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

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

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

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

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 ……

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

Example 1 1 L1 2 L2 3

Kinematic Equations of The Arm

Target By comparison we get:

Kinematic Equations - Solution

Example 2 1 L1 2 3 L2 4 L3

Example 2 (cont.)

Example 2 (cont.)

Example 2 (cont.)

Example 2 (cont.)