Introduction to ROBOTICS
Forward Kinematics
University of Bridgeport
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Kinematic
• Forward (direct) Kinematics
• Given: The values of the joint variables.
• Required: The position and the orientation of the end
effector.
• Inverse Kinematics
• Given : The position and the orientation of the
end effector.
• Required : The values of the joint variables.
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Why DH notation
• Find the homogeneous transformation H
relating the tool frame to the fixed base frame
3
Why DH notation
• A very simple way of modeling robot links and
joints that can be used for any kind of robot
configuration.
• This technique has became the standard way
of representing robots and modeling their
motions.
4
DH Techniques
1. Assign a reference frame to each joint (x-axis
and z-axis). The D-H representation does not use
the y-axis at all.
2. Each homogeneous transformation Ai is
represented as a product of four basic
transformations
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DH Techniques
• Matrix Ai representing the four movements is found by: four
movements
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  Rotz ,i Transz ,di Transx,ai Rotx,i
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Rx ,
1 0
 Rot( x,  )  0 C
0 S
C i
 S
Ai   i
 0

 0
 S i
C i
0
0
ci
 s
Ai   i
 0

 0
0
0
1
0
0 1
0 0
0  0

1  0
-c isi
ci c i
s i
0
0
1
0
0
C
 Rot( z ,  )   S
 0
0 
 S 
C 
Rz ,
0 0  1
0 0  0
1 d i  0

0 1  0
0
1
0
0
s isi
-s i ci
c i
0
a i ci 
a isi 
di 

1 
0 ai  1 0
0 0  0 C i
1 0  0 S i

0 1  0 0
0
 S i
C i
0
 S
C
0
0
0
1
0
0
0

1
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DH Techniques
• The link and joint parameters :
• Link length ai : the offset distance between the Zi-1
and Zi axes along the Xi axis.
• Link offset di the distance from the origin of frame
i−1 to the Xi axis along the Zi-1 axis.
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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.
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DH Techniques
• The four parameters:
ai: link length, αi: Link twist , di : Link offset and
θi : joint angle.
• The matrix Ai is a function of only a single
variable qi , it turns out that three of the above
four quantities are constant for a given link,
while the fourth parameter is the joint
variable.
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DH Techniques
• With the ith joint, a joint variable is qi associated
where
All joints are represented by the z-axis.
• If the joint is revolute, the z-axis is in the direction
of rotation as followed by the right hand rule.
• If the joint is prismatic, the z-axis for the joint is
along the direction of the liner movement.
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DH Techniques
3. Combine all transformations, from the first joint
(base) to the next until we get to the last joint, to
get the robot’s total transformation matrix.
T  A1.A2 .......An
0
n
0
T
4. From n , the position and orientation of the tool
frame are calculated.
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DH Techniques
13
DH Techniques
14
DH Techniques
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DH Techniques
ci
 s
Ai   i
 0

 0
-c isi
ci c i
s i
0
s isi
-s i ci
c i
0
a i ci 
a isi 
di 

1 
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Example I
The two links arm
•
Base frame O0
•All Z ‘s are normal to the page
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Example I
The two links arm
Where (θ1 + θ2 ) denoted by θ12 and
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Example 2
19
Example 2
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Example 3
The three links cylindrical
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Example 3
The three links cylindrical
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Example 3
The three links cylindrical
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Example 3
The three links cylindrical
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Example 4
Spherical wrist
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Example 4
Spherical wrist
26
Example 4
Spherical wrist
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Example 4
Spherical wrist
28
Example 5
The three links cylindrical with Spherical wrist
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Example 5
The three links cylindrical with Spherical wrist
•
0
given
3
3
by
example
2,
and
T6 given by
T
example 3.
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Example 5
The three links cylindrical with Spherical wrist
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Example 5
The three links cylindrical
with Spherical wrist
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Example 5
The three links cylindrical
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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Roll Pitch Yaw
• The rotation matrix for the following
operations:
Z
Rotation about X axis{ROLL}
Rotation about fixed Y axis{PIT CH}
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



Y
X
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Example 4
The three links cylindrical with Spherical wrist
• How to calculate
 , , and
• Compare the matrix R with
Of the matrix T60
C C
R   S C
  S
 S S  C S S
C S S  C S
C S
S  r31
  Sin (r31 )
1
C S  r32
  Sin 1 (
r32
)
C
 r11 r12
r r
 21 22
 r31 r32


r13
r23
r33






C S C  S S 
C S  S S C 

C C
SC  r21
  sin 1 (
r21
)
C
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Module 1
RRR:RRR
Links
α
a
θ
d
1
90
0
*
10
2
0
10
*
0
3
-90
0
*
0
4
90
0
*
10
5
-90
0
*
0
6
0
0
*
0
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HW
• From Spong book: page 112
• 3.2, 3.3, 3.4, 3.6 , 3.7, 3.8, 3.9, 3.11
• No Class on next Tuesday
37
Module 1
• Where   1 ,  2 , and3 are Roll, Pitch, and Yaw
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Representing forward kinematics
• Forward kinematics
1   px 
   p 
 2  y
3   pz 
  
 4   
5   
   
 6   
• Transformation Matrix
 r11 r12
r r
T   21 22
 r31 r32

0 0
r13
r23
r33
0
dx 
d y 
dz 

1
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Remember
•
•
•
•
•
For n joints: we have n+1 links. Link 0 is the base
Joints are numbered from 1 to n
Joint i connects link i − 1 to link i.
Frame i {Xi Yi Zi} is attached to joint i+1.
So, frame {O0 X0 Y0 Z0}, which is attached to the robot
base (inertial frame) “joint 1”.
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