Introduction to Robotics Tutorial II

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Introduction to RoboticsTutorial II

• Alfred Bruckstein
• Yaniv Altshuler

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Denavit-Hartenberg
Reminder

• Specialized description of articulated figures
• Each joint has only one degree of freedom
• rotate around its z-axis
• translate along its z-axis

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Denavit-Hartenberg

• Link length ai
• The perpendicular distance between the axes of
  jointi and jointi1

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Denavit-Hartenberg

• Link twist ai
• The angle between the axes of jointi and jointi1
• Angle around xi-axis

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Denavit-Hartenberg

• Link offset di
• The distance between the origins of the
  coordinate frames attached to jointi and jointi1
• Measured along the axis of jointi

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Denavit-Hartenberg

• Link rotation (joint angle) fi
• The angle between the link lenghts ai-1 and ai
• Angle around zi-axis

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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 ai
4. Compute the link offset di
5. Compute the joint angle fi
6. Compute the transformation (i-1)Ti which
   transforms entities from linki to linki-1

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Denavit-Hartenberg
This transformation is done in several steps

• Rotate the link twist angle ai-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 fi around the axis zi

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Denavit-Hartenberg
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Denavit-Hartenberg
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Denavit-Hartenberg
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Denavit-Hartenberg
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Denavit-Hartenberg
Multiplying the matrices
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DH Example
3 revolute joints Shown in home position
joint 1
R
Link 2
Link 3
Link 1
joint 2
joint 3
L1
L2
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DH Example
Shown with joints in non-zero positions
Z0
x3
z3
?3
?2
x2
x1
Z2
?1
x0
Z1
Observe that frame i moves with link i
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DH Example
?1 90o (rotate by 90o around x0 to align Z0
and Z1)
R
Z0
L2
L1
?1
x1
x2
x3
x0
?1
?3
?2
Z3
Z1
Z2
Link Var ? d ? a
1 ?1 ?1 0 90o R
2 ?2 ?2 0 0 L1
3 ?3 ?3 0 0 L2
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DH Example
Link Var ? d ? a
1 ?1 ?1 0 90o R
2 ?2 ?2 0 0 L1
3 ?3 ?3 0 0 L2
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DH Example
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DH Example
x1 axis expressed wrt 0
y1 axis expressed wrt 0
z1 axis expressed wrt 0
Origin of 1 w.r.t. 0
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DH Example
x2 axis expressed wrt 1
y2 axis expressed wrt 1
z2 axis expressed wrt 1
Origin of 2 w.r.t. 1
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DH Example
x3 axis expressed wrt 2
y3 axis expressed wrt 2
z3 axis expressed wrt 2
Origin of 3 w.r.t. 2
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DH Example
where
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Example the Stanford Arm
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Example the Stanford Arm
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Example the Stanford Arm
i ai di ?i ?i
1 0 d1 90 ?1
2
3
4
5
6
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Example the Stanford Arm
i ai di ?i ?i
1 0 d1 90 ?1
2 0 d2 90 ?2
3
4
5
6
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Example the Stanford Arm
i ai di ?i ?i
1 0 d1 90 ?1
2 0 d2 90 ?2
3 0 d3 (var) 90 90
4
5
6
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Example the Stanford Arm
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
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Example the Stanford Arm
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
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Example the Stanford Arm
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