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

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Presentation transcript:

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

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

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

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

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

{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

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

{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

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

Applications

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.

Thank you Júlia Borràs Sol Institut de robòtica i informàtica industrial. Barcelona Interactive visualizations done with GAViewer, developed by Daniel Fontijne - Amsterdam University