1. Federico Thomas Barcelona. Spain A Reconfigurable 5-DoF 5-SPU Parallel Platform Júlia Borràs, Federico Thomas, Erika Ottaviano, and Marco Ceccarelli

2. Outline 1 – Introduction 5 DoF 5-SPU parallel platform Previous works Goal 2 – Singularity-invariant leg rearrangements 3 – The proposed robot Architectural singularities The effect of reconfiguring Simulations 5 - Conclusions

3. 5-DoF 5-SPU Parallel Platform Let us consider a Stewart platform containing a line-plane subassembly We propose a reconfigurable 5-leg 5-SPU parallel plaform Robots with axisymmetric tool (Ex: 5-axis milling machine)

4. Previous works First closed-form solution for the forward kinematics of Stewart platforms containing a line-plane subassembly Algebraic architecturally singular condition for the line-plane subassembly Singularity-invariant leg rearrangements in line-plane subassemblies. Geometrical interpretation of the architectural singularities

5. Goal To change leg attachment locations in such a way that: The robot geometry is modified Singularity locus remains unaltered There exists a one-to-one mapping between the leg lengths before and after the reconfiguration Reconfigurations without increasing the control of the platform It can be adapted it to particular tasks Reducing the risk of collisions between legs Improving the stiffness of the robot, in a given region of its workspace

6. Singularity-invariant leg rearrangements Same singularities Different behavior even near a singularity Modify the location of the attachments Coefficients depend on the location of the attachments Singularities of a Stewart-Gough platform Unknowns: position and orientation parameters Singularity polynomial is the same (up to a constant multiple) Singularity-invariant leg rearrangement

7. Pose defined by Plücker coordinates of the leg lines Factorization of the jacobian determinant: Stewart platform containing a line-plane subassembly Singularity-invariant leg rearrangements

8. Hypersurface in Singularity polynomial of the line-plane subassembly Singularity-invariant leg rearrangements

9. (x1,y1,z1) (x2,y2,z2) (x3,y3,z3) (x4,y4,z4) (x5,y5,z5) Hypersurface in Singularity polynomial Singularity-invariant transformation!!! New point (x’,y’,z’) Singularity-invariant leg rearrangements

10. Giving values for z Giving values to (x,y) What does it mean? One-to-one correspondence points in the line lines in the plane Lines in the base plane Points in the line Singularity-invariant leg rearrangements

11. Base attachments Platform attachments Singularity-invariant leg rearrangements

12. B-lines as radial guides arranged passing though the vertices of a regular pentagon The proposed robot B point placed at the origin Reconfiguration of the attachments along the B-lines

13. Architectural singularities A line-plane subassembly is architecturally singular iff Any 4 base attachments are collinear or The 5 base attachments and B lie on a conic

14. Architectural singularities The design avoids all possible architectural singularities

15. The proposed robot Singularity locus Coordinates Cofactors

16. The proposed robot Singularity locus: For a fixed orientationFor a fixed position

17. The effect of reconfiguring Singularity locus: Multiplying factor:

18. The proposed robot

19. Prototype

20. Conclusions

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