Robot theory

1. Chapter 3: Inverse Kinematics
Instructor: Dr. Dang Xuan Ba
Email : badx@hcmute.edu.vn
Robotics

2. Content
3.1 Introduction
3.2 Planar robot
3.3 Spatial robot
Instructor: Dr. Dang Xuan Ba 2

3. 3.1 Introduction
Instructor: Dr. Dang Xuan Ba 3

4. What is inverse kinematics?
Instructor: Dr. Dang Xuan Ba 4
Fig. C3.1.1: A typical robot configuration
Joint-based state
variables (Joint
angles)
End-effector-based state
variables (Position and
orientation)
Forward kinematics
Inverse kinematics

5. What is inverse kinematics?
Instructor: Dr. Dang Xuan Ba 5
Fig. C3.1.2: Location of standard frames
B : Base
W: Wrist
T : Tool
S : Station
G : Goal
6 unknown joint angles
R: 9 eqs
P: 3 eqs
Example in 6DOF Robot:
0
0 6
6
0 1
R P
T
 
=  
 
R: 3 ind eqs
P: 3 eqs
Highly
Nonlinear
Solvability?

6. Solvability
Instructor: Dr. Dang Xuan Ba 6

7. Existence of Solution
Instructor: Dr. Dang Xuan Ba 7
Work space
Joint limitation Number of DOF
Volume of space which the end-effector of manipulator can reach.
Example:
1

2

0
x
0
y
( , )
ee ee
x y
1 2
l =
2 1
l =
1
x
1
y
2
x
2
y
Work space

8. Multiple Solution
Instructor: Dr. Dang Xuan Ba 8
Number of solutions
Number of joints
Function of link parameters
Range of motion of the joints
Examples:
Environments
2 Solutions 1 Solution
4 Solutions

9. Method of Solution
Instructor: Dr. Dang Xuan Ba 9
Inverse Kinematics Solution
Analytical (Closed-form) Solution
(Phương pháp phân tích)
Numerical Solution
(Phương pháp số)
Algebraic
(Đại số)
Geometric
(Hình học)
Necessary condition: Serial 6DOF robot is solvable.
Sufficient condition: Closed-form solution of a 6DOF robot is possible in cases of that joint axes of three consecutive
revolution joints intersect at a single point for all configuration.

10. 3.2 Planar robot
Instructor: Dr. Dang Xuan Ba 10

11. Algebraic solution
Instructor: Dr. Dang Xuan Ba 11
Fig. C3. 2. 1: A 3R Planar robot
Given: 0 0 0
( , , )
x y  Find: 1 2 3
( , , )?
  
D-H Table:
123
123
1 1 2 12
1 1 2 12
c c
s s
x l c l c
y l s l s


=


=


= +

 = +

Direct Reduction

12. Homework 1
Instructor: Dr. Dang Xuan Ba 12
Fig. C3. 2. 1: A 3R Planar robot
Given:
0 0 0 0
3 3
( 0.5, 1.4, 30 )
ORG ORG
x y 
= = =
1)Find:
2) Re-verify using Matlab.
1 2 3
( , , )?
  
1 2
l =
2 1
l =
3 1
l =

13. Geometric solution
Instructor: Dr. Dang Xuan Ba 13
Fig. C3. 2. 1: A 3R Planar robot
2

3

1

Given: 0 0 0
( , , )
x y  Find: 1 2 3
( , , )?
  

14. 3.3 Spatial robot
Instructor: Dr. Dang Xuan Ba 14

15. Pieper’s method
Instructor: Dr. Dang Xuan Ba 15
Application: 6DOF Robot with the last 3 axes intersecting

16. Pieper’s method
Instructor: Dr. Dang Xuan Ba 16

17. Pieper’s method
Instructor: Dr. Dang Xuan Ba 17
1 1
2 2
1
2
3 3
1 1
g f
g f
T
g f
   
   
   
=
   
   
   
1 1 1 1 2
2 1 1 1 2
0 0
4 1
3 3
1 1
ORG
g c g s g
g s g c g
P T
g g
−
   
   
+
   
= =
   
   
   

18. Pieper’s method
Instructor: Dr. Dang Xuan Ba 18

19. Pieper’s method
Instructor: Dr. Dang Xuan Ba 19
3
( )
 2
( )
 1
( )

1 1 1 1 2
2 1 1 1 2
0 0
4 1
3 3
1 1
ORG
g c g s g
g s g c g
P T
g g
−
   
   
+
   
= =
   
   
   
4 5 6
( , , )
  

20. Unimation method
Instructor: Dr. Dang Xuan Ba 20
Application: 6DOF Robot with the last 3 axes intersecting

21. Unimation method
Instructor: Dr. Dang Xuan Ba 21

22. Unimation method – Step 1
Instructor: Dr. Dang Xuan Ba 22
1
( )


23. Unimation method – Step 2
Instructor: Dr. Dang Xuan Ba 23
Square and add
(Bình phương và cộng) 3
( )


24. Unimation method – Step 3
Instructor: Dr. Dang Xuan Ba 24
2
( )


25. Unimation method – Step 4
Instructor: Dr. Dang Xuan Ba 25
1 2 3
( , , )
  
4 5 6
( , , )
  

26. Homework 3
Instructor: Dr. Dang Xuan Ba 26
Given:
0 0 0
6 6 6
0 0 0 0
( 0.5, 1, 0,
30 , 0)
ORG ORG ORG
x y z
x y z
  
= = =
= = =
1)Find:
2) Re-verify using Matlab.
1 2 3 4 5 6
( , , , , , )?
     
3 4 2 3
0.1, 1, 1; 0.1;
d d a a
= = = =
DH table of PUMA 560:

27. End of Chapter 3
Instructor: Dr. Dang Xuan Ba 27