:56 1 (Forward kinematics) (Global frame) G r = (X, Y, Z) (Local frame) L r = (x, y, z) 1 X Y, Z X Y, Z 1 ( ) ( ) 1.2 (Joint rotati
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- ぜんま いそみ
- 5 years ago
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1 :56 1 (Forward kinematics) (Global frame) G r (X, Y, Z) (Local frame) L r (x, y, z) 1 X Y, Z X Y, Z 1 ( ) ( ) 1.2 (Joint rotation) 3 P G r, L r G r L r Z α : G r Q L Z,α r (1) 1
2 G r X Y, Z L r x cos α sin α 0 y, Q Z,α sin α cos α 0 (2) z Y β X γ G r Q Y,β L r (3) G r Q X,γ L r (4) cos β 0 sin β Q Y,β (5) sin β 0 cos β Q X,γ cos γ sin γ (6) 0 sin γ cos γ 2 Z (A) (B) Z L r Q 1 Z,α r (7) L r Q 1 Y,β r (8) L r Q 1 X,γ r (9) Q 1, Q 2,, Q n G r G Q L L r (10) G Q L Q n Q 2 Q 1 (11) G Q L 2
3 1 L r 2 X 30 Y 45 X Y 3 G r X Y Q Y,45 Q L X,30 r (12) G r Y X Q X,30 Q L Y,45 r (13) X (Roll) Y (Pitch) Z (Yaw) - - G Q L Q Z,γ Q Y,β Q X,α TODO: p44 α tan 1 r32 r 33 β sin 1 (r 31 ) γ tan 1 r21 ( cos β 0) r 11 (14) (15) L r G d G R L : G r G R L L r + G d (16) G R L 3 3 G r, L r, G d 3 3
4 1.5 : G r G T L L r (17) r 11 r 12 r 13 X 0 G T L r 21 r 22 r 23 Y 0 r 31 r 32 r 33 Z ( G ) R G L d 0 1 X x X 0 G r Y Z, L r y z, G d Y 0 Z (18) G T L (Homogeneous transformation matrix) Denavit-Hartenberg (DH) n n + 1 (Base link) 0 n 1 n i + 1 i L 1 DH?? 1. z i i x i z i 1, z i 3. y i y i z i x i 1.8 Z 2 *1 *1 4
5 3 2 DH DH a i α i d i θ i 1 l θ 1 2 l θ 2 B i B i 1 cos θ 2 sin θ 2 0 l2 cos θ 2 1 T 2 sin θ 2 cos θ 2 0 l2 sin θ cos θ 1 sin θ 1 0 l1 cos θ 1 0 T 1 sin θ 1 cos θ 1 0 l1 sin θ (19) cos(θ 1 + θ 2 ) sin(θ 1 + θ 2 ) 0 l 1 cos θ 1 + l 2 cos(θ 1 + θ 2 ) 0 T 2 sin(θ 1 + θ 2 ) cos(θ 1 + θ 2 ) 0 l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ) (20) :
6 3 B 3 θ 3 l 3 2 T 3 0 T 3 0 B 3 3 r r 1 2 T 3 0 T 3 0 r 1.9 Nao Nao 1 #!/usr/local/bin/python 2 # -*- coding: utf-8 -*- 3 4 import sys 5 from naoqi import ALProxy 6 7 HOST "localhost" 8 PORT def main(): 11 motionproxy ALProxy("ALMotion", HOST, PORT) motionproxy.wakeup() """ """ motionproxy.rest() if name " main ": 22 main() Webots for Nao Nao PC Nao ALProxy localhost
7 Nao ALMotion 1 motionproxy ALProxy("ALMotion", HOST, PORT) Nao wakeup() Nao rest()) StandInit 1 #!/usr/bin/python 2 3 import sys 4 from naoqi import ALProxy 5 6 postureproxy ALProxy("ALRobotPosture", "localhost", 9559) 7 8 postureproxy.gotoposture("standinit", 1.0) ALMotion ALRobotPosture 1 motionproxy postureproxy Nao StandZero 1 #!/usr/local/bin/python 2 # -*- coding: utf-8 -*- 3 4 import sys 5 from naoqi import ALProxy 6 import time 7 import math 8 9 HOST "localhost" 10 PORT def main(): 13 motionproxy ALProxy("ALMotion", HOST, PORT) motionproxy.wakeup() postureproxy ALProxy("ALRobotPosture", "localhost", 9559) 18 postureproxy.gotoposture("standzero", 1.0) motionproxy.setangles("headyaw", -90.0*(2*math.pi/360.0), 1.0) 21 time.sleep(1.0) 22 motionproxy.setangles("headyaw", 90.0*(2*math.pi/360.0), 1.0) 23 time.sleep(1.0) 24 motionproxy.setangles("headyaw", 0, 1.0) 25 time.sleep(1.0) motionproxy.rest() 7
8 28 29 if name " main ": 30 main() wakeup() postureproxy StandZero HeadYaw setangles() 1 motionproxy.setangles("headyaw", -90.0*(2*math.pi/360.0), 1.0) -90.0*(2*math.pi/360.0) ( ) Nao 1 time.sleep(1.0) rest() HeadYaw Nao 1 #!/usr/local/bin/python 2 # -*- coding: utf-8 -*- 3 4 import sys 5 from naoqi import ALProxy 6 import math HOST "localhost" 9 PORT def main(): 12 motionproxy ALProxy("ALMotion", HOST, PORT) motionproxy.wakeup() bodynames motionproxy.getbodynames("body") targetanglesinrad motionproxy.getangles("body", False) 19 targetanglesindeg [ x * (360.0/2*math.pi) for x in targetanglesinrad ] for i in range(len(bodynames)): 22 print bodynames[i] + " " + str(targetanglesindeg[i]) motionproxy.rest() if name " main ": 27 main() getbodynames("body") getangles("body", False) (Radian) 1 targetanglesindeg [ x * (360.0/2*math.pi) for x in targetanglesinrad ] 8
9 (Degree) *2 for setangles 1 #!/usr/local/bin/python 2 # -*- coding: utf-8 -*- 3 4 import sys 5 from naoqi import ALProxy 6 import time 7 import math 8 9 HOST "localhost" 10 PORT def main(): 13 motionproxy ALProxy("ALMotion", HOST, PORT) motionproxy.wakeup() postureproxy ALProxy("ALRobotPosture", "localhost", 9559) 19 postureproxy.gotoposture("standzero", 1.0) names ["RShoulderRoll", "RElbowRoll", "RWristYaw"] 22 anglesindeg [15.0, 80.0, 90.0] 23 anglesinrad [ x * (2*math.pi/360.0) for x in anglesindeg ] 24 motionproxy.setangles(names, anglesinrad, 1.0) 25 time.sleep(2.0) motionproxy.rest() if name " main ": 30 main() : Nao (Nao ) 4 ( ) *2 Python Scheme Javascript map Lisp mapcar Ruby collect 9
10 StandZero Z X Y 0 T 1 1 T r T 1, 1 T 2 0 r 0 T 1 1 T 1 2 r (21) HandOffsetX 20mm ( ShoulderOffsetY ) cos(π/2) sin(π/2) Python numpy A sin(π/2) cos(π/2) x (1, 1) y Ax ( 1, 1) 1 #!/usr/local/bin/python 2 3 import numpy as np 4 import math 10
11 5 6 def main(): 7 8 PI math.pi 9 A np.array([[math.cos(0.5*pi), -math.sin(0.5*pi)], 10 [math.sin(0.5*pi), math.cos(0.5*pi)]]) 11 x np.array([[1.0], 12 [1.0]]) 13 y np.dot(a, x) print y if name " main ": 18 main() numpy import np np.array ( N 1 ) np.dot 0 T 1 1 T 2 11
12 2 (Inverse kinematics) ( )?? T 2 0 T 1 1 T 2 cos(θ 1 + θ 2 ) sin(θ 1 + θ 2 ) 0 l 1 cos θ 1 + l 2 cos(θ 1 + θ 2 ) sin(θ 1 + θ 2 ) cos(θ 1 + θ 2 ) 0 l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ) X l1 cos θ 1 + l 2 cos(θ 1 + θ 2 ) Y l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ) (22) (23) X 2 + Y 2 l l l 1 l 2 cos θ 2 (24) cos θ 2 X2 + Y 2 l 2 1 l 2 2 2l 1 l 2 (25) θ 2 cos 1 X2 + Y 2 l 2 1 l 2 2 2l 1 l 2 (26) cos 1 (x) ( x 1 acos,asin ) tan 2 θ 2 1 cos θ 1 + cos θ (27) θ 2 > 0 θ 2 ±atan2 (l 1 + l 2 ) 2 (X 2 + Y 2 ) (X 2 + Y 2 ) (l 1 + l 2 ) 2 (28) θ 1 atan2 Y X + atan2 l 2 sin θ 2 l 1 + l 2 cos θ 2 (29) 12
13 θ 2 < 0 θ 1 atan2 Y X atan2 l 2 sin θ 2 l 1 + l 2 cos θ 2 (30) : acos,asin acos, asin asin(x) x + 1 x x acos(x) π (31) 2 asin(x) x 1 x 1 x 1 acos(x) cos θ x cos 2 θ + sin 2 θ x 2 + sin 2 θ 1 ( ) θ asin 1 x 2 ( ) (32) 1 x 2 1 cos 1 (x) C acos (3) glibc double glibc/sysdeps/ieee754/dbl-64/e acos.c float flt-32/e acosf.c ( ) T q T T(q) (33) q q q δq (34) T T (q ) (35) δt T T T T (q) T (q + δq) T (q ) + T q δq + O ( δq 2) (36) δq δt T δq Jδq q 13
14 J T q J δq J 1 δt q q + δq q + J 1 δt (37) q (0) i ( q (i+1) q (i) + J 1 q (i)) ( δt q (i)) (38) (Newton-Raphson Method) 2 X l1 cos θ 1 + l 2 cos(θ 1 + θ 2 ) Y l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ) (39) ( θ1 ) q θ 2 (40) X T Y J(q) J(q) ( Ti ) q j ( X θ 1 Y θ 1 X θ 2 Y θ 2 ) ( l1 sin θ 1 l 2 sin(θ 1 + θ 2 ) l 2 sin(θ 1 + θ 2 ) l 1 cos θ 1 + l 2 cos(θ 1 + θ 2 ) l 2 cos(θ 1 + θ 2 ) ( ) J 1 1 l 2 cos(θ 1 + θ 2 ) l 2 sin(θ 1 + θ 2 ) l 1 l 2 sin θ 2 l 1 cos θ 1 + l 2 cos(θ 1 + θ 2 ) l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ) ) (41) (42) (i+1) [ (i) (X ) ] (i) θ1 θ1 + J 1 X θ 2 θ 2 Y Y (43) l 1 l 2 1 (44) T T X Y 1 1 (45) 14
15 q (0) (0) θ1 π/3 θ 2 π/3 (46) δt ( 1 1) ( cos π/3 + cos(π/3 π/3) 1 sin π/3 + sin(π/3 π/3) 1) ( 3 ) (47) 1 J (48) J ( θ1 ) (1) ( θ1 ) (0) + J 1 δt θ 2 θ 2 π/ π/ (49) 1 : 1 J ( 1 ) δt q (1) (50) 2 : J δt q (2) (51) 3 : 15
16 J δt q (3) (52) 4 : π/2 q (4) q π/ J δt q (4) (53) 2.3 θ 1 ( θ 2 ) X( Y ) *3 LU Python numpy.linalg.inv() 2 Python 1 1 #!/usr/bin/python 2 3 import numpy as np 4 import math 5 6 L L Epsilon 1.0e def X(theta1, theta2): 1 Python 11 return L1*math.cos(theta1) + L2*math.cos(theta1+theta2) def Y(theta1, theta2): *3 16
17 14 return L1*math.sin(theta1) + L2*math.sin(theta1+theta2) def Jacobian(q): 17 theta1 q[0] 18 theta2 q[1] 19 dxdtheta1 (X(theta1+Epsilon, theta2)-x(theta1, theta2))/epsilon 20 dxdtheta2 (X(theta1, theta2+epsilon)-x(theta1, theta2))/epsilon 21 dydtheta1 (Y(theta1+Epsilon, theta2)-y(theta1, theta2))/epsilon 22 dydtheta2 (Y(theta1, theta2+epsilon)-y(theta1, theta2))/epsilon 23 return np.array([[dxdtheta1, dxdtheta2], [dydtheta1, dydtheta2]]) def distance(u, v): 26 return math.sqrt((u[0]-v[0])*(u[0]-v[0])+(u[1]-v[1])*(u[1]-v[1])) def Tstar(q): 29 return np.array([[x(q[0],q[1])], [Y(q[0],q[1])]]) def main(): 32 T np.array([[1.0], [1.0]]) 33 q np.array([[np.pi/3.0], [-np.pi/3.0]]) while distance(t, Tstar(q)) > Epsilon: 36 th1 q[0] 37 th2 q[1] 38 deltat T - Tstar(q) 39 J Jacobian(q) 40 Jinv np.linalg.inv(j) 41 q + np.dot(jinv, deltat) 42 print q if name " main ": 45 main() ( )
18 2.4.3 ( ) 2/2 θ 1 π/4, θ /2 ( ) J 1 1 l 2 cos(θ 1 + θ 2 ) l 2 sin(θ 1 + θ 2 ) l 1 l 2 sin θ 2 l 1 cos θ 1 + l 2 cos(θ 1 + θ 2 ) l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ) (54) l 1 l 2 sin θ 2 (Singular configuration) : 2.5 3: 5 Nao ( ) Nao (StandInit ) ( Crouch ) StandInit (x, y, z) (x, y, z ) x x y y StandInit z 560 mm 450 mm mm 10 p 0, p 10 getangles() setangles() 5 StandZero θ 1 90 Nao θ StandZero 18
19 H 5 Nao A1: Python C Python *4 Hello world 1 #!/usr/local/bin/python 2 # -*- coding: utf-8 -*- 3 4 import sys 5 import time 6 7 def main(): 8 print "And now for something completely different..." 9 time.sleep(3.0) 10 for i in ["Graham Chapman", "John Cleese", "Terry Gilliam", 11 "Eric Idle", "Terry Jones", "Michael Palin"]: 12 print i if name " main ": 15 main() 1. 1 *4 19
20 2. 2 ( UTF8) C #include<stdio.h> 5. 5 time * main C main Python (:) * And now for something complete different... *7 C (;) C {} Python 9. 9 time sleep C sleep for C Python [] (:) import false main() for C range() 0 ( 1) 1 % python 2 Python (v2.7.5:ab05e7dd2788, May , 13:18:45) 3 [GCC (Apple Inc. build 5666) (dot 3)] on darwin 4 Type "help", "copyright", "credits" or "license" for more information. 5 >>> range(10) 6 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] 7 >>> for i in range(10): 8... print i 9... ^D *5 C *6 *7 Monty Python s Flying Circus ( ) The Olympic Hide-and-Seek Final ( YouTube ( ) 20
21 >>> 1 # # (C //) 2 """ 3 """ """ (C /*... */) 4 """ (/* foo */ """ foo """ ) 5 """ 21
22 A2: Nao Nao NAO Technical overview *8 6 *8 robots.html 22
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知能システム論 1 (9) 2015.6.17 情報システム学研究科情報メディアシステム学専攻知能システム学講座末廣尚士 13. アームモデルの Python による表現 理想ロボット :ArmWithHand 構造は関係なし move: 手先や持った物を動かす ハンド :Hand open, close, width アームのリンクの計算 :Link set_jparam シリアルリンクアーム :LinkedArm
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