structural and kinematic analysis and synthesis of parallel robots

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Structural and Kinematic Analysis and Synthesis of Parallel RobotsbyGrigore GOGU

Structural and Kinematic Structural and Kinematic Analysis and Synthesis of Analysis and Synthesis of Parallel RobotsParallel RobotsbybyGrigore GOGUGrigore GOGU

SSIR-Clermont-Ferrand - June 27 2008 - G. GOGU

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Structural and Kinematic Analysis and Synthesis of Parallel Robots

Structural and Kinematic Structural and Kinematic Analysis and Synthesis of Analysis and Synthesis of Parallel RobotsParallel Robots

Grigore GOGUGrigore GOGU

Mechanical Engineering Research Group (Mechanical Engineering Research Group (LaMILaMI))

French Institute of Advanced Mechanics (IFMA) French Institute of Advanced Mechanics (IFMA) and and BlaiseBlaise Pascal University (UBP) Pascal University (UBP)

ClermontClermont--FerrandFerrand, France, France


33SSIR-Clermont-Ferrand - June 27 2008 - G. GOGU

Structural and Kinematic Analysis and Synthesis of Parallel RobotsStructural and Kinematic Analysis and Structural and Kinematic Analysis and Synthesis of Parallel RobotsSynthesis of Parallel Robots

Abstract

• This lecture emphasizes on the application of the new formulae of mobility, connectivity overconstraint and redundancy, recently proposed by the author [Gogu 2005a,b,c, 2008a], in the kinematic analysis and synthesis of parallel robots. • The main structural parameters of the parallel mechanisms (connectivity, redundancy and overconstraint) can be easily associated with mobility calculation. • Singularity analysis of parallel robots can also be performed via this new mobility formula. • The new formulae for mobility, connectivity, redundancy and overconstraint are combined with an original evolutionary morphology approach for structural synthesis of parallel robots [Gogu 2008a,b].



Main references

1. GOGU, G. Structural Synthesis of Parallel Robots, Part 1: Methodology, Springer, Dordrecht, 2008a, ISBN: 978-1-4020-5102-9 (716 pages). 2. GOGU, G. Structural Synthesis of Parallel Robots, Part 2: Toplogies,Springer, Dordrecht, 2008b (in press, 650 pages).3. GOGU, G. Mobility of Mechanisms : A Critical Review. Mechanism and Machine Theory, vol. 40, pp. 1068-1097, 2005a.4. GOGU, G. Chebychev-Grubler-Kutzbach’s criterion for mobility calculation of multi-loop mechanisms revisited via theory of linear transformations. European Journal of Mechanics / A –Solids, vol. 24, pp. 427-441, 2005b. 5. GOGU, G. Mobility and spatiality of parallel robots revisited via theory oflinear transformations, European Journal of Mechanics / A –Solids, vol. 24, pp. 690-711, 2005c.

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Mobility, connectivity and overconstraints

New formulae for parallel manipulators

Kinematic criteria for structural synthesis of parallel manipulators

Mobility, connectivity and overconstraintsMobility, connectivity and overconstraints

New formulae for parallel manipulatorsNew formulae for parallel manipulators

Kinematic criteria for structural synthesis of Kinematic criteria for structural synthesis of parallel manipulatorsparallel manipulators

OverviewOverviewOverview



6

• Mobility, connectivity (spatiality) and overconstraints are the main structural parameters of a mechanism.

• Mobility indicates the number of independent joint parameters required to determine the motion of all links of the mechanism.

• Spatiality (connectivity) indicates the number of relative independent freedoms between two links of a mechanism.

• The number of overconstraints of a mechanism is given by the difference between the number of kinematic constraints introduced by the joints of the mechanism before and after closing the loops of the mechanism.

Mobility, connectivity and overconstraintsMobility, connectivity and overconstraintsMobility, connectivity and overconstraints


7

• Instantaneous mobility and global (full-cycle) mobility can be calculated.

• The global mobility has a single value for a branch of a given mechanism. This is a global parameter characterizing the mechanism in all configurations excepting singular ones.

• The instantaneous mobility is a local parameter characterizing the mechanism in a given configuration including singular ones.

• The global mobility is quickly calculated by using mobility formulae (criteria).

• The instantaneous mobility is calculated by developing the set of constraint equations.



8



9

• Many authors have emphasized, in the last 40 years, on the difference between general mobility and connectivity rather than on their connection.

(i)“The classical approach to the subject of number synthesis consists of an attempto find and classify all the mechanisms having a given mobility which obey Grübler’s,Kutzbach’s or a related equation. However, in the practical design of a machine, the primary requirement is usually that a mechanism have some pair of members which have a given number of degrees of freedom. Thus the constraint quantity which is actually required is the connectivity of the mechanism when it is consideredto be a joint between some chosen pair of members.”

K. J. Waldron, The constraint analysis of mechanisms, J. Mechanisms, vol. 1, pp. 112, 1966.

ContextContextContext



10


(ii) “The definition of linkage mobility is placed in perspective as secondary to a study of internal freedoms”– J. E. Baker, On mobility and relative freedoms in multiloop linkages and structures, Mech. Mach. Theory, vol. 16, n° 6, pp. 583, 1981.




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(iii) “However, we need to be cautious, since in devices like manipulators we are not generally concerned with the mechanism’s mobility, but rather the number of degrees-of-freedom between two specific links – usually, the freedom betweenthe ground link and the end-effector link.”– M. Shoham, B. Roth, Connectivity in open and closed loop robotic mechanisms, Mech. Mach. Theory, vol. 32, n° 3, pp. 279, 1997.




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(iv) “In the theory, we also point out that it is more appropriate to calculate the DOF of the mechanism with an output member rather than the DOF of the wholemechanism”J-S. Zhao, K, Zhou, Z-J. Feng, A theory of degree of freedom for mechanisms, Mech. Mach. Theory, vol. 39, pp. 622, 2004.




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• The development of a formula for mobility calculation has a history of about 150 years: Chebychev 1869, Sylvester 1874, Grübler 1883 and 1885, Somov 1887, Hochman 1890, Koenigs 1905, Grübler 1917, Malytsheff 1923, Kutzbach 1929.

• Versions of the initial Chebychev-Grübler-Kutzbach’s formula for multi-loop mechanisms were proposed all along the 20th century: Dobrovolski 1949 and 1951, Artobolevski 1953, Kolchin 1960, Rössner 1961, Boden 1962, Ozol 1963, Manolescu and Manafu 1963, Manolescu 1968, Bagci 1971, Hunt 1978, Hervé 1978-a and b, Tsai 1999

p

ii 1

M f bq=

= −∑p - total number of joints, fi - mobility of the i th joint bk – motion parameter of kth loop (the rank of the constraint equations of kth loop )q=p-m+1 - total number of independent closed loops in the sense of graph theorym – total number of links



14

• Important contributions in instantaneously mobility calculation of multi-loop mechanisms were also proposed by setting-up sequential computational approaches, but these contributions did not proposed new mobility formulae:Moroskine 1954 and 1958, Waldron 1966, Angeles and Gosselin 1988, Fayet1995, Huang and Li 2003, Huang 2004, Dai et al. 2004

• Important contributions in mobility determination of multi-loop mechanisms were also proposed by setting-up sequential equivalence approaches: J-S. Zhao et al. 2004, Kong and Gosselin 2006, 2007;



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• A slightly different formula that we call extended Chebychev-Grübler-Kutzbach’sformula was proposed at the middle of the 20th century by

• Voinea and Atanasiu (1960) • Manafu (Manolescu and Manafu 1963).

• The extended formula in the form proposed by Voinea and Atanasiu was developed later by Antonescu (1973), Freudenstein and Alizade (1975), Dudiţă and Diaconescu (1987).

• The extended formula in the form proposed by Manafu was developed later by Gronowicz (1981), Davies (1983), Agrawal and Rao (1987-b), Dudiţă and Diaconescu (1987), Rico and Ravani (2004-a).

p q

i ki 1 k 1

M f b= =

= −∑ ∑

1

q

i ci

M M M=

= −∑



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Today we can note that these mobility formulae do not fit for many classical mechanisms as, for instance, the mechanisms proposed by

Roberval (1670), Sarrus (1853), Delassus (1900, 1902, 1922), Bennett (1902), Bricard (1927), Myard (1931), Goldberg (1943), Altman (1952), Baker (1978), Waldron (1979), et al.,

or for many recent parallel robots.

•Special geometric conditions play a significant role in the determination of mobility of these mechanisms.



17

When these mechanisms were limited to special examples, consideredas “curiosities”, they were called:

• paradoxical mechanisms (Bricard, 1927),

• paradoxical chains (Hervé, 1978, 1999; Norton, 1999), • exceptions (Norton, 1999; Myszka, 2001),

• special cases (Mabie and Reinholtz, 1987),

• linkage with a paradox between “practical degrees of freedom”and “computed degrees of freedom” (Eckhardt, 1998),

• overconstrained yet mobile linkage (Phillips, 1984, 1990; Waldron and Kinzel, 1999),

• linkages with anomalous mobility (Waldron and Kinzel, 1999).



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Connectivity (spatiality) is also known in the literature as:• the rank of linear set of screws (Voinea and Atanasiu, 1959), • link mobility (Voinea and Atanasiu, 1960), • relative infinitesimal displacements of two bodies (Hunt, 1967), • connectivity of the complex joint between two bodies (Waldron, 1966), • connectivity of the instantaneous screw system of two bodies (Davies, 1971), • degree of freedom of the complex joint between two bodies (Hervé, 1978-a, b), • relative freedom between links (Baker, 1980), • complex connectivity (Baker, 1980), • internal freedom (Baker, 1981), • connectivity between a pair of members (Phillips, 1984), • freedom of the complex joint between a pair of members (Phillips, 1984), • rank of the kinematic space of the mechanism (Dudiţǎ and Diaconescu, 1987), • kinematic constraints between two bodies (Fanghella, 1988), • connectivity (Fanghella and Galletti, 1994), • dimension of the space of twists between two bodies (Fayet, 1995), • link connectivity (Shoham and Roth, 1997), • degree of freedom of the mechanism with an output member (Zhao et al., 2004).



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Various tools can be used to define and to calculate instantaneous values of connectivity for a given position of the mechanism: • Aronhold‘s (1872) and Kennedy’s (1886) theorem of three axes in planar motion of three bodies, • Phillips-Hunt’s theorem of three axes in spatial motion of three bodies (Phillips and Hunt, 1964), • Chasles’s (1830) and Ball’s (1900) theory of the instantaneous screw axes (Waldron,1966; Davies and Primrose, 1971; Dudiţǎ and Diaconescu, 1987), • Schoenflies’s (1893) geometry of linear complex (Hunt, 1967 and 1978), • Klein’s projective space of linear set of screws (Voinea and Atanasiu, 1959; Rico and Ravani, 2004), • reciprocal screw theory (Zhao et al., 2004), • screw motor (Baker, 1980), • twists theory and triangular projective method (Fayet, 1995), • mathematical group structure of the set of displacements (Hervé, 1978; Fanghella, 1988; Fanghella and Galletti, 1994; Rico and Ravani, 2003), • Lie group of rigid body displacement (Hervé, 1999; Rico and Ravani, 2003 and 2004)



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Hypothesis :General mobility, spatiality (connectivity) and overconstraints are closely relatedin parallel manipulators

Aim : a) to demonstrate and to propose for the first time in the literature general relations between:

• general mobility of a parallel manipulator, leg connectivity and mobile platform connectivity,• number of overconstraints of the parallel manipulator, leg connectivity and mobile platform connectivity.

b) to integrate these formulae in an original method for structural synthesis of parallel robots founded on an evolutionary morphology approach

Method: theory of linear transformations (Gogu, 2002)



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The mobility of a mechanism with m joints and q≥1 independent closed loops

1

m

ii

M f r=

= −∑

1

m

ii

f=∑ - the number of independent motion parameters of the joints before loop

closures provide further constraints

r - the number of joint parameters that have lost their independence after closing the loops

(Moroskine, 1954)

The number of overconstraints of a mechanism with mobility M and qindependent closed loops

6N q r= −



2222

Validity limitation of extended Chebychev-Grübler-Kutzbach (CGK) formula wasset up in (Gogu, EJM-A/Solids-2005):

q

kk 1

r b=

=∑if and only if the rank of the linear set of kinematic constraint equations of (k+1)th

loop is equal to the dimension of the range of the restriction of Fk+1 to the kernel of F1-2-…-k

Mobility calculation: 35 formulae/approaches developpend in the last 150 years andcritically reviewed in (Gogu, MMT-2005)



23

A fundamental mathematical property of a mechanism Q :

U WF

r =

New formulae for mobility, connectivity and overconstraints of parallel manipulatorsNew formulae for mobility, connectivity and New formulae for mobility, connectivity and overconstraints of parallel manipulatorsoverconstraints of parallel manipulators


2424

New formula for mobility calculation of PMs (Gogu, EJM-A/Solids-2005, 2008)



2525

New formulae for the structural parameters of PMs (Gogu, 2005)

k

Gi F li 1

r S S r=

= − +∑

p

ii 1

M f r=

= −∑

F F G1 G2 GkS dim( R ) dim( R R ... R )= = ∩ ∩ ∩

kGi

l li 1

r r=

= ∑k

Gii 1

p p=

=∑

N=6q-r T=M-SF

Mobility Overconstraint Redundancy

Connectivity



2626

Parallel mechanisms with simple legsExample – Isoglide3-T3 v1.1Parallel mechanisms with simple legsParallel mechanisms with simple legsExample Example –– Isoglide3Isoglide3--T3 v1.1T3 v1.1

33--PPRRRRRR--typetype



2727

Isoglide3-T3 v1.1Isoglide3Isoglide3--T3 v1.1T3 v1.1


p q

i ki 1 k 1

M f b 12 ( 5 5 ) 2= =

= − = − + =∑ ∑ ?!CGK:



2828

Isoglide3-T3 v1.1Isoglide3Isoglide3--T3 v1.1T3 v1.1




2929

Isoglide4-T3R1 v1.1Isoglide4Isoglide4--T3R1 v1.1T3R1 v1.1

33--PPRRRR+1RRRR+1--PPRRRRRR--typetype



3030

Example – Isoglide4-T3R1 v1.1Example Example –– Isoglide4Isoglide4--T3R1 v1.1T3R1 v1.1

33--PPRRRR+1RRRR+1--PPRRRRRR--typetype

p q

i ki 1 k 1

M f b 19 ? ?= =

= − = − =∑ ∑CGK: ?!



3131

Isoglide4-T3R1 v1.1Isoglide4Isoglide4--T3R1 v1.1T3R1 v1.133--PPRRRR+1RRRR+1--PPRRRRRR--typetype



3232

33--PPRRRR+1RRRR+1--PPRRRR*RRRR*--typetype




3333

Example – Isoglide4-T3R1 v3.1Example Example –– Isoglide4Isoglide4--T3R1 v3.1T3R1 v3.1k

Gii 1

p p=

=∑



3434

Isoglide4-T3R1 v3.1Isoglide4Isoglide4--T3R1 v3.1T3R1 v3.1CGK:

p q

i ki 1 k 1

M f b 18 ? ?= =

= − = − =∑ ∑ ?!



3535




3636

Isoglide3-T2R1Isoglide3Isoglide3--T2R1T2R1



3737

Isoglide4-T2R1 v3.1Isoglide4Isoglide4--T2R1 v3.1T2R1 v3.1 CGK: ?!p q

i ki 1 k 1

M f b 18 ? ?= =

= − = − =∑ ∑



3838

Isoglide3-T2R1Isoglide3Isoglide3--T2R1T2R1

G1lr 5=



3939

Types of PMs (Gogu EJM/A-2004):

• maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace,

• fully-isotropic PMs, if the Jacobian J is a diagonal matrix with identical diagonal elements throughout the entire workspace,

• PMs with uncoupled motions if J is a diagonal matrix,

• PMs with decoupled motions, if J is a triangular matrix

• PMs with coupled motions if J is neither triangular nor diagonal matrix.

[ ] [ ][ ]vA B q

ω⎡ ⎤

=⎢ ⎥⎣ ⎦

& [ ][ ]vJ q

ω⎡ ⎤

=⎢ ⎥⎣ ⎦

&[J]=[A]-1[B] IntroductionIntroductionIntroduction

Kinematic criteria for structural synthesis of parallel manipulatorsKinematic criteria for structural synthesis of Kinematic criteria for structural synthesis of parallel manipulatorsparallel manipulators


4040

PPM with coupled motions 3-TRR-type

Maximally regular PPM 3-TRR-type



4141

The following types of maximally regular and implicitly fully-isotropic PMs have been recently proposed:

• T3-type PMs with translational motion, Caricato and Parenti-Castelli (IJRR-2002); Kim and Tsai (ARK 2002), Kong and Gosselin (ARK-2002), Gogu (Robea-CNRS-2002, EJM/A-2004)

• planar PMs, Gogu (IROS-2004)

• PMs with Schönflies morions, Gogu (ARK-2004, ICAR-2005, EJM/A-2007), Caricato (IJRR-2005)

• T1R2-type PMs, Gogu (IDETC-2005)

• T2R2-type PMs, Gogu (IROS-2005)

• T3R2-type PMs, Gogu (ICRA-2006, RAM06)

• T3R3-type PMs, Gogu (ARK 2006)

• R2- and R3-type, Gogu (ICRA 2005, 2007)



4242

The following types of maximally regular and implicitly redundantly actuated fully-isotropic PMs have been recently proposed:

• T3R2-type RaPMs, Gogu (EUCOMES 2006)

• T1R3-type RaPMs, Gogu (ICAR 2007)

• R3-type RaPMs, Gogu (ASME-IDETC 2007)

• T2R3-type RaPMs, Gogu (IROS 2007)



4343

Methods for structural synthesis of PMs known in the literature :

• methods based on displacement group theory (Hervé and Sparacino, 1991; Hervé, 1995, 2004)

• methods based on screw algebra (Frisoli et al., 2000; Kong and Gosselin, 2001; Fang and Tsai, 2002; Huang and Li 2003; Hunag)

• methods based on constraint and direct singularity investigation (Caricato and Parenti-Castelli, 2002)

• method based on the theory of linear transformations and evolutionary morphology (Gogu, 2002, 2008)



4444

EM (Gogu, 2004)

EM can be defined as 6-tuple

applicable to each generation t of the morphological evolutionary process,

is the set of final objectives,

- the set of primary elements,

- the set of morphological operators applicable at generation t,

- the set of evolution criteria from generation t to generation t+1,

- the set of solutions (morphologies) at each generation t,

- a termination criterion for EM.

t t t tEM ( ,E, ,T , , )Φ Ο Σ ι=

1 n( ,..., )Φ φ φ=

1( ,..., )nE ε ε=

1( ,..., )t no oΟ =

1( ,..., )t nT τ τ=

1( ,..., )t nσ σΣ =

{ }: ,i true falseι ϕ →


Evolutionary morphologyEvolutionary morphologyEvolutionary morphology


4545

Set of design objectives1 n( ,..., )Φ φ φ=




4646

Set of constituent elements 1( ,..., )nE ε ε=




4747

Set of constituent elements 1( ,..., )nE ε ε=




4848

Set of evolution criteria 1( ,..., )t nT τ τ=



Set of morphological operators 1( ,..., )t no oΟ =


4949

Set of morphological operators 1( ,..., )t no oΟ =

MOBILITY OF MECHANISMS: an open problem in mechanism designMOBILITY OF MECHANISMS: MOBILITY OF MECHANISMS: an open problem in mechanism designan open problem in mechanism design



5050

Set of morphological operatorsStructural synthesisStructural synthesisStructural synthesis

1( ,..., )t no oΟ =

MOBILITY OF MECHANISMS: an open problem in mechanism designMOBILITY OF MECHANISMS: MOBILITY OF MECHANISMS: an open problem in mechanism designan open problem in mechanism design


5151

Isoglidex – TyRz (x=y+z) x=4, y=3, z=1IsoglideIsoglidexx –– TTyyRRzz ((xx==yy++zz) ) xx=4, =4, yy=3, =3, zz=1=1

KINEMATIC CRITERIA FOR STRUCTURAL SYNTHESIS OF MAXIMALLY REGULAR PM3KINEMATIC CRITERIA FOR STRUCTURAL KINEMATIC CRITERIA FOR STRUCTURAL SYNTHESIS OF MAXIMALLY REGULAR PMSYNTHESIS OF MAXIMALLY REGULAR PM33


5252

Isoglidex – TyRz (x=y+z)IsoglideIsoglidexx –– TTyyRRzz ((xx==yy++zz))

Comparaison Interféromètre - Vision

-100

-50

0

50

100

150

200

250

300

350

400

0 100 200 300 400 500 600 700 800 900

position axe X ( en mm )

( en

µm )

Verticale Interféromètre Horizontale Interféromètre

Horizontale Caméra Verticale Caméra

PhD Thesis: T. Cano; R. Rizk



5353


PhD Thesis: N. Rat



5454

Isoglidex – TyRz (x=y+z) x=4, y=3, z=1IsoglideIsoglidexx –– TTyyRRzz ((xx==yy++zz) ) xx=4, =4, yy=3, =3, zz=1=1PhD Thesis: R. Rizk



5555


PhD Thesis: R. Rizk



56

Concluding remarks Concluding remarks Concluding remarks

Acknowledgement

This work was sustained by CNRS (The French National Council of Scientific Research) in the frame of the projects ROBEA-MAX 2002-2003 and ROBEA-MP2 2003-2006.

We demonstrated and proposed for the first time novel formulae bridging the mobility, leg and mobile platform connectivity and the number of overconstraints in parallel robotic manipulators.

We integrated these new formulae in an original structural synthesis method for parallel robots founded on evolutionary morphology.



57

“Many solutions for parallel robots obtained throughthis systematic approach of structural synthesis are presented here for the first time in the literature. The author had to make a difficult and challenging choice between protecting these solutions throughpatents, and releasing them directly into the public domain. The second option was adopted by publishingthem in various recent scientific publications and mainly in this book. In this way, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products.”

Concluding remarks Concluding remarks Concluding remarks


Gogu, G. Structural Synthesis of Parallel Robots, Part 1: Methodology, Springer, 2008, ISBN 978-14020-5102-9, 714 pages




Grigore GOGU, Professor, Dr.Mechanical Engineering Research GroupHead of the Research Group Machines, Mechanisms and Systems French Institute of Advanced Mechanics and Blaise Pascal University Tel. + 33 4 73 28 80 22, Fax 33 4 73 28 81 00; e-mail: [email protected]://www.ifma.fr

59

.

www.strategic-innovation.dk

INNOVATIVEOR

CLASICAL PARALLEL ROBOTS- that’s the question !

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structural and kinematic analysis and synthesis of parallel robots

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