Introduction to Robotics cpsc - 460

1. Introduction to Roboticscpsc - 460 Lecture 3A – Forward Kinematics

2. DH Techniques • A link j can be specified by two parameters, its length aj and its twist αj • Joints are also described by two parameters. The link offset dj is the distance from one link coordinate frame to the next along the axis of the joint. The joint angle θjis the rotation of one link with respect to the next about the joint axis.

3. DH Techniques • Link twistαi :the angle from the Zi-1 axis to the Ziaxis about the Xi axis. The positive sense for α is determined from zi-1 and ziby the right-hand rule. • Joint angle θithe angle between the Xi-1 and Xi axes about the Zi-1 axis.

4. DH Techniques

5. DH Techniques • The four parameters for each link ai: link length αi: Link twist di: Link offset θi : joint angle • With the ith joint, a joint variable is qiassociated where

6. Transformation Matrix • Each homogeneous transformation Ai is represented as a product of four basic transformations

7. Transformation Matrix

8. Transformation Matrix • The matrix Ai is a function of only a single variable, as three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable, depending on whether it is a revolute or prismatic link

9. DH Notation Steps

10. DH Notation Steps

11. DH Notation Steps • From , the position and orientation of the tool frame are calculated.

12. Transformation Matrix

13. Example I - Two Link Planar Arm • Base frame O0 • All Z ‘s are normal to the page

14. Example I - Two Link Planar Arm Where (θ1 + θ2 ) denoted by θ12 and

15. Example 2 15

16. Forward Kinematics of Example 2 16

17. Example 3 - Three Link Cylindrical Manipulator

18. Example 3 - Three Link Cylindrical Manipulator

19. Example 3 - Three Link Cylindrical Manipulator

20. Example 3 - Three Link Cylindrical Manipulator

21. Example 4 – The Spherical Wrist

22. Example 4 – The Spherical Wrist

23. Example 4 – The Spherical Wrist

24. Example 4 – The Spherical Wrist

25. Example 5 - Cylindrical Manipulator with Spherical Wrist derived in Example 2, and derived in Example 3.

26. Example 5 - Cylindrical Manipulator with Spherical Wrist

27. Example 5 - Cylindrical Manipulator with Spherical Wrist

28. Example 5 - Cylindrical Manipulator with Spherical Wrist • Forward kinematics: 1. The position of the end-effector: (dx ,dy ,dz ) 2. The orientation {Roll, Pitch, Yaw }

29. Rotation – Roll, Pitch, Yaw • The rotation matrix for the following operations: Z Y X

30. Example 4The three links cylindrical with Spherical wrist • How to calculate • Compare the matrix R • With the rotation part of