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

1 0 0 L1
2 L2 0
3 0 0
8
Kinematic Equations of The Arm
9
Target

• By comparison we get

10
Kinematic Equations - Solution
11
Example 2

1 0 0 0
2 0 0
3 L2 0 0
4 L3 0 0
12
Example 2 (cont.)
13
Example 2 (cont.)
14
Example 2 (cont.)




15
Geometric Solution

• IDEA Decompose spatial geometry into several
  plane geometry problems

?2
Applying the law of cosines x2y2l12l22 ?
2l1l2cos(180??2)
16
Geometric Solution (II)

• Then

y
?
?
The LoC gives l22 x2y2l12 - 2l1? (x2y2) cos
? So that cos ? (x2y2l12 - l22 )/2l1?
(x2y2) We can solve for 0? ? ? 180, and then
?1???
x