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The L-E (Torque) Dynamical Model:

The L-E (Torque) Dynamical Model:. Gravitational Forces. Inertial Forces. Coriolis & Centrifugal Forces. Frictional Forces. Lets Apply the Technique -- . Lets do it for a 2-Link “Manipulator”. Link 1 has a Mass of m1; Link 2 a mass of m2. Before Starting lets define a L-E Algorithm:.

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The L-E (Torque) Dynamical Model:

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  1. The L-E (Torque) Dynamical Model: Gravitational Forces Inertial Forces Coriolis & Centrifugal Forces Frictional Forces

  2. Lets Apply the Technique -- Lets do it for a 2-Link “Manipulator” Link 1 has a Mass of m1; Link 2 a mass of m2

  3. Before Starting lets define a L-E Algorithm:

  4. Before Starting lets define a L-E Algorithm:

  5. We Start with Ai’s Not Exactly D-H Legal (unless there is more to the robot than these 2 links!)

  6. So Let’s find T02 • T02= A1*A2

  7. I’ll Compute Similar Terms back – to – back rather than by the Algorithm

  8. C2(bar) Computation:

  9. Finding D1 • Consider each link a thin cylinder • These are Inertial Tensors with respect to a Fc aligned with the link Frames at the Cm

  10. Continuing for Link 1

  11. Simplifying:

  12. Continuing for Link 2

  13. Now lets compute the Jacobians

  14. Finishing J1 Note the 2nd column is all zeros – even though Joint 2 is revolute – this is the special case!

  15. Jumping into J2 This is 4th column of A1

  16. Continuing:

  17. And Again:

  18. Summarizing, J2 is:

  19. Developing the D(q) Contributions • D(q)I = (Ai)TmiAi + (Bi)TDiBi • Ai is the “Upper half” of the Ji matrix • Bi is the “Lower Half” of the Ji matrix • Di is the Normalized Inertial Tensor of Linki defined in the Base space but acting on the link end

  20. Building D(q)1 • D(q)1 = (A1)Tm1A1 + (B1)TD1B1 • Here:

  21. Looking at the 1st Term (Linear Velocity term)

  22. Looking at the 2nd Term (Angular Velocity term) • Recall that D1 is: • Then:

  23. Putting the 2 terms together, D(q)1 is:

  24. Building the Full Manipulator D(q) • D(q)man= D(q)0 + D(q)1+ D(q)2 • Where • D(q)2 = (A2)Tm2A2+ (B2)TD2B2 • And recalling (from J2):

  25. Building the 1st D(q)2Term:

  26. How about term (1,1) details!

  27. Building the 2rd D(q)2 Term: • Recall D2(nor. In.Tensor): • Then:

  28. Combining the 3 Terms to construct the Full D(q) term:

  29. Simplifying then D(q) is: NOTE: D(q)man is Square in the number of Joints!

  30. This Completes the Fundamental Steps: • Now we compute the Velocity Coupling Matrix and Gravitation terms:

  31. For the 1st Link

  32. Plugging ‘n Chugging • From Earlier: • THUS:

  33. P & C cont:

  34. Finding h1: • Given: gravity vector points in –Y0direction (remembering the model!) • gk =(0, -g0, 0)T • g0is the gravitational constant • In the ‘h’ model Akij is extracted from the relevant Jacobianmatrix (– for Joint i) • Here:

  35. Continuing: Note: Only k = 2 has a value for gk which is -g0!

  36. Stepping to Link 2

  37. Computing h2

  38. Building “Torque” Models for each Link • In General:

  39. For Link 1: • The 1st terms: • 2nd Terms:

  40. Writing the Complete Link 1 Model

  41. And, Finally, For Link 2:

  42. Ist 2 terms: • 1st Terms: • 2nd Terms:

  43. Finalizing Link 2 Torque Model:

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