1998 Summer Q2 Forward Kinematics

1. 1998 Summer Q2 Forward Kinematics

2. Use the DH Algorithm to assign the frames and kinematic parameters

3. 4-Tool Pitch 3 2 5-Tool Pitch 1 Number the joints 1 to n starting with the base and ending with the tool yaw, pitch and roll in that order. The location of frame origins is ascertained only when the staps of the algorithm are carried out. Note: There is no tool yaw in this case

4. Frame 0 z0 y0 x0 Assign a right-handed orthonormal frame L0 to the robot base, making sure that z0 aligns with the axis of joint. Set K=1

5. Frame 1 z1 z0 y0 x0 Align zk with the axis of joint k+1. Locate the origin of Lk at the intersection of the zk and zk-1axes

6. y1 Frame 1 x1 z1 z0 y0 x0 Select xk to be orthogonal to both zk and zk-1. Select yk to form a right handed orthonormal co-ordinate frame Lk y1 will be hidden after this for the purpose of clarity

7. z2 Frame 2 x1 z1 z0 y0 x0 Align zk with the axis of joint k+1. This is the line of action of the prismatic joint

8. x2 z2 Frame 2 y2 x1 z1 z0 y0 x0 Select xk to be orthogonal to both zk and zk-1. Select yk to complete the right handed orthonormal co-ordinate frame

9. x2 Frame 3 z2 y2 x1 z1 z3 z0 y0 x0 Align zk with the axis of joint k+1. Locate the origin of Lk at the intersection of the zk and zk-1axes

10. x2 y3 z2 Frame 3 y2 x1 x3 z1 z3 z0 y0 x0 Select xk to be orthogonal to both zk and zk-1. Select yk to complete the right handed orthonormal co-ordinate frame

11. x2 Frame 4 y3 z2 y2 x1 x3 z1 z3 z4 z0 y0 x0 Align zk with the axis of joint k+1, the tool roll joint The origin is actually at the same point as that of the tool pitch joint

12. x2 y3 Frame 4 z2 y2 x1 x3 y4 z1 z3 z4 x4 z0 y0 x0 Select xk to be orthogonal to both zk and zk-1. Select yk to complete the right handed orthonormal co-ordinate frame

13. x2 y3 Frame 5 z2 z5 y2 x1 x3 y5 z1 z3 x5 y4 z4 x4 z0 y0 x0 Set the origin of Ln at the tool tip. Align zn with the approach vector of the tool. Align yn with the sliding vector of the tool. Align xn with the normal vector of the tool.

14. x2 y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 x0 With the frames assigned the kinematic parameters can be determined.

15. x2 b5 y3 k = 5 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 x0 Locate point bk (b5) at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal between xk and zk-1

16. x2 5 bk y3 k = 5 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 x0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from xk-1 to xk about zk-1 is -90 degrees i.e. 5 = -90º But this is only for the soft home position, 5 is the joint variable.

17. d5 k = 5 x2 5 b5 y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 x0 Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1 Remember the roll joint frame is just moved out of position for clarity Compute ak as the distance from point bk to the origin of frame Lk along xk In this case these are the same point therefore a5=0

18. d5 k = 5 x2 5 b5 y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 Compute k as the angle of rotation from zk-1 to zk measured about xk It can be seen here that the angle of rotation from z4 to z5 about x5is zero i.e. 5 = 0º

19. d5 k = 4 x2 5 y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 b4 z4 x4 z0 y0 Locate point bk at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal between xk and zk-1

20. d5 k = 4 x2 5 y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 b4 z4 x4 z0 y0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from xk-1 to xk about zk-1 is zero i.e. 4 = 0º

21. d5 k = 4 x2 5 y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 b4 z4 x4 z0 y0 As the origin of both frames are at the same point the a and d values are zero in this case, ie a4=0 and d4=0 Compute k as the angle of rotation from zk-1 to zk measured about xk It can be seen here that the angle of rotation from z3 to z4 about x4is -90º i.e. 4 = -90º

22. d5 k = 3 x2 5 bk y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 Locate point bk at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal between xk and zk-1

23. d5 k = 3 x2 5 bk y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from x2 to x3 about z2 is 180º i.e. 3 = 180º

24. d3 d5 k = 3 x2 5 b3 y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1 This is the joint variable for joint 3 which is the prismatic joint Compute ak as the distance from point bk to the origin of frame Lk along xk In this case these are the same point, therefore a3=0

25. d3 d5 k = 3 x2 5 b3 y3 z2 z5 y2 x5 x1 z3 x3 y5 z1 y4 z4 x4 z0 y0 Compute k as the angle of rotation from zk-1 to zk measured about xk It can be seen here that the angle of rotation from z2 to z3 about x3 is 90º i.e. 3 = 90º

26. d3 d5 k = 2 x2 5 y3 z2 z5 y2 x5 x1 z3 x3 y5 b2 z1 y4 z4 x4 z0 y0 Locate point bk at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal between xk and zk-1

27. d3 d5 k = 2 x2 5 y3 z2 z5 y2 x5 x1 z3 x3 y5 bk z1 y4 z4 x4 z0 y0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from xk-1 to xk about zk-1 is 90º i.e. 2 = 90º

28. d3 d5 k = 2 x2 5 y3 z2 z5 a2 y2 x5 x1 z3 x3 y5 b2 z1 y4 z4 x4 z0 y0 Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1 This is zero in this case, as bk is at the origin of frame Lk-1 therefore d2 =0 Compute ak as the distance from point bk to the origin of frame Lk along xk

29. d3 d5 k = 2 x2 5 y3 z2 z5 a2 y2 x5 x1 z3 x3 y5 b2 z1 y4 z4 x4 z0 y0 Compute k as the angle of rotation from zk-1 to zk measured about xk It can be seen here that the angle of rotation from z1 to z2 about x2 is 90º i.e. 2 = 90º

30. d3 d5 k = 1 x2 5 y3 z2 z5 a2 y2 x5 x1 z3 x3 y5 z1 b1 y4 z4 x4 z0 y0 x0 Locate point bk at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal between xk and zk-1

31. d3 d5 k = 1 x2 5 y3 z2 z5 a2 y2 x5 x1 z3 x3 y5 z1 b1 y4 z4 x4 z0 y0 x0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from xk-1 to xk about zk-1 is zero i.e. 1 = 0º

32. d3 d5 x2 5 y3 z2 z5 a2 y2 x5 x1 z3 x3 y5 z1 bk y4 d1 z4 x4 z0 y0 x0 Compute dk as the distance from the origin of frame Lk-1to point bk along zk-1 Compute ak as the distance from point bk to the origin of frame Lk along xk This is zero in this case, therefore, a1 = 0.

33. d3 d5 x2 5 y3 z2 z5 a2 y2 x5 x1 z3 x3 y5 z1 b1 y4 d1 z4 x4 z0 y0 x0 Compute k as the angle of rotation from zk-1 to zk measured about xk-1 It can be seen here that the angle of rotation from zk-1 to zk about xk-1 is 90º i.e. 1 =90º

34. d3 d5 x2 5 y3 z2 z5 a2 y2 x5 x1 z3 x3 y5 z1 y4 d1 z4 x4 z0 y0 x0 From this drawing a table of D-H parameters can be compiled

35. d3 d5 Link θ a d α Home q o o 1 q 0 d = 0.8m 90 0 1 1 o o 2 q a =0.15m 0 90 90 2 2 o o 3 180 0 q =d =0.6+l 90 0.6+l 3 3 1 1 o o 4 q 0 0 - 90 0 4 o 5 q 0 d =0.55m 0 - 90 5 5 x2 5 y3 z2 z5 a2 y2 x5 x1 z3 x3 y5 z1 y4 d1 z4 x4 z0 y0 x0 o