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

Spatial parallel manipulator manipulator Platform articulated by changing the 6 leg lengths Platform articulated by changing the 6 leg lengths Platform not fully controllable singular configuration Platform not fully controllable singular configuration Kinematics Singularities of a class of parallel Manipulators Kinematics Singularities of a class of parallel Manipulators platform 6 legs base

Kinematics Singularities of a class of parallel Manipulators Kinematics Singularities of a class of parallel Manipulators Direct finding location of platform with Direct finding location of platform with Kinematics respect to base from 6 leg lengths problem finding preimages of K problem finding preimages of K configuration space leg lengths space configuration space leg lengths space Singular locus Jacobian of K Branching locus of the number of ways of assembling the platform Branching locus of the number of ways of assembling the platform

Technical problems at Singularities Derivative of K loses degree(s) of constraint Derivative of K loses degree(s) of constraint gains degree(s) of freedom gains degree(s) of freedom Platform becomes uncontrollable Platform becomes uncontrollable not able to support weights not able to support weights Actuator forces in the legs may become very large breakdown of the robot Actuator forces in the legs may become very large breakdown of the robot

Parallel manipulators Forced to operate in reduced workspaces to avoid singular configurations. Alternative: adding redundant actuators to remove singularities. Issues: Issues: Requires a complete and precise characterization of the singularity loci  Applications to control and manipulator design Requires a complete and precise characterization of the singularity loci  Applications to control and manipulator design how to plan trajectories? how to plan trajectories? where to place the extra leg? where to place the extra leg?

Goal: characterization of the singularity loci (nature and location) Each pair of assemblies separated by Three assemblies coalesce P. Candidate of singular region never-singular motion is shown from A1 to A3.

Two-strategy approach Simplification of the manipulator design: Simplification of the manipulator design: collapse leg endpoints leaving singular locus invariant Attaching geometrical objects, Attaching geometrical objects, whose parameter space has good properties easily transferred to configuration space

Basic flagged manipulator 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

Basic flagged manipulator Why flagged? Because their singularities can be described in terms of incidences between two flags:

Singularities of the basic flagged manipulator Where the tetrahedra involved in the computation of its direct kinematics vanish

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

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

The family of flagged manipulators

Substituting 2-leg groups by serial chains

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

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

Classical results on the flag manifold

From projective to affine flags

From affine flags to manipulator configurations

Stratification of the flag manifold

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

Why x4?

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 flagged manipulators from the basic redundant one Local transformation on the leg endpoints that leaves 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. A local transformation that preserves singularities permit deriving whole families of non-redundant and redundant flagged manipulators. A local transformation that preserves singularities permit deriving whole families of non-redundant and redundant flagged manipulators.

Presentation based on publications: Carme Torras, Federico Thomas, and Maria Alberich-Carramiñana. Stratifying the Singularity Loci of a Class of Parallel Manipulators. IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 1, FEBRUARY Maria Alberich-Carramiñana, Federico Thomas, and Carme Torras. On redundant Flagged Manipulators. Proceedings of the 2006 IEEE International Conference on Robotics and Automation (ICRA). Orlando, Florida - May 2006 Maria Alberich-Carramiñana, Federico Thomas, and Carme Torras. Flagged Parallel Manipulators. To appear in IEEE TRANSACTIONS ON ROBOTICS

Parallel manipulators Forced to operate in reduced workspaces to avoid singular configurations. Alternative: adding redundant actuators to remove singularities. ( Merlet, Dasgupta and Mruthyunjaya, Notash and Podhorodeski, Kock and Schumacher, Voglewede,…) Issue: where to place the extra leg? Issue: where to place the extra leg? Requires a complete and precise characterization of the singularity loci  Flagged manipulators Requires a complete and precise characterization of the singularity loci  Flagged manipulators

Kinematics Singularities of a class of parallel Manipulators Kinematics Singularities of a class of parallel Manipulators Direct kinematics problem finding location of platform with respect to base from 6 leg lenghts finding preimages of finding preimages of??? configuration space leg lenghts space Singular locus Jacobian of Branching locus of the number of ways of assembling the platform Branching locus of the number of ways of assembling the platform

Parallel manipulators Forced to operate in reduced workspaces to avoid singular configurations. Alternative: adding redundant actuators to remove singularities. Issues: how to plan trajectories? Issues: how to plan trajectories? where to place the extra leg? Requires a complete and precise characterization of the singularity loci  Applications to control and manipulator design Requires a complete and precise characterization of the singularity loci  Applications to control and manipulator design

Kinematics Singularities of a class of parallel Manipulators Kinematics Singularities of a class of parallel Manipulators platform Spatial parallel manipulator 6 legs 6 legsbase Platform articulated by changing the 6 leg lengths Platform articulated by changing the 6 leg lengths Platform not fully controllable singular configuration Platform not fully controllable singular configuration

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. A local transformation that preserves singularities permit deriving whole families of non-redundant and redundant flagged manipulators. A local transformation that preserves singularities permit deriving whole families of non-redundant and redundant flagged manipulators. Application to wire-based tracking devices. Application to wire-based tracking devices.

Stratification of the flag manifold

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