1. ROBOTICS 01PEEQW Basilio Bona DAUIN – Politecnico di Torino

2. Dynamics

3. task space joint  Dynamics studies the relations between the task space forces/torques and the joint forces/torques in non-static equilibrium, i.e., when the robot moves  The dynamic model equation can be obtained applying two main approaches Lagrange equations based on energy functions Newton-Euler equations based on the equilibrium of the vector forces  The first approach is conceptually simpler and will be adopted here  The second approach is more efficient for implementation of recursive computer algorithms; only a brief review of this approach will be presented here 3 ROBOTICS 01PEEQW - 2015/2016 Dynamics – 1

4.  The dynamic equations of the robot can be obtained adopting the Lagrange approach  The derived state-space differential equations represent the robot dynamical model  Why state equations are necessary? Used for control design Used for robot simulation Used to implement model identification or parameter estimation algorithms 4 ROBOTICS 01PEEQW - 2015/2016 Dynamics – 2

5. 5 ROBOTICS 01PEEQW - 2015/2016 Newton-Euler approach – 1

6. 6 ROBOTICS 01PEEQW - 2015/2016 Newton-Euler approach – 2

7. 7 ROBOTICS 01PEEQW - 2015/2016 Newton-Euler approach – 3

8. 8 ROBOTICS 01PEEQW - 2015/2016 Newton-Euler approach – 4

9. 9 ROBOTICS 01PEEQW - 2015/2016 Newton-Euler approach – 5

10. 10 ROBOTICS 01PEEQW - 2015/2016 Newton-Euler approach – 6

11. 11 ROBOTICS 01PEEQW - 2015/2016 Newton-Euler approach – 7

12. 12 ROBOTICS 01PEEQW - 2015/2016 Lagrange equations – 1

13. 13 ROBOTICS 01PEEQW - 2015/2016 Lagrange equations – 2

14. 14 ROBOTICS 01PEEQW - 2015/2016 Lagrange equations – 3

15. 15 ROBOTICS 01PEEQW - 2015/2016 Lagrange equations – 4

16. 16 ROBOTICS 01PEEQW - 2015/2016 Kinetic Energy – 1

17. 17 ROBOTICS 01PEEQW - 2015/2016 Kinetic Energy – 2

18. 18 ROBOTICS 01PEEQW - 2015/2016 Kinetic Energy – 3 First form for the Kinetic Energy

19. 19 ROBOTICS 01PEEQW - 2015/2016 Kinetic Energy – 1 Second form for the Kinetic Energy

20. 20 ROBOTICS 01PEEQW - 2015/2016 Potential Energy – 1

21. 21 ROBOTICS 01PEEQW - 2015/2016 Potential Energy – 2

22. 22 ROBOTICS 01PEEQW - 2015/2016 Potential Energy – 3

23. 23 ROBOTICS 01PEEQW - 2015/2016 Generalized forces – 1

24. 24 ROBOTICS 01PEEQW - 2015/2016 Generalized forces – 2

25. 25 ROBOTICS 01PEEQW - 2015/2016 Generalized forces – 3

26. 26 ROBOTICS 01PEEQW - 2015/2016 Final equations – 1

27. 27 ROBOTICS 01PEEQW - 2015/2016 Final equations – 2

28. 28 ROBOTICS 01PEEQW - 2015/2016 Final equations – 3

29. 29 ROBOTICS 01PEEQW - 2015/2016 Physical interpretation – 1 2 1 43 5

30. 30 ROBOTICS 01PEEQW - 2015/2016 Physical interpretation – 2 1 2 3 4 5

31. 31 ROBOTICS 01PEEQW - 2015/2016 Properties of the Lagrange Equations – 1

32. 32 ROBOTICS 01PEEQW - 2015/2016 Properties of the Lagrange Equations – 2

33. 33 ROBOTICS 01PEEQW - 2015/2016 Properties of the Lagrange Equations – 3

34. 34 ROBOTICS 01PEEQW - 2015/2016 Dynamic calibration – 1

35.  Collecting all data one obtains 35 ROBOTICS 01PEEQW - 2015/2016 Dynamic calibration – 2  The linear least square solution is then computed, as follows

36. 36 ROBOTICS 01PEEQW - 2015/2016 State equations – 1

37. 37 ROBOTICS 01PEEQW - 2015/2016 State equations – 2

38. 38 ROBOTICS 01PEEQW - 2015/2016 Direct and inverse dynamics

39. 39 ROBOTICS 01PEEQW - 2015/2016 Numerical recursive algorithms – 1

40. 40 ROBOTICS 01PEEQW - 2015/2016 Numerical recursive algorithms – 2

41. 41 ROBOTICS 01PEEQW - 2015/2016 Numerical recursive algorithms – 3

42. 42 ROBOTICS 01PEEQW - 2015/2016 Numerical recursive algorithms – 4

43. 43 ROBOTICS 01PEEQW - 2015/2016 Numerical recursive algorithms – 5

44. 44 ROBOTICS 01PEEQW - 2015/2016 Numerical recursive algorithms – 6

45.  Dynamics equations are essential for modeling and control purposes  Modeling is easier to understand adopting the Lagrange energy function  Computer program are more efficient if they implement recursive Newton-Euler approach  Nonlinear state equations have this form 45 ROBOTICS 01PEEQW - 2015/2016 Conclusions Nonlinearities Products, squares, trigonometric functions here