1. Relations Between Engineering Fields

2. Duality between engineering systems
Since graph representations are mathematical entities, mathematical relations can be established between them, such as duality relations:
(giG1)=Tdual(gj G2) G1 G2
Tdual gj gi

3. Duality between engineering systems
Duality between graph representations yield the duality relations between the represented engineering systems. G1 G2 Db Da D T’ gj gi sj T si

4. Dual Representation Design Technique

5. Duality between Linkages and Structures
Duality between Stewart platforms and Robots
Duality between Planetary and Beam Systems

6. Duality relation between linkages and structures

7. Consider a kinematical linkage and its graph representation

8. Consider a kinematical linkage and its graph representation

9. Now, consider a static structure and its graph representation
Kinematical Linkage
Static Structure

10. There exist a mathematical relation between the representations of the two systems
Kinematical Linkage
Static Structure

11. Therefore there is the duality relation between the represented engineering systems
Kinematical Linkage
Static Structure

12. The relative velocity of each link of the linkage is equal to the internal force in the corresponding rod of the structure
Kinematical Linkage
Static Structure

13. The equilibrium of forces in the structure is thus equivalent to compatibility of relative velocities in the linkage
Kinematical Linkage
Static Structure

14. Checking system stability through the duality
Stable ???? 8 5 9 2 4 7 10 11 1 12 6 3 11 7 3 4 12 2 1 5 8 9 10 6
Due to links 1 and 9 being located on the same line
Definitely locked !!!!! 8 12 ’ 2 1 10 6 7 3 5 9 R’ 4 12 ’ 9 10 R’ 11 6 7 8 2 3 5 1 4

15. Duality relation between Stewart platform and serial robot

16. Consider a Stewart platform system.1 2 3 4 5 6 P

17. BUILDING THE GRAPH REPRESNTATION OF THE STEWART PLATFORM3 3 2 2 4 4 1 1 5 5 6 6

18. BUILDING THE DUAL GRAPH REPRESNTATION1 2 3 4 5 6 P 1 P 2 3 4 6 5 P 3 2 4 1 5 6

19. BUILDING THE DUAL SPATIAL LINKAGE (SERIAL ROBOT)1 2 3 4 5 6 P 1 P 2 3 4 5 6 1 2 3 4 5 6 P

20. BUILDING THE DUAL SPATIAL LINKAGE (SERIAL ROBOT)1 2 3 4 5 6 P 1 P 2 3 4 5 6 1 2 3 4 5 6 P

21. VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMS The correspondence in geometric positions of the kinematical pairs of the linkage and the legs of the Stewart platform. P

22. VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMS Each force in the Stewart platform leg is directed in the same direction as the relative angular velocity of the corresponding kinematical pair in the dual serial robot. P

23. Duality relation between planetary and beam systems

24. Consider a simple two-gear system.B 1 2

25. BUILDING THE GRAPH REPRESNTATION OF THE GEAR SYSTEMA C B 1 2 2 C rC B 1 rB A rA

26. VERIFYING THE ISOMORPHISM OF THE REPRESENTATIONA C B 1 2 2 C
w1/0 rC B
rA x w1/0 1 rB A rA

27. VERIFYING THE ISOMORPHISM OF THE REPRESENTATIONA C B 1 2 2 C
w1/0+w2/1 rC B
rA x w1/0+rB x w2/1 1 rB A rA

28. VERIFYING THE ISOMORPHISM OF THE REPRESENTATIONA C B 1 2 2 C
w1/0+w2/1+w0/2=0 rC B
rA x w1/0+rB x w2/1+rC x w0/2=0 1 rB A rA

29. VERIFYING THE ISOMORPHISM OF THE REPRESENTATIONA C B 1 2 2 C
w1/0+w2/1+w0/2=0 rC B
rA x w1/0+rB x w2/1+rC x w0/2=0 1 rB A rA

30. VERIFYING THE ISOMORPHISM OF THE REPRESENTATIONA C B 1 2 2 C
w1/0+w2/1+w0/2=0 rC B
rA x w1/0+rB x w2/1+rC x w0/2=0 1 rB A rA

31. DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH KNOWLEDGEA C B 1 2 2 C
DA+w2/1+w0/2=0 rC B
rA x DA+rB x w2/1+rC x w0/2=0 1 rB A rA

32. DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH KNOWLEDGEA C B 1 2 2 C
DA+DB+w0/2=0 rC B
rA x DA+rB x DB+rC x w0/2=0 1 rB A rA

33. DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH KNOWLEDGEA C B 1 2 2 C
DA+DB+DC=0 rC B
rA x DA+rB x DB+rC x DC=0 1 rB A rA

34. DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH KNOWLEDGEA A C B I B 1 2 2 C
DA+DB+DC=0 rC B
rA x DA+rB x DB+rC x DC=0 1 rB A rA

35. BUILDING THE DUAL GRAPH REPRESENTATIONC C I A A A C B B B 1 2 2 C
DA+DB+DC=0 rC B
rA x DA+rB x DB+rC x DC=0 1 rB A rA

36. DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL REPRESENTATIONA A C I B B 1 2 2 C
FA+DB+DC=0 rC B
rA x FA+rB x DB+rC x DC=0 1 rB A rA

37. DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL REPRESENTATIONA A C I B B 1 2 2 C
FA+FB+DC=0 rC B
rA x FA+rB x FB+rC x DC=0 1 rB A rA

38. DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL REPRESENTATIONA A C I B B 1 2 2 C
FA+FB+FC=0 rC B
rA x FA+rB x FB+rC x FC=0 1 rB A rA

39. DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL REPRESENTATIONA A C I B B 1 2 2 C
FA+FB+FC=0 rC B
rA x FA+rB x FB+rC x FC=0 1 rB A rA

40. BUILDING THE DUAL BEAM SYSTEMC I A A C B B 1 2 2 C C rC
FA+FB+FC=0 rC B B
rA x FA+rB x FB+rC x FC=0 rB 1 rB A A rA rA

41. VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMSC I A A C B B 1 2 2 C C rC
PA+FB+FC=0 rC B B
rA x PA+rB x FB+rC x FC=0 rB 1 rB A A rA rA

42. VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMSC I A A C B B 1 2 2 C C rC
PA+RB+FC=0 rC B B
rA x PA+rB x RB+rC x FC=0 rB 1 rB A A rA rA

43. VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMSC I A A C B B 1 2 2 C C rC
PA+RB+RC=0 rC B B
rA x PA+rB x RB+rC x RC=0 rB 1 rB A A rA rA

44. VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMSC I A A C B B 1 2 2 C C rC
PA+RB+RC=0 rC B B
rA x PA+rB x RB+rC x RC=0 rB 1 rB A A rA rA