Fast and Precise Point-to-point and Trajectory Control for Positioning Systems of Galvanometer Scanner 215

i 1 1 1.1................................ 1 1.2...................... 3 1.2.1............................ 3 1.2.2........................... 5 1.3................................ 7 2 11 2.1......... 11 2.1.1......................... 11 2.1.2............... 14 2.1.3............... 14 2.1.4...................... 15 2.1.5............... 16 2.2.......... 17 2.2.1........................ 18 2.2.2.................... 18 2.3 2.................... 21 2.3.1 FB............... 23 2.3.2 FSC FF................. 25 2.4 2............... 3 2.4.1 FB............... 31 2.4.2 2............ 32 2.5............. 36 2.5.1............ 36 2.5.2............. 38 2.5.3....... 4

ii 2.6................................... 42 3 45 3.1.............. 45 3.1.1........................ 45 3.1.2............ 47 3.2........ 5 3.3................................... 53 4 55 4.1............................ 55 4.2 NST............................... 56 4.2.1 NST............................ 57 4.2.2 2................... 6 4.3 NST......................... 61 4.4................................... 63 5 67 5.1...................... 67 5.1.1............... 68 5.1.2........................ 68 5.1.3............ 71 5.1.4......... 71 5.2...................... 73 5.2.1 FF.................. 76 5.2.2......... 77 5.3................................... 79 6 81 6.1........................... 81 6.2............................. 83 85 87 95

iii 1.1.................................. 8 2.1 CO2.......................... 12 2.2.......... 12 2.3......................... 14 2.4.......... 15 2.5.................. 16 2.6................... 16 2.7......... 18 2.8 ( 1)................... 19 2.9 ( 2)................... 19 2.1 ( 3)................... 19 2.11 ( 4)................... 2 2.12 ( 5)................... 2 2.13 ( 6)................... 2 2.14....................... 21 2.15 2................ 25 2.16 C(z) ( 1)..................... 27 2.17 ( 1)............... 27 2.18 ( 1)................... 27 2.19 C(z) ( 2)..................... 28 2.2 ( 2)............... 28 2.21 ( 2)................... 28 2.22 C(z) ( 3)..................... 29 2.23 ( 3)............... 29 2.24 ( 3)................... 29 2.25 ( )... 3

iv 2.26 ( )................................. 3 2.27 FSC......................... 31 2.28 FSC.................. 31 2.29 2........... 32 2.3 y u ( 4).......... 33 2.31 ( 4)............... 33 2.32 ( 4)................... 33 2.33 y u ( 5).......... 34 2.34 ( 5)............... 34 2.35 ( 5)................... 34 2.36 y u ( 6).......... 35 2.37 ( 6)............... 35 2.38 ( 6)................... 35 2.39 ( 5)................. 37 2.4 ( 6)................. 37 2.41 ( 2).......... 39 2.42 ( 2).............. 39 2.43........... 4 2.44.... 41 2.45.......... 42 2.46..................... 42 3.1 ( 3)........................ 48 3.2 / ( 3)...... 49 3.3 FB............... 5 3.4 ( 3 )................... 5 3.5 ( 3B )................. 53 3.6 ( 3B )................. 53 3.7 ( 3B No.1 ) 54 3.8 ( 3B No.4 ) 54 4.1 ( ).................. 57 4.2 ( ).................... 57 4.3 (NST ).................. 59

v 4.4 (NST ).................... 59 4.5....................... 6 4.6 2...................... 61 4.7......................... 62 4.8 x ( )......... 64 4.9 y ( )......... 64 4.1 ( ).................. 64 4.11 x (NST )........ 65 4.12 y (NST )........ 65 4.13 (NST ).................. 65 5.1............. 69 5.2...................... 7 5.3 ( 4)....................... 71 5.4 ( 4)....................... 72 5.5 ( 4)........................ 72 5.6 ( 4A)...................... 74 5.7 ( 4B)...................... 74 5.8 ( 4A)....................... 75 5.9 ( 4B)....................... 75 5.1 ( 4A)................. 76 5.11 ( 4B)................. 76 5.12 FF 2.... 77 5.13 FF................ 77 5.14 ( 3A FF )............. 78 5.15 ( 3A FF )............. 78

vii 2.1................................ 17 2.2 ( 1)....................... 22 2.3 ( 2)....................... 22 2.4 ( 3)....................... 22 2.5 ( 4)....................... 22 2.6 ( 5)....................... 23 2.7 ( 6)....................... 23 2.8 FB ( 1).................. 26 2.9 FB ( 2).................. 26 2.1 FB ( 3).................. 26 2.11 ( 1 3)....................... 26 2.12 ( 4 6)....................... 32 3.1 ( 3)............ 49 3.2 ( 3B )...... 52 3.3 ( 3B )...... 52 3.4 ( 3C )... 52 3.5 ( 3D )... 52 4.1 NST........................ 63 4.2 NST........................... 63 5.1 5.6 ( 4).......................... 72 5.2 5.6 ( 4A 4B)...................... 74

1 1 1.1 ( ) (ICT; Information and Communication Technology) [1] PC [2] ICT (IoT; Internet of Things) (Figure of merit) (Via hole) [3] 214 1 µm 12 µm 5 µm [4]

2 1 [5,6] (Mechanics) (Electronics) (Informatics) (Control Engineering) FA(Factory Automation) (CAD; Computer Aided Design) ( ) [7] (Hardware) (Software) ( ) [8, 9] [1] (1) ( ) (2) ( ) (3) ( ) (4) [11] (1) Bang-bang (FF; Feedforward FF) [12] FF (FSC; Final State Control) [13, 14] FSC FF [15] (2) 2 [16] 2 2 [17 19]

1.2 3 (ZPETC; Zero Phase Error Tracking Control) [2] (PTC; Perfect Tracking Control) [21] (3) [22] MVNLS(Multi Variable Natural Length Spring) [23] [24] (4) 1 [25] [26] [27] (FB; Feedback FB) [28] 1.2 1.2.1 ( ) ( ) [29] (HDI; High Density Interconnect)

4 1 ( ) [3] 1, Hz 2,4 Hz ± 1 µm [31] ( ) (1) (2) (3) (1) 2 16 ( 4 ) (2) (PTP; Point-to-point PTP) PTP (CP; Continuous path CP) (3)

1.2 5 (1) (3) (1) (2) (3) (2) (3) 1.2.2 FSC [32] FSC [33] [34] [35] [36] [37] (FFT; Fast Fourier Transform) [38] (STFT; Short Time Fourier Transform) [39] (WT; Wavelet Transform) [4] SFFT WT [41] [42, 43] [44] [45] [46] [47]

6 1 (1) (2) 2 [48] 2 2 [49] FB [5] (Fretting) (False brinelling) [51 53] [54, 55] [56] [57] [58, 59] ( 3 ) 3 [6] ( 4 5 ) 4 2

1.3 7 4 PTP CP (NST; Non Stop Trepanning NST) [62] NST 5 [63] FB FF FB 1.3 6 1.1 1 2 PTP 2 CP 2 FF FB FB FB

8 1 1.1: (The MathWorks MATLAB/Simulink) 3

1.3 9 4 2 NST PTP CP (Trepannning ) FF FB NST 5 2 FF 6

11 2 FB 2 [13, 14] 2 [18] FF 2.1 2.1.1 2.1 [3] 1.6 µm 9.4 µm CO2 Nd(Neodymium) YAG(Yttrium-Aluminum-Garnet) 355 nm YAG UV

12 2 4,2 mm 2.1: CO2 galvano scanner (CH1) galvano scanner (CH2) aperture collimation lens Mirrors laser oscillator f- lens printed-circuit board X-Y positioning table 2.2: YAG [64] 2.2 X Y ( ) f-θ

2.1 13 X Y X-Y (CH1) 2 (CH2) D λ f-θ F d d. =. 4λF πd (2.1) 2.1 CO2 UV UV CO2 CH2 CH1 UV CH1 CH2 f-θ y F θ y = F θ (2.2) 2.2 (Scanning Area ) 65 65 mm 61 51 mm 51 51 mm X-Y X-Y (Step and Repeat) GT (Galvano-table Snchronousness) X-Y 9

14 2 (a) punching (b) trepanning 2.3: 2.1.2 2.3 PTP (Punching ) PTP CP 9 1 UV 2.1.3 2.4 (CNC; Computer Numerical Control)

2.1 15 y m supervisor controller mirror x digital servo controller u i' D/A A/D i ref current control amplifier i galvano scanner angular sensor y' interface y 2.4: x T s u D/A(Digital Analog) i ref i i A/D(Analog Digital) FB y y FB y y m (Microprocessor ) 2.1.4 2.5 2.6

16 2 mirror part bearing permanent magnet outer yoke bearing shaft join part moving coil inner yoke joint part angular sensor 2.5: mirror bearing yoke case bearing shaft coil permanent magnet angular sensor 2.6: 2.1.5 3 CO2 ( 3) 4 UV ( 5 6) 5

2.2 17 2.1: Name Scanner type Mirror type 1 st resonance Using at... frequency (fpu) Scanner1 MM CO2 (CH1) 1.7 Chapter 2 Scanner2 MM CO2 (CH2) 1.52 Chapter 2 Scanner3 MM CO2 (CH2) 1. Chapter 3, 6 Scanner4 MC CO2 (CH2).74 Chapter 6 Scanner5 MC UV (CH1) 1.31 Chapter 5 Scanner6 MC UV (CH2) 1.32 Chapter 5 CO2 ( 3 4) 1 2 2 2.1 MM (Moving Magnet type) MC (Moving Coil type) 3 1 fpu 2.7 ± 2 mppu ppu ( ppu 1/1 mppu ) 1 ppu 37 µs 2.2 1 6

18 2 1 Position (ppu) Settling tolerance Settling time t Time (s) 2.7: 2.2.1 2.4 u y 1 6 2.8 2.13 FB 2.2.2 2.14 P (z) zoh C cur (s) P mec (s) ( ) FB (.3 fpu) C cur (s) 2.3 k c L c C cur (s) = i i r ef = k ce L cs (s)i ref (s) (2.3) 3 ( 2.1) 1 (1. fpu) 2 (2.35 fpu) 3

2.2 19 Gain [db] Phase [degree] -2-4 -6-8 -1 1-1 1 18 9 Frequency [fpu] Measurement result -9 Mathematical model -18 1-1 1 Frequency [fpu] 2.8: ( 1) Gain [db] Phase [degree] -2-4 -6-8 -1 1-1 1 18 9 Frequency [fpu] Measurement result -9 Mathematical model -18 1-1 1 Frequency [fpu] 2.9: ( 2) Gain [db] Phase [degree] -2-4 -6-8 -1 1-1 1 18 9 Frequency [fpu] Measurement result -9 Mathematical model -18 1-1 1 Frequency [fpu] 2.1: ( 3)

2 2 Gain [db] Phase [degree] -2-4 -6-8 -1 1-1 1 18 9 Frequency [fpu] Measurement result -9 Mathematical model -18 1-1 1 Frequency [fpu] 2.11: ( 4) Gain [db] Phase [degree] -2-4 -6-8 -1 1-1 1 18 9 Frequency [fpu] Measurement result -9 Mathematical model -18 1-1 1 Frequency [fpu] 2.12: ( 5) Gain [db] Phase [degree] -2-4 -6-8 -1 1-1 1 18 9 Frequency [fpu] Measurement result -9 Mathematical model -18 1-1 1 Frequency [fpu] 2.13: ( 6)

2.3 2 21 P (z) P (s) zoh C cur (s) P mec (s) T s 2.14: (2.96 fpu).5 fpu.1 fpu.1.5 fpu FB 3 2.4 P m ec(s) ( P mec (s) = y(s) i(s) = K pe Ls 1 3 s 2 + l=1 k l s 2 + 2ζ l ω l + ω 2 l ) (2.4) K p ( ) L ( D/A ) ω l l ζ l l k l l [65] 2.4 1 6 2.2 2.7 2.8 2.13.1 fpu 3.5 fpu 2.3 2 2 2

22 2 2.2: ( 1) Plant gain K p 12.34 1 5 Resonant frequency ω 1, ω 2, ω 3 fpu 1.7, 2.4, 3.73 Damping coefficient ζ 1, ζ 2, ζ 3.5,.9,.3 Influence coefficient k 1, k 2, k 3.456, 1.64,.193 Delay time L 1/fpu.8 2.3: ( 2) Plant gain K p 11.27 1 5 Resonant frequency ω 1, ω 2, ω 3 fpu 1.52, 2.4, 3.48 Damping coefficient ζ 1, ζ 2, ζ 3.5,.9,.2 Influence coefficient k 1, k 2, k 3.63, 1.927,.155 Delay time L 1/fpu.7 2.4: ( 3) Plant gain K p 11.83 1 5 Resonant frequency ω 1, ω 2, ω 3 fpu 1., 2.36, 2.96 Damping coefficient ζ 1, ζ 2, ζ 3.2,.6,.4 Influence coefficient k 1, k 2, k 3.675,.581, 1.84 Delay time L 1/fpu.1 2.5: ( 4) Plant gain K p 5.5 1 5 Resonant frequency ω 1, ω 2, ω 3 fpu.74, 1.51, 2.2 Damping coefficient ζ 1, ζ 2, ζ 3.11,.7,.4 Influence coefficient k 1, k 2, k 3.44, 1.832,.49 Delay time L 1/fpu.17

2.3 2 23 2.6: ( 5) Plant gain K p 6.79 1 5 Resonant frequency ω 1, ω 2, ω 3 fpu 1.31, 1.57, 2.23 Damping coefficient ζ 1, ζ 2, ζ 3.5,.1,.1 Influence coefficient k 1, k 2, k 3.152, 1.56,.61 Delay time L 1/fpu.19 2.7: ( 6) Plant gain K p 6.79 1 5 Resonant frequency ω 1, ω 2, ω 3 fpu 1.32, 1.54, 2.29 Damping coefficient ζ 1, ζ 2, ζ 3.5,.9,.4 Influence coefficient k 1, k 2, k 3.84, 1.587,.47 Delay time L 1/fpu.19 FB FSC FF 2 2 FF PTP PTP CO2 1 3 4 CO2 CP 2.4.1 2 2.3.1 FB FB u y (= P (s)) u y m (= P m (s)) 2 2.15 u f FF F F (z) FF r err C(z) FB u h H S

24 2 FF F F (z) 2.3 2.4 P n (z) FB (NF; Notch Filter NF) FB [25] NF 1 FB [26] NF 9 9 NF NF FB 2.5 3 (Biquad Notch Filter) C(s) = K c s + ω c1 s s + ω c2 s + ω c3 3 k=1 s 2 + 2ζ nk ω nk + ω 2 nk s 2 + 2ζ nk ω dk + ω 2 dk (2.5) 3 NF 1 2 [66] (APF; All Pass Filter APF) 1 3 C(s) 2.8 2.1 C(s) C(z) (IIR; Infinite Impulse Response Filter) 1 3 C(z) 2.16 2.24 2.11.42 fpu 4.5 db 25 PTP.79 fpu FF FB 2.25 3 1 ±3 FB

2.3 2 25 P m (s) u f r err + u h u y FF (z) C (z) H P (s) + + i ref y m y' S 2.15: 2 NF 2.26 FB FB FB 2.3.2 FSC FF FSC [13] FF FSC(FFSC; Frequencyshaped Final State Control) FF B FFSC 2.15 P (s) F F (z)(= P n (z)) y r err FB 2.27 3 P n (z) B FF U U 2.15 FF u f 3 u f [] = u f [N] = 1 N = 41 2.28 U 3 1. fpu 2.36 fpu 2.96 fpu FSC FF PTP PTP

26 2 2.8: FB ( 1) Compensator gain K c 3.69 1 4 Cut off frequency ω c1, ω c2, ω c3 fpu.5,.5, 3.68 Numerator damping coefficient ζ n1, ζ n2, ζ n3.12,.1,.2 Denominator damping coefficient ζ d1, ζ d2, ζ d3.12,.4,.1 Frequency of notch filter ω n1, ω n2, ω n3 fpu 1.77, 2.29, 3.68 2.9: FB ( 2) Compensator gain K c 3.69 1 4 Cut off frequency ω c1, ω c2, ω c3 fpu.5,.5, 3.68 Numerator damping coefficient ζ n1, ζ n2, ζ n3.16,.1,.2 Denominator damping coefficient ζ d1, ζ d2, ζ d3.16,.45,.1 Frequency of notch filter ω n1, ω n2, ω n3 fpu 1.53, 2.21, 3.42 2.1: FB ( 3) Compensator gain K c 3.55 1 4 Cut off frequency ω c1, ω c2, ω c3 fpu.4,.4, 3.68 Numerator damping coefficient ζ n1, ζ n2, ζ n3.1,.5,.1 Denominator damping coefficient ζ d1, ζ d2, ζ d3.32,.25,.2 Frequency of notch filter ω n1, ω n2, ω n3 fpu.91, 1.4, 2.36 2.11: ( 1 3) Scanner Number 1 2 3 Gain margin db 5.18 4.96 6.25 Phase margin degrees 29.6 28. 28. Zero cross frequency fpu.37.33.33 Sensitivity (.42 fpu) db 6.93 8.7 8.84

2.3 2 27 8 Gain [db] 6 4 Phase [degree] 2 1-2 1-1 1 Frequency [fpu] 18 9-9 -18 1-2 1-1 1 Frequency [fpu] 2.16: C(z) ( 1) 4 3 2 Imaginary axis 1-1 -2-3 -4-4 -3-2 -1 1 2 3 4 Real axis 2.17: ( 1) 1-1 Gain [db] -2-3 -4-5 1-2 1-1 1 Frequency [fpu] 2.18: ( 1)

28 2 8 Gain [db] 6 4 Phase [degree] 2 1-2 1-1 1 Frequency [fpu] 18 9-9 -18 1-2 1-1 1 Frequency [fpu] 2.19: C(z) ( 2) 4 3 2 Imaginary axis 1-1 -2-3 -4-4 -3-2 -1 1 2 3 4 Real axis 2.2: ( 2) 1-1 Gain [db] -2-3 -4-5 1-2 1-1 1 Frequency [fpu] 2.21: ( 2)

2.3 2 29 8 Gain [db] 6 4 Phase [degree] 2 1-2 1-1 1 Frequency [fpu] 18 9-9 -18 1-2 1-1 1 Frequency [fpu] 2.22: C(z) ( 3) 4 3 2 Imaginary axis 1-1 -2-3 -4-4 -3-2 -1 1 2 3 4 Real axis 2.23: ( 3) 1-1 Gain [db] -2-3 -4-5 1-2 1-1 1 Frequency [fpu] 2.24: ( 3)

3 2 4 3 2 nominal f1: +3 % f1: -3 % Imaginary axis 1-1 -2-3 -4-4 -3-2 -1 1 2 3 4 Real axis 2.25: ( ) 4 3 2 nominal f1: +3 % f1: -3 % Imaginary axis 1-1 -2-3 -4-4 -3-2 -1 1 2 3 4 Real axis 2.26: ( ) 2.4 2 2 PTP CP PTP CP UV 2 4 6 FB CP FB FB

2.4 2 31 6 4 FF command input (normalized) 2-2 -4-6 1 2 3 4 5 Steps 2.27: FSC -2 Gain [db] -4-6 -8-1 1-1 1 Frequency [Hz] 2.28: FSC 2.4.1 FB 2.29 F F 1 (z) FF 1 F F 2 (z) FF 2 ŷ C P I (z) ob(z) FB ẋ = Ax + Bu u = Kx(K FB ) ẋ = (A BK)x FB 2.29

32 2 r FF 2 (z) + FF 1 (z) err C PI (z) + + u h u f P lp (z) + u H i ref P m (s) P (s) y m y ob (z) y' S 2.29: 2 2.12: ( 4 6) Scanner Number 4 5 6 Gain margin db 5.17 5.2 4.89 Phase margin degrees 31.1 35.9 35.8 Zero cross frequency fpu.22.27.27 Sensitivity (.35 fpu) db 8.63 5.15 5.17 FB LQI 2 4 6 FB y u 2.3 2.38 2.12.35 fpu 2.32 2.38 2.4.2 2 2 2.29 FB P lp (z)

2.4 2 33 8 Gain [db] 6 4 Phase [degree] 2 1-2 1-1 1 Frequency [fpu] 18 9-9 -18 1-2 1-1 1 Frequency [fpu] 2.3: y u ( 4) 4 3 2 Imaginary axis 1-1 -2-3 -4-4 -3-2 -1 1 2 3 4 Real axis 2.31: ( 4) 1-1 Gain [db] -2-3 -4-5 1-2 1-1 1 Frequency [fpu] 2.32: ( 4)

34 2 8 Gain [db] 6 4 Phase [degree] 2 1-2 1-1 1 Frequency [fpu] 18 9-9 -18 1-2 1-1 1 Frequency [fpu] 2.33: y u ( 5) 4 3 2 Imaginary axis 1-1 -2-3 -4-4 -3-2 -1 1 2 3 4 Real axis 2.34: ( 5) 1-1 Gain [db] -2-3 -4-5 1-2 1-1 1 Frequency [fpu] 2.35: ( 5)

2.4 2 35 8 Gain [db] 6 4 Phase [degree] 2 1-2 1-1 1 Frequency [fpu] 18 9-9 -18 1-2 1-1 1 Frequency [fpu] 2.36: y u ( 6) 4 3 2 Imaginary axis 1-1 -2-3 -4-4 -3-2 -1 1 2 3 4 Real axis 2.37: ( 6) 1-1 Gain [db] -2-3 -4-5 1-2 1-1 1 Frequency [fpu] 2.38: ( 6)

36 2 P lp (z) F F 1 (z) F F 2 (z) 2.6 P lp (z) = num(z) den(z) F F 1(z) = den(z) F (z) F F 2(z) = num(z) F (z) (2.6) (2.7) (2.8) F (z) [18] 2 num(z) den(z) FF err u h r u f u y 2.9 y r = den(z) F (z) num(z) den(z) = num(z) F (z) 2.39 2.4 5 6 2 ( r y ).27 fpu.1 fpu.1 fpu.59 db 3.77 2.5 2.5.1 f-θ (= )

2.5 37 Gain [db] Phase [degree] -2-4 -6 18 1-2 1-1 1 Frequency [fpu] 9-9 -18 1-2 1-1 1 Frequency [fpu] 2.39: ( 5) Gain [db] Phase [degree] -2-4 -6 18 1-2 1-1 1 Frequency [fpu] 9-9 -18 1-2 1-1 1 Frequency [fpu] 2.4: ( 6) (1) (2) (1) FF FB FF FB (2.3 )

38 2 (2) 2.41 2 u 2.3 2 1 ppu ( ppu) 37 µs ±1.5 % 1.12 % ± 1.5 % ± 12.5 1.2 % 6 mppu 2.42 2.41 2 8 5 e x y e = x y 8 12 mppu 2.41 12 mppu 3 % 8 25 2.5.2 FF FB

2.5 39 8 6 +1.5 % Positioning error [mppu] 4 2-2 -4-6 -1.5 % -8.5 1 1.5 Time [ms] 2.41: ( 2) 15 1 Positioning error [mppu] 5-5 -1 5 s 8 s -15.1.2.3.4.5.6.7.8 Time [ms] 2.42: ( 2) (PTP ) (CP ) 5 6 2.43 2.4 2 x y 1 ppu 4 mppu.5 ms PTP CP 9 CP

4 2 1.3 1.2 scanner5 scanner6 Position [ppu] 1.1 1.99.98.97.5 1 1.5 2 Time [s] 1-3 2.43: ( 6 ) 4 2.5.3 ( 4 ) 2.44 1

2.5 41 8 6 Positioning error [mppu] 4 2-2 -4-6 -8.76ms.5 1. 1.5 2. 2.5 3. 3.5 4. Time [ms] 2.44: ( ) 2.45 2.46 5

第 2 章 ガルバノスキャナ位置決め装置の概要と本研究の課題 42 Rolling direction 2µm 図 2.45: フレッチング損傷が生じた転がり軸受の内輪軌道面写真 ball oil inner ring Scooping out the oil Contacting 図 2.46: フレッチング損傷発生のメカニズム 2.6 まとめ 本章では まず制御対象の周波数応答特性を取得し 計測結果をよく表現する数学モデ ルを獲得した 次に PTP 動作が主であるガルバノスキャナに対して 応答性のよいト ルク指令型 2 自由度制御系を構築し PTP 動作と CP 動作を切り替えて用いるガルバノ スキャナに対して 追従性のよい既約分解表現に基づく 2 自由度制御系を構築した そし て それぞれの制御系において ロバスト安定性に優れた FB 補償器の設計と 外乱圧縮 特性が良好な状態量 FB 制御系を設計した 次に 第 1 章で問題提起した位置決め性能の 低下に関して 供試ガルバノスキャナ位置決め装置を用いた実験と数値シミュレーション により性能低下の要因を考察し 技術課題を明確化した 第 3 章では 位置決め精度低下

2.6 43 4

45 3 3 3.1 3.1.1 [43] (Exprimental Modal Analysis) (SDOF; Single Degree of Freedom Method) (MDOF; Multipule Degree of Freedom Method) 2

46 3 e(k) (k = 1 N 1) 3.1 n ê = B i Zi k (3.1) i=1 Z i = exp(s t t) s t = σ i + jω di n t s i B i σ i ω di Z i 3.2 n Z n i (a 1 Z n 1 i + a 2 Z n 2 i + + a n Z i ) = (3.2) 3.1 3.2 a 1 a n e(n 1) e(n 2) e() a 1 e(n) e(n 1) e(1) a 2.... e(n 2) e(n 3) e(n n 1) a n = e(n) e(n + 1). e(n 1) (3.3) 3.3 a 1 a n 3.2 Z i s i 3.1 B i Z1 Z2 Zn Z1 1 Z2 1 Zn 1... Z1 N 1 Z2 N 1 Zn N 1 B 1 B 2. B n = y() y(1). y(n 1) (3.4) 3.4 B i A i ϕ i ζ f di 3.5 3.8 A i = B i (3.5) ϕ i = arg B i (3.6) ζ i = σ i /ω di 1 + (σi /ω di ) 2 (3.7)

3.1 47 f di = ω di 2π 1 ζ 2 i (3.8) Z i 3.1.2 3.1 2 2 3 1 ppu 37 µs 5 1 µs (= FSC FF ) 2 ms 2 3.1 n 8 5 3.1 3.2 1 fpu No.2 No.5 ( ) No.1 No.3 FB 3.3 2.1 FB y /r.5 fpu 1. fpu 1.2 fpu ζ.1 ( 3.1) No.4 (.45 fpu) No.2 (1.3 fpu) No.5 (1.2 fpu) 3 No.4 FB No.2 No.5 1 FB NF APF PTP 2 FSC

48 3 1-3 8 6 Position error [ppu] 4 2-2 -4-6 37 µs -8.5 1 1.5 2 2.5 3 Time [sec] 1-3 3.1: ( 3) 3.1 No.1 No.3 PTP ( ) ( ) 3.4

3.1 49 3.1: ( 3) Mode Normalized Initial Decay Initial number frequency amplitude rate phase f di (fpu) A i (1 3 ppu) σ i (rad/s) ϕ i (rad) 1 1.27 511 2 1.3 3.1 1939 2.69 3.16 2.46 317.14 4.45 2.96 3989 2.8 5 1.2 1.28 2846 1.13 1 5-3 No.1 No.2 No.3 No.4 No.5-5.5 1 1.5 2 1-3 1 5-3 -5.5 1 1.5 2 1-3 1 5-3 -5.5 1 1.5 2 1-3 1 5-3 -5.5 1 1.5 2 1-3 1 5-3 -5.5 1 1.5 2 Time [sec] 1-3 3.2: / ( 3)

5 3 ζ=.6 ζ=.4 ζ=.2ζ=.1 ζ=.8 Imaginary axis 1.2fpu1.fpu.5fpu Pole Zero Real axis 3.3: FB Position error [ppu] 1-3 8 6 4 2-2 -4 Sampled waveform Identified waveform -6 37 µs -8.5 1 1.5 2 2.5 3 Time [sec] 1-3 3.4: ( 3 ) 3.2 2 3 3B

3.2 51 ( ) ( ) 3B x = ppu x = 1 ppu x = 1 ppu 9 ppu 3.5 3.6 5 3.2 3.3 ( 3.2 No.1 3.3 No.4) 1 3.7 3.8 3B ( 3C 3D) 3.4 3.5 3C 3D [67]

52 3 3.2: ( 3B ) Mode Normalized Initial Decay Initial number frequency amplitude rate phase f di (fpu) A i (1 3 ppu) σ i (rad/s) ϕ i (rad) 1 4.2 745 2.42 4.74 423 1.85 3 1.3 1.69 144 2.65 4.16 2.2 4121.18 5 1.66 2778 3.3: ( 3B ) Mode Normalized Initial Decay Initial number frequency amplitude rate phase f di (fpu) A i (1 3 ppu) σ i (rad/s) ϕ i (rad) 1.42 4.9 459 1.55 2.13 3.98 3448 1.26 3 1.3 1.66 1357 2.58 4.43 389 5 2.37.81 2371.65 3.4: ( 3C ) Initial amplitude Decay rate A i (1 3 ppu) σ i (rad/s) Normal area 1.72 1671 Damaged area 5.3 885 3.5: ( 3D ) Initial amplitude Decay rate A i (1 3 ppu) σ i (rad/s) Normal area 2.79 2436 Damaged area 7.73 964

3.3 53 1-3 8 6 Position error [ppu] 4 2-2 -4-6 37 µs -8.5 1 1.5 2 2.5 3 Time [sec] 1-3 3.5: ( 3B ) 1-3 8 6 Position error [ppu] 4 2-2 -4-6 37 µs -8.5 1 1.5 2 2.5 3 Time [sec] 1-3 3.6: ( 3B ) 3.3 5

54 3 Position error [ppu] 1 5-3 -5.5 1 1.5 2 Time [sec] 1-3 3.7: ( 3B No.1 ) Position error [ppu] 1 5-3 -5.5 1 1.5 2 Time [sec] 1-3 3.8: ( 3B No.4 )

55 4 2 (NST) PTP CP FF FB NST NST 5 6 4.1 9 CNC PTP CP

56 4 4.1 4.2 4.1 4.2 A B O n 4.2 4.1 1 PTP ( ) 2 CP ( ) 3 ( ) 2 3 ( T trs ) 4.2 NST (NST) NST NST

4.2 NST 57 START 1 st stage Move to center of hole 3 rd stage Shot the laser-pulse Finish settling? Output all data? 2 nd stage Output of orbit data Track orbit? Drilling other hole? END 4.1: ( ) A B O n 4.2: ( ) 4.2.1 NST NST 4.3 2 3 4.4 NST A B O n r A O n B

58 4 r ptp (t) = [x ptp (t), y ptp (t)] (4.1) O n+1 r trp (t) = [x trp (t), y trp (t)] (4.2) r(t) = r ptp (t) + r trp (t) = [x(t), y(t)] (4.3) O n r R ( D L P f P ) 1 T trp T trp = πd f P L P (4.4) r R T trp ( f trp ) f trp f trp rr [7] 4.5 L D = πd 12 P i (i = 1, 2, 3, 12) A L B O n P t P t NST NST P s P t β f trp (

4.2 NST 59 START 1 st stage Move to center of hole and output of orbit data Track orbit? 2 nd stage Shot the laser-pulse Output all data? Drilling other hole? END 4.3: (NST ) A B O n 4.4: (NST ) ) T dly β T ptp T dly 4.5 β = T ptp T dly T trp 36 (4.5) 4.5 P i ( P 11 ) NST T trs T ptp

6 4 P 3 P t Rolling direction P 2 A P 1 B O n P s P 11 4.5: 4.2.2 2 2 4.6 2 (x o, y o ) (x trp [k], y trp [k]) (x ptp [k], y ptp [k]) k T S r x 4.6 y { x ptp [k] = x ptp [k 1] + α x ptp ( x ptp < x lim ) x ptp [k] = x ptp [k 1] + x lim ( x ptp > = x lim ) (4.6) x ptp = x ptp [k] x ptp [k 1] α ( < α < 1) x lim 2 2 ( 2.29)

4.3 NST 61 x o Scanner (X-axis) interpolation algorithm x ptp + + x Scanner (X-axis) servomechanism Supervisor controller memory x trp y trp y o Scanner (Y-axis) interpolation algorithm y ptp + + y Scanner (Y-axis) servomechanism Galvano controller 4.6: 2 4.3 NST x 5 y 6 NST y 4.7 A.15 ppu B 25 1.9 D max D min J J = D min D max (4.7) J = 1 4.1 D.4 ppu f trp f P L P 4.4.1 fpu A 5 6 r

62 4 (ppu) S.15 (.15,.15) (, ) (a) Pattern A (b) Pattern B G 4.7: y 4.8 4.1 NST 4.11 4.13 Shot signal NST 1/f trp 4.8 x r x x ptp x trp T trs 9 ( ).5 ms NST 4.11 x x ptp x trp 4.13 4.2 NST 13 % NST.9

4.4 63 4.1: NST Conventioanl Proposed method method Hole diameter D mppu 4 Pitch of laser pulse L P mppu 7 Laser pulse frequency f P fpu 1.75 Trepaning frequency f trp fpu.1 Transition time T trs ms.5 4.2: NST Pattern A Pattern B Working time Ellipticity Working time Ellipticity (ms) (average) (ms) (average) Conventional 98.8.944 792.8.941 method Proposed 83.8.939 683.7.939 method ( 15.2 %) ( 13.8 %) 4.4 PTP CP PTP CP 2 1 %

64 4.2 r x = x ptp r x = x trp Waveform of x-axis scanner (mppu).15.1.5 Target angle Processing time Detected angle Shot signal.5 1 1.5 2 2.5 3 3.5 4 Time (ms) 4.8: x ( ).2 r y = y ptp r y = y trp Waveform of y-axis scanner (mppu).15.1.5 Target angle Detected angle Shot signal Processing time.5 1 1.5 2 2.5 3 3.5 4 Time (ms) 4.9: y ( ).19 Position of y-axis scanner (ppu).17.15.13 Rolling direction.11.11.13.15.17.19 Position of x-axis scanner (ppu) 4.1: ( )

4.4 65.2 r x = x ptp + x trp r x = x trp Waveform of y-axis scanner (mppu).15.1.5 Target angle Processing time Detected angle Shot signal.5 1 1.5 2 2.5 3 3.5 4 Time (ms) 4.11: x (NST ).2 r y = y ptp + y trp r y = y trp Waveform of y-axis scanner (mppu).15.1.5 Target angle Detected angle Shot signal Processing time.5 1 1.5 2 2.5 3 3.5 4 Time (ms) 4.12: y (NST ).19 Position of y-axis scanner (ppu).17.15.13 Rolling direction.11.11.13.15.17.19 Position of x-axis scanner (ppu) 4.13: (NST )

67 5 2 FF 3 4 5.1 2

68 5 5.1.1 [63, 71] 4 5.1 1 ( 5.1 step1 5) 2 ( 5.1 step6 7) 5.1.2 5.1.3 5.1.2 5.2 C(z) P (s) H S r err u h u i ref 2.15 d H(z) (BPF; Band-pass filter BPF) u f (z) BPF y f (z) BPF BPFH(z) 2 (Butterworth filter) (HPF; High-pass filter) (LPF; Low-pass filter) H(z) = H lpf (z) H lpf (z) H hpf (z) H hpf (z) (5.1) H lpf H lpf.18 fpu z (Bilinear Z-transform) θ m (θ = 1, 2,, i,, 1) ( 5.1 step1) ±8 ppu 1-8 ppu

5.1 69 Start Step1 Positioning at the measuring angle Step2 Starting disturbance input Step3 Step4 Measuring the gain of the controlled object Finished at all measurement angles? Measuring normalized torque proportion Step5 Calculating the normalized torque proportion Step6 Step7 Calculating the fourth-order polynomial of the torque proportion Calculating the torque variation Calculating torque variation End 5.1: ( step2) 2.11 4 (.1.4 fpu).18 fpu.33 ppu u f (z) y f (z) u f (z) = H(z) u(z) (5.2) y f (z) = H(z) y(z) (5.3)

7 5 d r err u h + u y C (z) H P (s) + + i ref y' S H (z) H (z) u f y f 5.2: θ i BPF BPF (. =..18 fpu ) g i ( step3) g i = u peak y peak θi u peak = sup u f (z) y peak = sup y f (z) (5.4) g i L i ( step5) L i = g i g c (5.5) g c L i step1 5 5.3 4

5.1 71 1.1 Torque proportion (normalized) 1.5 1.995.99.985.98.975.97-8 -6-4 -2 2 4 6 8 Angle of measurement position (ppu) 5.3: ( 4) 5.1.3 5.1.2 θ 4 K c =.. a 1 θ 4 + a 2 θ 3 + a 3 θ 2 + a 4 θ + a 5 (5.6) 5.4 4 a 1 a 5 5.1 4 5.5 4 1 ppu 1 76 µs 1 ±4 mppu 4 5.1.4 4 ( 4A 4B) 4A

72 5 1.5 1 Torque variation (%).5 -.5-1 -1.5-8 -6-4 -2 2 4 6 8 Angle of measurement position (ppu) 5.4: ( 4) 5.1: 5.6 ( 4) a 1 a 2 a 3 a 4 1.49 1 5 4.91 1 5 1.21 1 3 2.54 1 3 a 5 1. 8 6 Positioning error (mppu) 4 2-2 -4-6 -8.76ms.5 1. 1.5 2. 2.5 3. 3.5 4. Time (ms) 5.5: ( 4)

5.2 73 4B 4B 2.5.3 5.6 5.7 4A 4B 5.2 5.6 4A 4B 4B 1 % 4A 4B 5.5 5.8 5.9 5.9 2.44 4A 7 mppu 4B 12 mppu 1 ms 4B ±4 mppu 4B 5.1 5.11 4A 4B 5.11 2.45 4A 2 4B 8 25 µm 5.1 5.11 1 % 12 mppu 5.1 5.11 [72] 4A 5.2 5.1 4A 4B

74 5 5.2: 5.6 ( 4A 4B) scanner 4A scanner 4B a 1 2.84 1 5 6.57 1 7 a 2 5.16 1 6 1.43 1 5 a 3 1.51 1 3 1.6 1 3 a 4 7.88 1 4 2.71 1 4 a 5 1.4 1. 1.5 1 Torque variation (%).5 -.5-1 -1.5-8 -6-4 -2 2 4 6 8 Angle of measurement position (ppu) 5.6: ( 4A) 1.5 1 Torque variation (%).5 -.5-1 -1.5-8 -6-4 -2 2 4 6 8 Angle of measurement position (ppu) 5.7: ( 4B)

5.2 75 8 6 Positioning error (mppu) 4 2-2 -4-6 -8 7 mppu.76ms.5 1. 1.5 2. 2.5 3. 3.5 4. Time (ms) 8 6 5.8: ( 4A) 12 mppu Positioning error (mppu) 4 2-2 -4-6 -8.76ms.5 1. 1.5 2. 2.5 3. 3.5 4. Time (ms) 5.9: ( 4B) 4A

76 第 5 章 小振幅揺動動作による転がり軸受性能低下の定量化とその補償 Rolling direction 2µm 図 5.1: 転がり軸受の内輪軌道面写真 (供試体 4A) Rolling direction 2µm 図 5.11: 転がり軸受の内輪軌道面写真 (供試体 4B) 5.2.1 FF 補償による位置決め精度改善 転がり軸受性能低下時に位置決め誤差波形に発生する遅い応答を抑制する方法として FB 系の外乱圧縮特性を向上する方法も考えられるが ねじり振動モードの変動に対する ロバスト性やサーボ安定性とトレードオフの関係にあるため ここでは速度 FF 補償によ る改善方法を検討した 提案する位置決め精度改善法のブロック線図を 図 5.12 に示す 元のトルク指令型 2 自由度制御系 (図 2.15) と比較して Dv 速度補償ゲインが追加され ている点が異なる ここで vm 速度指令である 図 5.13 は 提案法をフレッチング損 傷が発生していない供試体 4 に適用し 開始点から x =.6 ppu 離れた点に向けて目標

5.2 77 u f FF (z) v m + + D v r err + C (z) u h + + u H i ref P m (s) P (s) y m y y' S 5.12: FF 2 8 6 Positioning error (ppu) 4 2-2 -4-1.5 +1.5-6 -8.5 1 1.5 2 Time (s) 1-3 5.13: FF.28 ms D v ±1.5.5 D v 3.7 fpu D v.6 2. ms D v.28.6 ms D v u ff 5.2.2 3 ( 3A) FF 5.14 5.15 x = 1.ppu.4 ms FF FF FF

78 5 8 6 Positioning error (mppu) 4 2-2 -4-6 -8.2.4.6.8 1. 1.2 1.4 1.6 1.8 Time (ms) 5.14: ( 3A FF ) 8 6 Positioning error (mppu) 4 2-2 -4-6 -8.2.4.6.8 1. 1.2 1.4 1.6 1.8 Time (ms) 5.15: ( 3A FF ) D v.32 FF.6 1.2 ms FF FF

5.3 79 5.3 FB FF FB

81 6 6.1 (1) (2) 2 PTP 2 CP 2 FF FB FB FB

82 6 3 2 4 2 PTP CP NST PTP CP FF FB 5 2 FB FF

6.2 83 6.2

85 215 12 1

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