1. Just a quick reminder with another example
Recap D-H
Just a quick reminder with another example

2. 6 DOF revolute robot

3. 6 DOF revolute robot

5. 6 DOF revolute robot

6. Inverse Kinematic solution
As previously, to solve for the 6 angles, successively premultiply both matrices with An-1 matrices, starting with A1-1 .
At each stage it is possible to yield useful results…
…from large matrices and complex trig simplification due to many coupled angles.
Hence, results summarised here.
If you really want to know read Niku p87-93 (76-80)
(who obtained it from Paul, R.P. “Robot Manipulators, Mathematics, Programming & Control” MIT Press, 1981)

7. Summary for 6 d.o.f revolute robot
Don’t panic this is all in the Kinematics Data sheet!

8. Jacobian & Differential Motions
Look at velocity relationships rather than position.

9. Robot Dynamics
Previously we have only considered static robot mechanisms but they move!
Need to consider the velocity relationships between different parts of the robot mechanism.
Like the engineering dynamics you have already studied, we consider small movements i.e. a differential motion which is over a small period of time.

10. Consider a 2 DOF Mechanism
VB/A
VB (if VA = 0) VA
VB = VA + VB/A

11. 2 DOF mechanism
This equation can be expanded and written in matrix form:
Reason why this differs in second line is because you need to take into account that VA/B coordinate system changes due to point A moving i.e. axes x-z at point A are moving hence VB = VA + ω x rB/A + (VB/A)xyz (see Hibbeler dynamics p356) where middle bit is the angular velocity effect caused by rotation of the second link relative to the reference frame.

12. 2 DOF mechanism
Or differentiate the equations that describe the location of point B:

13. If you didn’t understand the last slide…
All dx, dy and dθ should be:
Hence:

14. Jacobian
It is a representation of the geometry of the elements of a mechanism in time.
In robotics, it relates joint velocities to Cartesian velocities of a point of interest e.g. the end-effector.
End-effector differential motions
Joint differential motions
displacement motions
rotation motions

15. Jacobian
dx, dy, dz (in [D]) represent the differential motions of the end-effector along the x-, y-, z-axes respectively.
δx, δy, δz (in [D]) represent the differential rotations of the end-effector along the x-, y-, z-axes respectively.
End-effector differential motions
Joint differential motions
displacement motions
rotation motions

16. Jacobian [Dθ] represents the differential motions of the joints
If matrices [D] and [Dθ] are divided by dt, they will represent the velocities.
End-effector differential motions
Joint differential motions
displacement motions
rotation motions

17. Jacobian Matrix compared with kinematics and with an alternative notation
Forward
Jacobian Matrix
Kinematics
Inverse
Jacobian Matrix: Relationship between joint
space velocity with world space velocity
Joint Space
World Space

18. Jacobian
Jacobian is a function of q, a joint variable (e.g. length or angle), it is not a constant!

19. Example 1
J, the Jacobian of a robot at a particular time is given. Calculate the linear and angular differential motions of the robots “Hand” frame for the given joint differential motions Dθ

20. Example 1

21. Differential Motions of a Frame
Differential motions of a frame – relate to differential motions of the robot

22. Differential translation
Trans (dx, dy, dz)

23. Differential Rotations
Approximate:
sin δx ≈ δx (in radians)
cos δx ≈ 1

24. Differential Rotation @ some random Axis
Any order SAME ANSWER velocities “commutative”
Neglect higher order differentials, e.g. δxδy→0

25. Example 2
Find total differential transformation from 3 small rotations: δx = 0.1, δy = 0.05, δz = 0.02 radians.

26. Differential Transformations of a Frame
If you have an original frame T
Unit or Identity matrix I

27. Differential Transformations of a Frame
[dT ] expresses the change in the frame after the differential transformation:
Differential operator

28. Differential Transformations of a Frame
Differential operator Δ
This is not a transformation nor a frame, merely an operator.

29. Example (part A)

30. Example 3A - solution

31. Differential Change – Example 3B
Find the location & orientation of frame B

32. Differential changes between frames
Differential operator Δ
Is relative to a fixed reference frame, UΔ
Define a differential operator TΔ relative to the current frame T.
As TΔ relative to the current frame T, we need to post-multiply (as we did in lecture 1).

33. Differential changes between frames

34. After some long derivation:
TΔ is made to look like Δ matrix where:

35. Example 3 Again Part C Find BΔ: From Boriginal matrix
From the differential operator OR differential transformation description

36. Example 3C continued

37. Example 3 – part D, alternative
Alternatively BΔ can be found using:
Same result as before!

38. Differential Motion of robot
From earlier slide:
How are differential motions accomplished by the robot?
End-effector differential motions
Joint differential motions
displacement motions
rotation motions

39. To Calculate the Jacobian
Next lecture!

40. Jacobian Matrix Inverse Jacobian Singularity Avoid it
rank(J)<min{6,n},
Jacobian Matrix is less than full rank
Jacobian is non-invertible
Occurs when two or more of the axes of the robot form a straight line, i.e., collinear
Avoid it