1 / 43

Dynamic Modeling of a 2-Link Manipulator: Gravitational and Inertial Forces Analysis

In this paper, we present a detailed Lagrangian Electromechanical (L-E) dynamical model of a two-link manipulator, focusing on gravitational, inertial, Coriolis, centrifugal, and frictional forces. We define the L-E algorithm starting with link parameters and compute transformation matrices. The inertial tensor for each link is derived using their respective properties, leading to the development of the dynamic equation D(q) for our model. Finally, we analyze the velocity coupling matrix and gravitational effects to derive accurate torque models for each link.

urbain
Télécharger la présentation

Dynamic Modeling of a 2-Link Manipulator: Gravitational and Inertial Forces Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 an 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 Lets 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 Di • Considers each link a thin cylindrical bar • These are Inertial Tensors with respect to a Fc aligned with the link Framesi at the Cm

  10. Continuing for Link 1

  11. Simplifying:

  12. Continuing for Link 2

  13. Now let’s 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

  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 Inertial Tensor of Linki defined in the Base space

  20. Building D1 • 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: 2x2 because it is a 2-jointed robot!!

  24. Building the Full Manipulator D(q) • D(q)2 = D(q)1 +(A2)Tm2A2 + (B2)TD2B2 • Recalling (from J2):

  25. Building the ‘2nd’ Term (the Linear Vel. part from the second link):

  26. Specifics on term (1,1) ~ details!

  27. Building the 3rd Term: • Recall D2: • Then:

  28. Combining the 3 Terms to construct the Full D(q) Manipulator Inertial Tensor “Solution”: 2x2 because it is 2 Joints (still)!

  29. Simplifying then, the Manipulator inertial Tensor D(q) is: NOTE: D(q)man is Square in number of Joints! 2x2 here because this is a 2-jointed robot!!

  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 (1st link) • From Earlier:

  33. P & C cont – 1st Link:

  34. Finding h1: • Given: gravity vector points in –Y0 direction • gk =(0, -g0, 0)T • g0 is gravitational constant • In the h model Akij is extracted from the relevant Jacobian matrix • Here: 3-directions of Cartesian base space! Here, i=1

  35. Continuing (i =1): From Jacobianj Upper half – specified components Gravitational effect: Would be 3 terms (BUT) only K = 2 survives since gravitational vector points in –Y0 direction

  36. Stepping to Link 2 (i = 2)

  37. Computing h2 Would be 3 terms (BUT) only K = 2 survives as gravitational vector points in –Y0 direction

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

  39. For Link 1 (i=1): • The 1st terms: • 2nd Terms:

  40. Writing the Complete Link 1 Model

  41. And, Finally, For Link 2 (i=2):

  42. Initial Link 2 terms: • !st Terms: • 2nd Terms:

  43. Finalizing Link 2 Torque Model:

More Related