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American Institute of Aeronautics and Astronautics

1

STABILITY AND CONTROL OF A FLEXIBLE SPACECRAFT DURING A SHALLOW AEROASSIST

Joseph R. Schultz* and Darryll J. Pines† Department of Aerospace Engineering

University of Maryland, College Park, MD 20742

Abstract In order for a spacecraft to make scientific

measurements in the upper atmosphere, it may be necessary for the spacecraft to “dip” into the upper atmosphere. This aeroassist maneuver presents stability and control challenges that are compounded if the spacecraft contains long flexible booms. This paper examines several stability and control aspects that are unique to this flexible class of spacecraft undertaking an aeroassist. Using a detailed, linear model, unique buckling-like modes are uncovered when the spacecraft body has high drag characteristics. Also shown is how the spacecraft may become uncontrollable during the shallow aeroassist even though it may be controllable in free-space. Finally, proper positioning of the booms for stability is also discussed.

Nomenclature b = spring damp ing constant = body coordinate direction = molecular thermal speed k = spring constant m = mass = inertial coordinate direction x = location of connection point from boom

center of mass xb = location of connection point from body

center of mass xac = force and moment location from center

of mass CD = drag coefficient CL = lift coefficient F = force J = moment of inertia L = boom length M = moment _______________

*Graduate Student, Student Member AIAA, Department of Aerospace Engineering, University of Maryland, College Park, MD 20742, email: [email protected]

†Professor, Associate Fellow AIAA, Department of Aerospace Engineering, University of Maryland, College Park, MD 20742, email: [email protected]

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

R = position in inertial space S = surface area Vo = free stream velocity ε = momentum rebound coefficient ρ = atmospheric density θ = position angle θo = neutral position angle of the boom ω = angular velocity ξ = percent specular to diffuse interaction subscripts b = body n = boom number x = x-direction z = z-direction R = reaction force or moment W = momentum wheel diff = diffuse gas-surface interaction spec = specular gas-surface interaction superscripts b = body frame of reference n = inertial frame of reference

Introduction This paper will examine the dynamic

stability and control of a spacecraft with flexible booms undergoing an atmospheric pass, or aeroassist, through the upper atmosphere.

Most satellites and spacecraft fly in high enough orbits so that they will not experience significant aerodynamic drag in an orbital time period. However, there are some missions where a spacecraft may wish to enter the upper regions of the Earth’s atmosphere for a short period of time. These missions may be operational or scientific in nature such as to conduct an aeroassist for an orbital plane change or to take magnetosphere readings in the upper atmosphere. Much of the literature on aeroassist maneuvers focus on spacecraft with a space-shuttle-like configurations where the spacecraft is treated like a point-mass model.1 Because of the rigid-body nature of these spacecraft, internal flexibility modes are usually not of concern.

b̂c

n̂

AIAA/AAS Astrodynamics Specialist Conference and Exhibit5-8 August 2002, Monterey, California

AIAA 2002-4521

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

Spacecraft with long, flexible booms,

however, present an extra layer of dynamics. Even in free-space, instabilities may arise that would not normally be predicted by a point-mass model. Although the literature quite extensively covers the dynamics and control of these flexible spacecraft in free-space, there is little analysis for when those types of spacecraft undergo the aerodynamic effects of an aeroassist.

This paper hopes to bridge that gap by developing a detailed linear model of a flexible class of spacecraft. With the model, stability characteristics will be analyzed, and controllability issues will be addressed.

Motivation The Geospace Electrodynamics Mission

(GEC) spacecraft currently being designed by the National Aeronautics and Space Administration (NASA) at Goddard is a key example of a flexible spacecraft undergoing a shallow atmospheric pass. Its goal is to measure the Earth-Sun electro-magnetic interaction near the Earth.

The spacecraft has six long booms that hold instruments that measure the local electrical and magnetic field. See Figure 1. One of the key missions of the spacecraft is to dip into the upper atmosphere (approximately 130 km above the Earth’s surface) to take field readings at those altitudes. In this rarefied atmospheric region there

Figure 1: The GEC Spacecraft (courtesy NASA/Goddard)

will be significant aerodynamic loading on the booms and spacecraft. One of the concerns is that the resulting flexing of the booms may lead to instabilities or loss of controllability of the spacecraft.

To examine this concern, a flexible spacecraft model was developed with the GEC spacecraft in mind.

The Spacecraft Model The spacecraft model will consist of a

main body with a variable number of flexible booms extending from it. The only source of control will be from a momentum wheel in the main body of the spacecraft.

The geometry for the spacecraft model is shown in Figure 2 with just one boom. Any number, N, booms may be attached to the spacecraft.

Figure 2: The Spacecraft Model

Free-body diagrams of both the booms and body of the spacecraft are shown to help derive the equations of motion. See Figures 3 and 4.

nx

nz

bz

bx

Rb

Rn

xn

xbn

θon

θn

θb

nth boom

boom center of mass

body center of mass

unbent position of boom

bent position of boom


3

Figure 3: Free Body Diagram of the Booms

Figure 4: Free Body Diagram of the Main Body

Newton-Euler equations of motion are

developed for the body and N number of booms:2 Translational Equations:

nRnnn

n

bN

Rnbn

b

FFRm

FFRm

vv&&v

vv&&v

+=

+−= ∑ (1)

Rotational Equations:

( )

RnRnn

nnacnnnnnan

n

WN

RnN

Rnbn

bABacbbbbbb

b

MFx

MFxJJ

MMFx

MFxJJ

vvv

vvvvv&v

vvvv

vvvvv&v

+×+

+×=×+

+−×−

+×=×+

∑∑

ωωω

ωωω

(2)

Constraint Equations:

nn

nn

bnn

bn xRxR &&v&&v&&v&&v +=+ (3)

Expansion of Terms:

( )( )nnnnn

ann

n

bnbbbnbb

bnn

xxx

xxxvvvv&v&&v

vvvv&v&&v

××+×=

××+×=

ωωω

ωωω (4)

Equations 1-4 can be rearranged and

grouped into a simple matrix form:

=

′′ wqF

F

r

XRXJRM

R

vv

v

v&v&&v

ω0

00

(5)

where the following matrices are defined:

=

N

b

m

mm

m

M

LMO

000000

000000000000

2

1 (6)

where the m’s are the masses of the body and booms 1 to N multiplied by an identity matrix.

=

N

b

J

JJ

J

J

LMO

000000

000000000000

2

1 (7)

where, similarly, the J’s are the moment of inertia matrices in the inertial, or n-frame, of reference.

−

−−

=

I

II

III

R

LMO

L

0000

000000

(8)

where I is the identity matrix. [ ] [ ] [ ]

[ ][ ]

[ ]

−

−−

=

N

bNbb

x

xx

xxx

X

~0000

00~0000~

~~~

2

1

21

L

MO

L

(9)

where the elements are expressed in the n-frame of reference and are skew-symmetric cross-product matrices, i.e.:

[ ]babavvv ~=×

nx

nz

Ln

Dn

FRnx

FRnz

Mn

MRn

xacn

nx

nz

Db

Lb

FRnx

FRnz

MRn

Mb

xacb


4

The vectors are described as follows:

=

Nn

n

n

n

R

RRR

r

&&vM

&&v&&v&&v

&&v1

1

1

,

+

++

=

Nn

bn

nb

n

nb

nb

n

θθ

θθθθ

θ

ω

&&v&&vM

&&v&&v&&v&&v

&&v

&v2

1 (10), (11)

=

RN

R

R

R

F

FF

FvM

vv

v 2

1

,

=

N

b

F

FFF

F

vM

vvv

v2

1 (12), (13)

×−+×+

×−+×+×−+×+

+×−×+−

=

∑

NNNRNNacNN

Rac

Rac

WbbbbacbN

Rnb

JMFxM

JMFxM

JMFxM

MJFxMM

q

ωω

ωωωω

ωω

vvvvvvM

vvvvvvvvvvvv

vvvvvvv

v2222222

1111111

(14)

( ) ( )( ) ( )

( ) ( )

××+××−

××+××−××+××−

=

NNNbNbb

bbb

bbb

xx

xxxx

w

vvvvvvM

vvvvvvvvvvvv

v

ωωωω

ωωωωωωωω

2222

1111

(15)

From geometry:

( )( )

++−

=non

nonb

nn

Lx

θθθθ

sincos

2 v (16)

for the x and z coordinates in the body frame.

Note that some of the variables in Equations 1-16 will need to be expressed in the inertial coordinate system via a coordinate transform. The size of each vector and matrix will depend on whether a 2-D or 3-D model is desired.

Taking the inverse of Equation 5 :

′−−′+−′

′−′

′+=

−

−−−−

−−−

−

−−

−−−

wqF

HJXHXHJJXXHJJRHMJXRHM

MRHMRXHJ

MRRHMM

F

r

R

vv

v

v&v&&v

1

1111

111

1

11

111

ω (17)

where,

( ) 111 −−− ′−′−= XJXRMRH (18)

This paper will focus on the attitude

stability and control, so only the angular derivatives will be examined:

( ) wXHJqJXXHJJ

FMRXHJvv

v&v

1111

11

−−−−

−−

−′++

′=ω (19)

Simplifications

To analyze the system using classical control theory, a number of linearizations will be implemented.

Because the longitudinal stability of the spacecraft is of interest, the model will be simplified to that of a two-dimensional system in the x-z plane. In addition, angular position changes are expected to be small in stable flight so that a small angle assumption will be invoked. This will result in the following approximations:

( ) 00vvvvv =⇒≈×× wxωω

0≈× ωωvv

J 1cos ≈θ θθ ≈sin

The booms will be modeled as rigid

bodies that are connected to the main body by a linear spring. Therefore the reaction moment between a boom and the main body is:

nnnnRn kbM θθ −−= & (20)

Aerodynamic Force Modeling

Because the flight path of the spacecraft will be confined to 130 km above the Earth’s surface and higher, a rarefied gas-surface interaction model will be used. This aerodynamic model is appropriate for flows where the Knudson number is much greater than 1.

Lewis 3 and Bowman4 provide formulas for lift and drag coefficients under diffuse and specular gas-surface interactions:

( )αεα 2cos14

sin2 −

+=

oDspec V

cC (21)

αεα 2sin4

sin2

+=

oLspec V

cC (22)

+

+= αεα sin

321

4sin2

oDdiff V

cC (23)


5

αεα cos32

4sin2

+=

oLdiff V

cC (24)

( ) DdiffDspecD CCC ξξ −+= 1 (18)

( ) LdiffLspecL CCC ξξ −+= 1 (19)

where α is the angle between a spacecraft surface and the incoming flow.

Although the booms are typically cylindrical, they will be modeled as flat plates. With the high orbital speeds and small molecular thermal speed ratio that will be encountered with the spacecraft, it can be shown that this is a reasonable approximation. For the x-z plane, the average force on the n th boom can be derived as:

( )

( )

+

−

+

+

+

−=

nbLon

Donnz

o

bLon

Donbnz

o

nbnL

nD

Lon

Don

non

CC

xV

CC

xV

CC

CC

SVF

θθ

θ

θθ

ρ

α

α

&&

&v

2

221 2

(20)

where CLo and CDo are the lift and drag coefficients and CLα and CDα are the slopes of the lift and drag coefficients. Those coefficients are linearized about α = θo for the booms so that:

oDDo CC

θα == ,

oLLo CC

θα ==

o

DnD

CCθα

α α =∂∂= ,

o

LnL

CCθα

α α =∂∂= (21)

Assuming the tip mass on the boom is

negligible and the booms are homogeneous along their length, the aerodynamic force will be located at the boom’s center of mass with no aerodynamic moment. Therefore,

0vv =acnx and 0

vv=nM (22)

The lift and drag coefficients for the main

body are more complex to derive, but assuming the body is a symmetric, cylindrical shape, a suitable model is:

( )

+

+

−= nb

bL

bD

Lob

Dobbob C

CCC

SVF θθρα

α2

21v (23)

where the lift and drag coefficients, CLob and CDob, and their slopes, CLαb and CDαb , can be determined from a numerical model.

With the above assumptions, the following vectors can be written into a more convenient matrix form:

UQKBq o

vvv&vv +++= θθ (24)

oFFFFvv&vv

& ++= θθ θθ (25)

where,

−

−−

+−

=

∑

NN

NN

n

bb

bbbb

bbbqb

B

000000000000

22

11

21

OM

L&θ

(26)

−

−−

+−

=

∑

NN

NN

n

kk

kkkk

kkkqk

K

000000000000

22

11

21

OM

Lθ

(27)

=

0

00

M

v

o

o

q

Q,

=

0

00

M

v

WM

U (28), (29)

−

−

−

−=

NzoNN

zo

zbo

zbo

zozbo

b

o

xCS

xCS

xCS

xCSxCSxCS

F

VF

LMO

M

&

&

00

00000000

000

0

222

222

222

111111

θ

θ ρ (30)

−=

NN

bb

o

CS

CSCS

CS

VF

α

α

α

α

θ ρ

LMOM

00000

000000000000

21

22

112

(31)


6

−=

oNN

o

o

obb

oo

CS

CSCSCS

VFM

v22

112

21 ρ

(32)

with,

+

++

=

Nb

b

b

b

θθ

θθθθ

θ

θM

v2

1 (33)

=

Lo

Doo C

CC ,

=

α

αα

L

D

CC

C (34), (35)

where,

( )acbxLobacbzDobboo xCxCSVq −= 2

21 ρ (36)

( )

( )

−+

+−=

acbzbDLob

acbxbLDobbo xCC

xCCSVq

α

αθ ρ 2

21 (37)

Due to the symmetric nature of the main

body, the following assumptions will also be made:

0=θ&

q and 0=b

Fθ&

(38)

but they may be non-zero if aerodynamic damping from the body is present.

Equation 19 then becomes:

( ){ }( ){ }

( )( )UJXXHJJ

QJXXHJJFMRXHJ

KJXXHJJFMRXHJ

BJXXHJJFMRXHJ

oo v

vv

v&v&&v

&

111

11111

11111

11111

−−−

−−−−−

−−−−−

−−−−−

′++

′++′+

′++′+

′++′=

θ

θθ

θ

θ

(39)

Both X and H depend on θ, but they can be linearized about θ o in the form:

( )( ) ( )Nbb

bbo

N

b

XX

XXXX

θθθθ

θθθ

θθ

θθ

+++++

+++=

K2

1

2

1 (40)

( )

( ) ( )Nbb

bbo

N

b

HH

HHHH

θθθθ

θθθ

θθ

θθ

+++++

+++=

K2

1

2

1 (41)

Then the final state-space equation becomes:

′+

+

′=

000F

uU

IKD

θθ

θθ

v&v

&v&&v

(42)

where,

( )BJXHXJJFMRHXJD ooooo11111 −−−−− ′++′=

θ&

(43)

( )

′+

′+

′

′+

′+

′

′+

′+

′

′+

′+

′

+

′+

′

′+

′

′+

′

′+

′+

′++′=′

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−

−−−−−

ooo

ooo

ooo

ooo

ooo

ooo

ooo

ooo

ooo

ooo

ooo

ooo

oo

oo

o

oo

ooo

oo

oo

oo

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

QJXHXJ

FMRHXJ

FMRHXJ

FMRHXJ

FMRHXJ

FMRHXJ

FMRHXJ

FMRHXJ

FMRHXJ

KJXXHJJFMRXHJK

N

N

N

b

b

b

N

N

b

b

v

v

v

Lv

v

v

v

v

v

v

v

v

v

v

Lv

v

v

v

v

v

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11111

2

2

2

1

1

1

2

2

1

1

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

(44)

( )

′+= −−−

0

0111

M

I

JXXHJJU (45)

WMu = (46)

( ) ooooooo QJXHXJJFMRHXJF

vv 11111 −−−−− ′++′=′

(47)

Stability Analysis With the final state-space matrix created,

one can apply standard control techniques to determine the stability of the spacecraft.

Table 1 displays the nominal spacecraft characteristics used in the analysis. The physical parameters are a reasonable approximation of the GEC spacecraft. The ensuing analyses will use these nominal conditions or variations from these conditions.


7

Table 1: Nominal Spacecraft Parameters

Atmosphere properties,

smc

mkg

smVo

640

107.6

300,8

39−×ρ

For the body: For each boom:

mx

CCCC

kgmJmS

kgm

acb

bL

bD

Lob

Dob

b

b

b

00

5.0203

080,11

500

2

2

vα

α

( )

2.5.

02

5.

4deg45,45,135,135

333.82.0

101

41

2

2

n

n

bn

n

n

n

n

n

n

n

mx

radNmsb

radNmk

kgmJmSmL

kgm

ξε

θ

−

−−=

v

K

For a system modeled in the state space form,

duBxAxvvv&v ++= (48)

the criteria for open-loop stability is that the real part of all the eigenvalues of the A-matrix must be negative.5

To examine the behavior of a flexible spacecraft undertaking an aeroassist, a spacecraft with just two booms in a symmetric configuration will be used for simplicity. See Figure 5 where θo is the boom’s non-bent, or neutral, angle position.

Figure 5: Two-Boom Spacecraft

It would be expected that due to the oncoming flow, the booms would bend further back. To see if this is true, the boom deflection

amounts were calculated for various boom angles and body surface areas. See Figure 6.

Figure 6: Graph of Boom Deflections

It is seen that for some combinations of

boom angles and body surface area, the booms will actually bend forward in flight. Although perhaps not immediately intuitive, the reason for this is due to the deceleration of the main body compared to the booms. With a large body surface area, greater drag will be produced by the main body and the body will decelerate more than the booms. When this happens, the booms will actually bend forward relative to the body. The absolute deceleration of the main body doesn’t have to be large at all - in this simulation it was on the order of 0.01 m/s2 - but it just needs to be larger than the boom’s.

Similar results are also seen when the drag coefficients or masses are changed for either the booms or the bodies. (The depicted surface area values were just chosen for convenience.) Whether or not the booms bend forward or back really depends on a kind of ratio of the boom’s and body’s ballistic coefficients.

So, how does the deceleration affect the stability of the spacecraft? Given a main body without extra stabilizing characteristics, it would be expected that a forward-facing boom configuration would be unstable and that a backward-facing boom configuration would be stable. It is the same expectation as flying an arrow with the tail feathers in the back or flying it with the feathers in the front. To see if this is true, the eigenvalues were calculated for the same parameters as before. By looking at the real part of the maximum eigenvalue, one can determine the stability. Note that if the maximum eigenvalue is negative then all

θo

θo Forward Velocity


8

of the eigenvalues are negative and the system will be stable.

Figure 7 shows the value of the real part of the maximum eigenvalue for the various boom angles and body surface areas. Because the magnitude of the negative values is small, a magnified view of the same data is shown in Figure 8.

Figure 7: Graph of Maximum Eigenvalues

Figure 8: Magnified Graph of Maximum

Eigenvalues

When the body surface area is low, i.e. the main body experiences low decelerations relative to the booms, the graph behaves like one would expect. When the booms are bent past 90 degrees in a backward-facing configuration, the eigenvalues are negative and the system is stable.

However, as the body surface area increases, i.e. the main body exp eriences higher decelerations relative to the booms, the range of boom angles where the system is stable decreases.

One might expect that the farther back the booms are, the more stable the system should be, however, one sees from the graphs that for large relative body decelerations, the system is unstable for these far-back boom positions. In these cases, one is actually seeing a buckling-like mode from the booms. Although the booms may themselves not buckle, they are not able to keep themselves at a stable position when the decelerations are large.

Similarly, far-forward facing booms actually end up being stable with large relative body decelerations. In these cases the deceleration of the main body produces a stabilizing moment on the boom that exceeds the unstabilizing aerodynamic moment. Controllability

Spacecraft and satellites with multiple, similar, flexible members may have problems with controllability when they are floating along in free space. In the hub-like configuration represented in the model, a momentum wheel placed in the hub will not be enough to control the positions of the arms if the arms have similar vibration modes.

The momentum wheel will not be able to artificially dampen out any vibrations that may develop if the boom-arms are similar. If there is an attempt to do so, the momentum wheel could actually excite one of the booms and cause the system to be unstable. At best, one may just have to wait for any vibrations to dampen out naturally. This may or may not be acceptable.

One solution to the problem is to make sure that the booms have different natural frequencies. If all of the booms have a different spring constant, then the system will become controllable. In the context of this paper, will this new spacecraft system still be controllable during an aeroassist?

To measure controllability, the Degree of Controllability (DoC) figure of merit developed by Müller and Weber is used.6 In particular, the 3rd DoC measure is used. This value is calculated by taking the determinant of the controllability matrix. If the value is zero, the system is uncontrollable. As the value becomes larger than zero, the system is more controllable than one with a smaller value.


9

Using again the simple two-boomed

spacecraft shown in Figure 5, the DoC was calculated. This time, the booms were fixed at 135 degrees, but the spring constant of one of the booms is varied for different surface areas of that same boom. The results are shown in Figure 9. Note that the actual DoC values end up being quite small, so for comparison purposes, they are arbitrarily normalized.

Figure 9: Degree of Controllability

The nominal surface area of one of the booms is 0.20 m2 with a spring constant of 4. From the chart, if the surface area of the second boom is also 0.20 m2, then the DoC drops to zero as the spring constant approaches 4, i.e. the system becomes uncontrollable. As the spring constant moves away from 4, the system becomes controllable and the DoC increases the farther from 4 one gets. Such results are expected.

If the second boom had a spring constant of 3.7, one would expect the system to be stable because it is different than the other boom’s spring constant. However, if the second boom, for some reason, had a larger surface area, say 0.30 m2, then the DoC also drops to zero despite the non-matching spring constants. Although the natural frequencies of the booms are different, the system is not controllable.

What one sees is that the aerodynamics affects the effective spring constants of the booms. A system that is controllable in free space may become uncontrollable during an aeroassist.

Similar results are obtained when the second boom’s mass or length properties or lift and

drag coefficients are changed as well. The change in surface area was chosen for this example because it doesn’t directly affect the natural frequency of the boom provided the mass and length remain constant.

One important design concern arises when

changing the natural frequencies of the booms, however. A designer may try to stiffen one of the booms to change its spring constant so that the spacecraft is stable in free space. If no other physical parameter is changed, the system will still be controllable during the aeroassist (assuming a symmetric configuration). However, if by the process of changing the boom stiffness other properties are changed, such as mass or surface area, then the system may not be controllable during the aeroassist as seen by Figure 9. Boom Location

The location of the booms is important in determining constraints on the design of the main body of the spacecraft. In particular, a designer would like a lot of freedom in positioning the center of mass of the spacecraft. Without booms, the aerodynamic center of the spacecraft generally needs to be behind the center of mass of the spacecraft for the spacecraft to be stable. With booms, however, the aerodynamic center of the main body of the spacecraft can be allowed to move forward. Like the tail feathers on an arrow, the drag produced by the booms helps stabilize the spacecraft.

To investigate the relationship between the body’s aerodynamic center and the boom location, a 4-boom model was used where the booms were placed at angles of 135, -135, 45, and –45 degrees. With the same connection point, the booms were positioned forward and back along the centerline of the spacecraft body, and the location of the aerodynamic center was varied between -1.0 and 1.0 meters. Figure 10 shows the resulting stability results and Figure 11 shows the same data magnified.

In general, backward positioned booms allow for a more positive location of the body’s aerodynamic center while forward positioned booms require the aerodynamic center to place very far rearward.

In the GEC-like configuration where the booms are placed at the tail of the spacecraft, the aerodynamic center can be positioned almost 0.35 meters ahead of the center of mass, or looking at it


10

the other way, the center of mass can be 0.35 meters behind the aerodynamic center.

Figure 10: Stability vs. Boom Location

Figure 11: Magnified View of Stability vs. Boom

Location

For a neutrally stabilized body where the center of mass and aerodynamic center coincide, the booms would have to be placed at least 1.1 meters behind the center of mass for the entire spacecraft to be stable.

Conclusion This paper demonstrates the development of a viable model to address the stability and control concerns of a flexible spacecraft undergoing an aeroassist maneuver.

Unusual stability characteristics were uncovered by the model when the main body of the spacecraft has a large deceleration relative to that of the booms. Under such decelerations, unstable

buckling-like modes develop in previously stable boom positions.

The analysis also shows that controllability may also be affected during an aeroassist. It was shown that a spacecraft that is controllable in free space might become uncontrollable during the aeroassist. A designer needs to be cautious when adjusting the boom properties so that the controllability is maintained in all flight regimes.

Finally, the model provides a guide for boom placement and constraints on the spacecraft body’s center of mass to ensure stability.

Future Work Future work will investigate linear and non-linear control methodologies for a spacecraft with flexible booms during an aeroassist. Stability sensitivities will also be investigated for different rarefied gas-surface interaction models.

Acknowledgements The authors would like to thank the

NASA/Goddard Graduate Student Research Program, technical advisor, Mr. Thomas Stengle, and GEC program manager, Mr. Paul Buchanan, for their research opportunities and support.

References 1Vinh, N. X., Optimal Trajectories in

Atmospheric Flight, Elsevier Scientific Publishing Co., New York, NY, 1981.

2Huston, R. L., Multibody Dynamics, Butterworth-Heinemann, Boston, MA, 1990.

3Lewis, M. J., “Aerodynamic Maneuvering for Stability and Control of Low-Perigee Satellites,” Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, AAS Paper 01-239, Feb 2001.

4Bowman, D. S., “Numerical Optimization of Low-Perigee Spacecraft Shapes,” Masters Thesis, University of Maryland, College Park, MD, UM-AERO 01-03, 2001.

5Ogata, K., Modern Control Engineering, Prentice Hall, Inc., Upper Saddle River, NJ, 1997.

6Müller, P. C., and Weber, H. I., “Analysis and Optimization of Certain Qualities of Controllability and Observability for Linear Dynamical Systems,” Automatica, Vol. 8, 1972, pp. 237-246.

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