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

2. Workspaces Dextrous workspace – the volume of
space which the robot end-effector
can reach with all orientations
Reachable workspace – the volume of space which the robot end-effector can reach in at least one orientation
If L1=L2 then the dextrous space = {origin} and the reachable space = full disc of radius 2L1
If then the dextrous space is empty and the reachable space is a ring bounded by discusses with radiuses |L1- L2| and L1+L2
The dextrous space is a subset of the reachable space

3. Solutions
A manipulator is solvable if an algorithm can determine the joint variables. The algorithm should find all possible solutions.
There are two kinds of solutions: closed-form and numerical (iterative)
Numerical solutions are in general time expensive
We are interested in closed-form solutions:
Algebraic Methods
Geometric Methods

4. Algebraic Solution Kinematics equations of this arm:
The structure of the transformation:

5. Algebraic Solution (cont.)
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 ……

6. Algebraic Solution by Reduction to Polynomial
The actual variable is u :

7. Example 1 1 L1 2 L2 3

8. Kinematic Equations of The Arm

9. Target
By comparison we get:

10. Kinematic Equations - Solution

11. Example 2 (L1=0) 1 L1 2 3 L2 4 L3

12. Example 2 (cont.)

13. Example 2 (cont.)

14. Example 2 (cont.)

15. Geometric Solution
IDEA: Decompose spatial geometry into several plane geometry problems x y L1 L2 2
Applying the “law of cosines”:
x2+y2=l12+l22  2l1l2cos(1802)

16. Geometric Solution (II)
Then: y  
The LoC gives:
l22 = x2+y2+l12 - 2l1 (x2+y2) cos 
So that cos  = (x2+y2+l12 - l22 )/2l1 (x2+y2)
We can solve for 0   180, and then 1= x