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Pergmon PIk sooo5-1098( )00176-8Aulomdcn, Vol. 33, No. 3. pp. 331-346, 19970 1997lsevier Science Ltd. All rights reserved

Printed in Great Britainm5-1098/97 17.Gu + 0.00

Stability of a Bottom-heavy Underwater Vehicle*NAOMI EHRICH LEONARD?

Conditions for stabil ity of underwater vehicle dynamics are established as afunction of vehicle shape, mass distributi on and equi li bri um veloci ty usingmethods from geometric mechanics.Key Words-Stability; dynamic stability; underwater vehicle dynamics; marine systems; stabilization.

Abstract-We study stability of underwater vehicle dynamicsfor a six-degree-of-freedom vehicle modeled as a neutrallybuoyant, submerged rigid body in an ideal fluid. We considerthe case in which the center of gravity and the center ofbuoyancy of the vehicle are noncoincident such that gravityintroduces an orientation-dependent moment. Noting thatKirchhoffs equations of motion for a submerged rigid bodyare Hamiltonian with respect to a Lie-Poisson structure, wederive the Lie-Poisson structure for the underwater vehicledynamics with noncoincident centers of gravity andbuoyancy. Using the energy-Casimir method, we then deriveconditions for Lyapunov stability of relative equilibria, i.e.stability of motions corresponding to constant translationsand rotations. The conditions reveal for the vehicle stabilityproblem the relevant design parameters, which in some casescan be interpreted as control parameters. Further, theformulation provides a setting for exploring the stabilizingand destabilizing effects of dissipation and externally appliedcontrol forces and torques. 0 1997 Elsevier Science Ltd.

1. INTRODUCTIONThe study of submerged bodies has a longhistory, but is receiving renewed attention inlight of challenging, new problems in underwatermotion control that are emerging from a growingindustry in underwater vehicles for deep seaexploration. This type of vehicle, often referredto as a submersible to distinguish it from asubmarine, is typically small and versatile, withperformance requirements, design capabilitiesand even physical configuration differing fromthose of a traditional submarine. Further, manyof the currently active underwater vehicles areremotely operated, and there is a more recenttrend to make them completely autonomous.In order for unmanned underwater vehicles tosucceed, they must be able to control their own

*Received 12 October 1995; revised 24 May 19%; receivedin final form 4 September 1996. This paper was not presentedat any IFAC meeting. This paper was recommended forpublication in revised form by Associate Editor Albert0Isidori under the direction of Editor Tamer Ba$ar.Corresponding author Professor Naomi Ehrich Leonard.Tel. +1 609 258 5129; Fax +1 609 258 6109, [email protected] Department of Mechanical and Aerospace Engineering,Princeton University, Princeton, NJ 08544, U.S.A.

motion reliably and efficiently. Because under-water vehicle models are nonlinear, it ispromising to explore the use of modem controltechniques that exploit system nonlinearities. Forexample, in the event of an actuator failure onboard an unmanned underwater vehicle, non-linear control methods can be used for motioncontrol where techniques based on linearizationwould fail to be useful (Leonard, 1995a,b).Other nonlinear control methods have also beenexplored for adaptive and robust control ofunderwater vehicle motion (Cristi et al . , 1990;Fjellstad and Fossen, 1994; Yoerger and Slotine,1985). Further, nonlinear methods have beenused successfully for the related problem ofattitude control and stabilization of a rigidspacecraft; see Tsiotras er al. (1995) for a survey.In this paper we use a modem geometricframework to investigate the stabilizing effect ofgravity on the motion of a six-degree-of-freedom, submerged rigid body in an ideal fluid.Fjellstad and Fossen (1994) also consider asix-degree-of-freedom vehicle model; however,this is a departure from a typical assumption thatdynamics in the horizontal plane are decoupledfrom the remaining degrees of freedom. In orderto address problems such as actuator failures, itis desirable to consider the coupled six-degree-of-freedom model, since we may need to takeadvantage of the coupling as means ofcompensating for the failures.We model the underwater vehicle as aneutrally buoyant rigid body in an ideal fluid, sothat Kirchhoffs equations of motion, based onpotential flow theory, provide a dynamic modelof the body. In the case that there are noexternal forces or torques acting on the body,these equations are Hamiltonian with respect toa Lie-Poisson structure (Arnold et al., 1993).We consider the case in which the center ofbuoyancy and center of gravity of the body arenoncoincident, resulting in an orientation-

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332 N . E. Leonarddependent moment acting on the body This is ofpractical interest, since underwater vehicles arebuilt with the center of gravity below the centerof buoyancy (i.e. bottom-heavy) for stability. Weshow that in this case the equations of motionare still Hamiltonian, and we derive theassociated Lie-Poisson structure. The Lie-Poisson structure is determined by a reductionprocess that exploits symmetry in the system.Here symmetry corresponds to invariance of theequations of motion under reference coordinateframe translation in any direction and rotationabout the direction of gravity.The advantage of describing the equations ofmotion in a geometric framework is theavailability of a variety of analytical tools thatmake use of the geometric structure as a meansof determining nonlinear systems behavior. Forexample, in order to study nonlinear (Lyapunov)stability of motions of the body, we use theenergy-Casimir method described in Marsdenand Ratiu (1994). The energy-Casimir methodprovides a means to derive sufficient conditionsfor nonlinear stability of an equilibrium solution.Here equilibrium solutions, or relative equilibria,correspond to constant translations and rotationsof the vehicle.The results of our analysis provide conditionsfor stability that depend on the physicalconfiguration of the vehicle, the equilibriumvelocity of the vehicle, and the distance betweenthe center of gravity and the center of buoyancy.These describe the relevant design parameters,which in some cases can be considered as controlparameters. For example, suppose the vehicleneeds to suddenly move quickly away from aparticular location and suppose the distancebetween the centers of gravity and buoyancy isnot sufficiently large to ensure stability. Bymoving masses around on board the vehicle, onecould control this length on-line, potentiallystabilizing the motion.The results of this study are practically validonly in those circumstances in which potentialflow theory is a sufficiently accurate model of thevehicle dynamics. This will be the case forstreamlined motions, e.g. when an ellipsoidalvehicle is translating along (and/or rotatingabout) its major axis. In this case viscous effectsare small and our ideal fluid model captures thedominating dynamic effects. In the case ofnon-streamlined motions, e.g. motion along thebodys minor axis, the potential flow model doesnot capture effects that may be important, e.g.vortex shedding. Nonetheless, we note that it isactually the streamlined motions that are of mostinterest to us. First, these are the motions thatwe shall want the vehicle to perform, since,

because resistance is minimal, they can beaccomplished most efficiently. Secondly, as weshow in this paper, these are the motions thatare naturally unstable and require help fromspin, gravity or external forces and torques inorder to be stabilized, i.e. these are the motionsthat provide a need for nonlinear methods.The results of this paper constitute a first stepin laying the groundwork for addressing ourlarger goal of deriving control strategies forstabilization and motion control of underwatervehicles, including those that have a limitednumber of actuators. The Hamiltonian structurederived and used here provides a means of

studying stability of submerged rigid-bodydynamics in which we focus on the effect ofhydrodynamics and gravity. It further provides astarting point to explore the larger picture thatincludes dissipation and external forces andtorques. We note in this regard that there arerecent relevant results on the effect of adding asmall amount of dissipation to a Hamiltoniansystem (Bloch et al., 1994) and on the use of theenergy-Casimir method in deriving stabilizingcontrols for a rigid spacecraft (Bloch et al.,1992). New results on feedback stabilization ofunderwater vehicle dynamics using the Hamil-tonian structure and stability conditions derivedhere can be found in Leonard (1996b).Kozlov (1989) has studied the stability andmotion of a submerged rigid body that is notneutrally buoyant but falls as a result of agravitational force that is larger than the buoyantforce. In this work the motion of the body isrestricted to the vertical plane, i.e. thesix-degree-of-freedom problem is notconsidered.This paper is organized as follows. In Section 2we derive the equations of motion for theunderwater vehicle with noncoincident centersusing Newton-Euler balance laws, and usereduction to derive the associated Lie-Poissonstructure. In Section 3 we apply the energy-Casimir method to study stability of equilibriumsolutions. We apply the method first to the caseof the body with coincident centers. This is theclassical problem studied by Lamb (1932) forwhich he derived conditions for spectral stabilityusing linearization. Using the energy-Casimirmethod, we prove conditions for nonlinearstability that are stronger than Lambs result.Next, we apply the energy-Casimir method tothe case with noncoincident centers to revealconditions for the stabilizing effect of gravity.In Section 4 we summarize our results anddiscuss future work. For additional detailson the work described in this paper see Leonard(1996a).


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Stability of a bottom-heavy vehicle 3332. EQUATIONS OF MOTION

2.1. K i nema t i c sFollowing the development in Leonard(1995a), we identify the position and orientationof the underwater vehicle with SE(3), the groupof rigid-body motions in R3. An elementX E SE(3) is given by X = (R, b), whereR E SO(3) is a rotation matrix that describes theorientation of the vehicle and b E R3 is a vectorthat describes the position of the vehicle. That is,if xi, is any point on the vehicle described withrespect to a body-fixed coordinate frame and X,is the same point expressed with respect to theinertial coordinate frame, then

(;) =(; i)(T) = fi;+b).Let R = (Q,, f& R3)T be the angular velocity ofthe vehicle in body-fixed coordinates. Letu = (ul, u2, LJ~)~be the translational velocity ofthe origin of the body frame given in terms ofbody-fixed coordinates. Define -: R3+ SO(~),where &(3) is the space of 3 X 3 skew-symmetricmatrices, by &;P = (YX @ for (Y,p E R3. Then, Rand b satisfy

Z?=R@b=Ru. (1)

The pair (&, u) is an element in the vector spaceSe(3) which is the space of infinitesimal rotationsand translations in R3. This space becomes a Liealgebra on defining the Lie bracket on ge(3) as

[q, .] :Ge 3)X Ge 3)+ Ge 3),[ a;, pi), Sj, pj)] H ( i j - j i, Sipj - djPi)*

The Lie bracket is a formal way of describinghow rotations and translations commute. Given{A,,.. . , A6}, a basis for se(3), we define thestructure constants I: for this basis as

[Ai, Aj] = 5 r$A,.k=l2.2. Newton-Euler description of dynamicsIn this section we derive the dynamicequations of motion using Newton-Euler bal-ance laws. The derivation follows Lamb (1932)with additional reference to Fossen (1994). Weassume that the vehicle is submerged in aninfinitely large volume of incompressible, irrota-tional and inviscid fluid at rest at infinity, wherethe motion of the fluid can be characterized by asingle-valued velocity potential. We fix theorthonormal coordinate frame b,, b2, b3) to thevehicle with origin located at the vehicles centerof buoyancy. Then, as Kirchhoff showed, thekinetic energy of the fluid Tf is given as a

quadratic function of body velocity. Let W =[u f iTIT; hen

where 0 is a 6 X 6 constant, symmetric matrixwith elements determined by the configurationof the vehicle and the density of the fluid. 0 isdiagonal in the case of a vehicle that has threemutually perpendicular planes of symmetry, e.g.a vehicle that can be approximated as anellipsoid, if we set the body-fixed axes to be theprincipal axes of the displaced fluid.Similarly, the kinetic energy of the vehiclealone, Tb, can be expressed as

T = wT@w @ = mlb 2 ,miG

where m is the mass of the vehicle, Z is the 3 X 3identity matrix, Zb is the inertia matrix of thevehicle and r, is the vector from the origin of thebody-fixed frame to the vehicle center of gravity.The total kinetic energy of the body-fluid systemT = Tf + Tb S

T = $W (O + 0 )W= &-l JO + 2QTDu + uTMu), (2)

whereM=mZ+O: l, .f=.& ,+@;,, D=mi ,+@:,.

The matrices O:, and 05, are sometimes referredto as added mass and added inertia matricesrespectively. In Appendix B we provide formulasfor the elements of these matrices.The impulse of the body-fluid system, asdefined by Lord Kelvin, is that required tocounteract the impulsive pressures acting on thesurface of the vehicle and to generate themomentum of the vehicle itself. While thisimpulse is not equivalent to the total momentumof the system (roughly speaking, it excludes theinfinite term in the momentum), it can be shownto vary as the momentum of a finite dynamicalsystem would vary under the influence ofexternal forces and torques.Let p and a be the linear and angularcomponents of the impulse respectively withrespect to the inertial orthonormal coordinateframe. Let P and II be the analogouscomponents given with respect to the body-fixedframe. Then

p=RP, (3)x=Rl-I+bxp. (4)Let fi, i = 1,. . . , k, be the external forcesapplied to the vehicle given in inertial coordin-


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334 N. E. Leonardates, and let pi be the vector from the origin ofthe inertial frame to the line of action of theforce A, i = 1,. . . , k. Let f = Ef==,fi, and letF = RTf, i.e. the applied forces given in bodycoordinates. Similarly, let r be the net externaltorque applied to the vehicle in inertialcoordinates and Y = Rz. Then

d =f, (5)*=T+ ;xf ; . (6)

i =

Differentiating (3) and (4) with respect to timeand using (l), (5) and (6), we find thatP=PxQ+F, (7)II=IIxn+Pxu

+ 5 RT(p; - b)) x Rf + T. (8)i=lP and II can be computed from the total energyT given by (2) as

P=E=Mu+DTR,

rl=$=J*+Du. (10)Using (9) and (10) for P and II in (7) and (8)gives Kirchhoffs equations of motion for a rigidbody in an ideal fluid.It is interesting to note that, because of thefluid, the mass matrix M is not a multiple of theidentity (unless the body is symmetric, as in thecase of a sphere). As a result, Mu is genericallynot parallel to u, so that the term Mu X u in (8)is nontrivial.In this paper we study a neutrally buoyant,ellipsoidal vehicle and assume that the principalaxes of the displaced fluid are the same as theprincipal axes of the body. Neutral buoyancymeans that the buoyant force is exactly equal

Fig. 1. Vehicle with noncoincident centers of gravity andbuoyancy.

and opposite to the gravitational force. Ourassumptions imply that D = miG and J and M areeach diagonal.In the special case that the center of buoyancyis coincident with the center of gravity, i.e.ro = 0, D = 0, so thatP=M u, n=JQ

and gravity does not appear in the dynamicequations of motion.In the more general case, we let the centers ofbuoyancy and gravity be noncoincident asillustrated in Fig. 1. We assume that the centerof gravity (CG) lies along the b3 axis a distance 1from the center of buoyancy (CB) such thatr, = l e3 (and so D = Ml e^ ,). When 1 >O thevehicle is bottom-heavy, and when 1~ 0 thevehicle is top-heavy. Further, while the buoyantforce acts through the origin of the body-fixedframe, the gravitational force does not. Thismeans that gravity exerts an orientation-dependent moment on the vehicle. Let k be theunit vector that points in the direction of gravitywith respect to the inertial frame. Define

r=R-kas the direction of gravity with respect tobody-fixed coordinates. The gravitational force isgiven by fi = mgk. Let p, be the vector from theorigin of the inertial frame to the center ofgravity of the vehicle. Then the moment appliedto the body due to gravity is given in bodycoordinates byR.(p, - b) X RTfi = r , X RTmgk = -mgl(T X e3).Thus the equations of motion for this case are

P=PxR+F, (11)Ii = I I X Q + P X u - mgl(T X e3)

+ i (RT(pi - b)) x RTJ + T,i=2 12)r=rx A, (13)

where gravitational terms have been explicitlyrepresented.2.3. Lie-Poisson description of dynamicsIn this section we derive the Lie-Poissondescription of the dynamics of the underwatervehicle with noncoincident centers. We show thedynamics to be Hamiltonian with respect to aLie-Poisson structure, which is a generalizationof the canonical Hamiltonian structure. Thederivation uses reduction, for which our mainreference is Marsden and Ratiu (1994). Reduc-tion is a way of exploiting symmetry in thedynamics so that one can study system dynamics


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Stability of a bottom-heavy vehicle 335by considering a reduced (i.e. simplified) set ofequations of motion. For the underwater vehicle,the configuration space is Q = SE(3) and thephase space is T*Q = T*SE(3), the cotangentbundle of SE(3), which is a twelve-dimensionalspace.

2.3.1. Coincident cent ers of buoyancy andgravity. In the preparation for our derivation inthe noncoincident centers case, we tirst describethe Lie-Poisson structure for the neutrallybuoyant vehicle with coincident centers and noexternal forces or torques (Arnold et al ., 1993).In the case where the vehicle is ellipsoidal, theLagrangian L : TSE(3) + R is given from (2) by

L(R, b, Rii , Ru) = $(Q=JQ + u=M u).SE(3) acts on itself by left translation,Z(,,,,: SE(3) + SE(3), according to- - - - -~~IT.#, b) = (RR Rb + b), (R, b),

(R, b) E SE(3).Under the left action of SE(3), the Lagrangianbecomes

W~~RS,(R b, Rfi, Ru))- -= L(i ?R, i?b + b, RR& RRu)= $(Q=JQ + u=M u) = L(R, b, R ?& Ru),

where T-Fe(i.6):TSE(3) + TSE(3) is the deriva-tive of the left translation. This shows theLagrangian (and likewise the Hamiltonian andthe equations of motion) to be invariant underrotations or translations of the inertial frame,and so we say the system has SE(3) symmetry.Accordingly, Lie-Poisson reduction of thedynamics by the action of SE(3) induces aLie-Poisson system on ge(3)*, the dual of theLie algebra Ge(3). That is, because of symmetry,the equations of motion can be reduced fromthose on the twelve-dimensional phase spaceT*SE(3) to the six-dimensional space Ge(3)*. Letp = (II, P) E ge(3)*; then the reduced Hamil-tonian on r;e(3)* is

H(p) = H(II, P) = ~(I-I AI I + P=CP), (14)where we define

A = J- l , C = M- .Given two differentiable functions G,K onde(3)*, the Lie-Poisson bracket on Ge(3)* thatmakes ge(3)* a Poisson manifold is

{G, K)(e) = VGT&)VK (15)where

The Lie-Poisson equations of motion are givenby

r_ii={pi,H}(p), i=1,...,6. (17)Equivalently, by (14) and (15),

(~)=*(p)vH(p)=(nx~~~xu),18)which are the equations of motion (7) and (8)derived using Newton-Euler balance laws.

A Casimi r functi on Ci on Ge(3)* is one inwhich VCi is in the nullspace of A, or,equivalently, {Ci, K}(p) = 0 for any function K.This implies that Casimir functions are con-served quantities along equations of the form(17) for any H, i.e. along the equations (18) inparticular. The nullspace of A given by (16) hasrank 2 (for P ZO ), and two independentCasimirs are given by

C,(II, P) = n * P, C,(rI , P) = ll P112.Further, any Casimir can be expressed as asmooth function of these two.

Coadjoint orbits of Se(3)* are smoothimmersed submanifolds of 5e(3)* on which theCasimir functions are constant. Therefore flowsof (17) do not leave the coadjoint orbit on whichthey start. The coadjoint orbit through p =(II, P), where P # 0, is diffeomorphic to T*S$,,,the cotangent bundle to the two-sphere of radiusIIP 11,which is a four-dimensional manifold.2.3.2. Noncoincident cent ers of buoyancy andgravity. The Lie-Poisson description of theneutrally buoyant vehicle with noncoincidentcenters and no external forces of torques (seeFig. 1) is derived in this section. In this case theLagrangian is the kinetic energy of the systemminus the potential energy due to gravity, i.e.L,(R, b, Rfi, Ru) = & I=JQ + 2Q=Du

+ uTM u + 2mgZ(k * Re,)),where the subscript k indicates that theLagrangian is smoothly parametrized by k.Under the left action of SE(3), the LagrangianbecomesL,W&,,_,(R, b, Rfi, Ru))- -= L,@R, Rb -I - , RR& , l?Ru)= $(QJQ + 2QTDu + uTM u + 2mgl(ETk *Re,)).So,L,W~ ,w,(R, b, Rfi , Ru))

= L,(R, b, Rfi , Ru)w i?=k = k,


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N. E. Leonard36left-invariant under the.e. the Lagrangian is

action of the groupGk = ((R, b) c SE(3) 1Rk = k) = SE(2) x R.

(19)Gravity has broken the full SE(3) symmetry ofthe coincident-centers case, leaving the (left)symmetry group of this system to be SE(2) x Ft.This means that the Lagrangian (and conse-quently the equations of motion) are unchangedif we translate the inertial frame in any directionand rotate it about k , the direction of gravity.Because of gravity, reduction by SE(3) as in thecoincident-centers case is no longer possible.However, following Marsden et al. (1984) we canreduce the Hamiltonian system on T*SE(3),which can also be parametrized by k and is alsonecessarily left-invariant under the action ofSE(2) X IL ,o a Hamiltonian system on the dualof the Lie algebra of a semidirect product. In thiscase we shall reduce the equations of motionfrom those on the twelve-dimensional phasespace T*SE(3) to a nine-dimensional space.Define p: SE(3)-+ Aut(R3) by p(R, b) = R,where Aut(W3) is the space of automorphisms ofR3. Then S = SE(3) X, R3 s a semidirect product,where group multiplication is given by

((R b), w)((Rt b), w)= ((R b)(R, b), p(R, b)w + w)= ((RR, Rb + b), Rw + w ).

Let p* be the associated right representation ofSE(3) on R3*. Then

p*(R, b)k = k @RRTk = k,so that G, as defined by (19) is the stabilizer ofk E W3*under p*, i.e.

Gk = {(R, b) E SE(3) 1p*(R, b)k = k}.By Theorem 3.4 of Marsden et al. (1984) theHamiltonian dynamics on T*SE(3) can bereduced to a Lie-Poisson system on thenine-dimensional space 5*, the dual of the Liealgebra 6 of S. 5 = Ge(3) X,. R3, here p is theinduced Lie algebra representation, i.e. for(IY,p) E se(3), ~(a, p) = & . By performingmatrix commutation of two elements in d givenin matrix form, we find that the Lie bracket on 4is given by[(a, P, Y), (a, P, ?)I= (a x a, a x p - a x p, a x y - ff x y).The triplet cr. = (II, P, F) is an element in G*,whereII = JR + Dv, P = M u + DTR, 1- = Rrk. (20)

Alternatively, we can express Q and u in termsof TI and P:R=ATI+BTP, u=CP+BII, (21)

whereA = (J - DM -LI T)-,0 = -CDTJ- = -M-DT A, 22)C = (M - D=J-D)-.

From this, we can compute the reducedHamiltonian on g* asH(p) = $(I ITAI l + 2TITBTP + P=CP

- 2mgqr .e3))). (23)The Lie-Poisson bracket on g* can be found asfollows. Let {B,, . . , B,} be the standard basisfor 5 such that if BP, i = I, . . . ,9, is the dualbasis for ?j*, then /1 is given in local coordinatesas

i=l

where (CL,, . , pg) = (II,, . . . , r,).tensor A is computed according to

where Fz are the structure constantschosen basis of 6. Explicitly, we find

A(~)=A(~, P, r) = (24)Thus the Lie-Poisson bracket of two

The Poisson

for the

differentiable functions G and K in dualcoordinates on g* is

{G, K}(P) = VGT&)VK,and the equations of motion are given by

Ii0= NP)VH(P)I-i

IIxQ+Pxu-mglrxe,= PXRi

. (25)rm

These are the equations of motion (ll)-(13)derived using the Newton-Euler balance laws.The nullspace of the Poisson tensor A given by


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Stability of a bottom-heavy vehicle 337(24) has rank 3 for P, r # 0 and Ptr. Threeindependent Casimir functions on 5* are

C,(lI, P, r) = P * ,w , p, r) = I I PI I * ,c,(n, p, r) = vi i* .

Further, any Casimir can be expressed as asmooth function of these three. Note thatc,(n, P, r) = ~PI I lrll os = ~COS e,

where 8 is the angle between P and I.Coadjoint orbits of 5* are smooth immersedsubmanifolds of 5* on which the Casimirfunctions are constant. For details of thecoadjoint action and orbits for the Poissonstructure on 5* see Appendix A and Leonard(1996a). The coadjoint orbit through p =(II, P, r), where P, r # 0 and P 1 I (i.e., 0 # 0),is diffeomorphic to the six-dimensional manifoldT*S0(3). When (II, P, r) is such that P 11 (i.e.8 = 0) and P and I are not both 0, then A losesrank, i.e. the nullspace of A has rank 5. Pointssuch as these are called nongeneric. In this case,in addition to the three Casimirs defined above,II. P (and therefore II - r) are constants ofmotion, called sub-Casimirs. The coadjoint orbitthrough (II, P, r), where P and I are not bothzero and P 11 , is diffeomorphic to thefour-dimensional manifold T*S*.

3. STABILITYWe analyze the stability of relative equilibriaof the underwater vehicle equations of motionusing the energy-Casimir method for the vehicleboth with coincident and with noncoincident

centers of buoyancy and gravity. A relativeequil ibrium for the underwater vehicle is asolution in the phase space T*SE(3) thatcorresponds to a fixed point p., for the reduceddynamic equations, i.e., for equations (18) or(25). In the case of coincident centers p., ESe(3)*, while for noncoincident centers CL,E G*.Here relative equilibria correspond to constanttranslations and rotations.Using the energy-Casimir method, we canstudy stability of the relative equilibrium in thephase space T*SE(3) by studying stability of thecorresponding equilibrium pe in the reducedspace. The energy-Casimir method providessufficient conditions for nonlinear stability of anequilibrium pe in reduced space. Here nonlinearstability refers to stability in the sense ofLyapunov, i.e. for every neighborhood V of p,there is a neighborhood U of p, such thattrajectories that start in U never leave V. This isdistinct from spectral stability, which requires

only that the linearization of the reduceddynamics at pe have no eigenvalues with positivereal part. We recall that for Hamiltonian systemseigenvalues are symmetrically distributed underreflection about the real and imaginary axes, sothat spectral stability implies that all eigenvaluesof the linearization are on the imaginary axis.Thus spectral stability does not imply Lyapunovstability, and so linearization techniques are notsufficient for proving stability of motions.Linearization can, however, be used to proveinstability of pe, i.e. divergent behavior in thereduced dynamics near pu,, since an eigenvaluewith positive real part (spectral instability) doesimply instability in the sense of Lyapunov.Proving Lyapunov stability of p, in thereduced space using the energy-Casimir methodimplies stability of the corresponding relativeequilibrium in the phase space T*SE(3) modulothe symmetry group G (e.g. here G = SE(3) orG = SE(2) X R). Stability modulo G is just theusual notion of Lyapunov stability, except thatone allows arbitrary drift along the orbits of G(Patrick, 1992). For example, stability inT*SE(3) modulo SE(3) means that there is thepossibility that there will be drift in therotational and translational parameters but notin the linear and angular momentum parameters.

However, for the underwater vehicle problem,stronger results are possible (Leonard andMarsden, 1996). Indeed, Lyapunov stability ofpe in the reduced space will typically implystability of the corresponding relative equilib-rium in T*SE(3) modulo a smaller group thanthe symmetry group. In particular, there will bethe possibility of drift in the translationalparameters (i.e. in the directions correspondingto the noncompact part of the symmetry group),but drift in the rotational parameters will bepossible only about the axis of rotation and notaway from the axis of rotation. An extension toenergy methods, such as the energy-Casimirmethod, that can be used to prove this kind ofstronger stability conclusion is presented inLeonard and Marsden (1996). This work makesuse of reduction by stages and a result forcompact symmetry groups due to Patrick (1992).We note that Lamb (1932) does not extend hisclassical stability analysis of a submerged rigidbody to the full twelve-dimensional phase spaceT*SE(3), i.e. he does not indicate whether ornot translation and rotation parameters drift.

The energy-Casimir method gives a step-by-step procedure for constructing a Lyapunovfunction to prove stability. For a genericequilibrium of a Lie-Poisson system, theLyapunov function is a function of theHamiltonian, the Casimir functions and any


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338 N. E. Leonardother conserved quantities. Let Ha.+ = H +@(Ci ) + $J~(c~)where H is the Hamiltonian, Cjare the Casimirs, Cj are other constants ofmotion, and @(a) and ~j(.) are smooth functionsto be determined. The method then consists infinding a(.) and ~j(.) such that /_L, s a criticalpoint of Ha,+,. The condition that the secondderivation of Ha.+, be definite (positive ornegative) at pe is sufficient for nonlinear stabilityof pe. See Holm et al. (1985), Krishnaprasad(1989) and Marsden and Ratiu (1994) for furtherdetails. Stability of a relative equilibrium issometimes referred to as relat ive stabil i t y.

may be other nontrivial equilibrium solutions forparticular orderings of the mass and inertiaparameters, but we do not consider those here.)Without loss of generality (because of sym-metry), we shall study the two-parameter familyof equilibrium solutions

Aeyels (1992) gives an alternative interpreta-tion of the energy-Casimir method in terms of atopological criterion sufficient for Lyapunovstability of an equilibrium point pe for a systemwith constants of motion. Given that cl, . . . , ckare constants of motion, he shows that if p, is alocally isolated solution of c,(p) = cj(p,), j =19 . . , k, then pe is Lyapunov-stable. Theenergy-Casimir method is a sufficient conditionfor this topological criterion.

II, = (0, 0, IQ, P, = (0, 0, Pi),which corresponds to constant translation alongand rotation about the b3 axis (i.e. the thirdprincipal axis). We take Pi #O , so that theequilibrium solution is generic, i.e. it lives on amaximal (four-dimensional) coadjoint orbit. Weshall further assume (for ease of computation)that the b3 axis is an axis of symmetry, i.e. Z, = Z2and m, = m2. In this case we have Ii3 = 0, so thatany function 4(II,) is conserved.

To begin, we consider conserved quantities ofthe form

Here we emphasize the perspective fromgeometric mechanics, since it is the Lie-Poissonstructure that provides a means to find theCasimirs that are the conserved quantities thatwe need to use to prove stability. Thesignificance of Lie-Poisson structure and Casi-mir functions in the energy-Casimir method canbe further appreciated in recent related work onfeedback stabilization of underwater vehicledynamics (Leonard, 1996b).

H a,+ = H + W . P, II PII ) + WL),where, from (14), H = $(II TAII + PTCP), andcP(*), and 4(e) are to be chosen as necessary.Define

When evaluated at the equilibrium solution, thefirst derivative of H,,, is zero if and only if

3.1. Coincident centers of buoyancy and gravit yIn this section we analyze stability of anellipsoidal, neutrally buoyant vehicle withcoincident centers of gravity and buoyancy usingthe Lie-Poisson structure described in Section2.3.1. Additional details of the computations canbe found in Leonard (1996a). We assume thatthe principal axes of the vehicle and the principalaxes of the displaced fluid are coincident, so that

Next we check for definiteness of the secondvariation of Ha,+ at the equilibrium point. Let

a = @(II;), b = 4@(II;e, (P;)),c = & (I@$, (P;)), d = 2&(II:e, (P:)).

The matrix of the secondequilibrium point isM = diag (m,, m2, m3).

Recall that the equations of motion with noexternal forces or torques are

P=PXR,

where II = JQ and P = M u.Of interest is the stability of three sets oftwo-parameter families of equilibrium solutions,each family corresponding to constant transla-tion along and rotation about one of theprincipal axes of the vehicle. (We note that there

(1r,006

0\O

&=_ L d@qrI *P) @ = a( 11 /I ) .

& (II ;P;, (P;)) = - i? + @(II :)) & , (26)

:= -$+(T+@(lI:,)&. (27)

derivative at the

01;00

0

0a330

0a36

00

1+2wm2

00

0 0

6 00 a360 0

1+2w 0m2

0 a66 ,(28)


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Stability of a bottom-heavy vehicle 339where

a66- $ + 2 0sufficiently large, positive-definiteness is assuredif and only if

(29)Define

e =?+ c#qII$)and note that we have the freedom to choose eto be as large as necessary, since we can choose4(II$ arbitrarily. Using (26) and (27), thecondition (29) becomes

-- IJ%+;(-&.$o. (30)(l&e2 + 12 P$The left-hand side of (30) is a pol@omial in e.Since the leading coefficient is negative, thispolynomial will be positive for some e if and onlyif the discriminant is positive, i.e.

(31)Thus, by the energy-Casimir method, (31) is asufficient condition for nonlinear stability of theequilibrium solution (nonzero e).

Stable.Vehicle cm be in any arkntatianwith wspect to the inertialfrnmcsincegravityplaysna role).

Stable ijangulnr/?ranslationalmomentum is sujicicntly high, i.e.C~3/~o)2.412w?n3 -l/ m2)>0Unstable f@3/kj, mt (see Appendix Bfor details). Thus the right-hand side of thecondition (31) is negative, so the equilibriumsolution is stable for all values of IIt and Pi.That is, the vehicle moving in the direction of itsminor axis is stable. Now suppose that l3 > Z,, i.e.the vehicle is a prolate spheroid. Then m3 C m2and the right-hand side of (31) is positive. Thusthe equilibrium solution is stable if the ratio ofangular momentum to linear momentum is largeenough to satisfy (31). These results aresummarized in Fig. 2.We use spectral analysis of the linearizedequations to find conditions for instability of theequilibrium solutions. When evaluated at(II,, P,), the linearized equations evolve on thetangent space to the coadjoint orbit in Ge(3)*through (II,, P,) given by{XI, 6P E lR3 R3 1Sri *P, + n, .6P = 0

and SP *P, = 0) = {NI ,, 6112,6P,, 6P2} = R4.That is, at (II,, P,), SfI , = Sp3 = 0, i.e. thelinearization always has at least two eigenvaluesfixed at 0. A computation shows that at least oneof the other four eigenvalues of the matrix of thelinearization will have positive real part if andonly if (31) holds with the direction of theinequality sign reversed.The condition (31) for stability in this caseagrees with that described by Lamb (1992).However, Lamb derives this condition usinglinearization, and thus can only conclude spectralstability when the condition is met. The resultderived here proves nonlinear (Lyapunov)stability of the equilibrium when the condition ismet. This is stronger than Lambs result. We canstate the results as the following theorem.Theorem 1. Consider an ellipsoidal, neutrallybuoyant vehicle with coincident centers ofgravity and buoyancy and b3 an axis ofsymmetry. Constant (nonzero) translation alongand rotation about the b3 axis is (nonlinearly)stable if

If this relation holds with the direction of theinequality sign reversed, the motion is unstable.Note that, based on our analysis, we cannotconclude stronger than spectral stability for thecase in which the condition in Theorem 1 holdswith equality. For this, one would need toexamine higher-order derivations. Because of


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340 N. E. Leonardsymmetry, the analogous theorem holds forconstant translation along and rotation about thebl or b2 axis with the appropriate substitutionsfor Z2,m2 and m3. See Fig. 2 for a summary.3.2. Noncoinci dent cent ers of buoyancy andgravi tyNext, we analyze stability in the case withnoncoincident centers of buoyancy and gravityusing the Lie-Poisson structure described inSection 2.3.2. Additional detail on the computa-tions can be found in Leonard (1996a). Weassume that the vehicle is an ellipsoid withsemiaxes I,, l 2 and I,, where l i lies along the bjaxis. We assume that the principal axes of thevehicle and the principal axes of the displacedfluid are coincident and rG = le,, so that

J = diag (I,, 4, M,M = diag (ml, m2, m3),D = mle^,.

From (22), this implies that

a, = m2 ii2 = mlm21, - m212 m, I 2 m212c , =

12 c ; = 1,m, I 2 m212 m I - m2121b = - m1m, I 2 m212 h2= _ m1m I - m2121

We have mzI, -m212>0 and m,t, - m212>0.Thus a,, Z2, C,, C2>0 and b,, b,


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Stability of a bottom-heavy vehicle 341The conditions (35) and (36) imply that

and consequently that these equilibrium solu-tions are nongeneric in that they live on afour-dimensional coadjoint orbit of G*. SinceI, 11R,, the spin axis will always coincide withthe direction of gravity. One set of equilibriumsolutions with spin is a two-parameter family,parameterized by Q,, and ue3, of the form

(37)where Q = 0, , a,,) and u, = 0, 0, u,~)~. Thiscorresponds to constant translation along androtation about the b3 axis, which coincides withthe direction of gravity, i.e. a rising or fallingvehicle. A second set of equilibria is charac-terized by P, = 0. In the general case in whichm, # m2, I1 # Z2, i.e. the body is not symmetricabout its b3 axis, this second set of equilibriamust satisfy 52elL&2 0. For either &, #O ora, # 0, the set of equilibria is a one-parameterfamily when we impose the additional constraintthat IlI,II = 1. For example, when R,, # 0, thefamily of equilibria is parametrized by SL,, andgiven by

r = (I/& - Z3)QT3e

4 ( 1

;I,

a3

where Q = (Q,,, 0, R,3)T, u, = (0, mlR,Jm2, O)T,and Re3 is computed such that llI,II = 1. Furtherequilibrium solutions with spin may be possible,depending upon the relative magnitudes of themass and inertia parameters.3.2.1. Stabil ity of equil ibri um solution with nospin. In this section we study the equilibriumsolution given by (34), which we shall denote asII, = (II:, 0, o)T, P, = (0, e, o)T, r, =(0,0, i)T,where I: = m2ue2and II: = -mlue2 = -mlP~lm,.We shall interpret II: = IIy(P$, so that we keepin mind that this is a single-parameter family ofequilibria. It is useful to study stability of thisfamily of equilibria, because it corresponds totranslation along a principal axis perpendicularto the direction of gravity, which is a practicalmotion of interest. Stability of the more general

two-parameter family of no-spin equilibriumsolutions can be analyzed in a similar manner.We begin by considering conserved quantitiesof the form

Z-Z* H + *(P - r, vi15 iirii7,where H is given by (23). Define

When evaluated at the equilibrium solution, thefirst derivative of Z& is zero if and only if

2@(IIyP$ (fi)2, 1) = -l/m2, (38)b(n:P:, (E)2, 1) = 0, (39)

2@+(@PX, (e)2, 1) = mgl . (40)Next we check for definiteness of the secondvariation of Ha at the equilibrium solution. Let

a=& , b=2& , c=4W ,d = 2&+, e = 4Wt, f = 4Qtt

with partial derivatives evaluated at the equilib-rium. In the case that we choose b = d = e = =0 and we use the conditions (38)-(40), forpositive-definiteness of the second variation ofHa we need

m2--mlm2(mI Z2 m212)> 0, c(P$%, > 0

L-L+a>O, mglB0,m3 m2

mgl + a(en( k - 2 + a) - (ae) > 0.We note that b=d=e=f=O is the bestsimplifying choice, since the above inequalitieswould have been necessary regardless, and thechoice introduces no extra conditions forpositive-definiteness. The last inequality issatisfied by taking a > 0 sufficiently large and

(41)Therefore if we choose a > 0 sufficiently largeand c > 0 then positive-definiteness is assured ifand only if (41) holds and

mgl > 0, (P ) > 0, m2 > ml. (42)By the energy-Casimir method (41) and (42) aresufficient conditions for nonlinear stability of theequilibrium solution.These conditions can be interpreted as follows.Recall that lj is the length of the semiaxis of theellipsoidal vehicle along the bi axis. For stable(nonzero) translation along the b axis, if I2 < l3then we also need l2 < I, and mgl >O, i.e. themotion should be along the minor axis and thecenter of gravity should be below the center ofbuoyancy. If l2 > 13, we also need l2 < 1, and mgl


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342 N. E. Leonard

Stable.

Stable i jsuf ic ient ly bot tom-heavy.

-E

02p20 mgl > (l/Q - 1 m P*) 0

Unstable ifl3 < l2 < II mgl< l/m,- 1/~,w3*

Spectrally stable I suf ic ient lytop-heavy and P{ is suf ic ient lylarge. Otherwise, unstab le (seeTheorem 2).

Fig. 3. Summary of stability with noncoincident centers andno spin (Theorem 2).

should satisfy (41) i.e. the motion should bealong the intermediate axis, the vertical b3 axisshould be the minor axis and the center ofgravity should be sufficiently far below the centerof buoyancy. These results are summarized inFig. 3.If one of the conditions (41) or (42) fails thenthe equilibrium solution is formally unstable.Using a spectral analysis of the linearizedequations, we now investigate when the equilib-rium solution is in fact unstable due to failure ofone of these conditions. When evaluated at(II,, P,, r,), the linearized equations evolve onthe tangent space to the coadjoint orbit in G*(3)through (n,, P,, r._) given by

{(HI, SP, ST) E 0316~. r, + 6r. P, = 0,~P,P,=o, sr.r,=o)=W

This means that the linearization has threeeigenvalues fixed at 0. The characteristicpolynomial of the matrix of the linearization is

where91 =

+ G2mgl,42= -ii*mgl( -.yy.

From the polynomial factor of degreesee that the chacteristic polynomialunstable eigenvalue if

From the polynomial factor of degree four, we

(43)(44)

two, wehas an

compute that the characteristic polynomial hasan unstable eigenvalue if (i) m2 > ml andmgl m3 and that the translational velocity ue2 behigh). In this latter exceptional case theequilibrium solution will be spectrally stable.This is very curious, since it implies that, for anappropriate vehicle configuration, constant tran-slation along the intermediate axis (ml > m2 >m3) of the vehicle looking like an invertedpendulum is spectrally stable (although notnecessarily nonlinearly stable). It is likely that,with the introduction of dissipation as in Bloch etal. (1994), this top-heavy vehicle motion will beunstable.We can state the results of this stabilityanalysis of the no-spin equilibrium solution asthe following theorem.Theorem 2. (St abil it y of bottom-heavyvehicle). Consider an ellipsoidal, neutrallybuoyant vehicle with noncoincident centers ofgravity and buoyancy. Constant (nonzero)translation along the b2 axis is (nonlinearly)stable if

m2>ml, mgl 0, mgl>($-$)(P$.If these relations hold with the direction of atleast one of the inequality signs reversed, thetranslation is spectrally unstable except underthe following special conditions, for which thetranslation is spectrally stable:

m2


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Stability of a bottom-heavy vehicle 343result refers to a top-heavy vehicle. Note alsothat equality conditions are not covered by ouranalysis. Because of symmetry the analogoustheorem holds for constant translation along thebl axis with the roles of ml and m2, etc.,swapped. See Fig. 3 for a summary.3.2.2. Stabi l i t y of equi l i br ium solut ion w it hspin. In this section we examine the two-parameter family of equilibrium solutions givenby (37) which corresponds to a vehicle rising (orfalling) along and rotating about the vertical axisand is denoted asII, = (O,O, II?), P, = (0, 0, fi )T, r, = (O,O, )T,

(45)where IIt = Z,Q_, and e = m3ue3. We shallassume (for ease of computation) that the b3axisis an axis of symmetry, i.e. Z1= Z2 and ml = m2.From the equations of motion, (25), this impliesthat H3 = 0, so that any function 4(113) isconserved.This equilibrium solution is nongeneric, sincethe equilibrium lives on a four-dimensionalcoadjoint orbit (rather than a generic six-dimensional coadjoint orbit). As a result, oneneeds to be careful with applying the energy-Casimir method in this case (Krishnaprasad,1989).

We consider conserved quantities of the formH a,d, = H + @(P * r, llPl12, llrl12 + W-I,),

where H is given by (23) and @(a)and 4(e) canbe chosen as necessary. A straightforwardcomputation as in the previous cases shows thatthe equilibrium solution (45) is a critical point ofHa,,, and the second variation of H*,+ evaluatedat the equilibrium is (positive-) definite if andonly if

mgl>($-$)(P )2. (46)Interpret Ha,+ as a Lyapunov function, which weknow to be constant along the equations ofmotion. If (46) holds, the equilibrium solution isa strict minimum of Ha,+, and is therefore stableby Lyapunovs direct method. Thus (46) is asufficient condition for stability of the equilib-rium solution.The condition (46) can be interpreted asfollows. If l3 < I1 = 12, i.e. the vehicle is an oblatespheroid, then m3 > m 2 so that the right-handside of (46) is negative. Therefore verticalmotion along the minor axis is stable if thevehicle is bottom-heavy or even slightly top-heavy, i.e. mgl may be slightly negative. IfI3 > II, i.e. the vehicle is a prolate spheroid, thenm3


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N. E. Leonard

Fig. 4. Summary of stability with noncoincident centers andspin (Theorem 3).

stability in terms of equilibrium velocity andsystem parameters such as masses and momentsof inertia for the body-plus-fluid system and thedistance between the centers of gravity andbuoyancy. These parameters can be consideredas design parameters. For example, given thedesired operating range of the vehicle in terms ofvelocity, one can design the vehicle for goodstability properties by applying the conditions ofthe theorems. Alternatively, these parameterscould be interpreted as control parameters, i.e.parameters to be changed on-line if necessary forstability. For example, redistribution of massinside the vehicle will affect the center of gravitywithout affecting the center of buoyancy.Consequently, one could change the distancebetween the two centers as a means ofpreventing unstable behavior in circumstancesthat required unusually high-velocity motions.For example, to stabilize a vehicle that mustsuddenly rise very quickly, one could lower massto the bottom of the vehicle so that the vehicle issufficiently bottom-heavy.Remark 5 As noted in Section 1, the potentialflow model that we use in this paper is onlypractically valid for streamlined motions, i.e. formotions with small viscous effects. For anellipsoidal body these correspond to translationsalong or rotations about the bodys major axis.In the case of a vehicle with coincident centers ofgravity and buoyancy the practically relevantstability result is the condition for stability oftranslation along the bodys major axis. FromTheorem 1, pure translation along the major axisis unstable, but can become stable by spinning

Stable

Stable i suficiently bottom-hea vy or suficien tly high- spin.i.e.

mg [> ( l /m, - ~lmdV~)*- nfj5q4

Unstable if

wl< ( l /m, - l /m Y- MYw

the body about this axis (assumed to be an axisof symmetry) such that the ratio of angularvelocity to translational velocity is sufficientlyhigh. In the case of a bottom-heavy vehicle, fromTheorem 3, pure translation along the major axisof the vehicle moving parallel to gravity is stablewith sufficient spin velocity and/or a sufficientlylow center of gravity. However, from Theorem2, pure translation along the major axis of thevehicle moving perpendicular to gravity isunstable no matter how far the center of gravityis located below the center of buoyancy. It istherefore of interest to find control forces andtorques that will stabilize this motion.

4. CONCLUSIONSUnderwater vehicles for deep sea exploration

typically have different performance require-ments compared with more traditional sub-marines. Accordingly, they may be quitedifferently shaped and equipped. This hasconsequences for stable and accurate control ofmotion. On the one hand, requirements for thesubmersible may preclude including surface suchas fins and a tail that on a submarine providestability. On the other hand, one may havegreater opportunity and flexibility in designingvarious other features such that the vehicle ismore easily and stably maneuvered and moreefficiently controlled. We believe that nonlinearmethods can be used to tackle some of therelevant issues.

In this paper we have provided somegroundwork for studying the six-degree-of-freedom underwater vehicle modeled as a rigidbody. We have investigated stability of thevehicle submerged in an ideal fluid, focusing onthe effects of hydrodynamics and gravity. Wehave derived the Hamiltonian structure andapplied the energy-Casimir method to findconditions for nonlinear stability of equilibriumsolutions corresponding to constant translationsand rotations.Since viscous effects are left out of the model,the results are only practically valid for thoseequilibrium solutions corresponding to stream-lined motions. These, however, are the solutionsof interest, since low resistance is desirable forhigh-efficiency motion. We have briefly men-tioned the potentially destabilizing effect ofdissipation; however, a deeper investigation ofthe effect of dissipation is warranted. Forexample, it is of interest to know, in the caseswhere we have shown stability, when theaddition of dissipation yields asymptotic stability.Results along the lines of Bloch et al. (1994) mayprovide direction.


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Stability of a bottom-heavy vehicleFor an ellipsoidal bottom-heavy vehicle,

motion along the major axis in the planeperpendicular to the direction of gravity wasshown to be unstable independent of how far wedrop the center of gravity relative to the centerof buoyancy. As a result, in order to stabilize thismotion, we need to consider applying externalforces and torques. In recent work we investigatethe use of external forces and torques forstabilization (Leonard, 1996b). In this work theenergy-Casimir method proves as useful as itwas in deriving a stabilizing control law for therigid spacecraft (Bloch ef al., 1992).Acknowledgement-It is a pleasure to thank P. S.Krishnaprasad and J. E. Marsden for helpful comments andsuggestions during the preparation of this work. I am alsograteful to H. C. Curtiss, P. J. Holmes and S. Vatuk forcommenting on an earlier draft of this paper.This research was supported in part by the NationalScience Foundation under Grant BES-9502477 and by theOffice of Naval Research under Grant NOOO14-96-1-0052.

REFERENCESAeyels, D. (1992). On stabilization by means of theenergy-Casimir method. Syst. Control Len., 18,325-328.Arnold, V. I., V. V. Kozlov and A. I. Neishtadt (1993).

Dy namical Systems II I: M athematical A spects of Classicaland Cel esti al M echanics, 2nd ed. (ed. V. I. Arnold).Encyclopedia of Mathematics, Vol. 3, Springer-Verlag,Berlin.Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden and G.Sanchez de Alvarez (1992). Stabilization of rigid bodvdynamics by internal and external torques. Autom&a, 2&745-756.

Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden and T. S.Ratiu (1994). Dissipation induced instabilities. Arm. Inst.Henri Poincark, Anal. Nonl int aire, 11,37-WCristi, R., F. A. Papoulias and A. J. Healey (1990). Adaptivesliding mode control of autonomous underwater vehicles inthe dive plane. IEEE J. Oceanic Engng, 15,152-160.Fjellstad, 0. and T. I. Fossen (1994). Position and attitudetracking of AUVs: A quaternion feedback approach.I EEE J. Oceani c Engng, 19,512-518.Fossen, T. I. (1994). Gui dance and Contr ol of OceanVehi cles. Wiley, New York.Holm, D., J. Marsden, T. Ratiu and A. Weinstein (1985).Nonlinear stability of fluid and plasma equilibria. Phys.Rep., U& 1-116.Kochen, N. E., I. A. Kibel and N. V. Roze (1964).Theoretical Hydromechanics. Wiley, New York.

Kozlov, V. V. (1989). Heavy rigid body falling in an idealfluid. Mech. Soli ds, 24,9-17.Krishnaprasad, P. S. (1989). Eulerian many-body problems.In Contemporary M athematics, Vol. 97, pp. 187-208.AMS, Providence, RI.Lamb, H. (1932). Hydrodynamics, 6th ed. CambridgeUniversity Press.Leonard, N. E. (1995a). Control synthesis and adaptation foran underactuated autonomous underwater vehicle. IEEEJ. Oceani c Engng, Z & 211-220.Leonard. N. E. (1995bj. Periodic forcing. dvnamics and,control of underactuated spacecraft -and* underwatervehicles. In Proc. 34th IEEE Con& on Decision andControl, New Orleans, LA, pp. 3980-3985.Leonard, N. E. (1996a). Stability of a bottom-heavyunderwater vehicle. MAE Technical Report 2048,Princeton University. Available electronically viahttp://www.princeton.edu//-naomi/.Leonard, N. E. (1996b). Stabilization of steady motions of anunderwater vehicle. In Proc. 35th I EEE Conf on Decisionand Control, Kobe, Japan.Leonard, N. E. and J. E. Marsden (1996). Stability and drift

of underwater vehicle dynamics: mechanical systems withrigid motion symmetry. MAE Technical Report 2075,Princeton University. To appear in Physica D.Marsden, J. E. (1992). Lectures on M echani cs. LondonMathematical Society Lecture Note Series, No. 174,Cambridge University Press.Marsden, J. E., T. Ratiu and A. Weinstein (1984). Semidirectproducts and reduction in mechanics. Trans. Am. M ath.Sot., 281, 147-177.Marsden, J. E. and T. S. Ratiu (1994). Introduction toM echanics and Symmetr y. Springer-Verlag, New York.Patrick, G. W. (1992). Relative equilibria in Hamiltoniansystems: the dynamic interpretation of nonlinear stabilityon a reduced phase space. J. Geom. Phys., 9,111-119.Tsiotras, P., M. Corless and J. M. Longuski (1995). A novelapproach to the attitude control of axisymmetricspacecraft. Automatica, 31, 1099-1112.Yoerger, D. R. and J.-J. E. Slotine (1985). Robust trajectorycontrol of underwater vehicles. I EEE J. Oceani c Engng,10,462-470.

APPENDIX A. COADJOINT ACTIONS AND ORBITSFOR SE(3) X, 88From Appendix A of Leonard (1996a), the coadjoint

action of S on d* = W3X R3 X R3 has the expressionAdfR,h.w&, y, z) = (Rx + 6Ry + @Rt , R y, Rz).

The coadjoint orbit through p E 6*, denoted by O,, isdefined as0, = (AdfR.h,w)-I(p) 1 R b>w) E Sl.

Since g* is nine-dimensional, there can be zero-, two-, four-,six- and eight-dimensional coadjoint orbits. It turns out thatthere are no eight-dimensional coadjoint orbits. The onlyzero-dimensional orbit is the origin. We find the four- andsix-dimensional coadjoint orbits by applying the semidirectproduct reductiop theorem (Marsden et al., 1984). LetG = SO(3) and V = R3 X R3, and let p: SO(3)+ Aut(W3 XR3) be defined by

p(R). (b, w ) = (Rb, Rw), (R, b, w ) E SO(3) X 8a3 R3.Then 3 = SO(3)X,-(W3 X R3) is a semidirect product andS =S. The dual of v =R3 X W3 can be identified with vitself. e = SO(3) acts on the dual of v as

R .(B, y) = UW, Ry), P, Y E R3 X R3.The isotropy_ for the action of SO(3) on v* at(/ I, y) E P* = V is then given by

Gc~.v) = (R SO(3) 1 R/3, RY) = (By Y), B, Y o R3 X R31.In the generic case in which /3 X y # 0 (i.e. ply and p,Y#tJ),

r&v) = 1 SO(3),which is trivially an Abelian group. Following Marsden(1992), the coadjoint orbit through generic p = (o, 8, y) EGO(~)* X R3 X R3 is diffeomorphic to

T*(C/&+,) = T*S0(3),which is a six-dimensional space. In the case that p 11 (ora=Oor y=O),

%.r, ={RESO(~)IR~~=/~}=S,which is again an Abelian group. The coadjoint orbit throughp = (a, B, y) E SO(~)* X R3 X W3, 8 11 , is then diffeomor-phic to - -T*(G /G(a,,,) = T*(S0(3)/S) = T*S,which is a four-dimensional space.


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346 N. E. LeonardAPPENDIX B. ADDED MASS AND INERTIA

In this appendix we give the formulas for the elements 01the added mass and inertia matrices for a submergedellipsoidal body. We assume, as in Section 3, that the ineritamatrix for the body, Jb, is diagonal. ThenM = diag (m,, m2, m3) = ml + MA urdrJ = diag (I,, 12. ) = 4, + JAuld, (B.1)

Mauid= @i = diag (A, B, C),Jnu,d = 05, = diag (P, Q, R).

Following Lamb (1932), we definecc dhatI = 111213 (f i + h)A FI PO = 1112h I = dho (I: + A)A (B.2)

Yo= i1r13 A = v(/cr: + A)([: + A)(/: + A) ,where I i is the length of the semiaxis of the ellipoidal bodyalong the axis b i = 1.2, 3. Then

A= ePv,-poB = _ p,, POV.

02 (1: I:)(% - Yo)5w: - I?)+ (1:+ MY" - ad PO&L (1: ~:)@cl-45 2(1: - 12,) (1: + I$)(a m,. For example, by (B.l) and (B.3)

If I, < l 3 then, by (B.2) a0 > y,,, and so ml > m3. Similarly, if1, > I3 then aa < yO, and so m, < m3.

The ordering of the moments of inertia, I,, I2and 4, is notso straightforward. Fix 1, and I 2 and suppose that I, > I2 > 13.For different values of 1, the ordering of the inertia termsdepends upon the ratio 1,/12. When 13/12 is close to one.1, > I2 > I,. However, when 13/12 1, > 13. For intermediate values of 13/12,12>13>l,. Note that it will always be true that I,>I,;however, 1s can be the largest, intermediate or smallestmoment of inertia. For physical intuition consider anellipsoid that rotates about its long axis. When this is an axisof symmetry, i.e. l2 = 13, there is no added inertia aboutthe long axis (p = 0). So, I, is equal to the moment of inertiaof the body alone. Now suppose we shrink 13, i.e. we flattenthe ellipsoid. In this case it is harder to spin it about the longaxis. In fact, the added inertia P about this axis can growarbitrari ly large (and thus I, grows very large) as l 3 decreases.At the same time 1, grows even faster. See Kochen et al.(1964) for parametric plots of added inertia terms.

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