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INTRODUCTION TO ROBOTICSCPSC - 460
Lecture 3A – Forward Kinematics
DH TECHNIQUES
A link j can be specified by two parameters, its length aj and its twist αj
Joints are also described by two parameters. The link offset dj is the distance from one link coordinate frame to the next along the axis of the joint. The joint angle θj is the rotation of one link with respect to the next about the joint axis.
DH TECHNIQUES
•Link twist αi :the angle from the Zi-1 axis to the Zi axis about the Xi axis. The positive sense for α is determined from zi-1 and zi by the right-hand rule.
•Joint angle θi the angle between the Xi-1 and Xi axes about the Zi-1 axis.
DH TECHNIQUES
4
DH TECHNIQUES
The four parameters for each linkai: link length
αi: Link twist
di : Link offset
θi : joint angle
With the ith joint, a joint variable is qi
associated where
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TRANSFORMATION MATRIX
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Each homogeneous transformation Ai is represented as a product of four basic transformations
, , , ,i i i ii z z d x a xA Rot Trans Trans Rot
1. Rotation of about current Z axis
2. Translation of d along current Z axis
3. Translation of a along current X axis
4. Rotation of about current X axis
TRANSFORMATION MATRIX
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i i i i i i i
i i i i i i i
i i i
c -c s s s a c
s c c -s c a s
0 s c d
0 0 0 1
iA
A
1000
00
00
0001
1000
0100
0010
001
1000
100
0010
0001
1000
0100
00
00
ii
ii
i
i
ii
ii
i CS
SC
a
d
CS
SC
A
, , , ,i i i ii z z d x a xA Rot Trans Trans Rot
TRANSFORMATION MATRIX
The matrix Ai is a function of only a single variable, as three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable, depending on whether it is a revolute or prismatic link
i i i i i i i
i i i i i i i
i i i
c -c s s s a c
s c c -s c a s
0 s c d
0 0 0 1
iA
DH NOTATION STEPS
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DH NOTATION STEPS
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DH NOTATION STEPS
From , the position and orientation of the tool frame are calculated.
i i i i i i i
i i i i i i i
i i i
c -c s s s a c
s c c -s c a s
0 s c d
0 0 0 1
iA
0nT
TRANSFORMATION MATRIX
11 12 13
21 22 23
31 32 33
0 0 0 1
x
y
z
r r r d
r r r dT
r r r d
EXAMPLE I - TWO LINK PLANAR ARM
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• Base frame O0
•All Z ‘s are normal to the page
EXAMPLE I - TWO LINK PLANAR ARM
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Where (θ1 + θ2 ) denoted by θ12
and
EXAMPLE 2
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FORWARD KINEMATICS OF EXAMPLE 2
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EXAMPLE 3 - THREE LINK CYLINDRICAL MANIPULATOR
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EXAMPLE 3 - THREE LINK CYLINDRICAL MANIPULATOR
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EXAMPLE 3 - THREE LINK CYLINDRICAL MANIPULATOR
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EXAMPLE 3 - THREE LINK CYLINDRICAL MANIPULATOR
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EXAMPLE 4 – THE SPHERICAL WRIST
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EXAMPLE 4 – THE SPHERICAL WRIST
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EXAMPLE 4 – THE SPHERICAL WRIST
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EXAMPLE 4 – THE SPHERICAL WRIST
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EXAMPLE 5 - CYLINDRICAL MANIPULATOR WITH SPHERICAL WRIST
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derived in Example 2, and
derived in Example 3.
03T
36T
EXAMPLE 5 - CYLINDRICAL MANIPULATOR WITH SPHERICAL WRIST
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EXAMPLE 5 - CYLINDRICAL MANIPULATOR WITH SPHERICAL WRIST
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EXAMPLE 5 - CYLINDRICAL MANIPULATOR WITH SPHERICAL WRIST
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Forward kinematics:
1. The position of the end-effector: (dx ,dy ,dz )
2. The orientation {Roll, Pitch, Yaw }Rotation about X axis{ROLL}
Rotation about fixed Y axis{PITCH}
Rotation about fixed Z axis{YAW}
ROTATION – ROLL, PITCH, YAW
The rotation matrix for the following operations:
X
Y
Z
axis{YAW} Zfixedabout Rotation
}axis{PITCH Y fixedabout Rotation
axis{ROLL} Xabout Rotation
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CCSCS
CSSSCSCSSCCS
SSCSCSSCSSCC
CS
SCCS
SC
xRotyRotzRotR
0
0
001
C0S-
010
S0C
100
0
0
),(),(),(
EXAMPLE 4THE THREE LINKS CYLINDRICAL WITH SPHERICAL WRIST
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How to calculate
Compare the matrix R
With the rotation part of
, ,and
06T
C C S S C S S C S C S S
R S C C S S C S C S S S C
S C S C C
31S r 32C S r 21S C r
131( )Sin r 1 32( )
rSin
C
1 21sin ( )
r
C
11 12 13
21 22 23
31 32 33
r r r
r r r
r r r
