Inverse Kinematics

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Inverse Kinematics

• Professor Nicola Ferrier
• ME 2246, 265-8793
• ferrier_at_engr.wisc.edu

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Forward Kinematics

• position equation
• Where the transformation for each link is built
  from our DH parameters

End-effector w.r.t. e
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Forward Kinematics

• Forward Kinematic Transformation
• Has an rotation and translation

Roll,pitch,yaw
Angle-axis
Euler Y-Z-Y
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Forward Kinematics

• Pick one rotation description
• Decompose R, i.e. solve

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Forward Kinematics

• Pick one rotation description

Find terms that you can easily solve equations
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Decomposition of Rotation Matrices

• Roll, Pitch, and Yaw
• The inverse solution for ? in (-?/2,?/2)

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Decomposition of Rotation Matrices

• Roll, Pitch, and Yaw
• The inverse solution for ? in (?/2,3?/2)

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Decomposition of Rotation Matrices

• Euler Angles
• Inverse solution for r13? 0 and r23? 0
  (singularity when sin?0)

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Decomposition of Rotation Matrices

• OR

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Decomposition of Rotation Matrices

• Angle Axis
• Inverse solution (singularity at ?0,?,... and
  there are two solutions for ?)

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Forward Kinematics

• Pick one rotation description
• Decompose T

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Cartesian Space Joint Space
Forward kinematics
Joint Space (q1,q2,,qN)
Cartesian Space (x,y,z,?,?,?)
Inverse kinematics (arm solution)
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Inverse Relationship?

• Find joint positions for a given end-effector
  pose
• Also find joint positions, speeds and
  accelerations for a sequence of poses

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Inverse Relationship?

• Analytic Inverse
• Robot specific
• Multiple solutions
• Once found, computationally simple
• Geometric
• Robot specific
• Insight?
• Iterative Computation (ME739)
• Computationally expensive
• General solution

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A Heuristic for Inverse Kinematics

• The solution to the inverse kinematic equations
  (called the arm solution) can often be found
  using a heuristic approach.
• Does not guarantee a solution
• Solution may not be unique
• Some solutions may be redundant

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IK Heuristic Algorithm

• Perform forward kinematics to find the general
  transformation matrix
• Equate the transformation matrix to the
  manipulator transformation matrix.
• For a particular solution the manipulator
  transformation matrix contains numbers
• For a general solution the manipulator
  transformation matrix contains variables

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IK Heuristic Algorithm

• Look at both matrices for
• Elements which contain only one joint variable
• Pairs of elements which will produce an
  expression in only one joint variable when
  divided (look for divisions that result in the
  atan2 function)
• Elements, or combinations of elements, that can
  be simplified using trigonometric identities
• Having selected an element, equate it to the
  corresponding element in the other matrix to
  produce an equation. Solve this equation to find
  a description of one joint variable in terms of
  the elements of the manipulator transformation
  matrix.

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IK Heuristic Algorithm

• Repeat step 4 until all the elements identified
  in step 3 have been used.
• If any of these solutions suffer from
  inaccuracies, undefined results, or redundant
  results, set them aside and look for better
  solutions.

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IK Heuristic Algorithm

• If there are more joint angles to be found,
  pre-multiply both sides of the matrix equation by
  the inverse of the A1 matrix to produce a new set
  of equivalent matrix elements.

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IK Heuristic Algorithm

• Repeat steps 3 to 7 until either solutions to all
  the joint variables have been found, or you have
  run out of A matrices to pre-multiply.
• If a suitable solution cannot be found for a
  joint variable, choose one of those discarded in
  step 6, taking note of regions where problems may
  occur.
• If a solution cannot be found for a joint
  variable in terms of the elements of the
  manipulator transform, it may be that the
  manipulator cannot achieve the specified position
  and orientation the position is outside the
  manipulators workspace.

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IK Heuristic Algorithm

• Note theoretical solutions may not be physically
  attainable because of the mechanical limits on
  the range of joint variables.

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DH Example academic manipulator
Z0
L2
?1
x1
x2
Z2
L1
?1
Z1
x0
?2
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Steps 1 and 2

• The general transformation matrix
• Set equal to

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Steps 3 and 4

• Look for elements and select terms

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More steps.

• Pre-multiply for a new equation

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DH Example two link planar arm
DH table for two link arm
L2
L1
x0
x1
x2
?2
?1
Z2
Z0
Z1
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Forward Kinematics planar 2-link manipulator
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Forward Kinematics planar 2-link manipulator
Heuristic solution? Cant select an obvious set
of equations
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Trignometric Solutions

• Sometimes simple (but usually not)!

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Trig Solutions

• For first angle revisit equations

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Trig Solutions

• A trig hint
• Let
• Then
• So

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Solution to 2 link planar arm

• Arm solution is
• where

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Multiple Solutions?!
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Geometric Solutions

• Intuition, insight.?

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Geometric Solutions

• Intuition, insight.?

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Geometric Solutions

• Intuition, insight.?

Get same solution as trig method
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Solution to 2 link planar arm

• Arm solution is
• where

Here we know the sign by observation