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Flagged Parallel Manipulators

Flagged Parallel Manipulators. F. Thomas (joint work with M. Alberich and C. Torras) Institut de Robòtica i Informàtica Industrial. Talk outline. PART I. Trilatelable Parallel Robots Forward kinematics Singularities Formulation using determinants

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Flagged Parallel Manipulators

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  1. Flagged Parallel Manipulators F. Thomas (joint work with M. Alberich and C. Torras) Institut de Robòtica i Informàtica Industrial

  2. Talk outline PART I • Trilatelable Parallel Robots • Forward kinematics • Singularities • Formulation using determinants • Singularities as basic contacts between polyhedra • Generalization to serial robots

  3. Talk outline PART II • Technical problems at singularities • The direct kinematics problem and singularities • The singularity locus • How to get rid of singularities? • Goal: Characterization of the singularity locus • Stratification of the singularity locus • Basic flagged parallel robot

  4. Talk outline PART II • Why flagged? • Ataching flags to parallel robots • Equilalence between basic contacts and volumes of tetrahedra • Deriving the whole family of flagged parallel robots • Local transformations • Substituting of 2-leg groups by serial chains • Examples

  5. Talk outline PART II • The direct kinematics of flagged parallel robots • Invariance of flags to certain transformations • Classical result from the flag manifold • Stratification of the flag manifold • From projective flags to affine flags • From afine flags to the configuration space of the platform • Strata of dimension 6 and 5 • Redundant flagged parallel robots

  6. Forward kinematics of trilaterable robots

  7. Forward kinematics of trilaterable robots

  8. Forward kinematics of trilaterable robots

  9. Forward kinematics of trilaterable robots

  10. Forward kinematics of trilaterable robots

  11. 0 r12 r13 r14 1 r12 0 r23 r24 1 r13 r23 0 r34 1 r14 r24 r34 0 1 1 1 1 1 0 = 288 V2 Cayley-Menger determinants Of four points p4 p1 p3 p2 rij = squared distance between pi and pj

  12. p3 p1 p2 Of two points: 0 r12 1 p2 = 2 d2 r12 0 1 p1 1 1 0 Cayley-Menger determinants Of three points: 0 r12 r131 r12 0 r23 1 = 16 A2 r13 r23 0 1 1 1 1 0

  13. Cayley-Menger determinants Notation D(1 2 ... n) Cayley-Menger determinant of the n points p1, p2, ... , pn

  14. D(123) D(123) D(234) - D(1234) D(23) D(1234) 2 D(123) Forward Kinematics using CM determinants p4 p1 Position of the apex: p3 p4= α1 p1 + α2 p2 + α3 p3 + β n p2

  15. Singularities in terms of CM determinants Singularity if and only if D(1234) = 0 If, additionally, D(123) = 0, the apex location is undetermined.

  16. 4 4 7 4 3 1 7 2 9 5 6 8 Singularities in terms of CM determinants D(1234) = 0 D(4567) = 0 D(4789) = 0

  17. Singularities in terms of basic contacts between polyhedra vertex - face contact edge - edge contact face - vertex contact

  18. Family of parallel trilaterable robots

  19. Singularities in the configuration space of the platform Each contact defines a surface in C-space, of equation: det(pi , pj , pk , pl ) = 0 C-space 6 7 1 8 5 2 4 3

  20. Generalization to serial robots A 6R robot can be seen as an articulated ring of six tetrahedra involving 12 points

  21. A PUMA robot… … and its equivalent framework 5 3 5 3 4 2 4 2 6 6 8 1 7 7 1 8 Generalization to serial robots

  22. 3 5 2 4 5 3 4 2 6 7 1 8 Generalization to serial robots

  23. Generalization to serial robots

  24. Generalization to serial robots

  25. Generalization to serial robots

  26. Generalization to serial robots

  27. Generalization to serial robots

  28. Talk outline PART II • Technical problems at singularities • The direct kinematics problem and singularities • The singularity locus • How to get rid of singularities? • Goal: Characterization of the singularity locus • Stratification of the singularity locus • Basic flagged parallel robot

  29. Talk outline PART II • Why flagged? • Ataching flags to parallel robots • Equilalence between basic contacts and volumes of tetrahedra • Deriving the whole family of flagged parallel robots • Local transformations • Substituting of 2-leg groups by serial chains • Examples

  30. Talk outline PART II • The direct kinematics of flagged parallel robots • Invariance of flags to certain transformations • Classical result from the flag manifold • Stratification of the flag manifold • From projective flags to affine flags • From afine flags to the configuration space of the platform • Strata of dimension 6 and 5 • Redundant flagged parallel robots

  31. platform 6 legs base Technical problems at singularities The platform becomes uncontrollable at certain locations • It is not able to support weights • The actuator forces in the legs may become very large. Breakdown of the robot

  32. The Direct Kinematics Problem and Singularities Direct finding location of platform with Kinematics respect to base from 6 leg lengths problem finding preimages of the forward kinematics mapping configuration spaceleg lengths space

  33. The Singularity Locus Rank of the Jacobian of the kinematics mapping Singularity locus Branching locus of the number of ways of assembling the platform

  34. How to get rid of singularities? • By operating in reduced workspaces • By adding redundant actuators Problems: • how to plan trajectories? • where to place the extra leg? In both cases we need a complete and precise characterization of the singularity locus

  35. Stratification of the singularity locus Exemple: 3RRR planar parallel robot with fixed orientation

  36. Goal: characterization of the singularity locus(nature and location) Configuration space Configuration space Two assembly modes are always separated by a singular region Branching locus Branching locus Leg lengths space Leg lengths space Two assembly modes can be connected by singularity-free paths

  37. Basic flagged parallel robot • Three possible architectures for 3-3 parallel manipulators: octahedral flagged 3-2-1

  38. Basic flagged parallel robot • One of the three possible architectures for 3-3 parallel manipulators: Trilaterable octahedral flagged 3-2-1

  39. Attaching flags vertex - face contact edge - edge contact face - vertex contact

  40. Attaching flags Attached flag to the platform Attached flag to the base

  41. Why flagged? Because their singularities can be described in terms of incidences between two flags. But, what’s a flag?

  42. Flags attached to the basic flaggedmanipulator Its singularities can be described in terms of incidences between its attached flags

  43. Implementation of the basic flaggedparallel robot [Bosscher and Ebert-Uphoff, 2003]

  44. Deriving other flagged parallel robots from the basic one Local transformation on the leg endpoints that leaves singularities invariant

  45. Local Transformations Composite transformations

  46. 2-2-2 3-2-1

  47. 2-2-2 3-2-1

  48. 2-2-2 3-2-1

  49. 2-2-2 3-2-1

  50. 2-2-2 3-2-1

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