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Solar Sail
Department of Aerospace Engineering and Mechanics
AEM 4332W – Spacecraft Design
Spring 2007
9
Recall Orbital Mechanics
• The state of a spacecraft can be described by a vector of 6 orbital elements.– Semi-major axis, a– Eccentricity, e– Inclination, i– Right ascension of the ascending node, Ω– Argument of perihelion, ω– True anomaly, f
• Equivalent to 6 Cartesian position and velocity components.
11
Equations of Motion
vr
nnrr
rr
v2^
2
^
2
^^^^
sinsincossincos rpprn
^
r
^
p
^^
rp
n
linesun
sail
= Sail Lightness Number = Gravitational Parameter
12
Problem: Minimize Transfer Time
1),,(2^
2
^
2
nnr
rr
rvuxH vvr
^
r
^
p
^^
rp
n
linesun
sail
^^^
353)(2))((2)(3 rnrnnnr
rrr
rr vrvr rv
^^
}max{ vv nn
By Inspection:
Transversality:
fttv
ttv npnr
rnpnr
r
2
^
22
^
2)()(
0
13
Solution
• Iterative methods are needed to calculate co-state boundary conditions.
• Initial guess of the co-states must be close to the true value, otherwise the solution will not converge.
• Difficult• Alternative: Parameter Optimization.
– For given state boundary conditions, maximize each element of the orbital state by an appropriate feedback law.
14
Orbital Equations of Motion
r
pTfSe
e
pr
df
dasin
)1(
222
2
e
p
rTf
p
rTfS
r
df
decos1sin
2
Wfp
r
df
di)cos(
3
Wfip
r
df
d)sin(
sin
3
f
p
rTfS
e
ri
df
d
df
dsin1coscos
2
12
2sin1cos1
f
p
rTfS
e
r
r
p
dt
df
)1( 2eap fe
pr
cos1
32
cosr
S sinsincos22r
T cossincos22r
W
),,( xgx
= Sail Lightness Number = Gravitational Parameter
15
Maximizing solar force in an arbitrary direction
^^^^
sinsincossincos rpprn ^^~~^~~^~
sinsincossincos rpprq
^
r
^
p
^^
rp
n
linesun
sail
Maximize:
qnnr
raq
2^
2
~
~
~2
tan4
tan893tan
Sail pointing for maximum acceleration in the q direction:
16
Locally Optimal Trajectories• Example: Use parameter optimization method to derive
feedback controller for semi-major axis reduction.
• Equations of motion for a:
r
pTfSe
e
pr
df
dasin
)1(
222
2
3
2cos
rS
sinsincos22r
T
fe
fe
cos1
sintan
~
fe
pr
cos1 )1( 2eap
2
~
~2
tan4
tan893tan
Feedback Law:
Use this procedure for all orbital elements
17
Method of patched local steering laws (LSL’s)
• Initial Conditions: Earth Orbit
• Final Conditions: semi-major axis: 0.48 AU inclination of 60 degrees
0
0
0
0
0
1
0tt
i
e
a
free
free
free
AU
i
e
a
tft
60
0~
48.0
20
Global Optimal Solution– Although the method of patched LSL’s is not ideal, it is a solution that is
close to the optimal solution.
– Example: SPI Comparison of LSL’s and Optimal control.
21
Conclusion
• Continuous thrust problems are common in spacecraft trajectory planning.
• True global optimal solutions are difficult to calculate.
• Local steering laws can be used effectively to provide a transfer time near that of the global solution.
28
Primary Structural Material
Weight and Volume Constraints• Delta II : 7400 Series • Launch into GEO
– 3.0 m Ferring» Maximum payload mass: 1073 kg» Maximum payload volume: 22.65 m3
– 2.9 m Ferring» Maximum payload mass: 1110 kg» Maximum payload volume: 16.14 m3
29
Primary Structural Material
Aluminum Alloy Unistrut– 7075 T6 Aluminum
Alloy• Density
– 2700 kg/m3
– 168.55 lb/ft^3
• Melting Point– ? Kelvin
Picture of Unistrut
30
Primary Structural Material
• Density
• Mechanical Properties– Allowing unistrut design
• Decreased volume
• Thermal Properties– Capible of taking thermal loads
31
Design Layout
• Constraints– Volume– Service task– Thermal consideration– Magnetic consideration– Vibration– G loading
32
Design Layout
• Unistrut Design– Allowing all inside surfaces to be bonded to
• Titanium hardware
– Organization• Allowing all the pointing requirements to be met with
minimal attitude adjustment
Kory Jenkins• Sail Support Structure• Anticipated Loading•Stress Analysis• Materials•Sail Deployment
47
Attitude Control
• Conducted trade between tip thrusters and sliding mass as primary ACS
• Considerations– Power required– Torque produced– Weight– Misc. Factors
48
Attitude Control
• Tip Thrusters (spt-50)– Pros
• High Torque Produced ~ 1.83 N-m• Low weight ~ 0.8 kg/thruster
– Cons• Large Power Requirement ~ 310 Watts• Lifetime of 2000 hrs• Requires a fuel, either a solid or gas
49
Attitude Control
• Attitude Control System Characteristics– Rotational Rate– Transfer Time– Required Torque– Accuracy– Disturbance compensation
50
Attitude Control
• Requirements– Orbit
• Make rotation rate as fast as possible
• Roll spacecraft as inclination changes
– Communications– Within Maximum Torque
• Pitch and Yaw Axis
~ 0.34 N-m
• Roll Axis
~ 0.2 N-m
M
mFzU
m – sliding massF – solar forcez – distance from cgM – spacecraft mass
51
Attitude Control
• Pitch and Yaw Axis • Rotation Rate = 0.144 rad/hr
~ 8.25 deg.
• Transfer Time = 5300s ~ 1.47 hrs
• Required Torque = 0.32 N-m
~ 98.8% of maximum produced
• Converges to desired angle
Slope = 0.00004 rad/s
Torque Req.
Transfer Time
52
Attitude Control
• Roll Axis • Rotation Rate = 0.072 rad/hr
~ 4.12 deg
• Transfer Time = 7000s ~ 1.94 hrs
• Required Torque = 0.15 N-m
~ 75% of maximum produced
• Converges to desired angle
Torque Req.
Slope = 0.00002 rad/s
Transfer Time
Raymond HaremzaThermal Analysis
•Solar Intensity and Thermal Environment•Film material•Thermal Properties of Spacecraft Parts•Analysis of Payload Module•Future Work