INTRODUCTION TO ROBOTICS
CPSC - 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
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DH TECHNIQUES

The four parameters for each link
ai: 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

Each homogeneous transformation Ai is represented
as a product of four basic transformations
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
Ai  Rot z ,i Transz ,di Transx ,ai Rot x ,i
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TRANSFORMATION MATRIX
Ai  Rot z ,i Transz ,di Transx ,ai Rot x ,i
C i
 S
Ai   i
 0

 0
 S i
C i
0
0
0 0 1
0 0 0
1 0  0

0 1  0
ci
 s
Ai   i
 0

 0
0  1
1 0 0  0
0 1 d i  0

0 0 1  0
0 0
0 0 ai  1 0
1 0 0  0 C i
0 1 0  0 S i

0 0 1  0 0
-c isi
s isi
ci c i
s i
-s i ci
c i
0
0
0
 S i
C i
0
0
0
0

1
a i ci 
a isi 
di 

1 
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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
ci
 s
Ai   i
 0

 0
-c isi
s isi
ci c i
s i
-s i ci
c i
0
0
a i ci 
a isi 
di 

1 
DH NOTATION STEPS
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DH NOTATION STEPS
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DH NOTATION STEPS
ci
 s
Ai   i
 0

 0

-c isi
s isi
ci c i
s i
-s i ci
c i
0
0
a i ci 
a isi 
di 

1 
From Tn , the position and orientation of the tool
frame are calculated.
0
TRANSFORMATION MATRIX
 r11 r12
r r
21
22

T
 r31 r32

0 0
r13
r23
r33
0
dx 

dy 
dz 

1
EXAMPLE I - TWO LINK PLANAR ARM
•
Base frame O0
•All Z ‘s are normal to the page
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EXAMPLE I - TWO LINK PLANAR ARM
Where (θ1 + θ2 ) denoted by θ12 and
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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
22
EXAMPLE 4 – THE SPHERICAL WRIST
23
EXAMPLE 4 – THE SPHERICAL WRIST
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EXAMPLE 5 - CYLINDRICAL MANIPULATOR
WITH
SPHERICAL WRIST
T30 derived in Example 2, and
T63 derived in Example 3.
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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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EXAMPLE 5 - CYLINDRICAL MANIPULATOR
WITH

SPHERICAL WRIST
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}
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ROTATION – ROLL, PITCH, YAW

The rotation matrix for the following operations:
Z
Rotation  about X axis{ROLL}
Rotation  about fixed Y axis{PITCH }

Rotation  about fixed Z axis{YAW}
R  Rot ( z ,  ) Rot ( y, ) Rot ( x, )
0 
C  S 0  C 0 S  1 0
  S C 0  0 1 0  0 C  S 
 0
0
1 - S 0 C  0 S C 
CC  SS  CSS CSC  SS 
  SC CSS  CS  CS  SSC 
  S

CS
CC


X
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Y
EXAMPLE 4
THE THREE LINKS CYLINDRICAL WITH
SPHERICAL WRIST

How to calculate  , , and
 r11
r
 21
 r31


r12
r13
r22
r32
r23
r33

Compare the matrix R

With the rotation part of T60
CC
R   SC
  S
 S  r31
  Sin (r31 )
1
 S S  C S S
C S S  C S
C S
C S  r32
  Sin 1 (
r32
)
C






C S C  S S 
C S  S S C 

C C
SC  r21
  sin 1 (
r21
)
C
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