1. KINEMATICS ANALYSIS OF ROBOTS (Part 5)

2. This lecture continues the discussion on the analysis of the forward and inverse kinematics of robots. After this lecture, the student should be able to: Solve problems of robot forward and inverse kinematics analysis using transformation matrices Kinematics Analysis of Robots V

3. Y3Y3 X3X3 Z3Z3 A 6 DOF Robot Y 0, Y 1, Z 2 X 0, X 1, X 2 Z 0, Z 1 Y2Y2 A2A2 d3d3 Z 4, Y 5, Z 6 X 4, X 5, X 6 Y 4, Z 5, Y 6 d4d4 A3A3

4. Link i Twist  i Link length a i Link offset d i Joint angle  i i=000…… i=1-9000 11 i=20A2A2 0 22 i=3-90A3A3 d3d3 33 i=4900d4d4 44 i=5-9000 55 i=6……0 66 Summary of D-H parameters

5. Transformation matrices

12. Forward Kinematics Given all the joint angles (  1,  2,  3,  4,  5, and  6 ) we can use the overall transformation matrix to solve for the position and orientation of frame {6}. The orientation of frame {6} w.r.t. frame {0} is defined by the rotational matrix: The translation of the origin of frame {6} w.r.t. frame {0} is defined by the vector

13. Inverse Kinematics We are now given the desired orientation and position of frame {6}, i.e. We now want to solve for all the joint angles (  1,  2,  3,  4,  5, and  6 )

14. Inverse Kinematics We are now given the desired orientation and position of frame {6}, i.e. We now want to solve for all the joint angles (  1,  2,  3,  4,  5, and  6 ). First, we get the following:

15. Inverse Kinematics

16. Equate elements (2,4) from both sides: Letwhere Therefore

17. Inverse Kinematics Equate elements (1,4), (2,4) and (3,4) from both sides: Square all equations and add them to get:

18. Letwhere Therefore Inverse Kinematics

22. Equate elements (1,4) and (2,4) from both sides: Rearranging: We can solve for (  2 +  3 ) from the above two equations:

23. Inverse Kinematics Equate elements (1,3) and (3,3) from both sides: Provided s 5  0

24. Inverse Kinematics

25. Equate elements (1,3) and (3,3) from both sides:

26. Inverse Kinematics To find  6, we use Equate elements (1,1) and (3,1) from both sides:

27. Inverse Kinematics Given the orientation and position of frame {6}, i.e. given all the joint angles (  1,  2,  3,  4,  5, and  6 ) can be found.

28. Summary This lecture continues the discussion on the analysis of the forward and inverse kinematics of robots. The following were covered: Robot forward and inverse kinematics analysis using transformation matrices