1. Kinematics – Frame Assignment using Denavit-Hartenberg Convention
Professor Nicola Ferrier
ME Room 2246,

2. Coordinate Transformations
End-effector Z
Base
Supply
Table
Goal X Y

3. Coordinate Transformations
End-effector
Base
Supply
Goal
Table

4. Coordinate Transformations
Robot forward kinematic model

5. Manipulator Forward Kinematics
Motion is composition of elementary motions for each link
End-effector
Base

6. Relative Pose between 2 links

7. Relative Pose between 2 links
Frames can be chosen arbitrarily
Denavit-Hartenberg convention is used to assign frames – described in §3.2.2 of Spong, Hutchinson, Vidyasagar Text
Iterative process (start at base, assign frames for each link from base to end-effector)

8. DH Frame assignment
Frame {i} moves with link i when joint i is actuated
Zi axis is along joint axis i+1
Zi is axis of actuation for joint i+1 Zi
Link i-1
Link i+1
Link i
Zi-1

9. DH convention: Assign Z axes
Use actuation as a guide
Prismatic – joint slides along zi
Revolute – joint rotates around zi
Establish base frame {0}:
Nearly arbitrary
Start at base and assign frames 1,…,N
Pick x-axis and origin
y-axis chosen to form a right hand system

10. Robot Base
Often base is “given” or some fixed point on the work-table is used.
z0 is along joint axis 1
Original:
any point on z0 for origin
Modified DH:
{0} is defined to be completely co-incident with the reference system {1}, when the variable joint parameter, d1 or q1 , is zero.

11. DH convention: Assign X axes
Start at base and assign frames 1,…,N
Pick x-axis and origin
y-axis chosen to form a right hand system
Consider 3 cases for zi-1 and zi:
Not-coplanar
Parallel
Intersect

12. zi-1 and zi are not-coplanar
DH convention: x axis
zi-1 and zi are not-coplanar
Common normal to axes is the “link” axis
Intersection with zi is origin
Usually, xi points from frame i-1 to i
zi-1 Xi zi

13. DH convention: x axis zi and zi-1 are parallel zi-1 zi
Infinitely many common normals
Pick one to be the “link” axis
Choose normal that passes through origin of frame {i-1} pointing toward zi
Origin is intersection of xi with zi Xi
zi-1 zi

14. DH convention: x axis zi zi-1
If joint axes zi-1 and zi intersect, xi is normal to the plane containing the axes
xi = (zi-1  zi )
zi-1
link i Xi

15. DH convention: Origin non-coplanar Z
Origin of frame {i} is placed at intersection of joint axis and link axis zi xi

16. Yi is chosen to make a right hand frame
DH convention: y axis
Yi is chosen to make a right hand frame Zi
xi points from frame i-1 to i Yi xi

17. DH convention: Origin parallel Z
zi and zi-1 are parallel
Origin is intersection of xi with zi
zi-1 zi xi

18. DH convention: x axis - parallel Z
zi and zi-1 are parallel
Origin is intersection of xi with zi
Yi is chosen to make a right hand frame yi
zi-1 zi xi

19. DH convention: origin zi zi-1
If joint axes intersect, the origin of frame {i} is usually placed at intersection of the joint axes zi
zi-1
link i xi

20. Yi is chosen to make a right hand frame
DH convention: y axis
Yi is chosen to make a right hand frame zi
zi-1 yi
link i xi

21. End-Effector Frame Frame to which the gripper is attached Z4 Ze Xe
Sometimes {n} is used
denoted by {e} (or {n+1} in many texts)
Often simple translation along Xn axis Z4 Ze Xe

22. End-Effector Frame Frame to which the gripper is attached – Often: Z4
denoted by {e} (or {n+1} in many texts)
Often simple translation along Xn axis
Often:
Origin between grippers
Z points outward (approach)
Y points along pinch direction (sliding)
X points normal Z4 ye xe ze

23. Link Parameters
ai+1 Zi
Z’i
Zi-1
Zi+1
Link i ai
ai+1 ai

24. Joint Parameters i
di+1
i+1 di i

25. Original DH Frame is placed at distal end of link xi screw motion-1
Frame is placed at distal end of link
xi screw motion
zi-1 screw motion

26. DH Frames and Parameters

27. Robot Revolute Joint DH

28. Prismatic Joint DH

29. Described by 4 parameters:
Link Transformations
Described by 4 parameters:
ai : twist
ai : link length
di : joint offset
qi : joint angle
Joint variable is di or qi
Build Table with values for each link:
Link
Var  d  a 1 1
90o L1 2 d2 

30. Described by 4 parameters:
Link Transformations
Described by 4 parameters:
ai : twist
ai : link length
di : joint offset
qi : joint angle
Joint variable is di or qi
Link Transformation is
zi-1 screw motion
xiscrew motion

31. Ai = A-matrices contains only one variable or
Equation 3.10 in Spong, Hutchinson, Vidyasagar

32. ! Original DH Frame is placed at distal end of link zi-1 screw motion
xi screw motion

33. ! Modified DH xi zi yi Zi+1 Zi Zi+2
Frame is placed at proximal end of link
xi-1 screw motion
zi screw motion

34. Modified DH – text figure

35. DH Example: “academic manipulator”
3 revolute joints
Shown in home position
joint 1 R
Link 2
Link 3
Link 1
joint 2
joint 3 L1 L2

36. DH Example: “academic manipulator”
Zi is axis of actuation for joint i+1 Z0
Z0 and Z1 are not co-planar
Z1 and Z2 are parallel 1 3 2 Z1 Z2

37. DH Example: “academic manipulator”
Z0 and Z1 are not co-planar:
x0 is the common normal Z0 1 x1 x2 x3 x0 3 2 Z1 Z3 Z2

38. DH Example: “academic manipulator”
Z0 and Z1 are not co-planar:
x0 is the common normal Z0 1 x1 x2 x3 x0 3 2 Z1 Z3 Z2
Z1 and Z2 are parallel :
x1 is selected as the common normal that lies along the center of the link

39. DH Example: “academic manipulator”
Z0 and Z1 are not co-planar:
x0 is the common normal Z0 1 x1 x2 x3 x0 3 2 Z1 Z3 Z2
Z2 and Z3 are parallel :
x2 is selected as the common normal that lies along the center of the link

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

41. DH Example: “academic manipulator”
Link lengths given
1 = 90o (rotate by 90o around x0 to align Z0 and Z1) R Z0 L1 L2 x1 x2 x3 1 x0 Z1 Z3 Z2

42. DH Example: “academic manipulator”
Build table 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

43. DH Example: “academic manipulator”
Link
Var  d  a 1 1
90o R 2 2 L1 3 3 L2

44. DH Example: “academic manipulator”

45. DH Example: “academic manipulator”z1 z0 z2 1 2 3 x0 x1 x2 z3 x3
x1 axis expressed wrt {0}
y1 axis expressed wrt {0}
z1 axis expressed wrt {0}
Origin of {1} w.r.t. {0}

46. DH Example: “academic manipulator”z1 z0 z2 1 2 3 x0 x1 x2 z3 x3
x2 axis expressed wrt {1}
y2 axis expressed wrt {1}
z2 axis expressed wrt {1}
Origin of {2} w.r.t. {1}

47. DH Example: “academic manipulator”z1 z0 z2 1 2 3 x0 x1 x2 z3 x3
x3 axis expressed wrt {2}
y3 axis expressed wrt {2}
z3 axis expressed wrt {2}
Origin of {3} w.r.t. {2}

48. DH Example: “academic manipulator”
where

49. DH Example: “academic manipulator” – alternate end-effector frame
Zi is axis of actuation for joint i+1 Z0
Z0 and Z1 are not co-planar
Z1 and Z2 are parallel 1
Pick this z3 3 2 Z1 Z2

50. DH Example: “academic manipulator” – alternate end-effector frameZ0 y2 1 x1 x2 1 x0 Z3 3 2 Z1 Z2
Would need to rotate about y2 here!

51. DH Example: “academic manipulator” – alternate end-effector frameZ0
x’2 1 x1 x2 1 x0 Z3 3 2 Z1
Solution: Add “offset” to rotation about z2
(q3+90o )

52. DH Example: “academic manipulator” – alternate end-effector frameZ0
x’2 L2 x3 1 x1 x2 1 x0 Z3 3 2 Z1 Z2
Now can rotate about x’ to align z2 and z3

53. DH Example: “academic manipulator” – alternate end-effector frame
Link
Var  d  a 1 1
90o R 2 2 L1 3 3
3 +90o e - L2

54. DH Example: “academic manipulator” – alternate end-effector frameZ0
x’2 L1 L2 Z3 1 x1 x2 1 x0 3 2 Z1 Z2
Link
Var  d  a 1 1
90o R 2 2 L1 3 3
3 +90o

55. DH Example: “academic manipulator” – alternate end-effector frameZ0
x’2 L1 L2 Z3 1 x1 x2 1 x0 Z3 3 2 Z1 Z2

56. DH Example: “academic manipulator” – alternate end-effector frameZ0
x’2 L1 L2 Z3 1 x1 x2 1 x0 Z3 3 2 Z1 Z2

57. DH Example: “academic manipulator” – alternate end-effector frameZ0
x’2 L1 L2 Z3 1 x1 x2 1 x0 Z3 3 2 Z1 Z2