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(1)

Study on Application of Controlling a Plate with a Dual-arm Robot

Wei WU*, Toshiki HIROGAKI** and Eiichi AOYAMA***

(Received January 15, 2010)

Recently, new tendency has emerged to control not only in linear motion but also rotational motion with high accuracy manufacturing fields. Many five-axis-controlled machining centers are currently in use. However, it is difficult to achieve the flexible manufacturing systems by means of methods based on the use of these machine tools. An industrial dual-arm robot has gained attention as a new method to control both linear motion and rotational motion accurately, in attempt to control a working plate like a machine tool table. In the present report, we attempt a motion controlling of a working plate with a dual-arm robot. As a result, the cooperating dual-arm motion is demonstrated to make it feasible to perform a stable operation control, such as an application of controlling the working plate to keep a ball rolling around a circular path.

Key words㧦autonomic, cooperating control, dual arm robot, working plate, industrial robot, automation 䉨䊷䊪䊷䊄䋺⥄ᓞ㧘ද⺞೙ᓮ㧘෺⣨ࡠࡏ࠶࠻㧘૞ᬺࡊ࡟࡯࠻㧘↥ᬺ↪ࡠࡏ࠶࠻㧘⥄േൻ㧚

෺⣨ࡠࡏ࠶࠻ࠍ↪޿ߚ૞ᬺࡊ࡟࡯࠻ߩᠲࠅേ૞ߣᔕ↪ߦ㑐ߔࠆ⎇ⓥ

๓ 㝵࡮ᑝ၂ବ᮸࡮㕍ጊᩕ৻

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↢ ↥ ࠪ ࠬ ࠹ ࡓ ߩ ⃻ ⁁ ߪ 㧘 ዋ ຠ ⒳ ᄙ ㊂ ↢ ↥ ߆ ࠄ (/5(NGZKDNG/CPWHCEVWTKPI5[UVGOߥߤߦઍ⴫ߐࠇࠆᄙ

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೙ᓮߦࠃࠆ↢↥ߩ૞ᬺࠍㆀⴕߔࠆ⥄േൻᛛⴚߣ⠨߃ࠄࠇ ࠆ㧚Ꮏ૞ᯏ᪾ߩ %0% ᛛⴚൻߪ ᐕߦࠕࡔ࡝ࠞ /+6 ߢౕ

⃻ൻߐࠇ㧘ߘߩᓟߪਗㅴ㧟ゲߩㆇേࠍ೙ᓮߔࠆ㧟ゲ೙ᓮࡑ

ࠪ࠾ࡦࠣ࠮ࡦ࠲ߣߒߡ⊒ዷߒ㧘⃻࿷ߪᢙǴO ࠝ࡯࠳ߩ㜞޿

ㆇേ♖ᐲ߇ታ↪⊛ߦ㆐ᚑߐࠇߡ޿ࠆ㧚ߘߩ৻ᣇ㧘ᨵエߥ↢

↥ࠪࠬ࠹ࡓࠍౕ⃻ൻߔࠆߚ߼ߦ㧘Ꮏ૞ᯏ᪾ߦ߽↥ᬺ↪ࡠࡏ

࠶࠻ߣหߓࠃ߁ߦᣓ࿁ㆇേᯏ⢻ߩౕ஻ࠍ᳞߼ࠆ࠾࡯࠭߇ 㜞߹ࠅ㧘ㄭᐕߪਗㅴ㧟ゲߦ࿁ォ㧞ゲߩㆇേࠍ೙ᓮߔࠆ㧡ゲ

೙ᓮࡑࠪ࠾ࡦࠣ࠮ࡦ࠲߇ታ↪ൻߐࠇߡ߈ߡ޿ࠆ㧚ߒ߆ߒߥ

߇ࠄ㧘ࡑࠪ࠾ࡦࠣ࠮ࡦ࠲ߪࠟࠗ࠼᭴ㅧࡌ࡯ࠬߦߒߚ࠹

࡯ࡉ࡞ߩㆇേ೙ᓮߢ޽ࠆߚ߼㧘᭴ㅧ⊛ߦᣓ࿁ゲߩᣓ࿁

ਛᔃߩ૏⟎߇࿕ቯߐࠇ㧘૞ᬺߩ⥄↱ᐲ߇ᄢ߈ߊ೙㒢ߐ ࠇߡ޿ࠆ㧚ࠟࠗ࠼᭴ㅧࠍᜬߚߥ޿ࡑࠪ࠾ࡦࠣ࠮ࡦ࠲

ߣߒߡߪ㧘ࡏ࡯࡞ࡀࠫߣᄙ⥄↱ᐲߥ⃿㕙࡚ࠫࠗࡦ࠻ࠍ

⚵ߺวࠊߖߡ࠹࡯ࡉ࡞ࠍᡰᜬߔࠆᣇᑼ߽৻ㇱታ↪ൻߐ ࠇߡ޿ࠆ߇㧘࡚ࠫࠗࡦ࠻ߩᐓᷤߦࠃࠆࡑࠪࡦߩᄢဳൻ

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ዷߦࠃࠅᄙゲߩหᦼ೙ᓮᛛⴚ߇ౕ⃻ൻߐࠇ㧘ᄙ㑐▵ࡠ ࡏ࠶࠻ࠕ࡯ࡓࠍฦ⣨ߣߒߡ᭴ᚑߐࠇߚ෺⣨ࡠࡏ࠶࠻߇ ታ↪ൻߐࠇ㧘ᣂߒ޿ (# ᛛⴚ߳ߩᔕ↪߇ᦼᓙߐࠇߡ߈ߡ

* Graduate Student, Doshisha University, Kyoto

Telephone: +81-774-65-6445, E-mail: dtg0386@mail4.doshisha.ac.jp

**Department of Mechanical Systems Engineering, Doshisha University, Kyoto Telephone/Fax: +81-774-65-6503, E-mail: thirogak@mail.doshisha.ac.jp

***Department of Energy and Mechanical Engineering, Doshisha University, Kyoto Telephone: +81-774-65-6506, Fax: +81-774-6829, E-mail: eaoyama@mail.doshisha.ac.jp

(2)

Fig.1 Dual-arm robot

޿ࠆ㧚෺⣨ࡠࡏ࠶࠻ߦ߅޿ߡ㧘෺⣨ߩవ┵ࠍ⚿วߔࠆߣ㐽

࡝ࡦࠢᯏ᭴ࠍ᭴ᚑߔࠆߎߣߦߥࠅ㧘ᡰᜬ೰ᕈߩะ਄߇น⢻

ߢ޽ࠆ㧚෺⣨ߩද⺞૞ᬺߦࠃࠅ⦡ޘߥ૞ᬺ߇⠨߃ࠄࠇࠆ߇㧘

૞ᬺⓨ㑆ߦ߅ߌࠆ㓚ኂ‛࿁ㆱ߿ᨵエ‛ߩᛠᜬߐࠄߦߪ

⚻〝⸘↹ߥߤߩ⎇ⓥߪߥߐࠇߡ޿ࠆ߇㧘ᡰᜬ೰ᕈߣᄙゲ ߩද⺞೙ᓮ․ᕈࠍࠛࡦ࠼ࠛࡈࠚࠢ࠲ߣߒߡⓍᭂ⊛ߦᵴ↪

ߒߚ߽ߩߢߪߥ߆ߞߚ㧚

ߘߎߢᧄ⎇ⓥߢߪ㧘෺⣨ࡠࡏ࠶࠻ߦ૞ᬺࡊ࡟࡯࠻ࠍᡰᜬ ߔࠆߎߣߢ㐽࡝ࡦࠢᯏ᭴ࠍ᭴ᚑߒߡ㧘ࠛࡦ࠼ࠛࡈࠚࠢ࠲ߣ ߒߡචಽߥᡰᜬ೰ᕈࠍ⛽ᜬߒߥ߇ࠄ㧘૞ᬺࡊ࡟࡯࠻ࠍᠲࠆ ߎߣࡊ࡟࡯࠻਄ߢᨵエߥ૞ᬺⓨ㑆ࠍታ⃻ߔࠆߎߣࠍ⹜ߺ ߚ㧚ᧄႎߢߪ૞ᬺࡊ࡟࡯࠻ߩᡰᜬᣇᴺࠍ⠨ኤߒ㧘ߐࠄߦᡰ ᜬߒߚ૞ᬺࡊ࡟࡯࠻ߦ෺⣨ߩද⺞೙ᓮߦࠃࠆᣓ࿁ㆇേࠍ ਈ߃ࠆߎߣߢ㧘ࡊ࡟࡯࠻਄ߦ⃿ߩߥ߼ࠄ߆ߥォ߇ࠅㆇേࠍ ታ⃻ߢ߈ࠆߎߣࠍ␜ߒߚ㧚

㧞㧚ታ㛎ⵝ⟎߅ࠃ߮ᣇᴺ ෺⣨ࡠࡏ࠶࠻ߩၮᧄᕈ⢻

૶↪ߒߚࡠࡏ࠶࠻ߪ቟Ꮉ㔚ᯏ⵾ߩ෺⣨ࡠࡏ࠶࠻

/161/#0&+#㧔࿑ 㧕ߢ޽ࠆ㧚ੱߩ਄ඨりࠍࠗࡔ࡯ࠫߒ ߡ㐿⊒ߐࠇ㧘ᚑੱ↵ᕈߣ߶߷หߓࠨࠗ࠭ߢ޽ࠆ㧚ฦ⣨ߦ 㑐▵ࠍᜬߜ㧘ੱߩ⣨ߣหߓࠃ߁ߥേ߈ࠍታ⃻ߔࠆߎߣ߇น

⢻ߢ޽ࠆ㧚ߐࠄߦฦ㑐▵ߩേ૞▸࿐ߪੱߩߘࠇࠍᄢ߈ߊ਄

࿁ࠅ㧘ขࠅᓧࠆᆫ൓ߩ⥄↱ᐲߪੱએ਄ߢ޽ࠆ㧚⢵૕ߩ߭ߨ ࠅࠍ⠨߃ߚࡏ࠺ࠖᣓ࿁ゲࠍ ゲⵝ஻ߒ㧘ว⸘ ゲ೙ᓮ㧔࿑

㧝ਛߩ ,㨪,㧕ߢ޽ࠆ㧚ฦ⣨ߪ MIH㧔MI㧕㧘ਔ⣨ MIH

Table 1 Specifications of Dual-arm robot

㧔MI㧕ߩ㊀㊂‛ࠍᜬߟߎߣ߇ߢ߈ࠆ㧚ࡠࡏ࠶࠻ߩฦㇱ ಽߦ㑐ߔࠆᕈ⢻ࠍ⴫ ߦ␜ߔ㧚

෺⣨ࡠࡏ࠶࠻ߩ ⣨ㆇേ♖ᐲ

♖ᐲ᷹ቯߢ૶↪ߒߚ᷹ቯⵝ⟎ߪ࡟࠾࡚ࠪ࡯␠⵾ߩ&$$

㧔&QWDNG$CNN$CT㧕ࡏ࡯࡞ࡃ࡯ࠪࠬ࠹ࡓ 3% ߢ޽ࠆ㧚

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Power Consumption Approximate Mass Repeatability

1,440 mm Vertical Reach

1,100 mm Horizontal Reach per Arm

10 kg/arm(98kgf) Payload

15(7 axes per arm and 1 rotary axis) ControlledAxes

Floor Mounting

Articulated Structure

Maximum Motion Range

Maximum Speed

Allowable Moment

Allowable Moment of Inertia

mm 1 . r0

$

$

$

$

$

$

$

$

180 100 180 130 180 120 180 180

rr rr rr rr

s s s s s s s s

/ 500

/ 250

/ 250

/ 170

/ 170

/ 170

/ 170

/ 170

$

$

$

$

$

$

$

$

Rotation-Axis (Waist) S-Axis (Lifting) L-Axis (Lower Arm) 7th-Axis (Lower Arm Twist) U-Axis (Upper Arm) R-Axis (Upper Arm Twist) B-Axis (Wrist Pitch/Yaw) T-Axis (Wrist Twist)

Turning-Axis S-Axis L-Axis 7th-Axis U-Axis R-Axis B-Axis T-Axis

R-Axis B-Axis T-Axis R-Axis B-Axis T-Axis

220kg 4.2KVA 31.4 N䊶m 31.4 N䊶m 19.6 N䊶m 1kg䊶m2 1kg䊶m2 0.4kg䊶m2

(3)

㧔a㧕

Set –up for DBB evaluation

㧔b㧕

Result of DBB experiment Fig.2 Motion accuracy of robot arm

Ball Z

Y X

Working plate Ball

Z

Y X

Working plate

Fig.3 Definition of coordinate in supporting working plate

)

1( ' ' '

Z y

x YC ZC

C R X

R

' 㧔㧕

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૞ᬺࡊ࡟࡯࠻ߩᡰᜬᣇᴺߣ೰ᕈ

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(4)

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೰ᕈߪ ୚㧘 ୚ߦߥࠆ㧚ߒߚ߇ߞߡ㧘ਔ┵ᡰᜬߦㄭ޿

೰ᕈߩะ਄ߢ޽ࠆ㧚એ਄ࠃࠅ㧘෺⣨ᡰᜬߦߔࠆߎߣߢ㧘Ყ セ⊛೰ᕈ߇ૐ޿ᧁ⵾ࡊ࡟࡯࠻ߢ߽චಽߦ೰ᕈߩ㜞޿૞ᬺ

ⓨ㑆߇ታ⃻ߢ߈ߚߎߣ߇ࠊ߆ࠆ㧚

㧟㧚ォ߇ࠅㆇേߩࠪࡒࡘ࡟࡯࡚ࠪࡦ ၮᧄࡕ࠺࡞

࿑ ߩࠃ߁ߦᢳ㕙㧔଀߃߫㧘: ᣇะ㧘௑ᢳⷺǰ㧕ߩࡏ࡯

࡞ߩォ߇ࠅㆇേߩࡕ࠺࡞ࠍ⠨߃ࠆ㧚

ᢳ㕙ߦᴪߞߡਅะ߈ߦ :㧔;㧕ゲ㧘ߎࠇߣု⋥ߦ <

ゲࠍߣࠆ㧚⃿ߦ૞↪ߔࠆജߪ㧘៺ᡂജ㨒ޔ㊀ജ /I ߩ ߺࠍ⠨߃ࠆ㧚

6

mߪࡊ࡟࡯࠻ߦ࿕ቯߐࠇߚᐳᮡ♽㧘

6

Bߪኻ⽎‛ߩ㊀ᔃ૏⟎ߦ࿕ቯߐࠇߚᐳᮡ♽㧔

z

B ߪࡊ࡟࡯࠻ߣߩធ⸅㕙ߦ⋥੤ߔࠆ㧕㧘

B m B

mX , Y ߪ

6

m

߆ࠄ⷗ߚ

6

B૏⟎㧘ǰߪࡊ࡟࡯࠻㨥ゲ߹ࠊࠅߩ᳓ᐔߦ ኻߔࠆ࿁ォⷺ㧔㧯㨃ᣇะࠍᱜ㧕ߣߔࠆ㧚ࡊ࡟࡯࠻ߩ

ⷺᐲǰࠍᤨ㑆㨠ߩ㑐ᢙǰVߣߒ㧘ࡏ࡯࡞߇ࡊ࡟࡯࠻

਄ࠍォ߇ࠆㆇേᣇ⒟ᑼߪㅦᐲߦᲧ଀ߒߚᷫ⴮㗄ࠍട ߃ߡᑼ㧔㧕ߣߒߡ⠨߃ࠆ㧚ᷫ⴮ଥᢙ &㧘ࡏ࡯࡞ߩ⾰㊂ ࠍ/ ߣߒߡ㧘&/㧩&/ ߢ޽ࠆ㧚

0 7 sin

5 x

x

xX D mXB g T

B M

m 㧔㧕

ߎߎߢ㧘:; ᐔ㕙ߦ߅ߌࠆォ߇ࠅ౞ㆇേߪ㧘<: ᐔ㕙߹

ߚߪ <; ᐔ㕙߆ࠄ⷗ࠆߣ㧘ߘࠇߙࠇߦォ߇ࠅߩනᝄേߔࠆ ߣ⠨߃ࠄࠇࠆ㧚ߘߎߢ㧘ታ㓙ࡠࡏ࠶࠻ߢࡊ࡟࡯࠻ࠍᠲ૞ߔ ࠆᤨߪ : ゲ߹ࠊࠅ߅ࠃ߮ ; ゲ߹ࠊࠅߢ࿁ォⷺǰVߩ૏⋧

ߪ ᐲߕࠄߒߡ޿ߚേ߈ࠍ⚵ߺวࠊߖߡࡊ࡟࡯࠻਄ߦࡏ

࡯࡞ߩ౞゠㆏ࠍ↢ߓߐߖࠆ㧚 ࠳ࡦࡇࡦࠣ㗄ߩ▚಴

ࡏ࡯࡞ߩォ߇ࠅㆇേਛߦߪ࠳ࡦࡇࡦࠣ㗄߇ήⷞߢ߈ߥ޿

ߣ⠨߃ࠄࠇࠆߩߢ㧘ᷫ⴮ଥᢙ& ࠍታ᷹ߔࠆ㧚

ታ㛎ߪ࿑ ߦ␜ߔ᭽ߦ㧘ࡊ࡟࡯࠻ߣหߓ᧚ᢱߩࠕ

ࠢ࡝࡞ࡄࠗࡊ㧔⋥ᓘǞ㧩OO㧕ߣࡑ࠙ࠬࡏ࡯࡞㧔Ǟ 㧩OO㧕ࠍ૶ߞߚォ߇ࠅᝄࠅሶߣߒߚ㧚ೋᦼ૏⟎㨔

3RVLWLRQRI< PP .LM­P1

'XDODUP[

'XDODUP[ PP 2QHDUP[

2QHDUP[ PP

Fig.4 Compliance of working plate

z

m

Frictional force

Gravitational force

¦

m

T

X

m

X

B

Fig.5 Model of ball rotating on working plate

h1=64mm h9=13mm

h2=53mmh8=16mm

h7=21mm h6=27mm

h5=32mm

h4=37mm h3=48mm

mm I 128

h15=0.8mm

h14=1.3mm h12=5mm

h11=8mm

h10=11mm

h13=2.6mm

Fig.6 Experimental set-up for measuring damping

(5)

㪄㪈㪌㪇 㪄㪈㪇㪇 㪄㪌㪇

㪌㪇 㪈㪇㪇 㪈㪌㪇

㪘㫉㪺㩷㫃㪼㫅㪾㫋㪿 㪟㪼㫀㪾㪿㫋

Height (mm)

Time (s)

Fig.7 Variation of amplitude

㪈㪇

㪈㪇 㪈㪉

㪫㫀㫃㫋㫀㫅㪾㩷㪸㫅㪾㫃㪼㩷㩿㪻㪼㪾㫉㪼㪼㪀

㪩㫌㫅㫅㫀㫅㪾㩷㫋㫀㫄㪼㩷㩿㫊㪀 㪫㪿㪼㫆㫉㪼㫋㫀㪺㪸㫃㩷㫍㪸㫃㫌㪼

㪜㫏㫇㪼㫉㫀㫄㪼㫅㫋㪸㫃㩷㫍㪸㫃㫌㪼

Fig.8 Influence of tilting angle on running time

5 10 15 20

1 2 3 4

X (m)

Time (s)

0

Fig.9 Single harmonic motion with no damping

㧩4㧩OO ߢᚻࠍ᡼ߒߡ㧘㨔㧘㨔࡮࡮࡮㨔ࠍ᷹ቯ ߒߚ㧚࿑ ߪ㧘᷹ቯߒߚ㜞ߐߣ౞╴㕙ߦᴪߞߚ౞ᒐ 㐳ߐߩᝄ᏷ߢ޽ࠆ㧚౞ᒐ㐳ߐߩฦᝄ᏷ DPࠍᑼ㧔㧕㧘 ᑼ㧔㧕ߦઍ౉ߒߡ

' ' 0

ln

0

1

n

n b

b b b

n

G

㧔8㧕

mkk

D

2 S 2

[ G

㧔9㧕

㧔ࡏ࡯࡞ߩඨᓘT㧘Ǭߪኻᢙᷫ⴮₸㧘Ǯߪᷫ⴮Ყ㧘MM

/I㧔4T㧕㧘D㧘D̈㧚㧚㧚ߪᝄ᏷㧘& ߪᷫ⴮ଥᢙߢ޽

ࠆ㧕⸘▚ߒߚᷫ⴮ଥᢙ& ߪ ߢ޽ࠆ㧚ߎߎߢ㧘ߎ ߩォ߇ࠅᝄࠅሶߩℂ⺰๟ᦼߪ 㨟ߢ޽ࠅ㧘ታ㛎ߢ

᷹ቯߐࠇߚ๟ᦼߪ 㨟ߢ㧘ਔ⠪ߪ߶߷৻⥌ߒߡ޿ࠆ㧚 ߒߚ߇ߞߡ㧘ᝄേㆇേਛߪߔߴࠆߎߣߥߊォ߇ࠅㆇേ

ࠍߒߡ޿ߚ߽ߩߣ⠨߃ࠄࠇࠆ㧚

ォ߇ࠅㆇേߣ࠹࡯ࡉ࡞௑ᢳⷺᐲߩ⸳ቯ ታ㛎ߦ↪޿ߚࡑ࠙ࠬࡏ࡯࡞㧔⋥ᓘ OO㧘⾰㊂ 㨓㧕ߣォ߇ࠅ㕙ߩ៺ᡂࠍ৻ቯߦߔࠆߚ߼ߦ㧘ࠕ

ࠢ࡝࡞᧼ࠍᡰᜬࡊ࡟࡯࠻ߩ਄ߦ࿕ቯߒߚ㧚࠹࡯ࡉ

࡞ߩ௑ᢳⷺǰ߇৻ቯߩ႐วߦߪ㧘ೋᦼㅦᐲࠍ࠯ࡠ ߣߒߚᤨ㧘ࡏ࡯࡞ߩ⒖േ〒㔌.㧘㊀ᔃߩᢳ㕙ᣇะ ߩടㅦᐲ

a

㧘ᤨ㑆㨠ߪᑼ㧔㧕㧘ᑼ㧔㧕ߣߥࠆ㧚

T 7 sin 5

g

a 㧔㧕

a

t

2

L 㧔㧕

࿑ ߩ࠹࡯ࡉ࡞ ; ゲᣇะ㧔.㨙㧕ߦታ㓙ߦࡏ

࡯࡞ࠍォ߇ߒ㧘ᑼ㧔㧕ࠃࠆℂ⺰୯ߣォ߇ࠅߦⷐ ߒߚታᤨ㑆ߣᲧセߒߚ⚿ᨐࠍ࿑ ߦ␜ߔ㧚

࿑ ࠃࠅ㧘௑ᢳⷺ߇ዊߐ޿႐วߪࡏ࡯࡞߇Ⱜⴕ ㆇേߔࠆߥߤߒߡ㧘ਔ⠪ߪ৻⥌޿ߥ޿ߎߣ߇ࠊ߆ ࠆ㧚ߒ߆ߒߥ߇ࠄ㧘௑ᢳⷺ ᐲએ਄ߢߪ቟ቯߒߚ ォ߇ࠅㆇേߦࠃࠅਔ⠪ߪ৻⥌ߒߚ㧚ߘߎߢᧄႎߢ ߪ㧘ᧄ⚿ᨐࠍ⋡቟ߦߒߡ࠹࡯ࡉ࡞ߩᦨᄢ௑ᢳⷺߪ qએ਄ࠍ↪޿ࠆߎߣߦߒߚ㧚

㧠㧚⚿ᨐ߅ࠃ߮⠨ኤ

නᝄേࡕ࠺࡞ߩࠪࡒࡘ࡟࡯࡚ࠪࡦ

ᑼ㧔㧕ߦ߅޿ߡ㧘ᷫ⴮㗄ࠍ࠯ࡠ㧘ǰ㧩ǰ5KPȁV ߣ ߒߚ႐วߩ:ᣇะߩࡏ࡯࡞ߩォ߇ࠅㆇേࠍ࿑ߦ␜ߔ㧚 ǰ㧩q㧘๟ᦼ 6㧩U㧘ߔߥࠊߜȁ㧩Ǹ㧩 TCFU ߣߒߡ⸘▚ߒߚ଀ߢ޽ࠆ㧚ᤨ㑆ߦᲧ଀ߒߡᝄ

േߩਛᔃ߇ : ᣇะߦ⒖േߔࠆ᭽ሶ߇ࠊ߆ࠆ㧚߹ߚ․ᱶ

⸃ࠃࠅᝄേߩ ᝄ᏷ࠍ᳞߼ࠆߣ 㧔㨙㧕ߣߥࠆߎߣ

߽ࠊ߆ߞߚ㧚ߔߥࠊߜ㧘ᷫ⴮㗄߇࠯ࡠߢߪ㧘․ᱶߥೋ

ᦼ᧦ઙએᄖߢߪනᝄേߒߥ޿⁁ᘒ㧔࿑ ߦ␜ߔ㧕ߦߥ ࠆߎߣ߇ࠊ߆ࠆ㧚

ᰴߦ㧘ᷫ⴮㗄ࠍᰴ╙ߦᄢ߈ߊߔࠆߣ㧘ਛᔃߩ ߕࠇࠍዋߒ↢ߓࠆ߇නᝄേ㧔࿑ 㧕ߩㆇേࠍታ

⃻ߢ߈ࠆߎߣ߇ࠊ߆ࠆ㧚

(6)

5 10 15 20

0.1

0.05

0

-0.05

X (m)

Time (s)

0

Fig.10 Influence of damping on single harmonic motion.

X (m) Y (m)

(m)

(m)

(a) Emitting motion

(m)

X (m) Y (m)

(m)

(b) Convergent circular motion Fig.11 Predicted motion on working plate

X Y

(m)

(m)

0.05

-0.05 0.05

-0.05

Fig.12 Motion error of rotating ball on working plate

ォ߇ࠅ౞ㆇേࡕ࠺࡞ߩࠪࡒࡘ࡟࡯࡚ࠪࡦ ▵ࠃࠅᷫ⴮㗄㗄߇ᄢಾߥᓎഀߦߥࠆߎߣ ߇ࠊ߆ߞߚ㧚․ߦᷫ⴮㗄ߩ୯߇ዊߐ޿ᤨ㧘ࡏ࡯࡞

ߩේὐ߆ࠄߩ゠〔ߪ࿑ 㧔C㧕ߩࠃ߁ߥ⊒ᢔ⃻⽎

ࠍ↢ߓࠆߎߣ߇ࠊ߆ߞߚ㧚: ゲ߹ࠊࠅ߅ࠃ߮ ; ゲ

߹ࠊࠅߩ࿁ォⷺㆇേߩ૏⋧ࠍ qߕࠄߔߎߣߢ ࡊ࡟࡯࠻਄ߦࡏ࡯࡞ߩォ߇ࠅ౞゠〔ࠍታ⃻ߢ߈ ࠆ߇㧘଀߃߫㧘: ゲ߹ࠊࠅߩㆇേਛᔃߩ⒖േ߇ᱷ ߞߡ޿ࠆߣ౞ߩ゠〔ߦ෼᧤ߒߥ޿㧚

ߘߎߢ㧘 ▵ߩታ㛎ߢ᷹ቯߒߚᷫ⴮㗄ߩ୯ࠍᑼ㧔㧕 ߦઍ౉ߒߡ⸘▚ߒߚ㧚⚿ᨐ߇࿑ 㧔D㧕ߢ޽ࠆ㧚ᧄ᧦

ઙࠍ↪޿ࠆߣࡊ࡟࡯࠻਄ߦォ߇ࠅ౞゠㆏㧔ඨᓘ㧩 OO㧕ࠍ↢ᚑߢ߈ࠆߎߣ߇ࠊ߆ߞߚ㧚

෺෺⣨ࡠࡏ࠶࠻ߦࠃࠆᠲࠅ⚿ᨐ

▵ߩ⚿ᨐࠍၮᧄߦߒߡ㧘෺⣨ࠍද⺞ߐߖߡ

࿑ ߦ␜ߔ : ゲ߹ࠊࠅ߅ࠃ߮ ; ゲ߹ࠊࠅߩㆇേࠍ หᦼߒߡࡊ࡟࡯࠻ߦਈ߃ߚ㧚ࡏ࡯࡞ߩォ߇ࠅߩታ

゠〔ߣࠪࡒࡘ࡟࡯࡚ࠪࡦ⚿ᨐߩᲧセ㧔௑ᢳⷺǰ 㧩q㧘๟ᦼ 6U㧘ඨᓘ4㧩O ߩ଀㧕ࠍ࿑

ߦ␜ߔ㧚ታ゠〔ߪࡊ࡟࡯࠻਄ࠍォ߇ࠆࡏ࡯࡞ࠍࡕ

࠾࠲࡯ࠞࡔ࡜ߢ᠟ᓇߒߡ࠺࡯࠲ࠍขᓧߒߚ㧚

࿑ ࠃࠅ㧘; ゲ߹ࠊࠅߩ⺋Ꮕߪᄢ߈ߊޔ: ゲ߹

ࠊࠅ߽ቢోߦኻ⒓ߥ゠〔ߣߥࠄߥ޿ߎߣ߇ࠊ߆ ࠆ㧚߹ߚォ߇ࠅ⺋Ꮕߪᦨᄢ OO ⒟ᐲߢ޽ࠆߎ ߣ߽ࠊ߆ࠆ㧚ߘߎߢࡊ࡟࡯࠻ߩේὐઃㄭߦⷺᐲ࠮

ࡦࠨ࡯ࠍขࠅઃߌ㧘ㆇേਛߩታ࠹࡯ࡉ࡞ⷺᐲࠍ⸘

᷹ߒߚ㧚⚿ᨐࠍ࿑ ߦ␜ߔ㧚; ゲ߹ࠊࠅߪ㧘ਥ ߦ࿑ ߩ ,ޔ, ߩㆇേߢ޽ࠆ߇㧘: ゲ߹ࠊࠅߩ

࿁ォㆇേߩ෻ォേ૞ᤨ㧔ਥߦ , ゲߩㆇേ㧕ߦᝄ

േ߇ടࠊࠆߥߤߣߒߡ޿ࠆ㧚ߔߥࠊߜ㧘; ゲ߹ࠊ

(7)

7LPHV

7LOWLQJDQJOHGHJUHH

X-axis circumference Y-axis circumference

(a) 2 Degree

㪄㪌 㪄㪋 㪄㪊 㪄㪉 㪄㪈

7LOWLQJDQJOHGHJUHH

7LPHV X-axis circumference Y-axis circumference

D

Degree

㪄㪍 㪄㪋 㪄㪉

7LOWLQJDQJOHGHJUHH

7LPHV

X-axis circumference Y-axis circumference

E

Degree

Fig.13 Tilting angle of working plate

ࠅߪ࿁ォᣇะߩേ೰ᕈ߇ૐߊ㧘േ⊛ߥ♖ᐲ߇ૐਅߒ ߡ޿ࠆߎߣ߇ࠊ߆ࠆ㧚৻ᣇ㧘: ゲ߹ࠊࠅ߽෻ォㆇേᓟ ߦਇ቟ቯߥᝄേ߇⷗ࠄࠇ㧘ⷺᐲߩᦨᄢߣᦨዊߩ⛘ኻ ୯ߦ⧯ᐓߩᏅ߇⷗ࠄࠇࠆ㧚ߎࠇࠄߩᓇ㗀ߦࠃࠅޔォ ߇ࠅㆇേߦ OO ⒟ᐲߩ⺋Ꮕ߇↢ߓߚ߽ߩߣ⠨߃ࠄࠇ ࠆ㧚߹ߚࡊ࡟࡯࠻ߩᠲ૞ⷺᐲࠍჇᄢߔࠆ߶ߤ㧘૏⟎

᳿߼♖ᐲ߇ᖡߊߥࠆߎߣ߽ࠊ߆ࠆ㧚

ߒ߆ߒߥ߇ࠄ㧘ࡊ࡟࡯࠻਄ߦታ⃻ߒߚࡏ࡯࡞ߩォ ߇ࠅㆇേߪ㧘ੱ߇૞ᬺߒߚ႐วߦᲧߴߡߪචಽߦ㜞

޿♖ᐲࠍ㆐ᚑߒߡ޿ࠆ㧚ߒߚ߇ߞߡ㧘ࡊ࡟࡯࠻ࠍࠛ

ࡦ࠼ࠛࡈࠚࠢ࠲࡯ߣ⠨߃ߚ႐ว㧘චಽߥ૞ᬺേ૞

ࠍታ⃻ߢ߈ߚߣ⠨߃ࠄࠇࠆ㧚 㧡㧚⚿⸒

(#㧔(CEVQT[#WVQOCVKQP㧕ߦ߅ߌࠆ෺⣨ࡠࡏ࠶

࠻ߩᣂߒ޿ᔕ↪ࠍ⋡ᜰߒߡ㧘෺⣨ߢ૞ᬺࡊ࡟࡯࠻

ࠍᡰᜬߒߡࠛࡦ࠼ࠛࡈࠚࠢ࠲࡯ߣߒߡᠲࠆᚻᴺ ࠍᬌ⸛ߒߚ㧚․ߦ૞ᬺࡊ࡟࡯࠻਄ߦ⃿ࠍォ߇ࠅㆇ

േߐߖࠆേ૞ߦขࠅ⚵ߺ㧘એਅߩ⚿ᨐࠍᓧߚ㧚 㧔1㧕 ෺⣨ߢᡰᜬߔࠆߎߣߢ㧘ᧁ⵾ߩ᭽ߥ೰ᕈ

߇Ყセ⊛ૐ޿૞ᬺࡊ࡟࡯࠻ࠍ↪޿ߡ߽㧘ࠛ

ࡦ࠼ࠛࡈࠚࠢ࠲࡯ߣߒߡචಽߥ૞ᬺⓨ㑆 ߇ౕ⃻ൻߢ߈ࠆ㧚

㧔2㧕 ᐔ㕙ߩ૞ᬺࡊ࡟࡯࠻਄ߦࡏ࡯࡞ࠍォ߇ࠅ

౞ㆇേߐߖࠆࠃ߁ߥ૞ᬺߦ߅޿ߡචಽߥ

♖ᐲ߇⏕଻ߐࠇߡ޿ࠆߎߣ߇್᣿ߒ㧘ᧄᚻ ᴺߪචಽߦታ↪ၞߦ޽ࠆߎ߇ࠊ߆ߞߚ㧚 㧔3㧕 ෺⣨ߩ㐽࡝ࡦࠢ⊛ߥද⺞೙ᓮࠍⓍᭂ⊛ߦ

೑↪ߔࠆᚻᴺߪ㧘ᰴ਎ઍߩ⥄േൻᛛⴚߣߒ ߡ᦭ᦸߢ޽ࠆߎߣ߇␜ߐࠇߚ㧚

ෳ⠨ᢥ₂

㧕 ᷦᎹື㇢㧘㆙ጊㅌਃ㧘᦯ㇱ๺਽㧘̍ࡄ࡜࡟࡞ࡔ

ࠞ ࠾ ࠭ ࡓ ᒻ ಾ ೥ ട Ꮏ ᯏ ̍㧘 ♖ ኒ Ꮏ ቇ ળ ⹹ 㧘 8QN㧘0Q㧘㧘㧚 㧕 ᧻ਅື਽㧘ᴒᔘᵗ㧘᧻ේෘ㧘̍࠹࡯ࡉ࡞࠴࡞

࠻ဳ㧡ゲ೙ᓮᎿ૞ᯏ᪾ߦࠃࠆ࠹࡯ࡄ࡯ࠦ࡯

ࡦടᎿ♖ᐲ̍㧘♖ኒᎿቇળ⹹㧘8QN㧘0Q㧘 㧘㧚

㧕 ᐔ㊁೰㧘ጊᧄరม㧘Ძ೑ᓆ㧘̍㧞บߩࡑ࠾ࡇ ࡘ࡟࡯࠲ࠍ↪޿ࠆಳኻ⽎‛ౝოߩ୮޿૞ᬺ

⸘↹̍㧘ᣣᧄᯏ᪾ቇળ⺰ᢥ㓸㧔% ✬㧕㧘8QN㧘 0Q㧘㧘㧚

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