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

Flagged Parallel ManipulatorsFlagged Parallel Manipulators

F. ThomasF. Thomas

(joint work with M. Alberich and C. Torras)(joint work with M. Alberich and C. Torras)

Institut de Robòtica i Informàtica IndustrialInstitut 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 outlineTalk outlinePART 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 outlineTalk outlinePART 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


•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


Forward kinematics Forward kinematics of of

trilaterable robotstrilaterable robots

Forward kinematics of Forward kinematics of trilaterable robotstrilaterable robots




0 r12 r13 r14 1

r12 0 r23 r24 1

r13 r23 0 r34 1

r14 r24 r34 0 1

1 1 1 1 0

p1

p2

p3

p4

rij = squared distance between pi and pj

288 V2

=

Of four pointsOf four points

Cayley-Menger determinantsCayley-Menger determinants

0 r12 r13 1

r12 0 r23 1

r13 r23 0 1

1 1 1 0

p1

p3

p2

= 16 A2

Of three points:Of three points:

Of two points:Of two points:0 r12 1

r12 0 1

1 1 0p1

p2 = 2 d2


D(1 2 ... n)

NotationNotation

Cayley-Menger determinant of the n pointsp1, p2, ... , pn


D(123)2

D(1234)D(123) D(234) - D(1234) D(23)

D(123)

p4 = α1 p1 + α2 p2 + α3 p3 + β n

p1

p2

p3

p4

Position of the apex:

Forward Kinematics using Forward Kinematics using CM determinantsCM determinants

Singularity if and only if D(1234) = 0

If, additionally, D(123) = 0, the apex location is undetermined.

Singularities in terms of Singularities in terms of CM determinantsCM determinants

D(1234) = 0

D(4567) = 0

D(4789) = 0

12

3

4

5

6

4

7

7

4

8

9

Singularities in terms of Singularities in terms of CM determinantsCM determinants

vertex - face contact

edge - edge contact

face - vertex contact

Singularities in terms of Singularities in terms of basic contacts between polyhedrabasic contacts between polyhedra

Family of parallel trilaterable Family of parallel trilaterable robots robots

Each contact defines a surface in C-space, of equation: det(pi , pj , pk , pl ) = 0

C-space

1

23 4

5

67

8

Singularities in the configuration Singularities in the configuration space of the platform space of the platform

Generalization Generalization to serial robotsto serial robots

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

A PUMA robot…

1

2

3

4

5

678

… and its equivalent framework

1

2

3

4

5

67

8


1

2

3

4

5

67

8

2

3

4

5







•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


•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


•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


Technical problems at Technical problems at singularitiessingularities

The platform becomes uncontrollable at certain locations The platform becomes uncontrollable at certain locations It is not able to support weightsIt is not able to support weights The actuator forces in the legs may become very The actuator forces in the legs may become very

large. Breakdown of the robot large. Breakdown of the robot

platformplatform

6 legs6 legs

basebase

The Direct Kinematics Problem and The Direct Kinematics Problem and SingularitiesSingularities

DirectDirect finding location of platform with finding location of platform with

KinematicsKinematics respect to base from 6 leg lengths respect to base from 6 leg lengths

problemproblem finding preimages of the forward finding preimages of the forward

kinematics mapping kinematics mapping

configuration spaceconfiguration space leg lengths spaceleg lengths space

The Singularity LocusThe Singularity Locus

Rank of the Rank of the Jacobian of the Jacobian of the

kinematics mappingkinematics mapping

Singularity locusSingularity locus

Branching locus of the Branching locus of the number of number of ways of ways of assembling the platformassembling the platform

How to get rid of How to get rid of singularities?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 In both cases we need a complete and precise

characterization of the singularity locuscharacterization of the singularity locus

Stratification of the Stratification of the singularity locussingularity locus

Exemple: 3RRR planar parallel robot with fixed orientation

Goal: characterization of theGoal: characterization of the

singularity locus singularity locus (nature and (nature and location)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 robotBasic flagged parallel robot

Three possible architectures for 3-3 parallel Three possible architectures for 3-3 parallel manipulators:manipulators:

octahedral flagged 3-2-1

Basic flagged parallel robotBasic flagged parallel robot

One of the three possible architectures for 3-3 One of the three possible architectures for 3-3 parallel manipulators:parallel manipulators:

octahedral flagged 3-2-1

Trilaterable

vertex - face contact

edge - edge contact

face - vertex contact

Attaching flags

Attaching flagsAttaching flags

Attached flag to the platform

Attached flag to the base

Why Why flaggedflagged??

Because their Because their singularities singularities can be described in terms can be described in terms of of incidencesincidences between two between two flagsflags. But, what’s a. But, what’s a flag? flag?

Flags attached to the basic Flags attached to the basic flaggedflagged manipulatormanipulator

Its singularities can be described in terms of incidences Its singularities can be described in terms of incidences between its attached flags between its attached flags

Implementation of the basic Implementation of the basic flaggedflagged parallel robotparallel robot

[Bosscher and Ebert-Uphoff, 2003]

Deriving other flagged parallel Deriving other flagged parallel robots from the basic onerobots from the basic one

Local transformation on the leg endpoints that leaves singularities

invariant

Local TransformationsLocal Transformations

Composite transformations

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

2-2-22-2-2 3-2-13-2-1

Example: the 3/2 Hunt-Example: the 3/2 Hunt-Primrose manipulator is Primrose manipulator is

flaggedflagged

The flags remain invariant under the transformations

Basic flagged manipulator

3/2 Hunt-Primrose manipulator

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

The family of flagged parallel The family of flagged parallel robotsrobots

The family of flagged parallel The family of flagged parallel robotsrobots

Substituting 2-leg groups by serial chains

The family of flagged The family of flagged manipulatorsmanipulators

Substituting 2-leg groups by serial chains

Remember the equivalence Remember the equivalence basic contact & volume of a basic contact & volume of a

tetrahedrontetrahedron

Plane-vertexPlane-vertex

contactcontact

Edge-edgeEdge-edge

contactcontact

Vertex-planeVertex-plane

contactcontact

Direct kinematicsDirect 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 lengths8 assemblies for a generic set of leg lengths

Stratification Stratification of the of the

flag manifoldflag manifold

Free Space

Vertex-

plane

contact

Edge-

edge

contact

Direct kinematicsDirect kinematics In general, 4 different sets of leg lengths In general, 4 different sets of leg lengths

lead to the same configuration of flagslead to the same configuration of flags

Invariance of flags to Invariance of flags to certain transformations certain transformations

The Abelian groupThe Abelian group

Classical results on the flag Classical results on the flag manifoldmanifold



Stratification Stratification of the of the

flag manifoldflag manifold

Free Space

Vertex-

plane

contact

Edge-

edge

contact

The topology of The topology of singularitiessingularities

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 From projective to affine flags flags

From projective to affine From projective to affine flags flags

From affine flags From affine flags to the robot C-spaceto the robot C-space

Strata of dimensions 6 and Strata of dimensions 6 and 55

X 2

Flag manifold

Affine flags

X 4

Strata of dimensions 6 and Strata of dimensions 6 and 55

X 4

Manipulator C-space

Redundant flagged Redundant flagged manipulatorsmanipulators

By adding an By adding an extra legextra leg and using and using switched switched controlcontrol, the 5D singular cells can be removed , the 5D singular cells can be removed workspace enlarged by a factor of workspace enlarged by a factor of 4.4.

Two waysTwo ways of adding an extra leg to the basic of adding an extra leg to the basic flagged manipulator:flagged manipulator:

Basic Redundant

Redundant flagged Redundant flagged manipulatorsmanipulators

The singularity loci of the The singularity loci of the two component basic two component basic manipulatorsmanipulators intersect intersect only on 4D sets.only on 4D sets.

Deriving other Deriving other redundant flagged manipulators redundant flagged manipulators

Again, we can apply our local transformations that leave singularities invariant

ConclusionsConclusions C-spaceC-space of flagged manipulatorsof flagged manipulators can be decomposed into can be decomposed into

8 connected components8 connected components (6D cells) separated by (6D cells) separated by singularities (cells of dimension 5 and lower).singularities (cells of dimension 5 and lower).

The topology of 6D and 5D cellsThe topology of 6D and 5D cells has been derived, and it has been derived, and it is is independent of the manipulator metricsindependent of the manipulator metrics..

Redundant flagged manipulatorsRedundant flagged manipulators permit permit removing 5D removing 5D singularitiessingularities by switching control between two legs. by switching control between two legs.

Local transformations that preserve singularities permit Local transformations that preserve singularities permit deriving whole deriving whole families of non-redundant and redundant families of non-redundant and redundant flagged manipulators.flagged manipulators.

Presentation based on: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. 23-32, 2006.

M. Alberich-Carramiñana, F. Thomas, and C. Torras. On redundant Flagged Manipulators. Proceedings of the IEEE Int. Conf. on Robotics and Automation, Orlando, 2006.

M. Alberich-Carramiñana, F. Thomas, and C. Torras. Flagged Parallel Manipulators. IEEE Trans. on Robotics, to appear, 2007.

Documents

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