How%20to%20Draw%20a%20Straight%20Line%20Using%20a%20GP

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How to Draw a Straight Line Using a GP

• Benchmarking Evolutionary Design Against 19th
  Century Kinematic Synthesis

Hod Lipson Computational Synthesis Lab Mechanical
Aerospace Engineering and Computing
Information Science, Cornell University
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Kinematic Synthesis is a Challenge

• Kinematics Science of pure motion
• Concerned with geometric displacement of
  connected rigid links without regard to forces or
  physical embodiment
• Kinematic analysis Predict motions of given
  mechanism
• Well understood
• Kinematic Synthesis Assemble of a mechanism that
  achieves a prespecified motion
• Poorly understood
• Analytical methods exist for some special cases
  (Chebychev)
• Evolutionary Methods Primarily in robotics
• Only tree-structures (Sims 1994, Komosinski 2000,
  Hornby Lipson, 2002)
• Compound mechanisms (kinematic loops) (Lipson
  Pollack, 2000)

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Background Long standing problem
One of the first synthesis challenges to be posed
A rational approach to synthesis is needed to
obtain, by direct and certain methods, all the
forms and arrangements that are applicable to the
desired purpose. At present, questions of this
kind can only be solved by that species of
intuition that which long familiarity with the
subject usually confers upon experienced persons,
but which they are totally unable to communicate
to others. When the mind of a mechanician is
occupied with the contrivance of a machine, he
must wait until, in the midst of his meditations,
some happy combination presents itself to his
mind which may answer his purpose. Robert
Willis, The Principles of Mechanism, 1841
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Background Degrees of Freedom

• Degrees of freedom (DoF)
• Number of independent parameters needed to
  describe the state of a mechanism
• For 2D Structure, DoF 2n-m-3
• nnodes mlinks 3rigid body DoF removed by
  ground
• Not applicable under singularities and
  degeneracies

1curve
2area
N/A (overconstrained)
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Mechanism Representation

• A mechanism could be represented as a graph
• Many graph encodings (Luke Spector, 1996)
• Cellular encoding (Gruau, 1994)
• Parse trees
• L-Systems
• Suitable for computational networks, not
  mechanical networks
• Highly connected graphs
• Vanishingly small number of them have one DoF
• Need a representation suitable mechanisms
• Retains DoF across variation
• Tree-based (hierarchical, for GP)
• Evolvable (less coupled, effective crossover)

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Background Top-down and Bottom-up Tree Encodings

Bottom up Composition of terminals, e.g. Symbolic expression
Top down Embryo variation operators. E.g. Circuit
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Top down encoding of a mechanism
Start with Embryo with desired of DoF, e.g. a
four-bar mechanism (1 DoF)
Two variation operators maintain DoF
Example A tree that constructs this 1-DoF
compound mechanism
E.g. Transform dyad into tryad
Operators provably DoF invariant
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Bottom-up encoding of a 1-DoF mechanism
1-DoF terminals
Join substructures hierarchically at exactly two
nodes (maintains DoF)
Example A tree that constructs this 1-DoF
compound mechanism
Operators provably DoF invariant
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Intermediate Conclusions

• Proposed two new DoF-invariant representations
  for kinematic mechanisms
• top-down and bottom-up tree encodings

But wait, theres more
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A hard test problem

• The straight line problem
• Devise a mechanism that traces a straight line
  without a reference to an existing straight line
• Human-competitive problem
• Of great practical importance in the 18th and
  19th century.
• has baffled the worlds greatest kinematic
  inventors for a century, many solutions put
  forward.
• Now forgotten
• with advent of precision manufacturing

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The Straight Line Problem
It is easy to think of a mechanism that traces an
exact circle without having a circle built in A
compass.
?
Can you think of a linkage mechanism that will
trace a straight line without reference to an
existing straight line?
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The Straight Line Problem
It is easy to think of a mechanism that traces an
exact circle without having a circle built in A
compass.
One solution The Peaucellier (1873)
The straightness of the links themselves does not
matter
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The Straight-Line problem

• Needed to guide the piston of the steam engine.
• The breakthrough that made steam engines a success

(b)
Though I am not over anxious after fame, yet I
am more proud of the parallel motion than of any
other mechanical invention I have ever made
James Watt, cf. 1810 15
Watts first straight line mechanism (1784)
James Watts original patents used racks and
sectors, and many other cumbersome solutions
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More established solutions
Peaucelier (1873)
Silverster-Kempes (1877)
Robert (1841)
Chebyshev (1867)
Source Kempe A. B., (1877), How To Draw A
Straight Line, London
See http//kmoddl.library.cornell.edu
Chebyshev-Evans (1907)
Chebyshev (1867)
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Considered fundamental technology


Cornell University acquired in 1882 about 40
straight-line mechanism models and used them in
the early engineering curriculum. See videos
at Cornell University Digital Library of
Kinematic Models http//kmoddl.library.cornell.ed
u


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Background Kinematic simulation

• Propagate motion using relaxation of elastic
  linkages
• Linkages stiffened to approximate rigid motion
• Good Simple, accurate and robust, handles
  singularities well
• Bad No dynamics (no accelerations, masses) slow

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Background Comparing Mechanisms

• Comparison of mechanisms can be difficult
• Equivalent mechanisms may appear very different
• Masked by excess and redundant topology
• Two transformations allow moving in neutral
  pathways of mechanisms
• Rigid diagonal swap
• Redundant dyad removal/addition

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Evolving Straight line mechanisms

• Used GP with Top-down tree encoding and 2-bar or
  4-bar embryo
• Population size 100
• Crossover 90
• Mutation 10 (Node positions, Operator types)
• Selection SUS
• Fitness Straightness
• Aspect ratio of tight bounding box of
  node traces

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Evolving Straight line mechanisms
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Some results
Linearity 14979
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Some results
Linearity 15300
Infringes on Roberts Linkage (1841) Published
Kempe A. B., (1877), How To Draw A Straight Line,
London
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Some results
Linearity 112819
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Some results
Linearity 128340
Many more solutions were produced
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Conclusions

• Open-ended Kinematics synthesis
• Long standing problem
• Many applications, e.g. Robotics
• Synthesis for computational networks inadequate
• Proposed two new representations for kinematic
  mechanisms
• Top-down and bottom-up tree encodings
• DoF-invariant operators
• Neutral-pathway operators
• Benchmarked against human-competitive problem
• Extensive documentation for a century of attempts
• Results infringe and outperform known solutions
• Next Step
• Improve search process, and
• Apply to contemporary challenges (multi DoF
  robotics)