Flagged Parallel Manipulators F. Thomas (joint work with M. Alberich and C. Torras) Institut de Robòtica i Informàtica Industrial Institut de Robòtica i Informàtica Industrial

Trilatelable Parallel Robots Forward kinematics Singularities Formulation using determinants Singularities as basic contacts between polyhedra Generalization to serial robots Talk outline PART I

Technical problems at singularities The direct kinematics problem and singularities The singularity locus How to get rid of singularities? Goal: Characterization of the singularity locus Stratification of the singularity locus Basic flagged parallel robot Talk outline PART II

Why flagged? Ataching flags to parallel robots Equilalence between basic contacts and volumes of tetrahedra Deriving the whole family of flagged parallel robots Local transformations Substituting of 2-leg groups by serial chains Examples Talk outline PART II

The direct kinematics of flagged parallel robots Invariance of flags to certain transformations Classical result from the flag manifold Stratification of the flag manifold From projective flags to affine flags From afine flags to the configuration space of the platform Strata of dimension 6 and 5 Redundant flagged parallel robots Talk outline PART II

Forward kinematics of trilaterable robots

0 r 12 r 13 r 14 1 r 12 0 r 23 r 24 1 r 13 r 23 0 r 34 1 r 14 r 24 r p1p1 p2p2 p3p3 p4p4 r ij = squared distance between p i and p j 288 V 2 = Of four points Cayley-Menger determinants

0 r 12 r 13 1 r 12 0 r 23 1 r 13 r p1p1 p3p3 p2p2 = 16 A 2 Of three points: Of two points: 0 r 12 1 r p1p1 p2p2 = 2 d 2 Cayley-Menger determinants

D ( n) Notation Cayley-Menger determinant of the n points p 1, p 2,..., p n Cayley-Menger determinants

D (123) 2 D (1234) D (123) D (234) - D (1234) D (23) D (123) p 4 = α 1 p 1 + α 2 p 2 + α 3 p 3 + β n p1p1 p2p2 p3p3 p4p4 Position of the apex: Forward Kinematics using CM determinants

Singularity if and only if D (1234) = 0 If, additionally, D (123) = 0, the apex location is undetermined. Singularities in terms of CM determinants

D (1234) = 0 D (4567) = 0 D (4789) = Singularities in terms of CM determinants

vertex - face contact edge - edge contact face - vertex contact Singularities in terms of basic contacts between polyhedra

Family of parallel trilaterable robots

Each contact defines a surface in C-space, of equation: det( p i, p j, p k, p l ) = 0 C-space Singularities in the configuration space of the platform

Generalization to serial robots A 6R robot can be seen as an articulated ring of six tetrahedra involving 12 points

A PUMA robot… … and its equivalent framework Generalization to serial robots

Technical problems at singularities The direct kinematics problem and singularities The singularity locus How to get rid of singularities? Goal: Characterization of the singularity locus Stratification of the singularity locus Basic flagged parallel robot Talk outline PART II

Why flagged? Ataching flags to parallel robots Equilalence between basic contacts and volumes of tetrahedra Deriving the whole family of flagged parallel robots Local transformations Substituting of 2-leg groups by serial chains Examples Talk outline PART II

The direct kinematics of flagged parallel robots Invariance of flags to certain transformations Classical result from the flag manifold Stratification of the flag manifold From projective flags to affine flags From afine flags to the configuration space of the platform Strata of dimension 6 and 5 Redundant flagged parallel robots Talk outline PART II

Technical problems at singularities The platform becomes uncontrollable at certain locations It is not able to support weights It is not able to support weights The actuator forces in the legs may become very large. Breakdown of the robot The actuator forces in the legs may become very large. Breakdown of the robot platform 6 legs base

The Direct Kinematics Problem and Singularities The Direct Kinematics Problem and Singularities Direct finding location of platform with Direct finding location of platform with Kinematics respect to base from 6 leg lengths problem finding preimages of the forward problem finding preimages of the forward kinematics mapping kinematics mapping configuration space leg lengths space configuration space leg lengths space

The Singularity Locus The Singularity Locus Rank of the Jacobian of the Rank of the Jacobian of the kinematics mapping kinematics mapping Singularity locus Singularity locus Branching locus of the number of ways of assembling the platform Branching locus of the number of ways of assembling the platform

How to get rid of singularities? By operating in reduced workspaces By adding redundant actuators Problems: Problems: how to plan trajectories? how to plan trajectories? where to place the extra leg? where to place the extra leg? In both cases we need a complete and precise characterization of the singularity locus

Stratification of the singularity locus Exemple: 3RRR planar parallel robot with fixed orientation

Goal: characterization of the singularity locus (nature and location) Two assembly modes are always separated by a singular region Two assembly modes can be connected by singularity-free paths Configuration space Branching locus Leg lengths space Configuration space Branching locus Leg lengths space

Basic flagged parallel robot Three possible architectures for 3-3 parallel manipulators: Three possible architectures for 3-3 parallel manipulators: octahedralflagged3-2-1

Basic flagged parallel robot One of the three possible architectures for 3-3 parallel manipulators: One of the three possible architectures for 3-3 parallel manipulators: octahedralflagged3-2-1 Trilaterable

vertex - face contact edge - edge contact face - vertex contact Attaching flags

Attached flag to the platform Attached flag to the base

Why flagged? Because their singularities can be described in terms of incidences between two flags. But, what’s a flag?

Flags attached to the basic flagged manipulator Its singularities can be described in terms of incidences between its attached flags Its singularities can be described in terms of incidences between its attached flags

Implementation of the basic flagged parallel robot [Bosscher and Ebert-Uphoff, 2003]

Deriving other flagged parallel robots from the basic one Deriving other flagged parallel robots from the basic one Local transformation on the leg endpoints that leaves singularities invariant

Local Transformations Local Transformations Composite transformations

Example: the 3/2 Hunt-Primrose manipulator is flagged The flags remain invariant under the transformations Basic flagged manipulator 3/2 Hunt-Primrose manipulator

Example: the 3/2 Hunt-Primrose at a singularity Example: the 3/2 Hunt-Primrose at a singularity

The family of flagged parallel robots The family of flagged parallel robots

Substituting 2-leg groups by serial chains

The family of flagged manipulators Substituting 2-leg groups by serial chains

Remember the equivalence basic contact & volume of a tetrahedron Plane-vertexcontact Edge-edgecontact Vertex-planecontact

Direct kinematics which, in general, lead to different configurations for the attached flags The four mirror configurations with respect to the base plane not shown 8 assemblies for a generic set of leg lengths 8 assemblies for a generic set of leg lengths

Stratification of the flag manifold Free Space Vertex- plane contact Edge- edge contact

Direct kinematics In general, 4 different sets of leg lengths lead to the same configuration of flags In general, 4 different sets of leg lengths lead to the same configuration of flags

Invariance of flags to certain transformations

The Abelian group

Classical results on the flag manifold

Stratification of the flag manifold Free Space Vertex- plane contact Edge- edge contact

The topology of singularities Flag manifold Subset of affine flags Manipulator C-space Schubert cells Ehresmann-Bruhat order Via a 4-fold covering map Restriction map splitted cells Refinement of the Ehresmann-Bruhat order

From projective to affine flags

From affine flags to the robot C-space

Strata of dimensions 6 and 5 X 2 Flag manifold Affine flags X 4

Strata of dimensions 6 and 5 X 4 Manipulator C-space

Redundant flagged manipulators By adding an extra leg and using switched control, the 5D singular cells can be removed  workspace enlarged by a factor of 4. By adding an extra leg and using switched control, the 5D singular cells can be removed  workspace enlarged by a factor of 4. Two ways of adding an extra leg to the basic flagged manipulator: Two ways of adding an extra leg to the basic flagged manipulator: Basic Redundant

Redundant flagged manipulators The singularity loci of the two component basic manipulators intersect only on 4D sets.

Deriving other redundant flagged manipulators Deriving other redundant flagged manipulators Again, we can apply our local transformations that leave singularities invariant

Conclusions C-space of flagged manipulators can be decomposed into 8 connected components (6D cells) separated by singularities (cells of dimension 5 and lower). C-space of flagged manipulators can be decomposed into 8 connected components (6D cells) separated by singularities (cells of dimension 5 and lower). The topology of 6D and 5D cells has been derived, and it is independent of the manipulator metrics. The topology of 6D and 5D cells has been derived, and it is independent of the manipulator metrics. Redundant flagged manipulators permit removing 5D singularities by switching control between two legs. Redundant flagged manipulators permit removing 5D singularities by switching control between two legs. Local transformations that preserve singularities permit deriving whole families of non-redundant and redundant flagged manipulators. Local transformations that preserve singularities permit deriving whole families of non-redundant and redundant flagged manipulators.

Presentation based on: C. Torras, F. Thomas, and M. Alberich-Carramiñana. Stratifying the Singularity Loci of a Class of Parallel Manipulators. IEEE Trans. on Robotics, Vol. 22, No. 1, pp , M. Alberich-Carramiñana, F. Thomas, and C. Torras. On redundant Flagged Manipulators. Proceedings of the IEEE Int. Conf. on Robotics and Automation, Orlando, M. Alberich-Carramiñana, F. Thomas, and C. Torras. Flagged Parallel Manipulators. IEEE Trans. on Robotics, to appear, 2007.