flexible manipulators

7/29/2019 Flexible Manipulators

1/34

Journal of Intelligent and Robotic Systems 34: 381414, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands.

381

Flexible Links Manipulators: from Modellingto Control

M. BENOSMAN, F. BOYER and G. LE VEYEcole des Mines de Nantes, Institut de Recherche en Communication et Cyberntique de Nantes,

UMR, 1 Rue de la No, B.P. 92101, 44321 Nantes Cedex 3, France;

e-mail: [email protected], [email protected]

D. PRIMAULTEcole Centrale de Nantes, Institut de Recherche en Communication et Cyberntique de Nantes,

UMR, 1 Rue de la No, B.P. 92101, 44321 Nantes Cedex 3, France

Abstract. This paper relates recent results obtained in the field of modelling and control of flexiblelink manipulators and proposes an investigation of the problem raised by this type of systems (atleast in the planar case). First, adopting the modal floating frame approach and the NewtonEulerformalism, we propose an extension of the models for control to the case of fast dynamics and finitedeformations. This dynamic model is based on a nonlinear generalisation of the standard EulerBernoulli kinematics. Then, based on the models recalled we treat the end-effector tracking problemfor the one-link case as well as for the planar multi-link case. For the one-link system, we proposetwo methods, the first one is based on causal stable inversion of linear non-minimum phase modelvia output trajectory planning. The other one is an algebraic scheme, based on the parametrizationof linear differential operators. For the planar multi-link case the control law proposed is basedon causal stable inversion over a bounded time domain of nonlinear non-minimum phase systems.Numerical tests are presented together with experimental results, displaying the well behaved ofthese approaches.

Key words: floating frame, NewtonEuler, finite deformation, Poincar equations, trajectory track-ing, non-minimum phase system, stable inversion, parametrization, two-point boundary value prob-lem.

1. Introduction

The dynamics of flexible multi-body systems, has got a great interest during thelast few years. In the field of robotics, two quite different types of systems motivatethis theory: the fast industrial manipulators, and the large space structures. The firsttype undergoes high frequency vibrations of small magnitudes while the second

one is rather characterized by slow oscillations of finite magnitude. To model thesetwo kinds of systems, two theories have been proposed.The first theory is well adapted to the first type of systems since it considers

the deformations of the links as small linear perturbations of some overall rigidmotions. In this approach the rigid motions are explicitly separated from the de-formations. Moreover, since the rigid motions are those of some fictitious bodies


2/34

382 M. BENOSMAN ET AL.

following the real ones, this approach is often called floating-frame approachin order to traduce the lake of materiality of the links reference frames (Canavinand Likins, 1977). As a consequence of the separation of rigid and deformationmotions, the nonlinearities of the dynamics appear via the inertia operator. Thiskind of approach can be divided into many subgroups depending on the reductionadopted to discretize the continuous field, if the model is looked for in an assembledor constrained form, and on the formalism adopted to describe the rigid motions.As far as the discretization operation is concerned, one can use a finite elementdiscretisation (Shabana, 1989; Theodor and Ghosal, 1995; Sharf, 1996) of theelastic deformation fields or alternatively a link by link RayleighRitz approach(Meirovitch, 1989; Agrawal and Shabana, 1985; Sharf and Damaren, 1992; Boyer,1994). In this last case the RayleighRitz functions of the links are most of the timethe natural modes of the single links under some specified boundary conditions.This modal approach gives models with the minimal number of state variables,a crucial advantage when we want to obtain models for control. The second choice

concerns the type of formulation. It is well known in mechanics that for any holo-nomic systems, it is always possible to solve the dynamic problem as a minimal setof equations, or alternatively by posing an augmented problem with constraints. Inthe first case, the final dynamic system is an explicit system of highly nonlinearequations in minimal number which we call assembled model (Boyer, 1994). Inthe second case we have to deal with an implicit system of algebraic-differentialequations (Wehage and Haug, 1982). Finally the formalism of the model is relatedto the type of kinematical variables which are adopted to describe the nonlineardynamics of the rigid overall motions of the chain (Boyer and Coiffet, 1996a). Ifwe adopt some positional kinematic variables, i.e. a set of generalised coordinates,the model is said Lagrangian. If we adopt some velocities which do not directly

derive from a set of parameters, the model will be said Eulerian. In the last case,the model is necessarily posed in an augmented form, while in the first one, themodel can be posed in an augmented or an assembled form. In the floating frameapproach, the most interesting formulations are the assembled Lagrangian modeland the augmented Eulerian one. We shall call the first models Lagrangian ones andthe second are known under the name of NewtonEuler models. These two kinds ofmodels have their own advantages and drawbacks. The Lagrangian ones are wellsuited to design closed-form control laws (De Luca and Siciliano, 1989), whilestarting from the NewtonEuler models we can design fast algorithms useful forcontrol and simulation (DEleuterio, 1992; Boyer and Khalil, 1998). Neverthelessthe Lagrangian model can be easily deduced from the NewtonEuler one by anoperation named assembling (Boyer and Coiffet, 1996b). Moreover, this way of

computing the Lagrangian model presents many advantages when compared to thestandard use of the Lagranges equations (Book, 1984). First it clearly separatesthe geometric, kinematic and dynamic modelling and gives a physical insight fromthe beginning of the modelling, contrarily to the alternative based on Lagrangeequations which replaces the dynamic modelling by complex non-intuitive compu-


3/34

FLEXIBLE LINKS MANIPULATORS 383

tations on energies. This lack of clarity of the pure Lagrangian approach becomesa drastic drawback when we want to improve the model of geometric nonlinearities(Boyer et al., 2001).

The second theory (called global one) was originally developed for the largespace structures submitted to finite deformations (Simo and Vu-Quoc, 1988; Car-dona and Gradin, 1988). It is based on the following basic remark: when theelastic motions are of the same order of magnitude as the rigid ones, it becomesartificial and too much involved to separate the two types of motions. Thus in thistheory the links deformations references are Galilean, the fields are global and themost of the nonlinearities appear through the stiffness operator. This theory beingmore dedicated to dynamic analysis of mechanisms than control we do not enterdeeper into details and focus on the modal floating frame approach. Practically,the modal reduction can be done following two ways. If the links have complexthree dimensional shapes, the only way is to start with a finite element model, thento conduct a numerical modal analysis and to deduce all the parameters required by

the dynamic model (NewtonEuler or Lagrangian at this step). In robotics the linksoften have a tubular shape. In this case they can be efficiently modelled by beams,and the modal analysis can be conducted analytically. Moreover the analyticaltreatment, allowed by the beam assumption, can be fruitfully used to refine theprecision of the dynamic model. By precision we mean that of the geometricnonlinearities due to the deformations of the links. This crucial point representsone of the original modelling aspects of the developments here related. It is worthnoting, for presently flexible industrial and laboratory testbed arms, controllerssynthesised on linear EulerBernoulli theory based models, are actually enough toperform precise fast motions. In (Bayo, 1987), an open loop control torque suitableto produce a desired tip trajectory has been computed by inversion in the frequency

domain, of a linear flexible arm joint torque to end effector transfer function. Thisresult has then been extended to an open chain flexible robot in (Bayo, 1988),where an iterative scheme, based on noncausal frequency domain calculus, hasbeen presented for end-effector motion tracking of a planar robots. Good trackinghas been obtained for the two-link case moving along specific desired trajectories(with Gaussian velocity). However the fact that this algorithm has been developedin the frequency domain, implies a heavy computation effort, thus, the same authorhas reformulated this algorithm in the time domain and has showed it efficiencyfor the one-link case (Bayo and Moulin, 1989). Passivity-based approaches, havebeen investigated in (Christoforoua and Damaren, 2000; Damaren, 1995), thesemethods based on modified input and output, have been successfully applied toflexible arms, under the assumption that they carry a large payload (relatively to

link mass). The problem of elastic zero dynamics instability (non-minimum phasecharacteristics), has been treated in (Zhao and Chen, 1998), using a linearizingscheme based on a non-minimum phase dynamics inversion procedure. A con-traction that converges to a bounded solution of the nonlinear unstable hyperbolicdynamics, has been build up. Roughly speaking, the tangent linearized system


4/34


of the nonlinear elastic dynamic is separated into two parts (stable and unstableparts), then two sets of boundary conditions are given. Initial conditions for stableequations and final one for unstable equations. The procedure consists then, on aniterative forward and backward integrations. Other algorithms for the computationof bounded link deformations (bounded inversion of the unstable elastic dynamics),have been proposed in (De Luca et al., 1998). These algorithms (based on time orfrequency domain calculus), generate noncausal nominal input torques, since anextended interval of definition is considered for the output reference trajectories,necessary to gives enough time to preload the flexible link to the proper initialstate, that enables stable reproduction of the desired trajectory. In fact, even forlarge space flexible arms, as Shuttle space arm, controls based on linear elastic fieldmodels shows to be efficient, when dealing with relatively slow motions (Damaren,1995). However precise models, fully based on nonlinear deformation fields mayactually be necessary for simulation purposes of large space arms (used for controlschemes validation or space engineers training). Furthermore, those fully nonlin-

ear models are definitely necessary to compute precise controls for space robotsperforming fast motion tasks, which to our knowledge, has not yet been done.Starting from the previous observations, the modal floating frame models, untiltoday restricted to the field of small deformations, have been recently extended tothe field of moderate deformations (Boyer et al., 2001).

In this paper, we report on recent results in modelling as well as control. InSection 2, we begin with the NewtonEuler model of a flexible link. Contrarily tothe common use, this model is obtained through the use of the Poincars equations(Boyer and Primault, 2001a, 2001b). After giving the NewtonEuler model of ageneral elastic body, the particular case of a beam is developed. Starting from anonlinear generalization of the EulerBernoulli kinematics, a modelling process is

proposed which allows one to extend the modal floating-frame approach to thecase of finite deformations and fast dynamics. Then, the kinematics of the chain isderived in order to obtain the NewtonEuler model of the manipulator. In Section 3,we establish the link between the NewtonEuler and the Lagrangian models. InSection 4, we follow on with the control problem, presenting new theoretical andexperimental results for the point to point end-effector motion problem for the one-link flexible arm. The method is based on end-effector trajectory planing leadingto causal stable inversion of the non-minimum phase joint torque to end-effectortransfer function (Benosman and Le Vey, 2000a). In the same section, we treatthe end-effector trajectory tracking, for the linear one-link model. The proposedmethod is based on the parametrization of linear differential operators (Benosmanand Le Vey, 2000b). Experimental results are reported. Eventually, we present in

Section 5, some results obtained, concerning the end-effector trajectory trackingproblem for planar multi-link flexible arms. The method is based on a causal stablenonlinear inversion of non-minimum phase systems, over a bounded time interval(Benosman and Le Vey, 2001c). We conclude this section by numerical tests carriedout for the planar two-link case.


5/34


2. NewtonEuler Model of an Elastic Body

In this section we consider an elastic body submitted to external forces and torquessimilar to those applied to a single link in a robotic open chain. The purpose of this

section is to compute the NewtonEuler model of such a system. In the floatingframe approach, the motion of the body is separated into that of a rigid fictitiousbody said of reference and an elastic motion responsible of the deformations. In theNewtonEuler formalism the rigid reference motion is described by the evolutionof a set of Eulerian unknowns, i.e. by the twist of the reference floating framewhile the deformations are parameterised by a set of RayleighRitz coordinates,i.e. in an Lagrangian manner. This mixed EulerianLagrangian formalism requiresa specific treatment. There are in fact two natural ways to obtain these equations,the principle of virtual powers, and the Poincars equations. We follow here thesecond way and refer the reader to (Boyer and Primault, 2001a, 2001b), for detailsabout the first one.

2.1. POINCARS EQUATIONS

In 1901 Poincar established a new set of equations for mechanics (Poincar,1901). These equations describe the dynamics of a system, the space of configura-tions of which realises a non commutative Lie group (Marsden and Ratiu, 1999).Moreover when the group is commutative, the configuration space becomes a flatvector space of parameters and the Poincars equations degenerate into the La-grange ones. These equations come from a Lagrangian which is not, as in theLagrangian mechanics, a functional of a set of parameters and their time deriva-tives, but directly a functional of the transformations of a Lie group G and of itsvelocities translated into the Lie algebra of the group. The transformations ofG aredefined as the maps applying a reference configuration o of the material systemonto any of its accessible transformed configuration , i.e. denoting X a materialparticle ofo, we have for any G:

: X o (X) .

Hence, in the language of transformations, a motion of the system is a curveof G parameterized by the time, i.e. a one dimensional continuous set of trans-formations applying at each instant the reference configuration onto the currenttransformed one (t). Moreover, in the cases studied here, the Lagrangian will beexpressed as a functional of the transformations of the form:

L: T G R,(, ) L(, ), (1)

where is the time derivative of named the velocity of the transformation. Thenthe Lie group properties allow one to replace the velocity vector of the transforma-


6/34


tion by its right or left translation into the Lie algebra of the group, denoted by G.These translations are defined respectively by:

1

=,

1

=.

For example, in the case of the rigid body, the configuration space can be identifiedto the group SE3 of affine transformations of the form:

X o, (X) = d + R X (t),where R and d are respectively the rotation and the translation part of the rigidtransformation applying o onto (t). At this point, let us notice that the transfor-mations can be applied not only onto material points but also on a frame (O, E1,E2, E3) ofo (named material frame). The current image of such a frame definesthe mobile frame: (O = (O), t1 = R E1, t2 = R E2, t3 = R E3), of thebody. Note that this group of transformations can be represented by the so calledhomogenous matrices group, and the composition law

by the matrix product.

Such an homogenous transformation is defined by (Brokett et al., 1993):

g =

R d

0 1

, with:

(X)

1

= g

X

1

.

With this matrix representation and can be interpreted as the matrices of com-ponents of the current twist of the body expressed in the fixed and mobile frames,respectively. We call them spatial and material twist respectively and write them as(the symbol designing a skew symmetric 3 3 matrix):

= g1 g =

V

0 0 , = g g1 =

v

0 0 .

Moreover, using the natural isomorphism between the skew-symmetric matricesand the pseudo-vectors ofR3, the spatial and material twists can be rewritten in themore standard form:

=

v

, =

V

.

With these definitions we can rewrite the Lagrangian (1) as:

L(, ) = L(, ) = l(,) = L(, ) = l#(, ),where l and l# are named reduced Lagrangian and are defined on the followingspaces:

l: G G R,(,) l(, ),

l#: G G R,(, ) l#(, ).

Moreover when the Lagrangian l (respectively l#) does not depend anymore of thetransformations, i.e. when l = l() (respectively l# = l#()) the Lagrangian L is


7/34


said left (respectively right) invariant. This property result of the isotropy of theLagrangian with respect to the ambient space (left invariance) or to the materialspace (right invariance). Since in all the cases here studied (with no gravity), theLagrangian will be left-invariant with respect to the rigid transformations, we willsystematically replace the Lagrangian L of (1) by the reduced one on the left: l.If the lagrangian is left-invariant, its dynamics on the group G is driven by thefollowing set of Poincars equations (Marsden and Ratiu, 1999):

d

dt

l

ad

l

= Wext, (2)

where ad is the co-adjoint action of the vector of the Lie algebra onto any vectorof its dual, it is a linear map G G (Arnold, 1988) and the right hand side Wextis a forced term given by the virtual power of the external forces:

Pext = WText

with is a virtual twist in the mobile frame. In the case ofSE3, Wext is the externalwrench applied onto the body and expressed in the mobile frame (in any case it isan element ofG).

Assuming a matrix representation of the group, if the Lagrangian L is not left-invariant, i.e., ifl is of the form: l = l(g,), then its dynamics is given by:

d

dt

l

ad

l

Xg(l) = Wext, (3)

with Xg(l) is a term taking into account the symmetry defect of the Lagrangian(Boyer and Primault, 2001a, 2001b). It is defined by:

Xg (l) = col=1,...,n dd =0l(,g exp(e)),where (e)=1,...,n is the canonical basis of the Lie algebra, exp is the exponen-tial map in the sense of matrices, and n is the dimension of the group. We invitethe reader to refer to (Boyer and Primault, 2001a, 2001b) for more details aboutthese considerations. For instance, let us apply these equations to the left-invariantLagrangian of the rigid body without gravity:

l() = 12

T, VT

J msmsT m1

V

, (4)

where = (T, VT)T, J and ms are the matrices of components of the body

twist, the body inertia tensor and the first order inertia moments expressed in themobile frame. In such a case, since the co-adjoint tangent map of SE3 is definedby (Marsden and Ratiu, 1999):

ad

V

CF

=

C + F VF

, (5)


8/34


where (CT, FT)T is an element ofSE3, i.e. a wrench. The application of (2) and (5)to (4), gives the so called NewtonEuler model of the rigid body:

J msmsT m1 + (J ) ( ms) = FextCext , (6)where we introduced the matrix of components of the acceleration vector in themobile frame:

= V + V .

2.2. APPLICATION TO THE ELASTIC BODY IN THE FLOATING FRAMEAPPROACH

In the case of the elastic body, a transformation mapping the reference config-uration o onto any accessible configuration can be written as the composition

of two transformations (Boyer and Primault, 2001a, 2001b). The first one is a puredeformation mapping o onto o . It is denoted by e. The second one, denotedr , is a rigid displacement mapping o onto . We thus have the sequence oftransformations:

= r e: oe o

r ,transforming a point X ofo as:

(X) = r (e(X)).As in the case of the rigid body, the rigid transformation is written as:

r (X) = d + R X,where X is now a current point of o and d is the displacement of the origin ofthe reference floating frame. As far as the elastic transformation is concerned, wecan write it:

e(X) = X = X + de(X),where de is the matrix components of the field of displacement in the floating framemapping the position of the particle in the reference configuration onto its imageby the pure deformation. This context is summed up Figure 1. As previously, theset of all rigid transformations realises the Lie group SE3, whereas the field eis an element of the infinite dimensional group of diffeomorphisms ofR3

R

3

restricted to o: Diff(o). Hence, the configuration space of the elastic body is thegroup G = SE3 Diff(o). For what concerns Diff(o), we are going to replaceit by a finite group thanks to a Lagrangian parameterisation of the displacementfield:

de(X) = (X,qe), (7)


9/34


Figure 1. The elastic body.

where qe = col=1,...,m(qe, ) is a vector of elastic generalised coordinates intro-duced below. Under this parameterisation, the group Diff(o) is geometricallyreplaced by the linear space Rm, i.e., by a commutative Lie group and each element of the group G = SE3 Rm of the elastic body is then represented by thetransformation:

g =

R d

0 1

qe

,

where we use the homogeneous matrices to represent SE3. Moreover, the internalcomposition law of this group is:

g1, g2 G, g1 g2 = R1 d10 1

qe,1

R2 d20 1

qe,2

= R1 R2 R1 d2 + d10 1

qe,1 + qe,2

,and its Lie algebra is the space of vectors:

= V

qe

,where and V are the matrices of components of the angular and linear velocitiesof the floating frame expressed in itself. Computing the kinetic and the potential en-


10/34


ergies in a standard way, we find after few computations the following Lagrangianleft reduced in the Lie algebra:

l(,qe) =1

2T, VT, qTe J m

s

msT M aT aT mee

Vqe

12

qTe kee(qe) qe, (8)

where we introduced the following tensors, expressed in the mobile floating frame:

The 1 m matrices of 3 1 vectors: = row=1,...,m( + ), a = row=1,...,m(a),

with:

a

= o qe dm, = o X qe dm, =

o

qe dm, the m m matrix of generalized elastic inertia:

mee = mat,=1,...,m(m), with: m =

o

qe T qe dm,

the mass of the body:

M =

o

dmI3,

the 3 1 vector of the first inertia moments of the deformed body:ms =

o

e(X) dm =

o

X dm +

o

dm = msr + mse,

the 3 3 inertia matrix of the deformed body:

J =

o

Te e dm = o

(X + )T (X = ) dm

= Jrr +

Jre + JTre+ Jee,

the m m stiffness matrix: kee(qe).Now, remarking that the subgroups SE3 and Rm are kinematically uncoupled,

the co-adjoint tangent map ofG = SE 3 Rm is defined by:

ad Vqe

CF

Q

= C + F VF

0

.


11/34


12/34


the dynamics which will be used in the following:

MNE Tr

qe + CNE + QNE = Wext, (10)

where MNE is the mass matrix of the NewtonEuler model, Tr is the twist of thefloating frame expressed in itself (sometimes called reference twist of the body),CNE is the vector of Coriolis and centrifugal forces and QNE is the vector ofrestoring forces. Finally, Wext is the vector of external forces applied onto the body.

2.3. THE PARTICULAR CASE OF A BEAM

We explicit here the parameterisation (7) of the deformation when the links canbe modeled by beams (which is often the case in robotics). Concerning the kine-matics of the beams, most of the models used today in robotics are based on

the rigidity of cross sections (Cosserat medium). At this point, different beammodels can be adopted depending if we neglect the transverse shearing (EulerBernoullis beams), if we consider it (Timoshenkos beams), if we neglect stretch-ing (non-extensible beams), if we take into account the rotary inertia of crosssections (Rayleighs beams). In any case, all the beam kinematics used today inthe floating frame approach are linear and cannot model finite deformations. Inorder to overcome this limit we proposed in (Boyer and Khalil, 1999) a nonlinearextension of the modal parameterisation of a non-extensible cantilevered EulerBernoulli beam. Before to recall this result let us introduce a few notations. Wedenote by (O,E1, E2, E3) an orthonormed frame attached to o (material frame)and X = (X1, X2, X3)T denotes the coordinates of a point of o (a material

particle) in this frame. The image of (O,E1, E2, E3) by the rigid component rof the global transformation defines the mobile floating frame (O , t1, t2, t3) (seeFigure 1). Moreover the frame (O,E1, E2, E3) is placed such that E1 supportsthe beam axis, and O is the first point of the beam when we progress along therobot from the basis to the tool. This point satisfies the cantilever conditions (seeFigure 2):

(0, qe) = 0, X(0, qe) = 0,

where is the nabla operator. Under these constraints, the floating frame is a framesupported by the tangent at the origin of the beam and can be considered rather asan embedding frame than a really floating one. Since the cross sections of the

beam are rigid, the deformation field can be written as:

(X,qe) = de(X1, qe) + (Re(X1, qe) 1) (X2E2 + X3E3), (11)

where X1 [0, L] de(X1, qe) R3 is the deformation field of the referenceline of the beam, and X1 [0, L] Re(X1, qe) SO3 is the rotation field of the


13/34


Figure 2. The nonlinear EulerBernoulli kinematics.

cross sections due to the deformation of the beam only. In the case of an EulerBernoulli kinematics the rotation and displacement fields are not independent, butlinked through the expression:

Re = g v wv h2(w2 + v2g) vwh2(g 1)

w vwh2(g 1) h2(v2 + w2g)

1 0 00 cos( ) sin( )

0 sin( ) cos( )

, (12)where: v, w are the transverse components ofde along E2 and E3 respectively, theprime denotes the derivative with respect to X1, h =

v2 + w2, g = 1 h2,

and finally (X1) is a field of torsion angle. Moreover if the beam is non-extensible,

the axial and transverse displacements are not independent, but linked through theexpression:

de(X1) =X1

0g d X1

E1 + v(X1)E2 + w(X1)E3. (13)

This kinematics is a generalisation of the linear kinematics on which are premisedthe standard floating frame models. This is notably attested by the fact that lineariz-ing (12) and (13) with respect to v, w and , and inserting the result into (11), wefind the so called linear EulerBernoulli field of strength materials theory. Finallythe three scalar fields v, w and are next parameterized as:

v(X1) =N

f=1

(X1)q, w(X1) =N

f=1

(X1)r,

(14)

(X1) =N

=1(X1),


14/34


where the Nfs and the Ns are the first linear bending and twisting modesof the cantilevered beam. It is worth noting that the material variable X1 is distin-guished from its spatial homologous in the expansions (14). Finally, this nonlinearextension of the modal approach turns out to be a nonlinear parameterisation ofthe deformation obtained by inserting (14), in the starting fields (12) and (13).Moreover, and this is a crucial point, this parameterisation gives the standard linearmodal parameterisation (Sharf and Damaren, 1992; Boyer, 1994) in the context ofsmall deformations.

3. Kinematic Models of a Robotic Open Chain

We now consider a multibody system constituted ofn beams of the precedent typedenoted by: B1, B2, . . . , Bn, from the base to the tip. The base Bo is supposedto be rigid. We suppose that the beams are connected by rotational joints centredat the points: O1, O2, . . . , On. The floating frames of the bodies B1, B2, . . . , Bnare denoted R1, R2, . . . , Rn. Each joint introduces one degree of freedom betweenBj1 and Bj parameterised by the angle qrj. These notations are summed up inFigure 3. Any vector, twist or wrench V of a body Bj is expressed in the floatingframe of the body and denoted jV. In order to achieve the NewtonEuler modelof the chain, we have to add to the NewtonEuler model of each link a kinematicmodel relating the reference frame of the chain, their twists and the time derivativeof their twists. We now give these three models without demonstration and referthe reader to (Boyer and Coiffet, 1996a; Boyer and Khalil, 1998) for more detailsabout these considerations.

Figure 3. Deformation of an open chain.


15/34


3.1. GEOMETRIC MODEL OF THE CHAIN

This model gives the relations for passing from any floating frame of the chainto its successor. Let us consider two contiguous bodies Bj and Bj+1. Using the

formalism of homogeneous transformations, the transformation which maps thefloating frame Rj onto Rj+1 is:

jTj+1 =

jRj+1 jPj+1

0 1

, (15)

where jRj+1 and jPj+1 define the rotation mapping the axis of Rj onto thoseofRj+1, and the translation mapping the origin Oj onto Oj+1, respectively. Theseparation of motions allows to detail these transformations as:

jTj+1 =

R

j

ej Rrj+1 jPrj+1 + dej0 1

.

Here the transformations

jTrj+1 =

jRrj+1 jPrj+1

0 1

define the rigid kinematics of the chain dej, Rej, are the elastic deformation and ro-tation endured by the extremity Oj+1 of the link Bj and expressed in the frame Rj.Their expressions are given by (12) and (13) or any of their approximations.

3.2. RECURRENCE ON VELOCITIES AND ACCELERATIONS

The recurrence on velocities links the twist of a floating frame to that of its succes-sor. For j = 1, . . . , n:

jTrj = jTj1 j1Trj1 + jRj1 j1Tej1 + jTaj. (16)The recurrence on accelerations is deduced from the preceding one by simpleGalilean derivation. For j = 1, . . . , n:

jTrj = jTj1 j1Trj1 + jRj1 j1Tej1 + jTaj + jgj, (17)where jTrj denotes the twist of the jth floating frame, jTej is the elastic twist of thelink Bj at Oj+1, jToj is the relative twist introduced by the jth joint, jTj1 is the6 6 operator carrying the twist jTrj from the frame Rj1 to Rj, while jRj1 is a6 6 matrix corresponding to a change of axis only (from those of Rj1 to thoseofRj). Finally jgj contains all the Coriolis and centrifugal accelerations resultingfrom the Galilean derivation of (16). Moreover, the elastic screw jTej is a maplinear in the elastic velocities:

jTej =

qejjTej

qej = qej,1 jTej, . . . , qej,Nj jTej qej,


16/34


where qej = colk=1,...,Nj(qek,j) is the elastic coordinates vector of the jth link,Nj being its total number of shape functions, and the qej,k

jTejs (k = 1, . . . , N j)denote the partial twists with respect to the generalized elastic velocities. Finally let

us note that the elastic twists and their derivatives appearing in (16) and (17) andexpressed in the floating frame are built from the nonlinear fields (12) and (13),following the relations: e Ve

0 0

=

Re(X1) RTe (X1) de(X1)0 0

,

e Ve0 0

=

Re(X1) RTe (X1) + RTe (X1) Re(X1) de(X1)0 0

,

with:

Te = dee and Te =

de

e .3.3. GLOBAL KINEMATICS

In the following we denote by qe = colj=1,...,n(qe,j) = colk=1,...,N(qek ), the globalvector of elastic coordinates, where: N = N1 + + Nn, is the total numberof elastic variables used to describe the chain deformations. The vector of jointvariables is denoted by: qr = colj=1,...,n(qrj) and the global vector by: qT =(qTr , q

Te ). The global kinematics relates the set of kinematic unknowns of a link

Bj i.e.: (jTrj, qej) to the set of global generalized coordinates of the chain via

the relations:

for j = 1, . . . , n:

jTrjqej

= j(q) q,

where j denotes the Jacobian of the jth link. Differentiating these equations withrespect to time yields the global model of accelerations:

for j = 1, . . . , n:

jTrjqej

= j(q) q + j(q) q,

where j(q) q is the matrix of Coriolis and centrifugal accelerations of link Bjand is due to the compositions of motions induced by the chain.

4. Dynamic Model of a Manipulator

We give now the dynamic model of a flexible manipulator under the NewtonEulerformalism and the Lagrangian one.


17/34


18/34


19/34


5. Control in the One-Link Case

Taking place in the framework of small deformations, the general parametriza-tion (7) can be approximated by the linear assumed modes expansion:

de(X1) = v(X1) E2 = m

=1(X1) q

E2, (23)

where the reference mobile frame is placed such that the deformations applied too are along E2. m is the number of retained modes, q the generalised modal co-ordinates corresponding to the bending clamped-mass modes (Schmitz, 1985).Following the modelling process detailed in the previous section leads to a modelof the form (21) where, in the one-link case, the nonlinearities are induced by thedeformations only. Then keeping only the linear part of the dynamics gives thelinear arm model:

M qrqe + 0Kee qe = 0 , (24)where M and Kee are constant matrices.

5.1. CAUSAL STABLE INVERSION VIA END-EFFECTOR TRAJECTORYPLANNING

In this subsection, we address the problem of planning a target tip trajectory, lead-ing to a finite smooth control torque (implying an exact reproduction of the com-puted trajectory), by using an inputoutput inversion technique applied to our non-minimum phase system. We know that the systems under study are characterisedby low frequencies elastic dynamics, then in practice a two first clamped modesapproximation is sufficient (De Luca and Siciliano, 1991; De Luca et al., 1998;Hastings and Book, 1986). Furthermore, for high frequencies the EulerBernoullimodel is not valid anymore (due to the apparition of unmodeled effect as angularcross sections dynamics), one should then turn out to the more refined model ofRayleigh (see 2.3). Consider then the following linear model in which we haveretained only two modes:

D1

M11qr + M12qe1 + M13qe2 = u,M21qr + M22qe1 + M23qe2 = 0,M31qr + M32qe1 + M33qe2 = 0.

(25)

This dynamics can be transformed into a transfer function, which leads to a lineardifferential equation of the form:

any(n)(t) + an1y(n1)(t) + + aoy(t)

= bmu(m) + bm1u(m1)(t) + + bou(t ), m n, (26)ai , bi R, i = 1, . . . , n, j = 1, . . . , m ,


20/34


with the appropriate initial conditions. One straightforward approach to computethe open loop necessary torque for tracking a desired output trajectory yd is toinvert the dynamic model. Unfortunately, this is not possible for a non-minimumphase system, where a direct inversion of the dynamic implies divergence of theinput control (see, e.g., (De Luca and Siciliano, 1989), for the nonlinear case).A simple analysis of Equation (26) solution (where yd is substituted for y) is asfollows: due to the linear nature of Equation (26), its solution is composed oftwo terms: the transient and the steady-state solutions. Since the system has non-minimum phase characteristics, the transient solution of (26) contains divergentterms (corresponding to the effect of unstable zeros of the system). The idea thatwe present here (introduce in (Benosman and Le Vey, 2000a)), is the planningof output trajectory leading to cancellation of these undesirable terms. Considerthen a polynomial trajectory yd(t). Solving Equation (26) (associated with initialconditions for u(t)), we obtain the following expression:

u(t)

=u1(t)

+u2(t) (27)

with u1(t) = mi=1 Ai eri t, where the ri s are the distinct roots of the characteristicequation of (26) (including the unstable zeros values) and Ai , i = 1, . . . , m, arelinear function of the yd polynomial coefficients (see the application paragraphbelow). u2 is a purely polynomial function (particular solution of Equation (26)with second member). To cancel the effect of unstable zeros, we just have to zerothe corresponding Ai coefficients in the expression (27). This corresponds to con-ditions that yd coefficients must satisfy, in addition to classical position, velocityand acceleration constraints. Finally, we obtain a bounded open loop control torque(through (27)), leading to exact reproduction of the planned output trajectory (com-putation details can be found in (Benosman and Le Vey, 2000a), we refer also to(Benosman and Le Vey, 2001a), where this scheme is presented in a more general

setting of causal stable inversion for SISO linear non-minimum phase systems, andis compared to presently available non-causal stable inversion methods).

Application

The above method has been used to compute a feasible tip trajectory and its cor-responding control law, for an experimental one-link flexible arm characterised bynumerical values presented in table 1 (first link), with: Jtip = 4.742102 kgm2 andMtip = 6.79 kg. The first two eigen-frequencies are: f1 = 6.03 Hz, f2 = 16.07 Hz.The arm is driven by a D.C. servo motor. For real time law computation a PC 486is used, with 12 bit A/D and D/A boards. The sampling period used is 5 ms. Theangular position is measured by an encoder of 14400 points. The angular velocity is

computed through basic numerical differentiation and filtering. Strain gauges areavailable for the tip deflections measurements. Through the procedure describedabove, we get the transfer function from the torque u to ytip:

F(s) = 0.295s4 + 98.736s2 + 3331956.636

s2(s4 + 19015.243s2 + 25179356.808) (28)


21/34


which has the following zeros: z1 = 59.4, z2,3 = 56.5i, z4 = z1. The transferF yields the following linear dynamics between input torque u and controlled tipposition ytip:

y(6)tip (t) + 19015, 243y(4)tip (t) + 25179356.808y(2)tip (t)= 0.295 u(t)(4) + 98.736 u(2)(t) + 3331956.636 u(t) (29)

with the initial conditions:

u(0) = u(1)(0) = u(2)(0) = u(3)(0) = 0,(30)

y(0) = y(1)tip (0) = y(2)tip (0) = y(3)tip (0) = y(4)tip (0) = y(5)tip (0) = 0.

The desired tip trajectory must satisfy the above initial conditions, in addition tothe classical one: ytip,d(tf) = yf, desired final end-effector position, reached att = tf y

(1)

tip,d(tf) = y(2)

tip,d(tf) = y(3)

tip,d(tf) = 0, velocity, acceleration and jerkfinales constraints. For experimental implementation reasons, the output trajectoryshould be symmetric to ensure symmetry for the control torque, avoiding outputovershooting (Benosman and Le Vey, 2001a) therefore we have extended to thefifth order the final conditions on the desired output trajectory. The remainingconstraints are the most important ones, since they cancel the effect of undesirablezeros (z1, z2, z3 leading to unstable and oscillating terms, respectively). Then ydmust satisfy three other equations Ai = 0, i = 1, 2, 3 (where Ai are the coeffi-cients of ezi t in u1(t) expression). Eventually, yd must satisfy thirteen constraints,then we choose the following polynomial trajectory:

yd(t) =12

i=0aiti . (31)

The initial conditions give the following values: ai = 0, i = 1, . . . , 5. Substitutingnow yd for y in (31) and solving for u(t) gives the closed-form solution for theopen-loop control torque:

uol =4

i=1Ai e

zi t + u2(t),

where u2(t) is a polynomial in time t of degree 10 and the Ais are linear functions

ofai , i = 6, . . . , 12. The full expression for uol can be found in (Benosman and LeVey, 2001a), together with the linear system of equations that the ais must satisfy.Resolution of this linear system yields the desired coefficients of the planned tiptrajectory. The open loop control torque leading to an exact tracking of the plannedtrajectory is then given by A4ez1 t + u2(t), which is substituted into (22) to obtainthe final form of control torque.


22/34


Figure 4. Actual output trajectory (dashedline) and desired trajectory (continuousline).

Figure 5. Output tracking error.

Figure 6. Feedback control torque. Figure 7. Actual tip deflection.

Experimental Results

We report here some experimental results obtained for the one-link flexible armdescribed in the above section . These results have been obtained with the followingvalues: tf = 7.36 s, yf = 1.57 m (values permitting comparison with the someexperimental results already obtained on the same testbed (Aoustin et al., 1994)).The applied control torque was obtained using (22), with the constant gains Kp =30, Kd = 2, corresponding to the poles: p1,2 = 63.13 120i, p3,4 = 3.81 40.45i, p5,6 = 0.11.8i. The obtained operational trajectory is given in Figure 4.Figure 5 shows the operational displacement tracking errors (emax = 2 cm). Onecan see a steady state error due to non modelled joint friction. The control torqueremains bounded and is given in Figure 6. In Figure 7 we can show that the tiposcillation nearly disappear when we reach the final position.

5.2. TRAJECTORY TRACKING: AN ALGEBRAIC APPROACH

We recall the method introduced in (Benosman and Le Vey, 2000b) on a one flex-ible mode model. This scheme is used for building a parametrization of the linear


23/34


differential operator represented by the dynamical equations of the model (24) inorder to solve the end-effector trajectory tracking problem. Parametrization hasto be understood in the same sense as one parametrizes curves and surfaces inR

n but it is done here for differential equations. Let us denote by D1, the operator

represented by the differential equation (24). We are looking for another differentialoperator D0 such that all its compatibility conditions are given by D1. This can bewritten: D1 D0 = 0 and there are no other compatibility conditions for D0. Inthat case, we say that D0 is a parametrization ofD1. IfD0 has more compatibilityconditions than the ones given by D1, then we get another operator D 1, that is,in some sense, greater than D1. It is shown in (Pommaret, 1994) that a systemadmits a parametrization if and only if it is controllable in the usual sense of linearsystems when in Kalman form. Consider our specific situation where the operatorD1 is described by Equation (25) (only two modes retained). The way one can geta parametrization goes as follows: first, consider this system without distinctionbetween input and output and built-up its adjoint operator by multiplying on the

left by the row vector (1, 2) and integration by parts:

D1:

M111 + M122 + M133 = 1,M211 + M222 + K12 = 2,M311 + M333 + K23 = 3,

1 = 4.

(32)

Now we compute the operator D0 giving the compatibility conditions of D1: usingthe third equation defining D1, we can write:

M114 + M122 + M133 = 1,

M214 + M222 + K12 = 2,M314 + M333 + K23 = 3,

(33)

so that, after elimination, we get the only compatibility condition giving D0:D0: A11 + A21 + A32 + A43 + A54 + A6(4)1 ++ A7(4)2 + A8(4)3 + A9(4)4 + A10(6)4 = 0, (34)

where the constants A1, . . . , A10 write as function of the model coefficients andare given in (Benosman and Le Vey, 2000c). Then, we compute D0: (qr , qe1,qe2, u), as the adjoint of

D0, by the same procedure of multiplying to the right by

the row vector and integration by parts:

D0

qr (t) = A1(t) + A2(2)(t) + A6(4)(t),qe1(t) = A3(2)(t) + A7(4)(t),qe2(t) = A4(2)(t) + A8(4)(t),u(t) = A5(2)(t) + A9(4)(t) + A10(6)(t).

(35)


24/34


One could verify that D0 is actually an effective parametrization ofD1, by check-ing the controllability of the initial system D1, using the usual Kalman test. Wesee that all the original variables are expressed as functions of one differentialparameter , that is: they are parametrized by this function. We underline at thislevel that, through this algebraic approach we obtain the same results holding forthe flatness scheme when dealing with linear controllable systems (see, e.g., (Fliesset al., 1998)). The parameter function (t) being considered as a flat output.

Application to Output Tracking

The controlled output is the linear position of the end-effector, given by

y(t) = Lqr (t) + v(L). (36)This gives in our specific two modes case:

y(t)

=Lqr (t)

+1(L)qe1(t)

+2(L)qe2(t). (37)

Then using (35):

y(t) = c1(t) + c2(2)(t) + c3(4)(t), (38)where: c1 = LA1, c2 = LA2 + 1(L)A3 + 2(L)A4, c3 = LA6 + 1(L)A7 +2(L)A8. Given a desired tip trajectory ydt(t), it is Possible to compute the corre-sponding parameter function d(t), by substituting ydt(t) for y in (38) and solvingfor (t). We know that our system associated to the y output is a non-minimumphase system (De Luca and Siciliano, 1989). Unfortunately the effect of non-minimum phase characteristic appears in (38) (since this equation admits an unsta-ble transient solution). Then keeping the complete solution (transient plus steadystate terms) will cause divergence of(t) and then ofu(t) (last equation of (35)). To

avoid this phenomena, we propose to keep just the steady state ds (t) term ofd(t)(computing by inversion of (38)), since this term is a particular solution of the dif-ferential equation with second member ydt(t) (i.e., substituting ds (t) for in (38)will automatically gives the desired trajectory ydt(t)). The only consequence of thiswill be the loose of the initial conditions fixed on (t) and its derivatives, but thiswill not disturb the initial behaviour ofuol(t), which is directly computed using thelast equation of (35) and the obtained parameter function ds (t) (see the followingsubsection).

Application

Consider a classical fifth order desired tip-trajectory:

yd(t) = yf

6 t

tf

5 15

ttf

4+ 10

ttf

3, (39)

then solving (38), leads to

d(t) = dt(t) + ds (t),


25/34


where dt(t) is the unstable transient part and

ds (t) =yf

c31t5f

360c1c3tf 360c22 tf + (720c22 720c1c3 60c1c2t2f)t+

+ 180c1c2tft2 + (10c21 t2f 120c1c2)t3 15c21tft4 + 6c21 t5. (40)Using (35) we can then directly compute the open-loop torque u1 as well as thejoint qrd(t) and elastic qe1d(t),qe2d(t) trajectories, corresponding to the desiredtip-motion.

uol(t) =yf

c21t5f

6A8c2tf 6A8c1tf + (A8t2fc1 12A8c2 + 12A9c1)

3A8tfc1t2 + 2A8c1t3

,

qrd(t)

=

60yf

t

5f

6tfc3

c

2

1

6tfc22

c

3

1 A1 +6tfc2

c

2

1

A2

6tf

c1

A3++

12 c3c21

c2t2f

c21+ 12c

22

c31

A1 +

t2f

c1 12c

22

c1

A2 + 12

A3

c1

t+

+

c2tf

c21A1 3

A2tf

c1

t2 +

+

t2f

6c1 2 c2

c1

A1 + 2 A2

c1

t3 + A1tf

4c1t4 + A1

10c1t5

,

qe1d(t) =60yfc21t

5f

6A4c2tf 6A5c1tf + (A4c1t2f 12A4c2)t 3A4c1tft2 +

+ 2A4c1t3,

qe2d(t) =60yfc21t

5f

6A6c2tf 6A7c1tf + (A6c1t2f 12A6c2)t 3A6c1tft2 +

+ 2A6c1t3

.

(41)

Experimental Results

We report here some experimental results obtained for the one-link flexible armdescribed above. Figure 11 shows the closed-loop torque for tf = 3 s, yf =1.58 m, with the gains Kpqr = 25, Kvq r = 2, corresponding to the poles: p1,2 =

62.98 118.61 i, p3,4 = 3.95 40.23 i, p5,6 = 0.09 1.67 i. It is clear thatthis torque is feasible without abrupt transient response. However, one can see asteady state effort, due to non modelled joint friction. Figure 8 displays the trackingerror which remains small (maximum value 2.5 cm), keeping in mind the fact thatthe practical system is somehow different from the theoretical model (i.e. higherfrequency modes, constant friction and coulomb friction have not been modelled).


26/34


Figure 8. End-effector tracking error. Figure 9. Actual tip-deflection.

Figure 10. Actual tip trajectory (dashedline) and desired trajectory (continuousline).

Figure 11. Feedback control torque.

The actual deflection of the end-effector is represented in Figure 9. We canverify that the oscillations are completely damped out at the end of the joint motion.The obtained tip-motion is compared to the desired one in Figure 10. The actualend-effector motion does not display any overshoot or high steady state error. Thesmooth feedback law is displayed in Figure 11.

6. The Planar Multi-Link Case

6.1. PLANAR MULTI-LINK NONLINEAR MODEL INVERSION

We now consider a planar multi-link manipulator as described in Figure 3. The

conventions adopted are those of the Sections 3 and 4.Since we restrict our modelling investigations for control to the standard planarmodel (SM), the field of deformation of any link Bi has just one transverse nonnull component (along E2,i ), i.e.:

dei (X1) = vi (X1)E2,i , (42)


27/34


that we discretize through the modal expansion:

vi (X1)

=

mi

=1 i,(X1)qi, = i qei (43)with i the row vectors of shape functions (for detailed discussion about shapefunctions choice see (Meirovitch, 1989; De Luca and Siciliano, 1991)) and qei thecolumn vector of elastic coordinates. Applying the modelling process defined inSections 24 with (42) and (43) as starting point, gives the dynamic model in theLagrangian or NewtonEuler formalism. We point out at this level, that we do nottake into account the elastic damping terms, this yielding more robustness to theobtained results concerning the tip oscillation cancellation.

6.2. PATH TRACKING PROBLEM

The goal is to reproduce a desired path (xd(t),yd(t)) in the manipulator operationalspace. First we define some virtual joint variables, called elastic angles (DeLuca and Siciliano, 1989), given by the vectors:

Y = (y1, . . . , yn)T = (qr1, qr2, . . . , qrn )T +

+

tan1

v1(L1)

L1

, . . . , tan1

vn(Ln)

Ln

T, (44)

where vi , i {1, . . . , n}, are given by Equation (43). The vector Y coordinatescorrespond to the tips displacements. For small elastic motion, this vector writes:

Y = qr + qe, (45)where

=

1,1

L1. . .

1,m1

L10 . . . . . . 0

0 . . . 02,1

L2. . .

2,m2

L20 . . . 0

.... . .

. . . . . .

0 . . . . . . 0n,1

Ln. . .

n,mn

Ln

.

Any operational motion is then directly related to elastic angular motion, throughthe geometric model of a virtual arm, made of rigid links relating each joint toits tip flexible link. Thus we obtain the desired vector Yd, corresponding to aprescribed operational path (xd(t),yd(t)) by inversion of this geometric model.Using Equations (21) and (45), we can transform the system dynamic equationsinto functions of the desired tips motion Yd and elastic coordinates vector qe. More


28/34


specifically, the dynamics can be rewritten separately for the rigid and flexible partas follows:

Mrr Yd

+(Mre

Mrr )

qe

+ Cr (Yd, qe, Yd, qe) = u, (46)MTre Yd + Mre MTre qe + Ce(Yd, qe, Yd, qe) + Keeqe = 0. (47)To compute the feed-forward torque from Equation (46), one needs the desiredelastic vector qed which must satisfy Equation (47). Unfortunately, the system ofordinary differential equations (47), with classical Cauchy data qed(to) = qed(to)= 0 is definitely unstable (De Luca and Siciliano, 1989). Another choice of theelastic constraints, can simply overcome this problem. In fact, instead of startingthe integration from the natural constraints (vanishing positions and velocities), weformulate the problem as a two-point boundary-value problem (TPBVP), stated as:

MTre Yd +

Mre MTre

qe +

Ce(Yd, qe, Yd, qe) + Keeqe = 0,

qe(to) = 0,qe(tf) = 0.

(48)

Doing so, we can force the elastic coordinates to remain bounded during all thetime motion (see (Zhao and Chen, 1998) for a linear boundary value problem for-mulation, involving boundary constraints on both displacements qe and velocitiesqe, yielding to two sets of initial value decoupled problems, forward and backwardproblems). For existence of bounded solutions for the TPBVP (48), we refer theinterested reader to (Benosman and Le Vey, 2001c). We underline at this levelthat , contrary to the presently available technique (Zhao and Chen, 1998), theformulation presented above, permits to compute causal bounded elastic trajectoryvector without the restrictive hypothesis of initial equilibrium point hyperbolicity,nor on the desired end-effector motion amplitude (we refer to (Benosman and Le

Vey, 2001b) for the presentation of this method in a general setting case, and acomparison with the non-causal available schemes). After the resolution of (48),one can obtain the desired feed-forward control law using (46). The final controllerstructure (22), is obtained by adding to the feed-forward torque obtained, jointfeedback terms, avoiding then all the problems due to elastic measure feedback(spillover problems (Balas, 1982)). For the matrices gain selection, we have usedthe gain scheduling technique (Khalil, 1992). At each instant in [t0, tf], the gainhas been computed such that the spectrum of the linearized system is localized inthe negative half plan (this ensures local stability of the nonlinear dynamic alongthe desired trajectories, the proof following standard Lyapunov arguments (Khalil,1992)).

6.3. NUMERICAL RESULTS FOR A TWO-LINK FLEXIBLE MANIPULATOR

We have tested the proposed approach, through simulation of the two-link flexiblemanipulator to our disposal at IRCCyN laboratory. The arm is characterised by themechanical properties summarised in Table I.


29/34


Table I. Mechanical properties of the arm

Properties First link Second link

Length: L (m) 1.005 0.52

Mass density: (kg/m) 2.032 0.706

Flexural rigidity: EI (Nm2) 47.25 1.749

Tip mass: mtip (kg) 7.66 0.51

Tip inertia: Jtip (kg m2) 171e3 623e7Hub inertia: Jhub (kg m2) 1.8e3 225e6

Figure 12. Desired first link tip-trajectory.Figure 13. Desired second link tip-trajecto-ry.

The desired circular operational path shown in Figure 17 is defined by:

xd(t) = 0.2cos(d(t)) + 0.5,yd(t) = 0.2sin(d(t)) + 0.5,

d(t) = f

6

t

tf

5 15

t

tf

4+ 10

t

tf

3,

f = 2 rad, tf = 10 s,

where d is the desired angle of polar coordinates.The corresponding desired tip motion, are presented in Figures 12 and 13.We recall that we have not any constraint concerning the choice of the pre-

scribed operational motion, except the classical velocity and acceleration con-

straints (simply satisfied by a fifth-order polynomial trajectory). The proceduredescribed in Section 6.2 has been run to compute causal bounded elastic trajecto-ries. To solve the nonlinear TPBVP we have chosen the collocation points codesoftware colnew (Ascher et al., 1981). Figures 15 and 16 display the feed-forwardcontrol torques obtained. They are bounded, smooth with zero initial and finalvalues (causal control). The associated feedback gain is displayed in Figure 14.


30/34


Figure 14. Feedback scheduled gain. Figure 15. Open-loop torqueforthe first joint.

Figure 16. Open-loop torque for the secondjoint.

Figure 17. Desired (continuous line) andactual (dashed line) paths.

The results obtained by the application of the feedback control law, given by Equa-tion (22), to the nominal plan, are given below. Figures 1719 convey informationabout the tracking errors, which remain small, with maximum values of about:3.2 103 m for X axes (Figure 18) and 3.5 103 for Y displacements (Figure 19).In Figure 17 one can see clearly the good matching between the desired and the ob-tained paths. The tip deflections given in Figures 20 and 21 show a smooth responseand are damped out at the end of the desired motion (this is partly due to the finalconstraints imposed to the elastic variables in Equation (48). The feedback laws aredisplayed in Figures 22 and 23, they show nice performance, without exceeding themaximum nominal torques of our system ( u1nom = 3.5 Nm, u2nom = 1.05 Nm).

7. Conclusion

The purpose of this paper is to give some recent theoretical, numerical and ex-perimental results obtained in the field of modelling and control of flexible linkmanipulators. The proposed modelling process is based on the modal-floating-frame approach which is the most adapted to the control. It is based on the gener-


31/34


Figure 18. X-position tracking error. Figure 19. Y-position tracking error.

Figure 20. First link tip-deflection. Figure 21. Second link tip-deflection.

Figure 22. Closed-loop first controltorque.

Figure 23. Closed-loop second controltorque.

alised NewtonEuler model of flexible manipulators. Unlike the standard uses, thismodel is deduced from a set of equations discovered by H. Poincar one hundredyears ago. These equations realise a generalisation of Lagrange equations on anon commutative Lie group. Following this way, we inscribed the floating frameapproach in the framework of the differential geometry. Starting from the gener-alised NewtonEuler model, we can design fast recursive algorithms for control


32/34


and simulations. Moreover, the NewtonEuler model is a good starting point ofthe Lagrangian modelling. This aspect has been detailed through the assemblingoperation. Until today the modal-floating-frame approach is restricted to the fieldof small deformations. This is due to the fact that the kinematics of the link de-formations are always linear. In this paper we recall a nonlinear generalization ofthe EulerBernoulli kinematics. These kinematics, once inserted in the modellingprocess allowed us to obtain a new model called MDM, capable of describing theevolution of a flexible manipulator enduring moderate but finite deformations.

Based on the above mentioned models, we have proposed different control lawsdealing with the end-effector trajectory control problem. First we have consideredthe one-link case and have proposed two approaches synthesized on linear models.One method is based on causal stable inversion of linear non-minimum phase mod-els, through output trajectory planning. The other one deals with parametrization oflinear differential operators. We have eventually treated the operational path track-ing problem for planar multi-link flexible manipulators. The method proposed is

based on causal stable inversion over a bounded time interval of this non-minimumphase nonlinear system through a two-point boundary value problem formulation.Numerical as well as practical tests have been reported to display the efficiency ofthese laws.

References

Agrawal, O. P. and Shabana, A.: 1985, Dynamical analysis of multibody systems systems usingcomponent modes, Comput. Struct. 21, 13031312.

Aoustin, Y., Chevallereau, C., Glumineau, A., and Moog, C. H.: 1994, Experimental results for theend-effector control of a single flexible robotic arm, IEEE Trans. Control Systems Technology 2,

371381.Arnold, V. I.: 1988, Mathematical Methods in Classical Mechanics, Springer, New York.Ascher, U., Christiansen J., and Russell R. D.: 1981, Collocation software for boundary-value ODEs,

ACM Trans. Math. Software 7(2), 209222.Balas, M. J.: 1982, Trends in large space structure control theory: Fondest hopes, wildest dreams,

IEEE Trans. Automat. Control 27(3), 522535.Bayo, E.: 1987, A finite-element approach to control the end-point motion of a single link flexible

robot, J. Robotic Systems 4, 6375.Bayo, E.: 1988, Computed torque for the position control of open-chain flexible robots, in: IEEE

Internat. Conf. on Robotics and Automation, Philadelphia, pp. 316321.Bayo, E. and Moulin, H.: 1989, An efficient computation of the inverse dynamics of flexible

manipulators in the time domain, in: IEEE Internat. Conf. on Robotics and Automation, pp.710715.

Benosman, M. and Le Vey, G.: 2000a, End-effector motion planning for one-link flexible robot, in:IFAC, Symposium on Robot Control, Vol. 2, pp. 561566.

Benosman, M. and Le Vey, G.: 2000b, Exact articular trajectory tracking for a one-link flexiblemanipulator: An approach through the parametrization of differential operators, in: 31th Internat.Symposium on Robotics (ISR2000), Montral, Canada, pp. 493498.

Benosman, M. and Le Vey, G.: 2000c, Accurate trajectory tracking of flexible arm end-point, in:IFAC Symposium on Robot Control, Vol. 2, pp. 569572.


33/34


Benosman, M. and Le Vey, G.: 2001a, Stable inversion of SISO non-minimum phase linear dy-namic through output planning: An experimental application to the one-link flexible manipulator,Technical Report 01/3/AUTO, Ecole des Mines de Nantes.

Benosman, M. and Le Vey, G.: 2001b, Output trajectory tracking for a particular class of nonlinear

non-minimum phase systems, Technical Report 01/6/AUTO, Ecole des Mines de Nantes.Benosman, M. and Le Vey, G.: 2001c, Model inversion for a particular class of nonlinear non-minimum phase systems: An application to the two-link flexible manipulator, in: 40th IEEECDC, pp. 11741180.

Book, W. J.: 1984, Recursive Lagrangian dynamics of flexible manipulators arms, Internat. J.Robotics Res. 3, 87101.

Boyer, F.: 1994, Contribution la modlisation et la commande dynamique des robots flexibles,PhD Thesis, Paris 6 University.

Boyer F. and Coiffet, P.: 1996a, Generalisation of NewtonEuler model for flexible manipulators,J. Robotic Systems 13(1), 1124.

Boyer, F. and Coiffet, P.: 1996b, Symbolic modelling of a flexible manipulator via assembling of itsgeneralized NewtonEuler model, J. Mechanism Machine Theory 31, 4556.

Boyer, F. and Khalil, W.: 1998, An efficient calculation of flexible manipulator inverse dynamics,Internat. J. Robotics Res. 17(3), 282293.

Boyer, F. and Khalil, W.: 1999, Kinematic model of a multi-beam structure undergoing large elasticdisplacement and rotations. Part one: Model of an isolated beam, J. Mechanism Machine Theory34, 205222.

Boyer, F., Glandais, N., and Khalil, W.: 2002, Flexible multibody dynamics based on a nonlinearEulerBernoulli kinematics, Internat. J. Numer. Methods Engrg. 54, 2759.

Boyer, F. and Primault, D.: 2001a, Poincars equations and multibody dynamics, part I: Mathemat-ical fundamentals, Technical Report 01/1/AUTO, Ecole des Mines de Nantes.

Boyer, F. and Primault, D.: 2001b, Poincares equations and multibody dynamics, part II: Applicationto multibody dynamics, Technical Report 01/2/AUTO, Ecole des Mines de Nantes.

Brokett, R. W., Stokes, A., and Park, F.: 1993, A geometrical formulation of the dynamical equationsdescribing kinematic chains, IEEE Internat. Conf. on Robotics and Automation, pp. 637641.

Canavin, J. R. and Likins, P. W.: 1977, Floating reference frames for flexible spacecrafts, J. Space-craft14(12), 724732.

Cardona, A. and Gradin, M.: 1988, A beam finite element nonlinear theory with finite rotations,Internat. J. Numer. Methods Engrg. 26, 24032438.Christoforoua, E. G. and Damaren, C. J.: 2000, The control of flexible-link robots manipulating large

payloads: Theory and experiments, J. Robotic Systems 17(5), 255271.Damaren, C. J.: 1995, Passivity analysis for flexible multilink space manipulators, J. Guidance

Control Dynamics 18(2), 272279.DEleuterio, G. M. T.: 1992, Dynamics of an elastic multibody chain. Part C-recusive dynamics,

J. Dyn. Stability Systems 7(2), 6189.De Luca, A. and Siciliano, B.: 1989, Trajectory control of a nonlinear one link flexible arm, Internat.

J. Control 50(5), 16991715.De Luca, A. and Siciliano, B.: 1991, Closed-form dynamic model of planar multilink lightweight

robots, IEEE Trans. Systems Man Cybernet.De Luca, A., Panzieri, S., and Ulivi, G.: 1998, Stable inversion control for flexible link manipulators,

in: IEEE Internat. Conf. of Robotics and Automation, Leuven, Belgium, pp. 799801.

Fliess, M., Sira-Ramirez, H., and Marquez, R.: 1998, Perspectives in Control, Theory and Appli-cations, Chapter: Control design methodologies. Regulation of non-minimum phase outputs:A flatness based approach, pp. 143163.

Hastings, G. G. and Book, W. J.: 1986, Verification of linear dynamic model for flexible roboticmanipulators, in: IEEE Internat. Conf. on Robotics and Automation, New York, pp. 10241029.


34/34


Hughes, P. C. and Sincarsin, G. B.: 1989, Dynamics of elastic multibody chains: Part B Globaldynamics, Dyn. Stability Systems 4, 227243.

Khalil, H.: 1992, Nonlinear Linear Systems, Macmillan, New York.Marsden, J. E. and Ratiu, T. S.: 1999, Introduction to Mechanics and Symmetry, Springer, New York.

Meirovitch, L.: 1989, Dynamics and Control of Structures, Wiley, New York.Paden B., Chen, D., Ledesma, R., and Bayo, E.: 1993, Exponentially stable tracking control formulitjoint flexible-link manipulators, J. Guidance Control Dynamics 115, 5359.

Poincar, H.: 1901, Sur une forme nouvelle des quations de la mcanique, C. R. Acad. Sci. Paris132, 369371.

Pommaret, J. F.: 1994, Partial Differential Equations and Group Theory, Mathematics and itsApplications, Vol. 293, Kluwer Academic, Dordrecht.

Schmitz, E.: 1985, Experiments on the end-point position control of a very flexible one-linkmanipulator, PhD Thesis, Standford University.

Shabana, A. A.: 1989, Dynamics of Multibody Systems, Wiley, New York.Sharf, I.: 1996, Geometrically non-linear beam element for dynamics of multibody systems, Internat.

J. Numer. Methods Engrg. 39, 763786.Sharf, I. and Damaren, C.: 1992, Simulation of flexible-link manipulators: Basis functions and non-

linear terms in the motion equations, in: IEEE Internat. Conf. on Robotics and Automation, Nice,

France, pp. 19561962.Simo, J. C. and Vu-Quoc L.: 1988, On the dynamics in space of rods undergoing large motions

A geometrically exact approach, Comput. Methods Appl. Mech. Engrg. 66, 125161.Spivak, M.: 1976, Differential Geometry, Perish, Houston.Theodor, R. J. and Ghosal, A.: 1995, Comparison of the assumed modes and finite elements models

for flexible multilink manipulators, Internat. J. Robotics Res. 14(2), 91111.Wehage, R. A. and Haug, E. J.: 1982, Generalized coordinate partitioning for dimension reduction

in analysis of constrained dynamic systems, J. Mech. Design. 104.Zhao, H. and Chen, D.: 1998, Tip trajectory tracking for multilink flexible manipulators using stable

inversion, J. Guidance Control Dynamics 21(2), 314320.

flexible manipulators

Documents