1. Introduction to Robotics Tutorial 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. This transformation is done in several steps :
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. Multiplying the matrices :
Denavit-Hartenberg
Multiplying the matrices :

14. DH Example 3 revolute joints Shown in home position R L1 L2 joint 1
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 2 3 x21 x0 Z1
Observe that frame i moves with link i

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

17. DH Example
Link
Var  d  a 1 1
90o R 2 2 L1 3 3 L2

18. DH Example

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

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

21. DH Example z0 x3 z3 2 3 x2 x1 z2 1 x0 z1 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. Example – the Stanford ArmY1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7

25. Example – the Stanford ArmY1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1
90° 1 2 3 4 5 6

26. Example – the Stanford ArmY1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1
90° 1 2 d2 2 3 4 5 6

27. Example – the Stanford ArmY1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1
90° 1 2 d2 2 3
d3 (var) 4 5 6

28. Example – the Stanford ArmY1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1
90° 1 2 d2 2 3
d3 (var) 4 d4 4 5 6

29. Example – the Stanford ArmY1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1
90° 1 2 d2 2 3
d3 (var) 4 d4 4 5 d5 0° 5 6

30. Example – the Stanford ArmY1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1
90° 1 2 d2 2 3
d3 (var) 4 d4 4 5 d5 0° 5 6 d6 6