continuous newton-euler algorithms for

8/3/2019 Continuous Newton-Euler Algorithms For

1/6

Continuous Newton-Euler Algorithms for

Geometrically Exact Flexible Beams

Georges Le Vey

IRCCyN-UMR CNRS 6597 and Ecole des Mines de Nantes4, rue A. Kastler, 44307 NANTES Cedex 1 - FRANCE

Email: [email protected]

Abstract A Newton-Euler formalism is derived for geometri-cally exact Cosserat beam theory in a purely deductive manner,thanks to an analogy with optimal control theory 1. The methodrelies upon a joint use of Gauss least constraint principle, Appellsequations and optimal control theory, that was used in a previouswork for the classical case of discrete Newton-Euler backwardand forward recursions for multibody systems. Motivating ap-plications are hyperedundant manipulators or biomimetic robotssuch as eel-robots, undergoing large displacements.

I. INTRODUCTION

The Newton-Euler approach to obtain dynamical equations

for complex mechanical systems has been for a long time

widely used for modelling in multibody systems and robotics

(see [17] and references therein and classical references on

the subject [3], [14], [24]), as an efficient alternative to

Lagrange approach. This approach does not make explicit the

dynamical equations but, instead, relies upon considering the

relative positions, velocities and accelerations of each body

with respect to its neighbours, in a recursive manner, leading to

low complexity algorithms. Two questions are then addressed :

either the joint accelerations are known and one searches

for the corresponding joint torques (backward algorithm, for

control objectives) or the converse ( forward algorithm, for

simulation purposes). These two questions are addressed here

in the case of a Cosserat continuous beam theory, on the

illustrative example of a Kirchoff, inextensible model, albeit

not limiting the generality of the method itself. The obtained

results as well as the purely deductive way of getting them

are new, to the best knowledge of the author.

Flexible multibody systems have been the object of exten-

sive research, in modelling, identification and control, during

the past decades. Thus the renowned efficiency of recursive

Newton-Euler algorithms for rigid multibody systems and

robotics manipulators has motivated their extension whenflexibilities should not be neglected ( [4][6], [10], [16], [18],

[27]). All these works differ on a number of significant points

from the present work : first, they deal with manipulators

that are actuated at a finite number of joints, while taking

into account flexibility of the links. Here, the purpose is

to derive algorithms that compute accelerations or torques

as functions of a continuous space variable, i.e. densities.

One motivation for such developments comes from some

of the targeted applications : hyper-redundant manipulators

1This article is the development of a short preliminary note [22]

and continuous biomimetics robots that can undergo large

displacements, making continuous models attractive when one

wants e.g. to simulate dynamical evolutions in a realistic

fashion. Notice that [23] argued that continuous beam models

are quite adequate to represent a living eel body. As a conse-

quence : whereas in most of existing works a small strain and

displacements assumption is made, hence making approximate

first order, linear elastic models sufficient, and allowing for

separating rigid body motions from elastic ones, the modelling

approach that is used in the present work is geometrically

exact in the sense of [26] : an important consequence is that

no approximation is made from the very beginning, while

deriving the algorithms. Only the unavoidable time and space

discretizations at the very end of the process lead to approx-

imations for numerical integration. Nevertheless, most of the

modelling approaches for very flexible biomimetical robots

generally use multibody models with a certain -possibly large-

number of rigid links [15]. Thus the use of continuous models

still has to be developed and the results of the present work can

then be seen as a building block towards modelling complex

flexible mechanical systems undergoing large displacements,in the spirit of Newton-Euler formalism.

On another hand, analogy of recursive Newton-Euler equa-

tions for multibody systems with optimal filtering [25] and

discrete-time optimal control [13] has been evidenced. This

interesting analogy was not really used as a principle for

deriving equations but simply pointed out while using the trick

of separating linear from nonlinear effects in [13]. A slight

improvement of this approach was given in [21] for the usual

case of Newton-Euler formalism for multibody systems, in

the sense that the Newton-Euler recursions were derived in

a purely deductive way. In the present work, this approach

is extended in order to obtain a Newton-Euler formalismfor continuous beam theory, where recursions are replaced

by ordinary differential equations in the continuous space

variable, as one can expect. It is worth noting that only such

a deductive approach can give a general means for deriving

analogous Newton-Euler equations, that would be otherwise

very hard to exhibit. The concrete motivation for deriving

a formalism in the continuous case comes from the recent

derivation of a continuous 3D computed torque algorithm for

a swimming eel-robot [7], where the analogy with (backward)

Newton-Euler has been pointed out but derivation of a for-

ward analogous is far from being evident. The present work

1-4244-0259-X/06/$20.00 2006 IEEE4163

Proceedings of the 2006 IEEE/RSJ

International Conference on Intelligent Robots and Systems

October 9 - 15, 2006, Beijing, China


2/6

makes derivation of both continuous algorithms systematic as

was done in the classical case, using Gauss least constraint

principle along the line evidenced by P. Appell [1]. It is worth

mentionning that all the ingredients on which the present work

rely upon have been known for a long time but that their

conjunction leads to derivations from first principles that are

new. The presentation is organized as follows : after having

fixed the main notations in section II, section III describes thegeometric model of the beam, together with a velocity and

acceleration analysis. Then, section IV, after having quickly

recalled the essence of Appells method for motion equations

derivation, exhibits the acceleration energy (Gibbs function) of

the considered system, leading to a quadratic function of the

acceleration to be minimized. In section V a related optimal

control problem is posed and solved that straightforwardly

leads, in section VI, to the continuous backward and forward

Newton-Euler algorithms. Eventually, some conclusions and

perspectives are drawn.

II. NOTATIONS

In the sequel, dots over some quantity will indicate dif-

ferentiation with respect to time t and primes, differentiationwith respect to the space variable X. The beam is supposedto have its length normed to one so that X [0, 1]. Thedependence on time and space variables will not, as a rule, be

made explicit in the curse of the paper. For a vector y R3,y is the antisymmetric matrix such that z R3, yz = y zand is the usual vector product. The identity matrix ofR3

is denoted I3 ; thus some care is in order for not confusingwith the inertia of one section, denoted I.

III. KINEMATICAL MODEL OF A CONTINUOUS COSSERAT

BEAM

The chosen geometry of the three dimensional model for

a geometrically exact continuous Cosserat beam is given for

granted and is recalled here only for the sake of fixing

notations. One can find details about its derivation in [7]. The

fundamental idea of the Cosserat theory [11] is to consider

a beam as a set of stacked microsolids, named hereafter

sections, labelled by the material abcissa, X, along someabstract neutral line (the line of centroids). To each section is

rigidly attached an orthonormal frame. In the case of an eel-

robot e.g. this line would represent the spine, this being in very

accordance with observations made by biologists on living

eels [23]. Also, it is assumed that there is neither shearing(Kirchoff beams hypothesis) nor extensibility of the body but

this does not prevent the generality of the method. Every cross

section, labelled X, of the beam has configuration describedby r(X) (i.e. an element ofR3, see Fig. 1), the position of itsmass center in a reference frame, and R(X), its attitude, underthe appearance of a rotation (i.e. an element of the rotation

group SO(3)) that transforms one section at rest into itself inthe deformed configuration. Thus the configuration space of

the beam is the principal bundle R3 SO(3).The kinematics of the beam is described first by a twist-

curvature tensor field . Actually is the Darboux vector as

X=1X=0 X

r(X)

R(X)

t (X)1

Fig. 1. Configuration variables of the beam model

defined in [19], so that the torsion is the material torsion of the

beam and not the geometric torsion of the centroid line. isdefined by : R/X= R

= R. One has also the constraintimposed by the inextensibility assumption : r/X= r

= t1where (see Fig. 1) t1 is the unit tangent vector to the neutralline of the beam, R is the rotation matrix mapping the X

mobile basis before deformation onto that after. Summarizingthese considerations, from now on, a geometric model is in

force : R

= R

r

= t1(1)

A velocity and acceleration analysis can then be done in the

usual way. A first time differentiation gives the kinematical

model for velocities :

= k + k

r

= t1(2)

with the axial vector corresponding to the matrix =

RRT

, and k, the axial vector corresponding to the matrixk = RRT. A second time differentiation gives the kinematicmodel for accelerations :

= k + k + k

r

= t1 + ( t1)(3)

For the ease of subsequent analogy with optimal control, it is

worth rewriting this system in matrix form as :

r

=

0 |t1|

0 | k

r

+

0

k

+

( t1)

k

(4)

Thus the system of equations for the accelerations is avariable coefficients, linear inhomogeneous differential one,

with independent variable X, It is imposed by the modellinghypothesis, i.e. it represents chosen design constraints : such a

choice is up to the designer hence any other geometric model

of a beam would be convenient too.

IV. GAUSS LEAST CONSTRAINT PRINCIPLE AND

GIBBS-APPELL APPROACH TO M OTION EQUATIONS

DERIVATION

The approach (also known as Gibbs-Appell) of P. Ap-

pell [1], [2] to deriving motion equations of a mechanical

4164


3/6

system is quickly summarized here. It is based on the consid-

eration of an acceleration energy [12] (or Gibbs function)

instead of kinetic energy that is used for deriving Lagrange

equations. Define S, this acceleration energy of the mechanicalsystem under consideration, the following way : let (P, q)be the acceleration of particle with mass dmP located at P.Then : S =

12

|(P, q)|2dmP where the integral extends

to the whole system, and q is the configuration parameter(generalized coordinate). Let Q be the vector of applied effortsand E the analytical expression of the constraint [2] in thesense of Gauss principle with E= SQTq. Then P. Appellhas shown that the motion equations take the form :

S

q= Q (5)

But, much more important in the present situation, P. Appell

observed that this acceleration energy has tight connections

with Gauss least constraint principle. More precisely, P. Appell

has shown [1] that the motion equations are those obtained

when searching for the minimum of the analytical expression

of the constraint, E, a quadratic function of q.For the Cosserat beam model, the generalized coordinates are

r(X), R(X),X [0, 1] from which the corresponding vectorof accelerations is : (rT, T)T. The acceleration energy ofone section at X is straightforwardly computed, thanks to aKoenigs theorem for the acceleration energy [2], quite similar

to the well-known Koenigs theorem for kinetic energies, to

be :

S(X) =1

2

rTT

AI3 00 I

r

+

0

( (I))

Tr

(6)

where for fixed X, is the density of the beam, A is the

section area and I its inertia. As for the applied efforts, itis necessary to consider the nonvanishing work of all active

efforts. External efforts (forces and torques) are summarized in

the vector :fText|c

Text

T. These applied efforts are not detailed

here as it is outside the scope of the present work but gravity

as well as a fluid model for a swimming robot e.g. can be taken

into account. Consider now the torque atX, denoted , and thecorresponding generalized coordinate, denoted k, the curvaturedensity at X : either k or can be considered as an unknowncontrol input, introducing external energy to the system. Thus,

remembering the above definition ofE, the contribution ofk to the constraint is Tk. Eventually, gathering results, theexpression of the constraint at section

Xis :

E(X) = S(X) Tk (7)

Define the analytical expression of the constraint on the whole

body as the sum : Eb =10E(X)dX. Gauss least constraint

principle stipulates that for obtaining motions equations, one

has to search for the minimum of this quadratic form (in rand ), subject to the dynamical (in space) equation (4).The supplementary fact now is that there remain unknown

variables : either k or , each playing the role of controlinput when the other set is given as data. Each situation

gives rise to the backward or forward Newton-Euler algorithm,

that is obtained in the next section as the solution of an

evolutionary quadratic optimal control problem.

V. A NON HOMOGENEOUS, SINGULAR, VARIABLE

COEFFICIENTS, OPTIMAL CONTROL PROBLEM

The conclusions of section IV, considering (6)-(7) together

with (4), suggest considering the following optimal control

problem2

. Firstly, let :

J(X,U) =

xfx0

(1

2XTX + bTX + cTU)dX (8)

be some quadratic functional of some given state X andcontrol U.

Problem

Solve :

minUJ(X,U) (9)

under the evolutionary constraint:

X

= FX +GU + h (10)

All the quantities either in the functional J or the dynamicalequation are functions of the independent space variable Xthus it is a variable coefficients problem. It is apparent that,

because the control appears linearly in the functional, the

optimization problem to be solved is a singular one [9].

Although it is a rather routine task, detailed derivation will

be given for the first order necessary conditions, because of

this singularity and also because of the inhomogeneous part

in the dynamical equations and this not usually exposed in

textbooks, to the authors knowledge.

It is usual, introducing Lagrange multiplier , to define thefollowing hamiltonian for such a problem :

H=1

2XTX + bTX + cTU + T(FX +GU + h) (11)

A first remark is that H does not depend explicitely on time.Thus it will remain constant along optimal trajectories. Now,

the first order necessary conditions for optimality [9] write :

X

= H

= FX +GU + h

= HX

T= FT X b

0 = HU

T= c+GT

(12)

The last condition (H

U = 0) does not give an expressionfor the optimal control, making it a singular problem ( [9],

ch. 8). In fact, in the setting of algebraic theory of differ-

ential equations, the above first order necessary conditions

for optimality constitute a differential system of equations in

(X, ) that is not formally integrable3 because of the presence

2All considerations in this paragraph are at a formal level, with no analyticalconsiderations, hence no mention is made e.g. of the spaces where the differentobjects live

3This means that the set of equations is unsufficient to determine thesolution through a Taylor power series, as conditions on successive derivativesare not explicit in the system and can appear after subsequent differentiations

4165


4/6

of a zero order equation (the last one). In the language of

DAE [8] (Differential Algebraic Equations), one would say

that the differential system is higher index and more precisely

an index two system. The usual way of dealing with this is

to compute successive derivatives of HU with respect to the

independent variable, until the control U explicitely appears,in order that, after substitution, the system will actually be

formally integrable.Starting thus from : Hu = G

T+ c = 0, a first differentiationgives :

H

u = GT

+GT

+ c

= GT

GTFTGTX GTb+ c

= (GT

GTFT)GTX GTb+ c

= G1G2X +G3

(13)

with G1 = GT

GTFT, G2 = GT, G3 = c

GTb. As Udoes not appear in this expression, another differentiation has

to be taken (that is why the system is index two) :

H

u=G

1+G1

G

2X G2X

+G

3

=(G

1 G1FT) (G1 +G

2 +G2F)X

+(G

3 G1bG2h) GTGU = 0

(14)

upon using the state and adjoint equations and collecting

terms. From this, the optimal control writes :

U = (GTG)1(K1+K2X +K3) (15)

where the intermediate quantities K1,K2, K3 are defined as :

K1 = G

1 G1FT (16)

K2=(G1 +G

2 +G2F) (17)

K3=G

3 G1bG2h (18)

The results so far obtained can be summarized now as :H =

Hu = GT+ c = 0

H

u = G1G2X +G3

(19)

This constitutes a system of equations in (X, ) that definethe locus of possible singular arcs in the space of(X, ) andthe so far obtained expression (15) for the optimal control U

is valid on such a singular arc, being a constant traducing

the fact that H is constant on the optimal trajectories.Now, one popular method of solution is the sweep method(see [9]). It is founded on the fact that, as the state and adjoint

equations are linear (and inhomogeneous here), we can seek

as an affine function of the state X : = SX + , whereS and have to be determined by some means, in orderthat the optimality system can be integrated. More precisely,

adding the preceding zero order equation to the original

system makes it nonformally integrable so that new equations

have to be found that will make the resulting system formally

integrable. It is worth noting that searching an expression for

, that is affine in X, simply amounts to finding an affine

feedback law.

Thus, starting from = SX + , differentiation and suitablesubstitutions lead to :

=S

X + SX

+

=S

X + S(FX +GU + h) +

=S

X + SFX + SG(GTG)1(K1+K2X +K3)

+Sh+

(20)

Using the adjoint equation (second equation in (12)), this is

equal to : FT X b. From this it is deduced that :

(S

+ SF+ FTS+ SG(GTG)1(K1S +K2) + )X

+(

+ Sh+ FT+ SG(GTG)1(K1+K3) + b) = 0(21)

As this equality must hold for any X (i.e. it must be anidentity), we get the following system of two differential

equations (for S and ) :

S

+ SF+ FTS+ SG(GTG)1(K1S +K2) + =0

+ Sh+ FT+ SG(GTG)1(K1+K3) + b=0

(22)which can be rewritten as :

S

+SG(GTG)1K1S+ S(F+G(GTG)1K2)

+FTS + = 0

+(FT + SG(GTG)1K1)+ Sh+SG(GTG)1K3 + b = 0

(23)

The equation for S is clearly a Riccati equation, as it isquadratic in S.In conclusion, to formally solve the optimal control problem

given at the beginning we proceed along the following steps :

1) Given ,b ,c,F,G,h

2) solve equation (23) for S and 3) solve the state equation, i.e. the first equation of sys-

tem (12) for X while simultaneously computing theoptimal control with equation (15)

The next section makes use of these four steps to derive the

continuous forward algorithm for the considered beam model.

VI. THE NEWTON-EULER BACKWARD AND FORWARD

ALGORITHMS

It is an easy task now, by identifying the data in section IV

and those in section V, to write down the backward and

forward algorithms for the continuous Newton-Euler model.

Comparing (9) to (6)-(7) and (10) to (4), respectively, and

looking at (12), let :

X =

r

; =

nM

;U = k;F=

0|t10| k

;G =

0I3

;

h =

( t1)

k

; =

1| 00 |2

;

b =

fext

cext + ( (I))

; c = ;

1 = AI3;2 = I(24)

With these definitions, some expressions are very simple :

definition ofG implies e.g. that (GTG)1 = 12 .

4166


5/6

S

+S

0 | 0

21(t

1 + kt1)|21(k

+ kk)

S+ S

1

1

1| 0

211t1|2

1

2 k

+

0 | 0

t1|k

S+

1| 00 |2

= 0

+

0 | 0

t1|k

+ S

0 | 0

12 (t

1 + kt1)|12 (k

+ kk)

+12 S

0

+ (cext (I))

t1fext + k(cext (I))

+

fext

cext + ( (I))

= 0

(25)

A. Backward algorithm

In the language of Newton-Euler formalism for multibody

systems, the backward algorithm takes the accelerations as

inputs and aims at giving the necessary torques as outputs.

Firstly, writing down the optimality condition of the hamilto-

nian H (11) with respect to the control (third equation of (12))gives : + M = 0, as expected, i.e. the adjoint state, aLagrange multiplier, gives the necessary efforts. Getting the

searched efforts thus simply amounts to solving the adjoint

equation of the state equation (4), i.e., using the secondequation in (12) :

nM

=

0 |0

t1|k

nM

AI3 0

0 I

r

fext

cext + ( (I))

(26)

One concludes that (26) is the backward algorithm for the

considered continuous beam model. Notice that (26) gives

exactly the continuous computed torque (i.e. backward) al-

gorithm as given in [7] when modelling a 3D eel-robot while

taking into account a simple Morison model for the fluid-

structure interaction (see [20] for explicit computations). Itwas obtained in [7] through the use of the principle of virtual

work. As the backward algorithm is used for control purposes,

r and are given as known control inputs so that, fixing e.g.n and M at one end allows for integrating (26) for X]0, 1[,i.e. one computes this way the distributed efforts along the

beam and this, for each time t. As for boundary conditions,for a beam free at both ends, the wrenches (n,M) have tovanish there. For a manipulator with fixed basis and one free

end, one would impose a vanishing wrench at this end and

vanishing accelerations at the basis.

B. Forward algorithm

The forward algorithm is known, when dealing with

classical serial multibody systems with usual methods, to

be significantly more difficult to obtain than the backward

one and to necessitate one recursion more. In the continuous

case, it would be a really difficult and huge task to obtain an

analogous algorithm when proceeding along the same path.

It is in that respect that the analogy with optimal control

shows real powerfulness, making these derivations systematic

and straightforward, as one can see hereafter. Recall that the

forward algorithm is to compute joint accelerations given

joint torques. In the present continuous case, one will in fact

compute accelerations as functions of the space variable X,given distributed torques for X ]0, 1[. Thus, following thesteps enumerated at the end of section V, this algorithm is

as follows, being understood that it can be included, when

necessary, in a time loop for a dynamical analysis, i.e. each

of the following steps is valid for all t :

1) Given ,A(X), I(X),(X) and suitable boundaryconditions (see below)

2) X]0, 1[

2-a) Compute ,b ,F,h2-b) Solve the differential system (25) for S and

2-c) Solve the state equation (see (4)) for

r

:

r

=

0|t10| k

r

+

0

k

+

( t1)

k

(27)

while computing at the same time the accelerations

through (15) as :

1k=

(t

1 + kt1)S11 + (k

+ kk)S21 + 1t1

r

+

(t

1 + kt1)S12 + (k

+ kk)S22

2I 22k

(t

1 + kt1)1 + (k

+ kk)2

+ (cext (I))

t1fext + k(cext (I)) 2k (28)

where a block decomposition of S is done that fits thedefinition of the state vector X with r and It is worthnoting that the differential system (in space) made of (27)

and (28) is actually a differential algebraic one, due to the

zero-order equation for k. But, due to the explicit expressionfor k, direct substitution of k into (27) makes it an ordinarydifferential system that is integrated without difficulty. Then,

the searched after field of accelerations are computed thanksto (28). Suitable boundary conditions are needed for the

algorithm to start. Let us mention some possible choices, as

illustrations : in the case of a free beam or of an eel-robot in a

fluid, both ends are free so that the wrench vanishes at X= 0and X= 1 :

nM

|(X=0)

=

nM

|(X=1)

=

00

(29)

In the case of a hyper-redundant manipulator, with a fixed

basis e.g., one will impose that the acceleration at X= 0 and

4167


6/6

the wrench at X= 1 vanish :r

|(X=0)

=

00

;

nM

|(X=1)

=

00

(30)

but one could have chosen to impose some motion to the

basis either. Such boundary conditions give starting values

for integrating (25), remembering that the adjoint state is

an affine function of the state X. Other situations such e.g.assigning a known motion to one end can model a hyper-redundant manipulator on a mobile platform and are taken

into account by the above considerations exactly the same way,

although in that case nonholonomic constraints on velocities

should be included at an earlier stage, in the derivations of

section III. One sees that a continuous forward algorithm for

the chosen beam model is straightforwardly obtained. A key

issue in the implementation of the algorithm is the solution

of the matrix-vector differential system (25) : thanks to the

availability of powerfull tools in Matlab-like programming

softwares and standard numerical libraries such as IMSl, such

matrix-vector equations are easily implemented and solved,

making the above algorithm attractive.

VII. CONCLUSION

In this work, a Newton-Euler formalism has been exhibited

for a continuous, geometrically exact, Cosserat beam model

thanks to a joint use of optimal control theory, Gauss least

constraint principle and Gibbs-Appell method for motion

equations derivation. It can be seen as an extension of a

method from the classical discrete case to the continuous case.

The most interesting result is a systematic derivation of the

forward algorithm that would be otherwise very difficult to

obtain. Also, the followed approach gives a unifying perspec-

tive of the two algorithms (backward and forward) of theNewton-Euler formalism. On the concrete side, the previous

derivations can be useful as well when one wants to develop

analogous algorithms for robots that have to be made of a -

possibly large- number of rigid links. Possible developments

using this approach could be made for manipulators on mobile

platforms or other mechanical systems that fit into the Newton-

Euler formalism, i.e. a great variety of mechanical structures. It

is likely that the framework developped here will facilitate the

derivations of backward and forward Newton-Euler algorithms

in these situations. Being recursive in nature, the method can

easily be implemented as the core of the forward algorithm is

a matrix Riccati equation, for which efficient solvers exist in

standard numerical libraries and software. The achievementsof the present work can be seen as a possible building block

for analysis and control of more complex flexible structures

undergoing large displacements.

REFERENCES

[1] P. Appell. Sur une forme generale des equations de la dynamique.Comptes rendus de lacademie des sciences de Paris, t. 129:423427,459460, 1899.

[2] P. Appell. Traite de Mecanique Rationnelle. Gauthier-Villars, 1921.[3] W. W Armstrong. Recursive solution to the equation of motion of a

n-links manipulator. In Proc. 5th World congress on theory of machinesand mechanisms, pages 13431346, Montreal, 1979.

[4] W.J. Book. Recursive lagrangian dynamics of flexible manipulatorsarms. Int. J. Robotics Research, 3:87101, 1984.

[5] F. Boyer and P. Coiffet. Symbolic modelling of a flexible manipulatorvia assembling ot its generalized newton-euler model. J. Mechanismand Machine theory, 1995.

[6] F. Boyer and W. Khalil. Recursive inverse and forward dynamics offlexible manipulators. In Proc. European Control Conf. (ECC95) Conf.,Rome, Italy, 1995.

[7] F. Boyer, M. Porez, and W. Khalil. Macro-continuous computed torquealgorithm for a 3d eel-like robot. IEEE Trans. Robotics, 2006. In press.

[8] C. Brenan, S.C. Campbell, and L. Petzold. The numerical solution ofdifferential-algebraic equations. North-Holland, 1989.

[9] A.E. Bryson and Y.C. Ho. Applied optimal control. Hemisphere Pub.Corp., 1975. Revised printing.

[10] S. Cetinkunt and W.J. Book. Symbolic modelling of flexible manipula-tors. In Proc. IEEE Int. Conf. Robotics and Automation, pages 20742080, 1987.

[11] E Cosserat and F. Cosserat. Theorie des corps d eformables. Hermann,1909.

[12] A. de St Germain. Sur la fonction S introduite par M. Appell dans lesequations de la dynamique. Comptes rendus de lacad emie des sciencesde Paris, t. 130:11741177, 1900.

[13] G.M.T. dEleuterio and C.J. Damaren. The relationship between recur-sive multibody dynamics and discrete-time optimal control. IEEE Trans.

Robotics and Automation, 7(6):743749, 1991.[14] R. Featherstone. The calculation of robot dynamics using articulated

body inertias. Int. Jal Robotics Research, 2(1):1330, 1983.[15] S. Hirose. Biologically Inspired Robots : Snake-like locomotors and

manipulators. Oxford University Press, 1993. Translated by P. Caveand C. Goulden.

[16] P.C. Hughes and G.B. Sincarsin. Dynamics of elastic multibody chains.J. Dyn. Stability Syst., 4:209226, 1989.

[17] W. Khalil and E. Dombre. Modelling, Identification and Control ofRobots. Hermes, London, 2002.

[18] S.S. Kim and E.J. Haug. Recursive formulation for flexible multibodydynamics. part 1 : open-loop systems. Journal of computer methods inapplied mechanics and engineering, 71:293314, 1988.

[19] L.D. Landau and E.M. Lifshitz. Theory of elasticity. Pergamon Press,1959.

[20] G. Le Vey. Hyperredundant manipulators, continuous newton-euleralgorithms and optimal control theory. Technical Report 05/3/AUTO,IRCCyN/Ecole des Mines de Nantes, 2005.

[21] G. Le Vey. The newton-euler formalism for general multibody systemsas the solution of an optimal control problem. Technical Report05/4/AUTO, IRCCyN/Ecole des Mines de Nantes, 2005.

[22] G. Le Vey. Optimal control theory and newton-euler formalism forcosserat beam theory. Comptes Rendus de lAcad emie des Sciences deParis, CR-Mecanique 334:170175, 2006.

[23] J.H. Long, B. Adcock, and R.G. Root. Force transmission via axialtendons in undulating fish : a dynamical analysis. Comparative bio-chemistry and physiology, 133(4):911929, 2002.

[24] J.Y.S. Luh, M.W. Walker, and R.C.P. Paul. On-line computation schemefor mechanical manipulators. In Proc. 2nd IFAC/IFIP Symp. InformationControl problems in manufacturing technology, pages 165172, 1979.

[25] G. Rodriguez. Kalman filtering, smoothing and recursive robot armforward and inverse dynamics. IEEE Trans. Robotics and Automation,3(6):624639, 1987.

[26] J.C. Simo and L. Vu-Quoc. On the dynamics in space of rods undergoinglarge motions - a geometrically exact approach. Computational Methods

in Applied Mechanics and Engineering, 66:125161, 1988.[27] R.P. Singh, J. Van der Voort, and R.J. Likins. Dynamics of flexible

bodies in tree topology. In Proc. AIAA Dyn. Specialist Conf., pages327337, Palm Springs, California, 1984.

ACKNOWLEDGMENT

This work, as part of an Eel-Robotproject, was supported by

french Centre National de la Recherche Scientifique (CNRS)

through the ROBEA program (RObotique et Entites Artifi-

cielles), and by Region Pays de la Loire, Institut de Recherche

en Communication et Cybernetique de Nantes (IRCCyN),

Ecole des Mines de Nantes.

4168

continuous newton-euler algorithms for

Documents