Control of Lagrange point orbits using

solar sail propulsion

John Bookless

Contents

• Solar Sail Propulsion

• Hill’s Approximation of the three-body problem

• Quasi-periodic Orbit

• Optimal Controller

• Orbit Control near L2

• Orbit Control near L1

• Geostorm Mission

• Conclusions

na 22

cosR

s

s

s

c

L

2

a – sail accelerationsail lightness number

s – solar gravitational parameter

– reflectivity

R – distance between Sun and sail

c speed of light

Ls – Solar Luminosity

– sail loading parameter– pitch angle

Solar sail Propulsion

Non-dimensionalised with L = 1 RE (Earth Radii)

Sun r

Earth . La

R

Rs

i

j k

xxr

xyx 23

32

yr

yxy

32

zzr

zz 2

3

Hill’s Equations

222 zyxr

EL 3

Separation distance between Earth and Sail

and characteristic time

• Libration points identified by setting

• For on-axis libration points y=z=0

• The Lagrange points are located at

• Artificial Libration points can be generated Sunward of L1

or Earthward of L2 using solar sail acceleration

• Acceleration variation due to position relative to Sun

0 zzyyxx

3123

x

o

o

oo x

r

x 23 3

2)()( tRRt oo

Quasi-periodic Orbit

Hill’s equations can be linearised about the libration point (xo, yo, zo)using new coordinates

xzxyxx fff 2

yzyyyx fff 2

zzzyzx fff

oxx oyy ozz

Linearised equations have the form

where the pseudo-potential function

xzxr

zyxf

)3(2

1),,( 22

2

For oscillatory motion, real eigenvalues 1,2are suppressed. The resultingsolution can be expressed as

xyyvA sin

xyyA cos

zzA sin

where Ay is the y-axis amplitude, Az is the z-axis amplitude, the eigenvalues

zzz f

yyf

xxf

yyf

xxf

yyf

xxf

xy4

22424

2

1

and the eigenvector

xy

xyv

2

3 22

The orbit is quasi-periodic as the ratio of the in-plane and out-of-planefrequencies is non-rationalzxy

Optimal Controller

Orbit can be controlled by applying trims to solar sail area or pitching the sail relative to the Sun-line

)()()( tBtAt uxx )()( tCt xy

x(t) – state vector

y(t) – output vector

u(t) – control vector

A – linear coefficient matrix

B – control matrix

C – output matrix

t

dNQV )()(')()(' uuxx

The optimal controller selects a gain matrix which minimises the performance function

where Q – state weighting matrixN – control weighting matrix

QMBMBNMAMAM '' 1

A suitable gain matrix is calculated by solving the Ricatti equation for M

where M is the performance matrix and is related to the performance function MxxV '

The optimal gain matrix is evaluated as MBNG '1

Sail area control

)()()(

000ttt

B zyx

33 coscos)(tx sincoscos)( 23ty

sincoscos)( 22tz

Components of sail acceleration

Sail area variation control matrix

Partial derivatives with respect to acceleration

33 coscos

)(

tx

sincoscos)(

23

ty

sincoscos)(

22

tz

Control matrix 001000B evaluated at

zyx

zyx

B000

000

Sail pitch and yaw angle control

Sail pitch and yaw angle control matrix

Partial derivatives with respect to pitch angle

32 cossincos)(3 tx

sincossincos)(3 22ty

223 tan21coscos)(

tz

and with respect to yaw angle

sincoscos)(3 23tx

233 tan21coscos)(

ty

sincossincos)(2 2tz

00000

00000

o

oB

Control matrix evaluated at

For solar sail area control, the acceleration variation is calculated using

654321 GGGGGGx

The error between the actual position and desired position

)sin( xyyA

)cos( xyyA

)sin( zyA

)cos( xyxyyvA

)sin( xyxyyA

)cos( zzyA

For sail pitch and yaw angle control

161514131211 GGGGGG

262524232221 GGGGGG

Orbit control near L2

xo = 230RE = 0.008mms-2

Ay = 20REC = -0.0131 Manifold insertion at 19.1RE from Earth

Solar sail winds onto nominal orbitwithin ~91 days

Orbit Insertion

Area Control

Linear relationship between required acceleration and required area

Acceleration varies between 0.0068→0.0114 mms-2

For a 200kg sail + payload mass, area varies between 152→254 m2

Figure below based on sail loading =12gm-2

A 100 kg payload could be controlled using a 129 m2 solar sailAverage V=280 ms-1 per year

Pitch and Yaw angle control

xo = 230RE = 0.01mms-2

Ay = 20RE

Pitch angle -42.9o→2.9o

Yaw angle -0.69o→0.78o

A 200kg sail + payload mass could be controlled using a 222 m2 sail

Sail area / payload mass gradient = 1.1268 for loading =12gm-2

Orbit control near L1Orbit Insertion

xo = -240RE = 0.014mms-2

Ay = 20REC = -0.0120

Manifold insertion at 1.15RE from Earth

Solar sail winds onto nominal orbitwithin ~320 days

Area Control

Acceleration varies between 0.0115→0.0159 mms-2

For a 200kg sail + payload mass, area varies between 246→340 m2

Average V=395 ms-1 per year

D = 93,500km at libration point

Telemetry Exclusion Region

For area control, solar sail spends approximately 3.89 years within exclusion zone

Pitch and Yaw angle control

xo = -240RE = 0.025mms-2

Ay = 20RE

Pitch angle -52.3o→8.3o

Yaw angle -1.78x10-6 o→1.89x10-6 o

Solar sail spends about 1/5th of orbit duration within telemetry exclusion zone

A 200kg sail + payload mass can be controlled using a 529 m2 solar sail withloading =12gm-2

Geostorm Mission

ACE (Advanced Composition Explorer)Launched in 1997

Earth’s Magnetosphere

Mission proposed by JPL to use solar sail to position a science payloadSunward of L1

xo = -234.46RE = 0 mms-2Ay = 50REC = -0.01226

Manifold insertion at 11.15RE from Earth

Solar sail winds onto nominal orbitwithin ~186.5 days

• Un-deployed solar sail remains at L1 for 186.5 days

• Solar sail is gradually deployed and spirals onto new orbitwithin 560 days

• Area variation control used to prevent escape from new orbit

100 kg sail + payload massSail acceleration 0.24 mms-2

Sail area 2517 m2

Sail loading =12gm-2

Sail mass = 30kgPayload mass = 70kg

Average V=6.184 kms-1 per year

Telemetry exclusion radius 150,000km at libration point

Conclusions

• Demonstrated orbit insertion and solar sail control techniques ata quasi-periodic orbit

o Mission duration near Lagrange points not limited by lifetimeof stored reaction mass

o Low acceleration achievable with present day technology highlights a possible near-term application for solar sails

• Possible trajectory identified for the proposed Geostorm mission

o Initial delivery of un-deployed solar sail to L1 enables piggy-back transfer

o Large V requirement indicates that solar sails are the only viable propulsion option for a long-term mission