1. KINEMATICS ANALYSIS OF ROBOTS (Part 3)

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 kinematics analysis using transformation matrices Kinematics Analysis of Robots III

3. Example: A 3 DOF RRR Robot Link and Joint Assignment Link (2)Link (3)Link (1) Revolute joint Link (0) Revolute joint

4. Example: A 3 DOF RRR Robot Frame Assignment Z1Z1 Z1Z1 Y1Y1 Y1Y1 X1X1

5. Example: A 3 DOF RRR Robot Frame Assignment Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 Z2Z2 X2X2 Y2Y2 Y2Y2

6. Example: A 3 DOF RRR Robot Frame Assignment Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 Z3Z3 X3X3 Y3Y3 Y3Y3

7. Example: A 3 DOF RRR Robot Frame Assignment Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 11 22 33

8. Example: A 3 DOF RRR Robot Tabulation of D-H parameters Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 A B  0 = (angle from Z 0 to Z 1 measured along X 0 ) = 0° a 0 = (distance from Z 0 to Z 1 measured along X 0 ) = 0 d 1 = (distance from X 0 to X 1 measured along Z 1 )= 0  1 = variable (angle from X 0 to X 1 measured along Z 1 )  1 = 0° (at home position) but  1 can change as the arm moves

9. Example: A 3 DOF RRR Robot Tabulation of D-H parameters Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 A B  1 = (angle from Z 1 to Z 2 measured along X 1 ) = 90° a 1 = (distance from Z 1 to Z 2 measured along X 1 ) = A d 2 = (distance from X 1 to X 2 measured along Z 2 ) = 0  2 = variable (angle from X 1 to X 2 measured along Z 2 )  2 = 0° (at home position) but  2 can change as the arm moves

10. Example: A 3 DOF RRR Robot Tabulation of D-H parameters Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 A B  2 = (angle from Z 2 to Z 3 measured along X 2 ) = 0° a 2 = (distance from Z 2 to Z 3 measured along X 2 ) = B d 3 = (distance from X 2 to X 3 measured along Z 3 ) = 0  3 = variable (angle from X 2 to X 3 measured along Z 3 )  3 = 0° (at home position) but  3 can change as the arm moves

11. Link i Twist  i Link length a i Link offset d i Joint angle  i i=000…… i=190°A0  1 (  1 =0° at home position) i=20B0  2 (  2 =-0° at home position) i=3……0  3 (  3 =-0° at home position) Summary of D-H parameters

12. Example: A 3 DOF RRR Robot Tabulation of Transformation Matrices from the D-H table

13. Example: A 3 DOF RRR Robot Tabulation of Transformation Matrices from the D-H table

14. Example: A 3 DOF RRR Robot Tabulation of Transformation Matrices from the D-H table

15. Example: A 3 DOF RRR Robot Forward Kinematics

16. Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 A=3 B=2C=1 P Example: A 3 DOF RRR Robot What is the position of point “P” at the home position? Solution:

17. Example: A 3 DOF RRR Robot  1 =  2 =  3 =0, A=3, and B=2:

18. Example: A 3 DOF RRR Robot Inverse Kinematics Given the orientation and position of point “P”:

19. Example: A 3 DOF RRR Robot Inverse Kinematics Equate elements (1,3) and (2,3): Provided that Equate elements (1,4) and (2,4):

20. Example: A 3 DOF RRR Robot Inverse Kinematics If cos(  1 )  0, then use p x to find cos(  2 ). Afterwards, find Otherwise use p y to find sin(  2 ) and then solve using

21. Example: A 3 DOF RRR Robot Inverse Kinematics Equate elements (3,1) and (3,2): Now find  1,  2, and  3 given the orientation and position of point “P”:

22. Example: A 3 DOF RRR Robot Inverse Kinematics Now cos(  1 )=1  0. We use p x to find cos(  2 ):

23. Example: A 3 DOF RRR Robot Inverse Kinematics Y0Y0 X0X0 Z0Z0 Z3Z3 X3X3 Y3Y3 A=3 B=2 P C=1

24. Forward & Inverse Kinematics Issues Given a set of joint variables, the forward kinematics will always produce an unique solution giving the robot global position and orientation. On the other hand, there may be no solution to the inverse kinematics problem. The reasons include: The given global position of the arm may be beyond the robot work space The given global orientation of the gripper may not be possible given that the gripper frame must be a right hand frame For the inverse kinematics problem, there may also exist multiple solutions, i.e. the solution may not be unique.

25. Forward & Inverse Kinematics Issues Example of multiple solutions given the same gripper global position and orientation: First solution Second solution First solution Second solution Some solutions may not be feasible due to obstacles in the workspace

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