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.
㩷
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↪ߚᬺࡊ࠻ߩᠲࠅേߣᔕ↪ߦ㑐ߔࠆ⎇ⓥ
๓ 㝵ᑝ၂ବ᮸㕍ጊᩕ৻
㧝㧚✜⸒
↢ ↥ ࠪ ࠬ ࠹ ࡓ ߩ ⁁ ߪ 㧘 ዋ ຠ ⒳ ᄙ ㊂ ↢ ↥ ߆ ࠄ (/5(NGZKDNG/CPWHCEVWTKPI5[UVGOߥߤߦઍߐࠇࠆᄙ
ຠ⒳ዋ㊂↢↥߳⒖ⴕߒߡ߈ߚ㧚ߘߩਛߢ㧘(#(CEVQT[
#WVQOCVKQPߩᛛⴚߩ㊀ⷐᕈ߇⋉ޘჇᄢߒߡࠆ㧚ߣߎࠈ ߢ㧘%0% Ꮏᯏ᪾ߩᓎഀߪᎿ‛ࠍタߖߚ࠹ࡉ࡞ߩㆇേ
ᓮߦࠃࠆ↢↥ߩᬺࠍㆀⴕߔࠆ⥄േൻᛛⴚߣ⠨߃ࠄࠇ ࠆ㧚Ꮏᯏ᪾ߩ %0% ᛛⴚൻߪ ᐕߦࠕࡔࠞ /+6 ߢౕ
ൻߐࠇ㧘ߘߩᓟߪਗㅴ㧟ゲߩㆇേࠍᓮߔࠆ㧟ゲᓮࡑ
ࠪ࠾ࡦࠣࡦ࠲ߣߒߡ⊒ዷߒ㧘ߪᢙǴO ࠝ࠳ߩ㜞
ㆇേ♖ᐲ߇ታ↪⊛ߦ㆐ᚑߐࠇߡࠆ㧚ߘߩ৻ᣇ㧘ᨵエߥ↢
↥ࠪࠬ࠹ࡓࠍౕൻߔࠆߚߦ㧘Ꮏᯏ᪾ߦ߽↥ᬺ↪ࡠࡏ
࠶࠻ߣหߓࠃ߁ߦᣓ࿁ㆇേᯏ⢻ߩౕࠍ᳞ࠆ࠾࠭߇ 㜞߹ࠅ㧘ㄭᐕߪਗㅴ㧟ゲߦ࿁ォ㧞ゲߩㆇേࠍᓮߔࠆ㧡ゲ
ᓮࡑࠪ࠾ࡦࠣࡦ࠲߇ታ↪ൻߐࠇߡ߈ߡࠆ㧚ߒ߆ߒߥ
߇ࠄ㧘ࡑࠪ࠾ࡦࠣࡦ࠲ߪࠟࠗ࠼᭴ㅧࡌࠬߦߒߚ࠹
ࡉ࡞ߩㆇേᓮߢࠆߚ㧘᭴ㅧ⊛ߦᣓ࿁ゲߩᣓ࿁
ਛᔃߩ⟎߇࿕ቯߐࠇ㧘ᬺߩ⥄↱ᐲ߇ᄢ߈ߊ㒢ߐ ࠇߡࠆ㧚ࠟࠗ࠼᭴ㅧࠍᜬߚߥࡑࠪ࠾ࡦࠣࡦ࠲
ߣߒߡߪ㧘ࡏ࡞ࡀࠫߣᄙ⥄↱ᐲߥ㕙࡚ࠫࠗࡦ࠻ࠍ
⚵ߺวࠊߖߡ࠹ࡉ࡞ࠍᡰᜬߔࠆᣇᑼ߽৻ㇱታ↪ൻߐ ࠇߡࠆ߇㧘࡚ࠫࠗࡦ࠻ߩᐓᷤߦࠃࠆࡑࠪࡦߩᄢဳൻ
߿♖ᐲᩞᱜߩਇ⏕ታߐߥߤߩ㗴ὐ߇㗼ൻߒߡ
ࠆ㧚৻ᣇ㧘ᄙ㑐▵ࡠࡏ࠶࠻ߦࠃࠆㆇേߪᬺߩ⥄↱ᐲ ߇ᄢ߈߇㧘ᡰᜬᕈ߇ૐߚߦ࠹ࡉ࡞ࠍᜬߒ㧘 ߘߩ࠹ࡉ࡞ߢᬺࠍㆀⴕߔࠆࠃ߁ߥേߦߪะ߆ ߥߣߐࠇߡ߈ߚ㧚ߒ߆ߒߥ߇ࠄㄭᐕ㧘%0% ᛛⴚߩ⊒
ዷߦࠃࠅᄙゲߩหᦼᓮᛛⴚ߇ౕൻߐࠇ㧘ᄙ㑐▵ࡠ ࡏ࠶࠻ࠕࡓࠍฦ⣨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
Fig.1 Dual-arm robot
ࠆ㧚⣨ࡠࡏ࠶࠻ߦ߅ߡ㧘⣨ߩవ┵ࠍ⚿วߔࠆߣ㐽
ࡦࠢᯏ᭴ࠍ᭴ᚑߔࠆߎߣߦߥࠅ㧘ᡰᜬᕈߩะ߇น⢻
ߢࠆ㧚⣨ߩද⺞ᬺߦࠃࠅ⦡ޘߥᬺ߇⠨߃ࠄࠇࠆ߇㧘
ᬺⓨ㑆ߦ߅ߌࠆ㓚ኂ‛࿁ㆱ߿ᨵエ‛ߩᛠᜬߐࠄߦߪ
⚻〝⸘↹ߥߤߩ⎇ⓥߪߥߐࠇߡࠆ߇㧘ᡰᜬᕈߣᄙゲ ߩද⺞ᓮ․ᕈࠍࠛࡦ࠼ࠛࡈࠚࠢ࠲ߣߒߡⓍᭂ⊛ߦᵴ↪
ߒߚ߽ߩߢߪߥ߆ߞߚ㧚
ߘߎߢᧄ⎇ⓥߢߪ㧘⣨ࡠࡏ࠶࠻ߦᬺࡊ࠻ࠍᡰᜬ ߔࠆߎߣߢ㐽ࡦࠢᯏ᭴ࠍ᭴ᚑߒߡ㧘ࠛࡦ࠼ࠛࡈࠚࠢ࠲ߣ ߒߡචಽߥᡰᜬᕈࠍ⛽ᜬߒߥ߇ࠄ㧘ᬺࡊ࠻ࠍᠲࠆ ߎߣࡊ࠻ߢᨵエߥᬺⓨ㑆ࠍታߔࠆߎߣࠍ⹜ߺ ߚ㧚ᧄႎߢߪᬺࡊ࠻ߩᡰᜬᣇᴺࠍ⠨ኤߒ㧘ߐࠄߦᡰ ᜬߒߚᬺࡊ࠻ߦ⣨ߩද⺞ᓮߦࠃࠆᣓ࿁ㆇേࠍ ਈ߃ࠆߎߣߢ㧘ࡊ࠻ߦߩߥࠄ߆ߥォ߇ࠅㆇേࠍ ታߢ߈ࠆߎߣࠍ␜ߒߚ㧚
㧞㧚ታ㛎ⵝ⟎߅ࠃ߮ᣇᴺ ⣨ࡠࡏ࠶࠻ߩၮᧄᕈ⢻
↪ߒߚࡠࡏ࠶࠻ߪᎹ㔚ᯏߩ⣨ࡠࡏ࠶࠻
/161/#0&+#㧔࿑ 㧕ߢࠆ㧚ੱߩඨりࠍࠗࡔࠫߒ ߡ㐿⊒ߐࠇ㧘ᚑੱ↵ᕈߣ߶߷หߓࠨࠗ࠭ߢࠆ㧚ฦ⣨ߦ 㑐▵ࠍᜬߜ㧘ੱߩ⣨ߣหߓࠃ߁ߥേ߈ࠍታߔࠆߎߣ߇น
⢻ߢࠆ㧚ߐࠄߦฦ㑐▵ߩേ▸࿐ߪੱߩߘࠇࠍᄢ߈ߊ
࿁ࠅ㧘ขࠅᓧࠆᆫߩ⥄↱ᐲߪੱએߢࠆ㧚⢵ߩ߭ߨ ࠅࠍ⠨߃ߚࡏ࠺ࠖᣓ࿁ゲࠍ ゲⵝߒ㧘ว⸘ ゲᓮ㧔࿑
㧝ਛߩ ,㨪,㧕ߢࠆ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&$$
㧔&QWDNG$CNN$CT㧕ࡏ࡞ࡃࠪࠬ࠹ࡓ 3% ߢࠆ㧚
ᒐ㑆ㆇേ߇น⢻ߥ 0% Ꮏᯏ᪾㧔ᧄႎߢߪ⣨ࡠ ࡏ࠶࠻ߢࠆ㧕ߦ߅ߡ㧘ࠆὐ 1㧔㧕ࠍਛᔃߣ ߔࠆඨᓘ4 ߩᒐࠍឬߊࠃ߁ߥὐ 3̈㧔:̉ޔ;̈ޔ<̉㧕ߩ⒖
േᜰࠍਈ߃ࠆ㧚ߒ߆ߒ 0% ߩᜰ୯ߣᯏ᪾ߩታ㓙ߩ
⟎ߪㆇേ⺋Ꮕߦࠃࠅ৻⥌ߒߥ㧚ߘߎߢታ㓙ߦߪ⒖േὐ ߇ὐ 3:;<ߦߞߚߣߔࠆ㧚ߘߎߢਔ⠪ߩߕࠇࠍࡌ
ࠢ࠻࡞␜ߒ ( , , )
z y
x C C
C C
o ߣߒ㧘
R2 X'2Y'2Z'2 㧔㧕 ߣߥࠆ㑐ଥ߇ࠆ㧚
⺋Ꮕࡌࠢ࠻࡞ࠍේὐ 1 ߢ 0
( 0 , 0 , 0 )
o
C 㧘ὐ 3 ߢ )
, , (Cx Cy Cz C
o ߣߔࠆߣ㧘ਛᔃ 1ߣὐ 3:;<ߩ 㑆ߩ〒㔌ߪ4 ߣߪߥࠄߕ㧘ේὐ 1 ߣ 3̈ߩ㑆ߩ〒㔌ࠍ᷹
ቯߔࠆࡃߪિ❗ߔࠆ㧚ߘߩિ❗㊂ࠍǍ4 ߣߒߡ㧘⺋Ꮕ ߩੑਸ਼㗄ࠍήⷞߔࠆߣ
, , ,
, , ,
, , , ,
, ,
, ,
,
Power Consumption Approximate Mass Repeatability
1,440 mm Vertical Reach
1,100 mm Horizontal Reach per Arm
10 kg/arm(98kgf) Payload
15(7 axes per arm and 1 rotary axis) ControlledAxes
Floor Mounting
Articulated Structure
Maximum Motion Range
Maximum Speed
Allowable Moment
Allowable Moment of Inertia
mm 1 . r0
$
$
$
$
$
$
$
$
180 100 180 130 180 120 180 180
rr rr rr rr
s s s s s s s s
/ 500
/ 250
/ 250
/ 170
/ 170
/ 170
/ 170
/ 170
$
$
$
$
$
$
$
$
Rotation-Axis (Waist) S-Axis (Lifting) L-Axis (Lower Arm) 7th-Axis (Lower Arm Twist) U-Axis (Upper Arm) R-Axis (Upper Arm Twist) B-Axis (Wrist Pitch/Yaw) T-Axis (Wrist Twist)
Turning-Axis S-Axis L-Axis 7th-Axis U-Axis R-Axis B-Axis T-Axis
R-Axis B-Axis T-Axis R-Axis B-Axis T-Axis
220kg 4.2KVA 31.4 N䊶m 31.4 N䊶m 19.6 N䊶m 1kg䊶m2 1kg䊶m2 0.4kg䊶m2
㧔a㧕
Set –up for DBB evaluation
㧔b㧕
Result of DBB experiment Fig.2 Motion accuracy of robot arm
Ball Z
Y X
Working plate Ball
Z
Y X
Working plate
Fig.3 Definition of coordinate in supporting working plate
)
1( ' ' '
Z y
x YC ZC
C R X
R
' 㧔㧕
ߣߥࠆ㧚ᑼ㧔㧕ߩේℂߦࠃࠅ㧘&$$ ᴺߢ :; ᐔ㕙㧘ඨ ᓘ OO ߢㆇേߒߚᤨߩㆇേ⺋Ꮕࠍ᷹ቯߒߚ㧚࿑ 㧔C㧕 ߪ᷹ቯߩ᭽ሶߢ㧘࿑ 㧔D㧕ߪ᷹ቯ⚿ᨐߢࠆ㧚
ߎߩࡠࡏ࠶࠻ߩૐㅦᦨዊ⋥✢ㆇേߩࠞ࠲ࡠࠣ♖ᐲߪ ǴO ߢ㧘㜞ㅦㆇേߔࠆᤨߩࠛࡦ࠼ࠛࡈࠚࠢ࠲ߩ⺋
Ꮕߪ OO ߣߐࠇߡࠆ㧚࿑ 㧔D㧕ㅍࠅ OOU ߩㅦ ᐲ㧔ੱ߇ߔ߫߿ߊേࠍⴕߞߚᤨ⒟ᐲߩㅦᐲ㧕ߢ :;
ᐔ㕙ࠍㆇേߐߖߚ႐วߩหᦼ♖ᐲߪ㧘ห࿑ࠃࠅᦨᄢߢ OO ⒟ᐲߢࠆߎߣ߇ࠊ߆ߞߚ㧚ߔߥࠊߜ㧘ᄙ㑐▵
ဳߩ↥ᬺ↪ࡠࡏ࠶࠻ߣߒߡߪචಽߥ♖ᐲࠍߒߡࠆ ߎߣ߇ࠊ߆ࠆ㧚
ᬺࡊ࠻ߩᡰᜬᣇᴺߣᕈ
ੱ߇ᬺࡊ࠻ࠍᠲߔࠆᆫࠍෳ⠨ߒߡ㧘ᧁࡊ
࠻㧔OO㧕ࠍᡰᜬߒߚࡠࡏ࠶࠻ߩ᭽ሶߣ ᐳᮡ♽ࠍ࿑ ߦ␜ߔ㧚
ࡊ࠻ߩࠆὐߩ < ᣇะߩ⩄㊀ࠍ2㧘ᄌࠍǬߣ ߔࠆߣછᗧߩ⩄㊀ὐ K ߣᄌὐ L ߦ߅ߌࠆᕈ㨗ߪ P
[
N/
m]
k
j i
ij
P
G
㧔㧕ߣߥࠅ㧘ߘߩࠦࡦࡊࠗࠕࡦࠬ
Kij㧩
kijߣߐࠇࠆ㧚
ij ij k
K 1 㧔㧕
ߘߎߢ㧘ᄌࡌࠢ࠻࡞ ( , , , )
6 5 4
3 U U U
U U
o 㧘ജࡌࠢ࠻࡞
) , , ,
(P3 P4 P5 P6 P
o ߣߔࠆߣ㧘⣨ᡰᜬߐࠇߚࡊ࠻ߩᄌ
ߪએਅߩᑼࠍḩ⿷ߔࠆ㧚
) 6 ...
3 , (
6 5 4 3
66 56 46 36
65 55 45 35
64 54 44 34
63 53 43 33
»»
»
¼ º
««
«
¬ ª
»»
»
¼ º
««
«
¬ ª
o
j i
P P P P
K K K K
K K K K
K K K K
K K K K
Ui 㧔㧕
ߎߎߢ㧘࿑ ߦ␜ߔฝ⣨ߩࡊ࠻ᡰᜬὐࠍࡊ࠻
ᐳᮡߩ⛘ኻේὐߣߒߡ㧘KL㧩 ߪ㧔㧕㧘KL㧩 ߪ㧔㧕㧘KL㧩 ߪ㧔㧕㧘KL㧩 ߪ㧔㧕 ߩ⟎ࠍ␜ߔ㧚ᰴߦ㧘ห᭽ߦߒߡ ⣨㧔Ꮐ⣨ߛߌᡰᜬ㧕 ߩ႐วߪ
o o
o o
P K U
j i P P P
U U U
ij j i
) 6 ...
1 , ( ), ,...
(
) ,....
(
6 1
6 1
㧔㧕
ߎߎߢ KL㧩 ߪ㧔㧕㧘KL㧩 ߪ㧔㧕ߢࠆ㧚
ታ㓙ߦ⩄㊀ࠍട߃ߡ-KLࠍ᷹ቯߒߚᑼ㧔㧕߅ࠃ߮ᑼ㧔㧕 ߦ߅ߌࠆ⣨ᡰᜬߣ ⣨ᡰᜬߩᕈߩઍ୯ߣߒߡ -KLߩ ୯ࠍ࿑㧠ߦ␜ߔ㧚࿑㧠ࠃࠅ㧘 ⣨ᡰᜬߢߪ ; ߩ⟎ߢ Ǵ㨙0 ࠍ߃ࠆ୯ࠍ␜ߒߡࠆߎߣ߇ࠊ߆ࠆ㧚৻ᣇ㧘ਔ⣨
ᡰᜬߢߪ ; ߩ⟎ߩ ǴO0 એਅߩ୯ߢ એߩ 㜞ᕈࠍߒߡࠆߎߣ߇ࠊ߆ࠆ㧚৻⥸⊛ߥන⚐ߥ ᜬ ߜߪࠅߣਔ┵ᡰᜬߪࠅ㧘ਔ┵࿕ቯߪࠅࠍ⠨߃ࠆߣ㧘ਔ⠪ߩ
ᕈߪ 㧘 ߦߥࠆ㧚ߒߚ߇ߞߡ㧘ਔ┵ᡰᜬߦㄭ
ᕈߩะߢࠆ㧚એࠃࠅ㧘⣨ᡰᜬߦߔࠆߎߣߢ㧘Ყ セ⊛ᕈ߇ૐᧁࡊ࠻ߢ߽චಽߦᕈߩ㜞ᬺ
ⓨ㑆߇ታߢ߈ߚߎߣ߇ࠊ߆ࠆ㧚
㧟㧚ォ߇ࠅㆇേߩࠪࡒࡘ࡚ࠪࡦ 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ゲ ߪࡊ࠻ߣߩធ⸅㕙ߦ⋥ߔࠆ㧕㧘B m B
mX , Y ߪ
6
m߆ࠄߚ
6
B⟎㧘ǰߪࡊ࠻㨥ゲ߹ࠊࠅߩ᳓ᐔߦ ኻߔࠆ࿁ォⷺ㧔㧯㨃ᣇะࠍᱜ㧕ߣߔࠆ㧚ࡊ࠻ߩⷺᐲǰࠍᤨ㑆㨠ߩ㑐ᢙǰVߣߒ㧘ࡏ࡞߇ࡊ࠻
ࠍォ߇ࠆㆇേᣇ⒟ᑼߪㅦᐲߦᲧߒߚᷫ㗄ࠍട ߃ߡᑼ㧔㧕ߣߒߡ⠨߃ࠆ㧚ᷫଥᢙ &㧘ࡏ࡞ߩ⾰㊂ ࠍ/ ߣߒߡ㧘&/㧩&/ ߢࠆ㧚
0 7 sin
5 x
x
xX D mXB g T
B M
m 㧔㧕
ߎߎߢ㧘:; ᐔ㕙ߦ߅ߌࠆォ߇ࠅㆇേߪ㧘<: ᐔ㕙߹
ߚߪ <; ᐔ㕙߆ࠄࠆߣ㧘ߘࠇߙࠇߦォ߇ࠅߩනᝄേߔࠆ ߣ⠨߃ࠄࠇࠆ㧚ߘߎߢ㧘ታ㓙ࡠࡏ࠶࠻ߢࡊ࠻ࠍᠲߔ ࠆᤨߪ : ゲ߹ࠊࠅ߅ࠃ߮ ; ゲ߹ࠊࠅߢ࿁ォⷺǰVߩ⋧
ߪ ᐲߕࠄߒߡߚേ߈ࠍ⚵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ࠍߞߚォ߇ࠅᝄࠅሶߣߒߚ㧚ೋᦼ⟎㨔
3RVLWLRQRI< PP .LMP1
'XDODUP[
'XDODUP[ PP 2QHDUP[
2QHDUP[ PP
Fig.4 Compliance of working plate
z
mFrictional force
Gravitational force
¦
mT
X
mX
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
㪄㪈㪌㪇 㪄㪈㪇㪇 㪄㪌㪇
㪇 㪌㪇 㪈㪇㪇 㪈㪌㪇
㪇 㪈 㪉 㪊 㪋 㪌 㪍 㪎 㪏 㪐
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
㪇 㪈 㪉 㪊 㪋 㪌 㪍 㪎 㪏 㪐 㪈㪇
㪇 㪉 㪋 㪍 㪏 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 ߢᚻࠍߒߡ㧘㨔㧞㧘㨔㧟㨔㨚ࠍ᷹ቯ ߒߚ㧚࿑ ߪ㧘᷹ቯߒߚ㜞ߐߣ╴㕙ߦᴪߞߚᒐ 㐳ߐߩᝄߢࠆ㧚ᒐ㐳ߐߩฦᝄ DPࠍᑼ㧔㧕㧘 ᑼ㧔㧕ߦઍߒߡ
' ' 0
ln
01
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ǮߪᷫᲧ㧘MM㧩
/I㧔4T㧕㧘D㧘D̈㧚㧚㧚ߪᝄ㧘& ߪᷫଥᢙߢ
ࠆ㧕⸘▚ߒߚᷫଥᢙ& ߪ ߢࠆ㧚ߎߎߢ㧘ߎ ߩォ߇ࠅᝄࠅሶߩℂ⺰ᦼߪ 㨟ߢࠅ㧘ታ㛎ߢ
᷹ቯߐࠇߚᦼߪ 㨟ߢ㧘ਔ⠪ߪ߶߷৻⥌ߒߡࠆ㧚 ߒߚ߇ߞߡ㧘ᝄേㆇേਛߪߔߴࠆߎߣߥߊォ߇ࠅㆇേ
ࠍߒߡߚ߽ߩߣ⠨߃ࠄࠇࠆ㧚
ォ߇ࠅㆇേߣ࠹ࡉ࡞ᢳⷺᐲߩ⸳ቯ ታ㛎ߦ↪ߚࡑ࠙ࠬࡏ࡞㧔⋥ᓘ OO㧘⾰㊂ 㨓㧕ߣォ߇ࠅ㕙ߩᡂࠍ৻ቯߦߔࠆߚߦ㧘ࠕ
ࠢ࡞᧼ࠍᡰᜬࡊ࠻ߩߦ࿕ቯߒߚ㧚࠹ࡉ
࡞ߩᢳⷺǰ߇৻ቯߩ႐วߦߪ㧘ೋᦼㅦᐲࠍࡠ 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
ga 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ᢳⷺ߇ዊߐ႐วߪࡏ࡞߇Ⱜⴕ ㆇേߔࠆߥߤߒߡ㧘ਔ⠪ߪ৻⥌ߥߎߣ߇ࠊ߆ ࠆ㧚ߒ߆ߒߥ߇ࠄ㧘ᢳⷺ ᐲએߢߪቯߒߚ ォ߇ࠅㆇേߦࠃࠅਔ⠪ߪ৻⥌ߒߚ㧚ߘߎߢᧄႎߢ ߪ㧘ᧄ⚿ᨐࠍ⋡ߦߒߡ࠹ࡉ࡞ߩᦨᄢᢳⷺߪ qએࠍ↪ࠆߎߣߦߒߚ㧚
㧠㧚⚿ᨐ߅ࠃ߮⠨ኤ
නᝄേࡕ࠺࡞ߩࠪࡒࡘ࡚ࠪࡦ
ᑼ㧔㧕ߦ߅ߡ㧘ᷫ㗄ࠍࡠ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⸘▚ߒߚߢࠆ㧚ᤨ㑆ߦᲧߒߡᝄ
േߩਛᔃ߇ : ᣇะߦ⒖േߔࠆ᭽ሶ߇ࠊ߆ࠆ㧚߹ߚ․ᱶ
⸃ࠃࠅᝄേߩ ᝄࠍ᳞ࠆߣ 㧔㨙㧕ߣߥࠆߎߣ
߽ࠊ߆ߞߚ㧚ߔߥࠊߜ㧘ᷫ㗄߇ࡠߢߪ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
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߇ࠅㆇേࡕ࠺࡞ߩࠪࡒࡘ࡚ࠪࡦ ▵ࠃࠅᷫ㗄㗄߇ᄢಾߥᓎഀߦߥࠆߎߣ ߇ࠊ߆ߞߚ㧚․ߦᷫ㗄ߩ୯߇ዊߐᤨ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゠〔ࠍታߢ߈ ࠆ߇㧘߃߫㧘: ゲ߹ࠊࠅߩㆇേਛᔃߩ⒖േ߇ᱷ ߞߡࠆߣߩ゠〔ߦ᧤ߒߥ㧚
ߘߎߢ㧘 ▵ߩታ㛎ߢ᷹ቯߒߚᷫ㗄ߩ୯ࠍᑼ㧔㧕 ߦઍߒߡ⸘▚ߒߚ㧚⚿ᨐ߇࿑ 㧔D㧕ߢࠆ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
゠〔ߣࠪࡒࡘ࡚ࠪࡦ⚿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
࠾࠲ࠞࡔߢᓇߒߡ࠺࠲ࠍขᓧߒߚ㧚
࿑ ࠃࠅ㧘; ゲ߹ࠊࠅߩ⺋Ꮕߪᄢ߈ߊޔ: ゲ߹
ࠊࠅ߽ቢోߦኻ⒓ߥ゠〔ߣߥࠄߥߎߣ߇ࠊ߆ ࠆ㧚߹ߚォ߇ࠅ⺋Ꮕߪᦨᄢ OO ⒟ᐲߢࠆߎ ߣ߽ࠊ߆ࠆ㧚ߘߎߢࡊ࠻ߩේὐઃㄭߦⷺᐲ
ࡦࠨࠍขࠅઃߌ㧘ㆇേਛߩታ࠹ࡉ࡞ⷺᐲࠍ⸘
᷹ߒߚ㧚⚿ᨐࠍ࿑ ߦ␜ߔ㧚; ゲ߹ࠊࠅߪ㧘ਥ ߦ࿑ ߩ ,ޔ, ߩㆇേߢࠆ߇㧘: ゲ߹ࠊࠅߩ
࿁ォㆇേߩォേᤨ㧔ਥߦ , ゲߩㆇേ㧕ߦᝄ
േ߇ടࠊࠆߥߤߣߒߡࠆ㧚ߔߥࠊߜ㧘; ゲ߹ࠊ
7LPHV
7LOWLQJDQJOHGHJUHH
X-axis circumference Y-axis circumference
(a) 2 Degree
㪄㪌 㪄㪋 㪄㪊 㪄㪉 㪄㪈 㪇 㪈 㪉 㪊 㪋 㪌
㪇 㪉 㪋 㪍 㪏
7LOWLQJDQJOHGHJUHH
7LPHV X-axis circumference Y-axis circumference
D
Degree
㪄㪍 㪄㪋 㪄㪉 㪇 㪉 㪋 㪍
㪇 㪉 㪋 㪍 㪏
7LOWLQJDQJOHGHJUHH
7LPHV
X-axis circumference Y-axis circumference
E
Degree
Fig.13 Tilting angle of working plate
ࠅߪ࿁ォᣇะߩേᕈ߇ૐߊ㧘േ⊛ߥ♖ᐲ߇ૐਅߒ ߡࠆߎߣ߇ࠊ߆ࠆ㧚৻ᣇ㧘: ゲ߹ࠊࠅ߽ォㆇേᓟ ߦਇቯߥᝄേ߇ࠄࠇ㧘ⷺᐲߩᦨᄢߣᦨዊߩ⛘ኻ ୯ߦ⧯ᐓߩᏅ߇ࠄࠇࠆ㧚ߎࠇࠄߩᓇ㗀ߦࠃࠅޔォ ߇ࠅㆇേߦ OO ⒟ᐲߩ⺋Ꮕ߇↢ߓߚ߽ߩߣ⠨߃ࠄࠇ ࠆ㧚߹ߚࡊ࠻ߩᠲⷺᐲࠍჇᄢߔࠆ߶ߤ㧘⟎
♖ᐲ߇ᖡߊߥࠆߎߣ߽ࠊ߆ࠆ㧚
ߒ߆ߒߥ߇ࠄ㧘ࡊ࠻ߦታߒߚࡏ࡞ߩォ ߇ࠅㆇേߪ㧘ੱ߇ᬺߒߚ႐วߦᲧߴߡߪචಽߦ㜞
♖ᐲࠍ㆐ᚑߒߡࠆ㧚ߒߚ߇ߞߡ㧘ࡊ࠻ࠍࠛ
ࡦ࠼ࠛࡈࠚࠢ࠲ߣ⠨߃ߚ႐ว㧘චಽߥᬺേ
ࠍታߢ߈ߚߣ⠨߃ࠄࠇࠆ㧚 㧡㧚⚿⸒
(#㧔(CEVQT[#WVQOCVKQP㧕ߦ߅ߌࠆ⣨ࡠࡏ࠶
࠻ߩᣂߒᔕ↪ࠍ⋡ᜰߒߡ㧘⣨ߢᬺࡊ࠻
ࠍᡰᜬߒߡࠛࡦ࠼ࠛࡈࠚࠢ࠲ߣߒߡᠲࠆᚻᴺ ࠍᬌ⸛ߒߚ㧚․ߦᬺࡊ࠻ߦࠍォ߇ࠅㆇ
േߐߖࠆേߦขࠅ⚵ߺ㧘એਅߩ⚿ᨐࠍᓧߚ㧚 㧔1㧕 ⣨ߢᡰᜬߔࠆߎߣߢ㧘ᧁߩ᭽ߥᕈ
߇Ყセ⊛ૐᬺࡊ࠻ࠍ↪ߡ߽㧘ࠛ
ࡦ࠼ࠛࡈࠚࠢ࠲ߣߒߡචಽߥᬺⓨ㑆 ߇ౕൻߢ߈ࠆ㧚
㧔2㧕 ᐔ㕙ߩᬺࡊ࠻ߦࡏ࡞ࠍォ߇ࠅ
ㆇേߐߖࠆࠃ߁ߥᬺߦ߅ߡචಽߥ
♖ᐲ߇⏕ߐࠇߡࠆߎߣ߇್ߒ㧘ᧄᚻ ᴺߪචಽߦታ↪ၞߦࠆߎ߇ࠊ߆ߞߚ㧚 㧔3㧕 ⣨ߩ㐽ࡦࠢ⊛ߥද⺞ᓮࠍⓍᭂ⊛ߦ
↪ߔࠆᚻᴺߪ㧘ᰴઍߩ⥄േൻᛛⴚߣߒ ߡᦸߢࠆߎߣ߇␜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ᓮᎿᯏ᪾ߦࠃࠆ࠹ࡄࠦ
ࡦടᎿ♖ᐲ̍㧘♖ኒᎿቇળ㧘8QN㧘0Q㧘 㧘㧚
㧕 ᐔ㊁㧘ጊᧄరม㧘Ძᓆ㧘̍㧞บߩࡑ࠾ࡇ ࡘ࠲ࠍ↪ࠆಳኻ⽎‛ౝოߩ୮ᬺ
⸘↹̍㧘ᣣᧄᯏ᪾ቇળ⺰ᢥ㓸㧔% ✬㧕㧘8QN㧘 0Q㧘㧘㧚
㧕 ዊ⩲৻ᒄ㧘ศ↰⧷ඳ㧘↰ᢅ↵㧘Ⳝ⼱ᷡ㧘㈬
ൎ㧘̍⣨ࡑ࠾ࡇࡘ࠲ߦࠃࠆ⭯᧼ߩࡑ
࠾ࡇࡘ࡚ࠪࡦ̍㧘ᣣᧄᯏ᪾ቇળ⺰ᢥ㓸㧔%
✬㧕㧘8QN㧘0Q㧘㧘㧚
㧕 ฎᎹ⍮ᚑ㧘᧘ኅੳ㧘/9/)࠺ࠖࠨ࠽ࡗ
ࠞ㧘̍౬㐳⣨ࡑ࠾ࡘࡇ࠲♽ߩḰᦨ⍴ᤨ
㑆゠⸘↹̍㧘ᣣᧄࡠࡏ࠶࠻ቇળ㧘8QN㧘 0Q㧘㧘㧚