vipsymetries of vacanomic dynamics application to op cont

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Symmetries in Vakonomic Dynamics.

Applications to optimal control

Sonia Martnez, Jorge Cortes, Manuel de Leon

Laboratory of Dynamical Systems, Mechanics and Control, Instituto de

Matematicas y Fsica Fundamental, Consejo Superior de Investigaciones

Cientficas, Serrano 123, 28006 Madrid, SPAIN

Abstract

Symmetries in vakonomic dynamics are discussed. Appropriate notions are intro-duced and their relationship with previous work on symmetries of singular La-grangian systems is shown. Some Noether-type theorems are obtained. The resultsare applied to a class of general optimal control problems and to kinematic locomo-tion systems.

Key words: constrained variational problem, symmetry, optimal control1991 MSC: 37J15, 49J15, 49K15, 70F25, 70G65PACS:03.20.+i

1 Introduction

The existence of symmetries for a dynamical system is of major theoreticaland practical importance. In fact, there are plenty of works devoted to de-velop methods and algorithms to find symmetries for a given problem or tocharacterize the different types of symmetries it can admit.

A paradigmatic example of the utility of symmetry properties is the Noetherstheorem for Hamiltonian systems, which asserts that if one has a certain typeof symmetry (called Noether symmetry), then a conservation law for the equa-

tions of motion can be directly obtained. The relevance of this result is obviousin Marsden-Weinstein theorem via the momentum map, where one can reducethe number of degrees of freedom of the system even preserving the symplecticstructure.

In Geometric Mechanics, there has been a considerable effort on the descrip-tion of the symmetry properties of general Lagrangian and Hamiltonian sys-tems, even with nonholonomic constraints. A classification of infinitesimal

Preprint submitted to Journal of Geometry and Physics 6 August 2004


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symmetries of a given dynamical system was given in [30,31]. Making use ofthe constraint algorithm [13,14], one can extend many of the results obtainedfor regular Lagrangian systems to singular Lagrangians [7,19].

In this paper, we focus our attention on symmetries in vakonomic dynamics. A

vakonomic system consists of a Lagrangian L : T Q R on the tangent bun-dle of a n-dimensional configuration manifold Q and a (2n m)-dimensionalsubmanifold of constraints M T Q. The point is to extremize the func-tional defined by the Lagrangian among all the curves c(t) onQ which satisfythe constraints, that is, c(t) M. This constrained variational problem isthe natural setting of many optimization problems encountered in economics,control theory, motion of microorganisms, etc. ([16,33,34]). A thorough dis-cussion of the relationship between optimal control problems and constrainedvariational problems can be found in [4]. We would like to stress that the rele-vant equations describing the dynamic behaviour of systems subject to generalconstraints are obtained through Lagrange-dAlembert principle, which is not

atrulyvariational principle. This gives rise to the so-called nonholonomic me-chanics, which has been a field of intensive research in the last years. Weuse the term vakonomic dynamics to refer to the use of typical tools fromGeometric Mechanics (such as the ones described below) in the study of opti-mization problems subject to constraints, which we feel can bring new insightsto these problems [4].

There are several geometric descriptions of the vakonomic problem [6,8,12,20,23].Some of them are based on the fact that, under certain regularity conditions,the vakonomic equations of motion can be obtained as the Euler-Lagrange

equations for an extended Lagrangian L: T(Q R

m

)R

, L= L+

,where= 0, 1 m, describe locally the constraint submanifold M andthe are Lagrange multipliers. The Lagrangian Lis obviously singular andthe vakonomic system can be studied as a presymplectic system. This will bethe point of view adopted in Section 2.

This approach will allow us to adapt the theory of symmetries developed inthe general presymplectic setting to vakonomic dynamics. This is done in thefirst part of the paper, where the notions of vakonomic symmetry, vakonomicinfinitesimal symmetry and vakonomic Noether symmetry are introduced (Sec-tion 3). These concepts are developed both in the Lagrangian and the Hamil-

tonian formalisms, where corresponding versions of the Noethers theorem areobtained (Sections 4 and 5). Section 6 is devoted to the case of a Lie groupacting by vakonomic symmetries on a vakonomic system.

The developments of the first part are exploited in the second one, where wehave dealt with some applications to control theory (Section 7). In particu-lar, we have considered a general optimal control problem consisting of a setof differential equations xi = fi(x(t), u(t)), where xi are the states and the

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ua are the control variables, and a cost function L = L(x, u) which must beextremized during the motion. In a geometrical setting, this problem is mod-elled on an affine bundle CB , where B is the manifold of states. Then,the controls ua are seen as the fibers of the affine bundle. We can consider avakonomic problem whose solutions exactly correspond to the solutions of the

general optimal control problem and we can make use of the results obtainedin the first part generalizing some of the results stated in [10].

We have also treated another application: an optimal control problem for kine-matic locomotion systems [16,25,34]. Such systems (which include, among oth-ers, robotic devices and microorganisms) are modelled on a principalG-bundleQB endowed with a principal connection.Qis the space of configurationsof the system, B is the shape space and G is the (Lie group) manifold of allpossible positions of the device in its environment. In this case, the controlsare precisely the shape velocities, which are the variables the device can af-fect directly. There is also a cost function to minimize, generally associated to

the energy expenditure of the manouevers the device is making. This costfunction is accordingly defined on T B from a Riemannian metric on B. Theproblem is then the following: given two points in Q, find the optimal controlsuwhich steer the system from one point to the other minimizing the cost func-tion while satisfying the constraints provided by the horizontal distributionof the connection. Again, this can be seen as a vakonomic system and we canapply the results for symmetries. This leads us to obtain Wongs equations[24] via a Poisson reduction in a rather straightforward way.

Finally, we have included in an appendix the basic definitions concerning liftsof vectors and functions as well as symmetries of presymplectic systems.

2 Vakonomic Dynamics

Unlike what happens in nonholonomic mechanics [27], in vakonomic mechan-ics the equations of motion for systems in the presence of nonholonomic con-straints are obtained through the application of a variational principle.

The starting point is a n-dimensional configuration manifold Q, a (2n m)-

dimensional constraint submanifoldMofT Q, locally defined by the indepen-dent equations = 0, 1 m, and a Lagrangian L: T Q R. If (q

A)are coordinates in Q with (qA, qA) the induced coordinates in T Q, then wewrite L = L(qA, qA). In general, M will be a subbundle ofT Q over Q. Forexample, in the following sections we will treat the case of a vector subbundleof T Q, M D, defined by a distribution D on Q, or the case of an affinesubbundle Mmodelled on the vector subbundle D ofT Q with an additionalvector fieldon Q.

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Now, according to the theory of the calculus of variations, we extremize thefunctional

J(c(t)) =

10

L(c(t),c(t))dt ,

defined by L on the set of twice piecewise differentiable curves c(t) joiningc(0) =q0 and c(1) =q1, and satisfying the constraints c(t) Mc(t),t.

We denote the space of such curves by C(q0, q1) and will assume that it is anon-empty manifold. A curve in C(q0, q1), cs : (, ) R C(q0, q1), is afunction such that cs(t) is a curve in Q joining q0 andq1 for all s. A curve cswill be a variation of c if c0(t) = c(t), t. The tangent space ofC(q0, q1) atc(t) consists of the infinitesimal variations ofc; i.e. given s cs(t), c0 = c

then

X : [0, 1]T Q , X (t) = d

ds |s=0cs(t) ,

is an infinitesimal variation ofc. We will assume that there are enough varia-tions and non-trivial infinitesimal variations for each c C(q0, q1) (see [1,26]for a discussion of the contrary situation or abnormal case).

Now, we set up the equation

dJc(X) = 0 , XTc C(q0, q1) ,

and use the Lagrange Multipliers Theorem in an infinite dimensional context,to state (see [1,2,22]) that c is an admissible motion if and only if there existmfunctions1, . . . , m, : [0, 1] Rsuch that

d

dt

L

qA

L

qA =

d

dt

qA

qA

d

dt

qA

, 1 A n ,(1)

and(qA, qA) = 0, 1 m. From (1) we deduce that a curve c = (qA(t))

in C2(q0, q1) is a solution of the vakonomic equations if and only if there existlocal functions 1, . . . , m on R such that c(t) = (qA(t), (t)) is an extremalfor the extended Lagrangian

L: T(Q Rm) R , L= L + ,

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i.e. it satisfies the Euler-Lagrange equations

d

dt

L

qA

L

qA = 0 , 1 A n ,

d

dt

L

L

(q

A, qA) = 0 , 1 m ,

(see [1,22] for details).

From the extended LagrangianL we can construct the system (T P , L, dEL),whereL= dLis the Poincare-Cartan 2-form, L= S

(dL) is the Poincare-

Cartan 1-form, andS=

qA dqA +

d is the canonical almost tan-

gent structure onT P.EL= L Lis the energy associated with L, which isdefined using the Liouville vector field = qA

qA+

. We will assume

that (T P , L, dEL) is presymplectic, i.e. L has constant rank.

Within this geometrical framework we can pose the equation

iL= dEL , (2)

which codifies the vakonomic equations (1). In [23], this point of view for

vakonomic dynamics was developed for a natural Lagrangian L (Lagrangianequal to kinetic minus potential energy) and linear constraints. will be asecond order differential equation (SODE) to be found on T Pwhose integralcurves (qA(t), (t)) are the vakonomic solutions (qA(t)) together with thecorresponding Lagrange multipliers ((t)).

Equation (2) will not have in general a global well defined solution on T P.Applying the Gotay-Nester algorithm [13,14] for presymplectic systems, wegenerate a sequence of submanifolds as follows (this is valid for general presym-plectic systems). First put P1= T P. Then, consider the set

P2= {x P1| Zx TxP1 solution of (2)} .

Assume that P2 is a submanifold of P1. It may happen that the obtainedsolutions are not tangent to P1. Then, we restrict P2 to the submanifold

P3= {x P2 | Zx TxP2solution of (iZL= dEL)|P2} .

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Proceeding further, we construct a sequence

. . . Pk . . . P 3P2 P1 .

Alternatively, the constraint submanifolds can be described by

Pk ={x P1 | dEL(x)(v) = 0 , v TxPk1 } ,

where

TxPk1= {v TxP1| L(x)(u, v) = 0 , u TxPk1 } .

We say that P2 is the secondary constraint submanifold, P3 is the tertiaryconstraint submanifold, and so on.

In the most favourable case, the algorithm will stabilize at some step k and afinal constraint submanifold Pk = Pfwill exist where there is a well definedvector field T Pf such that

(iL = dEL)|Pf. (3)

This solution is not necessarily unique and we will usually have a set of solu-tions XL(Pf), where

XL

(Pf) ={ T Pf |(iL= dEL)|Pf} .

See the Appendix for other notations and basic definitions that will be usedalong the paper.

Remark 1 In [8] an alternative geometric description of vakonomic dynamicsin the extended phase space TQ QMwas described. This formulation wasused to compare the solutions of vakonomic dynamics with the solutions ofnonholonomic mechanics for nonholonomic Lagrangian systems.

3 Symmetries

In this section we study the general symmetries of a vakonomic system ( L, M)on T Q and their relationship with the symmetries of L, an extended La-grangian of the form L = L + , where { | 1 m} is a global basisof functions defining the submanifold of constraintsM.

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We will consider that Mis an affine subbundle ofT Qmodelled on the vectorsubbundleD T Q, dim M= dim D= 2nm, with an additional vector field :Q T Q. We say that a vector Xq is inMq if and only ifXq q Dq. Inother words, if the annihilator Do ofD is spanned by{(q) =(q)dq

A | 1 m} and (q) =h(q) ,1 m, then

Xq Mq (Xq q) =AXA + h= 0 , 1 m ,

i.e. the constraint functions defining M are

(q, q) = A(q)qA + h(q) , 1 m .

In the sequel, 1 : Q Rm Q and 2 : Q R

m Rm will denote the

projections onto each factor ofQ Rm

.

3.1 Vakonomic symmetries

Definition 2 ([1]) A vakonomic symmetry for (L, M) will be a diffeomor-phisms : Q Q such thatT sleavesMandL|M invariant, i.e.T s(M) =Mand(L T s)|M=L|M.

In this way, we assure that the constrained variational problem is preservedbysand so will be its solutions.

The condition (L T s)|M =L|Mis equivalent to say that there exist mlocalfunctions{0 :T Q R | 1 m} such that L T s L=

0 , while

the conditionT s(M) =Mmeans that the transformation { T s= | 1 m}, gives rise to new independent constraint functions defining M.

In fact, ifD T Q is the distribution modelling Mwe have

(i) SinceM, then,Tqs(q) Ms(q), or, equivalently,Tqs(q) s(q) Dq.

(ii) Let Xq be a vector in Dq. Then, Xq+ q Mq and T s(Xq) +T s(q) Ms(q). But again, this means T s(Xq) +T s(q) s(q) Ds(q). By (i) wededuce that T s(Xq) Ds(q).

That is,Dis invariant byT sand so isDo. Thus, a basis {}m=1ofD

o is trans-formed into a new oneTs() = . Then, there exists a non-singular matrix-valued function on Q, (s) : Q GL(m,R) such that =

(s). In

other words, if = AqA +h, = Aq

A +h, 1 m, are local

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expressions for the constraint functions corresponding to these basis, we get,

(Y ) =(Y) ,

(Y ) = (s)(Y ) =

(s)(Y) ,

for a given Y T Q, i.e. = (s), or equivalently, if

(s) denotes the

entries of the inverse matrix of ((s)), we have (s)=.

WhenL T s= L, we can extend the diffeomorphism s to P =Q Rm as

s: P P

(qA, ) (sA(q),(s)(q)) ,

so that TsleavesL invariant. Indeed,

L Ts= L += L +

=L .

This systematic procedure allows us to translate all the vakonomic symmetriessinto symmetries sof the singular Lagrangian L and viceversa, we can recoverthem just by projecting sto Q.

3.2 Vakonomic infinitesimal symmetries

Definition 3 A vakonomic infinitesimal symmetry (from now on VIS) for(L, M) is a vector field X on Q such that its complete lift Xc X(T Q) istangent to Mand satisfiesXc(L)|M=X

c|M(L|M) = 0.

In other words, X is a VIS if and only if its flow, {st :Q Q}, consists ofvakonomic symmetries for (L, M).

For simplicity, we will consider those Xsuch that Xc(L) = 0. Then, from aVIS X X(Q), one can obtain an infinitesimal symmetry ofL, X X(P).Indeed, since Xc(L) = 0 and Xc()|M= 0, 1 m, the flow ofX, {st}verifies for all < t < ,

L T st = L ,

T st = t = (t) .

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We can then define the one-parameter group

st : P P

(q, )(st(q), (t)(q)

) ,

and take the vector field whose flow is given by {st}(its infinitesimal genera-tor), XX+ YL, where

YL=

d

dt |t=0(t)(q)

.

SinceL Tst = L, for all < t < , it is immediate that Xc(L) = 0.

Conversely, given an infinitesimal symmetry ofL, X=XA(q)

qA+ f(q)

,

we have

Xc(L) = Xc(L) + (f + Xc()) = 0 .

Since this is valid for every , we obtain

Xc(L) = 0 , Xc() =f .

That is, X projects onto a vector field on Q, X = XA(q)

qA, which is a

VIS for (L, M). For this reason, we will focus our attention on infinitesimal

symmetries ofL given by X=X+ f(q)

, where (f) is a matrix-valued

function on Q, (f) :Q gl(m,R). We will call to this type of symmetrya VIS for (L, M) on P.

Definition 4 A vakonomic Noether symmetry (VNS) for (L, M) will be avector field X on Q such that Xc|M X(M) and X

c(L)|M = Fc|M for some

associated functionF :Q R.

Observe that, although the flow of a Noether symmetry preserves M, it doesnot consists of vakonomic symmetries in the sense of Definition 2. Its role willbe explained in the next section.

In case Xc(L) = Fc on the whole ofT Q, the above-defined extension X=X+ YLgives rise to a Noether symmetry ofL; that is, Xc(L) =1(F

c) Fc.

Conversely, ifX=XA(q)

qA+ f(q)

=X+ f(q)

is a Noether

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symmetry forL, say,

Xc(L) = Xc(L) + (f + Xc()) = F

c , (4)

for some F : P R. Since F

= Xc(L)

= 0, equating () = (0), we

have

Xc(L) = Fc ,

and being (4) valid for all , we also have

Xc() =f .

Thus, Fmust be the pullback of a function F :Q R, and Xprojects toX, a VNS for (L, M) with associated function F. These type of symmetriesXwill be referred as VNS for (L, M) on P.

Example 5 (Closed von Neumann model) In economics, the variational cal-culus is an indispensable tool when dealing with typical optimization prob-lems. The following example was taken from [8,32,33]. The n capital goodsK1, . . . , K n and the respective capital formations K1, . . . , Kn can be consid-ered as coordinates (K1, . . . , K n, K1, . . . , Kn) inTR

n.

Given the Lagrangian L : TRn R, L(K1, . . . , K n, K1, . . . , Kn) = Kn andthe constraint function

(K1, . . . , K n, K1, . . . , Kn) =K11 K

22 K

nn [ K

21 + + K

2n]

1

2 ,

withn

i=1 i = 1, which defines the submanifold M = { 0} ofTRn, the

von Neumann problem consists of maximizing

T0

Kndt , subject to 0 ,

for some T 0 and appropriate initial conditions.

Alternatively, we can formulate the von Neumann problem in terms of theextended LagrangianL(K1, . . . , K n, , . . . , Kn,) = Kn+ .

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LetXbe a vector field on Rn,X=n

j=1

Xj(K1, . . . , K n)

Kj. Then,

Xc(L) =n

i=1

KiXnKi

= (Xn)c ,

and

Xc() =n

i=1

iK

11 K

i1i1 XiK

i1i K

i+1i+1 K

nn

( K21 + + K2n) 12

nj=1

Ki KjXiKj

.

IfX=Cn

i=1

Ki

Ki, withCa non-zero constant, we have

Xc() =C

ni=1

i

K11 K

nn C(K

21 + + K

2n)

1

2 =C ,

or, equivalently, Xc()|M= 0. Thus, X = Cn

i=1

Ki

Kiis a VNS for (L, M)

with associated functionXn = C Kn. We have shown above that Xgives riseto a Noether symmetry Xof the extended Lagrangian L. In fact, we have

X=C n

i=1Ki

Ki

.

3.3 Symmetries given by the action of a Lie group

Finally, let :G QQ be a free and proper left action of a Lie group Gon the configuration space Q. Denote by g the diffeomorphism ofQ, q (g, q), for eachg G. The groupG will be a group of vakonomic symmetriesfor (L, M), if each g is a vakonomic symmetry, that is, if the lifted action

T :G T QT Q satisfiesL|M Tg =L|M andTg(M) =M,g G.We can make use of the procedure described before to extend a symmetryfrom Q to P = Q Rm. Given a fixed Lagrangian, L = L+ , let usassume that L Tg =L for all g G. Then we define the new action

:G P P

(g, (q, )) (g(q),(g)(q)

) .

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It is easy to check that this is indeed a free action and, when G is compact,one can assure that it is also proper.

4 Constants of the motion

One is commonly interested in studying the symmetry properties of a dy-namical problem because this can yield, via a Noethers theorem for example,information about conservation laws or reduction of the number of degrees offreedom. In the following three sections we shall explore this topic. Some ofthe work developed in [19] for symmetries of singular Lagrangian systems willbe helpful in the context we have exposed for vakonomic mechanics. We referto the Appendix for a review of several definitions of symmetries that will beused in the sequel.

Lemma 6 Let(N, , ) be a presymplectic system and: NNa diffeo-morphism such that

= , = .

Consider . . . N k . . . N2 N1 the sequence of constraint submanifoldsobtained applying the Gotay-Nester algorithm. Then, restricts to diffeomor-phismsk :Nk Nk, k.

PROOF. See [19].

Now, we are in a position to prove

Proposition 7 (Noethers theorem). Assume that the sequence of submani-folds obtained through the application of the Gotay-Nester algorithm stabilizesat some step kf f. LetX X(P) be a VNS for (L, M) with associated

functionF :P R. Then,

(i) Xc|Pk X(Pk) 1 k kfandXc|Pf

is a dynamical symmetry ofXL(Pf).

(ii) (Fv iXcL)|Pf :Pf R is a constant of the motion forXL(Pf).

PROOF. Let

X=XA(q)

qA+ g(q)

,

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be the local expression ofX. Since

Xc(L) =XAL

qA+ g

L

+ qB

XA

qBL

qA =Fc ,

andF is the pullback of a function on Q, then

Xc(EL) = Xc

L

qBqB

Xc(L) = Xc

L

qB

qB +

L

qBXB

qCqC Fc

=

XA

2L

qBqA+ g

2L

qB+ qC

XA

qC2L

qAqB+

L

qAXA

qB

qBFc

=

qB

Xc(L)

qB Fc = 0 ,

and similarly,

LXcL= LXc

L

qBdqB

= Xc

L

qB

dqB +

L

qAXA

qBdqB

=

qB(Xc(L))

dqB =dFv .

In particular, these computations imply

iXcL= d(iXcL) LXcL= d(iXcL Fv) , (5)

and

LXcL= iXcdL+ diXcL= dd(iXcL Fv) = 0 .

Therefore, the presymplectic structure of (P1, L, dEL) is invariant along theflow ofXc,{Tst : P1P1},

(Tst)L= L , (Tst)

(EL) =EL .

By Lemma 6, the flow {Tst}restricts to eachPk and Xc|Pk

X(Pk),1 k

kf. In particular [Xc|Pf

, ] X(Pf) for every XL(Pf), and

i[Xc|Pf

,]L|Pf =LXc|Pf(iL) i(LXc

|Pf

L) = 0 .

Thus, Xc|Pf is a dynamical symmetry ofXL(Pf) (see the Appendix).

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To prove (ii) take XL(Pf). Using (5), we get

(iXcL Fv)|Pf= d(iXcL F

v)|Pf() = (iXcL)|Pf()

= (iL)|Pf(Xc) =dEL(X

c)|Pf = 0 .

Therefore, (iXcLFv)|Pfis constant along the integral curves of X

L(Pf).

Example 8 (Closed von Neumann model, revisited)As a consequence of Propo-sition 7, we are able to find a constant of the motion for the von Neumann

problem in a systematic way. We have that X = C

ni=1

Ki

Ki

is

a VNS for (L, M) on Rn R with Noether function CKn. Consequently weobtain the conservation law

(CKn)v iXcL= (CKn)

v iXc

L

KjdKj

=C

Kn Kn

nj=1

Kj

Kj

= C

K21 + . . .+ K2n

n

j=1

Kj Kj

,

on the final submanifold of constraints.

5 Relationship with the Hamiltonian formulation and the SODEs

problem

If the extended Lagrangian L is almost-regular, then the vakonomic problemadmits an equivalent formulation which is Hamiltonian. In this case, as theconstraint functions are linear or affine, it can be proven that L is almost-regular if and only ifL is almost-regular (see [23]).

Let FL : T P TP be the Legendre mapping ofL. If (qA, , pA, p) are

local coordinates inT

P, then the Legendre mapping is locally written as

FL(qA, , qA,) =

qA, ,

L

qA, L

=

qA, ,

L

qA, 0

.

We say that L is almost-regular ifM1 = FL(T P) is a submanifold ofTP,

j1 : M1 TP, and FL : T P M1 is a submersion whose fibers are

connected. When this holds true, it can be assured that EL is constant along

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the fibers ofFL and a Hamiltonian h1 :M1 R can be defined implicitlyas h1 FL = EL. Taking 1 = j

1(P) the pullback to M1 of the canonical

symplectic form P of TP, we obtain a presymplectic system (M1, 1, h1).

The equations of motion are then,

i1= dh1 . (6)

To solve it we apply the Gotay-Nester algorithm and get the sequence ofsubmanifolds,

. . . Mk . . . M3 M2 M1 .

The Gotay-Nester equivalence theorem [14] relates this sequence with the for-mer one{Pk}k1 of (T P , L, EL). Denote byPf,Mfthe final submanifolds ofconstraints (if they exist). Then, the theorem asserts

(i) FL|Pk FLk : Pk Mk is a fibration and Mk is diffeomorphic toPk/Ker(FLk),k.

(ii) If the sequence {Pk}k1 terminates at step kf so will {Mk}k1 and thesolutions of the systems are equivalent in the following sense. Given XL(Pf) which is FLf-projectable, then TFLf() = is a solution of

(i1= dh1)|Mf. (7)

On the other hand, if is a solution of (7) and X(Pf) projects byTFLfonto , then is a solution of (3).

Thus, solving (7) we will obtain a set X1(Mf) of vector fields such that theirintegral curves, (qA(t), (t), pA(t), 0), give the vakonomic solutions (q

A(t), (t)).

Now we study how the vakonomic symmetries can be seen as symmetries of(M1, 1, h1).

Proposition 9 (Noethers theorem). LetX : P T P a VNS for (L, M)with associated functionF :P R. Then,

(i) Xc|Pk isFLk-projectable ontoXc|Mk +

F

qB

pB|Mk

X(Mk), k 1.

(ii) Xc|Mf +

F

qB

pB

|Mf

is a Cartan symmetry forX1(Mf)and X()|Mf

Fv|Mf is a constant of the motion.

PROOF. We extend a result of [19] for infinitesimal symmetries. We considerhere the more general case of Noether symmetries.

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It is easy to show that X is FL1-projectable and TFL1(Xc) = Y FL1,

with Y X(M1). Let us see what the expression for TFL1(Xc) is in local

coordinates. On the one hand, we have

Xc(L) =XA(q)L

qA

+ g(q)L

+ qCXA

qC

L

qA

= F

qB

qB , (8)

while

TFL1(Xc) = XA

qA+ g

+

XA

2L

qAqB+ g

2L

qB+ qC

XA

qC2L

qAqB

pB.

Taking the derivative

qB

in (8), and substituting into the last expression we

get

TFL1(Xc) =XA

qA+ g

+

F

qB pA

XA

qB

pB,

which is just Xc|M1+

F

qA

pA

|M1

. Now, it is clear that

TFLk(Xc|Pk

) = Xc|Mk +

F

qA

pA

|Mk

=Y|Mk, with Y|Mk X(Mk) , k 1 .

Finally, (ii) can be proven using similar arguments as in Proposition 4.2.Firstly, ifZis a vector field on M1and Ua vector field on T Pprojecting ontoit byFL, then for anyz M1 and arbitraryx FL

1(z),

iTFL(Xc)1(z)(Zz) = 1(z)(TFL(Xcx), TFL(Ux)) = (FL

1)(x)(Xcx, Ux)

= L(x)(Xcx, Ux) =d(iXcL F

v)(x)(Ux)

= d(FL(X() Fv))(Ux) =d(X() Fv)(TFL(Ux))

= d(X() F

v

)(Zz) .

Secondly,h1 is invariant by TFL(Xc) due to

LTFL(Xc)(h1) =LXcFL(h1) =LXc(EL) = 0 .

Therefore,

Xc +

F

qA

pA

|Mf

is a Cartan symmetry for X1(Mf), with

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X() Fv the associated constant of the motion.

It is possible to find a submanifold S Pf on which there exists a tangentsolution

L TS, satisfying the SODE condition [14]. Let be a vector field

onMf satisfying (7) and X(Pf) a vector field which projects onto . Nowdefine the mapping

: Mf Pf

y T P((x)) ,

where P : T P P is the canonical projection and FLf(x) = y. Observethat is well-defined as it does not depend on the choice ofx FL1f (y),because is FL-projectable. In fact,is a section ofFLf, FLf= id|Mfand

its image(Mf) =Sis a submanifold ofPf. The vector field L = T ()satisfies

(iLL= dEL)|S,

and the SODE condition, S(L) = .

Locally,

L= qA

qA+

+ CA

qA+ D

,

for certain functionsCA =CA(q,,q, ) and D =D(q,, q, ).

Notice that S and Mf are diffeomorphic by and FLf|S = 1 and that

the dynamics on them are equivalent. Then, we have a complete equivalencebetween symmetries and constants of the motion on both the Lagrangian andthe Hamiltonian sides via andFLf|S.

6 Constants of the motion given by the action of a Lie group

We particularize now the results given in Sections 4 and 5 to the case of a LiegroupG which acts onQ, :G QQ, freely and properly. We will makeuse of it in the applications that follow.

As we have seen, if :G QQ verifiesL Tg =L and Tg(M) =M,g G, then we can build an action on P, : G P P as g(q, ) =

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(g(q),(g)(q)

) such that its complete lift to T P, T : G T P T Psatisfies L Tg = L, g G. This implies that

T restricts to well-definedactions T|Pk

Tk :G Pk Pk,k.

Let be an element in g, the Lie algebra ofG. Denote by P (respectively

Pk ,Q) the vector field generated by the flow exp(t)(respectively (T

k )exp(t),exp(t)). Then, as a consequence of Proposition 7 we have thatPis a VIS for(L, M) onP,Pfis a dynamical symmetry for X

L(Pf) and

Jf : Pf g

x Jf(x) : g R

Jf()(x) = iPfL(x)

is a momentum map for the presymplectic system (Pf, Pf, dEL|Pf). We will

call it the vakonomic momentum map [1,12]. Therefore, we have that Jf() :Pf R,xJf()(x) =Jf(x)() is a constant of the motion.

IfQ(q) = AQ(q)

qA andP(q, ) =

AQ(q)

qA+ (q)

, then, givenx

Pf, we have,

J(x) = iPk L(x) = L

qA(x)AQ(x) =

L

qA+

qA

(x)AQ(x) .

Using this momentum map, one can perform a presymplectic reduction asdeveloped in [11].

On the other hand, the action : GP Pcan be lifted toTP, T :GTP TPas follows. Let (q,) be a 1-form in T

(q,)P. Then

Tg ((q,))

Tg(q,)Pwill be such that

Tg ((q,))(v) = (q,)(Tg1(v)) ,

for everyv Tg(q,)P. In coordinates Tg reads as

Tg (qA, , pA, p) =(

Ag(q),

(g)(q)

,

pBBg1

qA +p

(g

1)

qA , p

(g

1)(q)) .

This action restricts to M1 and it is theFL1-projection of T. To check this,

observe that, since L T =L,g G, after a straightforward computation,

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we have

L

qA(x) =

L

qB(Tg(x))

BgqA

(x) ,

forx P1. Then,

FL(x) =FL(qA, , qA,) = (qA, , L

qB(Tg(x))

BgqA

(x), 0) ,

and

Tg (FL(x)) = (Bg(q),

(g)(q)

, L

qA(Tg(x)), 0) .

But

FL(Tg(x)) = (Bg(q),

(q)

, L

qA(Tg(x)), 0) .

That is, Tg (FL(x)) =FL(Tg(x)) M1, for all x P1, so for all g G the

following diagrams

PkFL

Mk

(Tk )g (Tk )g (

T|Mk

)g

PkFL

Mk

are conmutative. Therefore, the actions Tk and Tk are equivalent for all k,

through the Legendre transformation.

If Mk denotes the vector field generated by (Tk )exp(t), g, then, by

Proposition 5.1 we have that FL(Pk) =Mk k and

Jf : Mk g

z Jf(z) : g R

Jf()(z) =P(z)

is a momentum map associated to Tf such thatFLJf=Jf, which gives the

constants of the motion Jf() :Pf R, z Jf()(z). Locally, Jf simplyreads as Jf()(z) =pA(z)

AQ(z).

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7 Applications to control theory

In this section we describe optimal control problems in terms of vakonomicmechanics and apply the theory of symmetries developed in the preceding

sections.

7.1 General Optimal Control Problems

A general optimal control problem consists of a set of differential equations

xi =fi(x(t), u(t)) , 1 i n , (9)

where the x

i

denote the states and the u the control variables, and a costfunctionL(x, u). Given initial and final states x0,xf, the objective is to find aC2-piecewise smooth curvec(t) = (x(t), u(t)) such thatx(t0) =x0,x(tf) =xf,satisfying the control equations (9) and minimizing the functional

J(c) =

tft0

L(x(t), u(t))dt .

In a global description, one assumes an affine bundle structure :N B ,whereB is the configuration manifold with local coordinates x i and N is thebundle of controls, with local coordinates (xi, ua).

The ordinary differential equations (9) on B depending on the parameters ucan be seen as a vector field along the projection map , that is, is asmooth map :NT B such that the diagram

N T B

B

B

is conmutative. This vector field is locally written as = fi(x, u)

xi.

In the following, similarly to [15], we show how this kind of problems admita formulation in terms of vakonomic dynamics. Consider the cost functionL: N R and its pullback NLtoT N. Let us define the set,

M={v T N| (v) = (N(v))} ,

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and assume that it is a submanifold ofT N. Locally this submanifold is definedby the conditions xi = fi(x, u), 1 i n, which are just the differentialequations (9). Then, to solve the vakonomic problem with Lagrangian NL:T N R and constraint submanifold M T N is equivalent to solve theoriginal general optimal control problem. Moreover, one can make use of the

already developed theory of the dynamics of vakonomic systems in the singularLagrangian framework and of the different types of symmetries associated tosuch systems, to analyze general control problems.

Remark 10 An alternative way of rephrasing the general optimal controlproblem in terms of a constrained variational problem is considered in [3,4].Assuming that equation (9) determinesu as a function of (x, x), one can posethe vakonomic problem with Lagrangian L = L(x, u(x, x)) and constraintsxf(x, u(x, x)) onT B. In particular it can be shown that the condition foundin [3] to be able to generalize the Legendre transformation arises naturally inthe vakonomic setting as the compatibility conditionbetween the Lagrangian

and the constraints [8,23], provided L is regular.

If one performs the Gotay-Nester algorithm with the extended LagrangianL= L + i(x

i fi(x, u)), one finds that the second constraint submanifold P2is the final constraint manifold if and only if the matrix

Wab= 2L

uaub i

2fi

uaub

is invertible, which is exactly the characterization found in [10] for the so-called

regular optimal control problem.

On the other hand, one can easily state a version of the Noethers theorem forgeneral optimal control problems.

Proposition 11 (Noethers theorem) Consider a regular optimal control prob-lem. LetX X(N) be a vakonomic Noether symmetry (VNS) for (NL, M)with associated function F : N R. Then Fv iXcL : P2 R is aconstant of the motion along any optimal trajectory.

Locally, ifX=Xi(x, u)

xi+ Xa(x, u)

ua, then X=X+ gij(x, u)i

j, for

some (gij) :Ngl(n,R), and the constant of the motion reads locally as

Fv iXcL= Fv

L

xiXi

L

uaXa =Fv

ni=1

iXi .

This result is a corollary of Proposition 7 when applied to general optimalcontrol problems. This theorem generalizes the results obtained in [10].

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7.2 Optimal control problems for kinematic locomotion systems

In the following, we shall focus our attention on a more concrete type ofoptimal control problems associated to kinematic locomotion systems.

Robotic locomotion can be described via a trivial principal bundle (Q,B,,G)equipped with a connectionA. Examples of locomotion systems described inthis framework include legged robots, snakelike robots and wheeled mobilerobots [16]. Even the motion of paramecia in fluids at very low Reynoldsnumber can be understood by calculating the geometric phase with respect toa certain connection, determined by the underlying fluid dynamics [34]. Weremark here that there is also another important type of problems which donot fall into this category, the so-calleddynamiclocomotion systems. A well-known example of this kind of problems is the snakeboard [5,29]. A treatmentof the optimal control problem of such systems using Lagrangian reduction

has been developed in [17] and has also been treated within a vakonomicperspective in [9].

Basically,B = Q/Grepresents the shape space of the robot,G is the manifoldconsisting of all the possible positions and orientations of the robot in itsenvironment andQ is the systems space of configurations. The rigid motionsof the system are given by a free and proper left action ofG on Q, :GQQ(left multiplication onQ). The relevant kinematics of the locomotion systemis modelled in terms of a connection A : T Q g. Indeed, the horizontalsubspace of A will be the set of velocities for which the constraints on thesystem (usually of non-slipping type) are satisfied.

Now, a closed path in the shape space B induces a net motion in the groupvariables, which is nothing but the holonomy of the connection Aassociatedto the concrete path.

Denote by (ra) the local coordinates in the quotient Band by (ra, g) the fibredcoordinates on Q such that the surjective submersion reads as (ra, g) =(ra). Then, if g is a Riemannian metric on B, define the cost function C ofthe problem,

C(r,r) =1

2gabr

arb ,

where gabare the components of the metric on B in the local chart (ra). Now,

consider the following control problem,

Strong Optimal Control Problem for Robotic Locomotion Given twopointsq0,q1 inQ, find the optimal controls u() which steer the system fromq0 to q1 and minimize

10 C(r, u)dt subject to the constraints r = u, g

1g =

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Aloc(r)u.

Observe that the statement of the optimal control problem is equivalent tothe vakonomic problem on Q corresponding to the Lagrangian function L :

T Q R defined as L(vq) =1

2

g(vq, vq), or, in fiber coordinates,

L(r,g,r, g) =1

2gab(r)r

arb ,

and the constraint submanifoldM=HT Q, the horizontal subspace of theconnection A. That is,M={vq | A(vq) = 0}= {(r,g, r, g) |g+gAloc(r)r= 0}.In what follows, we fix a basis {e1, . . . , em}of the Lie algebra g. Then we haveA(vq) = A

(vq)e, where the A are functions on T Q defining M globally.

Alternatively, we can write

Aloc(r)r= Aa (r)r

a

e , T Lge = M (g)

g ,

and consider the constraint functions = g +Aa(r)raM(g), 1

m. These functions define M locally (because of the choice of coordinates).However, we will use them in our description, for reasons that will be moreclear later.

Now, we apply the theory explained in former sections to the extended La-grangianL = L +

defined onT P =T(Q Rm), and see what happens.By the equivalence theorem, we can perform the constraint algorithm eitherin the Lagrangian context or in the Hamiltonian one, provided that our La-grangianL is almost regular.

In our case, the local expression of the Legendre transformation is

(ra, g, ,ra,g,)

FL(ra, g, , gabr

b + M(g)A

a(r), , 0) .

Observe that

z= (ra, g, , pa, p, p) FL(T P)p = 0 , p = , .

The right implication is clear, while the left one is equivalent to say that

FL1(z) ={(ra, g, , gab(pb M

A

b ),g

,) |g, R

m} ,

which can be identified with R2m. Then, by the rank theorem, M1 = FL(T P)is the manifold with atlas given by the local charts {(ra, g, , pa)}, embedded

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intoTP as

j : M1 TP

(ra, g, , pa)(ra, g, , pa, , 0) .

Besides,FL1(z) R2m, are connected submanifolds ofP1 zM1. Hence,Lis almost regular and so we can perform the constraint algorithm on M1.

Notice that the dimension ofM1is just 2nand that we can regard the Lagrangemultipliers as the generalized momenta corresponding tog. Now, considerthe pullback of the canonical 1-form P ofT

P toM1. We obtain,

1 = jP =dr

a dpa+ dg d ,

which is clearly symplectic. (M1, 1, h1) is then a symplectic system, whereh1: M1 R is the pushforward byFLof the energy EL. The local explicitexpression for h1 is

h1=1

2gcd(r)(pc A

c (r)M

(g))(pd A

d (r)M

(g)) .

Therefore, the Gotay-Nester algorithm for this system will stop at the firststep and we have a well defined solution of the vakonomic problem, and of theoptimal control problem, on M1.

Remark 12 In [25], the author proves a nice theorem which essentially assertsthat a curveq(t) inQ is a solution of the optimal control problem if and onlyif there is a curve z(t) in TQ satisfying Q(z(t)) = q(t), which is a solutioncurve of a Hamilton differential equation with Hamiltonian H0. To defineH0,we first take h : (T(Q/G)) T Q the horizontal lift operator associatedto the connectionAand consider the dual operator

h : TQ(T(Q/G))

pq

hq(p

q) T

(q)(Q/G) .

Next, using the vector bundle isomorphisms induced by g, : T(Q/G) T(Q/G) and : T(Q/G) T(Q/G), where (X)(Y) = g(X, Y) and =1, defineH0 as

H0(q, p) =1

2g(hqp, h

qp) .

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The relation between this result and what we have obtained here is the fol-lowing. We can embed the manifold M1 into T

Qsimply as

i: M1 TQ

(ra, g, , pa) (ra, g, pa, ) ,

regarding the as the generalized momenta corresponding tog. Now, some

easy computations prove that the next diagram

R

M1 TQ

i

h1 H0

is commutative. The point we want to stress here is that the formulation ofour problem in vakonomic terms and the use, via the extended Lagrangian, oftools of singular Lagrangian theory has led us to the same results of [18,25] ina straightforward way.

The action of the Lie group G on T Qleaving invariant the Lagrangian L andthe constraint submanifoldMcan be lifted to an action on T P leaving invari-ant the extended LagrangianL. Takeg G and consider the diffeomorphismg : Q Q. Since L = L(r,r), we have that L Tg = L. After somecomputations it can be verified that

Th= (g, h)

,

where TgLh

g

g

= (g, h)

g

hg

. In addition, as Lh = Lhg Lg1,

we get (g, h) = M(hg)M

(g), where M(g) denotes the inverse matrix of

M(g).

Consequently, we have an action of the Lie group on the manifold M1 given

by

G M1 M1

(h, (r,g,,p)) (r, hg,(g, h1), p) .

It is easy to see that this action is free and proper, and leaves invariant thesymplectic form1 and the hamiltonian h1.

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Now, we perform a Poisson reduction on (M1, 1, h1). We choose local coor-dinates (ra, , pa) on M1/G, just taking the representative (r

a, e , , pa) ofeach equivalence class. Then, the equations of motion on M1/G are

ra

= gad

(pd A

d (r))pa= g

cd Ac

ra(pd A

d (r))

1

2

gcd

ra(pc A

c (r))(pd A

d (r))

= rcAc c

, (10)

where c are the estructural constants of the Lie algebra g. Through thechange of coordinates pa = pa A

a (r), the equations of motion (10) take

the simpler form

ra = gad pd

pa= Bcar

c 12

gcd

rapc pd

= rcAc c

, (11)

where Bca =Ac

ra

Aarc

cAaA

c are the local coefficients of the curvature

of the connectionA. Equations (11) are precisely Wongs equations [17,24,25].Here it is the reason why in the beginning of this section we chose the con-straint functions and not the A. Although both formulations are clearlyequivalent, the derivation of Wongs equations is more straightforward withthe choice done.

If we perform a symplectic reduction on the symplectic manifold (M1, 1), weindeed obtain reduced symplectic manifolds. A standard result in the theoryof Hamiltonian systems with symmetry (see Theorem 6.48 in [28]) states thatthe reduced symplectic manifolds obtained by using the momentum map canbe seen as submanifolds ofM1/G; more precisely, they constitute the canonicalsymplectic foliation of the Poisson structure.

8 Appendix

We briefly review in this appendix the basic notions of vertical and completelifts of vector fields and functions. We refer to [21,35] for a comprehensivetreatment of the subject.

Let (ya) be local coordinates of a manifold N and let (ya, ya) (respectively(ya, pa)) be the induced fibred coordinates inT N(respectivelyT

N). Consider

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a vector fieldY X(N) with local expressionY =Ya

yaand letF :N R

be a function on N.

A 1-form inNcan be regarded as a function in T N. We denote it as . If

is locally written as =ady

a

, thenreads as =ay

a

. Similarly,Y isthe function given as

Y : TN R

(ya, pa) Y(ya, pa) =paY

a .

The complete lift ofF toT N is another functionFc :T N R, defined as

Fc =(dF). Locally,Fc = F

yaya.

Thecomplete liftofY toT Nis the unique vector field Yc X(T N) such thatF C(N),Yc(Fc) = (Y F)c. The local expression for Y is

Yc =Ya

ya+ yb

Ya

yb

ya.

Thecomplete liftofY toTN,Yc X(TN), is the Hamiltonian vector fieldassociated to the function Y, i.e.,

iYc= d(Y) ,

whereis the canonical 2-form onTN, defined as = dfrom the Liouville1-form = pady

a. Locally, we obtain

Yc =Ya

yapb

Ya

yb

pa.

The vertical lift ofF toT Nis its pullback to T Nby the canonical projectionN :T N N. We denote it as F

v =NF. On the other hand, the verticallift ofF toTNis the pullbackFv =NF byN :TNN.

Finally, we recall some different types of symmetries associated to a presym-plectic system (N, , ).

In first place, consider the presymplectic equation

iZ= , (12)

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and the sequence of submanifolds that results of applying the Gotay-Nesteralgorithm,

. . . Nk . . . N2N1 .

We use the following notation,

X(N) ={ ZX(N)| Zis a solution of (12) } ,

X(Nk) ={ ZX(Nk)| Z is a solution of (iZ= )|Nk } .

A dynamical symmetryofX(N) (respectively ofX(Nk)), is a vector fieldY X(N) (resp. Y X(Nk)) such that

[Y, Z] Ker , (resp. [Y, Z] Ker T Nk) ,

for all Z X(N), (resp. Z X(Nk)). In this way we assure that the flowof Y preserves solutions, i.e., the integral curves of Z are transformed intosolutions of the system, (see [19]).

A Cartan symmetryof (N, , ) will be a Y X(N) such that

(i) iY= dF for some F :N R.(ii) iY= 0.

Acknowledgements

This work was partially supported by grant DGICYT (Spain) PB97-1257. S.Martnez and J. Cortes wish to thank the Spanish Ministerio de Educaciony Cultura for FPI and FPU grants respectively. We would like to thank F.Cantrijn for helpful conversations and critical reading of preliminary versionsof this paper and the referee for pointing out us references [3] and [4].

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vipsymetries of vacanomic dynamics application to op cont

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