Inverse Kinematics

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 L1L2 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 L1L2
• 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
8
Kinematic Equations of The Arm
9
Target

• By comparison we get

10
Kinematic Equations - Solution
11
Example 2
12
Example 2 (cont.)
13
Example 2 (cont.)
14
Example 2 (cont.)




15
Example 2 (cont.)