federico thomas barcelona. spain may, 2009 computational kinematics 2009 straightening-free...
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Federico ThomasBarcelona. Spain
May, 2009
Computational Kinematics 2009
Straightening-Free Algorithm for the Singularity Analysis of Stewart-Gough Platforms
with Collinear/Coplanar Attachments
Júlia Borràs, Federico Thomas, and Carme Torras
Outline
Ben-Horin & Shoham’s algorithm
Introduction: Grassmann-Cayley algebra and the Pure Condition
Straightening-free algorithm
Examples
Conclusions
Introduction
=
The columns of the Jacobian Matrix associated with a Gough-Stewart platform are the Plücker coordinates of the leg lines
Superbracket
Neil White proved that a superbracket can be expressed as the sum of terms involving the product of three 4 × 4 determinants
The Pure Condition
The singularities correspond to those locations in which it vanishes
Grassmann-Cayley Algebra provides tools to operate with geometric entities in a coordinate-free fashion
The pure condition
Brackets
The pure condition
The three 3-3 architectures.
Simplifications are not always direct and one needs to use syzygies to obtain the simplest expressions
Existing algorithm
Multilinear properties of brackets were used to simplify the pure condition of platforms with collinear attachments on the base and/or the platform
The straightening procedure needs tbe applied to sort them again
Straightening procedure:
- 3-bracket terms are put in a tableaux (each row is a bracket). - Sorted in a lexicographic order by rows and columns by applying syzygies.- Brackets with two equal elements vanish.
After the application of a decomposition
Order is broken
The main idea of the proposed algorithm
A superbracket is, like an ordinary determinants, multilinear.
We apply the decompositions directly to the superbracket
Output is a linear combination of superbrackets.
The straightening algorithm is avoided.
Applying the pure condition formula to each superbracket, the same result as in the B&S algorithm is obtained.
: composite point
, : characteristic points
The algorithm: expandSB(sb)
Given a superbracket
Its zero? (pure condition
formula) Does it contain a
composite point?
Yes Return 0.
No Sort the elements of the superbracketReturn it sorted (with corresponding sign).
No
Yes Split the superbracket
sb1=expandSB( )
sb2=expandSB( )
Return sb1 sb2
Recursive algorithm
To compare them, they must be sorted.
Application I
The pure condition of any double planar Stewart platform can be expressed a as the linear combination of the pure conditions of 3-3 platforms.
The shortest expression for each superbracket in terms of brackets can be obtained by applying syzygies.
Example I
Example:
p. flagged p. flagged flagged flagged flagged
Input:
Output:
Example II
Example2:
p. flagged p. flagged octahedral
Input:
Output:
p. flagged p. flagged
After computing the pure condition, it contains no common factor.
Common factors Rigid components
If the octahedral topology appearsin the decomposition
The manipulator has no rigid components.
Applications II: Singularity equivalences
Case 1
coplanar
Applications II: Singularity equivalences
Case 2
Applications II: Singularity equivalences
Case 3
Applications II: Singularity equivalences
Architectural singularities
Cross-ratio condition of the Line-Plane component.
Griffis-Duffy architectural Condition.
Conclusions
An important simplification with respect to the Ben-Horin & Shoham’s algorithm has been obtained.
The structure of the solution provides other applications for the algorithm
Detect platforms with the same singularity locus
Express the pure condition of any double planar Stewart platform as the linear combination of pure conditions of 3-3 platforms
The straightening procedure is avoided.
Detect rigid components
Obtain algebraic conditions for architectural singularities in a straightforward way
Thank youFederico Thomas ([email protected])
Institut de robòtica i informàtica industrial.Barcelona