Transcript Presentation

A Family of Quadratically-Solvable
5-SPU Parallel Robots
Júlia Borràs, Federico Thomas and Carme Torras
Júlia Borràs Sol
Barcelona. Spain
Thursday May 6, 2010
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
5 length leg equation  5 sphere equations
One equation can be use to simplify the others
{C1, C2, C3, C4, C5}
Quadratic system
Associated Linear system
Forward Kinematics
Input: 5 leg lengths
Output: Position and orientation of the platform
5 length leg equation  5 sphere equations
One equation can be use to simplify the others
Quadratic system
Associated Linear system
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 = C 5 = 0
{C1, C2, C3, C4, C5}
Quadratic polynomial
Quadratically-solvable 5-SPU
C4 = C 5 = 0
B point at infinity
All base lines are parallel.
Applications
Conclusions
- Family of manipulators whose forward kinematics are greatly simplified:
From
Solve a 4th degree polynomial
and a 2-degree polynomial.
8 assembly modes (16)
To
Solve 2 quadratic polynomials
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 ([email protected])
Institut de robòtica i informàtica industrial.
Barcelona
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