Figure 4. theGenerally, resonance the behavior resonance in thebehavior position in loopthe position cannot beloop suppressed cannot be because suppressed the position because controller the position is a controller is a position gain and has no notch filter. As the result, the cannot be adjusted position gain Kpp and has no notch filter. As the result, the Kpp cannot be adjusted too high to avoid too high to avoid exciting the resonance. exciting the resonance. 𝐾𝐾𝑝𝑝𝑝𝑝 𝐾𝐾𝑝𝑝𝑝𝑝 AlthoughAlthough using using the the traditional traditional method method could could allow allow the controller the controller parameters parameters to be easily to be designed, easily thedesigned bandwidth, the ofbandwidth the position of loopthe isposition quite limited. loop is Onequite systematical limited. One approach systematical is to consider approach both is theto positionconsider and both velocity the position loops and simultaneously velocity loops when simultaneously designing the when parameters designing of the notch parameters filter. of notch filter. 3. The Notch Filter Tuning Approach Using the Optimization 3. The Notch Filter Tuning Approach Using the Optimization In this section, an optimization design methodology for the notch filter is proposed such that the resonanceIn this behaviors section, ofan theoptimization motor and design transmission methodology models for in the position notch loopfiltercan is proposed be suppressed such that as well. the Theresonance range of behaviors the center of frequency the motor of and the notchtransmission filter is firstlymodels determined in position by loop Nyquist can be plot suppressed of the motor as modelwell. The as follows. range of the center frequency of the notch filter is firstly determined by Nyquist plot of the motor model as follows. 3.1. The Determination of the Range of the Center Frequency by Nyquist Plot

Using the parameters listed in Table1, the Bode plot of Gm is shown in Figure5a where the two resonances occur at about 1000 and 1500 (rad/s). The Nyquist diagram shown in Figure5b indicates that the frequency curves around each resonance will form a circle due to the phase changes within the J. Manuf. Mater. Process. 2020, 4, x FOR PEER REVIEW 6 of 11

3.1. The Determination of the Range of the Center Frequency by Nyquist Plot J. Manuf. Mater. Process. 2020, 4, 21 6 of 11 Using the parameters listed in Table 1, the Bode plot of is shown in Figure 5a where the two resonances occur at about 1000 and 1500 (rad/s). The Nyquist diagram shown in Figure 5b indicates 𝐺𝐺𝑚𝑚 regionthat the 1 and frequency 2. The ω curvess1 and aroundωe1 in Figure each resonance5a correspond will toform the a same circle point due ofto theP1 inphase Figure changes5b. Taking within regionthe region 1 as an 1 example,and 2. The by searching and the in intersection Figure 5a pointcorrespondP1, one to can the locate same the point ﬁrst of resonance in Figure range. 5b. TheTaking searching region of ω1 asand anω example,is to let theby followingsearching function the intersection equal to 0.point , one can locate the first s e𝜔𝜔𝑠𝑠1 𝜔𝜔𝑒𝑒1 𝑃𝑃1 resonance range. The searching of and is to let the following function equal to 0. q 𝑃𝑃1 2 2 L== ( (Re(G(m(jω)𝑠𝑠s) Gm( (𝑒𝑒jω)e)))) ++( (Im(P((jω)s) P((jωe))))) (3)(3) 𝜔𝜔 − 𝜔𝜔 − 2 2 In this case, the is equal𝑚𝑚 to𝑠𝑠 534𝑚𝑚 (rad/s𝑒𝑒 ) and is equal𝑠𝑠 to 1100𝑒𝑒 (rad/s). The resonance in In this case, the𝐿𝐿ωs1 is� equal𝑅𝑅𝑅𝑅�𝐺𝐺 to𝑗𝑗 534𝜔𝜔 (rad− 𝐺𝐺/s)𝑗𝑗 and𝜔𝜔 �ωe1 is equal𝐼𝐼𝐼𝐼�𝑃𝑃 to𝑗𝑗𝜔𝜔 1100− (rad𝑃𝑃 𝑗𝑗𝜔𝜔/s).� The resonance in region region 1 occurs within (534,1100) (rad/s). Using the same approach, one can determine the and 1 occurs within (534,1100)𝜔𝜔𝑠𝑠1 (rad/s). Using the same approach,𝜔𝜔𝑒𝑒1 one can determine the ωs2 and ωe2 equal equal to 1400 and 1628 (rad/s) for region 2. to 1400 and 1628 (rad/s) for region 2. 𝜔𝜔𝑠𝑠2 𝑒𝑒2 𝜔𝜔 0 𝜔𝜔𝑠𝑠1 𝜔𝜔𝑒𝑒1 𝜔𝜔𝑠𝑠2 0.04 -20 region 2 𝜔𝜔𝑒𝑒2 0.02

0 -40 region 1

magnitude (dB) region 1 -0.02 region 2 -60 -0.04 2 0 1 2 3 4 10 10 10 10 10 𝑃𝑃

frequency (rad/s) -0.06

0 1 Imaginary Axis -0.08 𝑃𝑃 𝑠𝑠1 𝑒𝑒1 𝑠𝑠2 𝜔𝜔 𝜔𝜔 𝜔𝜔 -0.1

𝑒𝑒2 -50 region 2 -0.12 region 1 𝜔𝜔

-0.14 phase (deg)

-0.16 -100 0 1 2 3 4 10 10 10 10 10 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Real Axis frequency (rad/s) (a) (b)

FigureFigure 5. 5.The The dynamic dynamic behavior behavior of ofG m.( .a ()a The) The Bode Bode plot plot of ofG m;(b; ()b the) the Nyquist Nyquist plot plot of ofG m. . 𝐺𝐺𝑚𝑚 𝐺𝐺𝑚𝑚 𝐺𝐺𝑚𝑚 3.2.3.2. Optimal Optimal Approach Approach for for Tuning Tuning Notch Notch Filter Filter

AfterAfter locating locating the the range range of of the the center center frequency, frequency, the the optimization optimization approach approach is is to to search search the the parameterparameter of of the the notch notch ﬁlter filter such such that that the the transfer transfer function functionG vp become become a smootha smooth curve curve since since the theG vp containscontains both both the the dynamics dynamics of theof velocitythe velocity loop loop and positionand pos loop.ition loop. The tuning The tuning process process is given isas given three as 𝐺𝐺𝑣𝑣𝑣𝑣 𝐺𝐺𝑣𝑣𝑣𝑣 majorthree steps. major steps.

Determine the start frequency ωs and end frequency ωe of each closed path using the searching • • Determine the start frequency and end frequency of each closed path using the method in Section 3.1. searching method in Section 3.1. 𝑠𝑠 𝑒𝑒 Smooth Gvp by applying the moving𝜔𝜔 average ﬁlter H given as𝜔𝜔 the following. • • Smooth by applying the moving average filter given as the following.

𝑣𝑣𝑣𝑣 (s) = ( ) ( ), ( ) = sTN 𝐺𝐺 𝐻𝐻 1 1 −e𝑠𝑠𝑠𝑠𝑠𝑠− (4) G (s) = G (s)H(s) H(s) = 1 1−𝑒𝑒 scs vp , − sT (4) 𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 N 1 −e𝑠𝑠𝑠𝑠 Here is the smoothed𝐺𝐺 transfer function𝐺𝐺 𝑠𝑠 𝐻𝐻 of𝑠𝑠 𝐻𝐻 𝑠𝑠. The𝑁𝑁 magnitude1−−𝑒𝑒 − plot of and three cases of with different are shown in Figure 6. In the figure, it can be observed that with Here Gscs𝐺𝐺𝑠𝑠𝑠𝑠is𝑠𝑠 the smoothed transfer function of Gvp𝐺𝐺𝑣𝑣𝑣𝑣. The magnitude plot of Gvp𝐺𝐺𝑣𝑣𝑣𝑣and three cases smaller has a worse result; however, with larger are changed completely and also of Gscs𝐺𝐺𝑠𝑠𝑠𝑠with𝑠𝑠 different N are𝑁𝑁 shown in Figure6. In the figure, it can be observed that G𝐺𝐺scs𝑠𝑠𝑠𝑠𝑠𝑠with have an unsuitable result. Hence, here the is selected as 500 to remain the main profile of smaller N has𝑁𝑁 a worse result; however, Gscs with𝐺𝐺𝑠𝑠𝑠𝑠𝑠𝑠 larger N are𝑁𝑁 changed completely and also have and smooth the peak parts of . an unsuitable result. Hence, here the N is selected𝑁𝑁 as 500 to remain the main profile of Gvp and𝐺𝐺𝑣𝑣𝑣𝑣 smooth the peak parts of Gvp. 𝑣𝑣𝑣𝑣 𝐺𝐺 The notch filter such that the resulting Gvp can approach the Gscs by minimizing the following • objective function. The lower and upper bounds are obtained from step 1.

Z ωe J = G (j ) G (j ) d scs ω vs ω ω (5) ωs − J. Manuf. Mater. Process. 2020, 4, 21 7 of 11 J. Manuf. Mater. Process. 2020, 4, x FOR PEER REVIEW 7 of 11

-20 G v p G s c s (N=150) -40 G s c s (N=500)

magnitude (dB) G s c s (N=2000) -60 0 1 2 3 4 10 10 10 10 10 frequency (rad/s) FigureFigure 6. 6. TheThe magnitude magnitude plot plot of of Gvp and three cases of Gscs withwith di differentﬀerent N . .

𝐺𝐺𝑣𝑣𝑣𝑣 𝐺𝐺𝑠𝑠𝑠𝑠𝑠𝑠 𝑁𝑁 • Bythe applying notch filter the such particle that swarmingthe resulting optimization can approach [17], the the optimal by depth, minimiz dampinging the and following center frequency of the notch filter can be obtained. According to the content of Section 3.1, the resonance objective function. The lower and upper 𝑣𝑣𝑣𝑣bounds are obtained from𝑠𝑠𝑠𝑠𝑠𝑠 step 1. behavior of the plant occurs in two regions𝐺𝐺 and they are within (534,1100)𝐺𝐺 and (1400,1628) (rad/s), = | ( ) ( )| respectively. Because the resonance behavior𝜔𝜔𝑒𝑒 is more serious and its frequency range is wider, we chose(5) two notch filters (notch filter 1 and notch filter𝑠𝑠𝑠𝑠𝑠𝑠 2) in the𝑣𝑣𝑣𝑣 first region and adopt one notch filter (notch By applying the particle swarming𝐽𝐽 �𝜔𝜔𝑠𝑠 𝐺𝐺optimization𝑗𝑗𝑗𝑗 − 𝐺𝐺 [17],𝑗𝑗𝑗𝑗 the𝑑𝑑𝑑𝑑 optimal depth, damping and center filter 3) in the second region, and Table2 expresses the sets of the three notch filters for determining frequency of the notch filter can be obtained. According to the content of Section 3.1, the the optimalresonance parameter behavior through of the PSO. plant In Tableoccurs2, thein two center regions frequencies and the ofy the are notch within filters (534,1100 are limited) and within (534,1100) or (1400,1628) (rad/s), the other parameters such as damping and depth of the notch (1400,1628) (rad/s), respectively. Because the resonance behavior is more serious and its filters are within a wider range, and the initial values are the same as the minimum values. frequency range is wider, we chose two notch filters (notch filter 1 and notch filter 2) in the first region and adopt one notch filter (notch filter 3) in the second region, and Table 2 expresses the Table 2. The sets of the notch filters for PSO optimization. sets of the three notch filters for determining the optimal parameter through PSO. In Table 2, the Transfercenter Functionfrequencies Parameterof the notch Initial filters Value are limited Minimum within Value (534,1100 Maximum) or (1400,1628 Value) (rad/s), Unit the

other parameters suchω as1 damping and534 depth of the 534notch filters are within 1100 a wider range, rad/s and theNotch initial Filter values 1 are theζ1 same as the0.1 minimum values. 0.1 10 - κ1 0.1 0.1 50 -

Tableω2 2. The sets of534 the notch filters for 534 PSO optimization. 1100 rad/s Notch Filter 2 ζ2 0.1 0.1 10 - Transfer Function Parameter Initial Value Minimum Value Maximum Value Unit κ2 0.1 0.1 50 - 534 534 1100 rad/s ω3 1400 1400 1628 rad/s Notch Filter 1 0.1 0.1 10 - Notch Filter 3 ζ𝜔𝜔3 1 0.1 0.1 10 - κ 0.10.1 0.10.1 5050 - - 3𝜁𝜁1 534 534 1100 rad/s 𝜅𝜅1 Notch Filter 2 0.1 0.1 10 - Because the notch design𝜔𝜔2 could approach our desired smooth Gscs, the advantage is that both Gp 0.1 0.1 50 - and Gv are included in the design𝜁𝜁2 process. In the next section, the comparisons of using traditional 1400 1400 1628 rad/s and optimal approaches in time𝜅𝜅2 and frequency domains are discussed. Notch Filter 3 0.1 0.1 10 - 𝜔𝜔3 4. Results and Discussion 0.1 0.1 50 - 𝜁𝜁3 In this section, a comparison𝜅𝜅3 is given for the frequency response functions and the dynamic Because the notch design could approach our desired smooth , the advantage is that both behavior of the feed drive system for the traditional and optimal approaches. and are included in the design process. In the next section, the comparisons of using 𝐺𝐺𝑠𝑠𝑠𝑠𝑠𝑠 traditional and optimal approaches in time and frequency domains are discussed. 4.1.𝐺𝐺𝑝𝑝 The Comparisons𝐺𝐺𝑣𝑣 of Frequency Response Functions Using Traditional and Optimal Approaches 4. ResultsThe parameters and Discussion for traditional and optimal approaches are shown in Table3. Figure7 shows the Bode plots of the velocity closed loop Gv and the function Gvs with traditional values (TVs) and In this section, a comparison is given for the frequency response functions and the dynamic optimal values (OVs). It is shown that the Gv with TVs could suppress the resonance of the motor behavior of the feed drive system for the traditional and optimal approaches. model Gm completely. However, the resonances of Gvp might hinder the increase of position gain. By using the OVs, the notch ﬁlter is designed to be deeper in order to compensate the resonance eﬀect 4.1. The Comparisons of Frequency Response Functions Using Traditional and Optimal Approaches of Gp. Therefore, the Gvp is smoother than Gvp using the traditional approach. The parameters for traditional and optimal approaches are shown in Table 3. Figure 7 shows the Bode plots of the velocity closed loop and the function with traditional values (TVs) and optimal values (OVs). It is shown that the with TVs could suppress the resonance of the motor 𝐺𝐺𝑣𝑣 𝐺𝐺𝑣𝑣𝑣𝑣 model completely. However, the resonances of might hinder the increase of position gain. 𝐺𝐺𝑣𝑣 𝐺𝐺𝑚𝑚 𝐺𝐺𝑣𝑣𝑣𝑣

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By using the OVs, the notch filter is designed to be deeper in order to compensate the resonance effect of . Therefore, the is smoother than using the traditional approach.

𝑝𝑝 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝐺𝐺 Table𝐺𝐺 3. The parameter values𝐺𝐺 of optimal and traditional approaches.

Parameter Traditional Value Optimal Value Unit 92.47 298.41 1/s 223.46 280.65 Nm s /rad 𝐾𝐾𝑝𝑝𝑝𝑝 J. Manuf. Mater. Process. 2020, 4, 21 1.78 4.46 Nms/rad2 8 of 11 𝐾𝐾𝑣𝑣𝑣𝑣 1012.13 1099.92 rad/s 𝐾𝐾𝑣𝑣𝑣𝑣 Table 3. The parameter0.10 values of optimal and traditional0.17 approaches. - 𝜔𝜔1 Parameter Traditional13.28 Value Optimal4.60 Value Unit - 𝜁𝜁1 Kpp 92.47 298.41 1/s 1 884.33 988.31 2 rad/s 𝜅𝜅 Kvi 223.46 280.65 Nm s /rad Kvp 1.111.78 4.460.05 Nms/rad - 𝜔𝜔2 ω1 0.101012.13 1099.922.97 rad/s - 𝜁𝜁2 ζ1 0.10 0.17 - κ 13.28 4.60 - 2 1 1570.05 1485.52 rad/s 𝜅𝜅 ω2 884.33 988.31 rad/s 0.1 0.03 - 𝜔𝜔3 ζ2 1.11 0.05 - κ2 0.10 2.97 - 3 1.2 5.48 - 𝜁𝜁 ω3 1570.05 1485.52 rad/s ζ 0.1 0.03 - 𝜅𝜅3 3 κ3 1.2 5.48 -

-20

-40 G v (OVs) -60 magnitude (dB) 10 0 10 1 10 2 10 3 frequency (Hz)

-20

-40 G v (TVs) -60 magnitude (dB) 10 0 10 1 10 2 10 3 frequency (Hz)

-20

-40 G (OVs) v p -60 magnitude (dB) 10 0 10 1 10 2 10 3 frequency (Hz)

-20

-40 G (TVs) v p -60 magnitude (dB) 10 0 10 1 10 2 10 3 frequency (Hz) Figure 7. Magnitude plot of G and G with TVs and OVs. Figure 7. Magnitude plot of v andvp with TVs and OVs. To further investigate the eﬀects on the position loop, the Bode plots of position loop for both cases 𝐺𝐺𝑣𝑣 𝐺𝐺𝑣𝑣𝑣𝑣 To furtherare shown investigate in Figure8 the. Here, effects the K pponfor the both position cases shown loop, in Table the3 areBode chosen plots to beof the position maximum loop for both cases are shownvalues such in Figure that the overshoot 8. Here is, lessthe than 5%. for It isboth shown cases that the shown bandwidth in ofTable position 3 are closed-loop chosen to be the with TVs is 90.47 (rad/s), and the bandwidth of position closed-loop with OVs is 177.75 (rad/s). maximum valuesThe bandwidth such that is increased the overshoot by 96% through is less using than OVs. 5%. It is shown that the bandwidth of position 𝐾𝐾𝑝𝑝𝑝𝑝 closed-loop with TVs is 90.47 (rad/s), and the bandwidth of position closed-loop with OVs is 177.75 (rad/s). The bandwidth is increased by 96% through using OVs.

J. Manuf. Mater. Process. 2020, 4, 21 9 of 11 J. Manuf. Mater. Process. 2020, 4, x FOR PEER REVIEW 9 of 11 J. Manuf. Mater. Process. 2020, 4, x FOR PEER REVIEW 9 of 11

90.47 90.47

177.75 177.75

FigureFigure 8. Magnitude8. Magnitude plots plots of ofG p with with traditional traditional and and optimal optimal parameters.parameters. Figure 8. Magnitude plots of with traditional and optimal parameters. 𝐺𝐺𝑝𝑝 4.2.4.2. The The Comparision Comparision of Simulationof Simulation Results Results with with𝐺𝐺 Traditional𝑝𝑝 Traditional and and Optimal Optimal Parameters Parameters 4.2. The Comparision of Simulation Results with Traditional and Optimal Parameters TheThe time time domain domain simulations simulations for for one one step step response response and and one one trajectorytrajectory are conducted to to compare compare The time domain simulations for one step response and one trajectory are conducted to compare the performancesthe performances of theof the cases cases using using the the TVs TVs and and OVs. OVs. the performances of the cases using the TVs and OVs. As shownAs shown in Figure in Figure9, the 9, stepthe step responses responses for thefor the input inputθc are are shown shown in Figurein Figure4. The 4. T outputhe output is the is As shown in Figure 9, the step responses for the input are shown in Figure 4. The output is linearthe scale linearX s.cale It could . It be could found be found that the that rising the rising time oftime the of system the system with with TVs T isV 0.03s is 0.03 s and second that withand the linear scale . It could be found that the rising time of the𝜃𝜃𝑐𝑐 system with TVs is 0.03 second and OVsthat is 0.02 with s. O TheVs is time 0.02 tosecond reach. The the time steady to reach state the with steady TVs isstate 0.11𝜃𝜃𝑐𝑐 with s and TVs with is 0.11 OVs second is 0.03 and s wherewith OV thes that with OVs is 𝑋𝑋0.02𝑠𝑠 second. The time to reach the steady state with TVs is 0.11 second and with OVs caseis using 0.03 second OVs could where𝑋𝑋𝑠𝑠 improve the case by using 72%. O BecauseVs could theimprove bandwidth by 72%. of Because the closed the loopbandwidth using TVsof the is closed higher, is 0.03 second where the case using OVs could improve by 72%. Because the bandwidth of the closed loop using TVs is higher, a quick response time can be expected. To validate that the algorithm can a quickloop response using TV times is higher, can be a expected. quick response To validate time can that be the expected. algorithm To can validate be applied that the in realalgorithm machining, can be applied in real machining, the results of the trajectory are performed. thebe results applied of the in real trajectory machining, are performed. the results of the trajectory are performed.

0.11 0.11

0.03 TVs 0.03 TVs

0.03 0.03

0.02 0.02 OVs OVs

Figure 9. The step responses of optical scale feedback using TVs and OVs. FigureFigure 9. 9.The The step step responses responses of of optical optical scalescale feedbackfeedback Xs usingusing TVsTVs and OVs. 𝑋𝑋𝑠𝑠 𝑋𝑋𝑠𝑠

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PositionPosition commands commands of of the the trajectory trajectory under givengiven velocity velocity and and accelerations accelerations are shownare shown in the in the upperupper subfigure subfigure of ofFigure Figure 10. 10 .The The trajectory trajectory isis givengiven with with a displacement,a displacement, velocity velocity and accelerationand acceleration equalsequals to 20 to 20(mm), (mm), 3000 3000 (mm/min), (mm/min), and and 5000 (mm(mm//s2s),) respectively., respectivel They. The simulation simulation results results of using of using traditionaltraditional and and optimal optimal parameters parameters are are shown in in the the lower 2lower subfigure subfigure of Figureof Figure 10. It10. can It becan observed be observed thatthat the thesystem system with with OVs OVs has has a ahigher higher response response andand less less resonance resonance behavior behavior as compared as compared to the to case the case withwith TVs. TVs. Again, Again, it v italidates validates the the advantages advantages ofof usingusing the the optimal optimal design design approach approach for a for notch a notch filter. filter.

TVs

15 OVs

position feedback (mm) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time (s)

3000 TVs OVs 2000

1000

velocity feedback (mm/min) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time (s) FigureFigure 10. 10. TheThe simulation simulation results results of the velocityvelocity and and position position feedback feedback with with TVs TVs and OVs.and OVs. 5. Conclusions 5. Conclusions Generally, the parameter determination of the notch filter is based on trial and error. In this paper, aGenerally, systematical the approach parameter is proposed determination to obtain of the the optimal notch parameters filter is based of the on notch trial filter. and Theerror. first In this paper,contribution a systematical of the approach paper is to is deriveproposed the transformedto obtain the block optimal diagram, parameters which couldof the separate notch filter. the The firstdynamics contribution within of the the velocity paper loopis to and derive outside thethe transformed velocity loop. block In order diagram, to locate which the resonance could separate range the dynamicsof center within frequency, the theveloci Nyquistty loop diagram and outside is applied the to findvelocity the intersection loop. In order point whichto locate could the be resonance used rangein theof center optimization frequency, process. the The Nyquist second diagram contribution is applied of the paper to find is to the use intersection the moving average point which filter to could specify the desired loop transfer function of the position loop. By applying the PSO method, one can be used in the optimization process. The second contribution of the paper is to use the moving determine the center frequency, damping and depth of notch filter at one time. The frequency response average filter to specify the desired loop transfer function of the position loop. By applying the PSO function of position closed loop shows that the bandwidth could increase compared to the tradition method,parameter one can approach determine and optimalthe center approach. frequency, Furthermore, damping the and rising depth time of andnotch steady filter state at one time time. of The frequencythe step response response arefunction improved of byposition 33% and cl 72%,osed respectively.loop shows In that order the to validatebandwidth that thecould optimal increase comparedapproach to can the be tradition applied toparameter real machining, approach the trajectory and optimal with equal approach. velocity Furthermore, and acceleration the is tested.rising time andIt steady is shown state that time the systemof the withstep hasresponse higher responseare improved and less by resonance 33% and behavior 72%, respectively. as compared In to theorder to validatecase usingthat the the optimal traditional approach approach. can be applied to real machining, the trajectory with equal velocity and acceleration is tested. It is shown that the system with has higher response and less resonance Author Contributions: The individual author contributions follow: conceptualization, C.-C.L.; methodology, behaviorC.-C.L.; as software, compared C.-C.L., to the M.-Q.H. case using and P.-Y.T.;the traditional validation, approach. C.-C.L., M.-S.T., M.-Q.H. and P.-Y.T.; formal analysis, M.-S.T.; investigation, C.-C.L., M.-S.T., M.-Q.H. and P.-Y.T.; resources, M.-S.T.; data curation, C.-C.L.; Authorwriting—original Contributions: draft The preparation, individual C.-C.L. author and contributions M.-S.T.; writing—review follow: conceptualization, and editing, C.-C.L., C.-C.L.; M.-S.T., methodology, M.-Q.H. C.- and P.-Y.T.; visualization, C.-C.L.; supervision, M.-S.T.; project administration, C.-C.L.; funding acquisition, M.-S.T. C.L.;All software, authors haveC.-C.L., read M. and-Q.H. agreed and to theP.-Y.T. published; validation, version C. of- theC.L. manuscript., M.-S.T., M.-Q.H. and P.-Y.T.; formal analysis, M.-S.T.Funding:; investigation,This research C.-C.L. was, M. funded-S.T., M. by-Q.H. the Ministryand P.-Y.T.; of Scienceresources, Technology, M.-S.T; data R.O.C. curation, under C. the-C.L. contract; writing— original107-2218-E-002-071 draft preparation, and by C. the-C.L. Ministry and M. of- EconomicS.T.; writing Affair,—review R.O.C under and editing, the Contract C.-C.L., of 105-EC-17-A-05-5-001. M.-S.T., M.-Q.H. and P.- Y.T.; visualization, C.-C.L.; supervision, M.-S.T.; project administration, C.-C.L.; funding acquisition, M.-S.T. Alls authors have read and agreed to the published version of the manuscript.

Funding: This research was funded by the Ministry of Science Technology, R.O.C. under the contract 107-2218- E-002-071 and by the Ministry of Economic Affair, R. O. C under the Contract of 105-EC-17-A-05-5-001.

J. Manuf. Mater. Process. 2020, 4, 21 11 of 11

Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study, in the collection, analyses, or interpretation of data, in the writing of the manuscript, or in the decision to publish the results.

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