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# Workspace analysis of stewart platform

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### Workspace analysis of stewart platform

1. 1. Workspace Increase of StewartPlatform Using Movable Base By: Marzieh Nabi Supervisor: prof. Malaek
2. 2. Introduction of Stewart PlatformApplications:Simulators Aircraft Car MotorcycleManufacturing ToolsMedicine Physio-Trappy Eye-SurgeryEntertainment Entertainment Skate-Learning
3. 3. Workspace Limitations1. Actuator Length Limit2. Joints Universal Joint3. Collision Between Actuators4. Dexterity
4. 4. Two Questions1. 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. 5. One SolutionDegree of Freedom
6. 6. Dodekapode Hybrid Stewart Platform (HSP)
7. 7. Comparison Between Optimization of SP and HSP SP Comparison Between OPTIMIZED SP and HSP
8. 8. Optimized SP: Workspace OptimizationCost Function
9. 9. Constraints of the Optimizationa. Actuators Length Limitationsb. Interference of Actuatorsc. Joint Angle Limitationsd. Dexterity
10. 10. Optimization Methodsa. Genetic Algorithmb. Adaptive Simulated Annealing
11. 11. Translational WorkspaceResults of the Simulation 3.5Desired 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. 12. Translational WorkspaceResult of the Simulation 3.5Desired 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. 13. Case 1 & 2No Joint Angle Limitation & min  2.7 m Covered Workspace = 96 %  20     20  min  2.3 m Covered Workspace = 82.1 %
14. 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 0.4Fitness Value 0.35 Convergence of GA 0.3 Case 2 0.25 0.2 0 10 20 30 40 50 60 Generation
15. 15. Case 3No 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. 16. 0.45 0.4 0.35 0.3Fitness 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. 17. Case 4  max Pla tfo rm 160Joint Angle Limitations  max Ba see  160  min  2.3 m Covered Workspace = 45.21 %
18. 18. Case 5  max Pla tfo rm 160Joint Angle Limitations  max Ba see  160   min 16    min 25 As Optimization Variables    min 34
19. 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 0.7Convergence 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. 20. Optimization With Adaptive Simulated Annealing1. Running time is more than the genetic algorithm (GA)2. Percentile of the Covered Workspace is Less than GA3. Slower Convergence rate compare to GA
21. 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. 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. 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. 24. Next Step of the Comparison Between SP and HSP Length of SP actuators Inverse Kinematic of OPTIMIZED SPCoordinate ofEnd Effector Comparison Inverse Kinematic of HSP Length of HSP actuators So we need to solve the inverse kinematic of HSP
25. 25. Inverse Kinematic of HSP
26. 26. The DOF of Lower Robot Lower Robot Screw Theory 3 Translational DOF
27. 27. Dividing Strategies 1. Translational Coordinates of the Middle Plate  k 0  1  Translational Coordinates of End Effector k  0.5, 0.7
28. 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. 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) SPActuator 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 3Actuator 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. 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. 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 3.5Actuator 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. 32. l  lminMinimization 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. 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 3Actuator 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. 34. lMinimization 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. 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 SPActuator 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. 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. 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 lmaxActuator 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. 38. SummeryOptimization of the Stewart platform Genetic Algorithm Adaptive Simulated AnnealingInverse Kinematic of Hybrid Stewart PlatformComparison between the optimized SP and HSP
39. 39. Submitted papers1. S.M. Malaek, M. Nabi-A, “Optimal Design of Stewart Platform Using Genetic Algorithm”, ICINCO, 9-12 May 2007, Angers, France2. S.M. Malaek, M. Nabi-A, “Optimal Design of Stewart Platform Using Adaptive Simulated Annealing”, ICINCO, 9-12 May 2007, Angers, France
40. 40. Thanks for Your Attention