18 A Study on the Running and Jumping by a One-leg Robot ( )
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- しょうり ひがき
- 4 years ago
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1 18 A Study on the Running and Jumping by a One-leg Robot ( )
2 CAD CAD
3 A...41 A A CAD...46 A B...49 C...50
![6[km/h] 1.](/docs-images/99/140888400/images/4-2.jpg "1 5) Fig.1.1 The")
4 Raibert 1) Raibert 1996 P ) QRIO 3) ASIMO 4) [km/h] [km/h] 1.1 5) Fig.1.1 The difference between a robot (top) and a human (bottom) in running 3
5 1. ASIMO 1. ( ) 6) Fig.1.Motion of jumping 1 (left) and jumping (right) 1.3 ZMP(Zero Moment Point) 4
6 loss of contact. bent knee Fig..1 Walking (left) and running (right) half cycle 7) Fig.. Motion of Runningone cycle 8) 5
7 ..0.5[m/s]7.9.0[km/h] 9) [m/s](43.38[km/h]) [m/s]0.7[km/h].0 1[m/s] 6.0[m/s] Fig..3Velocity curve of 100 meter dash 10) 6
8 3 CAD ( A ) 3.1 CAD MSC.visualNastran 4D 003( vn4d MATLAB6.5 Simulink5.0 vn4d Nastran Simulink vn4d 3.1 Simulink vn4d S-Function C Control LawvN4D Model Fig. 3.1Simulink model 3.1.CAD Pro/ENGINEER Pro/E Pro/E vn4d vn4d 3 Y Y 3.3 0[deg] -9090[deg] [deg] 7
![Z Z 0.08[m] X Y X Y Knee Joint 0.36[m] Ankle Joint 0.](/docs-images/99/140888400/images/9-0.jpg "39[m] 0.05[m] Fig. 3.CAD model 0.185 [m] Table 3.")
9 Z Z 0.08[m] X Y X Y Knee Joint 0.36[m] Ankle Joint 0.39[m] 0.05[m] Fig. 3.CAD model [m] Table 3.1 Specification of CAD Model Number Parts Quantity Weight[kg] Motor and Gear Thigh Shin Foot Pulley Ankle Shaft Knee Shaft Set Collar TOTAL 4.74 Fig. 3.3Joint of plus and minus 8
10 3.1.3 vn4d Fig. 3.4Gravity Fig. 3.5Permitted value [m/s ] 3.5 Fig. 3.6 Integration time [s] Kutta-Merson 3.6 Euler Kutta-Merson Euler Kutta-Merson 9
![3.1.4 Fig. 3.7 Control block (Simulink model) 3.7 vn4d vn4d 15[Nm] 0.0[s] γ 15[Nm] CAD γ 11) 3.](/docs-images/99/140888400/images/11-1.jpg "8 γ a L a = L1 sinγ 1 + L sin( γ1 + γ ) 3-1 γ γ b b = L1 cosγ 1 + L cos( γ1 + γ ) γ = tan 1 a ( ) b 3-3-3 γ 1 L 1 Fig.")
11 3.1.4 Fig. 3.7 Control block (Simulink model) 3.7 vn4d vn4d 15[Nm] 0.0[s] γ 15[Nm] CAD γ 11) 3.8 γ a L a = L1 sinγ 1 + L sin( γ1 + γ ) 3-1 γ γ b b = L1 cosγ 1 + L cos( γ1 + γ ) γ = tan 1 a ( ) b γ 1 L 1 Fig. 3.8 Definition of Gamma 10
![3.1.5 10 0 30 35[deg]](/docs-images/99/140888400/images/12-0.jpg "(γ=10[deg]) (γ=0[deg])")
![(γ=30[deg]) (γ=35[deg]) Fig.](/docs-images/99/140888400/images/12-1.jpg "3.9 Motion of jumping Table")
![3.9 γ 1035[deg] 3.](/docs-images/99/140888400/images/12-3.jpg "X Z (15[Nm] ) W z γ=10")
12 [deg] (γ=10[deg]) (γ=0[deg]) (γ=30[deg]) (γ=35[deg]) Fig. 3.9 Motion of jumping Table 3.Result of Simulation γ[deg] X[m] Z[m] W z [deg/s] 3.9 γ 1035[deg] 3. X Z (15[Nm] ) W z γ=10 0[deg]γ Thigh X Z 0 γ 30 35[deg] 11
![14[deg] 3.](/docs-images/99/140888400/images/13-1.jpg ".1 A Z Z X Y X Y Knee Joint")
13 3.1.6 γ W z γ Y Z [deg/s]γ 10[deg] 0[deg] γ 14[deg] 3..1 A Z Z X Y X Y Knee Joint Ankle Joint Fig. 3.10One-leg Robot 1
14 Table 3.3 Specification of One-leg Robot Number Parts Quantity Weight[kg] Motor and Gear 0.81 Thigh Shin Foot Pulley Ankle Shaft Knee Shaft Set Collar Total 5.1 Fig. 3.11Joint of plus and minus [kg] 18[V] ±40[V]±15A[V] BWS40-15 PC CPU:Pentium4.8[GHz] Memory:51MBAD DAPIO OS RedHat9 RTLinux(.4) AD RTLinux PC DA 18[V] ±10[A] [deg] PC PotentiometerMotor P M PowerSupply RTLinux DA AD LED PIO Fig. 3.1Control System 13
15 3..3 Fig.3.13 Motion of jumping18[v] Fig.3.14Ankle joint angle18[v] 14
![4[deg/s]](/docs-images/99/140888400/images/16-1.jpg "300[deg/s] 0.")
![06[] 0.](/docs-images/99/140888400/images/16-2.jpg "15[] 15[Nm]")
![30[V] 3.](/docs-images/99/140888400/images/16-3.jpg ".4 .5 Fig.3.")
16 γ 14[deg] 4[deg/s] 300[deg/s] 0.06[] 0.15[] 15[Nm] 30[V] [V] 3..5 Fig.3.15 Motion of jumping30[v] [deg/s] 300[deg/s] 0.10[] 0.4[] B 30[V] Z 15
![Fig.3.16Ankle joint angle30[v] 3.](/docs-images/99/140888400/images/17-0.jpg ".6 γ 14[deg] 3.")
17 Fig.3.16Ankle joint angle30[v] 3..6 γ 14[deg] 3..7 Fig. 3.17Motion of jumping (γ=14[deg]) 16
18 Fig. 3.18Ankle joint angle (γ=14[deg]) [V] 30[V] 0.13[m] 0.55[m] 17
19 4 4.1 Z X Y θ M,J L β L 1 α α 1 B A Fig. 4.1Motion of model Fig. 4.Simple model B X Z Z F z Mx& Mx & 1 = F dt x (4-1) Mz& Mz = F dt (4-) & 1 z θ & & θ 1,θ J & θ J & θ1 = L cosα F dt L sinα F dt (4-3) z x 18
20 (4-1) (4-) (4-3) J & θ & ( & & & & α (4-4) Jθ1 = Mz Mz1) L cosα ( Mx Mx1 ) L sin θ x, z B x & = & θ L sinα, z& = & θ L cosα (4-4) ( J + ML & & & α + J & θ (4-5) ) θ = Mz1L cosα + Mx1L sin 1 θ 1 x 1, z 1 A x & 1 = & θ1l1 sinα1, z& 1 = & θ1l1 cosα1 (4-5) ( J + ML & + & θ (4-6) ) θ = ( ML1 L cosα cosα1 + ML1L sinα sinα1 J ) 1 (4-6) & θ & θ ML1 L cos( α α1) = 1 J + ML + J (4-7) & θ 1,θ & x & 1, x& x& x& ML1 L cos( α α1) + J L1 sinα1 ML1 L cos( α α1) = ( ) = 1 J + ML L sinα J + ML + J (4-8) (4-7) (4-8) X L sinα L sinα β=0[deg] A B α =90[deg] α 90[deg] X X β=0deg 19
![4. 1 (XYZ) X Z 4.3 XZ MSC.visualNastran 4D 003 1 ( ) (J 1 ) (J 3 ) ( ) 4.1 (J 1 ) (J ) (J 3 ) (J 4 ) J 4 J 1 4[Nm]J J 3 15 [Nm] (J 4 )030[deg] (J 3 )-9090[deg] (J )-180180[deg] (J 1 )-180180[deg] 4.](/docs-images/99/140888400/images/21-1.jpg "4 0[deg] 6 4.5 =0.")
21 4. 1 (XYZ) X Z 4.3 XZ MSC.visualNastran 4D ( ) (J 1 ) (J 3 ) ( ) 4.1 (J 1 ) (J ) (J 3 ) (J 4 ) J 4 J 1 4[Nm]J J 3 15 [Nm] (J 4 )030[deg] (J 3 )-9090[deg] (J ) [deg] (J 1 ) [deg] 4.4 0[deg] =0.001[Nm/deg] -K [deg] C=0[Nms/deg] -C [deg/s] Z Mass Thigh J 1 Jumping Leg Supporting Leg X φ 1 Y J Toe φ Shin L φ 3 Leg Z J 4 Instep J 3 Ball Boot X Y (Side view) (Birds-eye view) Fig. 4.3Mathematical model 0
22 J 1 Toe φ J 4 J φ φ Instep J 3 Fig. 4.4Joint of plus and minus Fig. 4.5 Standing posture and feet bottom Ball Table 4.1 Specification of mathematical model Fig. 4.6Mathematical model footless Parts Shape Size(x y z)[m] Weight[kg] Mass RS Thigh RS Shin RS Instep RS Toe RS Ball S R Leg RS Boot C R0.01H Coefficient of friction 0.4 Coefficient of restitution 0.1 R: radiush: heightrs: rectangular solids: sphere C: cylinder 1
![4.3 4. 4.3.1 4.1 L 1 =L=0.7[m] (J 1 ) φ 1 =60[deg] (J ) φ =-50[deg] (J 3 ) 0.](/docs-images/99/140888400/images/23-0.jpg "15[m] J 3 φ 3 10~70[deg] L φ 3 J 1 0[Nm]J J 3 0[Nm] Toe InstepBall 4.6 4.3. 4.7 4.9 4.10 Mass X 4.")
![φ 3 A B stride α 90[deg] X V x /V x1 KE /KE 1 1 φ 3 =10[deg] α φ 3 =0[deg] φ 3 =60[deg] α V x /V x1 KE /KE](/docs-images/99/140888400/images/23-1.jpg "1 φ 3 =0[deg] 30[deg] 0[deg] 40[deg] X 40[deg] 90[deg]α Z X Y Fig. 4.7Behavior of model (φ 3 =10[deg]) Fig.")
23 L 1 =L=0.7[m] (J 1 ) φ 1 =60[deg] (J ) φ =-50[deg] (J 3 ) 0.15[m] J 3 φ 3 10~70[deg] L φ 3 J 1 0[Nm]J J 3 0[Nm] Toe InstepBall Mass X 4. φ 3 A B stride α 90[deg] X V x /V x1 KE /KE 1 1 φ 3 =10[deg] α φ 3 =0[deg] φ 3 =60[deg] α V x /V x1 KE /KE 1 φ 3 =0[deg] 30[deg] 0[deg] 40[deg] X 40[deg] 90[deg]α Z X Y Fig. 4.7Behavior of model (φ 3 =10[deg]) Fig. 4.8Result of simulation 1 (footless)
![(φ 3 =0[deg]) (φ 3 =30[deg]) (φ 3 =40[deg]) (φ 3 =50[deg]) (φ 3 =60[deg]) (φ 3 =70[deg]) Fig. 4.9Result of simulation 1 Table 4.](/docs-images/99/140888400/images/24-0.jpg "Ratio of the velocity and angle α φ 3 L [m] stride [m] α [deg] Vx /Vx 1 KE /KE 1 10 0.771 0.360 66.986 0.51 0.37 0 0.755 0.331 85.79 0.847 0.789 30 0.79 0.187 81.71 0.961 0.889 40 0.737 0.14 78.087 0.")
24 (φ 3 =0[deg]) (φ 3 =30[deg]) (φ 3 =40[deg]) (φ 3 =50[deg]) (φ 3 =60[deg]) (φ 3 =70[deg]) Fig. 4.9Result of simulation 1 Table 4.Ratio of the velocity and angle α φ 3 L [m] stride [m] α [deg] Vx /Vx 1 KE /KE ToeInstepBall α stride φ 3 =40[deg] X V x /V x1 KE /KE 1 φ 3 =40[deg] 1 3
25 Fig.4.10Velocity of X direction [deg] Boot Mass 4
![15[Nm] J 1 4.](/docs-images/99/140888400/images/26-1.jpg "3.1 4.")
![4. (φ 3 =0[deg]) (φ 3](/docs-images/99/140888400/images/26-2.jpg "=30[deg]) (φ 3 =40[deg]) Fig.")
26 φ 3 0[deg]30[deg]40[deg] J 3 15[Nm] J (φ 3 =0[deg]) (φ 3 =30[deg]) (φ 3 =40[deg]) Fig. 4.11Motion of running 1 Fig. 4.1Velocity of X direction 5
27 Mass X J 3 0[Nm] Mass J 3 J 3 15[Nm] 1.75[m/s] 0.9[m/s] 0.48[m] F (4-9) F = (4-9) F=1 F=1 F=1 0.7[m] 0.9[m/s] F=0.34 F=1.66[m/s] 4.3 φ 3 =30[deg] Vx Vz Mass φ 3 =40[deg] Vz Mass Lx φ 3 =30[deg] Table 4.3Mass of velocity and distance φ 3 [deg] Vx[m/s] Vz[m/s] Jx[m] Jz[m] Lx[m] Lz[m]
28 Mass 0[Nm] ( ) 1) J3 φ 3 10~70[deg] L φ 3 J1 0[Nm]J J (φ 3 =0[deg]) (φ 3 =30[deg]) (φ 3 =40[deg]) (φ 3 =50[deg]) (φ 3 =60[deg]) (φ 3 =70[deg]) Fig. 4.13Result of simulation 7
29 Fig. 4.14Velocity of X direction Table 4.4Ratio of the velocity and angle α φ 3 L [m] stride [m] α [deg] Vx /Vx 1 KE /KE φ 3 =10[deg] Mass X 8
30 φ 3 =30[deg] ( ) ) ZMP φ 3 =30[deg] 4.16 Mass Fig. 4.15Alteration of rotation center 7) Fig. 4.16Movement of fulcrum (φ 3 =30[deg]) 9
![view) (Birds-eye view) Fig. 4.17Mathematical model 0.6[m] 0.](/docs-images/99/140888400/images/31-3.jpg "05[] 0.01[] Fig. 4.18Foot 1 4.17 1[kg]Toe Instep 4.")
31 Z Mass J 1 X Thigh φ 1 Jumping Leg Support Leg Y J φ Shin L J 3 φ 3 Leg Z Foot Ball Boot X Y (Side view) (Birds-eye view) Fig. 4.17Mathematical model 0.6[m] 0.05[] 0.01[] Fig. 4.18Foot [kg]Toe Instep 4.18 Ball R 30
![4.6. 4.17 Foot 4.5.1 J3 φ 3 0~40[deg] φ 3 J1 0[Nm]J J3 4.18 R 0.4~10[m] 4.6.3 (R=0.4[m]φ 3 =0[deg]) (R=0.6[m]φ 3 =0[deg]) (R=0.8[m]φ 3 =0[deg]) (R=1[m]φ 3 =0[deg]) (R=0.](/docs-images/99/140888400/images/32-0.jpg "4[m]φ 3 =30[deg]) (R=0.6[m]φ 3 =30[deg]) (R=0.8[m]φ 3 =30[deg]) (R=1[m]φ 3 =30[deg]) (R=0.4[m]φ 3 =40[deg]) (R=0.6[m]φ 3 =40[deg]) (R=0.8[m]φ 3 =40[deg]) (R=1[m]φ 3 =40[deg]) Fig. 4.")
32 Foot J3 φ 3 0~40[deg] φ 3 J1 0[Nm]J J R 0.4~10[m] (R=0.4[m]φ 3 =0[deg]) (R=0.6[m]φ 3 =0[deg]) (R=0.8[m]φ 3 =0[deg]) (R=1[m]φ 3 =0[deg]) (R=0.4[m]φ 3 =30[deg]) (R=0.6[m]φ 3 =30[deg]) (R=0.8[m]φ 3 =30[deg]) (R=1[m]φ 3 =30[deg]) (R=0.4[m]φ 3 =40[deg]) (R=0.6[m]φ 3 =40[deg]) (R=0.8[m]φ 3 =40[deg]) (R=1[m]φ 3 =40[deg]) Fig. 4.19Result of simulation 3 31
33 Fig. 4.0Velocity of X direction Table 4.5Ratio of the velocity and angle α R[m] φ 3 [deg] L [m] stride [m] α [deg] Vx /Vx 1 KE /KE
34 Fig. 4.1Velocity of Z direction (φ 3 =40[deg]) R ~10[m] C 4.0 R φ 3 40[deg] Foot Foot R=1[m]φ 3 =40[deg] φ 3 =40[deg] R Foot Mass 4.1 φ 3 =40[deg] Z Z Foot 0.5[s] Foot R Mass vn4d 4.5 φ 3 30[deg] R φ 3 =0[deg] L strideα Foot 4.1 φ 3 33
![J1 4.6 Foot 8[deg] 0[Nm] 4.7. (R=0.](/docs-images/99/140888400/images/35-1.jpg "4[m]φ 3 =30[deg]) (R=0.")
![6[m]φ 3 =30[deg]) (R=1[m]φ 3](/docs-images/99/140888400/images/35-2.jpg "=30[deg]) Fig. 4.")
35 R 0.4~1[m]φ 3 30[deg]40[deg] J3 15[Nm] J1 4.6 Foot 8[deg] 0[Nm] 4.7. (R=0.4[m]φ 3 =30[deg]) (R=0.6[m]φ 3 =30[deg]) (R=1[m]φ 3 =30[deg]) Fig. 4.Motion of running Fig. 4.3Velocity of X direction 34
![4. 4.6.3 4.3 Mass X 15[Nm] 1.75[m/s] 0.9[m/s] 0.](/docs-images/99/140888400/images/36-0.jpg "48[m] 0.99[/] 1.[m/s] 15[m] R 0.")
![4[m] R 1[m] C R [m] R R 0.6[m] 4.8 4.4 4.](/docs-images/99/140888400/images/36-1.jpg "7 ( ) Z Spring Z X Y X Y (Side view) (Birds-eye")
36 Mass X 15[Nm] 1.75[m/s] 0.9[m/s] 0.48[m] 0.99[/] 1.[m/s] 15[m] R 0.4[m] R 1[m] C R [m] R R 0.6[m] ( ) Z Spring Z X Y X Y (Side view) (Birds-eye view) Fig. 4.4Mathematical model 3 35
![4.8.1 1 4.4 4.4 K K 1000~4000[N/m] (J 1 ) φ 1 =50[deg] (J ) φ =-60[deg] (J 3 ) φ 3 =40[deg] ( ) Mass 0.](/docs-images/99/140888400/images/37-0.jpg "7[m] 15[Nm] 4.8. (K=1000[N/m]) (K=000[N/m]) (K=4000[N/m]) Fig. 4.5Motion of running 3 Fig. 4.6Velocity of X direction 36")
37 K K 1000~4000[N/m] (J 1 ) φ 1 =50[deg] (J ) φ =-60[deg] (J 3 ) φ 3 =40[deg] ( ) Mass 0.7[m] 15[Nm] 4.8. (K=1000[N/m]) (K=000[N/m]) (K=4000[N/m]) Fig. 4.5Motion of running 3 Fig. 4.6Velocity of X direction 36
38 Mass X α 90[deg] K K=500[N/m] 0.7[m] Mass X Z K Mass Mass Mass 37
39 5 1) ) 0.1[] 0.3[] 3) 1 α 90deg X Z β0[deg] 4) 1.75[m/s] 0.9[m/s] 5) 6)
40 39
41 1) MITMIT Leg Laboratory 007/0/08 ( ) HONDAP 007/0/08 ( 3) SONYQRIO 007/0/08 ( 4) HONDAASIMO 007/0/08 ( 5) Tom F.Novacheck The biomechanics of running Gait and Posture 7pp ), AEM Vol.14No.1pp ) 8) 9) T.A.McMahonMechanics of Locomotion, International Journal of Robotics ResearchVol.3No.pp ) 11) pp ) 16 40
42 A A-1 3 ( ) 15[Nm] 9[kg] ( 5[kg] 4[kg] ) 0.8[m] ( ) maxon maxon maxon DC motor RE mm,, 70 Watt 18[V] maxon gear GP 4 C 4 mm, 315 Nm( ) [rpm]81 50[rpm] [] 57.0 [mnm]15000[mnm] [rpm] [mNm] 57.0 [mnm] / n/ M [rpm/mnm] n/ M / 9.054[rpm/mNm] (RE36 18[V] / 9.3[rpm/mNm]) 41
43 M[mNm] / n/ M [rpm/mnm] n [rpm] n [rpm] n [rpm] n 0 n [rpm] n 0 [rpm] U [V] [rpm/v] [rpm/v] (RE36 kv 375[rpm/V]) a) 81[mNm] 57.0 b) n[rpm] = kv[rpm/v] U[V] n/ M[rpm/mNm] [mnm] n[rpm] = [rpm] 41 [rpm] M[mNm] kn[mnm/a] I[A] I = M kn = = [A] [rpm]91 50[rpm] 8.94[mNm]15000[mNm] [rpm] 4550[rpm] 81[mNm] 8.94[mNm] / n/ M9.054[rpm/mNm] n [rpm] n [rpm] [rpm/v] (RE36 375[rpm/V]) a) 81[mNm] 8.94[mNm] b) n[rpm] = [rpm] 4550[rpm] II = M kn = = 8.98 [A] 4
44 1 Fig. A.1Timing Belts & Pulleys selection software 4P8M15AF P8M 4 8 UP8M 480[mm]800[mm] 43
45 A- 1[mm] 1[mm] SUS304 T q [Nm] τ[n/m ] Z p [m 3 ] T q = Z p τ (1) d(m) Z p [m 3 ] Z π 3 p = 16 d () τ τ a d 16T 16 τ = (3) πd q T q d 3 3 πτ a T q =15000[Nmm] τ[n/mm] 16T q τ = = = = πd π (1) π 178 SUS [N/mm ] τ 1[mm] M[Nm] σ[n/m ] Z[m 3 ] M σ = (4) Z d[m] Z[m 3 Z π 3 = 3 d (5) σ (6) σ a d σ 3M 3M = d 3 (6) 3 πd πσ a 44
46 F/ F/ L 1 L 1 L L Fig. A.Bending of a Beam Element F M (7) F F M = L L1 (7) F 1000[N] σ F=1000[N]L1=48[mm]L=66.5[mm]M=950[Nmm] F=1000[N]L1=8[mm]L=45.5[mm]M=8750[Nmm] 3M σ = = = πd π (1) SUS [N/mm ] σ 1[mm] 1[mm] 45
47 CAD CAD Pro/ENGINEER001 Fig. A.3 CAD Model 46
48 Fig. A.4 Block diagram Fig. A.5 Block diagram 47
49 A-3 3 Fig. A.6 One-leg Robot ( ) 48
50 B 0.185[m] 0.07[m] 0.1[m] 0.41[m] 0.336[m] Fig. B.1 Motion of jumping 30 Volt (Zoom) 49
51 C Table C.1Ratio of the velocity and angle α (R=~10[m]) R[m] φ 3 [deg] L [m] stride [m] α [deg] Vx /Vx 1 KE /KE Fig. C.1 Velocity of X direction 50
52 Fig. C.Velocity of Z direction 51
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66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI
65 8. K 8 8 7 8 K 6 7 8 K 6 M Q σ (6.4) M O ρ dθ D N d N 1 P Q B C (1 + ε)d M N N h 2 h 1 ( ) B (+) M 8.1: σ = E ρ (E, 1/ρ ) (8.1) 66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3)
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
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( ) 2017 2 23 : 1998 1 23 ii All Rights Reserved (c) Yoichi OKABE 1998-present. ( ) ( ) Web iii iv ( ) (G) 1998 1 23 : 1998 12 30 : TeX 2007 1 27 : 2011 9 26 : 2012 4 15 : 2012 5 6 : 2015 4 25 : v 1 1
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vol5-honma (LSR: Local Standard of Rest) 2.1 LSR R 0 LSR Θ 0 (Galactic Constant) 1985 (IAU: International Astronomical Union) R 0 =8.5
2.2 1 2.2 2.2.1 (LSR: Local Standard of Rest) 2.1 LSR R 0 LSR Θ 0 (Galactic Constant) 1985 (IAU: International Astronomical Union) R 0 =8.5 kpc, Θ 0 = 220 km s 1. (2.1) R 0 7kpc 8kpc Θ 0 180 km s 1 270
1.500 m X Y m m m m m m m m m m m m N/ N/ ( ) qa N/ N/ 2 2
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