iac-12-c1.8.1 coupled orbit and attitude dynamics for

63rd International Astronautical Congress, Naples, Italy. Copyright c©2012 by the International Astronautical Federation. All rights reserved.

IAC-12-C1.8.1

COUPLED ORBIT AND ATTITUDE DYNAMICS FOR SPACECRAFTCOMPRISED OF MULTIPLE BODIES IN EARTH-MOON HALO ORBITS

Amanda J. Knutson∗

Purdue University, United States of [email protected]

Kathleen C. Howell†

Purdue University, United States of [email protected]

The coupled orbit and attitude motion of a spacecraft in three dimensional (halo) referenceorbits is explored in this investigation within the context of the Circular Restricted Three BodyProblem (CR3BP). The motion of a spacecraft, comprised of two rigid bodies connected by a singledegree of freedom joint, is examined. The nonlinear variational form of the equations of motion isemployed to mitigate numerical effects caused by large discrepancies in the length scales. Severalnonlinear, periodic reference orbits in the vicinity of the collinear libration points are selected forcase studies and the effects of the orbit on the orientation, as well as the orientation on the orbit,are examined. Additionally, the model is extended to include solar radiation pressure and themotion of a solar sail type spacecraft is considered.

I. INTRODUCTION

The Circular Restricted Three Body Problem(CR3BP) has been examined extensively in recentyears; however, considerably fewer studies incorpo-rate the effects of spacecraft attitude in this regime.Numerous missions in the vicinity of the collinear li-bration points have been completed, are ongoing, orare planned for the near future.1–3 For example, theJames Webb Space Telescope (JWST), successor tothe Hubble Telescope, is to be deployed in a Sun-EarthL2 Lissajous orbit within the next decade.4 Due tothe sensitive nature of the dynamics at the librationpoints, a better understanding of the attitude motionof spacecraft in the vicinity of these points is desired.The ability to predict the attitude motion of a space-craft as orbital parameters and spacecraft characteris-tics are varied can be very useful. Once the behavioris investigated, the possibility of exploiting the naturaldynamics for passive attitude control in these regimesis an option. A recent investigation examined the mo-tion of a spacecraft constructed as two rigid bodies; thevehicle is located in various planar Lyapunov referenceorbits, in the vicinity of the Earth-Moon collinear li-bration points.5 One goal of the current investigationis an examination of the effects of incorporating atti-tude dynamics into the model for the nonlinear threedimensional orbits within the context of the CR3BP,

∗Ph.D. Candidate, School of Aeronautics and Astronautics†Hsu Lo Professor of Aeronautical and Astronautical Engi-

neering

particularly the impact on vehicles in halo referenceorbits. Additionally, the model is extended to includeother types of perturbations, for example, a spacecraftmodel for which solar radiation pressure is significant;the analysis is further expanded to explore the motionof solar sails.

Kane and Marsh first consider the attitude stabil-ity of a axisymmetric satellite, located exactly at theequilibrium points.6 In their analysis, the satellite isartificially maintained at each of the equilibrium pointsand only the attitude motion is considered. Robin-son continues this work with an examination of theattitude motion of both a dumbbell satellite7 and anasymmetric rigid body8 artificially maintained at theequilibrium points. Abad et al. introduce the use ofEuler parameters in the study of a single rigid bodylocated at L4.9 More recently, Brucker et al. ex-plore the dynamics and stability of a single rigid bodyspacecraft in the vicinity of the collinear points in theSun-Earth system, using Poincare maps.10 The effectsof the gravity gradient torque are explored by the au-thors, but the spacecraft is still artificially fixed to theequilibrium point.

Wong et al. offer a detailed examination of the mo-tion of a single rigid body near the vicinity of the Sun-Earth libration points, using linear Lyapunov and haloorbit approximations.11 The response of the spacecraftis detailed by Wong et al. in a meaningful manner, of-fering orientation results for a number of numericalsimulations in terms of a body 3-2-1 (θ, φ, ψ) Eulerangle rotation sequence. However, since the Lyapunov

IAC-12-C1.8.1 of 15


and halo orbits are represented in a linear form, thesimulations are only valid for relatively small orbitsclose to the equilibrium points.

The main goal of the current research effort is anexpansion of the results from Wong et al., that is,to examine the motion of a spacecraft comprised ofmultiple rigid bodies in three dimensional orbits (ha-los) in the vicinity of the libration points. Knutsonand Howell explore the behavior of multibody space-craft in the circular restricted three body problem,using nonlinear Lyaponov orbits as reference orbits,where the coupled orbit and attitude equations of mo-tion are formulated using Kane’s dynamical method.5

The equations of motion governing the orbit, as wellas the spacecraft attitude, are blended in a cohesivemanner that minimizes computational difficulties andallows for the propagation of meaningful results. Oneof the major computational issues in this process isthe management of the disparity in length scales inthis problem. The distance from the barycenter to thespacecraft is on the order of hundreds of thousandsof km (depending on the system), and the spacecraftdimensions are typically measured in terms of meters.The differences in the lengths present a computationalchallenge, as only finite precision is possible. Thus, anonlinear variational form of the equations of motion isemployed to mitigate these numerical effects. The mo-tion of the spacecraft is also monitored relative to someknown reference orbit. The equations for the timerate of change of linear momentum first describe theabsolute motion of the spacecraft and are then againemployed to describe the motion of the point mass ref-erence system. The contributions from the referenceare subtracted from the absolute motion, eliminatingthe difficulties of significantly different lengths cou-pled within the same differential equations. The sameformulation is applied in the current work to analysemotion in three dimensional orbits in the Circular Re-stricted Three Body Problem.

Guzzetti examines the coupled motion of a space-craft in nonlinear Lyapunov reference orbits, consid-ering the out-of-plane attitude motion.12 Guzzettiformulates the equations of motion using a Lagrangianapproach and also incorporates the effects of solarradiation pressure and flexible bodies. Guzzetti com-pares the out-of-plane attitude response for several testcases to those generated from the model proposed byKnutson and Howell.12 The results indicate good cor-relation between the attitude response using the twomodels.

Another goal of the current work is the assessmentof the departure from the vicinity of the reference or-bit that is predicted by the coupled orbit and attitudemodel and the potential to employ this deviation todetermine the required attitude configuration to con-trol the orbital motion. A sail is employed to repre-sent the coupled system and it is oriented to remain

close to the reference. First, a family of solar sail or-bits in the vicinity of the L2 libration point in theEarth-Moon system is computed using a single shoot-ing algorithm as described by Wawrzyniak.13 Themethod is an extension of the vertically shifted linearapproximation for elliptical orbits near the librationpoints originally introduced by Simo and McInnes.14

Each member in the family of orbits computed nu-merically by the current authors maintains a fixed sailorientation with respect to the Sun frame in the Earth-Moon system. A multiple shooting scheme is thenemployed to determine the required corrections to theattitude motion based on an offset in orbital quanti-ties from the desired reference at some instant in time.A turn-and-hold strategy, similar to one proposed byWawrzyniak,15 is used to determine attitude correc-tions. In the turn-and-hold strategy, the spacecraft isconsidered to maintain a fixed orientation over eachmultiple shooting segment. Finally, the attitude cor-rections are applied to the coupled orbit and attitudemodel. If successful, a satisfactory attitude historymaintains the configuration in the vicinity of the ref-erence orbit. The corrections are computed iteratively,as required, based on the departure of the state fromthe reference path.

II. SPACECRAFT MODEL

The spacecraft model is comprised of two generallyasymmetric rigid bodies, that is, body B and body R.Two coordinate frames are assigned to the spacecraft.The first coordinate frame is fixed in body B, denotedthe b-frame. The second set of unit vectors is fixed inbody R and is denoted the r-frame. The coordinatesystems align with the principal directions of the in-dividual bodies for each center of mass. The mass ofbody B is mb and the center of mass location is de-noted B∗. The mass of body R is mr and the centerof mass location is denoted R∗. The inertia matricescorresponding to the two bodies are expressed in termsof b-frame and r-frame coordinates, respectively,

IB/B∗=

Ib1b1 0 00 Ib2b2 00 0 Ib3b3

[1]

IR/R∗=

Ir1r1 0 00 Ir2r2 00 0 Ir3r3

[2]

The two bodies are connected by a revolute joint witha single degree of freedom parallel to the b3 direc-tion. Thus, the b3 and r3 directions are aligned forall time. The location of the joint, denoted J , is spec-ified relative to the centers of mass of each body. Inthe b-frame, the vector from B∗ to J is defined asrJ/B

∗= s1b1 + s2b2 + s3b3. In the r-frame, the vector

from J to R∗ is expressed by rR∗/J = j1r1+j2r2+j3r3.

A schematic representation of the spacecraft model isillustrated in Fig. 1.

IAC-12-C1.8.1 of 15


b1

b2

b3

r1

r3

r2

B*

R*

J

Fig. 1 Spacecraft Model

For certain applications, it may be desirable to uti-lize a single rigid body spacecraft. In this situation,the rotational degree of freedom between body B andbody R is fixed, which acts in the same manner as aspacecraft comprised of a single body.

III. DYNAMIC FORMULATION

The formulation of Kane’s equations is exploredto model the problem. Kane’s method is based onthe principles of linear and angular momentum. Themethod partitions the general formulas for the timerate of change of linear and angular momentum intoseparate terms in the directions of a set of specifiedgeneralized speeds.16,17 This approach is employed forseveral reasons. Kane’s method is well suited for de-riving equations of motion for systems, such as space-craft, that are composed of many individual bodiesconnected by joints in a chain or tree-like structure.Kane’s method employs simple matrix multiplicationmethods to derive the equations of motion. Assumingworkless constraints, the contact forces are automati-cally eliminated.16

As an alternative to basing the analysis on configu-ration coordinates (position variables), Kane’s methoduses velocity variables identified as generalized speeds.The generalized speeds, denoted ui, are selected inany manner, but it is usually desirable to select anindependent set such that changes to one speed donot affect any other speed. If minimal variables arenot assigned, constraint equations are incorporated toreduce the order of the system. If a set of minimalvariables is selected, one generalized speed is requiredfor each degree of freedom. The total number of gen-eralized speeds is denoted n∗. Some configurationcoordinates may be required to fully describe the sys-tem. The derivatives of the configuration coordinatesare formulated by some combination of a subset of thegeneralized speeds.

Kane’s equations rely on the principles of linear andangular momentum. The following relationship pro-

duces the rate of change of linear momentum, for anybody, S, in a system,∑

FS − dn

dt

(ms vn S∗

)= 0 [3]

The sum of all the external forces on body S is thenequal to the resultant

∑FS . The mass for any body,

S, is defined by the scalar quantity ms. The singleoverbar denotes a vector. A similar expression yieldsthe rate of change of angular momentum,∑

MS/S∗− dn

dt

(IS/S∗

· ωn s

)= 0 [4]

The sum of all external moments on body B about thecenter of mass, S∗, is represented by the resultant sum∑MS/S∗

. The double overbar represents a dyadic.

Then, IS/S∗

is the inertia dyadic relative to S∗ forany three dimensional body. The components of theinertia dyadic are frequently collected in a matrix as,

IS/S∗

=

Is1s1 Is1s2 Is1s3Is2s1 Is2s2 Is2s3Is3s1 Is3s2 Is3s3

[5]

where the vector basis is identified prior to computa-tion. The mass moments of inertia of body S, aboutthe center of mass, S∗, are described by Is1s1 , Is2s2 ,and Is3s3 , and the remaining terms are products of in-ertia. Equations [3] and [4] are then used to formulateKane’s equations.

IV. EQUATIONS OF MOTION

The model formulation previously proposed byKnutson and Howell5 is employed to develop the equa-tions of motion governing the behavior of the space-craft. An overview of the key concepts that are de-tailed in Knutson and Howell5 is presented. To developthe equations of motion governing the behavior of thespacecraft, generalized speeds are first assigned. Thetwo body spacecraft possesses seven degrees of free-dom; therefore seven independent generalized speeds,ui, are selected. The angular velocity of the b-framewith respect to the a-frame is expressed,

ωa b = u1b1 + u2b2 + u3b3 [6]

In this application, the relative motion of the twobodies is restricted to be one-dimensional. Thus, anexpression for the angular velocity of the r-frame withrespect to the b-frame is formulated,

ωb r = u4b3 [7]

The remaining generalized speeds are used to describethe translational velocity of B∗, i.e, the center of massof body B, relative to the current location in the ref-erence orbit, denoted by O

IAC-12-C1.8.1 of 15


vn B∗/O = u5b1 + u6b2 + u7b3 [8]

Several other configuration coordinates are also re-quired to fully describe the system translation andorientation. Euler parameters, as elements in a four-vector, are selected to describe the orientation of theb-frame with respect to the frame associated with thereference orbit, denoted the a-frame. The Euler vectoris defined,

εa b = ε1b1 + ε2b2 + ε3b3 [9]

with the constraint,

ε12 + ε2

2 + ε32 + ε4

2 = 1 [10]

For visualization purposes, it is sometimes desirableto convert the Euler parameter representation of theorientation to an Euler angle rotation sequence. Tworotation sequences, a body 3-2-1 (θ, φ, ψ) and a body3-2-3 (α, ζ, γ) are employed to represent the rotationfrom the a-frame to the b-frame for the various testcases that are examined. Kinematic differential equa-tions are employed to determine the Euler parameterrates and are formulated from the following expres-sion,17

˙εa b =1

2ωa bET [11]

whereωa b =

[ωa b

1 ωa b2 ωa b

3 0]

[12]

and

E =

ε4 −ε3 ε2 ε1ε3 ε4 −ε1 ε2−ε2 ε1 ε4 ε3−ε1 −ε2 −ε3 ε4

[13]

The angular velocity of the r-frame, fixed in the sec-ond body of the spacecraft, relative to the b-frame isintegrated directly to yield the angle between theseframes. The direction cosine matrix to convert fromthe b-frame to the r-frame is described as a rotationabout b3 as follows,

Cbr =

cosβ − sinβ 0sinβ cosβ 0

0 0 1

[14]

The position of R∗ with respect to B∗ is eventuallyrequired and is written in b-frame coordinates andevaluated as,

rR∗/B∗

= rJ/B∗

+ rR∗/JCT

br [15]

To determine the effects of the attitude motion on theorbit, it is assumed that the spacecraft system centerof mass, denoted G∗, is offset from the reference trajec-tory. The following set of scalar variables, as definedin the rotating orbit frame, is used to track the mo-tion of G∗ with respect to the current position alongthe reference orbit, O, that is,

δ = g1a1 + g2a2 + g3a3 [16]

The current position in the reference orbit is deter-mined by propagating the point mass system forwardin time, or by interpolating a known reference path.The motion of the center of mass of the spacecraft sys-tem, G∗, is tracked as the simulation proceeds. Thisquantity is required to determine the contributions ofthe gravitational potential from the two primaries.

The momentum principles introduced in Equations[3] and [4] are now applied. Equation [3] is first em-ployed to form an equation for the total motion of eachof the bodies that comprise the multibody spacecraft,and then to write the equation governing the referencepoint mass system. To remove the undesirable effectsof the discrepancies in the length scales, the point massequation is subtracted from the total motion of bodyB, yielding the following variational form,

∑(FB − FO) − dn

dt

(mb vn B∗/O

)= 0 [17]

Similar expressions are derived for the second body,R. Equation [4] is applied as written since the pointmass system includes no inertia properties with re-spect to the spacecraft center of mass and, therefore,no external moments are applied. External forces andmoments are discussed separately below.

This model assumes that the net force derived fromthe gravitational potential due to the two primariesis the only external force applied to the system. Theexternal field force is assumed to be applied to thecenter of mass of the system, i.e, G∗. An equivalentsystem is derived for the two individual bodies, B andR by replacing the force applied at G∗, with a forceand moment pair applied at each of the centers of massof the individual bodies. The difference between thelocation of the center of mass of the system, rG

∗, and

the point on the reference, rO is represented by δ. BothrG

∗and rO are defined with base points fixed in the

inertial frame at the barycenter of the Earth-Moonsystem. The gravitational force exerted on the centerof mass of the system G∗ is represented by FG∗

and thegravitational force exerted on the point mass referencesystem at O, is represented by FO. The total mass ofthe spacecraft system of two bodies is defined as mt

and the difference between the two forces is denoted,

∆F = FG∗− FO [18]

Since the gravitational force applied at the center ofmass of the system, G∗, is nearly identical to thegravitational force exerted on the point mass refer-ence system at O, a variational representation ensuresthat the appropriate precision is maintained. A gen-eral implementation of Encke’s method is employed toachieve this task.18 The details of this procedure are

IAC-12-C1.8.1 of 15


presented in Knutson and Howell.5 This approach pro-duces results that are accurate given that the actualspacecraft remains close to the reference path. The ref-erence may require rectification for certain simulationsif the spacecraft center of mass is observed to shiftsufficiently far from the reference path. The contribu-tions applied to each body individually is determinedby partitioning the total mass, mt, into the mass of thetwo bodies, mb and mr. All force terms are written inthe b-frame vector basis.

The gravity gradient moment in this problem is de-livered by the two primary bodies. For each primary,this moment is derived from the cross product of thedistance from the attracting body to the spacecraftand the second-order force expansion.19 These mo-ments are computed in terms of the b-frame vectorbasis.

The partial velocity matrix is formed for each bodyfrom the partial derivative of the center of mass ve-locity with respect to each generalized speed. Thepartial derivative of the velocity vector is evaluatedfor each generalized speed and this vector quantity isstored as a row in the partial velocity matrix, defined

for body B as∂ vn B∗

∂u and body R as∂ vn R∗

∂u . Thepartial velocity matrix and partial angular velocitymatrix for both body B and body R are describedin terms of the b-frame vector basis. Each matrix hasseven rows, one for each generalized speed, and threecolumns, one for each spatial dimension of the b-frame.The partial angular velocity matrix for each body alsooriginates from the partial derivative of the angularvelocity vector with respect to each generalized speed.As described previously, these matrices also have sevenrows and three columns, and are described in terms ofthe b-frame vector basis. The matrices for body B

and body R are represented by∂ ωn B

∂u and∂ ωn R

∂u , re-spectively.

Using the quantities derived previously, the equa-tions of motion are now constructed using Kane’sformulation. The partial velocity matrices and themomentum equations are combined to form Kane’sequations of motion. The product of the partial ve-locity matrix and the time rate of change of linearmomentum in Equation [3] and the product of the par-tial angular velocity matrix and the time rate of changeof angular momentum in Equation [4] are determinedfor each body in the system. A coordinate frame fixedin body B is selected as the common vector basis, andthe partial velocity and partial angular velocity matri-ces and Equations [3] and [4] are represented in termsof this frame. For body B, the equations are summa-rized below,

∂ vn B∗

∂u

∑FB − ∂ vn B∗

∂u

dn

dt

(mb vn B∗

)= 0 [19]

∂ ωn b

∂u

∑MB/B∗

− ∂ ωn b

∂u

dn

dt

(IB/B∗

· ωn b

)= 0

[20]

These two vector equations for body B are nowsummed to create one vector equation. Kane splits thisequation into two parts, generalized inertia forces andgeneralized active forces. Generalized inertia forces,denoted Q∗, combine the contribution of the quan-tities that depend on the time rate of change of bothlinear and angular momentum, and the generalized ac-tive forces, denoted Q, collect the contributions fromexternal loads. The generalized inertia force and gen-eralized active force for body B are described as fol-lows,

Q∗B = − ∂ vn B∗

∂u

dn

dt

(mb vn B∗

)− ∂ ωn b

∂u

dn

dt

(IB/B∗

· ωn b

) [21]

QB =∂ vn B∗

∂u

∑FB +

∂ ωn b

∂u

∑MB/B∗

[22]

Similar expression are determined for body R. To ap-ply Kane’s method to a multibody system, the con-tributions from each body must be summed to yielda single equation. A coordinate frame fixed in bodyB remains as the common vector basis, and all contri-butions are expressed in terms of this frame. Kane’sformulation for a system composed of m∗ bodies isrepresented by the following simplified expression,

m∗∑i=1

Q∗Bi

+

m∗∑i=1

QBi= 0 [23]

The generalized inertia forces and generalized activeforces, described in Equations [21] and [22], are com-puted for each body using the partial velocities ma-trices and the appropriate parts of the momentumequations, from Equations [4] and [17]. These twoquantities are combined for each body, as describedin Equation [23]. Finally, the contributions from thetwo separate bodies are added to obtain a single vectorequation of motion corresponding to each generalizedspeed. Thus, the governing equations are produced inthe form A ˙u = w. This system of equations is com-bined with derivatives for the required configurationcoordinates and integrated to determine the motion ofthe individual bodies in the system. The equations ofmotion for the reference point mass orbit are also in-tegrated simultaneously and conserved quantities aremonitored to ensure accuracy of the solution.

IAC-12-C1.8.1 of 15


V. MOTION IN EARTH-MOON HALOS

The reference trajectory for this analysis is a paththat reflects the motion of a point mass spacecraftpropagated in the CR3BP, where the spacecraft ismoving under the gravitational influence of two pri-mary bodies, also modeled as point masses. There areseveral underlying assumptions that are significant inthis model. Since the particle is assumed to be ofinfinitesimal mass, the primary orbits are modeled asKeplerian. The two primary bodies then move in circu-lar orbits about their common barycenter at a constantrate Ω. A rotating coordinate system is assigned, de-noted (a1, a2, a3), as depicted in Fig. 2. The angularvelocity of the a-frame relative to an inertial frame, n,is denoted ωn a and is equal to Ωa3 .

n1

a1

a2

n2

P1

P2

P3

d r

p

t

Fig. 2 Circular Restricted Three Body Problem

The nondimensional form of the scalar equationsof motion for the spacecraft, in the rotating frame(a1, a2, a3), are described below. In nondimensionalform, the position vector from the barycenter to thespacecraft center of mass is written p = xa1+ya2+za3.The symbol µ represents the mass parameter for thespecific system and is defined as the ratio of the massof the smaller primary body to the total mass of thetwo primaries. The resulting scalar nondimensionalequations of motion are,

x = 2y + x− (1 − µ)(x+ µ)

d3− µ(x− 1 + µ)

r3[24]

y = −2x+ y − (1 − µ)y

d3− µy

r3[25]

z = − (1 − µ)z

d3− µz

r3[26]

The CR3BP allows five equilibrium solutions as pointsrelative to the rotating system where the gravitationalforces and rotating frame dynamical accelerations can-cel, assuming a vehicle modeled as a point mass. Oneof the collinear equilibrium points, L1, located alongthe line connecting the two primaries, and located be-tween the two primaries, is of particular interest inthis investigation. Both two and three dimensionalperiodic orbits exist in the vicinity of the libration

points. Three dimensional periodic orbits have beencomputed by differentially correcting initial conditionsin the vicinity of the L1 collinear libration point. TheEOM in Equations [24] -[26] comprise the model forthe motion of the point mass reference body. A fam-ily of northern Earth-Moon (E-M) halo orbits in thevicinity of L1 is depicted in Fig. 3.

3.23.4

3.63.8

x 105

−5

0

5

x 104

−2

0

2

4

6

x 104

a1 (km)

a2 (km)

a 3 (km

)

Az

(km

)

385

76500

Fig. 3 Family of Halo Orbits in the Vicinity of L1

in the E-M System

Members of this orbit family are used as references forseveral test cases.

V.I PERTURBED ASYMMETRIC RIGID BODY

In a previous analysis,5 the motion of a perturbedasymmetric rigid body in a set L2 Earth-Moon Lya-punov orbits is determined and compared to the workby Wong et al.,11 for a similar rigid body moving ina similar set of L2 Sun-Earth Lyapunov orbits. Theresults indicate good correlation between the two in-vestigations, especially for smaller orbit sizes. As anexpansion of the previous contributions, the motion ofan asymmetric rigid body moving in the vicinity ofL1 in the Earth-Moon system is now examined. Twosets of nonlinear halo reference orbits are considered,a small set corresponding to orbits in the blue range ofthe family, as depicted in Fig 3, and a set of larger haloorbits, which corresponds to orbits in the red range.The small orbit set is displayed below in Fig 4. Thegreen orbit represents a halo orbit corresponding toAz = 500 km, the red orbit is larger such that Az

= 1000 km, and the blue orbit corresponds to Az =5000 km. Note that Az corresponds to the maximumout-of-plane excursion along the periodic orbit.

The spacecraft is modeled as a single asymmet-ric rigid body. The inertia properties in b-framecoordinates for the rigid body include the mo-ments Ib1b1=7.083 kg·m2, Ib2b2=11.333 kg·m2, andIb3b3=12.417 kg·m2, which are consistent with thequantities employed by Wong et al.11 All products ofinertia are equal to zero. A body fixed 3-2-1 (θ, φ, ψ)Euler angle sequence is employed to visualize theorientation response. A perturbation in (θ, φ, ψ) is

IAC-12-C1.8.1 of 15


3.2 3.24 3.28−2−1

01

2−4000

−2000

0

2000

4000

a1 (105 km)

a2 (104 km)

a 3 (km

)

Az=500kmAz=1000kmAz=5000km

Fig. 4 Small Set of L1 Halo Reference Orbits inEarth-Moon System

explored: (5, 5, 5) deg. The impact on the Eulerangles as a result of simulations initialized with theset of perturbations displayed in Figs. 5 - 7.

0 0.5 1 1.5 2 2.5−60

−40

−20

0

20

40

60

Revs

θ (d

eg)

___ unperturbed− − perturbed


Fig. 5 Asymmetric Orientation Response in θ forSmall Halos Reference Orbits in the E-M System

0 0.5 1 1.5 2 2.5−40

−30

−20

−10

0

10

20

30

40

Revs

φ (d

eg)



Fig. 6 Asymmetric Orientation Response in φ forSmall Halo Reference Orbits in the E-M System

0 0.5 1 1.5 2 2.5−80

−60

−40

−20

0

20

Revs

ψ (

deg)



Fig. 7 Asymmetric Orientation Response in ψ forSmall Halo Reference Orbits in the E-M System

The results that emerge when the vehicle is movingin a halo orbit from the larger set is displayed belowin Fig. 8. The green orbit represents a halo orbitcorresponding to Az = 10000 km, the red orbitpossess a maximum out-of-plane excursion equal toAz = 50000 km, and the path along the blue orbitreaches Az = 70000 km.

3.23.4

3.6−4

−20

24

−2

0

2

4

6

a1 (105 km)

a2 (104 km)

a 3 (10

4 km

)Az=10000kmAz=50000kmAz=70000km

Fig. 8 Large Set of L1 Halo Reference Orbits inEarth-Moon System

Then again, a perturbation in (θ, φ, ψ) of (5, 5, 5) degis explored for the large orbit references. The impacton the Euler angles as a result of simulations that areinitialized with the perturbations is displayed in Figs.9 - 11.

In general, the orientation responses are observed todiffer depending on the orbit type. For the small refer-ence orbit set, the orientation response in terms of thefirst Euler angle, θ, appears in Fig. 5 to be oscillatoryand relatively periodic over the length of the simula-tion. The perturbation is noted to shift and increasesthe response in θ for all three small reference orbits.

IAC-12-C1.8.1 of 15


0 0.5 1 1.5 2 2.5−1200

−1000

−800

−600

−400

−200

0

200

Revs

θ (d

eg)

___ unperturbed

− − perturbed


Fig. 9 Asymmetric Orientation Response in θ forLarge Halo Reference Orbits in the E-M System

0 0.5 1 1.5 2 2.5

−100

−50

0

50

100

Revs

φ (d

eg)

___ unperturbed

− − perturbed


Fig. 10 Asymmetric Orientation Response in φ forLarge Halo Reference Orbits in the E-M System

0 0.5 1 1.5 2 2.5−400

−200

0

200

400

600

800

Revs

ψ (

deg)

___ unperturbed

− − perturbed


Fig. 11 Asymmetric Orientation Response in ψ forLarge Halo Reference Orbits in the E-M System

The response in θ for the large halo orbit set, as ap-parent in Fig. 9, does not exhibit similar behavior.As the orbit size increases, the motion becomes un-bounded and the divergence appears relatively quickly.The motion in the second angle, φ, for the small orbitset is displayed in Fig. 6; the response appears to beoscillatory although the amplitude is increasing withtime. The magnitude of the response for a vehicle inone of the orbits from the large set in Fig. 10, diverges

more quickly than the response when the vehicle is inan orbit from the small orbit set, but overall, gener-ally similar characteristics appear. The exception isthe response for the largest orbit, with an Az ampli-tude of 70000 km, where the influence of the close passto the Moon at 0.5 rev and 1.5 rev locations is visible.In comparing Figs. 7 and 10, the magnitude in theamplitude response of the third angle ψ, between thesmall and large reference orbits, is noted to increasemuch more significantly. The test cases are exploredto further expand on the analysis offered in Wong etal.11 by considering the orientation history resultingfrom selecting nonlinear, L1, halo orbits as the refer-ence. While the results presented by Wong et al.11

display bounded, oscillatory motion for the first twoEuler angles, θ and φ, for the length of the simulation,the above sample cases indicate that for larger orbitsize, the motion may become unbounded, less oscilla-tory, and less predictable in general.

V.II PERTURBED AXISYMMETRIC BODY

The general model that has been developed is ca-pable of exploring the behavior of a range of vehicles,and is now employed to predict the behavior of anaxisymmetric body under a perturbation in the ini-tial orientation. The spacecraft is comprised of tworigid rods, each of length 10 meters and radius 0.1meter. The rods are stacked on top of each other, con-figured so that the axial direction is aligned with b3.The top rod can spin about b3 relative to the bottomrod. For visualization purposes, the orientation is ex-pressed in terms of a body fixed 3-2-3 (α, ζ, γ) Eulerangle sequence, for a variety of test cases. The an-gle α represents precession of the b-frame relative tothe a frame, the angle ζ is nutation, and spin aboutthe axisymmetic axis is denoted by the angle γ. A 5degree perturbation in the nutation angle, ζ, is consid-ered for four periodic L1 halo reference orbits in theEarth-Moon system with Az values of 500 km, 5000km, 10000 km, and 50000 km. The orbital motionover 2.5 revolutions of the reference trajectory is firstconsidered for the three smaller Az values. In Fig. 12,the departure of the spacecraft from the reference is ev-ident. It is observed that smaller values of Az actuallycorrespond to larger motion of the spacecraft center ofmass, G∗, away from the isochronous location alongthe reference path, O. In any family of L1 halo orbits,the orbits with smaller Az values are unstable. Theperturbations in these smaller, more unstable orbitsappear to induce greater effect than in larger, less un-stable orbits, based on the magnitude of the unstableeigenvalue of the associated monodromy matrix.

The attitude response, in terms of the body fixed3-2-3 (α, ζ, γ) Euler angle sequence is also considered.A perturbation in the nutation angle of ζ = 5 degrees,for the four halo reference orbits is introduced. In Fig.13, the precession angle of the b-frame with respect

IAC-12-C1.8.1 of 15


3

3.2

3.4 −3

0

3−10

−5

0

5

10

a2 (104 km)

a1 (105 km)

a 3 (10

3 km)


Fig. 12 Orbital Response to perturbation of ζ =5 degrees in E-M Halos

to the a-frame, α, is observed to be oscillatory andbounded for motion in the orbit with the smallestAz value, irregular for intermediate values of Az, andincreasing at a fairly constant rate for the largest Az

of 50000 km. The nutation angle, ζ, for rods movingin halo orbits of all sizes is observed in Fig. 14 tooscillate around ζ = 90 degrees, which is consistentwith knowledge of attitude motion in the two bodyproblem; in terms of the nutation angle, a long thinrod is unstable in the vertical configuration andstable while lying horizontally. The frequency ofoscillations appears to increase with the size of thehalo reference trajectory. In Fig. 15, the spacecraftspin angle, γ, is noted to be increasing a greater ratefor the relatively smaller halo reference orbits than forreference orbits possessing larger values of Az. Withthe exception of the largest halo orbit however, thespin angle, γ, appears to be fairly irregular. Along thereference trajectory at Az = 50000 km, the overall re-sponse in γ is observed to be oscillatory and increasing.

0 0.5 1 1.5 2 2.5−500

0

500

1000

1500

2000

Rev

α (d

eg)

Az=500kmAz=5000kmAz=10000kmAz=50000km

Fig. 13 Orientation Response in α to Perturbationζ = 5 degrees in E-M Halos

Spin stabilization of the spacecraft, under theperturbation ζ = 5 degrees, is explored for one E-Mhalo reference orbit of Az = 5000 km and is attemptedby spinning body R with respect to body B at two

0 0.5 1 1.5 2 2.5−150

−100

−50

0

50

100

150

200

Rev

ζ (d

eg)


Fig. 14 Orientation Response in ζ to Perturbationζ = 5 degrees in E-M Halos

0 0.5 1 1.5 2 2.5−200

−100

0

100

200

300

400

500

600

700

Rev

γ (d

eg)


Fig. 15 Orientation Response in γ to Perturbationζ = 5 degrees in E-M Halos

rates: 0.1 rad/s (approximately 1 rpm) and 10 rad/s(approximately 95 rpm). In Fig. 16 through Fig. 18,the perturbed response is displayed in green, the redline indicates motion corresponding to a spin rate of0.1 rad/s, and the blue line reflects to a spin rate of10 rad/s.

0 0.5 1 1.5 2 2.5−600

−500

−400

−300

−200

−100

0

100

200

Rev

α (d

eg)

perturbedw spin (0.1 rad/s)w spin (10 rad/s)

Fig. 16 Orientation Response in α to Spin Stabi-lization in the E-M System

It is apparent in Fig. 17 that spin stabilization to

IAC-12-C1.8.1 of 15


0 0.5 1 1.5 2 2.5−100

−50

0

50

100

150

200

Rev

ζ (d

eg)


Fig. 17 Orientation Response in ζ to Spin Stabi-lization in the E-M System

0 0.5 1 1.5 2 2.5−100

0

100

200

300

400

500

600

Rev

γ (d

eg)


Fig. 18 Orientation Response in γ to Spin Stabi-lization in the E-M System

accommodate this particular perturbation depends onthe spin rate employed, when focused on the nutationangle ζ. In the simulations of the perturbed motion,the spacecraft exhibits oscillatory behavior in α andunbounded motion in the other two angles which con-firms that the spacecraft is not maintaining the initialorientation. Employing a spin rate of 0.1 rad/sec, thenutation angle response is observed to be damped;however, the spacecraft is still oscillating about ζ =90 degrees. This response is not unexpected; in ad-dition, recall that this orbit possesses an out-of-planeexcursion of 5000 km The precession of the b-framewith respect to the a-frame appears in Fig. 16 to bebounded for this spin rate, and the spin angle for bodyB of the spacecraft is growing less rapidly in compar-ison to the perturbed case, as apparent in Fig. 18.When the spin rate of body R is increased to 10 rad/s,nearly all effects of the initial perturbation on the nuta-tion angle are eliminated and the body is maintainingthe initial orientation in ζ. Not surprisingly, the space-craft still precesses with respect to the orbit frame ata comparable rate to the perturbed case, but the spinangle response is improved with respect to the per-turbed case.

Finally, the impact of spin stabilization on the

orbit is considered. The drift in the center of masslocation relative to the reference orbit is displayedin Fig. 19. The orbital motion for the perturbedspacecraft is plotted in green, and the spin stabilizedspacecraft, with spin rate equal to 0.1 rad/sec, inred; spin rate of 10 rad/sec is plotted in blue. Thisfigure illustrates that spinning the spacecraft resultsin only very marginal improvement in maintaining thedesired orbital motion relative to the reference orbit.

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Rev

Mag

nitu

de o

f M

otio

n of

G*

wrt

0 (

km)


Fig. 19 Orbital Motion Response to Spin Stabi-lization in the E-M System

This test case serves two purposes. First, the re-sults confirm that the spacecraft is displaying motionconsistent with previous results and serves to verifythe model formulation. Secondly, it demonstrates anapplication of spin stabilization, at least for some ori-entation parameters, in the coupled orbit and attitudemodel in the CR3BP.

VI. APPLICATION TO SOLAR SAILS

One further objective in the development of thecoupled orbit and attitude model is the capability todirectly influence the orbit with the orientation. A so-lar sail offers a potentially significant coupling. Thus,it is also desired to consider the departure from thevicinity of the reference orbit, as predicted by the cou-pled orbit and attitude model, and to determine if theattitude configuration can control the orbital motionof the solar sail.

VI.I REFERENCE FAMILY

A solar sail immediately introduces the challenge ofa more complex reference trajectory. Thus the firststep is the construction of a family of periodic solarsail orbits in the vicinity of the L2 libration point. Thecomputation is accomplished using a single shootingalgorithm as described by Wawrzyniak.13 The schemeis an extension of the vertically shifted linear approx-imation for elliptical orbits near the libration pointsthat is explored in Simo and McInnes.14 In the in-vestigation here, an ideal solar sail model is assumed,

IAC-12-C1.8.1 of 15


where the nondimensional acceleration associated withthe sail is,

as(t) = ac

(l1(t) · us

)2

us [27]

where ac is the characteristic sail acceleration, l1 is thesunlight direction, t is the nondimensional time and usis the direction of the sail normal. In the Earth-Moonrotating frame, the sunlight direction frame, denotedthe l-frame, is defined,

l1 =cos(Ωlt)a1 − sin(Ωlt)a2

l2 =sin(Ωlt)a1 + cos(Ωlt)a2

l3 =a3

[28]

where Ωl is the rate at which the sunlight directionis changing in the Earth-Moon reference frame, de-scribed as the ratio of the sidereal to synodic periodsof the Earth-Moon system.15

The orientation of the solar sail normal direction,us, with respect to the Earth-Moon rotating frame,the a-frame, can be described by a body 3-2-1 (θs −Ωlt, αs, γs) Euler angle rotation sequence. The thirdangle, γs, set to zero, as the sail normal can be fully de-scribed by solely two angles. The sequence is noted toconsistently convey the sail orientation with respect tothe Earth-Moon rotating frame. The family of orbitscomputed by Wawrzyniak is based on the assumptionthat the vehicle maintains a fixed sail orientation withrespect to sunlight direction in the Earth-Moon sys-tem. To generate this particular family, the value of θsis held constant at 0 degrees and the value of αs is heldconstant at −35.264 degrees. This value for αs corre-sponds to the angle that generates the maximum out ofplane force, based on the linear equations of motion asdescribed by Simo and McInnes.14 Using Equations[27] and [28] and the Euler angle rotation sequence,the contributions of the sail are incorporated into thetraditional Circular Restricted Three Body Problemequations of motion. Note that the characteristic ac-celeration is permitted to vary over the family.

x = 2y + x− (1 − µ)(x+ µ)

d3− µ(x− 1 + µ)

r3

+ ac cos(αs)3 cos(Ωlt)

[29]

y = −2x+y− (1 − µ)y

d3−µyr3

−ac cos(αs)3 sin(Ωlt) [30]

z = − (1 − µ)z

d3− µz

r3− ac cos(αs)

2 sin(αs) [31]

From these equations of motion, a family of peri-odic, three dimensional, solar sail reference orbits iscomputed via a single shooting differential correctionsprocess. The family of orbits is displayed in Fig. 20.

4.4 4.43 4.46

−1

0

1

0

2000

4000

6000

8000

10000

a2 (104 km)

a1 (105 km)

a 3 (km

)

Cha

ract

eris

tic A

ccel

erat

ion

(nd)

0

0.09

Fig. 20 Family of Solar Sail Reference Orbits inthe Vicinity of L2 in the E-M System

The color variation in Fig. 20 indicates the range invalues of the nondimensional characteristic accelera-tion across the family of trajectories. The smallestorbit, displayed in blue possesses a characteristic ac-celeration of 0.0001 (0.3 nm/s2). The largest orbit,displayed in red, corresponds to a value of 0.0894(245.0 nm/s2).

VI.II COUPLED MODEL MODIFICATIONS

The model framework that was previously developedto formulate the coupled orbit and attitude equationsof motion is now modified slightly to incorporate theeffects of solar radiation pressure. A particular ref-erence orbit is selected, and accordingly, the charac-teristic acceleration corresponding to that referenceis, thus, available. The rotational degree of freedombetween the two bodies in the spacecraft model is as-sumed to be fixed, creating one rigid body. Realisticparameters are selected to reflect the mass and inertiaof an approximate solar sail type spacecraft. The pa-rameters are described such that the sail normal, uscorresponds to the body fixed b1 direction. The mo-tion of the sunlight direction, as described by l1 in inEquation [28] is monitored and the force contributiondue to the solar sail is computed by incorporating themass of the solar sail and the acceleration described inEquation [27], dimensionalizing as appropriate. Theinitial conditions are consistent with the selected solarsail reference orbit. As expected, due to the orbit andattitude coupling, when the system equations of mo-tion are propagated forward in time, the solar sail doesnot remain in the vicinity of the reference solution.

VI.III TURN-AND-HOLD MODEL

A multiple shooting scheme is employed in an at-tempt to control the orbital motion of the spacecraftso it remains in the vicinity of the desired reference.The applied control is not optimal. At some pointin time, ti, it is noted that the orbital motion of thecoupled orbit and attitude model deviates sufficiently

IAC-12-C1.8.1 of 15


from that described by the reference orbit. The posi-tion and velocity state of the coupled system at ti isdenoted xi. A turn-and-hold strategy, similar to oneproposed by Wawrzyniak,15 is then employed to de-termine the attitude corrections that are required toguide the solar sail back to the vicinity of the referenceorbit at some final time, tf . Intermediate patch pointsare introduced at times, ti+1, ti+2, ... ti+np , where nprepresents the number of intermediate patch points.The position and velocity states at each intermediatepoint are described by, xi+1, xi+2, ... xi+np

and theposition and velocity at the final state, is denoted xf ,and corresponds to a state at tf on the reference tra-jectory. Initial conditions for the position and velocitystates are selected to be close to those associated withthe reference trajectory; however, the position and ve-locity state at each patch point is part of the designvector and vary during the multiple shooting process.In the turn-and-hold strategy, the spacecraft is con-sidered to maintain a fixed orientation with respectto a frame associated with the sunlight direction overeach multiple shooting segment. This simplification tothe model facilitates the computation of discrete cor-rections to the spacecraft orientation at a number ofintermediate patch points. Finally, if the attitude cor-rections are successfully designed, their application tothe coupled orbit and attitude model produces a con-figuration that remains in the vicinity of the referenceorbit. The corrections are actually determined iter-atively as required, based on the departure from thereference orbit.

A schematic of the process is displayed in Figs. 21to 23, using three internal patch points (depicted ingreen). The starting location at t1 is denoted by thered point and the desired final location at t5 is theblack point. The orientation corrections are appliedat t1 to t4, computed via the turn-and-hold model,to the coupled orbit and attitude model. These cor-rections should aid the spacecraft in maintaining anorbital response that remains in the vicinity of the ref-erence path.

t i

t 2t 3

t 4

t 5

Reference Trajectory

Coupled Orbit and Attitude Trajectory

Multiple Shooting with Turn-and-Hold Model

Fig. 21 Initial Condition for Multiple ShootingAlgorithm

The sail normal direction at the initial point andat each intermediate patch point is defined as partof the design vector. Over the multiple shooting seg-

t it 2

t 3

t 4

t 5



Multiple Shooting with Turn-and-Hold Model

Fig. 22 Corrections Determined Using Turn-and-Hold Model

t i

t 2

t 3

t 4

t 5



Apply Corrections from Turn-and-Hold Model to Coupled Model

Fig. 23 Corrections Applied to Coupled Orbit andAttitude Model

ments, the sail normal direction is fixed with respectto a frame associated with the sunlight pointing direc-tion, l1 as described in Equation [28]. At the initialtime,ti, the sail normal direction is expressed,

usi = Us1il1 + Us2i

l2 + Us3il3 [32]

Since the sail normal direction is a unit vector, thefollowing equality constraint equation is applied at theinitial point.

Us1iUs1i + Us2iUs2i + Us3iUs3i − 1 = 0 [33]

Similar sail normal directions are described for eachpatch point, ui+1 to ui+n, and the corresponding con-straint equations are applied similarly to Equation[33]. The description of the sail normal from Equation[32] is applied to Equation [27] and the following up-dated equations of motion are derived that govern thebehavior under the turn-and-hold model. Note that inthis formulation, the characteristic acceleration is fixedand specified by the selection of the reference orbit.

x = 2y + x− (1 − µ)(x+ µ)

d3− µ(x− 1 + µ)

r3

+ ac(U3s1 cos(Ωlt) + U2

s1Us2 sin(Ωlt)) [34]

y = −2x+ y − (1 − µ)y

d3− µy

r3

+ ac(U2s1Us2 cos(Ωlt) − U3

s1 sin(Ωlt)) [35]

z = − (1 − µ)z

d3− µz

r3− acU

2s1Us3 [36]

IAC-12-C1.8.1 of 15


Consistent with the analysis by Wawrzyniak, continu-ity is enforced in position and velocity at each of theintermediate patch points. The equations of motionin Equations [34-36]are propagated forward using theinitial conditions at times ti, ti+1 ... ti+np until reach-ing the time associated with the next patch point orthe final time in the case of the last patch point, i.e.ti+1, ti+2 ... tf . The position and velocity states at theend of the integration interval are denoted, xi+1(usi),xi+2(xi+1, usi+1

), ... xf (xi+np, usi+np). Therefore, the

position and velocity states at the end of the propa-gation segments are functions of both the state andsail normal vectors delivered at the start of the prop-agation, except for the first arc where the final statesare merely functions of the sail normal vector since theinitial position and velocity state are fixed at time ti.To force continuity in position and velocity, the statesdetermined by the propagation must be equal to thestates that define the next patch point. Inequality con-straints are also implemented at the initial point andeach subsequent patch point to ensure that the sailremains oriented with the appropriate side facing theSun. Slack variables are employed, denoted ηi, ηi+1 ...ηi+n.15 Additionally, six inequality constraints mustbe added for this particular application, to allow forsome variation in the position and velocity at the finaltime, yet remain within some ’box’ around the finalstate along the reference.

xflb < xf(xi+np, usi+np

)< xfub

[37]

where xflb defines the lower bound and xfubspecifies

the upper bound. Twelve additional scalar slack vari-ables are required to enforce the inequality constraintsin Equation [37], six associated with the lower boundconstraint, denoted slow and six associated with theupper bound constraint, denoted supp.

After the problem is completely formulated, thevariables and constraints are collected in vectors. Thedesign vector for the multiple shooting scheme, X iscomposed of the sail normal directions, slack variables,and the position and velocity states of the patch pointsas follows,

X =

usiηixi+1

usi+1

ηi+1

...xi+np

usi+np

ηi+np

slowsupp

[38]

The constraints, F (X), applied are follows,

F(X)

=

u2si − 1

−Us1i+ η2

i

xi+1(usi) − xi+1

u2si − 1

−Us1i+1 + η2i+1

xi+2(xi+1, usi+1) − xi+2

...−(xf(xi+np, usi+np

)− xflb

)+ s2

low

−(xfub

− xf(xi+np, usi+np

))+ s2

upp

[39]

The remainder of the corrections scheme is imple-mented in the same manner as Wawrzyniak.15 Oncethe corrections are determined for the orientation viathe turn-and-hold model, the prescribed orientationsare applied to the coupled orbit and attitude model.

VI.IV APPLICATION

The multiple shooting control strategy is nowapplied to a solar sail type system. As previouslynoted, a rigid body system representative of a solarsail type spacecraft is created by fixing the degree offreedom between body B and body R. Each body isassigned a sail thickness of 7.5 µm in the b1 direction,and dimensions of 14 m in the b2 direction and 7 m inthe b3 direction. This creates a single rigid body withinertia properties on the same order of magnitude asthe IKAROS sailcraft.20 A single reference orbit isselected from the solar sail family displayed in Fig.20 in blue. This reference orbit is constructed witha nondimensional characteristic acceleration of ac =0.01 (28.2 nm/s2). Given the sail reference orbit, thestates are then employed in the coupled orbit andattitude solar sail model. The response, in terms ofthe center of mass motion, appears in red in Fig.24; the path of G∗ is propagated for 0.25 rev of thereference path and clearly departs the vicinity ofthis orbit. Employing spin stabilization, the sail isgiven a spin rate of 0.1 rad/s20 about the b1 axisin anticipation that spinning the sail improves theorbital motion response. In Fig. 24, the spinning sailresponse, depicted in black, is observed to remainnear the reference for approximately 0.25 rev, afterwhich it departs the vicinity. The multiple shootingstrategy is now employed, in conjunction with thespin stabilized results, to further improve the sailorbital response.

The turn-and-hold multiple shooting scheme is nowapplied. After a time interval corresponding to ap-proximately 0.35 rev of the reference orbit, the re-sponse in the coupled, spinning model response is de-parted from the vicinity of the reference. The locationof the spacecraft at this particular time is depicted bythe red dot in Fig. 25. Seven patch points, repre-sented by the green dots in Fig. 25, are assigned. Theinitial paths of all the intermediate multiple shooting

IAC-12-C1.8.1 of 15


4.44 4.4425 4.445 −2

−1

0

1

2

400

600

y (104 km)

x (105 km)

z (k

m)

referencecoupled modelcoupled model w spin (0.1 rad/s)

Fig. 24 Coupled Orbit and Attitude Response forSolar Sail in the Vicinity of L2 in the E-M System

arcs are plotted in magenta, and the full reference or-bit is displayed in blue. The location at the final time,indicated by the black dot, is selected to occur at apoint located after 0.7 rev along the reference.

4.44 4.4425 4.445−2

−1

0

1

2400

600

x (105 km)

y (104 km)

z (k

m)

referencestart pt.stop pt.patch pt.uncorrected

Fig. 25 Multiple Shooting Initial Conditions

Using the turn-and-hold strategy and applyingcorrections or updates to the orientation of thespacecraft at the initial point and each subsequentpatch point, a new path is determined that allows thespacecraft to return to the vicinity of the referenceby the final time corresponding to 0.7 rev alongthe reference trajectory. The resulting trajectoryis displayed in Fig.26 in magenta, where again thereference is represented in blue, the initial positionis indicated by the red dot, the final position ishighlighted by the black dot; then, the convergedpatch point locations are represented by the greendots along the final corrected arc.

The corrections to the sail orientation that are de-termined for the initial point and all intermediatepatch points are then applied to the vehicle in thecoupled, spinning, orbit and attitude model. Onceimplemented, in the coupled model, the sail still de-parts as evidenced by the black trajectory in Fig. 27.

4.44 4.44 4.44−2

−1

0

1

2400

600

x (105 km)

y (104 km)

z (k

m) reference

start pt.stop pt.patch pt.corrected

Fig. 26 Corrected Trajectory Using Turn-and-Hold Strategy; Decoupled Model

The converged turn-and-hold trajectory is again rep-resented in magenta, located near the blue referencepath, the green dots indicate patch point locations, thered dot highlights the starting location, and the blackdot indicates the desired position at the final time.The control strategy consisting of the combination ofspin stabilization and applying corrections determinedby the multiple shooting technique demonstrates vastimprovement over simply allowing the sail natural dy-namics to dominate the response. However, at thefinal time, the location of the center of mass G∗ in thecoupled, spinning, orbit and attitude model does notreach the desired position, although the path of thereference is maintained for a longer time interval thanthe response to spin stabilization alone, in red. Fur-ther iterations to update the corrections determinedby the multiple shooting technique do improve the or-bital response but minimally. Further investigation isrequired to incorporate the coupled orbit and attitudemodel into the multiple shooting corrections process.

4.44 4.44 4.44−2

0

2

−2

0400

600

x (105 km)

y (104 km)

z (1

03 km

)

referenceturn−and−holdcoupled − no corcoupled − with cor

Fig. 27 Orientation Corrections Applied in theCoupled Orbit and Attitude Model

IAC-12-C1.8.1 of 15


VI. SUMMARY

A framework was previously developed the couplethe orbit and attitude dynamics in the CR3BP. Themain goal of this investigation is the application ofthe coupled model i) to explore attitude dynamicswithin the context of the circular restricted three bodyproblem, and ii) to examine the motion of spacecraftcomprised of multiple rigid bodies in three dimensionalorbits (halos) in the vicinity of the libration points inthe Earth-Moon System. The motion of a spacecraft,comprised of two rigid bodies connected by a single de-gree of freedom joint, is examined. The coupled orbitand attitude nonlinear variational equations of mo-tion are formulated using Kane’s method. Test casesdemonstrate the effects of orbit size on the attituderesponse and the effects of some perturbations. Spinstabilization is also introduced to validate the modeland preliminarily examine the spin rates in the initialorientation, with results indicating that spin stabiliza-tion is possible for some orientation parameters forsome test cases.

Another goal of the current work is an initial inves-tigation of the center of mass behavior in a coupledmodel. In particular, the departure of the center ofmass from the vicinity of the reference orbit as pre-dicted by the coupled orbit and attitude model can beused to determine the required attitude configurationto control the orbital motion of the sail to remain closeto the reference. Both spin stabilization and controlthrough a turn-and-hold scheme is effective but ad-ditional development is necessary to determine if thecoupled orbit and attitude model can be integratedinto a multiple shooting algorithm. The ultimate ob-jective is an orientation corrections strategy that re-sults in the sail spacecraft returning to the vicinity ofthe reference solution, given some initial perturbation.

ACKNOWLEDGEMENTS

The support of the following organizations is appre-ciated: Canadian Space Agency (CSA), Zonta Inter-national, and the College of Engineering at PurdueUniversity (School of Aeronautics and Astronauticsand School of Engineering Education). Many thanksto Geoff Wawrzyniak for sharing his expertise in theturn-and-hold approach for solar sail trajectories.

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