Rigid Body Equations of Motion for Modeling and Control Of

8/14/2019 Rigid Body Equations of Motion for Modeling and Control Of

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Rigid Body Equations of Motion for Modeling and Control ofSpacecraft Formations - Part 1: Absolute Equations of Motion.

Scott R. Ploen, Fred Y. Hadaegh, and Daniel P. ScharfJet Propulsion Laboratory

California Institute of Technology4800 Oak Grove Drive, Pasadena, CA, 91109-8099

AbstractIn this paper, we present a tensorial (Le., coordinate-free)derivationof the equations of motion of a formation consist-ing of N spacecraft each modeled as a rigid body. Specifi-cally, using spatial velocities and spatial forces we demon-strate that the equations of motion for a single free rigidbody (Le., a single spacecraft) can be naturally expressedin four fundamental forms. The four forms of the dynamicequations include (1) motion about the system center-of-mass in terms of absolute rates-of-change, (2 ) motion aboutthe system center-of-mass in terms of body rates of change,(3) motion about an arbitrary point fixed on the rigid bodyin terms of absolute rates-of-change, and (4) motion aboutan arbitrary point fixed on the rigid body in terms of bodyrates-of-change. We then introduce the spatial Coriolisdyadic and discuss how a proper choice of this non-uniquetensor leads to dynamic models of formations satisfying theskew-symmetry property required by an important class ofnonlinear tracking control laws. Next, we demonstrate thatthe equations of motion of the entire formation have thesame structure as the equations of motion of an individ-ual spacecraft. The results presented in this paper formthe cornerstone of a coordinate-free modeling environmentfor developing dynamic models for various formation flyingapplications.

1 IntroductionThe ability to accurately capt ure the dynamic behaviorof separated spacecraft formations in both deep spaceand in orbit around a central body is critical to thesuccess of many planned and future NASA missions.For example, the development and assessment of high-precision formation flying control laws will require aspectrum of spacecraft dynamic models ranging frompoint mass models to multi-flexible body models.To this end, we develop a tensorial formulation of theequations of motion of formations consisting of N sep-ara ted spacecraft. In addition, by utilizing the con-cept of a spatial vector (viz., a vectrix consisting ofboth rotational and translational vector quantities) itis possible to unify formation translational and rota-

tional dynamics into a single framework. The coordi-nate free approach using spatial vectors allows one tohave maximal physical insight into the structure of for-mation dynamics with a minimum of notational over-head. The coordinate-free approach discussed here isbased on the use of direct tensor notation to formu-late the equations of motion of the system. This ap-proach is especially powerful in applications where alarge number of observers (Le., reference frames) areinvolved in the dynamic analysis. Further, once a spe-cific set of generalized coordinates has been chosen, thetensorial equations admit a concise matrix form whichis amenable to computer simulation. The coordinate-free modeling architecture developed in this paper alsofacilitates the design of nonlinear tracking control lawsfor separated spacecraft formations.The primary focus of this paper is on the applicationof the coordinate-free approach to develop the equa-tions of motion for formations consisting of N space-craft, where each spacecraft is modeled as a single rigidbody. Much of the research done to date in the areaof formation flying dynamics has concentrated on thedevelopment of 3 degree-of-freedom (3DOF) transla-tional equations of motion along with associated dis-turbance models. For example, the linearized trans-lational motion equations of one spacecraft relativeto another spacecraft in a circular orbit (commonlycalled the Clohessy-Wiltshire-Hill equations) has beenaddressed by many researchers; e.g., see [ lo] . The as-sumption of a circular reference orbit has been relaxedin a number of papers; e.g., see [I ] or an overview. Us-ing coordinate-free notation, [ l l] iscusses the relativetranslational dynamics of formations in deep space andprovides insight into the validity of utilizing linear dy-namic models ( double integrator models) fox controllaw design. A unified 6 DOF description of formationflying dynamics (as well as guidance and control) hasbeen elusive; notable exceptions are [SI and [14].The rest of th is paper is organized as follows. Firs t, wediscuss material from rigid body kinematics and ten-sor analysis tha t are needed in the sequel. Then, usingthe concept of spa tial velocities and forces (i.e., com-bining linear/angular velocities and forces/torques into


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a single entity), we demonstrate that the equations ofmotion of a rigid body can be naturally expressed infour distinct forms. Although each form is an exactdescription of the dynamics of a single rigid body, eachform is not equivalent for developing dynamic modelsand control laws for separated spacecraft formations.To this end, we then discuss the proper selection of thespatial Coriolis dyadic, which is required in an impor-tant class of adaptive control laws used for the controlof systems of bodies including underwater vehicles [2],flexible space structures [7],and robotic systems [12].Next, we demonstrate that the absolute equations ofmotion of the N individual spacecraft can be concate-nated to form the motion equations of the entire for-mation. Moreover, it is shown that the equations ofmotion at the formation-level have the same struc tureas the equations of motion of a single rigid spacecraft.The final form of the absolute equations of motion pro-vide the first step toward a complete description of thedynamics of formations and can be tailored to applica-tions in various dynamic environments.

2 Differentiation of Vectors in RotatingFrames of Reference

In this section we discuss background material fromrigid body kinematics and tensor analysis that is re-quired in the sequel; see [4]and [6] for further informa-tion. Consider a geometric vector s describing somephysical quantity (e.g. velocity, force, angular momen-tum, etc.) of interest. Here we are making the impor-tant distinction between geometric or Gibbsian vectorsand column matGces or 3-tuples of real numbers. A ge-ometric vector Q is a quantity possessing magnitude,direction, and obeying the parallelogram law of addi-tion in three dimensional Euclidean point space, de-noted E 3 . A geometric vector should be thought of asan arrow or directed line segment in E3 . In particular,a vector 0 s a geometric object that exists indepen-dently of any particular basis chosen for E3 while acolumn matrix of numbers Q = [QI, 2 , Q3 ] E !R3'lis the representation of a geometric vector in a partic-ular basis. It follows that a column matrix Q dependson both the underlying vector and the particular frameof reference.

.i.

In rigid body kinematics we limit our attention to aspecial class of basis vectors for E3 called referenceframes. A reference frame consists of a right-handedset of three mutually orthonormal vectors located atan arbitra ry point (called the origin of the frame) inE 3 . The basis vectors associated with a reference framecan be easily accounted by defining a vectrix as fol-

Although this notational device apparently goes back to the1960's, the phrase vectrix (i .e. , part vector, part matrix) waspopularized by Hughes in [4]. pecifically, a vectrix is a matrix

- Alows: FA = [a'l a'2 & I T . In other words, the elementsof the vectrix f~ re the basis vectors characterizingthe given frame of reference. In the sequel, a referenceframe will be denoted as and the vectrix associatedwith the frame as ?A.Once a reference frame 34 has been defined, a-ge-ometric vector can be represented uniquzly as& =Qla'l + Qza'2 + Q & or equivalently as Q = ~ Q AThe real numbers Qi = . i;i are called the compo-nents of s elative to FA. n short, Q A is the col-8umn matrix whose entries are the components of Qin FA. Note that when a different reference frameF B is chosen, the same underlying geometric object

will admit a different column matrix representationQ B = [Q : ,Qh ,&S l T E !X3'l.Physically, a reference frame can be identified with anobserver who is rigidly mounted along the three mutu-ally orthogonal axes of FA. In th is paper, all observersare assumed to measure the same absolute time irre-spective of their state of motion.A fundamental result that will be used in the sequel isthe Transport Theorem for geometric vectors [6]:

A , ?Q=Q + [ A i j B]0fHere 0 enotes an arbitrary geometric vector, [a']b=

a' X b', A 3 B denotes the angular velocity of 3 B in FA ,A@ Qla'l+ Q&2 +Q&, and Q= Q',& +Q$2 +Q&&.The term Q (resp. Q ) can be interpreted physically asthe rat e of change of 0 s seen by an observer rigidlymounted to the axes of FA (resp. FB). As a conse-quence, if s is a vector fixed in FA (resp. FB) henA,Q= 0' (resp. s=8) .In the sequel we will also consider tensors of secondrank, called dyadics. In complete analogy with a vec-tor, a dyadic T is a geometric object that is indepen-dent of any observer. For our purposes, we regard adyadic as a linear operator T: 3 H E3; .e., a dyadicis a linear mapping on the space of geometric vec-tors. However, once a reference frame FA has beenintroduced, a dyadic can be represented uniquely as?=E T A ~ Ahere the elements of the 3 x 3 matrixTA are called the components (or matrix representa-tion) of T relative to FA. n short, TA is the 3 x 3matrix whose entries Tij are the representation of Tin FA, ote that when a different reference frame 3 Bis chosen, the same underlying geometric object T willadmit a different matrix representation, given by a dif-ferent 3 x 3 matrix TB E !R3x3with entries Tlj.whose elements are geometric vectors.

B-aA, B,

B

t)

+-+

*t)

*


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Table 1: Four Forms of the Dynamic EquationsI Form 11 Reference Point I Vector Derivatives IInertial

Th e following generalization of the Transport Theoremfor dyadics will also be used in the sequel:

A B

A* *AHere T denotes an arbitrary second rank tensor, T=T i j i i i Z j , T= Tlj&6j and the notation &S;j (resp. b i b j )denotes the dyadic (or tensor) product. Note that wehave employed the summat ion convention in the aboveexpressions. The reader should consult [5] for furtherinformation.

-.-.-A

Spacecraft Equations of Motion:Rigid Body Models

In this section we show that the equations of motionof a spacecraft modeled as a single rigid body can benaturally expressed in four different forms. To obtainmaximum insight into the structure of the equations ofmotion, coordinate-free vector/dyadic notation will beutilized throughout. The four forms of the equations ofmotion are classified as follows: (1)motion equationsabou t the system center-of-mass in terms of absoluterates-of-change, (2) motion equations about the systemcenter-of-mass in terms of body rates of change, (3)motion equations about a n arbi trary point fixed on therigid body in terms of absolute rates-of-change, and (4 )motion equations about an a rbitrary point fxed on therigid body in terms of body rates-of-change. The fourforms2 of the equations of motion are summarized inTable 1 .

Euler's Fundamental Laws of MechanicsThe following independent laws of mechanics, due toEuler in 1775, characterize the momentum balance of

2Note that other forms of the equations of motion result wheninertial derivatives are expressed with respect t o an obsever hav-ing arbitrary motion relative to the body. See Greenwood [3] forfurther details.

a single rigid body3:5h, = 7', (3)5P c = f (4)

where fi C is the absolute linear momentum of the body,zc is the absolute angular momentum about the masscenter of the system, f i s the resultant external forceacting on the body, ?,, is the resultant torque aboutthe system center-of-mass, and ( 0 ) denotes the rate-of-change relative to an inertial frame (i.e., an inertiallyk e d observer). Equation (3) is called the balance ofangular momentum and (4) is called the balance of lin-ear momentum. We now define the spatial momentumand spatial force vectors4 as follows:

N

- AF, = 151 (5)L J J

As a result, Euler's Laws of Mechanics (3)-(4) can beexpressed in the concise formN,H , = Pc (7)

The use of spatial vectors (i.e., the combination of lin-ear and angular quantities) not only leads to a sim-plified set of motion equations and deeper insight intothe dynamic behavior of rigid bodies, but also allowsthe unification o f translational and rotational motionwithi n a single framework. The spatial momentum vec-tor (about the center-of-mass) of a rigid body is relatedto the spatial velocity as follows

where the spatial velocity is defined as

and the spatial inertia dyadic5 is* *

0 m l

(9)

Here m denotes the (assumed constant) mass of thebody, 5, denotes the absolute velocity of the center-of-mass of the body, J , denotes the inertia dyadic of thebody abou t its center of mass, 1 is the unit dyadic, 0is the null dyadic, and w' = N3Bs the angular velocityof the body in FN. ote that the spatial inertia dyadicM , is symmetric.

*tt CI

*

3More generally, Euler's laws can be used to describe the mo-tion of a finite, arbitrarily deforming body; see [5] and [13] forfurther information.

4Note that the spatial velocity, spatial momentum, and spa,tial force are vectrices.5A patial dyadic is a matrix of dyadics.

P. 3


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. .First (Fundamental) Form of the Equations ofMotionIn this section we obtain the equations of motion of asingle rigid body about the center-of-mass in terms ofabsolute derivatives.Substituting the expression for the spatial momentum(8) into the momentum balance (7) and performing theinertial derivative we find

Nt iIn order to determine M c we generalize (2) for use withspatial dyadics:

N BGc=Gc p]iic- Gcp] (12)where

t+

p]& O ][GIBt + *Noting that Mc=O for a rigid body of constant mass,

we find that the equations of motion are

where

Expanding out (14) we find

Equation (17) (resp. (14)) will be called the fundamen-ta l form of the equations of motion for a single rigidbody.Second Form of the Equations of MotionIn this section we develop the equations of motionabout the center-of-mass in terms of body rates-of-change.Applying the transport formula (1) to the vectors L3and v', we find 3=3 and Gc=v', +[L3]GC. Note that interms of spatial velocities the above equations can be

N B N B

where R is as defined in (13) . Also [3]3 6 has beenused in (18) .Substituting (18) into (14) results in

[ 1G, (;c +p]E)+El vc= Pc (19)

Rearranging we find

where

= p]iic (22)Expanding out (20) the explicit form of the equationsof motion are

u

[ '++I [ ] + [ m [w']c ' ]', = [ 71 (23)V Cm lThird Form of the Equations of MotionIn this section we derive the equations of motion of asingle rigid body about an arbitra ry point fixed on thebody in terms of inertial rates-of-change.For an arbitrary point, denoted 0 , k e d o a rigid body

v', = 77 + [ 3 ] F c / , (24)7, = 7 c + [FC/,]S (25)

where FC/, denotes the vector from point o t o pointc. Using spatial velocities and forces (24)-(25) can beexpressed concisely as follows

- 4gc = T v, (26)(27)

t-+4F, = TT Pc

where(28)1

andthat 3, = 3, = L3 and f, = f7, = f i n (26)-(27).

= - [FC/ , ] . +Note that have used the fact

The absolute derivative of (26) is given byN N!Y * 4Vc=T vo+ ?v7, (29)

Substituting (29) and (26) into (11) and pre-multiplying by T we findt+ T

- 2 uM,V, + Cs Vo= Z', (30)P. 4


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uT u tN+-+ u T - u * U T * + +where M , = T M c T , C3=T M , T + T CIT, andUT Ur', =T 2c. Expanding out the expression for M ,we find

UTo find an explicit formula for C3 note that

Bu uwhere T= 0 since Fclo s fixed in the body. Substitut ing(32) into the above expression for c3 and expandingyie ds

H

We immediately find

Fourth Form of the Equations of MotionIn this section we derive two useful representations ofthe equations of motion of a rigid body about an ar-bitrary point fixed to the body in terms of body-fixedrates of change.

First Representation: Applying the trans-

[ 1eport formula Vo=V,+ R v, to (30) we obtain- B , * +M o V o + C4 Vo= Po (37)

wheret+

c 4 = c3 +Gop] (38)Substituting (33) into (38) and expanding we find

(39)It follows that

The following fact will be used to simplify (34):

Proposition 1 If a' an d b are arbitrary vectors then[Z][b7[qZ -[b][Z][a']G.- - -Proof: For any vector 2, [@'= 0 . Letting z'= Zx b =[ q g we find [Z x q[a' ]g= 0'. Upon using the identity[ Z x b] = [a '][q q[qand the fact that [Z]6= - [ qZ theresult follows.

-4

Applying Proposition 1 to (34LWe find m[Fc/,]GI [GI? = -m [GI [F,~,,][Fc~,]G. Asa result,

U HNote that J,=Jc -m [FC/,] F,..,] by the parallel axistheorem.Collecting together (31) and (35) , the equations of mo-tion of a rigid body about an arbitrary point fixed onth e body in terms of inertial rates of change is

Applying Proposition 1 to the term m[FCc/,]GI [W ]?,,I:in (40) , we find that the equations of motion about anarbitrary point on a rigid body in terms of body-fixedrates-of-change are

Note that (41) can be also be derived by applying thetransport formula directly to (36) .

Second Representation: An alternate repre-sentation of (41) results from applying the Jacobi iden-tity a x (6 + b x (Z x a') + c' x (a' x b') = 0' to theterm m[ F, /,] [GIG,:

m[Fc/o][G]Go = -[3][50]Fc/0 [50][Fc/o]3 (42)= [w'l[Fc/o]Go [50][Fc/o]w' (43)

We immediately find

P* 5


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Adding the zero vector in the form of m[i70]i70 o (44)yields

m l

After some manipulation the above equation can beexpressed as follows

- 2 *M o V o+ C4 Vo= Po (45)where

and(47)

Equation (45) is a coordinate-free version of the Liegroup based equat ions of motion of a single rigid bodydescribed in [9].Kinematic EquationsIn order to provide a complete description of the mo-tion of a single rigid body (i.e., a single spacecraft), aset of kinematic equations for each body is required.For a single unconstrained rigid body the kinematicdifferential equations (relative to the center-of-mass ofthe body) are given by r:= GC and R= [3 ] R. HereR denotes the rotation dyadic describing the orienta-tion of .FB (with origin at the center-of-mass of thebody c) relative to FN, C denotes the position of thecenter-of-mass of the rigid body relative to the origin ofthe inertial frame, and GC enotes the absolute velocityof the center-of-mass of the rigid body relative to a ninertially fixed observer. Note that a set of kinematicequations similar to those given above can be developedabout an arbitrary point o fixed on the rigid body.

ttNN *

Ct

The Spatial Coriolis DyadicIn the last section we developed the following four al-ternate forms of the equations of motion of a singlerigid body:

* * u * c ) Hwhere M l = M 2 = M c and M3=M4=Mo. Although theproducts C1 VclC2 Vc,C3 V o l C 4 Vo are certainlyunique, the spatial Coriolis dyadic Ci i = 1 , 2 , 3 , 4is not. As will be shown, this is a consequence of thefact that Cg is itself a function of the spatial velocity.

* tt * ttc)

tt

For the purpose of dynamic modeling and simulation,the non-uniqueness of the spatial Coriolis dyadic is notan important issue as any admissible choice of Ci leadsto the correct linear/angular accelerations. However,when developing nonlinear tracking control laws forseparated spacecraft (see [2 ] for applications to under-water vehicles) the choice of the spatial Coriolis dyadicis critical. Specifically, Ci must be defined in such away tha t it renders M i -2 Ci skew-symmetric. For ex-ample, in [12] a globally stable adaptive control law forrobotic vehicles is designed that results in asymptotictracking of a desired reference trajectory q d ( t ) E !Rnwhere qd denotes specific generalized coordinates. Th estability proof of the adaptive control law requires thatsT(U- C )s = 0 where s E !Rn is a function of bothq and q and M is the system mass matrix [12]. As aresult, the matrix representation of the Coriolis dyadic,denoted C, ust be constructed in such a way to ren-der the matrix (U - 2C ) skew-symmetric6. The ex-plicit relationship between the matrices M and C andthe dyadics M and C for a multibody spacecraft is dis-cussed in [9].It is also known [9] th at the equations of motion of amultibody spacecraft (i.e., a spacecraft consisting of acollection of hinge connected rigid bodies) inheri t theskew-symmetry property from the equations of motionat the individual body level. As a result, i t is importantto define the appropriate spatial Coriolis dyadic Ci atthe level of each individual rigid body. To this end, thefollowing result is useful.

c)

*

e-*NC I

c) *

c)

N* *Proposition 2 If Ci s skew-symmetr ic then M i-2 Ci i s skew-symmetr ic .c)

NProof: Recalling Gi= d] Gi - Gi p] t fol-

N* Tlows immediately that M i = - M i Observing thatthe difference of two skew-symmetric tensors is skew-svmmetric establishes the result.

N*

'It is important to note that if s = q , then qT(k 2C)q = 0irrespective of the skew-symmetry of A4 - C. This statement isa property of finite-dimensional natural systems; see [9] for addi-tional details and references. As a result, it is only in situationswhere (M - 2C) is pre- and post-multipl ied by a column vectordifferent from q (the typical case in control design) that C shouldbe carefully defined.


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We now discuss some specific choices of the spa tia lCoriolis dyadic tha t render M i -2 Ci skew-symmetric.Beginning with the fundamental form of the equationsof motion (14) we find from expanding (16) that

uNu

*Noting that C1 is skew-symmetric, it follows that thefundamental form of the equations of motion are man-ifestly skew-symmetric. We will denote C1 in (52) byc1 *

u

C I ss

The Coriolis dyadic associated with the second formof the equations of motion (20) is not skew-symmetric.In order to develop a skew-symmetric representationwe expand (22) and find

(53)A skew-symmetric Coriolis dyadic results by modifying(53) as follows

[w']J1: - J1: [d] 0- ] (54)2 = [ uu ss 0 77431Note that although we have modified the Coriolisdyadic, the second form of the equations of motion (20)has not changed since C2 V, =C2 V,. The above tech-nique of adding the zero vector in a judicious fashion isthe key to developing the appropriate spatial Coriolisdyadic for use in nonlinear spacecraft control.The fourth form of the equations of motion leads toseveral admissible skew-symmetric forms. For example,although (39) is not skew-symmetric, a skew-symmetricrepresentation of the fourth form of the equations ofmotion results from the following modification of (39)

*s s u

The skew-symmetric spatial Coriolis dyadic given in(55) can be used to construct another skew-symmetricform of the equations of motion. To this end considerthe product

Applying the identity [Z][@ = [Z x 6]Z to the term[w'] [Fc /O] , using the identity [Z]b= -[b]Z, and sub-tracting the zero term m[GO]GOields

... ...

To simplify (57) the following result is required:Proposition 3 If Z, 6 an d c' are arbitrary vectors then[Z][[b7c'= -[6x dc'+ [Zx 216Proof: We find from rearranging the Jacobi ide_ntitya x (6x E') +6 x (Z x Z)4+c'x (a' x 6) = 0' that... [Z][b]c'=- [ q [ q Z - [ q [ Z ] b= - [ b ] [ q Z + [ q [ q Z = -[b](:c'x Z) +[ q 6 x Z) = [Zx Z ] L 6x Z]Z. a

+

Applying Proposition 3 to the term [Fc/o][d]fjon (57) ,we find after some rearranging

Th e form of the Coriolis dyadic given in (58) is similarto the result obtained by Fossen in [2].

3 Equations of Motion for FormationsIn this section we demonstrate that the equations ofmotion of an entire formation of N rigid spacecraft hasthe same structure as the equations of motion of a sin-gle rigid spacecraft. In order to discuss collections ofrigid bodies, the previous notation introduced for a sin-gle rigid body (cf. equation (48)) must be modified. Tothis end, the equations of motion of the ith spacecrafti = 1 , 2 , .. ,N are denoted

EGi ( P I vi (PI+ Ei (p)E(p)= F t ( p ) (59)where point p is either the center-of-mass c or a generalpoint on the body o of the ith spacecraft7. Here theCoriolis dyadic Ci ( p ) is assumed to be any admissibleskew-symmetric dyadic (consistent with the point p ) asdiscussed in the previous section.

u

we immediately find tha t th e global equation s of motionof the format ion can be expressed as:

- 2 u _ .M V + C V = T (64)7Strictlv p = p ; because the equations of motion of each bodv

G- J , [GI W' +m [GIC - m[ Co ] Coc, VI =-m [w' x W' - m [go] - - - ~.(57) can be expressed with respect t o different reference points.

P. 7


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. _Note that equation (64), describing the global dynam-ics of the formation, has the same st ruct ure as the equa-tions of motion of a single rigid body. Although we haveused inertial derivatives in (64), body fixed derivativescan also be used. The following property of the forma-tion equations of motion is of interest for the design offormation control laws:

uProposition 4 If ci ( p ) is skew-symmetric for i =1,2 , . . ,N then M -2 c is also skew-symmetric.N* t-)Proof: The proof follows from applying Proposition2 directly t o t he definitions of M and c given above.u et

As will be demonstrated in a later paper, the global(absolute) formation equations of motion (64) formthe starting point for describing a formation as a vir-tua l multi-body system with a branched-chain (or tree)topology . For example, once a particular spacecrafthas been designated as the leader (i.e., the basebody ofthe virtual multibody chain), the remaining spacecraftare analogous to the rigid links of a multibody chain.Each follower spacecraft is attached to the basebodyspacecraft via a 6DOF free-free joint. Here, a free-freejoint represents unconstrained motion between bodies;i.e., a free-free joint consists of a free 3 DO F prismaticjoint and a free 3DOF spherical joint. As a result, inthe virtual multi-body framework, the relative equa-tions of motion of th e formation become the primaryconcern. Due to space limitations, the relative equa-tions of motion will be developed in a future paper.

4 ConclusionsIn this paper we have developed a tensorial (i.e.,coordinate-free) derivation of the equations of motionof a formation consisting of N spacecraft each mod-eled as a rigid body. The results presented here arethe first step toward developing a coordinate-free ar-chitecture for formation flying dynamic modeling andcontrol. Future work will address (1) characterizing therelative dynamics of formations using the concept of avirtual multi-body system, (2) developing the appropri-ate environmental d isturbance models, (3) developingexplicit techniques to linearize and perform sensitiv-ity analysis on the formation equations of motion, and(4) eveloping the equations of motion for formationswhere each individual spacecraft is itself a multi-bodysystem.

*A multibody system is called branched-chain or tree topologyif it consists of a set of serial chains of bodies, each connected toa central basebody.

AcknowledgmentsThe research described in this article was performed a tthe Jet Propulsion Laboratory, California Institute ofTechnology, under contract with th e National Aeronau-tics and Space Administration. The authors would liketo thank Dr. A. Behcet Acikmese of JPL for valuabletechnical discussions regarding this paper.

References[I ] T.E . Carter , State transition matrices for termi-nal rendezvous studies: brief survey and new example,J . Guidance, Control, and Dynamics. Vol. 21, No. 1,[2] T.I. Fossen, Guidance and Control of Ocean Ve-hicles. Chichester: John Wiley, 1994.[3] D.T. Greenwood, Advanced Dynamics. Cam-bridge: Cambridge University Press, 2003.[4] P.C. Hughes, Spacecraft Attitude Dynamics. NewYork: John Wiley, 1986.[5] W. Jaunzemis, Continuum Mechanics. NewYork: MacMillan, 1967.[6] T.R. Kane, and D.A. Levinson, Dynamics: The-ory and Applications. New York: McGraw-Hill, 1985.[7] A. Kelkar, and S. Joshi, Control of NonlinearMultibody Flexible Space Structures: Lecture Notes inControl and Information Sciences, Volume 221. NewYork, Springer, 1996[8] H. Pan, and V. Kapila, Adaptive nonlinear con-trol for spacecraft formation flying with coupled trans-lational and attitude dynamics, Proc. IEEE Conf.on Decision and Control. Orlando, FL, pp. 2057-2062,2001[9] S.R. Ploen, A skew-symmetric form of the recur-sive Newton-Euler algorithm for the control of multi-body systems, Proc. American Control Conf. SanDiego, pp. 3770-3773, 1999.[IO] J.E. Prussing, and B.A. Conway, Orbital Me-chanics. New York, Oxford University Press, 1993.[ l l ] D. Scharf, F. Hadaegh, and B. Kang, On thevalidity of th e double integrator approximation in deepspace formation flying, Proc. Int. Symp. FormationFlying Missions and Technology. Toulouse, France,2002.[I21 J.J.E. Slotine, and W. Li, Applied NonlinearControl. Englewood Cliffs, NJ: Prentice Hall, 1991.[13] C. Truesdell, Whence the law of moment ofmomentum?, Essays in the History of Mechanics.Springer-Verlag, New York, pp. 239-271, 1968.[14] P.K.C. Wang, and F.Y. Hadaegh, Coordinationand control of multiple microspacecraft moving in for-mation, J . of the Astronautical Sciences. Vol. 44, No.3, pp. 315-355, 1996.

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Rigid Body Equations of Motion for Modeling and Control Of