From Position to Angles I n v e r s e K i n e m a t i c s From Position to Angles

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

Inverse Kinematics of a Two Link Manipulator (x , y) Given: l1, l2 , 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. 2 l2 l1 1 l2 l1 (x , y) l2 l1

The Geometric Solution (x , y) Using the Law of Cosines: l2 2 l1  1 Using the Law of Cosines: Redundant since 2 could be in the first or fourth quadrant. Redundancy caused since 2 has two possible values

The Algebraic Solution (x , y) l2  2 l1 1 Only Unknown

We know what 2 is from the previous slide. We need to solve for 1 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 c1 and simplifying many times Notice this is the law of cosines and can be replaced by x2+ y2