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