1. KINEMATICS ANALYSIS OF ROBOTS (Part 1) ENG4406 ROBOTICS AND MACHINE VISION PART 2 LECTURE 8

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: Put into practice the concept of forward kinematics analysis of robots Derive transformation matrix between coupled links Formulate the kinematics of articulated robots in terms of the link transformation matrices Solve problems of robot kinematics analysis using transformation matrices Kinematics Analysis of Robots I

3. The position of gripper can be viewed as the origin of frame {3}, i.e. the distances along X 3,Y 3 and Z 3 are 0, i.e. X0X0 Y0Y0 Y3Y3 X3X3 Position of the gripper w.r.t. frame {3}

4. Position of the gripper w.r.t. frame {0} The same home position for the robot can be described w.r.t. frame {0} Obviously, the position of gripper w.r.t. frame {0} at the home position is at a distance of A 2 in the X 0 direction, a distance of A 1 in the Y 0 direction, and 0 distance in the Z 0 direction, i.e. X0X0 Y0Y0 A2A2 A1A1

5. Y3Y3 X3X3 Positioning of the gripper Notice that the robot arm will move as the joints rotate. The gripper position w.r.t. frame {3} is always fixed at no matter where the arm had moved. However, the gripper position w.r.t. frame {0} may be different! For example, let  2 rotates  90° from the home position: Home position  2 =  90° remains constant Y3Y3 X3X3

6. Positioning of the gripper After  2 has rotated  90° from the home position, the gripper position w.r.t. frame {0} has changed although its position w.r.t. frame {3} is fixed. The gripper position w.r.t. frame {0} is X0X0 Y0Y0  2 =  90° X0X0 Y0Y0 A 1 + A 2 The description of the gripper w.r.t. frame {0} as  1,  2 &  3 change is called the forward kinematics problem.

7. Forward Kinematics and Transformation Matrices  2 rotates  90° Y0Y0 X2X2 Y2Y2 Y3Y3 X3X3 The position and orientation of frame {3} w.r.t. {2} before and after  2 has rotated  90° are different. The matrix relates this changes of frame {3} w.r.t. frame {2} as  2 changes. In general, the transformation matrix between frame {i} w.r.t. {i-1} is represented by. The transformation matrices can be used to solve the forward kinematics problem. X0X0 Y0Y0 Y3Y3 X3X3 X2X2 Y2Y2

8. The transformation matrix can be obtained using the D-H parameters as follow: Transformation Matrix between 2 Coupled links Use the summary of D-H parameters to derive the 3 transformation matrices for the planar robot, i.e.

9. Link i Twist  i Link length a i Link offset d i Joint angle  i i=000…… i=10A1A1 0  1 (  1 =90° at home position) i=20A2A2 0  2 (  2 =-90° at home position) i=3……0  3 (  3 =-90° at home position) Summary of D-H parameters

10. Transformation Matrix for The transformation matrix can be obtained using the D-H parameters as follow: You leave  1 as it is because the angle can change when the robot arm moves, i.e. joint is a revolute joint

11. The transformation matrix can be obtained using the D-H parameters as follow: You leave  2 as it is because the angle can change when the robot arm moves, i.e. joint is a revolute joint Transformation Matrix for

12. The transformation matrix can be obtained using the D-H parameters as follow: You leave  3 as it is because the angle can change when the robot arm moves, i.e. joint is a revolute joint Transformation Matrix for

13. The transformation matrix for is You should revise the trigonometric relations if you are having problems with the above Transformation Matrix for

14. Did you get the above overall transformation matrix? Transformation Matrix for

15. Transformation matrices and Forward Kinematics Remember, matrix relates the transformation between frame {j} w.r.t. {i}. For example, relates the transformation of frame {3} w.r.t. frame {0} as  1,  2 &  3 changes. If the changes in  1,  2 &  3 are known, then the gripper position w.r.t frame {0} can be found using Where. The gripper location is assumed to be fixed at the origin of frame {3}. Hence

16. Notice that at the home position,  1 = 90°,  2 = -90°, and  3 = -90° Finding the gripper home position using Transformation Matrix Or

17. The position of gripper w.r.t. frame {0} is obvious given by Visualization of the gripper home position The home position for the robot is shown below: X0X0 Y0Y0 A1A1 which is the same as that obtained from A2A2

18. Use the overall transformation matrix to find the gripper position w.r.t. frame {0} if the arm rotates (from the home position) another  1 = 0°,  2 = 90°, and  3 = 90°. This means that the total rotation of  1 about Z 1 axis is 90° the total rotation of  2 about Z 2 axis is 0° the total rotation of  3 about Z 3 axis is 0° Or Finding new gripper position using Transformation Matrix

19. After the arm rotates (from the home position) another  1 = 0°,  2 = 90°, and  3 = 90°. This means that the total rotation of  1 about Z 1 axis is 90° the total rotation of  2 about Z 2 axis is 0° the total rotation of  3 about Z 3 axis is 0° The arm should looks like this after rotating  1 = 0° from the home position Y1Y1 X1X1 X2X2 Y2Y2 Visualizing the new gripper position

20. After the arm rotates (from the home position) another  1 = 0°,  2 = 90°, and  3 = 90°. This means that the total rotation of  1 about Z 1 axis is 90° the total rotation of  2 about Z 2 axis is 0° the total rotation of  3 about Z 3 axis is 0° The arm should looks like this after rotating  1 = 0° and  2 = 90° from the home position Visualizing the new gripper position Y1Y1 X1X1 Y1Y1 X1X1 X2X2 Y2Y2 Y3Y3 X3X3

21. After the arm rotates (from the home position) another  1 = 0°,  2 = 90°, and  3 = 90°. This means that the total rotation of  1 about Z 1 axis is 90° the total rotation of  2 about Z 2 axis is 0° the total rotation of  3 about Z 3 axis is 0° The arm should looks like this after rotating  1 = 0° and  2 = 90° and  3 = 90° from the home position Visualizing the new gripper position X2X2 Y2Y2 Y3Y3 X3X3

22. The arm should looks like this after rotating  1 = 0° and  2 = 90° and  3 = 90° from the home position. The gripper position w.r.t. frame {0} is similar to that obtained from the overall transformation matrix: Y0Y0 X0X0 Visualizing the new gripper position A1A1 A2A2 After the arm rotates (from the home position) another  1 = 0°,  2 = 90°, and  3 = 90°. This means that the total rotation of  1 about Z 1 axis is 90° the total rotation of  2 about Z 2 axis is 0° the total rotation of  3 about Z 3 axis is 0°

23. Frame of reference for gripper Notice that frame {3} have been used as a frame of reference for the gripper. This means that the orientation of the gripper can also be described based on frame {3} w.r.t. frame {0}. To understand this, we will next re-examine the overall transformation matrix. X0X0 Y0Y0 Y1Y1 X1,X1, Y2Y2 X2X2 X3X3 Y3Y3

24. Transformation Matrix for This transformation of gripper w.r.t. base frame {0} has the form where Specify the orientation of the gripper w.r.t. frame {0} Specify the position of the gripper w.r.t. frame {0}

25. Example for orientation & position of gripper Notice that at the home position,  1 = 90°,  2 = -90°, and  3 = -90° Gives the orientation of X 3 w.r.t. frame {0} Gives the orientation of Y 3 w.r.t. frame {0} Gives the orientation of Z 3 w.r.t. frame {0} Gives the position of gripper w.r.t. frame {0}

26. Visualization of previous Example Notice that at the home position,  1 = 90°,  2 = -90°, and  3 = -90° X0X0 Y0Y0 Gives the direction of X 3 w.r.t. frame {0}. X 3 is in the negative Y 0 direction Gives the direction of Y 3 w.r.t. frame {0}. Y 3 is in the positive X 0 direction Gives the direction of Z 3 w.r.t. frame {0}. Z 3 is in the positive Z 0 direction Gives the position of the gripper w.r.t. frame {0} A2A2 A1A1 X3X3 Y3Y3

27. Another Example Find the gripper orientation & position w.r.t. frame {0} if the arm rotates (from the home position) another  1 = 0°,  2 = 90°, and  3 = 90°. This means that the total rotation of  1 about Z 1 axis is 90° the total rotation of  2 about Z 2 axis is 0° the total rotation of  3 about Z 3 axis is 0°

28. The arm should look like this after rotating (from the home position) Visualizing the new gripper position & orientation another  1 = 0°,  2 = 90°, and  3 = 90°. X 3 is in the positive Y 0 direction Y 3 is in the negative X 0 direction Z 3 is in the positive Z 0 direction Gives the position of the gripper w.r.t. frame {0} X3X3 Y3Y3 X0X0 Y0Y0

29. Forward Kinematics - Summary Robot forward kinematics involve finding the robot arm global position and orientation given the angles of rotation of the linkages This problem can be solved using the overall transformation matrix Although we have use a planar robot to illustrate the concept, the approach can be applied to any robot moving in 3-D space. The approach involves the following: 1.Frame assignment 2.Tabulation of the D-H parameters 3.Formulation of the individual transformation matrices 4.Derivation of the overall transformation matrix

30. This lecture continues the discussion on the analysis of the forward and inverse kinematics of robots. The following were covered: The concept of forward kinematics analysis of robots Transformation matrix between coupled links Kinematics of articulated robots in terms of the link transformation matrices Robot kinematics analysis using transformation matrices Summary