ROBOTICS 01PEEQW Basilio Bona DAUIN – Politecnico di Torino

Dynamics

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 /2016 Dynamics – 1

 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 /2016 Dynamics – 2

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

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

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

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

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

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

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

12 ROBOTICS 01PEEQW /2016 Lagrange equations – 1

13 ROBOTICS 01PEEQW /2016 Lagrange equations – 2

14 ROBOTICS 01PEEQW /2016 Lagrange equations – 3

15 ROBOTICS 01PEEQW /2016 Lagrange equations – 4

16 ROBOTICS 01PEEQW /2016 Kinetic Energy – 1

17 ROBOTICS 01PEEQW /2016 Kinetic Energy – 2

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

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

20 ROBOTICS 01PEEQW /2016 Potential Energy – 1

21 ROBOTICS 01PEEQW /2016 Potential Energy – 2

22 ROBOTICS 01PEEQW /2016 Potential Energy – 3

23 ROBOTICS 01PEEQW /2016 Generalized forces – 1

24 ROBOTICS 01PEEQW /2016 Generalized forces – 2

25 ROBOTICS 01PEEQW /2016 Generalized forces – 3

26 ROBOTICS 01PEEQW /2016 Final equations – 1

27 ROBOTICS 01PEEQW /2016 Final equations – 2

28 ROBOTICS 01PEEQW /2016 Final equations – 3

29 ROBOTICS 01PEEQW /2016 Physical interpretation –

30 ROBOTICS 01PEEQW /2016 Physical interpretation –

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

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

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

34 ROBOTICS 01PEEQW /2016 Dynamic calibration – 1

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

36 ROBOTICS 01PEEQW /2016 State equations – 1

37 ROBOTICS 01PEEQW /2016 State equations – 2

38 ROBOTICS 01PEEQW /2016 Direct and inverse dynamics

39 ROBOTICS 01PEEQW /2016 Numerical recursive algorithms – 1

40 ROBOTICS 01PEEQW /2016 Numerical recursive algorithms – 2

41 ROBOTICS 01PEEQW /2016 Numerical recursive algorithms – 3

42 ROBOTICS 01PEEQW /2016 Numerical recursive algorithms – 4

43 ROBOTICS 01PEEQW /2016 Numerical recursive algorithms – 5

44 ROBOTICS 01PEEQW /2016 Numerical recursive algorithms – 6

 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 /2016 Conclusions Nonlinearities Products, squares, trigonometric functions here