Introduction to Robotics Tutorial II

1. Introduction to RoboticsTutorial II Alfred Bruckstein Yaniv Altshuler

2. Denavit-Hartenberg Reminder • Specialized description of articulated figures • Each joint has only one degree of freedom • rotate around its z-axis • translate along its z-axis

3. Denavit-Hartenberg • Link length ai • The perpendicular distance between the axes of jointi and jointi+1

4. Denavit-Hartenberg • Link twist αi • The angle between the axes of jointi and jointi+1 • Angle around xi-axis

5. Denavit-Hartenberg • Link offset di • The distance between the origins of the coordinate frames attached to jointi and jointi+1 • Measured along the axis of jointi

6. Denavit-Hartenberg • Link rotation (joint angle) φi • The angle between the link lenghts αi-1 and αi • Angle around zi-axis

7. Denavit-Hartenberg • Compute the link vector ai and the link length • Attach coordinate frames to the joint axes • Compute the link twist αi • Compute the link offset di • Compute the joint angle φi • Compute the transformation (i-1)Ti which transforms entities from linki to linki-1

8. Denavit-Hartenberg This transformation is done in several steps : • Rotate the link twist angle αi-1 around the axis xi • Translate the link length ai-1 along the axis xi • Translate the link offset di along the axis zi • Rotate the joint angle φi around the axis zi

9. Denavit-Hartenberg

10. Denavit-Hartenberg

11. Denavit-Hartenberg

12. Denavit-Hartenberg

13. Denavit-Hartenberg Multiplying the matrices :

14. DH Example 3 revolute joints Shown in home position joint 1 R Link 2 Link 3 Link 1 joint 2 joint 3 L1 L2

15. DH Example Shown with joints in non-zero positions Z0 x3 z3 3 2 x2 x1 Z2 1 x0 Z1 Observe that frame i moves with link i

16. DH Example 1 = 90o(rotate by 90o around x0 to align Z0 and Z1) R Z0 L2 L1 1 x1 x2 x3 x0 1 3 2 Z3 Z1 Z2

17. DH Example

18. DH Example

19. z0 x3 z3 3 2 x2 x1 z2 1 x0 z1 DH Example x1 axis expressed wrt {0} y1 axis expressed wrt {0} z1 axis expressed wrt {0} Origin of {1} w.r.t. {0}

20. z0 x3 z3 3 2 x2 x1 z2 1 x0 z1 DH Example x2 axis expressed wrt {1} y2 axis expressed wrt {1} z2 axis expressed wrt {1} Origin of {2} w.r.t. {1}

21. z0 x3 z3 3 2 x2 x1 z2 1 x0 z1 DH Example x3 axis expressed wrt {2} y3 axis expressed wrt {2} z3 axis expressed wrt {2} Origin of {3} w.r.t. {2}

22. DH Example where

23. Example – the Stanford Arm

24. Z7 Z6 Z4 X7 Z5 X6 X4 X3 Z3 Z2 X5 Z1 X2 Y1 X1 Example – the Stanford Arm

25. Z7 Z6 Z4 X7 Z5 X6 X4 X3 Z3 Z2 X5 Z1 X2 Y1 X1 Example – the Stanford Arm

26. Z7 Z6 Z4 X7 Z5 X6 X4 X3 Z3 Z2 X5 Z1 X2 Y1 X1 Example – the Stanford Arm

27. Z7 Z6 Z4 X7 Z5 X6 X4 X3 Z3 Z2 X5 Z1 X2 Y1 X1 Example – the Stanford Arm

28. Z7 Z6 Z4 X7 Z5 X6 X4 X3 Z3 Z2 X5 Z1 X2 Y1 X1 Example – the Stanford Arm

29. Z7 Z6 Z4 X7 Z5 X6 X4 X3 Z3 Z2 X5 Z1 X2 Y1 X1 Example – the Stanford Arm

30. Z7 Z6 Z4 X7 Z5 X6 X4 X3 Z3 Z2 X5 Z1 X2 Y1 X1 Example – the Stanford Arm