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Algorithms for Identifying Rigid Subsystems in Geometric Constraint Systems Christophe Jermann, LINA, University of Nantes Bertrand Neveu, Gilles Trombettoni, INRIA/I3S/CERTIS, Sophia Antipolis Contents Definitions Geometric

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Algorithms for Identifying Rigid Subsystems in Geometric Constraint Systems

Christophe Jermann,

LINA, University of Nantes

Bertrand Neveu, Gilles Trombettoni,

INRIA/I3S/CERTIS, Sophia Antipolis

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Contents

Definitions

– Geometric Constraint Satisfaction Problems (GCSP) – Rigidity – Structural Approximations

Algorithms for Rigidity Detection

– Objects-Constraints Network – Distribute Function – ES_Rigid Algorithm – Other Rigidity Related Algorithms

Conclusion

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Contents

Definitions

– Geometric Constraint Satisfaction Problems (GCSP) – Rigidity – Structural Approximations

Algorithms for Rigidity Detection

– Objects-Constraints Network – Distribute Function – ES_Rigid Algorithm – Other Rigidity Related Algorithms

Conclusion

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A GCSP S=(O,C) is composed of :

– O = geometric objects

Geometric Constraint Satisfaction Problem

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A GCSP S=(O,C) is composed of :

– O = geometric objects (lines, …)

Geometric Constraint Satisfaction Problem

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A GCSP S=(O,C) is composed of :

– O = geometric objects (lines, points, …)

Geometric Constraint Satisfaction Problem

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A GCSP S=(O,C) is composed of :

– O = geometric objects (lines, points, …) – C = geometric constraints

Geometric Constraint Satisfaction Problem

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A GCSP S=(O,C) is composed of :

– O = geometric objects (lines, points, …) – C = geometric constraints (incidences, …)

Geometric Constraint Satisfaction Problem

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A GCSP S=(O,C) is composed of :

– O = geometric objects (lines, points, …) – C = geometric constraints (incidences, distances, …)

Geometric Constraint Satisfaction Problem

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A GCSP S=(O,C) is composed of :

– O = geometric objects (lines, points, …) – C = geometric constraints (incidences, distances, …) – A solution = position,

  • rientation, dimensions
  • f each object satisfying

all the constraints

Geometric Constraint Satisfaction Problem

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  • Relative to movements

Rigidity

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Rigidity

  • Relative to movements

– Displacements

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Rigidity

  • Relative to movements

– Displacements (translations, …)

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Rigidity

  • Relative to movements

– Displacements (translations, …)

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Rigidity

  • Relative to movements

– Displacements (translations, rotations)

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Rigidity

  • Relative to movements

– Displacements (translations, rotations)

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Rigidity

  • Relative to movements

– Displacements (translations, rotations)

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Rigidity

  • Relative to movements

– Displacements (translations, rotations)

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Rigidity

  • Relative to movements

– Displacements (translations, rotations) – Deformations

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Rigidity

  • Relative to movements

– Displacements (translations, rotations) – Deformations

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Rigidity

  • Relative to movements

– Displacements (translations, rotations) – Deformations

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Rigidity

  • Relative to movements

– Displacements (translations, rotations) – Deformations

Rigid = only displacements ~ well-constrained

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Rigidity

Studied in:

– Structural topology (points-distances) – Theory of mechanisms => robots – CAD

Motivation:

– Is a given structure rigid ? – What are the over/under-determined subparts ? – Why is there no solution to the GCSP ? – What are the redundant constraints ? – Find a geometric assembly of the GCSP. – ...

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Characterization principle:

– Count the number of movements M – Count the number of displacements D – A GCSP S=(O,C) is:

  • ver-rigid if ∃ S’⊆ S such that M(S’)<D(S’)

rigid if M(S) =D(S) and S is not over-rigid under-rigid if M(S) >D(S) and S is not over-rigid

Difficulty: – No polynomial and general way of counting

M and D

Rigidity

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Approximation based on a count of the

degrees of freedom (DOF)

– Intuition: 1 DOF = 1 independent movement

Number of Movements M

M M M

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Approximation based on a count of the

degrees of freedom (DOF)

– Intuition: 1 DOF = 1 independent movement – DOF(object)= number of independent variables

Number of Movements M

M M M

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Approximation based on a count of the

degrees of freedom (DOF)

– Intuition: 1 DOF = 1 independent movement – DOF(object)= number of independent variables – DOF(constraint)= number

  • f independent equations

Number of Movements M

M M M

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Approximation based on a count of the

degrees of freedom (DOF)

– Intuition: 1 DOF = 1 independent movement – DOF(object)= number of independent variables – DOF(constraint)= number

  • f independent equations

– DOF(GCSP) =

DOF(objects) – DOF(constraints)

Number of Movements M

M M M

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Number of Displacements D

D D D

Classical approximation: number of

displacements (rotations+translations) of a rigid-body in dimension

  • Structural Rigidity:

– S is over-s_rigid if ∃ S’⊆ S such that

  • – S is s_rigid if , and S is not over-

s_rigid

– S is under-s_rigid if , and S is not

  • ver-s_rigid
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Degree of rigidity (DOR), depending on the

geometric properties in the GCSP

  • Extended Structural Rigidity:

– S is over-es_rigid if ∃ S’⊆ S such that

  • – S is es_rigid if , and S is not over-

es_rigid

– S is under-es_rigid if , and S is not

  • ver-es_rigid

Number of Displacements D

D D D

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Structural Rigidities: Comparison

SubGCSP Rigidity DOF s_rigidity (++) DOR es_rigidity AB well 5

  • ver

5 well AF under 7 under 6 under ABCD under 6 well 5 under ACDE well 5

  • ver

5 well ACDEF

  • ver

5

  • ver

6

  • ver

ABCDEF

  • ver

6

  • ver

6

  • ver
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Contents

Definitions

– Geometric Constraint Satisfaction Problems (GCSP) – Rigidity – Structural Approximations

Algorithms for Rigidity Detection

– Objects-Constraints Network – Distribute Function – ES_Rigid Algorithm – Other Rigidity Related Algorithms

Conclusion

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Objects-Constraints Network

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  • Objects-Constraints Network
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  • Objects-Constraints Network
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  • Objects-Constraints Network
  • !
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  • Objects-Constraints Network
  • !
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  • Objects-Constraints Network
  • !
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  • Objects-Constraints Network
  • !
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Distribute Function - Principle

Maximum flow = optimal distribution of the

constraints’ DOFs on the objets’ DOFs

Non-saturated constraint = subGCSP with less than 0 DOFs (over-constrained)

Overload K = fix K independent

displacements

Non-saturated constraint = subGCSP with less than K DOFs

Apply K=d(d+1)/2+1 on a constraint "

Non-saturated constraint, capacity-flow=1 => s_rigid

subGCSP

Non-saturated constraint, capacity-flow>1 => over-

s_rigid subGCSP

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Distribute Function - Example

  • !
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  • !
  • Distribute Function - Example
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  • !
  • Distribute Function - Example
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  • !
  • Distribute Function - Example
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  • !
  • Distribute Function - Example
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  • !
  • #$% & '#()

$ *#$+,(-$,

Distribute Function - Example

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New Distribute Function

Problem of the Distribute function:

– Nothing ensures the subGCSP S induced by a single

constraint verifies D(S)=d(d+1)/2

2 modifications:

– A virtual constraint R is dedicated to the overload

distribution, and can be linked to any subset of objects

– The overload K is adaptive: it becomes DOR+1

R non-saturated = subGCSP well- or over-es_rigid

Results in:

– Geometrically meaningful – Detects extended structural rigidity

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New Distribute Function - Example

  • !
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  • !
  • New Distribute Function - Example
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  • !
  • New Distribute Function - Example
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  • !
  • New Distribute Function - Example
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  • !
  • New Distribute Function - Example
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  • !
  • New Distribute Function - Example
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  • !
  • New Distribute Function - Example
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  • !
  • New Distribute Function - Example

#$% & '#() $ $+,

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ES_Rigid Algorithm

Goal:

– Identify a well- or over-rigid subGCSP

Principle:

. ": apply Distribute constraint by constraint – New algorithm: apply new Distribute subGCSP by

subGCSP

Problem:

– The number of subGCSPs is exponential !

Solution:

– Apply only to DOR-minimal subGCSPs

DOR-minimal =

– GCSP containing no subGCSP with the same DOR

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Complexity

Distribute: same as Hoffmann et al.’s

– Dominated by flow distribution: /00122

ES_Rigid:

– Hoffmann: O(mn2(n+m)) in any dimension – New algorithm:

Compute the set of DOR-Minimal subGCSPs: /3

. 4%*, -&*,,56&$*7 . $ $$$89$-&:$&' . --$-%$ $ *-%$-&*,

Number of DOR-Minimal subGCSPs: /02 in dimension

Complexity /0/310122

Overhead ~linear in 2D, ~quadratic in 3D

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Other Algorithms

Over-ES_Rigid:

– Set overload to K=DDR instead of DDR+1

Well-ES_Rigid:

– Apply ES_Rigid, then Over-ES_Rigid until the

  • btained subGCSP is well-es_rigid

Minimize-ES_Rigid:

– Remove objects one by one as long as they are not

necessary, using ES_Rigid to determine this.

…

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Contents

Definitions

– Geometric Constraint Satisfaction Problems (GCSP) – Rigidity – Structural Approximations

Algorithms for Rigidity Detection

– Objects-Constraints Network – Distribute Function – ES_Rigid Algorithm – Other Rigidity Related Algorithms

Conclusion

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Contribution:

– A new family of algorithms for the main rigidity related

issues, based on a new and better characterization of rigidity.

Future Works

– Experimental validation of the approach:

Evaluate practical feasibility (complexity !) Measure gain in reliability and generality

– Deal with the DOR problem:

Identify new classes of GCSPs for which the problem is easy Propose new practical heuristics to avoid theorem proving as

much as possible

Conclusion