1. Review: Homogeneous Transformations
Homogeneous vector
Homogeneous transformation matrix

2. Review: Aim of Direct Kinematics
Compute the position and orientation of the end effector as a function of the joint variables

3. Review: Direct Kinematics
The direct kinematics function is expressed by the homogeneous transformation matrix

4. Review: Open Chain
Computation of direct kinematics function is recursive and systematic

5. Review: Denavit-Hartenberg Convention

6. Review : D-H Convention

7. Class Problem: Spherical Wrist
1. Fill in the table of D-H parameters for the spherical wrist.
2. write the three D-H transformation matrices (one for each joint) for the spherical wrist
3. Find the overall transformation matrix which relates the final coordinates (x6y6z6) to the “base” coordinates (x3y3z3) for the spherical wrist

8. Review : D-H Convention

9. Joint Space and Operational Space
Description of end-effector task
position: coordinates (easy)
orientation: (n s a) (difficult)
w.r.t base frame
Function of time
Operational space
Joint space
Prismatic: d
Revolute: theta
Independent variables

10. Joint Space and Operational Space
Direct kinematics equation
Three-link planar arm
(Pp )
M的最大值是？平面运动只需要3个自由度

11. Joint Space and Operational Space
Generally not easy to express

12. Joint Space and Operational Space
Workspace
reachable workspace
dexterous workspace
Factors determining workspace
Manipulator geometry
Mechanical joint limits
Mathematical description of workspace
Workspace is finite, closed, connected

13. Workspace Example

14. Performance Indexes of Manipulator
Accuracy of manipulator
Deviation between the reached position and the position computed via direct kinematics.
repeatability of manipulator
A measure of the ability to return to a previously reached position.

15. Kinematic Redundancy Definition
A manipulator is termed kinematically redundant when it has a number of degrees of mobility which is greater than the number of variables that are necessary to describe a given task.

16. Kinematic Redundancy Intrinsic redundancy m<n functional redundancy
relative to the task
Why to intentionally utilize redundancy?

17. Kinematic Calibration
Kinematic calibration techniques are devoted to finding accurate estimates of D-H parameters from a series of measurements on the manipulator’s end-effector location.
Direct measurement of D-H is not allowed.

18. Inverse Kinematics

19. Inverse Kinematics
we know the desired “world” or “base” coordinates for the end-effector or tool
we need to compute the set of joint coordinates that will give us this desired position (and orientation in the 6-link case).
the inverse kinematics problem is much more difficult than the forward problem!

20. Inverse Kinematics
there is no general purpose technique that will guarantee a closed-form solution to the inverse problem!
Multiple solutions may exist
Infinite solutions may exist, e.g., in the case of redundancy
There might be no admissible solutions (condition: x in (dexterous) workspace)

21. Inverse Kinematics
most solution techniques (particularly the one shown below) rely a great deal on geometric or algebraic insight and a few common “tricks” to generate a closed-form solution
Numerical solution techniques may be applied to all problems, but in general do not allow computation of all admissible solutions

22. Three-link Planar Arm
x is known, compute q

23. Three-link Planar Arm
W can be expressed both as a function of end-effector p&o, and as a function of a reduced number of joint variables

24. Class problem
Two-link planar arm
one-link planar arm

25. Three-link Planar Arm
Algebraic approach

26. Three-link Planar Arm
no admissible solution If c2 is out of this range
Elbow up and elbow down

27. Three-link Planar Arm
对c12和s12用三角公式展开，则得二元方程组，变量为s1和c1。

28. Three-link Planar Arm Geometric approach l
Feasible condition: a1+a2>l and |a1-a2|<l

29. Class Problem
what are the forward and inverse kinematics equations for the two-link planar robot shown on the right? Y0
2nd Joint:
Prismatic
90 deg
Attention: m= ?
1st Joint:
Revolute X0