Thesis Project
Singularity-Invariant Transformations
in Stewart-Gough Platforms:
Theory and Applications
Júlia Borràs Sol
Barcelona. Spain
Thesis Advisor : Federico Thomas
Tutor : Raúl Suárez
29/01/2009
Contents
1.
2.
3.
Introduction
State of the art
i.
Kinematics
ii.
Classifications
iii.
Singularities
iv.
Workspace and dexterity
Objectives and Contributions
i.
General Objectives
ii.
Achieved Objectives/Publications
4.
Work scheduling & Publications
5.
Conclusions
Introduction
Stewart-Gough Platform is a parallel manipulator consisting of a moving rigid
platform and a rigid base connected with six extensible legs.
In general, its forward kinematics and singularity analysis are difficult
State of the art
Stewart-Gough Platform: prototypes
[1] D. Stewart,“A platform with 6 degrees of freedom,” Proc. of The Institution of
Mechanical Engineers, Vol.180, No. 15, pp. 371-386, 1965/66.
[2] V.E. Gough and S.G. Whitehall,“Universal tyre test machine,” Proc. 9th Int.
Tech. Congr. FISITA., Instn Mech. Engrs, pp. 117, 1962.
Parallel manipulators (respect Serial Manipulators):
-Greater rigidity.
-Superior positioning capabilities.
-Much better load/mass ratio
(load is distributed on the legs).
Since 1990’s, there has been a steady increase in
research interest in the Stewart-Gough platform.
[20] B. Dasguptaa and T.S. Mruthyunjayab,“The Stewart platform manipulator: a review,” Mechanism and Machine Theory, Vol. 35,
pp.15-40, 2000.
i.Kinematics
qi
Configuration Space
Workspace
li
pi
FK
IK
Inverse kinematics problem: Given the position and orientation of the mobile platform,
find the legs lengths.
Easy. You have the coordinates of all the points in the same reference.
Compute the lengths with the Euclidean distance formula.
Forward kinematics problem: Given the lengths of the 6 legs, find the position and
orientation of the mobile platform.
Several configurations are possible for a given set of lengths.
No closed-form solution in general. Six sphere equations system must be solved. Intersection
theory is used to prove that the maximum number of solutions is 40.
[9] D. Lazard, “Stewart platforms and Grobner basis,” Proceedings of Advances in Robotics Kinematics, pp. 136-142, 1992.
[10] F. Ronga and T. Vust, “Stewart platforms without computer?”,Preprint, Université de Genève, 1992
[11] M. Raghavan, “The Stewart platform of general geometry has 40 configurations,” Journal of mechanical design,
Vol. 115, No. 2, pp. 277-282, 1993.
ii. Classifications
Combinatorial representation using graphs
Possible names:
3-3  3 platform points, 3 base points
3 / 2 / 1 tripod / 2 leg group / single leg
Each graph represents the combinatorial class of manipulators with the same
structure
[14] J.C Faugère and D. Lazard, “Combinatorial classes of Parallel Manipulators”, Mechanism and Machine Theory, 1995
Classification based on
Rigid Subcomponents
Point-Line
Point-Plane
Line-Line
Line-Plane
Examples of one of the classes presented in [23]
Existence of rigid subcomponents simplifies its
analysis.
Platforms in a class have
-neither the same forward kinematics
-nor the same singularity structure .
[23] X. Kong and C.M. Gosselin, “Classification of 6-SPS Parallel Manipulators According to Their Components,”
Proc. ASME Des. Eng. Tech. Conf., 2000.
iii.Singularities
[5] J-P. Merlet, “Singular Configurations of Parallel Manipulators and Grassmann
Geometry,” The International Journal of Robotics Research, Vol. 8, No. 5, pp. 4556,1989.
Plücker Coordinates of the leg lines
Type I
Type II
SINGULARITIES
[6] C. Gosselin and J. Angeles, “Singularity Analysis of Closed-Loop Kinematic Chains,” IEEE Transactions on Robotics and Automation,
Vol. 6, No. 3, pp. 281-290, 1990.
[24] B. Mayer St-Onge and C. Gosselin, “Singularity Analysis and Representation of the General Gough- Stewart Platform,” The
International Journal of Robotics Research, Vol. 19, No. 3, pp. 271-288, 2000.
Architectural singularities
Type III
[7] O. Ma and J. Angeles, “Architecture Singularities of Platform Manipulators,” Proc. IEEE Intl. Conf. on Robotics and
Automation, Vol. 2, pp. 1542-1547, 1991.
With the prismatic joints locked, architecturally singular
manipulators exhibit a self-motion.
When legs are forming a linear
complex independently of the leg
lengths.
When the attachments fulfill
some metric relation
Architectural singularities are directly related with projective invariants.
[22] M.L. Husty and A. Karger, “Architecture Singular Parallel Manipulators and Their Self-Motions,” in Advances in Robot Kinematics, J.
Lenarcic and M. M. Stanisic (eds.), Kluwer Academic Publishers, pp. 355-364, 2000.
[33] A. Karger, “Architecture Singular Planar Parallel Manipulators,” Mechanism and Machine Theory, Vol. 38, pp. 1149-1164, 2003.
Architectural singularities
Type III
iv.Workspace and dexterity
Well-conditioned workspace
Distance to singularities
Manipulability ellipsoid
Error amplifications
Accuracy/dexterity of the robot
Manipulability index
Condition number
Closeness to a singularity
Criteria for optimal design
Problems:
Strong coupling between position and orientation.
Must take into account singularity analysis and leg collision.
All indexes depend on the chosen norm and units, and they all rely on the Jacobian matrix.
[39] J-P. Merlet, “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots,” Journal of Mechanical Design, Vol.
128, pp. 199-206, 2006.
Contents
1.
2.
3.
Introduction
State of the art
i.
Kinematics
ii.
Classifications
iii.
Singularities
iv.
Workspace and dexterity
Objectives and Contributions
i.
General Objectives
ii.
Achieved Objectives/Publications
4.
Work scheduling & Publications
5.
Conclusions
i.Objectives
In general, moving the attachments modifies the singularity
locus in an unexpected way.
When the platforms contains rigid subassemblies, we
can rearrange the legs so that the singularity locus is
modified in a controlled way.
Key points:
- kinematics of the components are not modified.
- the resultant Jacobian determinant is not modified, up
to scalar multiplication.
Singularity-invariant transformations derived from rigid components:
Delta-transform
Line-Plane transform
Does any singularity-invariant Plane-Plane transformation exist?
Objectives: Applications of the SI transformations
* Generate families of Stewart-Gough platforms with the same singularity locus.
* Classify and characterize architectural singularities through singularity-invariant transforms.
* Decouple the two problems of
- location of the singularities in a given workspace
- improving the dexterity in a given region of the configuration space.
* Provide a tool to optimize in a design process
-the dexterity or
-avoid possible collisions between its legs
without modifying the singularities.
* Avoid multiple spherical joints.
* Design reconfigurable platforms whose attachments can be modified statically or
dynamically.
-If a reconfiguration is done with a singularity-invariant transformation, the control of
the platform is not increased by the fact of being reconfigurable.
* Make contributions in the study of mathematical tools to deal with singularities:
- Grassmann-Cayley algebra.
- Geometric Algebra
ii. Achieved objectives
Point-Line transformation:
J. Borràs, F. Thomas and C.Torras, "On Delta-transforms,'' submitted to the IEEE Transactions on Robotics
(second revision).
J. Borràs, F. Thomas and C. Torras, "Analyzing the Singularities of 6-SPS Parallel Robots Using Virtual
Legs,'' Proc. II International Workshop on Fundamental Issues and Future Research Directions for Parallel
Mechanisms and Manipulators, pp. 145-150, 2008.
J. Borràs, F. Thomas and C. Torras, "Architecture Singularities in Flagged Parallel Manipulators,'' IEEE
International Conference on Robotics and Automation, pp. 3844-3850, 2008.
Line-Plane transformation:
J. Borràs and F. Thomas, "Kinematics of the Line-Plane Subassembly in Stewart Platforms,'' accepted to
the IEEE International Conference on Robotics and Automation, 2009.
J. Borràs, F. Thomas, E. Ottaviano and M. Ceccarelli, "A Reconfigurable 5-DOF 5-SPU Parallel Platform,''
submitted to IEEE/ASME International Conference on Reconfigurable Mechanisms and Robots.
Contributions on Grassmann-Cayley algebra dealing with singularities:
J. Borràs, F. Thomas and C. Torras, "Straightening-Free Algorithm for the Singularity Analysis of StewartGough Platforms with Collinear/Coplanar Attachments," submitted to Computational Kinematics.
Point-Line Transformation
m
n
The length of the new leg can be
found uniquely
Point-Line Transformation
Point-Line Transformation
Point-Line Transformation
We got the first singularity invariant transformation!
Applying sequentially
Applying sequentially several Delta-transforms, we obtain complex transformations.
Applying simultaneously
Applying simultaneously
Architectural
Singularity
Can this factor be zero?
Condition
Line-Plane Component
Line-Plane Component
SINGULARITIES OF THE LINE-PLANE COMPONENT:
Line-Plane Component
Let us consider the hypersurface in
(x4,y4,z4)
(x1,y1,z1)
(x3,y3,z3)
(x2,y2,z2)
(x5,y5,z5)
The same
coefficients !!
Line-Plane Component
We got the second singularity invariant transformation!!
Line-Plane Component
Reconfigurable robot
J. Borràs, F. Thomas, E. Ottaviano and M. Ceccarelli, "A Reconfigurable 5-DOF 5-SPU Parallel Platform,''
submitted to IEEE/ASME International Conference on Reconfigurable Mechanisms and Robots.
Contents
1.
2.
Introduction
State of the art
I.
Kinematics
II.
Classifications
III.
Singularities
IV. Workspace and dexterity
3.
Objectives and Contributions
I.
General Objectives
II.
Achieved Objectives/Publications
4.
Work scheduling & Publications
5.
Conclusions
Work Scheduling
2007
Publications
J. Borràs, F. Thomas and C. Torras, "Straightening-Free Algorithm for the Singularity Analysis of Stewart-Gough Platforms with
Collinear/Coplanar Attachments," submitted to Computational Kinematics.
J. Borràs, F. Thomas, E. Ottaviano and M. Ceccarelli, "A Reconfigurable 5-DOF 5-SPU Parallel Platform,'' submitted to IEEE/ASME
International Conference on Reconfigurable Mechanisms and Robots.
J. Borràs, F. Thomas and C.Torras, "On Delta-transforms,'' submitted to the IEEE Transactions on Robotics (second revision).
J. Borràs and F. Thomas, "Kinematics of the Line-Plane Subassembly in Stewart Platforms,'' IEEE International Conference on
Robotics and Automation, 2009.
J. Borràs and R. Di Gregorio, "Polynomial Solution to the Position Analysis of Two Assur Kinematic Chains with Four Loops
and the Same Topology,'' Journal of Mechanisms and Robotics, to appear in 2009.
J. Borràs, F. Thomas and C. Torras, "Analyzing the Singularities of 6-SPS Parallel Robots Using Virtual Legs,'' Proc. II
International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, pp. 145150, 2008.
J. Borràs and R. Di Gregorio, "Direct Position Analysis of a Large Family of Spherical and Planar Parallel Manipulators with
Four Loops,'' Proc. II International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and
Manipulators, pp. 19-29, 2008.
J. Borràs, F. Thomas and C. Torras, "Architecture Singularities in Flagged Parallel Manipulators,'' IEEE International Conference
on Robotics and Automation, pp. 3844-3850, 2008.
J. Borràs, E. Ottaviano, M. Ceccarelli, and F. Thomas, "Optimal Design of a 6-DOF 4-4 Parallel Manipulator with Uncoupled
Singularities,'' Revista de la Asociación Española de Ingeniería Mecánica, Año 16, Vol. 2, pp. 1047-1052, 2008.
Conclusions
We think that it is a new and original approach to singularity analysis.
Singularity-invariant transformations can be useful to:
Increase the number of known manipulators with closed-form kinematics
Classify manipulators with respect to singularities
Design reconfigurable robots
Provide a tool to designers to optimize workspace or dexterity.
Characterize architectural singularities with full geometrical meaning.
Thank you!!