The_6-Dof_2-Delta_parallel_robot1

R obotica (1997) volume 15,pp 407 –416.Printed in the United Kingdom ÷1997 Cambridge University Press

T he 6-Dof 2-Delta parallel robot

https://www.360docs.net/doc/557520180.html,llemand,A.Goudali and S.Zeghloul

L aboratoire de Me′c anique des Solides ,U .M .R .C .N .R .S 6610,SP 2M I -B.P .17986960F UTUROSCOPE Ce′d ex (F rance )

S UMMARY

I n this paper,we will present a new 6-DOF parallel robot u sing a set of two Delta structures.An ef f ective method i s proposed to establish explicit relationships between the e nd ef f ector co-ordinates and the active and passive joint v ariables.A simulation of the 2-Delta robot on a C.A.D. R obotics system will also be presented.This simulation w ill allow us to validate the cohesion of our calculations, a nd to show the workspace depending on the mechanical l imits on passive joints variables.Finally,an approach is p roposed to study the in?uence of small clearances of the p assive joint on the precision of the position and rotation o f the ef f ector.This approach is based on a concept s imilar to that of Yoshikawa’s manipulability.

K EYWORDS :P arallel robot ;Uncoupled ;Modelization ; W orkspace ;Ellipsoid of clearance.

1.I NTRODUCTION

P arallel robots have been the subject of several studies d ue to the considerable interest they have demonstrated t hrough their lightness,rigidity and rapidity.These p articular properties open the door to all spatial a pplications where mass problems are particularly c rucial.They are also used as robot end ef f ector.The ?rst parallel robot structure dates from 1939 when P ollard1proposed a parallel structure to paint cars.In 1949,Gough proposed an articulated machine to test t ires.Next Stewart2suggested the utilisation of this s tructure as a movement generator for ?ight simulators.

I t was also used by Reboulet,3and by Merlet4as a c ompliant wrist of a robot.Amongst spatial operators of 3d.o.f.,the Delta structure designed by Clavel5 c onstitutes technological innovation.Other 3 d.o.f. p arallel structures have been developed.Three of these a re the parallel robot H-Star developed by Herve′,6the r obot Speed R-Man developed by Reboulet7and the s tructure proposed by Jacquet.8

W e will put forward the 2-Delta robot which in p rinciple,constitutes a new and original structure.This r obot is entirely parallel and possesses 6 d.o.f.Its f eatures lie in mechanical uncoupling of translation and o rientation motions.This property is what dif f erentiates i t from other traditional 6 d.o.f parallel robots or from t he one issued from the Delta structure.This paper p artially treats the dif f erent studies that have been c arried out on the 2-Delta robot and which have been p resented in reference 9.2.P RINCIPLE OF THE 2-DELTA STRUCTURE

I n Figure 1(a) we can see the well known Delta

a rchitecture proposed by Clavel.It is a 3 d.o.f. m echanism for which the moving platform remains p arallel to the base platform.

I t is easy to see that Delta mechanism is equivalent to

a n open loop kinematic chain composed of 3 orthogonal p rismatic joints as shown in Figure 1(b).This observation w ill be used later.

S tarting from Clavel’s discovery,many authors have p roposed translation devices shown,for example,in F igure 2 and Figure 3.The H-star robot by Herve which a llows for a very large displacement in X-direction is d epicted in Figure 2.And in Figure 3 the translation d evice by Jacquet in which the authors put double u niversal joints in the place of double parallel rods is s hown.

M any robotic tasks need the end ef f ector rotates in a ddition to the 3 d.o.f.in translation.The 4 d.o.f.Delta R obot in which the gripper is connected to the moving p latform by a revolute joint and to the actuator by means o f telescopic arm is seen in Figure 4.If 6 d.o.f.are n eeded we will use the Hexa Robot proposed by P ierrot10represented in Figure 5.

I n the case presented in Figure 4,we notice that the r otation of the end ef f ector does not depend on its t ranslation,which is a good result that does not exist in t he Hexa Robot,unfortunately the end ef f ector has only o ne d.o.f.in rotation.

T he problem that we seek to resolve may be f ormulated in the following way.How to ?nd a m echanism on Delta architecture that would allow us to o btain 3 d.o.f.rotation for the end ef f ector not d epending on its translation and actuated by motors ?xed o n the base of the Delta.

O ne solution has been studied using the following idea t hat is illustrated in Figure 6(a).

H ere,the gripper is connected to moving platform by a s pherical joint R n.A n ‘‘input rod’’ is also connected to t he ?xed base by spherical joint R b.T he question is :what k ind of links joined together between gripper and ‘‘input r od’’ must we use so that the rotation of the gripper will b e independent from the translation of the moving p latform.And so,this kinematic chain must allow free t ranslations and forbid rotations between ‘‘input rod’’a nd gripper.Such a kinematic chain could be constituted, a s shown in Figure 6(b),by a set of 3 links connected t ogether by 3 orthogonal prismatic joints.As it has been m entioned previously,this kinematic chain represents t he equivalent open loop kinematic chain for the Delta m echanism shown in Figure 1(b).Therefore,as shown in

F ig.1.(a) Three d.o.f.Delta architecture (b) Equivalent open loop kinematic chain.

F igure 7,we can use a Delta mechanism whose moving p latform and base platform are rigidly connected to the e nd ef f ector and to the ‘‘input rod’’,respectively.Thus, w e obtain the 6 d.o.f.Delta decoupled parallel robot. T he external Delta controls the translation movement of t he end ef f ector,whereas the internal Delta controls its r otation movement.

A ctually,in order to obtain the rotation movement of t he orientable base,we could put a spherical 3 d.o.f. p arallel manipulator proposed by Gosselin11in the place o f the input rod.

I n Figure 8 is shown a prototype without actuators of t he 2-Delta robot.This device allows us to verify that the r otation and translation motions of the gripper are i ndependent.The device is composed of 21 bodies c onnected together by 26 spherical joints and 6 revolute j oints.Due to the great number of bodies and joints we h ad to verify that the 2-Delta robot is not an o verconstraint mechanism.The device we developed is a ctually a non-overconstraint mechanism.

3.K INEMATICS MODELING OF THE 2-DELTA S TRUCTURE

T he modelization of parallel robots is quite speci?c. T here is no existing systematic and simple model, t herefore it is necessary to ?nd the well adapted methods o f each structure.In the case of the 2-Delta robot,we

F ig.2.H-Star robot.p ropose an analytic modelization of the direct kinematic p roblem as well as of the inverse kinematic problem.

3.1.N otations and mechanical limits on joints ?a riables

I n Figure 9 we show how to de?ne joint variables.Here, B i C i i s a virtual rod which represents the double parallel r ods.θi1,θi2,θi3,?i2,?i3are the joint angles for the k inematic chain numbered i f or the external Delta,while ?i1,?i2,?i3,?i2,?i3are the joint angles for the kinematic c hain numbered i f or the internal Delta.

T he ?i j a ngles depend on θi j a ngles through very simple r elations (1),soon,we will no longer consider them.

?i3 ? ?θi3;?i2?θi2?θi1(1) I n Figure 10 we can see how θi2an θi3spherical joint v ariables are depicted.

θi2is the angle between the arm A i B i a nd the plane c alled (?) de?ned by the double parallel rods.

θi3is the rotation angle of the double parallel rods in t he (?)plane.

I t is important to focus on the fact that the spherical j oints lead to mechanical limits on θi3(?i3) given by r elations (2) :

?θi3??θi3max,i?1,2,3??i3???i3max,i?1,2,3(2) T here is no limit on θi2(?i2) but we consider them to be

F ig.3.T ranslation device with double universal joints.

F ig . 4 .F our d . o . f . Delta robot .

d if f erent from zero in order to avoid singular c on?gurations .

θ i 2 ? 0 , i ? 1 , 2 , 3 ? i 2 ? 0 , i ? 1 , 2 ,3(3)

F inally , there are also mechanical limits on θ i 1 ( ? i 1)so t hat :

θ i 1 m in ? θ i 1 ? θ i 1 m ax , i ? 1 , 2 , 3

? i 1 m in ? ? i 1 ? ? i 1 m ax , i ? 1 , 2 , 3(4)

T he end ef f ector coordinates or task coordinates are :

—the position vector of the point V r eferred to the ?xed

f rame R 0. I t is written as O V 0

? ( X ? , Y ? , Z ? ) T ,t he index 0 recalls the reference frame . —the Bryant angles ? ? ( ? 1 , ? 2 , ? 3 ) T o f the frame R ? a ttached to the end ef f ector with respect to ?xed frame .

T he control variables are θ

i 1for i e quals 1 to 3 noted as

θ 1vector θ 1 ? ( θ 1 1 , θ 2 1 , θ 3 1 ) T

. T he three other active joint v ariables are the Bryant angles of the frame attached to t he orientable base with respect to the ?xed frame .These

a ngles are written as ? ? ( ? 1 , ? 2 , ? 3 )

T .S imilar notations a re used for passive joint variables as follows :

θ 2 ? ( θ 1 2 , θ 2 2 , θ 3 2 ) T , θ 3 ? ( θ 1 3 , θ 2 3 , θ 3 3 ) T , ? 1 ? ( ? 1 1 , ? 2 1 , ? 3

1 ) T , ?

2 ? ( ? 1 2 , ? 2 2 , ?

3 2 ) T , ? 3 ? ( ? 1 3 , ? 2 3 , ? 3 3 ) T

. D ue to the property of the Delta mechanism it is

o bvious that ? e quals ?. B ecause of the mechanical limit o f the spherical joint located at D , t he angle between Z 0a nd Z ? a xis is lower than ? m ax .Thus , the active joint v ariables ? 1and ? 2

are subject to the following

F ig . 5 . T he six d . o . f . Hexa robot .F ig . 6 . P rinciple of uncoupling movement between orientation

a nd translation of the gripper

.

F ig . 7 . T he 6 d . o . f . 2-Delta robot .

F ig.8.P rototype of the 2-Delta architecture.

c onstraint :

c os?1иc os?2?c os?m ax(5) I n order to obtain relationships between geometrical

t erms previously described we must use some results

f ound by Clavel for the Delta architecture.Clavel gave c losed-form relationships connectin

g cartesian coordin-

a tes of the point D t o the control variables θ1and v ice-versa.These relationships we will use are presented i n the following way :

D irect k inematicséO D0?f(θ1)(6)

I nverse kinematicséθ1?f?1(O D0)(7)

3.2S olution to the direct kinematic problem for the

2-D elta robot

T he solution to the direct kinematic problem gives the

p osition and the orientation of the gripper as well as

p assive joint variables of the external and the internal

s tructure when control joint variables are known.Given t erms are θ1and ?,u nknowns are O V0,?,θ2,θ3,?1,?2,?3.

S teps of the calculation are as follows :

—S tep 1 .T he location of point D i s calculated by means

o f Clavel’s relation (6)—S tep 2 .C alculation of the location of point V g iven by t he following relation :

O V0?O D0?R(?)D V0?(8) w here R(?) is the matrix transformation in Bryant a ngles from frame R0?with respect to frame R0and D V0??(0,0 , DV)T.

—S tep 3 .D etermination of the gripper orientation ?.I t i s obvious that :

???(9)—S tep 4 .C alculation of passive joint variables θ2and θ3 (external Delta).

C onsidering the kinematic chain numbered i,w e write t he vector A i C i i n two dif f erent ways :

A i C0i?A i O0?O D0?D C0i(10)

A i C0i?R(Z0,?i)R(Y A i,θi1)

?[A i B i i?R(Y i,θi2)R(Zπ,θi3)B i C r i i](11) R elation (10) involves points A i,O ,D a nd C i,i n which e ach term is known.Relation (11) involves points A i,B i a nd C i a nd rotation matrices between body frames d epending on unknowns θi2and θi3.T hese relations lead t o both of the following equations

s inθi3?k1

c osθi3(k2c osθi3?k3s inθi2)?k4

w hose solution results in θi2and θi3.W e complete this c omputation three times since i e quals 1 to 3.

—S tep 5 .C alculation of ?1(internal Delta).

F irstly,we determine the position vector of point E r eferred to the orientable base frame R0?by the following r elation :

O?E0??O?O0??R T(?)O D0?D E0?(13) w here O?O0??(0,0,?O O?)T,DE0??(0,0 , DE)T.T hen,?1is a given according to Clavel’s relation (7) :

?1?f?1(O?E0?)(14)—S tep 6 .C alculation of ?2and ?3(internal Delta).

W e use the same procedure as in step 4 after having e xpressed vector H i F i r eferred to orientable base frame R0?i nstead of vector A i C i r eferred to ?xed base frame R0.

3.3.S olution to the in?e rse kinematic problem for the

2-D elta robot

I n order to obtain the closed-form solution to the inverse k inematic we will use the following procedure.

K nown terms are O V0,t he position vector point V, a nd the gripper orientation ?.U nknowns are the control j oint variables θ1and ?,a nd the passive joint variables θ2,θ3,?1,?2,?3.

—S tep 1g ives ?b y

???(15)

F ig.9.D e?nition of joints parameters.

—S tep 2 .C alculation of position vector of point D r eferred to frame R0:

O D0?O V0?R(?)D V0?(16)w here R(?)and D V0?are previously de?ned.

—S tep 3 .C alculation of θ1from the Clavel’s relation (7).

F ig.10.D e?nition of angles θi2,θi3.

F inally , the passive joint variables θ 2, θ 3, ? 1, ? 2, ? 3 a re obtained from the procedure (s teps 4 ,5 ,6)described i n the direct kinematic problem .3.4.‘‘C hecking ’’ o f the modeling D ue to the complexity of kinematic modeling ,many m iscalculations were likely to appear . Thus it was n ecessary to check all the relationships . To this end ,we b uilt the 2-Delta architecture on a C . A . D .Robotic

S imulator developed in our Laboratory 1 2

as it is shown in F igure 11 . If the kinematic relationships are correct we w ill see that bodies remain connected together at joints w hen the mechanical structure is actuated . These ?gures a llow us to check that the kinematic modeling is right .4.G ENERATION OF POSITION WORKSPACE A ND ORIENTATION WORKSPACE S tarting from the knowledge of kinematic modeling we m ay obtain the workspace of the 2-Delta robot which d epends on the mechanical limits of both active and p assive joint variables . T o determine the position or the orientation w orkspace of the 2-Delta robot , we will use the

t echnique of discretization 1 3

which consists , in this case , o f the de?nition of the position workspace , of ?xing the o rientation of the end ef f ector and ?nally seeking all p ositions that can be reached by the end ef f ector , r egardless of the cross section chosen . In the case of the g eneration of workspace orientation , we will consider the e nd ef f ector position will be ?xed .Then , we will seek all p ossible orientations . We have separated this space into t wo distinct subspaces :

W orkspace without joint limits or theoretic space

. w orkspace with joint limits or space with constraints .

4.1.R elationship between space of theoretical position a nd space with constraints T he introduction of limits (relation (2) to (5)) on the p assive and active joint variables reduces the workspace o f the 2-Delta structure in comparison to the theoretical s pace . We have attempted to investigate the proportion t hat represents space with constraints in comparison to t heoretical space . These two spaces in a cross section d e?ned by plane X ? ? 0 and plane Z ? ?450m m are i llustrated in Figure 12 .

F or a given cross section of the workspace (for

e xample Z ? ?c onstant) , we can evaluate the reachable s urface by de?ning a workspace position factor ? z a s :

? z ?

S p ,c

S p ,t

(17)

w here , S p ,t i s the surface of theoretical space in the plane

Z ? ?c onstant and S p ,c i s the surface of space with c onstraints in the same plane . It is obvious that ? z d epends on Z ?a nd orientation ?.F or example , in Figure 13 , ? z v ersus Z ? f or dif f erent orientations of the end e f f ector are shown .4.2.C ross section of theoretical orientation space I n order to obtain a representation in the plane , we will ?x one of the three orientation angles of the gripper ( ? 3 ?0), t hen will seek all possible orientations for a g iven position of point V f or both cases , without and with m echanical limits on joints . In Figure 14 , (a) and (b) s how two examples of two dif f erent cartesian coordinates o f point V . A s we de?ned in (17) the position factor we could d e?ne the orientation factor ? 0as :

? 0 ?

S o ,c

S o ,t

(18)

w here , S o ,t i s the surface of the theoretical orientation

s pace and S o ,c i s the surface of the orientation space with c onstraints . It is obvious that ? 0 depends on the position c hosen for the point V . I n Figure 15 , ? 0versus Z ?w hen t he point V b elongs to Z 0 axis is shown . 5 . M ODELIZATION OF THE CLEARANCES IN P OSITION AND IN ORIENTATION I n this section , we will consider the in?uence of the small v ariations of the rod lengths of the 2-Delta around their f ace values on translation and rotation vectors of the end e f f ector . We will assume that these variations express the e xistence of localized clearance in the passive joints of t he rods . Afterwards we will de?ne the ellipso? ¨d es of c learance based on the concept of the manipulability e llipso?¨d s . 1

3

F ig .11 . S imulation of the 2-Delta robot .

F ig.12.S uperposition of the theoretical space and the space with constraints (a) cross section for plane X V?0 (b) cross section for

p lane Z

V?450m m

.

5.1.P roblem statement and simplifying hypotheses

I n order to reduce the problem complexity,it seems u seful to propose some simplifying hypotheses concern-i ng the geometry of the robot,as well as its physical p arameters.We suppose that :

–e ach parallelogram is made of one rod.

–a ll simulations are carried out taking into considera-t ion that active joints are without clearances in each c hosen con?guration,and that the variation of the p osition and the orientation of the ef f ector are due solely t o variations of rod lengths which act as linear actuators. W e will separate the modelization of clearance into t wo parts :

–p osition clearance model which gives position v ariation of the ef f ector solely for clearance at the p assive joint of each rod of the external structure.

–o rientation clearance model which gives orientation v ariation of the ef f ector solely for clearance at the p assive joint of the internal structure.

5.2.P osition clearance model

W e de?ne the position clearance model as,

?X V?J p?L1(19) w here :

?X??(?x V,?y V,?z V)T r epresents the variations of the p osition of the end ef f ector.

?L1?(?L11,?L21,?L31)T r epresents the elementary length v ariations of the rods found in the external structure.

J p i s the Jacobian matrix of partial derivatives of X V w ith r espect to L1,o f dimension 3?3.

T erms of J p h ave complex expressions,thus we must

F ig.13.P osition factor on the axis Z0.

F ig .14 . S uperposition of the orientation and with stops theoretical space for two sets of coordinates of point V .

u se the symbolic calculation software (Maplle V )to o btain these expressions . I n Figure 16 there is a representation of the ellipsoid

o f clearance de?ned by ? X

T ? (J J T ) ? 1 ? X ? ? 1.W hen the e f f ector moves along the Z 0axis , we see that this axis c onstitutes an axis of revolution for these ellipso?

¨d s the ?rst two eigenvalues of each matrix (J J T ) ? 1

always r emain equal ( ? 1 ? ? 2 ) whatever the position may be . T his leads to an equitable distribution of the clearance in t he plane [ x 0 , y 0 ].S imilarly , we observe that the volume o f the ellipso? ¨ d s of clearance increases considerably as t he position of the ef f ector moves along the axis z 0.T his i s due to the vicinity of the singular con?gurations .On t he other hand , for postures of the wrist following the a xis x 0 or the axis y 0 (that is not shown here) ,the e llipsoids change in dimension and in form depending on t heir relation to the robot con?guration . In fact ,the c learance in these postures does not have the same ef f ect i n all directions . T

he exploitation and the application of these results

F ig .15 . O rientation factor on the axis Z 0 .

a re very extensive . They allow , for example , a designer

t o know in advance the value of error in a position of the e nd ef f ector according to the structure con?guration . T hey also allow us to select the best con?gurations to e xecute a task inside the workspace .5.3.O rientation clearance model I n this section , we will deal with the clearance at the p assive joint of the internal structure on the gripper o rientation . To obtain this model , we deduce it from the d irect kinematic model which gives orientation coordin- a tes of the gripper in relation to the rod lengths of the i nternal structure . Therefore we obtain :

?? ? J 0 ? L 2

(20)

w here :?? ? (??1 ,??2 ) T r epresents the variation of the orienta- t ion of the ef f ector with ? 3 ? 0 .

? L 2 ? ( ? L 1 2, ? L 2 2 , ? L 3 2 ) T

r epresents elementary length v ariations of the rod for internal Delta . J 0 is the Jacobian matrix of partial derivatives of ? w ith r espect to L 2, o f dimension 2 ? 3 . W e have studied ellipses of clearance in dif f erent p ositions .Presently , in Figure 17 , we show the results of t wo ellipses obtained for two extreme positions on the a xis Z 0 . W e notice in general that eigenvalues which represent d imensions of clearance ellipse are roughly equal w hatever the con?guration . This shows that the u ncertainty in orientation is almost the same in all p ostures (except postures close to single con?guration) . I f we consider the obtained ellipses along the axis z 0for o rientation to be zero , we will observe that the surface of o rientation uncertainty is represented by a circle since t he two eigenvalues are equal , which shows that no d irection is penalized . On the other hand ,for c on?gurations away from the axis z 0, w e notice a d issymmetry represented by a change of ellipse

F ig .16 . D imensions and morphology of the ellipsoids of position clearance on the axis Z 0

.

F ig .17 . D imensions and morphology of clearance ellipses in orientation on the axis Z 0 .

m orphology , which explains the uncertainty dif f erence d ue to the clearance that is not the same in all directions . T his study shows that the existence of passive joints , w hich in general characterize parallel robots , can be a s ource of error and perturbation and susceptible to p roducing position and orientation imprecisions of the e f f ector . In the case of the 2-Delta robot , we have shown f or distanced postures of singular con?gurations that the c learance has more in?uence on the position of the e f f ector than on its orientation . In this section we have p resented a methodology that allows one to identify and s elect con?gurations so that the ef f ect of the clearance is m inimized . Similarly the introduction of the ellipso? ¨d es o f clearance (or ellipses of clearance) allows for a c omplete understanding of the general volume of the u ncertainty in function of the con?guration .6.C ONCLUSION T he main objective of this study consisted of examining a n ew parallel robot family , characterized by the u ncoupled mechanics between the translation and r otation of the wrist , and proposing approaches to solve p roblems in design ,modelization , joint limits and i n?uence of the clearance . The study that we expect to d o in the future will deal with the possible collisions

b etween the internal and external Delta . With this aim in

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