flagged parallel manipulators f. thomas (joint work with m. alberich and c. torras) institut de...
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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
Talk outlineTalk outlinePART 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 outlineTalk outlinePART II
Forward kinematics Forward kinematics of of
trilaterable robotstrilaterable robots
Forward kinematics of Forward kinematics of trilaterable robotstrilaterable robots
Forward kinematics of Forward kinematics of trilaterable robotstrilaterable robots
Forward kinematics of Forward kinematics 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
Cayley-Menger determinantsCayley-Menger determinants
D(1 2 ... n)
NotationNotation
Cayley-Menger determinant of the n pointsp1, p2, ... , pn
Cayley-Menger determinantsCayley-Menger determinants
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
Generalization Generalization to serial robotsto serial robots
1
2
3
4
5
67
8
2
3
4
5
Generalization Generalization to serial robotsto serial robots
Generalization Generalization to serial robotsto serial robots
Generalization Generalization to serial robotsto serial robots
Generalization Generalization to serial robotsto serial robots
Generalization Generalization to serial robotsto serial robots
Generalization Generalization to serial robotsto 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 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
Talk outlineTalk outlinePART 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 outlineTalk outlinePART II
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
Classical results on the flag Classical results on the flag manifoldmanifold
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.