Direct Kinematics- The Arm Equation
Link Coordinates and Kinematics Parameters (Cont‘d)
Another example: A 5-axis articulated robot (Rhino XR-3)
(refer to class notes for details)
Exercise: Determine the link coordinates and the
kinematics parameters of
• the Motoman robot arm
• the SONY arm (LAB-2)
COMP322/S2000/L9
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The Arm Equation
For a n-axis robot arm, the arm equation is
P0 = T Pn
where P0 is a point (vector) w.r.t. frame L0 , ie. The base frame;
Pn is a point (vector) w.r.t. frame Ln, ie. The tool frame;
T is the transformation matrix.
Question: What is T?
T is the composite transformation from one frame to another
and is expressed in terms of the kinematics parameters.
COMP322/S2000/L9
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Arm Equation
Consider the transformation from one frame to another, say
from frame Lk-1 to Lk.
Idea is find the transformation that will bring Lk-1 to align with
Lk .
4 fundamental motions:
First two motions:
• Rotate Lk-1 about zk-1 to bring xk-1 parallel to xk , ie. by an
angle of qk => pure rotation: Rot(qk , zk-1 )
• Translate Lk-1 along zk-1 to bring xk-1 align with xk , ie. by a
distance of dk => pure translation: Trans(dk , zk-1 )
=> Screw (dk , qk , zk-1 )
COMP322/S2000/L9
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Arm Equation
Second two motions:
• Translate Lk-1 along xk-1 (xk ) to bring Lk-1 and Lk (the two
origins) to coincide, ie. by a distance of ak => pure
translation: Trans(ak , xk-1 )
• Rotate Lk-1 about xk-1 (xk ) to bring zk-1 to align with zk , ie. by
an angle of ak => pure rotation: Rot(ak , xk-1 )
=> Screw (ak , ak , xk-1 )
Let Tk-1k denote the transformation from frame Lk-1 to frame Lk,
i.e.
Pk-1 = Tk-1k Pk
COMP322/S2000/L9
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Arm Equation
Question:
• Tk-1k = Screw (dk , qk , zk-1 ) Screw (ak , ak , xk-1 ) = A ?
or
• Tk-1k = Screw (ak , ak , xk-1 ) Screw (dk , qk , zk-1 ) = B ?
Note: Refer to class notes for the details to answer the above
question.
COMP322/S2000/L9
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