120 likes | 155 Views
This paper explores a family of manipulators with simplified forward kinematics, using quadratic equations and geometric interpretations. It discusses singularity reformulation, leg rearrangements, and practical applications in robotics. The study shows how these robots can optimize manipulability, stiffness, and workspace.
E N D
A Family of Quadratically-Solvable 5-SPU 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
Previous works Reformulation of singularities in terms of a matrix T det(J) = det(T) Singularity polynomial {C1, C2, C3, C4, C5}
Previous works Singularity-invariant leg rearrangements Leg rearrangements that preserve the 6 coefficients, up to constant multiple. {C1, C2, C3, C4, C5}
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}
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}
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}
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}
Quadratically-solvable 5-SPU C4= C5 = 0 B point at infinity All base lines are parallel.
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.
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