Solving Distance Constraints by Iterative Projections and Backprojections

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Solving Distance Constraints by Iterative
Projections and Backprojections

• F. Thomas, J.M. Porta, and L. Ros
• Institut de Robòtica i Informàtica Industrial
• Barcelona - Catalonia

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Outline

• The problem
• Realizability and its characterization
• The Cholesky decomposition
• The algorithm
• Examples

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The problem
Given a set of points and some pairwise
distances Find all point configurations
compatible with the distances Up to mirror and
rigid transformations
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The problem
Given a set of points and some pairwise
distances Find all point configurations
compatible with the distances Up to mirror and
rigid transformations
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Translating a 6R robot into a set of distance
constraints
An articulated ring of six tetrahedra 12 points,
30 known distances, 36 unknown distances
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Trilaterable robots
Sometimes, the unknown distances may be obtained
by a sequence of trilaterations
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Trilaterable robots
Similarly, the 3-2-1 and 3/2 parallel robots are
also trilaterable
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Outline

• The problem
• Realizability and its characterization
• The Cholesky decomposition
• The algorithm
• Examples

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Realizabilityof a distance matrix

• A distance matrix D is said to be realizable in
  d - space if there is a configuration of points
  in such space whose pairwise distances are those
  specified in D

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Necessary and suficient conditions of
realizability

• The elements of a distance matrix must satisfy
  certain algebraic conditions to guarantee that it
  is realizable (that it comes from a point
  configuration in 3-space)

Mainly two characterizations have been reported

• Theory of Cayley-Menger determinants
• Theory of positive semidefinite matrices

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Theory of Cayley-Menger determinants
Conditions that a set of distances between points
must satisfy so that the points have a valid
configuration in 3D

• All C-M determinants of 6 points 0
• " " " 5 points
  0
• " " " 4 points 0
• " " " 3 points 0

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Theory of semidefinite matrices (I)
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Theory of semidefinite matrices (II)
Theorem (Schoenberg 1935)
A distance matrix D, n ? n, is realizable in d
space if, and only if, its associated Gram
matrix G, n-1 ? n-1, is positive semidefinite of
rank d
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Outline

• The problem
• Realizability and its characterization
• The Cholesky decomposition
• The algorithm
• Examples

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The Cholesky decomposition
Provides
A consistency test for our distance matrix D
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Geometric interpretation
In general If the initial dimension is d, after
d projections the points of the configuration
will collapse to a single point
Iterating
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Geometric interpretation
(In general, after d projections, we get an n-d x
n-d zero matrix)
In sum
In general If the initial dimension is d, after
d projections the points of the configuration
will collapse to a single point
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Outline

• The problem
• Realizability and its characterization
• The Cholesky decomposition
• The algorithm
• Examples

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The algorithmBasic operations

• Projection
• Backprojection

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Projection
Uses the geometric interpretation of the Cholesky
decomposition to prune regions of the search
space that contain no solution. Best seen on an
example
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Projection (cont)
Contains the zero matrix
Detected does not contain the zero
matrix. 5.1,5.2 can be pruned from the search
space
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Backprojection
The projection scheme
can be reversed
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The basic algorithm in 2D
Projection phase
If the new bounds arent much smaller, bisect
the bounds
Backprojection phase
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Outline

• The problem
• Realizability and its characterization
• The Cholesky decomposition
• The algorithm
• Example

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ExampleThe 3RPR planar manipulator
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ExampleThe 3RPR planar manipulator
Choosing the previous leg lengths, it has rigid
and flexible configurations
flexible
rigid
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Solutions found
One box, for each isolated solution
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Conclusions

• An algorithm for solving systems of distance
  constraints based on projections and
  backprojections

It relies on a necessary and sufficient condition
of realizability, thus avoiding the cluster
effect
Its speed of convergence to the solutions depends
on the chosen sequence of projections that are
iteratively applied
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Future work
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Projection in detail
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