Contribution to generic modeling and vision-based control of a broad class of fully parallel robots

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Contribution to generic modeling and vision-based

control of a broad class of fully parallel robots

Tej Dallej, Nicolas Andreff, Philippe Martinet

To cite this version:

Tej Dallej, Nicolas Andreff, Philippe Martinet. Contribution to generic modeling and vision-based

control of a broad class of fully parallel robots. Robotica, Cambridge University Press, 2018, 36 (12),

pp.1874-1896. �10.1017/S0263574718000784�. �hal-01966655�

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Tej Dallej

⊳

,Ni olas Andre

⊳⊲

and Philippe Martinet

⊳

Institut Pas al, UBP/CNRS/SIGMA,Clermont-Ferrand,Fran e.

⊲

Institut FEMTO-ST, Univ.Fran he-Comté/CNRS/ENSMM/UTBM, Besançon,

Fran e.

IRCCYN,CNRS/LS2N,Nantes Cedex, Fran e.

)

SUMMARY

This paperdeals witha generi modeling and vision-based ontrol approa h for abroad lass of parallel me hanisms. First, a generi ar hite ture representing several families is proposed. Se ond, inspired by the geometry of lines, a generi dierential inverse kinemati modela ordingtotheproposedgeneri stru tureisintrodu ed.Finally,based on the image proje tion of ylindri al legs, a kinemati vision-based ontrol using legs observationispresented.Theapproa hisillustratedandvalidatedontheGough-Stewart and Par4parallel robots.

KEYWORDS: VisualServoing; Control;Modeling;ParallelRobots; Kinemati s.

1. Introdu tion

The oupling between visual servoing te hniques 1 3

and parallel robots is be oming in reasinglyimportantandhasbeenthesubje tofseveralstudies. Essentially,kinemati vision-based ontrol generatesa Cartesiandesired velo ity,whi h is onverted into joint velo ities by the dierential inversekinemati model.The latter isusually an analyti al modelinthe aseofparallelrobots.Thedierentialinversekinemati modelalsodepends on theCartesianpose,

4,5

whi h needsto beestimated.

Authors inrefs. [6,7℄ have been among the rst to introdu e vision-based ontrol for parallelrobots.Theinterestingpointoftheaforementionedapproa hesistheuseofvision in the feedba k ontrol for regulation.

8

However, the used parallel robots are designed with parti ular me hani al stru tures to have an analyti al forward kinemati model. Consequently,the dierential inverse kinemati model turnsout to bedependent onthe end-ee tor poseestimated usingtheforward kinemati modelandjoint values.

Ina tuality,theformulationoftheparallelrobotforwardkinemati model 9

isgenerally di ulttosolveandgivesseveralpossibleposesfortheend-ee tor.

10

Tomakethe ontrol robust with respe t to modeling errors, vision repla es the forward kinemati model in the feedba k with a amera measurement. This method an also be used to simplify dierent modelsof the ontrol s heme.

8

Amongstthevariousvisual servoingte hniques, 3D pose visual servoing

11,12

an be applied to parallel robots. In ref. [13℄,a generi 3D pose visual servoing of six degrees of freedom (DOF) Gough-Stewart platform (Fig. 1) wasproposed. In this kinemati ontrol,the end-ee tor pose, indire tly measured, was used inthefeedba k ontrol andinthedierential inverse kinemati model.

ImageBasedVisualServoing(IBVS)wasalsoappliedtoparallelrobots 14,15

using end-ee tor observation.This method doesnot require themodel ofthevisual target or the end-ee tor pose. However, the method has di ulties with large rotational motion.

16 The main limitation of visualservoing ofparallel robots fo used on theobservationof a visual target isthatitrequirestheestimation ofthe end-ee tor to tooltransformation, the worldtobasetransformation andtheentire kinemati parameter set.Moreover,itis

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Fig. 1.Fromleft to right: theGough-Stewartplatform(Institut Pas al), theH4 andthe Par4robots (LIRMM).

Fig.2.Fromlefttoright:theOrthoglide(IRCCyN),theIsoglide-4T3R1(IFMA)andtheI4L(LIRMM) robots.

notwiseto onsiderobservingtheend-ee torofama hiningtool.Itmaybein ompatible with various appli ations.

Itisnoti eablethatthestateofaparallelrobotisanyrepresentationoftheend-ee tor pose.A newwayto usevisionwhi hgathers theadvantagesof redundant metrology

17,18 and visual servoingwas presentedinref. [19℄. Thismethod was proposedto ontrol the well-knownsixdegreesoffreedom(DOF)Gough-Stewartplatformusinglegsobservation. This proposedapproa h hasaredu ed setofkinemati parameters anddoesnot require anyvisualtarget.Theorientation oftherobot'slegswas hosenasvisualprimitives.The ontrol law was based on their re onstru tion from the image whi h might not be very a urate for intrinsi reasons. To improve the pra ti al robustness by servoing the legs in the image, it was proposed in ref. [20℄ to servo leg edge rather than leg orientation. The proposed method was extended to Par4

21

and I4L 22

robots (Fig. 1 and Fig. 2). In ref. [23℄,theauthors suggestedadynami ontrol vialegedges ofaparallel robot taking into a ount allrelevant aspe tsof legedges observation.

Thenotionthatthelegs an ontainthestateofaparallelrobotleadsthewaytowards an innovative approa h in whi h modeling, alibration

24

and ontrol are interla ed. In addition, it an permit a better representation of a parallel robot. Referen es [25,26℄, show that observing the dire tion of the parallel robot's legs involves ontrolling the displa ement of a hidden robot. The latter has up to eight assembly modes that are dierentfromthoseoftherealliverobot.First,themethodshowninref.[19℄wasusedfor ontrol.Afterwards,basedonthekinemati modelingofthehiddenrobot,thesingularity problemofthe ontrollerwasstudiedand ontrollabilityinformationoftheparallelrobot wasgiven.

In our opinion, a generi solution for ontrolling any parallel robot should take into a ountthe spe i kinemati propertiesofparallel robotsandtheobservablekinemati elements.Thissolutiondoesnotrequireanyadditionalvisualtarget.Theterminologyofa kinemati elementwasrstintrodu edinref.[27℄inordertondgeometri andkinemati properties whi h an be applied to many parallel robots. The indi ated representative kinemati hain ismade up of at leastone kinemati element whi h is omposed of two rigid bodieslinked byaPrismati joint (xedor moving joint).

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Thenotionofakinemati elementisthenfullyformalizedinref.[28℄andaninteresting mathemati al model of a kinemati element was given. The proposed method gives a linear dynami modeling of parallel robot from observable kinemati elements. It relies onabody-orientedrepresentationofobservablere tilinearkinemati stru tures.However, the modelingand ontrolofparallelrobotsonakinemati levelwerenot learlyandfully studied. In addition, the proposed modeling method in ref. [28℄ was illustrated on a simple 2 degrees of freedom (DOF) ve-bar me hanism whi h is an RRR-RR stru ture planar parallel robot. Some dis ussion about the extension to a broad lass of parallel me hanisms wouldhave been more interesting, omplete and generi .

Condent that one an fuse kinemati s and proje tive geometry into a proje tive kinemati modelfor ontrolandinspiredbythegeometryoflinesandtheimageproje tion of ylindri al legs,this paperhasthreemain goals:

r

toproposeageneri stru tureofthekinemati hainsandthena elle 2931

(anarti ulated setofelementshavingxedlength and ontaining the movingplatform).Itisageneral way to nd generi parallel robot stru ture orresponding to a broad lass of fully parallel robots,e.g. theGough-StewartPlatform, the H4,

29 theI4R, 30 thePar4 31 (Fig. 1) and the I4L

32

(Fig. 2).

r

to provide a thorough study of modeling and kinemati vision-based ontrol derived a ording to a generi methodology takinginto a ount the proposed generi parallel robot stru ture. The edges of the last elements (referent elements) atta hed to the na elle are used in the servo loop and in the kinemati visual servoing law. We will onrm that, measuring by vision the referent elements edges, one an easily measure a proje tive kinemati model for ontrol without proprio eptive sensors, whi h might simplify therobot design.

r

to present a oherent representation of a broad lass of fully parallel robots, in whi h one an fuse a generi ar hite ture, the kinemati modeling and referent elements observation.

The paper is organised as follows: Se tion 2 presents a omparative study of some existing parallel robots. The aim is to nd similar points and to introdu e a generi ar hite ture. Based on some assumptions and the stru tures of dierent parallel robot families, se tion 3presentsa generi ar hite ture ofa broad lassoffully parallelrobots with one a tuated joint. It is omposed of the kinemati hains ar hite tures and the na ellein ludingthemovingplatform.Se tion 4introdu esthe on eptoftheobservable referentkinemati elements.Theproje tionof ylindri alelementsintheimagegivesthe visualprimitivesusedfortheedge-based ontrol.Se tion5explainsthegeneri dierential inversekinemati modela ordingtotheproposedgeneri stru ture.Akinemati vision-based ontrol using referent elements observation is thenpresentedin se tion6. Se tion 7 presentsexperimental validationson theGough-Stewart platform andthePar4robot. A dis ussionand on lusionsarepresentedinse tions 8and 9,respe tively.

2. Stru ture of some parallel robotfamilies

As we plan to propose a generi ar hite ture whi h unies several families of parallel robots, spe i kinemati properties of a broad lass of parallel robots must be found. The generi unied method must be able to be used for more than one type of parallel robot to prove the validityof theproposed approa hes. Consequently, this se tion deals with a omparative studyof some existingparallel robots. The mainobje tive isto nd a ommon representation and to introdu e a generi ar hite ture for a broad lass of parallelrobots.This lassi ationisbasedonthepresen eorabsen eofaPrismati joint inea h kinemati hain.

24,33

Throughout the paper,thenotationsgiven inTable Iwill be used.

2.1. First family

The rst family in ludes robots whose a tuated Prismati joint is lo ated between two rigid bodies of the kinemati hain. For example, the Gough-Stewart platform (Fig. 3)

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is a sixdegrees of freedom parallel me hanism. It has 6 kinemati hains

[A

1i

A

2i

]

with varyinglengths L

(r

1i

) = r

1i

, i ∈ 1..6

dueto the a tuatedPrismati jointslo atedat

P

1i

. The kinemati hainsofa standardGough-Stewart platformareatta hed tothebaseby Universaljoints(lo atedatpoints

A

1i

)andtothemovingplatformbyBalljointslo ated at points

A

2i

.The moving platform of theGough-Stewart robot is made up of a single element

S

atta hed to ea hkinemati hain at point

A

2i

.

A

1i

A

_2i

P

1i

r

1i

.

E

S

Fig.3.ArealGough-StewartplatformanditsrepresentativeSket h.

TableI.Notations.

r

Boldfa e hara tersdenoteve torsormatri es.Unitve torsareunderlined.

r

i

_T

j

=

_i

R

j

i

t

j

0

1

isthehomogeneousmatrixasso iatedtotherigidtransformationfromframe

F

_i

toframe

F

j

.

r

i

_v

isve tor

v

expressed inframe

F

i

.

r

_K

isthematrixofthe amera'sintrinsi parameters.

r

_q

0i

denesthersta tuatedjointofkinemati hain

i

.

r

_q

denestheve torofalla tuatedjoints.

r

j

τ

_i

=

j

V

_i

j

Ω

_i

T

istheCartesianvelo ity(linearandangularvelo ities)of

F

i

expressedinand withrespe tto

F

j

.It anbeminimallyrepresentedby

i

_V

j

whenthe omponentsoftheCartesian velo ityarelessthan6.

r

_M

+

isthepseudo-inverseof

M

.

r

c

m

= 0..1

denotes the kinemati elements whi h formthekinemati hainand

j

= 1..2

denotestheedgesofea hobservable ylindri al kinemati element.

r

_F

b

= (O, x

b

, y

_b

, z

b

)

,

F

e

= (E, x

e

, y

_e

, z

e

)

,

F

c

= (O

c

, x

c

, y

_c

, z

c

)

and

F

pmi

=

(P

mi

, x

pmi

, y

pmi

, z

pmi

)

denote the base, end-ee tor, amera and

i

th

kinemati element referen eframes,respe tively.

(α

ki

, β

ki

)

forUniversaljointand

(α

ki

, β

ki

, γ

ki

)

forBalljoint,with

k

= 0..2

.

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2.2. Se ondfamily

These ondfamilyin ludesrobotswhi hhave oneor two kinemati elementswithaxed length andana tuated Prismati jointbetween theseelementsandthebase.It in ludes, for instan e, the Orthoglide robot,

34

theIsoglide-4 T3R1 35

andtheI4L 32

(Fig. 2).

2.2.1. The Orthoglide robot. The Orthoglide is a 3-DOF translational parallel manipulator.It onsistsofthreeidenti alkinemati hainsatta hedtothebaseat points

P

0i

bythree a tuated andorthogonal Prismati joints.

Upon analyzing in more detail the ar hite ture of the Orthoglide (Fig. 4), one an onsider a rst kinemati element

[A

0i

A

1 1i

]

with variable length

l(r

0i

)

,

i ∈ 1..3

. It is onne ted to the arti ulated parallelogram omposed of two elements

[A

1 1i

A

1 2i

]

and

[A

2 1i

A

2 2i

]

witha xedlength L

(r

1i

) =

L (Fig.4).

Fig. 4.Sket hoftheOrthoglide.

or

A

2 1i

and

A

1 2i

or

A

2 2i

,respe tively.

Therotationofmovingplatform

S

isduetotherelativedispla ementofthetwona elle parts

S

01 = S

02

and

S

03 = S

04

(Fig. 5), using two ra k-and-pinion systems. The relative translation

T

is transformed into a proportional end-ee tor rotation

θ = T/K

, with

K

as the transmission ratio. It should be observed that ea h kinemati hain

i

is atta hed to a na elleelement

S

0i

whi h is dire tlyatta hed to moving platform

S

at

D

0

or

D

1

. 2.3. Third family

Inthethirdfamily,thereisnoa tuatedPrismati jointinthekinemati hains.Itin ludes theH4,theI4RandthePar4(Fig.1).Alltheserobotshaveonlya tuatedRevolutejoints. H4,I4RandPar4robotsarebasedonfouridenti alkinemati hains(Fig.6,Fig.7,Fig.8 andFig.9).Ea hrevolutemotorlo atedat

A

0i

movesarm

i

withaxedlength

l(r

0i

) = l

,

i ∈ 1..4

. Ea h arm is onne ted to a forearm(parallelogram) madeup of two stru tures equipped with ball joints

(A

1 1i

, A

1 2i

)

and

(A

2 1i

, A

2 2i

)

. The axis passing through the two upperends(orthelowerextremities)ofthisparallelogram keepsthesamedire tion.The state oftheforearm an be dened usingpoints

A

1 1i

or

A

2 1i

and

A

1 2i

or

A

2 2i

,respe tively.

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L

.

A

21 A

22 A

23 A

24 .

.

z

b

.

r

01 r

₀₂

r

₀₃

..

.

A

01 .

.

A

11 P

₀₂

.

P

03 A

₀₂

A

04 .

P

04 .

P

₀₁

.

A

12 .

.

A

13 _A

14 .

.

r

₀₄

S

01 ; S

02 .

.

A

23 A

24

2 A

23

1 A

23

2 A

₂₄

1 A

₂₄

.

A

22 A

21

2 A

₂₁

1 A

₂₁

1 A

₂₂

2 A

22 D

₀

.

S

03 ; S

04 D

1 .

Guidance system

.

E

x

e

y

e

Fig.5.TopviewoftheI4Lrobot(left)anditsna elle(right).

Ea h forearmis onne ted atea h endto the arti ulated na elle. Theend-ee tor frame lo atedat

E

an be translated inthreedire tionsand rotated arounda xedaxis

c

_z

e

.

A

_0i

P

1i

L

P

_0i

l

A

_2i

2 A

_1i

2 _A

1i

1 A

_2i

1

Fig.6.H4,I4RandPar4kinemati hainparameters.

2.3.1. The na elle of the H4 robot. The na elle of the H4 (Fig. 7) is omposed of two lateralpartsanda entral rod

S = [D

24 D

21 ]

.The enterofthisrodistheend-ee tor

E

whi h has4 DOF(3 translations and1 rotation).

O

x

b

y

b

x

p02

A

02 x

p01

x

p04

x

p03

A

01 A

04 A

03 z

p03

z

p01

z

_p04

z

p02

Fig.7.TopviewoftheH4robot(left)anditsna elle(right).

2.3.2. The na elleof the I4R robot.Thena elleof I4Rrobotismade upof3arti ulated elements.Therelativedispla ementofthetwoplateparts

S

01 = S

02

and

S

03 = S

04

(Fig.8)

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indu es therotationof theend-ee tor lo ated at the enter of

S = [D

0 E]

.The relative translation

T

is transformed into aproportional end-ee tor rotation

θ = T/K

.

)

z

b

x

b

y

b

O

α

A

01 x

p01

z

p01

.

A

02 A

03 A

04 x

p02

x

p03

x

p04

z

p02

z

p03

z

p04

.

γ

.

T

.

x

_e

y

_e

)

θ

x

_b

y

_b

h

0 A

₂₂

A

23 A

24 A

₂₁

2 A

21

1 A

21

1 A

22

2 A

22

2 A

23

1 A

23

2 A

24

1 A

24 D

0 .

S

E

S

01 ; S

02 S

₀₃

; S

₀₄

Fig. 8.TopviewoftheI4Rrobot(left)anditsna elle (right).

2.3.3. The na elle of the Par4 robot. As shown in ref. [31℄, the Par4 na elle (Fig. 9) is omposed of four parts: two parts dened by

[D

21 D

22 ]

and

[D

23 D

24 ]

and linked by two rods

[D

21 D

24 ]

and

[D

22 D

23 ]

withrevolute joints.

Thearti ulatedna elleisequippedwithanampli ationsystem(Fig.9)to transform the relative rotation

θ = ±

π

4

of theend-ee tor at

E

= D

24

into aproportional rotation (

β = −κθ

,

κ = 3

)of a newend-ee tor at

E

1

.However, this ampli ation systemis not very signi ant from a kinemati point of view. Consequently, one an hoose

D

24 = E

asthe end-ee tor position linked totheelement

[D

21 D

24 ]

.

)

z

b

x

b

y

b

O

α

A

01 x

p01

z

p01

.

A

02 A

03 A

04 x

p02

x

p03

x

p04

z

p02

z

p03

z

p04

.

E

1 D

21 D

22 D

23 A

21 )

d

h

A

23 A

22 A

21 A

24 .

.

x

_e

y

_e

D

24 (E)

1 A

21

2 A

22

1 A

22

2 A

23

1 A

23

2 A

24

1 A

24

2 )

X

e

Y

e

Fig. 9.TopviewofthePar4robot(left)anditsna elle (right).

3. Contribution to a generi ar hite ture of a broad lass of fullyparallel robots

To join together a broad lass of fully parallel me hanisms in the same generi ar hite ture,a ontribution to a generi stru ture of thekinemati hains and ageneri representation of an arti ulated na elle will be dis ussed in this se tion. Only the manipulators havingthe following hara teristi s willbe onsidered:

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r

Fully parallel robot: the number of ontrolled degrees of freedom of the end-ee tor is stri tly equal to the number of kinemati hains and just one a tuator exists in ea h kinemati hain.

5,36

r

There are onlytwo elementsin ea h kinemati hain. Ea helement is omposed of two rigid bodies linked by a Prismati joint (xed or a tuated moving joint).

The majority of existing parallel me hanisms are designed with a redu ed number of kinemati hain elements ompared to serial stru tures. Sin e ea h kinemati hain is usually short and based on the omparative study of some existing parallel robots showninSe tion2,wewillrestri ttheproposedgeneri ar hite tureto fourserialrigid bodies (twoserial kinemati elementsasdened inTableI)inea hkinemati hain. It is su efrom our pointof viewto des ribea broad lassoffully parallel robots.

r

The joint between two kinemati elements of the kinemati hain must be a Ball, Revolute or Universal joint.

[P

1i

A

2i

]

.

r

5 groups ofjoints showninTableII: two joints(Prismati or Fixed joint)at

P

mi

,two joints (Ball, Universal or Revolute joint) at

A

1i

and

A

2i

and one joint (Revolute or Fixed joint)at

provides the variation of the length L

(r

1i

)

.

Based inthe hara teristi s ofseveral familiesofparallel robots, following onstraints must be satised:

r

The a tuator of ea h kinemati hain is lo ated at

A

0i

or

P

0i

or

P

1i

. The two other passive jointsare xed.

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Fig. 10.Ageneri kinemati hainar hite tureofabroad lassofparallelrobots.

Thisis due to the fa tthat thea tuated joint mounted at or nearthe xedbase (and not themoving platform) ismore interesting for two reasons. First, this onguration redu es the moving platform mass asso iated with the a tuation. Se ond, it is also a more onvenient way to design a parallel robot with low inertia, high load and high speed.

r

Forea hkinemati hain, thea tuated jointat

A

0i

)

l(r

0i

)

or onstant onstant

L(r

1i

)

L(r

1i

)

or onstant

6=

0

L(r

l(r

0i

)

onstant

l(r

0i

)

L(r

1i

)

onstant onstant

Therefore, one an write thea tuated joint

q

0i

as:

q

0i

= λ

i

α

0i

+ λ

i

(µ

i

r

0i

+ µ

i

r

1i

)

(1) where

r

µ

i

= 1 − µ

i

and

λ

i

= 1 − λ

i

.

r

µ

i

= 1

ifjointat

P

0i

is a tuated .

r

µ

i

= 0

ifjointat

P

1i

is a tuated .

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r

λ

i

= 1

ifjoint at

A

0i

isa tuated .

r

λ

i

= 0

ifjoint at

P

0i

or at

P

1i

is a tuated.

In the ase of an una tuated joint at

P

0i

and

P

1i

,

µ

i

an take the value0 or 1.This ongurationis denotedby

N

.It denotes anindeterminate value.

3.2. Contribution to a generi representation of a na elle

Themajorityofexistingparallelrobotsaredesigned withana elle omposedofonerigid body,whi h isa ompa t solid ontainingtheend-ee tor

E

andthemoving platform

S

(Fig.11).It islinked toea h kinemati hainat

A

2i

(e.g.the aseoftheGough-Stewart and the Orthoglide robot). The na elle an be also omposed of several elements

29

S

0i

and

S

(Fig.11).Severalresear hstudieshaveproventhat ombiningaparallelrobotand anarti ulatedna elle(e.g.H4,

29 I4L,

32

theI4R 30

andthePar4 31

)isagoodalternativefor high-speedpi kandpla eoperationsandobtainingunlimitedrotationofthetoolaround one xed axis.

In order to propose generi representation of a fully parallel robot na elle, let us onsider the following lassi ations:

r

First family:

This family in ludes na elles omposed of one rigid body (e.g. Gough-Stewart, Orthoglide 34 , Agile Eye, 37 the Hexapteron 38

). This family also ontains the na elle of the fully-isotropi parallel robot.

35

r

Se ondfamily:

In this family one an nd arti ulated na elles of parallel robots with S hoenies motions

39,40

(also alled3T1Rparallel robotsorSCARAmotions).The usedna elleis made up of at least 2 arti ulated bodies (e.g. H4,

29 I4L,

32

the I4R 30

and thePar4 31

). In general ases, arti ulated elementsof the na elleare restri ted to be oplanar. The na elle has only translational motion. However, the relative motion of thearti ulated elements of the na elle turns the moving platform around an axis of xed dire tion, using anampli ation system.

E

A

_2i

S

0i

D

0i

S

A

_1i

A

_0i

Joints

Articulated Nacelle

E

A

2i

D

_0i

S

A

_1i

A

0i

=

Fig. 11.Generi representationofthe arti ulatedna elle (left) andthena elle omposedof onerigid body(right).

The proposedstudy islimitedto ana elle madeupof nbodies

S

0i

(i

=

1..n) and one entral rigid body

S

.For ea h kinemati hain

i

,the rst element

S

0i

ontains

A

2i

and these ond one

S

ontains theend-ee tor

E

.TableIIIpresents na ellesofa broad lass of parallel robotsa ordingto thegeneri representation inFig.11.

Asshown inFig.11,one an write:

c

_A

2i

=

c

E

+

c

−−−→

ED

0i

+

c

−−−−→

D

0i

A

2i

(2)

(12)

26 A

26 ]

[A

26 E]

A

26

The H4 or the Par4 robot

24 ]

[A

22 D

0 ]

[D

0 E]

D

0

3

[A

23 E]

[ED

0 ]

E

4

[A

24 E]

[ED

0 ]

E

The I4L robot

0i

and

S

entered at

D

S

( aseofgears,pulleyand belt, hainandgears orside byside movement transmissions).

Takingintoa ountthe hara teristi softhena elle,one anwriteinthexed amera frame

F

c

:







c

_V

D

0 i

∈S

=

c

_V

D

0 i

∈S

0 i

c

_Ω

0i

= ̺

c

Ω

e

(3)

where

̺

is thetransmissionratio between thetworotational velo ities of

S

and

S

0i

. It should be notedthat:

r

̺ = 1

ifthena elleis representing asingleand ompa t solid(

c

_Ω

0i

=

c

Ω

e

).

r

̺ = 0

ifthere is no rotational movement of

S

0i

withrespe t to xed amera frame

F

c

(

c

_Ω

0i

= 0

3X1

).

Applying thevelo ity omposition lawexpressedin

F

c

,one an formulate:

(

_dA

2 i

dt

=

c

_V

A

2 i

=

c

_V

D

0 i

∈S

0 i

+

c

_Ω

0i

×

c

−−−−→

D

0i

A

2i

dD

0 i

dt

=

c

_V

D

0 i

∈S

=

c

_V

e

+

c

Ω

e

×

c

−−−→

ED

0i

(4)

(13)

From(3)and (4), one an ompute:

c

_V

A

2 i

=

c

_V

e

+

c

Ω

e

×

c

−−−→

ED

0i

+

c

Ω

0i

×

c

−−−−→

D

0i

A

2i

(5)

Consequently, using(3)and (5), one an write thefollowing generi relation:

c

_V

A

2 i

=

I

3 −[

c

−−−→

ED

0i

+ ̺

c

−−−−→

D

0i

A

2i

]

×

c

_τ

e

= G

1 2i

c

τ

e

(6) where:

G

1 _2i

=

I

3 −[

c

−−−→

ED

0i

+ ̺

c

−−−−→

D

0i

A

2i

]

×

(7)

Two signi ant ases an be dedu ed:

r

The kinemati elements of na elle

S

and

S

0i

arerigidly xedto a single and ompa t solid (the ase of Gough-Stewart platform and the Orthoglide robot), whi h is the moving platform (

̺ = 1

). One an write:

c

_V

A

2 i

=

I

3 −[

c

−−−→

EA

2i

]

×

c

_τ

e

(8)

r

There isno rotational movement of

S

0i

withrespe tto xed amera frame

F

c

(

̺ = 0

). However,

S

has a rotational movement withrespe t to xed amera frame

F

c

(in the ase ofthe na elle ofH4,Par4, I4Rand I4Lrobots).Consequently,one an write:

c

_V

A

2 i

=

I

3 −[

c

−−−→

ED

0i

]

×

c

_τ

e

(9)

4. Vision-based kinemati using referent edges observation

Themajorityof existingparallel me hanisms aredesigned withslimand ylindri al legs between their base and their moving platform. Thus, one an onsider these legs as straightlinesfor kinemati analysis.

5

whi h istheorthogonal distan eof

L

to theorigin.

4.2. Proje tion of the ylindri al referent element in the image

The losest element

i

to the na elle is hosen as the referent element of ea h kinemati hain. By observing ea hreferent ylindri al element, one an extra t theedges

n

j

i

with

j = 1, 2

asso iated to the lines' proje tions in the image plane (Fig. 12). The line's image proje tion ouldberepresentedbythenormalve tor

n

j

i

(

j = 1, 2

)to theso- alled interpretation planeasso iated to ea hedge

j

.

Su h a proje ted line in the image plane, expressed inthe amera frame

F

c

, has the following equation:

c

_n

j

i

T c

p

j

i

= 0

(10)

(14)

where

c

_p

i

representsthe oordinatesinthe amera frameofanypointintheimageplane, lying onthe edge onsidered asastraight line.

Fig. 12.Proje tionofa ylinderintheimage(Left)andaproje tionofa3Dlinerepresentation.

Withtheintrinsi ameramatrix

K

,(10) analsobeexpressedintheimage frameas:

im

_n

j

i

T im

p

j

i

=

im

n

j

i

T

K

c

p







im

_n

j

i

=

K

−T c

n

j

i

kK

−T c

n

j

i

k

c

_n

j

i

=

K

T im

n

j

i

kK

T im

n

j

i

k

(12)

Consequently,one an dedu ethe dire tion

c

_u

i

of ea h ylinder 42 from(12):

c

_u

i

=

c

_n

1 i

×

c

n

2 i

k

c

_n

1 i

×

c

n

2 i

k

(13)

Moreover, onsider point

P

i

lying on the ylinder axis

i

(Fig. 12), the edge

j

of ea h ylinder is dened bythe following onstraints, expressedin the amera frame:











c

_n

j

i

T

_c

u

_i

= 0

c

_n

j

i

T

_c

n

j

_i

= 1

c

_n

j

i

T

_c

P

i

=

c

n

j

i

T

(

c

_p

j

i

+

c

−−−

→

p

j

_i

P

i

) =

c

n

j

i

T

(

c

_p

j

i

+ λ

j

R

c

n

j

i

) = λ

j

R

(14)

where

λ

1 = λ

2 = ±1

and

R

isthe radius of ylinder

i

.

Thelast onstraintmeansthattheedgesarelo atedatadistan e

R

fromthe ylinder's axis (Fig.12).

4.3. The observation of the referent element andthe state of a parallel robot

Two edges an be extra ted using ylindri al referent element observation. These edges are dire tlyusedinthe ontrol loopand ina dierential kinemati model.

Thisreferen eelementis hara terized bythereferen edire tion

u

i

andalengthL

(r

1i

)

whi h anbeeithervariableL

(r

1i

)

or onstantL

(r

1i

) =

L,butneverzero.

A

1i

and

A

2i

are thelowerandupperextremitiesofthiselement.Thelowerextremity anbedenedusing all joint positions of the kinemati hain

A

1i

(α

0i

, r

0i

, ξ

geom

)

, where

(

c

_n

j

i

T

_c

A

2i

(X, ξ

geom

) = λ

i

R

c

_A

2i

=

c

A

21 (X, ξ

geom

) +

c

−−−−→

A

21 A

2i

(X, ξ

geom

)

(15)

The end-ee tor

X

an be represented by the axis-angle representation (

c

_u

e

,

A unique solution of the following system an be omputed using onstraints in(15) applied toat least4 kinemati hains:











c

_n

1

1 T c

A

21 (X, ξ

geom

) = λ

1 R

c

_n

2

1 T c

A

21 (X, ξ

geom

) = λ

1 R

c

_n

1

2 T

(

c

_A

21 (X, ξ

geom

) +

c

−−−−−→

A

21 A

22 (X, ξ

geom

)) = λ

2 R

c

_n

2

2 T

(

c

_A

21 (X, ξ

geom

) +

c

−−−−−→

A

21 A

22 (X, ξ

geom

)) = λ

2 R

c

_n

1

3 T

(

c

_A

21 (X, ξ

geom

) +

c

−−−−−→

A

21 A

23 (X, ξ

geom

)) = λ

3 R

c

_n

2

3 T

(

c

_A

21 (X, ξ

geom

) +

c

−−−−−→

A

21 A

23 (X, ξ

geom

)) = λ

3 R

c

_n

1

4 T

(

c

_A

21 (X, ξ

geom

) +

c

−−−−−→

A

21 A

24 (X, ξ

geom

)) = λ

4 R

c

_n

2

4 T

(

c

_A

21 (X, ξ

geom

) +

c

−−−−−→

A

21 A

24 (X, ξ

geom

)) = λ

4 R

(16)

5. Generi dierential inverse kinemati model

A ording to the generi ar hite ture presented in se tion 3, the main obje tive of this se tionistofusekinemati s andproje tive geometryintoageneri proje tivedierential kinemati modelfor ontrol,whi h an havethe following form:





˙q

˙u

˙n





=





D

inv

e

M

inv

e

L

inv

e





τ

(17) where

D

inv

e

,

M

inv

e

and

L

inv

e

arethejointkinemati matrix,theCartesiankinemati matrix asso iated to thedire tionsand the Cartesiankinemati matrix asso iated to theedges, respe tively.

The kinemati hains' losurearound the referen e element

[A

1i

A

2i

]

(Fig. 10) yields, forea h kinemati hain

i

,thefollowing kinemati modelingeneri ve tor form:

L

(r

1i

)u

i

=

−−−−→

A

1i

A

2i

= A

2i

(X, ξ

geom

) − A

1i

(α

0i

, r

0i

, ξ

geom

)

(18)

where

X

isa representation of theend-ee tor pose.

Assuming that the kinemati parameters

ξ

geom

are onstants, time dierentiating of thekinemati model (18)gives:

˙

L

(r

1i

)u

i

+

L

(r

1i

) ˙u

i

=

(19)

d

dt

(A

2i

(X, ξ

geom

)) −

d

dt

(A

1i

(α

0i

, r

0i

, ξ

geom

))

(16)

Time dierentiating ofL

(r

1i

)

,

A

2i

and

A

1i

provides:











d

dt

(

L

(r

1i

)) =

∂

L

(r

1 i

)

∂r

1 i

˙r

1i

d

dt

(A

2i

) =

∂A

2 i

∂X

L

X

τ

d

dt

(A

1i

) =

∂A

1 i

∂α

0 i

˙α

0i

+

∂A

1 i

∂r

0 i

˙r

0i

(20)

where

L

X

isthematrixrelatingtimedierentiatingoftheCartesianposetotheCartesian velo ity.

4

One an rewrite (19)as:

L

(r

1i

) ˙u

i

= G

1 2i

τ

+ G

0 1i

˙α

0i

+ G

1i

1 ˙r

0i

+ G

2 1i

˙r

1i

(21) where

r

G

1 2i

=

∂A

2 i

∂X

L

X

istheintera tionmatrixasso iated tothe3Dpoint

A

2i

expressedin(7) (see Se tion.3.2for more details).

r

G

0 1i

= −

∂A

1 i

∂α

0 i

,

G

1 1i

= −

∂A

1 i

∂r

0 i

and

G

2 1i

= −

∂

L

(r

1 i

)

∂r

1 i

u

i

arejoint kinemati matri es.

5.1. Dierential inverse kinemati model asso iated tothe a tuated joints

In the ase of a parallel robot with one a tuated joint

q

0i

in ea h kinemati hain, the ve tor of the joint velo ities

˙q

is obtained from the individual joint velo ity

˙q

0i

. The dierential inverse kinemati model asso iated to all a tuated joints expressed in the amera frame

F

c

an have the following formula:

˙q = D

inv

e

c

τ

e

(22)

Assuming thatonly one joint variable(

α

0i

,

r

0i

or

r

1i

)is usedin(21), one anwrite: L

(r

1i

)

c

_˙u

i

=

(23)

G

1 2i

c

τ

e

+ G

0 1i

˙α

0i

+ G

1 1i

˙r

0i

+ G

2 1i

˙r

1i

≡ G

1 2i

c

τ

e

+ G

1i

˙q

0i

where

G

1i

dependson

q

0i

(seese tion3.1and(1))andhasthefollowinggeneri equation:

G

1i

= λ

i

G

0 1i

+ λ

i

(µ

i

G

1 1i

+ µ

i

G

2 1i

)

(24)

Sin e

c

_u

T

i

c

˙u

i

= 0

,(23) gives thedierential inverse kinemati modelasso iated to

q

0i

:

˙q

0i

= −

c

_u

T

i

G

1 2i

c

_u

T

i

G

1i

c

_τ

e

= D

inv

ei

c

_τ

e

(25) where

D

inv

ei

isthe individual matrix (rowve tor) of

D

inv

e

.

5.2. Dierential inverse kinemati model asso iated todire tions

Using (23)and (25), the dierential inverse kinemati modelasso iated to dire tions

u

i

an bewritten as:

c

_˙u

i

=

1

L

(r

1i

)

(I

3 −

G

1i

c

u

T

i

c

_u

T

i

G

1i

)G

1 2i

c

τ

e

= M

i

c

τ

_e

(26) where

M

i

=

L

(r

1

1 i

)

(I

3 −

G

1 i

c

u

T

i

c

u

T

i

G

1 i

)G

1 2i

is theindividualmatrix of

M

inv

(17)

5.3. Dierential inverse kinemati model asso iated toedges

Thetimederivative of onstraintsin(14)applied topoint

A

2i

indu es arstdierential inverse kinemati model asso iated to theedges expressedin the amera frame and has the following equation:

20

c

_˙n

j

i

= (R

j

1i

G

1 2i

+ R

j

2i

M

i

)τ =

c

L

j

i

c

_τ

e

(27) where











R

j

1i

= −

(

c

u

i

×

c

n

j

i

)

c

n

j

i

T

c

A

T

2 i

(

c

u

i

×

c

n

j

i

)

R

j

2i

= −(I

3 −

(

c

_u

i

×

c

_n

j

i

)

c

_A

T

2 i

c

A

T

2 i

(

c

u

i

×

c

n

j

i

)

c

_u

i

c

n

j

i

T

c

_L

j

i

= R

j

1i

G

1 2i

+ R

j

2i

M

i

(28) where

M

i

isdened in(26) and

G

1 2i

is dened in(7).

This proves that the movement of the platform depends linearly on the variation of themovement of

A

2i

representedby

G

1 2i

and onthedierential inverse kinemati model asso iated to dire tions

M

i

.

A se ond dierential inverse kinemati model asso iated to edges expressed in the image frame an also be omputed. As a results one needs to present the intera tion matrix

=

im

J

c

L

j

i

(30)

where

im

_J

c

is asso iated to the amera-to-pixel hange of frame:

im

_˙n

j

i

=

im

_J

c

˙n

j

i

(31) A ording to ref. [20℄,

im

_J

c

an be written as:

im

_J

c

=k K

T im

n

j

i

k (I

3 −

im

n

j

i

im

n

j

i

T

)K

−T

(32)

6. Visual servoing using referent elements observation

6.1. Error to servo

Thegeneri ontrollawisbasedontheobservationofreferentelementsandtheextra tion ofthevisualprimitives

s

j

i

=

im

n

j

i

usedforregulation.We hoosetominimizetheerror

e

j

i

n

j

i

×

im

n

j

i

∗

(33)

Taking into a ount that the proposed generi ar hite ture is omposed of

n

kinemati hains, theve torof allerrors is

e

=

e

1

1 T

e

2

1 T

... e

1 n

T

e

2 n

T

.

(18)

6.2. Generi ontrol law Time derivativeof (33)gives:

˙e

j

i

=

im

_˙n

j

i

×

im

_n

j

i

∗

= −[

im

_n

j

i

∗

]

×

im

˙n

j

i

(34) Using (29), one an ompute:

˙e

j

i

= N

j

i

c

τ

e

(35) where

N

j

_i

= −[

im

_n

j

i

∗

]

×

im

L

j

i

(36)

Taking into a ount anexponential behaviourof the error(

˙e = −λ

p

e

),(35) gives:

c

_τ

e

= −λ

p

N

b

+

e

(37)

where

N

is a ompound matrix from the asso iated individual intera tion matri es

N

j

i

and

λ

p

isa onstantparameter.

Inserting (37)into (22)deliversthe nalgeneri ontrol law:

˙q = −λ

p

c

\

D

inv

e

N

b

+

_e

(38)

7. Experimental validations

7.1. Experimental ontext

The proposed approa hes are validated on the Gough-Stewart Platform and the Par4 robot. By providing an interfa e withLinux-RTAI, the kinemati ontrol of the Gough-Stewart platform and the Par4 robot is ensured using ViSP library

43

for extra ting edges, tra king andmatrix omputation.Ea hrobotisobservedbyaperspe tive amera (1024x780 pixels,IEEE1394)xedwithrespe tto thebasereferen eframe.The amera is pla edinfront ofthe robot legs sothatit an overtherobot workspa e (Fig.13 and Fig. 14).

Fig. 13. A photographof aGough-Stewartplatform(left) and a urrent onguration seenfrom the amera(thedesired ongurationisintheba kground).

7.2. Experimental validation on a Gough-Stewart Platform

7.2.1. TheGough-Stewartplatformmodela ordingtothegeneri stru turemodel.Inthe ase of the Gough-Stewart platform, thetwo extremities ofthe referent element are

A

1i

and

A

2i

(see Fig. 3 and Fig.10). Noti e that

A

1i

is a onstant parameter (

d(

c

A

1 i

)

(19)

=

c

A

2i

−

c

A

1i

(39) TableIVgivesthe hara teristi softheGough-Stewartplatformmodelsdedu edfrom the proposedgeneri model(see se tion3 andse tion 5).

TableIV.TheGough-Stewartmodels.

A tuatorsinea h kinemati hain 1

µ

i

0

λ

i

0

q

0i

r

1i

A

1i

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7.2.2. Image-based visual servoing results. Toevaluatethe ontrol approa h,the Gough-Stewart platform is asked to rea h a desired pose (Fig. 13) obtained from an initial onguration. Therefore, Fig. 15 presents the evolution of the unit-less errors in the imageand theCartesian velo ities.

Theerrors onvergeexponentiallyto0withaperfe tde ouplingfromaninitialposition tothedesiredone.The onvergen e errorsareessentiallydue tothea ura ylevelofthe edges' extra tion. The position of the vision sensor (Fig. 13) does not allow the same pre isionon all referent kinemati elements, whi h generatesmeasurement sensitivity.

7.3. Experimental validation on the Par4 robot

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Contribution to generic modeling and vision-based control of a broad class of fully parallel robots

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Contribution to generic modeling and vision-based

control of a broad class of fully parallel robots

Tej Dallej, Nicolas Andreff, Philippe Martinet

To cite this version:

Tej Dallej, Nicolas Andreff, Philippe Martinet. Contribution to generic modeling and vision-based

control of a broad class of fully parallel robots. Robotica, Cambridge University Press, 2018, 36 (12),

pp.1874-1896. �10.1017/S0263574718000784�. �hal-01966655�

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