1. The document discusses direct kinematics and introduces the Denavit-Hartenberg (D-H) representation for assigning coordinate frames to robot links in a systematic way. It describes the D-H algorithm for determining link and joint parameters. 2. Kinematic parameters like link length, link twist angle, joint distance, and joint angle are defined. An example of applying D-H frames to a microrobot is provided along with its kinematic parameters. 3. Arm equations are developed to determine the transformation matrix between links using individual link transformation matrices. Examples of arm matrices for a microrobot and 5-axis robot are shown.

1. 1
Industrial Robotics
Lecture - 10
By
Prof. Punit Kumar
Department of Mechanical Engineering
National Institute of Technology Kurukshetra,

2. 2
CHAPTER 3
Direct Kinematics

3. 3
LINK
COORDINATES

4. 4
Denavit-Hartnberg
(D-H)
Representation
Systematic assignment of
coordinate frames to each link

5. 5
D-H Algorithm
1. Number the joints from 1 to n starting
from the base and ending with tool
yaw, pitch and roll in that order
2. Assign a right handed orthogonal
coordinate frame L0 (x0y0z0) to the base
such that z0 axis is aligned with the
axis of base joint (J1). Set k=1

6. 6
3. Assign zk to the axis of joint
(k+1)
4. Locate the origin of Lk at the
intersection of zk and zk-1. If they
do not intersect, use the
intersection of zk with the
common normal between
zk and zk-1

7. 7
5. Select xk to be orthogonal to both
zk and zk-1. If zk and zk-1 are
parallel, point xk away from zk-1
6. Select yk to form a right handec
cordinate frame Lk
7. Set the origin of Ln at the tool
tip.

8. 8
8. Align zk along the approach
vector, yk along the sliding vector
and xk along the normal vector of
the tool.

9. 9
Kinematic Parameters

10. 10
Link Length & Link Twist Angle

11. 11
Link Length ak
It is the distance to be covered by zk-1
along xk axis so as to intersect zk.
Link Twist Angle αk
It is the angle by which zk-1 should
be rotated about xk axis so as to
make it parallel to zk.

12. 12
Joint Distance & Joint Angle

13. 13
Joint Distance dk
It is the translation required along zk-1 axis
so that xk-1 intersects xk.
Joint Angle θk
It is the rotation about zk-1 axis
required to make xk-1 parallel to
xk.

14. 14
Example 1: Microrobot Alpha II

15. 15
Link Frames: Microrobot Alpha II

16. 16
Kinematic Parameters
Microrobot Alpha II

17. 17
Spherical Wrist

18. 18
ARM
EQUATION

19. 19
Arm Equation
Transformations to obtain Lk from Lk-1

20. 20
Link Trnasformation Matrix
























 


1
0
0
0
0
cos
sin
0
0
sin
cos
0
0
0
1
1
0
0
0
1
0
0
0
0
cos
sin
0
0
sin
cos
1
k
k
k
k
k
k
k
k
k
k
k
k
a
d
T









21. 21
Link Trnasformation Matrix

22. 22
 T
n
q
q
q
q
vector
coordinate
jo
d
Generalize
q
where
......
int
3
2
1


Arm Matrix
           
q
T
q
T
q
T
q
T
q
T
q
T Tool
Base
n
n
n
k
k
k
n

 
 1
1
2
2
1
1
1
0
0 ....
...

23. 23
Arm Equation

24. 24
Arm Matrix
             
q
T
q
T
q
T
q
T
q
T
q
T
q
T Tool
Base
n
n
n
n

 
 1
4
4
3
3
3
2
2
2
1
1
1
0
0 ....
.
     
n
Tool
Wrist
Wrist
Base
Tool
Base q
q
q
T
q
q
q
T
q
T ,....,
,
.
,
, 5
4
3
2
1

       
3
3
2
2
2
1
1
1
0
3
2
1 ,
, q
T
q
T
q
T
q
q
q
TWrist
Base 
     
n
n
n
n
Tool
Wrist q
T
q
T
q
q
q
T 1
4
4
3
5
4 ....
,....,
, 



25. 25
Example 2: Arm Matrix of Microrobot
Alpha II

26. 26
Link Frames: Microrobot Alpha II

27. 27
Kinematic Parameters
Microrobot Alpha II

28. 28

29. 29

30. 30
Example 3: Five Axes Articulated
Robot (Rhino-XR3)

31. 31
Link Frames: RHINO-XR3

32. 32
Kinematic Parameters
RHINO XR3

33. 33

34. 34

35. 35

36. 36
CHAPTER 4
Inverse Kinematics