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The Denavit-Hartenberg Algorithm

Dr. Sallehuddin Mohamed Haris

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Any two neighbouring frames can be brought into coincidence by aprescribed sequence of at most two rotations and two translations.

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Any two neighbouring frames can be brought into coincidence by aprescribed sequence of at most two rotations and two translations.

The D-H algorithm is a systematic matrix method based onHomogeneous Transformation Matrices to describe the positionand orientation of each link or the tool tip with respect to itsneighbouring link in a static situation.

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Steps to take when performing the D-H algorithm:

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Steps to take when performing the D-H algorithm:

1. Assign coordinate frames to all links and the tool tip.

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Steps to take when performing the D-H algorithm:

1. Assign coordinate frames to all links and the tool tip.

2. Derive 4 × 4 HTM to describe the position and orientation ofeach link or tool tip wrt its neighbouring link.

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Steps to take when performing the D-H algorithm:

1. Assign coordinate frames to all links and the tool tip.

2. Derive 4 × 4 HTM to describe the position and orientation ofeach link or tool tip wrt its neighbouring link.

3. Use the post-multiplication rule to compute the forwardkinematics.

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Steps to take when performing the D-H algorithm:

1. Assign coordinate frames to all links and the tool tip.

2. Derive 4 × 4 HTM to describe the position and orientation ofeach link or tool tip wrt its neighbouring link.

3. Use the post-multiplication rule to compute the forwardkinematics.

4. From the forward kinematic equation, determine the positionand orientation of the end effector (tool tip).

Arm Parameters

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Arm Parameters

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● Define the robot base as link 0.

Arm Parameters

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● Define the robot base as link 0.

● Number each link and the end effector consecutively inincreasing order.

Arm Parameters

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● Define the robot base as link 0.

● Number each link and the end effector consecutively inincreasing order.

● Joint 1 connects link 0 to link 1.

Arm Parameters

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● Define the robot base as link 0.

● Number each link and the end effector consecutively inincreasing order.

● Joint 1 connects link 0 to link 1.

● Joint i connects link i − 1 to link i.

Arm Parameters

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● Define the robot base as link 0.

● Number each link and the end effector consecutively inincreasing order.

● Joint 1 connects link 0 to link 1.

● Joint i connects link i − 1 to link i.

● Assign a Cartesian frame xiyizi to each joint.

Arm Parameter Assignment

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Arm Parameter Assignment

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● The zi axis is aligned with the axis of the rotary joint i + 1.

Arm Parameter Assignment

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● The zi axis is aligned with the axis of the rotary joint i + 1.

● The xi axis

Arm Parameter Assignment

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● The zi axis is aligned with the axis of the rotary joint i + 1.

● The xi axis

✦ is normal to both the zi−1 and zi axes.

Arm Parameter Assignment

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● The zi axis is aligned with the axis of the rotary joint i + 1.

● The xi axis

✦ is normal to both the zi−1 and zi axes.

✦ points from the intersection of the zi−1 and xi axes

Arm Parameter Assignment

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● The zi axis is aligned with the axis of the rotary joint i + 1.

● The xi axis

✦ is normal to both the zi−1 and zi axes.

✦ points from the intersection of the zi−1 and xi axes to theintersection of the zi and xi axes.

Arm Parameter Assignment

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● The zi axis is aligned with the axis of the rotary joint i + 1.

● The xi axis

✦ is normal to both the zi−1 and zi axes.

✦ points from the intersection of the zi−1 and xi axes to theintersection of the zi and xi axes.

● The yi axis is chosen from to form a right-handed xiyizi frame.

Link Parameters

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Link Parameters

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● ai (the link length) is the common normal between the zi−1 andzi axes.

Link Parameters

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● ai (the link length) is the common normal between the zi−1 andzi axes.

● αi (the twist angle) is the rotational angle of the zi−1 axis aboutthe xi axis such that the zi−1 will be parallel to the xi axis afterthe rotation.

Joint Parameters

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Joint Parameters

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● θi (the joint angle) is the rotational angle of the xi−1 axis aboutthe zi−1 axis such that the xi−1 axis will be parallel to the xi axisafter the rotation.

Joint Parameters

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● θi (the joint angle) is the rotational angle of the xi−1 axis aboutthe zi−1 axis such that the xi−1 axis will be parallel to the xi axisafter the rotation.

● d (the joint distance) is the translational distance from oi−1 (theorigin of the xi−1yi−1zi−1 frame) to bi (the intersection of the xi

and zi−1 axes).

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Arm Parameter Symbol Revolute Joint Prismatic JointJoint angle θ variable constant

Joint distance d constant variableLink length a constant constantTwist angle α constant constant

HTMs

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Tθi=

cos θi − sin θi 0 0

sin θi cos θi 0 0

0 0 1 0

0 0 0 1

Tdi=

1 0 0 0

0 1 0 0

0 0 1 di

0 0 0 1

Tai=

1 0 0 ai

0 1 0 0

0 0 1 0

0 0 0 1

Tαi=

1 0 0 0

0 cos αi − sin αi 0

0 sin αi cos αi 0

0 0 0 1

D-H Transformation Matrix

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T i

i−1= Tθi

TdiTai

Tαi

=

Cθi −CαiSθi SαiSθi aCθi

Sθi CαiCθi −SαiCθi aSθi

0 Sαi Cαi di

0 0 0 1