Motion and Manipulation

1. Motion and Manipulation Inverse Kinematics

2. Homogeneous Transformations • Recall that the general form is where R is a 3 x 3 rotation matrix and t is a translation vector of length 3. • If T is the transformation matrix for an entire arm, then R gives the orientation and t the position of its end-effector (or tip).

3. Inverse Kinematics • Given the configuration T, determine the joint variables that result in this configuration T. • Difficulty: • No general solution method • No unique solution

4. Inverse Kinematics The transformation matrix 0 Tncan be expressed in two different ways: • through the position and orientation of the tip • as a product of successive transformation matrices i Ti+1for 0 ≤ i ≤ n-1

5. Inverse Kinematics • Determine given

6. Inverse Kinematics • In the case of a 6 DOF arm: 12 nontrivial elements describe 6 unknown joint variables: only 3 elements of the 3 x 3 rotation submatrix are independent

7. Simplest Nontrivial Example • Determine joint variables θ1 and θ2 for a planar manipulator with two rotational degrees of freedom and tip at (X,Y) in world coordinates

8. Simplest Nontrivial Example • Solution for this simple example illustrates complexity of less trivial cases • Two solutions as expected:

9. Motion of the Manipulator • Simple tip trajectory results in complex parameter variations

10. Decoupling • Decompose transformation matrix into translation and rotation • Many arms have a wrist consisting of three revolute joint with orthogonal axes intersecting in a single point

11. Wrist with three orthogonal axes intersecting in a point: 3 t 6 = 0 and therefore 0 t 6 = 0 t 3 so Decoupling with the rotation matrix as follows

12. Decoupling Approach Assume the location of the wrist with respect to frame 3 is w and its desired location with respect to world frame 0 is p then • solve q1, q2, q3 from • solve q4, q5, q6 using

13. Decoupling Approach Note that • the first step determines the values of joint variables q1, q2, q3 to get the wrist in the right position • the second step determines the values of joint variables q4, q5, q6 to get the wrist in the right orientation

14. Example of First Step • Wrist with three degrees of freedom (q4, q5, q6) is assumed to be mounted at point P

15. Example of First Step • Assignment of frames and determining parameters according to the Denavit-Hartenberg algorithm yields the following transformation matrices:

16. Example of First Step • Assuming the wrist is at a distance l3 along the z3 axis and its desired placement is (dx, dy, dz) with respect to the world frame we must solve for θ1,θ2,θ3

17. Second Step • Substitute θ1,θ2,θ3 into0 R 3and solve q4, q5, q6 from

18. Inverse Transformation Technique • Recall that for a 6 DOF we have and assume all i Ti+1 are given.

19. Inverse Transformation Technique • A simple mathematical transformation leads from to

20. Inverse Transformation Technique resulting in • Attempt to solve by equating the 12 nontrivial matrix entries and identifying equation involving a single variable

21. Inverse Transformation Technique • Similar to we can obtain

22. Example of Inverse Transformation • A robot with 5 rotational and 1 prismatic DOF.

23. Example of Inverse Transformation • Assignment of frames and determining parameters according to the Denavit-Hartenberg algorithm yields the following transformation matrices:

24. Inverse Transformation Technique Recall All entries rij are given but note that they are not fully independent

25. Left-Hand Side We get and note that the left-hand side only depends on θ1

26. Right-Hand Side • We get where

27. Inverse Transformation Example Solve θ1 and θ2 and then proceed with

28. Iterative Solution • Recall that we must solve for joint variables q=(q1,q2,q3,…,qn)

29. Iterative Solution • If we substitute some joint variable values q’=(q’1,q’2,q’3,…,q’n) for q into then usually

30. Iterative Solution: Approach Repeatedly • substitute current best solution q* for q into T(q), that is, compute the pose of the tip for the joint variable values q* using forward kinematics • make a smart adjustment to the solution q* based on the deviation δT(q*) of T(q*) from the desired tip pose, so

31. Iterative Solution: Adjustment • A first-order Taylor expansion yields that the error δq in q satisfies δT = J δqwhere J is the Jacobian, which consists of the partial derivatives of all tip pose parameters with respect to all joint parameters: Alternatively we have that δq = J-1δT

32. Iterative Solution Repeatedly adjust solution for q according to until the error in the tip pose is small enough.