1. Inverse Kinematics Professor Nicola Ferrier ME 2246, 265-8793 ferrier@engr.wisc.edu

2. Forward Kinematics position equation: Where the transformation for each link is built from our DH parameters: End-effector w.r.t. {e}

3. Forward Kinematics Forward Kinematic Transformation: Has an rotation and translation: Roll,pitch,yaw Angle-axis Euler Y-Z-Y

4. Forward Kinematics Pick one rotation description: Decompose R, i.e. solve:

5. Forward Kinematics Pick one rotation description: Find terms that you can easily solve equations

6. Decomposition of Rotation Matrices Roll, Pitch, and Yaw The inverse solution for  in (-/2,/2):

7. Decomposition of Rotation Matrices Roll, Pitch, and Yaw The inverse solution for  in (/2,3/2):

8. Decomposition of Rotation Matrices Euler Angles Inverse solution for r 13  0 and r 23  0 (singularity when sin=0):

9. Decomposition of Rotation Matrices OR

10. Decomposition of Rotation Matrices Angle Axis Inverse solution (singularity at =0,,... and there are two solutions for § ):

11. Forward Kinematics Pick one rotation description: Decompose T :

12. Cartesian Space & Joint Space Joint Space (q 1,q 2,…,q N ) Cartesian Space (x,y,z,,,) Forward kinematics Inverse kinematics (arm solution)

13. Inverse Relationship? Find joint positions for a given end- effector pose Also find joint positions, speeds and accelerations for a sequence of poses

14. Inverse Relationship? Analytic Inverse –Robot specific –Multiple solutions –Once found, computationally simple Geometric –Robot specific –Insight? Iterative Computation (ME739) –Computationally expensive –General solution

15. A Heuristic for Inverse Kinematics The solution to the inverse kinematic equations (called the arm solution) can often be found using a heuristic approach. –Does not guarantee a solution –Solution may not be unique –Some solutions may be redundant

16. IK Heuristic Algorithm 1.Perform forward kinematics to find the general transformation matrix 2.Equate the transformation matrix to the manipulator transformation matrix. a)For a particular solution the manipulator transformation matrix contains “numbers” b)For a general solution the manipulator transformation matrix contains “variables”

17. IK Heuristic Algorithm 3.Look at both matrices for: a)Elements which contain only one joint variable b)Pairs of elements which will produce an expression in only one joint variable when divided (look for divisions that result in the atan2 function) c)Elements, or combinations of elements, that can be simplified using trigonometric identities 4.Having selected an element, equate it to the corresponding element in the other matrix to produce an equation. Solve this equation to find a description of one joint variable in terms of the elements of the manipulator transformation matrix.

18. IK Heuristic Algorithm 5.Repeat step 4 until all the elements identified in step 3 have been used. 6.If any of these solutions suffer from inaccuracies, undefined results, or redundant results, set them aside and look for better solutions.

19. IK Heuristic Algorithm 7.If there are more joint angles to be found, pre-multiply both sides of the matrix equation by the inverse of the A 1 matrix to produce a new set of equivalent matrix elements.

20. IK Heuristic Algorithm 8.Repeat steps 3 to 7 until either solutions to all the joint variables have been found, or you have run out of A matrices to pre- multiply. 9.If a suitable solution cannot be found for a joint variable, choose one of those discarded in step 6, taking note of regions where problems may occur. 10.If a solution cannot be found for a joint variable in terms of the elements of the manipulator transform, it may be that the manipulator cannot achieve the specified position and orientation: the position is outside the manipulator’s workspace.

21. IK Heuristic Algorithm Note: theoretical solutions may not be physically attainable because of the mechanical limits on the range of joint variables.

22. DH Example: “academic manipulator” Z1Z1 Z0Z0 Z2Z2 11 22 x0x0 x1x1 x2x2 LinkVar  d  a 1 11 11 L1L1 90 o 0 2 22 22 00L2L2 L1L1 L2L2 11

23. Steps 1 and 2 The general transformation matrix Set equal to:

24. Steps 3 and 4 Look for elements and select terms

25. More steps…. Pre-multiply for a new equation:

26. DH Example: two link planar arm Z0Z0 Z1Z1 11 22 DH table for two link arm x0x0 x1x1 x2x2 Z2Z2 LinkVar  d  a 1 11 11 00L1L1 2 22 22 00L2L2 L1L1 L2L2

27. Forward Kinematics: planar 2-link manipulator

28. Heuristic solution? Can’t select an “obvious” set of equations…

29. Trignometric Solutions Sometimes simple (but usually not)!

30. Trig Solutions For first angle – revisit equations

31. Trig Solutions A trig hint: Let Then So

32. Solution to 2 link planar arm Arm solution is –where

33. Multiple Solutions?!

34. Geometric Solutions Intuition, insight….?

35. Geometric Solutions Intuition, insight….?

36. Geometric Solutions Intuition, insight….? Get same solution as trig method

37. Solution to 2 link planar arm Arm solution is –where Here we know the sign by observation