1. Júlia Borràs Sol Barcelona. Spain Thursday May 6, 2010 A Family of Quadratically-Solvable 5-SPU Parallel Robots Júlia Borràs, Federico Thomas and Carme Torras

2. Contents Forward Kinematics  Geometric interpretation Previous work  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 ) {C 1, C 2, C 3, C 4, C 5 } Singularity polynomial

4. Singularity-invariant leg rearrangements Leg rearrangements that preserve the 6 coefficients, up to constant multiple. {C 1, C 2, C 3, C 4, C 5 } Previous works

5. Identification of relevant geometric entities {C 1, C 2, C 3, C 4, C 5 } B point location & Geometric interpretation of the 5 constants Yellow line &Distance between Red and Yellow line

6. {C 1, C 2, C 3, C 4, C 5 } Forward Kinematics 5 length leg equation  5 sphere equations One equation can be use to simplify the others Associated Linear system Quadratic system Input: 5 leg lengthsOutput: Position and orientation of the platform

7. Forward Kinematics 5 length leg equation  5 sphere equations One equation can be use to simplify the others Associated Linear system Quadratic system 4 linear equations in 5 unknowns {C 1, C 2, C 3, C 4, C 5 } Input: 5 leg lengthsOutput: Position and orientation of the platform

8. {C 1, C 2, C 3, C 4, C 5 } The linear system solution is used to generate a uni-variate 4 degree polynomial C 4 = C 5 = 0 Quadratic polynomial Forward Kinematics

9. Quadratically-solvable 5-SPU C 4 = C 5 = 0 B point at infinity All base lines are parallel.

10. Applications

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

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