1. Federico Thomas Barcelona. 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

2. Outline Ben-Horin & Shoham’s algorithm Introduction: Grassmann-Cayley algebra and the Pure Condition Straightening-free algorithm Examples Conclusions

3. 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

4. The pure condition Brackets

5. The pure condition The three 3-3 architectures. Simplifications are not always direct and one needs to use syzygies to obtain the simplest expressions

6. 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

7. 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

8. The algorithm: expandSB(sb) Given a superbracket Its zero? (pure condition formula) Does it contain a composite point? YesReturn 0. NoSort the elements of the superbracket Return it sorted (with corresponding sign). No YesSplit the superbracket sb1=expandSB( ) sb2=expandSB( ) Return sb1 sb2 Recursive algorithm To compare them, they must be sorted.

9. 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.

10. Example I Example: p. flagged flagged Input: Output:

11. Example II Example2: p. flagged octahedral Input: Output: p. flagged After computing the pure condition, it contains no common factor. Common factorsRigid components If the octahedral topology appears in the decomposition The manipulator has no rigid components.

12. Applications II: Singularity equivalences Case 1

13. coplanar Applications II: Singularity equivalences Case 2

14. Applications II: Singularity equivalences Case 3

15. Applications II: Singularity equivalences Architectural singularities Cross-ratio condition of the Line-Plane component. Griffis-Duffy architectural Condition.

16. 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

17. Thank you Federico Thomas (fthomas@iri.upc.edu) Institut de robòtica i informàtica industrial. Barcelona