Prof Charlton Lagrangian dynamics used in robotics

1. ROBOT DYNAMICS DYNAMIC ANALYSIS AND FORCES Prof. Charlton S. Inao Defence University DebreZeit , Ethiopia

2. Introduction • Analysis of forces, torques, inertias, loads, and accelerations. • Derivation of dynamic equations of motion. • Allows determination of important loads for design. • Assists in the selection of actuators.

3. Newtonian Mechanics • Easier for simpler systems. • Familiar to users

4. Lagrangian Mechanics • Easier for more complicated systems • Based on system’s energies • Systematic • Lagrangian is the difference between kinetic and potential energies of the system:

5. Lagrangian Relationships • Lagrangian relationships are:

6. Review of Basics • Potential energy is energy stored in an object due to its position or arrangement. • Kinetic energy is energy of an object due to its movement - its motion

7. Illustration

9. Lagrangian Mechanics • Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics • Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom. • Deals with energy.. The difference between Kinetic and Potential energy.

10. In this chapter, we’re going to learn about a whole new way of looking at things. • Consider the system of a mass on the end of a spring. We can analyze this, of course, by using F = ma to write downmm¨x =−kx. • The solutions to this equation are sinusoidal functions, as we well know. • We can, however, figure things out by using another method which doesn’t explicitly use • F = ma. • In many (in fact, probably most) physical situations, this new method is far superior to using F = ma.

11. Lagrangian mechanics • Please see examples in the book for the application of Lagrangian mechanics. For the following system we can derive equations of motion as:

12. Example 1

13. Kinematics Solution W=angular velocity of the wheel 2r=diameter X dot= linear velocity

14. Dynamics Solution

15. Notes:

16. Notes: velocity distance

17. Example 2 Newtonian mechanics VS Lagrangian mechanics

18. Solution a) Lagrangian Mechanics

19. b) Newtonian Mechanics

21. Example 3

22. Solution 3

23. a) KE=KE cart +KE pendulum 1

24. b) PE=PE spring + PE pendulum Pendulum Spring

25. b.1) PE pendulum

26. L=K-P

27. Solution 3 continued…

28. Review/Reference

29. Solution 3 continued…

30. Example 4 (For home study or work out exercise)

31. Solution # 4

32. Solution # 4 cont’d…..

33. Solution # 4 cont’d…..

34. Solution # 4 cont’d…..

35. Solution # 4 cont’d…..

36. Taking the derivatives of the Langrangian and substituting into Equation 4.4 yields to the ffg. Equation of Motion

37. Equation of motion in Matrix Form Solution # 4 cont’d…..

38. Example 5 (Class Discussion)

39. Solution # 5

41. Solution # 5 .. In matrix form

42. DETAILED SOLUTION No. 5 Solve for Kinetic Energy for each link Solve for K1

43. RECALL

44. Reference: Moment of InertiaThin rod about axis through end perpendicular to length

45. Solve for the X and Y component of M2 Horizontal component up to point m2 vertical component up to point m2

47. Com=center of mass

49. Differentiate XD and YD to get the velocity and the velocity of the arm distance up to point m2 U= r V=cosθ U V Recall

50. Calculate the VD ,up to Mass2and square it, for linear an d angular component of momentum expression, V2D