Transcript PowerPoint presentation

Straightening-Free Algorithm for the
Singularity Analysis of Stewart-Gough Platforms with
Collinear/Coplanar Attachments
Júlia Borràs, Federico Thomas, and Carme Torras
Federico Thomas
Barcelona. Spain
Computational Kinematics 2009
May, 2009
Outline
Introduction: Grassmann-Cayley algebra
and the Pure Condition
Ben-Horin & Shoham’s algorithm
Straightening-free algorithm
Examples
Conclusions
Introduction
Grassmann-Cayley Algebra provides tools to operate with geometric entities in a
coordinate-free fashion
The columns of the Jacobian Matrix associated with a
Gough-Stewart platform are the Plücker coordinates of
the leg lines
=
Superbracket
The singularities
correspond to those
locations in which it
vanishes
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 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
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 straightening procedure needs tbe
applied to sort them again
The main idea of the proposed algorithm
: composite point
,
: characteristic points
A superbracket is, like an ordinary determinants, multilinear.
We apply the decompositions directly to the superbracket
Applying the pure condition formula to each
Output is a linear combination of superbrackets.
superbracket, the same result as in the B&S
algorithm is obtained.
The straightening algorithm is avoided.
The algorithm: expandSB(sb)
Given a superbracket
Yes
No
Its zero?
(pure condition
formula)
Does it
contain a
composite
point?
Return 0.
Recursive
algorithm
No
Sort the elements of the superbracket
Return it sorted (with corresponding sign).
To compare them, they must be sorted.
Yes
Split the superbracket
Return
sb1
sb1=expandSB(
)
sb2=expandSB(
)
sb2
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:
Input:
Output:
p. flagged
p. flagged
flagged
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Example II
Example2:
Input:
Output:
p. flagged
p. flagged
octahedral
p. flagged
After computing the pure condition, it contains no common factor.
Common factors
If the octahedral topology appears
in the decomposition
Rigid components
The manipulator has no
rigid components.
p. flagged
Applications II:
Singularity equivalences
Case 1
Applications II:
Singularity equivalences
Case 2
coplanar
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 straightening procedure is avoided.
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
Obtain algebraic conditions for architectural singularities in a straightforward
way
Detect rigid components
Thank you
Federico Thomas ([email protected])
Institut de robòtica i informàtica industrial.
Barcelona