Introduction to Robotics Tutorial II
description
Transcript of Introduction to Robotics Tutorial II
Introduction to Robotics
Tutorial II
Alfred BrucksteinYaniv Altshuler
Denavit-Hartenberg
•Specialized description of articulated figures
•Each joint has only one degree of freedom
•rotate around its z-axis
•translate along its z-axis
Reminder
Denavit-Hartenberg• Link length ai
• The perpendicular distance between the axes of jointi and jointi+1
• Link twist αi
• The angle between the axes of jointi and jointi+1
• Angle around xi-axis
Denavit-Hartenberg
• Link offset di
• The distance between the origins of the coordinate frames attached to jointi and jointi+1
• Measured along the axis of jointi
Denavit-Hartenberg
• Link rotation (joint angle) φi
• The angle between the link lenghts αi-1 and αi
• Angle around zi-axis
Denavit-Hartenberg
Denavit-Hartenberg
1.Compute the link vector ai and the link length
2.Attach coordinate frames to the joint axes
3.Compute the link twist αi
4.Compute the link offset di
5.Compute the joint angle φi
6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1
Denavit-Hartenberg
This transformation is done in several steps :
•Rotate the link twist angle αi-1 around the axis xi
•Translate the link length ai-1 along the axis xi
•Translate the link offset di along the axis zi
•Rotate the joint angle φi around the axis zi
iiii xxzzi
i RotTransTransRotT 1
Denavit-Hartenberg
1000
0cossin0
0sincos0
0001
ii
iixi
Rot
Denavit-Hartenberg
1000
0100
0010
001 i
x
a
Transi
Denavit-Hartenberg
1000
100
0010
0001
iz d
Transi
Denavit-Hartenberg
1000
0100
00cossin
00sincos
ii
ii
ziRot
Denavit-Hartenberg
Multiplying the matrices :
iiii xxzzi
i RotTransTransRotT 1
1000
cossin0
sinsincoscoscossin
cossinsincossincos
1
iii
iiiiiii
iiiiiii
ii
d
a
a
T
DH Example3 revolute joints
Shown in home position
Link 1 Link 3
Link 2
joint 1
joint 2 joint 3
R
L1 L2
Shown with joints in non-zero positions
Z1
Z0
Z21
2 3
x0
x1
x2
z3
x3
Observe that frame i moves with link i
DH Example
Z1
Z0
Z2
1
23
x0
x1 x2 x3
Z3
Link Var d a
1 1 1 0 90o R
2 2 2 0 0 L1
3 3 3 0 0 L2
R
L1 L2
1
DH Example
1 = 90o (rotate by 90o around x0 to
align Z0 and Z1)
Link Var d a
1 1 1 0 90o R
2 2 2 0 0 L1
3 3 3 0 0 L2
DH Example
DH Example
z1
z0
z21
2 3
x0
x1
x2
z3
x3
Origin of {1} w.r.t. {0}
x1 axis expressed wrt {0}
y1 axis expressed wrt {0}
z1 axis expressed wrt {0}
DH Example
z1
z0
z21
2 3
x0
x1
x2
z3
x3
Origin of {2} w.r.t. {1}
x2 axis expressed wrt {1}
y2 axis expressed wrt {1}
z2 axis expressed wrt {1}
DH Example
z1
z0
z21
2 3
x0
x1
x2
z3
x3
Origin of {3} w.r.t. {2}
x3 axis expressed wrt {2}
y3 axis expressed wrt {2}
z3 axis expressed wrt {2}
DH Example
where
DH Example
Example – the Stanford Arm
Example – the Stanford Arm
X1Y1
Z1
X2
Z2
X3
Z3
X4
X5
X6
Z4
Z5
Z6
X7
Z7
Example – the Stanford Arm
X1Y1
Z1
X2
Z2
X3
Z3
X4
X5
X6
Z4
Z5
Z6
X7
Z7
i ai di i i
1 0 d1 90° 1
2
3
4
5
6
Example – the Stanford Arm
X1Y1
Z1
X2
Z2
X3
Z3
X4
X5
X6
Z4
Z5
Z6
X7
Z7
i ai di i i
1 0 d1 90° 1
2 0 d2 90° 2
3
4
5
6
Example – the Stanford Arm
X1Y1
Z1
X2
Z2
X3
Z3
X4
X5
X6
Z4
Z5
Z6
X7
Z7
i ai di i i
1 0 d1 90° 1
2 0 d2 90° 2
3 0 d3
(var)90° 90°
4
5
6
Example – the Stanford Arm
X1Y1
Z1
X2
Z2
X3
Z3
X4
X5
X6
Z4
Z5
Z6
X7
Z7
i ai di i i
1 0 d1 90° 1
2 0 d2 90° 2
3 0 d3
(var)90° 90°
4 0 d4 90° 4
5
6
Example – the Stanford Arm
X1Y1
Z1
X2
Z2
X3
Z3
X4
X5
X6
Z4
Z5
Z6
X7
Z7
i ai di i i
1 0 d1 90° 1
2 0 d2 90° 2
3 0 d3
(var)90° 90°
4 0 d4 90° 4
5 0 d5 0° 5
6
Example – the Stanford Arm
X1Y1
Z1
X2
Z2
X3
Z3
X4
X5
X6
Z4
Z5
Z6
X7
Z7
i ai di i i
1 0 d1 90° 1
2 0 d2 90° 2
3 0 d3
(var)90° 90°
4 0 d4 90° 4
5 0 d5 0° 5
6 0 d6 0° 6