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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

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

  2. 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 2 16/11/2005 Christophe Jermann, FJCP 2005

  3. 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 3 16/11/2005 Christophe Jermann, FJCP 2005

  4. Geometric Constraint Satisfaction Problem � A GCSP S=(O,C) is composed of : – O = geometric objects 4 16/11/2005 Christophe Jermann, FJCP 2005

  5. Geometric Constraint Satisfaction Problem � A GCSP S=(O,C) is composed of : – O = geometric objects (lines, …) � 5 16/11/2005 Christophe Jermann, FJCP 2005

  6. Geometric Constraint Satisfaction Problem � A GCSP S=(O,C) is composed of : – O = geometric objects (lines, points, …) � � � � � � 6 16/11/2005 Christophe Jermann, FJCP 2005

  7. Geometric Constraint Satisfaction Problem � A GCSP S=(O,C) is composed of : – O = geometric objects (lines, points, …) – C = geometric constraints � � � � � � 7 16/11/2005 Christophe Jermann, FJCP 2005

  8. Geometric Constraint Satisfaction Problem � A GCSP S=(O,C) is composed of : – O = geometric objects (lines, points, …) – C = geometric constraints (incidences, …) � � � � � � 8 16/11/2005 Christophe Jermann, FJCP 2005

  9. Geometric Constraint Satisfaction Problem � A GCSP S=(O,C) is composed of : – O = geometric objects (lines, points, …) – C = geometric constraints (incidences, distances, …) � � � � � � 9 16/11/2005 Christophe Jermann, FJCP 2005

  10. Geometric Constraint Satisfaction Problem � A GCSP S=(O,C) is composed of : – O = geometric objects (lines, points, …) – C = geometric constraints (incidences, distances, …) � � � – A solution = position, � orientation, dimensions of each object satisfying � all the constraints � 10 16/11/2005 Christophe Jermann, FJCP 2005

  11. Rigidity � Relative to movements � � � � � � 11 16/11/2005 Christophe Jermann, FJCP 2005

  12. Rigidity � Relative to movements – Displacements � � � � � � 12 16/11/2005 Christophe Jermann, FJCP 2005

  13. Rigidity � Relative to movements – Displacements (translations, …) � � � � � � 13 16/11/2005 Christophe Jermann, FJCP 2005

  14. Rigidity � Relative to movements – Displacements (translations, …) � � � � � � 14 16/11/2005 Christophe Jermann, FJCP 2005

  15. Rigidity � Relative to movements – Displacements (translations, rotations) � � � � � � 15 16/11/2005 Christophe Jermann, FJCP 2005

  16. Rigidity � Relative to movements – Displacements (translations, rotations) � � � � � � 16 16/11/2005 Christophe Jermann, FJCP 2005

  17. Rigidity � Relative to movements – Displacements (translations, rotations) � � � � � � 17 16/11/2005 Christophe Jermann, FJCP 2005

  18. Rigidity � Relative to movements – Displacements (translations, rotations) � � � � � � 18 16/11/2005 Christophe Jermann, FJCP 2005

  19. Rigidity � Relative to movements – Displacements (translations, rotations) – Deformations � � � � � � 19 16/11/2005 Christophe Jermann, FJCP 2005

  20. Rigidity � Relative to movements – Displacements (translations, rotations) – Deformations � � � � � � 20 16/11/2005 Christophe Jermann, FJCP 2005

  21. Rigidity � Relative to movements – Displacements (translations, rotations) – Deformations � � � � � � 21 16/11/2005 Christophe Jermann, FJCP 2005

  22. Rigidity � Relative to movements – Displacements (translations, rotations) – Deformations � � � � � � � Rigid = only displacements ~ well-constrained 22 16/11/2005 Christophe Jermann, FJCP 2005

  23. 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. – ... 23 16/11/2005 Christophe Jermann, FJCP 2005

  24. Rigidity � Characterization principle: – Count the number of movements M – Count the number of displacements D – A GCSP S=(O,C) is: � over-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 24 16/11/2005 Christophe Jermann, FJCP 2005

  25. Number of Movements M M M M � Approximation based on a count of the degrees of freedom (DOF) – Intuition: 1 DOF = 1 independent movement � � � � � � � 25 16/11/2005 Christophe Jermann, FJCP 2005

  26. Number of Movements M M M M � Approximation based on a count of the degrees of freedom (DOF) – Intuition: 1 DOF = 1 independent movement – DOF(object)= number of independent variables � � � � � � � � � � � 26 � 16/11/2005 Christophe Jermann, FJCP 2005

  27. Number of Movements M M M M � 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 � � of independent equations � � � � � � � � � � � � � � � � 27 � 16/11/2005 Christophe Jermann, FJCP 2005

  28. Number of Movements M M M M � 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 � � of independent equations � � � � � � � – DOF(GCSP) = � � DOF(objects) – � � � DOF(constraints) � � � � 28 � 16/11/2005 Christophe Jermann, FJCP 2005

  29. 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 over-s_rigid 29 16/11/2005 Christophe Jermann, FJCP 2005

  30. Number of Displacements D D D D � 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 over-es_rigid 30 16/11/2005 Christophe Jermann, FJCP 2005

  31. Structural Rigidities: Comparison Rigidity DOF s_rigidity (++) DOR es_rigidity SubGCSP AB well 5 over 5 well AF under 7 under 6 under ABCD under 6 well 5 under ACDE well 5 over 5 well ACDEF over 5 over 6 over ABCDEF over 6 over 6 over � � � � � � � � � � � � � � � � � � � � � 31 16/11/2005 Christophe Jermann, FJCP 2005

  32. 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 32 16/11/2005 Christophe Jermann, FJCP 2005

  33. Objects-Constraints Network ���������������������� � � � � � � � � � � � � � � � � � � � � � 33 16/11/2005 Christophe Jermann, FJCP 2005

  34. Objects-Constraints Network ���������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � 34 16/11/2005 Christophe Jermann, FJCP 2005

  35. Objects-Constraints Network ���������������������� � � � �� � � �� � � � � � � � �� � � � � �� � � � � �� � � � � � � �� � � �� � � � �� � � � �� � � 35 16/11/2005 Christophe Jermann, FJCP 2005

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