1. Introduction to Robotics Tutorial III Alfred Bruckstein Yaniv Altshuler

2. Inverse Kinematics Problem Input: the desired position and orientation of the tool Output: the set of joints parameters

3. A Simple Example 11 X Y S Revolute and Prismatic Joints Combined (x, y) Finding  : More Specifically : arctan2() specifies that it’s in the first quadrant Finding S :

4. 22 11 (x, y) l2l2 l1l1 Inverse Kinematics of a Two Link Manipulator Given: l 1, l 2, x, y Find:  1,  2 Redundancy: A unique solution to this problem does not exist. Notice, that using the “givens” two solutions are possible. Sometimes no solution is possible. (x, y) l2l2 l1l1 l2l2 l1l1

5. The Geometric Approach l1l1 l2l2 22 11  (x, y)Using the Law of Cosines: Redundant since  2 could be in the first or fourth quadrant. Redundancy caused since  2 has two possible values

6. The Algebraic Approach l1l1 l2l2 22 11  (x, y) Only Unknown

7. We know what  2 is from the previous slide. We need to solve for  1. Now we have two equations and two unknowns (sin  1 and cos  1 ) Substituting for c 1 and simplifying many times Notice this is the law of cosines and can be replaced by x 2 + y 2

8. Kinematics equations of this arm: The structure of the transformation: Example #2 - Algebraic Approach

9. We are interested in x, y, and (of the end-effector) By comparison of the two matrices above we obtain: And by further manipulations: and …… Example #2 - Algebraic Approach

10. 100L1L1 2L2L2 0 300 Example #3

12. By comparison we get: Example #3