The document describes research to increase the workspace of a Stewart platform using an optimized movable base. It introduces Stewart platforms and their applications. It then discusses limitations of workspace due to actuator length and joint constraints. Two questions are posed regarding platform architecture and guaranteeing continuous motion. Solutions explored include a hybrid Stewart platform with a movable middle plate, optimization of platform geometry using genetic algorithms and adaptive simulated annealing, and inverse kinematics analysis. The optimized hybrid platform is shown to increase workspace coverage compared to a standard Stewart platform.

1. Workspace Increase of Stewart
Platform Using Movable Base
By: Marzieh Nabi
Supervisor: prof. Malaek

2. Introduction of Stewart Platform
Applications:
Simulators Aircraft
Car
Motorcycle
Manufacturing Tools
Medicine Physio-Trappy
Eye-Surgery
Entertainment
Entertainment
Skate-Learning

3. Workspace Limitations
1. Actuator Length Limit
2. Joints
Universal Joint
3. Collision Between Actuators
4. Dexterity

4. Two Questions
1. What should be the architecture of the Stewart platform
for an specific application?
2. How to guarantee continues motion of the Stewart
platform at each step simulation considering the joints
and the actuators constraints?

5. One Solution
Degree of Freedom

6. Dodekapode
Hybrid Stewart Platform
(HSP)

7. Comparison Between Optimization of
SP and HSP SP
Comparison Between OPTIMIZED SP
and HSP

8. Optimized SP: Workspace Optimization
Cost Function

9. Constraints of the Optimization
a. Actuators Length Limitations
b. Interference of Actuators
c. Joint Angle Limitations
d. Dexterity

10. Optimization Methods
a. Genetic Algorithm
b. Adaptive Simulated Annealing

11. Translational Workspace
Results of the Simulation
3.5
Desired Workspace 3
2.5
Z
2
 20     20 
1.5
2
1 2
0
-1
L 0
1
-1
-2 -2
Y X
Rotational Workspace
 20     20 
 20     20 

12. Translational Workspace
Result of the Simulation
3.5
Desired Workspace 3
2.5
Z
2
1.5
Rotational Workspace
2
1 2
0
-1
L 0
1
-1
20 -2 -2
Y X
10
0

-10
-20
20
10 20
0 10
-10 0
-10
-20 -20
 

13. Case 1 & 2
No Joint Angle Limitation &
 min  2.7 m Covered Workspace = 96 %
 20     20 
 min  2.3 m Covered Workspace = 82.1 %

14. 0.25
0.2
Fitness Value
0.15
Convergence of GA
0.1
Case 1
0.05
0
0 5 10 15 20 25 30 35 40
Generation
0.5
0.45
0.4
Fitness Value
0.35 Convergence of GA
0.3
Case 2
0.25
0.2
0 10 20 30 40 50 60
Generation

15. Case 3
No Joint Angle Limitation &
  min 16

  min 25 As Optimization Variables

  min 34
 min 16  3.0 m


 min 25  3.0
Covered Workspace = 99.65 %
m

 min 34  3.0
 m

16. 0.45
0.4
0.35
0.3
Fitness Value
0.25
Convergence of GA
0.2
0.15
Case 3
0.1
0.05
0
0 10 20 30 40 50
Generation
4
3
Schematic of SP 2
Case 3 1
0
2 4
1 2
0 0
-1 -2
-2 -4

17. Case 4
 max Pla tfo rm
160
Joint Angle Limitations
 max Ba see
 160
 min  2.3 m
Covered Workspace = 45.21 %

18. Case 5
 max Pla tfo rm
160
Joint Angle Limitations
 max Ba see
 160
  min 16

  min 25 As Optimization Variables

  min 34

19.  min 16  3.0 m


 min 25  2.92 m Covered Workspace = 94.47 %

 min 34  2.48 m
 1
0.9
0.8
0.7
Convergence of GA 0.6
Fitness Value
0.5
Case 5 0.4
0.3
0.2
0.1
0
0 5 10 15 20 25 30 35 40 45 50
Generation

20. Optimization With Adaptive Simulated Annealing
1. Running time is more than the genetic algorithm
(GA)
2. Percentile of the Covered Workspace is Less than
GA
3. Slower Convergence rate compare to GA

21. Rate of Convergence: GA and ASA
1
0.9
0.8 ASA
0.7
ASA
Fitness Value
0.6
0.5
0.4
0.3 GA
GA
0.2
0.1
0
0 5 10 15 20 25 30 35 40
Iteration
Case 1

22. Rate of Convergence: GA and ASA
1
0.9
0.8
ASA
ASA
ASA
0.7
Fitness Value
0.6
0.5
GA
0.4
GA
0.3
0.2
0.1
0
0 10 20 30 40 50 60
Iteration
Case 2

23. Rate of Convergence: GA and ASA
1
0.95
ASA
0.9
ASA
0.85
Fitness Value
0.8
0.75
0.7
GA GA
0.65
0.6
0.55
0.5
0 5 10 15 20 25 30 35 40 45 50
Iteration
Case 4

24. Next Step of the Comparison Between SP and HSP
Length of SP actuators
Inverse Kinematic of
OPTIMIZED SP
Coordinate of
End Effector Comparison
Inverse Kinematic of
HSP
Length of HSP actuators
So we need to solve the inverse kinematic of HSP

25. Inverse Kinematic of HSP

26. The DOF of Lower Robot
Lower Robot
Screw Theory 3 Translational DOF

27. Dividing Strategies
1. Translational Coordinates of the Middle Plate  k 0  1 
Translational Coordinates of End Effector
k  0.5, 0.7

28. k  0.7 x1 y1 z1   k x y z 
Actuator 1&2(m) 4.5
4
3.5
3 Actuator 1 & 2
2.5
0 5 10 15 20 25 30
time (s)
k  0.7
2.6
2.4
Actuator 3(m)
2.2
2
Actuator 3
1.8
1.6
0 5 10 15 20 25 30
time (s)

29. 5 4
Stewart Platform Stewart Platform
Hybrid Stewart Platform Act. 4 & 9 Hybrid Stewart Platform
4.5
Act. 5 & 8
3.5
Actuator 5&8(m)
SP
Actuator 4&9 (m)
4
3.5
SP 3
HSP
HSP
3
2.5
2.5 0 5 10 15 20 25 30
0 5 10 15 20 25 30
time (s)
time (s)
3.6
Stewart Platform 6
HSP
3.4
Hybrid Stewart Platform
Act. 6 & 7 5
3.2
4
3
3
Actuator 6&7(m)
2.8
2.6
SP 2
2.4 1
2.2
HSP 0
4
2 2
0
1.8
-2
4 6
1.6 0 2
0 5 10 15 20 25 30 -4 -4 -2
-6
time (s)

30. k  0.5 x1 y1 z1   k x y z 
4
Actuator 1&2(m)
3.5
3
Actuator 1 & 2
2.5
0 5 10 15 20 25 30
time (s)
2
1.8
Actuator 3(m)
1.6
1.4 Actuator 3
0 5 10 15 20 25 30
time (s)

31. 5 4
Stewart Platform Stewart Platform
Hybrid Stewart Platform Act. 4 & 9 3.8
Hybrid Stewart Platform
4.5
3.6
Act. 5 & 8
3.4
HSP
Actuator 4&9 (m)
SP
Actuator 5&8(m)
4
3.2
SP 3
3.5 HSP
2.8
3 2.6
2.4
2.5
0 5 10 15 20 25 30 2.2
0 5 10 15 20 25 30
time (s)
3.6 time (s)
Stewart Platform
3.4
Hybrid Stewart Platform Act. 6 & 7 HSP
5
3.2
4.5
3 4
3.5
Actuator 6&7(m)
2.8
2.6
SP 3
2.5
2.4 HSP 2
1.5
2.2 1
0.5
2
0
4
1.8 2
0 6
4
-2 2
1.6 0
-2
0 5 10 15 20 25 30 -4 -4
-6
time (s)

32. l  lmin
Minimization of f1 
lmax
2.6
Actuator 1&2(m)
2.4
2.2
2
Actuator 1 & 2
1.8
0 5 10 15 20 25 30
time (s)
1.3
1.2
Actuator 3(m)
1.1
1
0.9
Actuator 3
0.8
0 5 10 15 20 25 30
time (s)

33. 4.8 4
Stewart Platform Stewart Platform
4.6
Hybrid Stewart Platform Act. 4 & 9 Hybrid Stewart Platform
3.5
Act. 5 & 8
4.4
4.2
Actuator 5&8(m)
Actuator 4&9 (m)
4
3
SP HSP
3.8 SP
HSP 2.5
3.6
3.4 2
3.2
1.5
3 0 5 10 15 20 25 30
0 5 10 15 20 25 30 time (s)
3.6 time (s)
Stewart Platform
4.5
Hybrid Stewart Platform
3.4
Act. 6 & 7 4 HSP
3.2
3.5
3 3
Actuator 6&7(m)
2.8 2.5
2.6 SP 2
1.5
2.4
1
2.2
0.5
2
0
1.8 HSP 5
0
1.6
0 5 10 15 20 25 30
-5
-6 -4 -2 0 2 4 6
time (s)

34. l
Minimization of f2 
lmax
3
Actuator 1&2(m)
2.8
2.6
Actuator 1 & 2
2.4
2.2
0 5 10 15 20 25 30
time (s)
2
1.8
Actuator 3(m)
1.6
1.4
Actuator 3
1.2
1
0 5 10 15 20 25 30
time (s)

35. 5 4
Stewart Platform Stewart Platform
Hybrid Stewart Platform Act. 4 & 9 Hybrid Stewart Platform
3.5 Act. 5 & 8
4.5
3
SP
Actuator 5&8(m)
Actuator 4&9 (m)
4
HSP
SP 2.5
3.5
HSP 2
3 1.5
1
2.5 0 5 10 15 20 25 30
0 5 10 15 20 25 30
time (s)
4 time (s)
Stewart Platform
Hybrid Stewart Platform 5
HSP
3.5 Act. 6 & 7 4.5
4
3.5
3
SP
Actuator 5&8(m)
3
2.5
2.5
2
1.5
2 1
0.5
1.5 0
HSP 4
2
0
-2 4 6
1 -2 0 2
0 5 10 15 20 25 30 -4 -4
-6
time (s)

36. Comparison Between Different 4.5
Hybrid Stewart Platform (K=0.7)
Hybrid Stewart Platform (K=0.5)
Hybrid Stewart Platform (|(l-l )/l |)
min max
Actuator 1&2(m)
4
K=0.7 Hybrid Stewart Platform (|l/lmax|)
Strategies 3.5
l  lmin
3 f1 
lmax
2.5
0 5 10 15 20 25 30
time (s)
2.5
K=0.7
2
Actuator 3(m)
1.5
l  lmin
1 f1 
lmax
5
Stewart Platform 0.5
0 5 10 15 20 25 30
Hybrid Stewart Platform (K=0.7)
time (s)
Hybrid Stewart Platform (K=0.5)
Hybrid Stewart Platform ( ||l-lmin/lmax || )
4.5
Hybrid Stewart Platform ( ||l/lmax || )
SP
Actuator 4 & 9
Actuator 4&9 (m)
4
l  lmin
f1 
lmax
3.5
3
K=0.7
2.5
0 5 10 15 20 25 30
time (s)

37. 4
Stewart Platform
Hybrid Stewart Platform (K=0.7)
3.8 Hybrid Stewart Platform (K=0.5)
Hybrid Stewart Platform ( || ( l-l ) /l || )
SP
min max
3.6 Hybrid Stewart Platform ( || l/lmax || )
l  lmin
3.4 f1  Actuator 5 & 8
lmax
Actuator 5&8(m)
3.2
3
2.8
K=0.7
2.6
2.4
3.6
Stewart Platform
2.2 Hybrid Stewart Platform (K=0.7)
0 5 10 15 20 25 3.4 30
Hybrid Stewart Platform (K=0.5)
time (s)
Hybrid Stewart Platform ( || ( l-l ) /l || )
3.2 min max
Hybrid Stewart Platform ( || l/l || )
max
3
SP
Actuator 6&7(m)
2.8
l  lmin
2.6 f1 
Actuator 6 & 7 2.4
lmax
2.2
2
1.8 K=0.7
1.6
0 5 10 15 20 25 30
time (s)

38. Summery
Optimization of the Stewart platform
Genetic Algorithm
Adaptive Simulated Annealing
Inverse Kinematic of Hybrid Stewart Platform
Comparison between the optimized SP and HSP

39. Submitted papers
1. S.M. Malaek, M. Nabi-A, “Optimal Design of Stewart Platform Using
Genetic Algorithm”, ICINCO, 9-12 May 2007, Angers, France
2. S.M. Malaek, M. Nabi-A, “Optimal Design of Stewart Platform Using
Adaptive Simulated Annealing”, ICINCO, 9-12 May 2007, Angers,
France

40. Thanks for Your Attention