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A Family of Quadratically-Solvable 5-S P U Parallel Robots

A Family of Quadratically-Solvable 5-S P U Parallel Robots. Júlia Borràs, Federico Thomas and Carme Torras. Contents. Previous work  Geometric interpretation. Forward Kinematics  Geometric interpretation. How to obtain a quadratically-solvable 5-SPU. Conclusions.

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A Family of Quadratically-Solvable 5-S P U Parallel Robots

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  1. A Family of Quadratically-Solvable 5-SPU Parallel Robots Júlia Borràs, Federico Thomas and Carme Torras

  2. Contents • Previous work  Geometric interpretation • Forward Kinematics  Geometric interpretation • How to obtain a quadratically-solvable 5-SPU • Conclusions

  3. Previous works Reformulation of singularities in terms of a matrix T det(J) = det(T) Singularity polynomial {C1, C2, C3, C4, C5}

  4. Previous works Singularity-invariant leg rearrangements Leg rearrangements that preserve the 6 coefficients, up to constant multiple. {C1, C2, C3, C4, C5}

  5. Identification of relevant geometric entities B point location & Yellow line & Distance between Red and Yellow line Geometric interpretation of the 5 constants {C1, C2, C3, C4, C5}

  6. Forward Kinematics Input: 5 leg lengths Output: Position and orientation of the platform Quadratic system 5 length leg equation  5 sphere equations Associated Linear system One equation can be use to simplify the others {C1, C2, C3, C4, C5}

  7. Forward Kinematics Input: 5 leg lengths Output: Position and orientation of the platform Quadratic system 5 length leg equation  5 sphere equations Associated Linear system One equation can be use to simplify the others 4 linear equations in 5 unknowns {C1, C2, C3, C4, C5}

  8. Forward Kinematics The linear system solution is used to generate a uni-variate 4 degree polynomial C4= C5 = 0 Quadratic polynomial {C1, C2, C3, C4, C5}

  9. Quadratically-solvable 5-SPU C4= C5 = 0 B point at infinity All base lines are parallel.

  10. Applications

  11. Conclusions - Family of manipulators whose forward kinematics are greatly simplified: From To Solve a 4th degree polynomial and a 2-degree polynomial. Solve 2 quadratic polynomials 8 assembly modes (16) 4 assembly modes (8) - Easy geometric interpretation of architectural singularities. - Full stratification of the singularity locus. - Direct applications on: - reconfigurable robots, with attachment placed on actuated guides. - Increase the workspace of manipulators. - Optimization of indexes like manipulability, stiffness and avoidance of leg collisions.

  12. Thank you Júlia Borràs Sol (jborras@iri.upc.edu) Institut de robòtica i informàtica industrial. Barcelona Interactive visualizations done with GAViewer, developed by Daniel Fontijne - Amsterdam University http://www.science.uva.nl/ga/viewer/content_viewer.html

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