1. 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

2. 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

3. 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

4. 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

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

6. 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