KINEMATICS 2

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KINEMATICS 2

• OSMAN PARLAKTUNA
• OSMANGAZI UNIVERSITY
• ESKISEHIR, TURKEY
• www.ogu.edu.tr/oparlak

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WHERE IS THE TOOL?
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DETERMINATION OF HOMEGENEOUS TRANSFORMATION
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Problem Determine the transformation which
defines frame i relative to the frame i-1
Note For any given link of a robot,
will be a function of only one variable out of
ai, ?i, di, ?i The other three parameters being
fixed by mechanical design. Revolute Joint -gt
?i Prismatic Joint -gtdi
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Solution The problem is further braked into 4
sub problem such that each of the
transformations will be a function of one link
parameters only Define three intermediate
frames P, Q, and R -Frame R is different
from i-1 only by a rotation of ?I-1 -Frame Q
is different from R only by a translation
ai-1 -Frame P is different from Q only by a
rotation ?i -Frame i is different from P only
by a translation di
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CONCATENATING LINK TRANSFORMATIONS
Define link frames Define DH parameters of each
link Compute the individual link transformation
matrix Relates frame N to frame 0 The
transformation will be a function of all n
joint variables. If the robots joint position
sensors are measured, the Cartesian position and
orientation of the last link may be computed by
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DIFFERENT SPACES
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PUMA 560 6R
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The robot position in which all joint angles are
equal to zero
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FRAME ASSIGNMENTS 0,1
Assign 0 to match 1 when the first joint
variable is zero. Frame 0 is coincident with
Frame 1 Assign Z1 the axis pointing along the
1st joint axis. Assign the X1 axis pointing
along the common perpendicular, or if the axes
intersect, assign X1 to be normal to the plane
containing the two axes Assign the Y1 axis to
the complete a right hand coordinate system.
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FRAME 2
Assign the Z2 axis pointing along the 2nd joint
axis. Assign the X2 axis pointing along the
common perpendicular Assign the Y2 axis to the
complete a right hand coordinate system.
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FRAME 3

• Assign the Z3 axis pointing along the 3rd joint
  axis.
• Assign the X3 axis pointing along the common
  perpendicular
• Assign the Y3 axis to complete a right hand
  coordinate system.

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FRAME 4

• Assign the Z4 axis pointing along the 4th joint
  axis.
• Assign the X4 axis pointing along the common
  perpendicular if the axes intersect, assign X4 to
  be normal to the plane containing the two axes
• Assign the Y4 axis to the complete a right hand
  coordinate system.

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FRAME 5
Assign the Z5 axis pointing along the 5th joint
axis. Assign the X5 axis pointing along the
common perpendicular if the axes intersect,
assign X5 to be normal to the plane containing
the two axes Assign the Y5 axis to the complete
a right hand coordinate system.
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FRAMES 6,(N)

• Assign the Z6 axis pointing along the 6th joint
  axis.
• For frame N (6) choose an origin location
  and direction X6 freely, but generally so as to
  cause
• as many linkage parameters as possible to be zero
• Assign the Y6 axis to the complete a right hand
  coordinate system.

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FRAMES
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DH PARAMETERS
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LINK TRANSFORMATION MATRICES
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Px c1 a2c2 a3c23 d4s23 d3s1 Py s1
a2c2 a3c23 d4s23 d3c1 Pz a3s23 a2s2
d4c23