Abstract
HAYS, SCOTT ALLEN. A Study of the Advantages of Using a Tether Ballast Mass as Part of the Control System in a Solar Sail based Solar Storm Warning Mission. (Under the direction of Dr. Andre Mazzoleni.)
The objective of the thesis is to analyze the dynamics and controls of a solar
sail operating on a Solar Storm Warning Mission. The Solar Storm Warning
Mission was chosen for analysis because it has a high priority mission to provide
data on solar storm activity. Solar storms can cause blackouts in power grids
and disable satellites that are currently orbiting the Earth. A payload extends
from the sailcraft by four tethers attached near the ends of the booms to provide
stability. The tether length that produced a stable sailcraft was equal to or
greater than 84.31 meters. A parametric study was completed to examine the
effects of the roll, pitch, and yaw disturbances on the sailcraft dynamics. The
roll and pitch coupling (in the equations of motion) had a large effect on the
sailcraft using short tether lengths, but had a much smaller effect at the longer
tether lengths of 84.31 meters or more. Thrusters were placed at the ends of
the sail booms to provide dampening to the sailcraft. Full-state feedback (pole
placement) provided settling times and number of maneuvers for each specific
thruster/tether length combination. If the disturbances (roll, pitch, and yaw)
expected to impact the sailcraft are known, an ideal tether length can be found
as a function of the settling time and number of maneuvers. For example, the
125 mN thruster requires a tether length of approximately 250 meters to provide
A Study of the Advantages of Using a
Tether Ballast Mass as Part of the
Control System in a Solar
Sail Based Solar Storm
Warning Mission
by
Scott A. Hays
A thesis submitted to the Graduate Faculty of North Carolina State University
in partial fulfillment of the requirements for the Degree of
Master of Science
Aerospace Engineering
Raleigh, North Carolina
December, 2007
APPROVED BY:
Dr. Fred R. DeJarnette Dr. Larry M. Silverberg
Biography
Acknowledgements
I am extremely grateful to my extended family Debbie Rowland, Patrick Hays, Robert Rowland, and Barbara Hays for their continuous support of my educa-tion. Their support has been unwavering and I cannot thank them enough. I would like to thank Lauren Clark and Alex Hartl for careful reading and helpful comments.
Contents
List of Figures . . . vii
List of Tables . . . xi
Nomenclature . . . xiii
1 Introduction and Background 1 2 Mission Design 5 2.1 Force Equations . . . 5
2.1.1 Gravitational Force . . . 5
2.1.2 Non-Perfectly Reflecting Solar Sail . . . 7
2.1.3 Sail-Loading . . . 9
2.1.4 Initial Sail Acceleration . . . 11
2.2 Mission Definition . . . 12
2.2.1 Initial Mass Budget . . . 13
2.2.2 Sail Size Determination . . . 13
2.2.3 Thruster Selection . . . 16
2.3 Final Design Decision . . . 19
2.4 Check of Point Mass Assumption . . . 22
3 Inertia Calculations 27 3.1 Orientation . . . 27
3.2 Torque Calculation . . . 28
3.2.1 Thruster Placement . . . 30
3.3 Moment of Inertia Background . . . 32
3.4 Component Description . . . 35
3.5.1 Sail Film Inertia . . . 40
3.5.2 Sail Boom Inertia . . . 41
3.5.3 Tether Inertia . . . 42
3.5.4 Payload Inertia . . . 49
3.5.5 Thruster Inertia . . . 51
3.5.6 Fuel Tank Inertia . . . 54
4 System Dynamics 55 4.1 Euler’s Moment Equations . . . 55
4.2 Orbital Dynamics . . . 56
4.3 Equations of Motion . . . 59
4.3.1 Nonlinear Equations of Motion . . . 59
4.3.2 Linear Equations of Motion . . . 63
4.4 State Space Equations . . . 65
5 Case Studies 69 5.1 Case Study . . . 70
5.2 Sailcraft Stability . . . 72
5.3 Case Study Parameters . . . 76
5.3.1 Case Study Definition (Tether Variation) . . . 76
5.3.2 Parametric Study Definition . . . 82
5.4 Dynamics Parametric Study for Linear Systems . . . 84
5.4.1 Unstable Linear Dynamics . . . 84
5.4.2 Stable Linear Dynamics . . . 88
6 Sailcraft Controls 109 6.1 System Poles . . . 109
6.2 Sailcraft Controllability . . . 111
6.2.1 Controllability and Observability . . . 112
6.3 Full-State Feedback . . . 115
6.4 Closed Loop Poles . . . 116
6.4.1 Poles for Thruster A . . . 117
6.4.2 Poles for Thruster B . . . 122
6.4.4 Controlled Response . . . 127 6.4.5 Change in Mass Assumption . . . 131 6.5 Ideal Mission Tether Length . . . 132
7 Conclusions 140
References 142
Appendix . . . 145
A Initial Sail Forces and Accelerations 146
B Example of Moment of Inertia Integration 149
C Calculation of the Transformation Matrix 152
D Calculation of the Equations of Motion 154
E Controlled Responses for Thruster A 160
F Controlled Responses for Thruster B 206
List of Figures
1.1 Solar Storm Warning Time Based on Orbit Radius (9) . . . 4
2.1 Force Diagram for Solar Sail . . . 6
2.2 Sail Length Required for 50kg Payload . . . 14
2.3 Sail Length Required for 75 kg Payload . . . 15
2.4 Sail Length Required for 2 and 5 kg/m2 Sail Film Area Densities 16 2.5 1 N Monopropellant Thruster Model No. CHT-1(12) . . . 18
2.6 Surrey Satellite Technology Ltd Resistojet(12) . . . 18
2.7 Sail-Loading Plot . . . 20
3.1 Orientation of the Sailcraft Model with respect to Frame C . . . . 27
3.2 Relationship between Frames C and B . . . 28
3.3 Sail Lengths . . . 29
3.4 Torque One Setup . . . 31
3.5 Overall Torque Setup . . . 32
3.6 Sail Film Sectioning . . . 36
3.7 Sail Boom 1 Sectioning . . . 36
3.8 Sail Boom 2 Sectioning . . . 37
3.9 System Setup With Tether . . . 37
3.10 2-D Payload Model . . . 39
3.11 Tether 1 Configuration . . . 43
3.12 Tether 2 Configuration . . . 43
3.13 Tether 3 Configuration . . . 44
3.14 Tether 4 Configuration . . . 44
4.1 Orbital Dynamics . . . 56
5.1 Unstable Dynamics Sailcraft 1- 4 Roll/Roll Rate for Cases 1-4 . . 85
5.2 Unstable Dynamics Sailcraft 1- 4 Pitch/Pitch Rate for Cases 1-4 . 86 5.3 Roll for Cases 1-8 . . . 89
5.4 Roll for Cases 9-16 . . . 90
5.5 Roll for Cases 1-8 . . . 91
5.6 Roll for Cases 9-16 . . . 92
5.7 Roll Rate for Cases 1-8 . . . 93
5.8 Roll Rate for Cases 9-16 . . . 94
5.9 Roll Rate for Cases 1-8 . . . 95
5.10 Roll Rate for Cases 9-16 . . . 96
5.11 Pitch for Cases 1-8 . . . 97
5.12 Pitch for Cases 9-16 . . . 98
5.13 Pitch for Cases 1-8 . . . 99
5.14 Pitch for Cases 9-16 . . . 100
5.15 Pitch Rate for Cases 1-8 . . . 101
5.16 Pitch Rate for Cases 9-16 . . . 102
5.17 Pitch Rate for Cases 1-8 . . . 103
5.18 Pitch Rate for Cases 9-16 . . . 104
5.19 Yaw for Cases 1-16 . . . 105
5.20 Yaw Rate for Cases 1-16 . . . 106
6.1 Sailcraft 1: Thrusters 1 and 2 Output For Cases 1 -16 . . . 118
6.2 Sailcraft 1: Thrusters 3 and 4 Output For Cases 1 -16 . . . 118
6.3 Sailcraft 1: Thruster 1 and 2 Thrust For Cases 1 - 16 . . . 119
6.4 Sailcraft 1: Thruster 3 and 4 Thrust For Cases 1 - 16 . . . 120
6.5 Sailcraft 1: Roll Response for Thruster A and B . . . 128
6.6 Sailcraft 1: Pitch Response for Thruster A and B . . . 129
6.7 Sailcraft 1: Yaw Response for Thruster A and B . . . 130
E.1 Sailcraft 1: Roll and Roll Rate . . . 161
E.2 Sailcraft 1: Pitch and Pitch Rate . . . 162
E.4 Sailcraft 1: Thruster 1 and 2 Thrust For Cases 1 - 16 . . . 164
E.5 Sailcraft 1: Thrusters 1, 2, 3, and 4 Torque For Cases 1 - 16 . . . 165
E.6 Sailcraft 1: Thruster 1 and 2 Thrust For Cases 1 - 16 . . . 166
E.7 Sailcraft 1: Thruster 3 and 4 Thrust For Cases 1 - 16 . . . 167
E.8 Sailcraft 2: Roll and Roll Rate . . . 168
E.9 Sailcraft 2: Pitch and Pitch Rate . . . 169
E.10 Sailcraft 2: Thrusters 1, 2, 3, and 4 Torque For Cases 1 - 16 . . . 170
E.11 Sailcraft 2: Thruster 1 and 2 Thrust For Cases 1 - 16 . . . 171
E.12 Sailcraft 2: Thruster 3 and 4 Thrust For Cases 1 - 16 . . . 172
E.13 Sailcraft 3: Roll and Roll Rate . . . 173
E.14 Sailcraft 3: Pitch and Pitch Rate . . . 174
E.15 Sailcraft 3: Thrusters 1, 2, 3, and 4 Torque For Cases 1 - 16 . . . 175
E.16 Sailcraft 3: Thruster 1 and 2 Thrust For Cases 1 - 16 . . . 176
E.17 Sailcraft 3: Thruster 3 and 4 Thrust For Cases 1 - 16 . . . 177
E.18 Sailcraft 4: Roll and Roll Rate . . . 178
E.19 Sailcraft 4: Pitch and Pitch Rate . . . 179
E.20 Sailcraft 4: Thrusters 1, 2, 3, and 4 Torque For Cases 1 - 16 . . . 180
E.21 Sailcraft 4: Thruster 1 and 2 Thrust For Cases 1 - 16 . . . 181
E.22 Sailcraft 4: Thruster 3 and 4 Thrust For Cases 1 - 16 . . . 182
E.23 Sailcraft 5: Roll and Roll Rate . . . 183
E.24 Sailcraft 5: Pitch and Pitch Rate . . . 184
E.25 Sailcraft 5: Thrusters 1, 2, 3, and 4 Torque For Cases 1 - 16 . . . 185
E.26 Sailcraft 5: Thruster 1 and 2 Thrust For Cases 1 - 16 . . . 186
E.27 Sailcraft 5: Thruster 3 and 4 Thrust For Cases 1 - 16 . . . 187
E.28 Sailcraft 6: Roll and Roll Rate . . . 188
E.29 Sailcraft 6: Pitch and Pitch Rate . . . 189
E.30 Sailcraft 6: Thrusters 1, 2, 3, and 4 Torque For Cases 1 - 16 . . . 190
E.31 Sailcraft 6: Thruster 1 and 2 Thrust For Cases 1 - 16 . . . 191
E.32 Sailcraft 6: Thruster 3 and 4 Thrust For Cases 1 - 16 . . . 192
E.33 Sailcraft 7: Roll and Roll Rate . . . 193
E.34 Sailcraft 7: Pitch and Pitch Rate . . . 194
E.35 Sailcraft 7: Thrusters 1, 2, 3, and 4 Torque For Cases 1 - 16 . . . 195
E.37 Sailcraft 7: Thruster 3 and 4 Thrust For Cases 1 - 16 . . . 197
E.38 Sailcraft 1 and 2: Yaw and Yaw Rate . . . 198
E.39 Sailcraft 3 and 4: Yaw and Yaw Rate . . . 199
E.40 Sailcraft 5 and 6: Yaw and Yaw Rate . . . 200
E.41 Sailcraft 7: Yaw and Yaw Rate . . . 201
E.42 Sailcraft 1 and 2: Thruster 5 and 6 Thrust For Cases 17 - 20 . . . 202
E.43 Sailcraft 3 and 4: Thruster 5 and 6 Thrust For Cases 17 - 20 . . . 203
E.44 Sailcraft 5 and 6: Thruster 5 and 6 Thrust For Cases 17 - 20 . . . 204
E.45 Sailcraft 7: Thruster 5 and 6 Thrust For Cases 17 - 20 . . . 205
F.1 Sailcraft 1: Roll and Roll Rate . . . 207
F.2 Sailcraft 2: Roll and Roll Rate . . . 208
F.3 Sailcraft 3: Roll and Roll Rate . . . 209
F.4 Sailcraft 5: Roll and Roll Rate . . . 210
F.5 Sailcraft 6: Roll and Roll Rate . . . 211
G.1 Sailcraft 1: Pitch and Pitch Rate . . . 213
G.2 Sailcraft 2: Pitch and Pitch Rate . . . 214
G.3 Sailcraft 3: Pitch and Pitch Rate . . . 215
G.4 Sailcraft 5: Pitch and Pitch Rate . . . 216
List of Tables
2.1 Optical Coefficients . . . 10
2.2 Initial Mass Budget Estimate . . . 13
2.3 Final Material Properties . . . 19
2.4 Final Mass Budget . . . 21
2.5 Overall System Mass . . . 22
3.1 Thruster Definition . . . 32
5.1 Fuel Tank Information . . . 71
5.2 Roll and Pitch Angles for the Linear Equations of Motion . . . 83
5.3 Yaw Angles for the Linear Equations of Motion . . . 83
5.4 Color Definition for Dynamic Parametric Study . . . 88
6.1 Thruster A Mass(g) Used Per Maneuver . . . 122
6.2 Thruster B Mass(g) Used Per Maneuver . . . 124
6.3 Thruster C Mass(g) Used Per Maneuver . . . 126
6.4 Percent Error for Thruster A . . . 132
6.5 Fuel Tank Selection . . . 133
6.6 Fuel Mass (kg) . . . 134
6.7 Number of Maneuvers for Thruster A . . . 135
6.8 Number of Maneuvers for Thruster B . . . 136
6.9 Number of Maneuvers for Thruster C . . . 137
6.10 Settling Time (sec) . . . 138
6.11 Ideal Tether Length for Thruster A (meters) . . . 139
A.2 Sail Acceleration for 0.6 AU . . . 147
A.3 Sail Force for 0.6 AU . . . 147
A.4 Sail Acceleration for 0.8 AU . . . 148
Nomenclature
1
Fsail/sun Force acting on the Sail from the Sun
P Pressure
A Area
Bf Material Properties for Non-Lambertian surfaces
Emissivity
~
n Normal Direction
~t Tangential Direction
σ Sail Loading
ρ Density
Ix Principal Moment of Inertia
Ixy Product of Inertia
θ1 Roll
θ2 Pitch
θ3 Yaw
τ Torque
s System Pole
Chapter 1
Introduction and Background
In ancient times, ships that used sails to catch the wind allowed man to explore new lands and expand the reach of their kingdoms. Today, man can use space-craft with sails to travel to new galaxies or to explore new worlds. The concept of sailing through space is called solar sailing. Solar sailing is the same concept as sailing. An outside force that is free (wind) impacts the sail and exerts a force on the sail. In space, the particles impacting the sail will be photons (light particles) that are emitted by the sun. A system orbiting at a constant distance from the Sun will have a constant force acting on the sail. The force acting on the sail will be proportional to the inverse of the orbiting radius squared (the radius from the Sun to the sail).
the Halley’s Comet rendezvous mission resulted in the first and most complete analysis for the design of solar sails. The Halley’s Comet mission produced three basic design setups for a solar sail, these being square sails, disk (or radial) sails, and heliogyro sails (all other concepts can be categorized as a variant of these three sails) (2). Each of these three concepts was formulated for different reasons. Heliogyro and disk sails would require an angular velocity on the system at all times to create the controls for the spacecraft. While this has disadvantages, the constant angular velocity provides stiffness to the outer members of the disk and heliogyro sails. The square sail could rotate; however, most controls devised for the square sail include extra components to provide control. The Halley’s Comet rendezvous mission team decided that the best design for solar sailing was a heliogyro sail. This design was selected primarily because of its capability to deploy in space. The first research designs for solar sails required extremely large areas because the thickness of the materials available caused weight problems. However, with the materials available today, the sails can be much thinner and do not require the large sail as they once did. This allows many different solar sail designs to become realistic means of propulsion for travel within our solar system. The thinner solar sail materials recently discovered have stimulated much of the current interest in the subject, although recent research has covered all different components of solar sailing.
of pressure, 2) reaction wheels, 3) gimbaled control booms, 4) control vanes, and 5) shifting and tilting of sail panels(6),(7).
Figure 1.1: Solar Storm Warning Time Based on Orbit Radius (9)
Chapter 2
Mission Design
2.1
Force Equations
Analyzing the orbital motion of the sailcraft involves assessing the effects of grav-itational forces and solar pressure forces. For the initial calculations the sail and payload are assumed to be a rigid body point mass located at a set distance from the Sun; that is, it is assumed that all of the mass is at one point and the tether extended payload will not have an effect on the gravitational force acting on the body.
2.1.1
Gravitational Force
Figure 2.1: Force Diagram for Solar Sail
The Solar Sail Solar Storm Mission will have three different forces acting on the sail (seen in figure 2.1). Two of the three forces are gravitational forces resulting from the attraction of the Sun and Earth.
Fg =
−GM m r2
~ r
r (2.1)
To simplify the equation the gravitational parameter (µ) will be defined. The gravitational parameter is equal to the gravitational constant times the mass of the planet(10).
Fg =
µm r2
~ r
r (2.2)
The notation used for the radius from the sail to the planet is expressed as a subscript of the force variable. For example, the force acting on the sail due to the sun is Fsail/sun. The resulting force equations for the Sun and Earth acting on the sail are:
Fsail/sun =
µsunms/c
r2 sail/sun
~ rsail/sun
rsail/sun
Fsail/earth =
µearthms/c
r2
sail/earth
~ rsail/earth
rsail/earth
(2.4)
The gravitational forces were calculated as a function of the mass of the sail and the distance the sail is located from the Sun. The gravitational parameter for each planet, or in this case the Sun and the Earth, is a constant value shown in many Astrodynamic tables(10).
2.1.2
Non-Perfectly Reflecting Solar Sail
The solar pressure force for a solar sail that does not have a perfectly reflective surface is dependent on the properties of the sail material. The solar sail force, or radiation pressure, is dependent on the force components of reflection, absorption, and emission. The following optical equations for a solar sail were derived by McInnes (Solar Sailing: Technology, Dynamics, and Mission Applications)(11).
Fsail =Fref lective+Fabsorption+Femission (2.5)
The solar sail forces are defined by the normal (n) and transverse (t) vectors of the sail. A portion of the photons that impact the sail will be reflected and another portion of the photons will be absorbed by the sail. The solar force is a function of the pressure of the photons and the area of the sail. Assuming the photons that impact the sail were perfectly reflected back and the sail was perpendicular to the Sun, then the equation for the solar force would be:
Fsail =PphotonsAsail (2.6)
Equation 2.6 is an ideal case that would never occur since the pressure acting on the sail will come at an angle, and the sail will not be perfectly reflective. All of the forces acting on the sail become a function of the pitch angle of the sail (this is the angle that the sail is aligned with the Sun). The resulting forces due to the reflective and absorption of the photons are:
Fref lective = PphotonsAsail pref lpspref lcos2α+Bf(1−pspref l)pref lcosα
~n
−PphotonsAsail(pref lpspref lcosαsinα)~t (2.8)
The force due to the emission of photons is a function of the sail temperature, the front and back surface emissivity, and the material properties. The material properties that have an effect on the system are due to the non-Lambertian surfaces (both front and back and shown as coefficients Bf and Bb) of the sail.
Femission =PphotonsAsail[(1−pref l)
fBf −bBb
f +b
cosα]~n (2.9)
Ftotalsail~n =PphotonsAsail
(1 +pref lpspref l) cos2α~n+Bf(1−pspref l)pref lcosα
~ n
+PphotonsAsail(1−pref l)
fBf −bBb
f +b
cosα~n (2.10)
Ftotalsail~t=PphotonsAsail[(1−pref lpspref l) cosαsinα]~t (2.11)
The total force equation described above can be used to find the center line angle. The center line angle is the angle between the normal and the transverse vectors of the sail.
φ= arctan Ftotalsail~t
Ftotalsail~n
(2.12)
φ= arctan (1−pref lpspref l) cosαsinα
(1+pref lpspref l) cos2α+Bf(1−pspref l)pref lcosα+(1−pref l)f Bf
−bBb
f+b cosα
(2.13)
of equations 2.10 and 2.11.
Fsimp~n=PphotonsAsail
(1+pref lpspref l)+Bf(1−pspref l)pref l+(1−pref l)f Bf
−bBb
f+b
~
n (2.14)
Fsimp~t = 0~t (2.15)
For more information on the equations above, see McInnes (Solar Sailing: Tech-nology, Dynamics, and Mission Applications)(11).
2.1.3
Sail-Loading
A sail loading study was completed for the solar sail on a circular orbit around the Sun. Newton’s Second Law will be used to find the acceleration of the sail. The equation for the acceleration of the sail, resulting from summing the normal forces (gravitational and solar pressure) acting on the sail, is:
F~n=msailcraf ta~n =Fsun−Fearth−Fsail (2.16)
This problem can be treated like the analysis of the kinematics of a particle. The acceleration vector can be broken down into the normal and tangential accelera-tions.
~a =atu~t+anu~n (2.17)
Since the particle (or sail) moves around a circular orbit with a constant velocity, the normal acceleration equation would be:
a~n=
v2
ρ =ω
2
s/cRsail/sun (2.18)
Using the normal acceleration equation 2.18, the angular velocity can be related to the forces acting on the sailcraft .
ωs/c2 = Fsun−Fearth−Fsail
ms/cRsail/sun
The solar sail is required to orbit the Sun at the same rate as Earth; therefore, the angular velocity of the sail is equal to the angular velocity of Earth. The equation for the angular velocity of the sail is a function of the mass of the sail, the area of the sail, the sail properties, and the distance the sail is located from the Sun. Therefore, the sail loading variable will be introduced to simplify equation 2.19. First, the entire equation is divided by the mass of the sail. The new variable sail loading (σ) will be defined as sail mass divided by sail area.
σ = ms/c
Asail
(2.20)
Solving equation 2.20 for the sail loading will result in an equation that is de-pendent only on gravitational parameters, the material properties of the sail, and the distance the sail is located from the Sun.
Pphotons[(1 +pref lpspref l) +Bf(1−pspref l)pref l+ (1−pref l)
fBf −bBb
f +b ] =
σ( µsun
R2 sail/sun
− µearth
R2
sail/earth
−ω2s/cRsail/sun2 ) (2.21)
The gravitational parameters and material properties of the sail are both known values (the material properties for the sail are based on previous tested mate-rial and the gravitational parameter is a constant for each planet or mass). The optical coefficients for the square sail will be based on the tested values from JPL. These values are from tests completed for the solar sail Halley’s Comet mission(11).
Table 2.1: Optical Coefficients
pref l 0.88
pspref l 0.9
f 0.05
b 0.55
Bf 0.79
Inserting the known values into the forcing equation will result in a third or-der equation based on the distance the sail is from the Sun and the sail-loading. The equation is now a function of two variables; therefore, if the distance to the Sun or the sail-loading is known, the other value can be found.
2.1.4
Initial Sail Acceleration
After the equations for the non-perfectly reflecting solar sail were derived, it was then possible to calculate the different characteristics of particular sized sails. These calculations were performed to give initial design parameters for any solar sail mission. In order to complete the design specifications, a mass matrix and area matrix were provided for standard solar sails.
mass= [50 100 150 200]kg (2.22)
Area= [752 1002 1502 2002]m2 (2.23)
The values for the mass matrix were determined based on different missions com-pleted in the past using masses of 50 kg and 100 kg plus two extreme mass values of 150 kg and 200 kg. The extreme values were provided in case mission would re-quire heavier payloads to be onboard. The area matrix produced realistic mission designs since all of the area values have been used in different mission designs. From these values, a sail loading matrix was calculated as follows:
σi, j = 4 X i=1 4 X j=1 ms/c(i) Asail(j) (2.24)
The sail acceleration and sail force were calculated using the sail loading matrix above and the assumption that the photons that impacted the sail were perfectly reflected (as seen in equation 2.7).
a~n(i,j)= 4 X i=1 4 X j=1 Pphotons σ(i,j)
F~n(i)= 4
X
i=1
2Asail(i)Pphotonscos2(α) (2.26)
The pressure terms in both equations is dependent on the solar constant (WE), the speed of light (c), and the ratio of the distance from the Sun to the Earth over the distance to the Sun from the sail. The equations for the sail force and sail accelerations without the photons’ pressure are:
a~n(i,j) = 4 X i=1 4 X j=1 2WE
c
1
σ(i,j)
R2E r2 cos
2(α) (2.27)
F~n(i) = 4
X
i=1
2Asail(i)WE
c
R2 E
r2 cos
2(α) (2.28)
The results of the sail acceleration calculations (Appendix A) show the small values produced from the acceleration or forces acting on the solar sail. The sail force calculations (Appendix A) give the force values that will be used to complete the final mission design.
2.2
Mission Definition
2.2.1
Initial Mass Budget
The mass budget is a table created to estimate the overall mass of the payload. The values seen in the table below were estimated after studying several different solar sailcraft mission designs.
Table 2.2: Initial Mass Budget Estimate Payload Mass
Thruster, Fuel Tanks, and Fuel Lines 10 kg Sensors/Communication Instrumentation 5 kg
Power Mechanisms (Solar etc) 10 kg
Thermal Control/Coating 1 kg
Total Mass 26 kg
Tether Mass
Tether Length up to 8000 m
Tether Diameter 0.002 m
Tether Density 970 kg/m3
Tether Mass/Length 0.00304 kg/m
Tether Mass up to 24.378 kg
Total Mass Excluding Booms and Sail Film 50.378 kg Round Because Values were estimates 50 kg
Table 2.2 does not take into account the mass of the sail structural components (sail film or sail boom). The values from the table were used to create different mission designs for a payload mass of 50 kg, which will be explained in the next section, and also for a mass of 75 kg which will allow for some error in estimation.
2.2.2
Sail Size Determination
50 and 75 kg, and boom length density of 20 kg/m. Lengths of fewer than 100 meters have already been constructed and tested to show that deployment meth-ods work. Therefore, during this analysis, a measuring tool for realistic mission designs will be a sail length of 100 meters or less.
Figure 2.2: Sail Length Required for 50kg Payload
Figure 2.3: Sail Length Required for 75 kg Payload
Figure 2.4: Sail Length Required for 2 and 5 kg/m2 Sail Film Area Densities
In this study, the sail film area density of 10 kg/m2 produced the least desir-able results (as expected). The construction and material cost savings for the 10
kg/m2 are not sufficient to make this sail film area density a legitimate alternative for use on a Solar Sail Solar Storm Mission. To give a better estimate of ideal mission designs, figure 2.4 was created using sail film area densities of 2 and 5
kg/m2. After removing the value for the higher sail film area density, figure 2.4 shows the closest possible distance to the sun for the previously described mission constraints (lengths of less than 100 meters). A sail film area density of 2 kg/m2 is limited to 0.93 AU to the Sun for the 50 kg payload and 0.95 AU for the 75 kg payload. To summarize, the sail film area density determines the distance from the Sun the sailcraft orbits, and the sail film area density is limited by the cost of material (thinner density has greater cost) and capability of deployment.
2.2.3
Thruster Selection
thrusters were put into three separate classes: large thrusters, medium thrusters, and small thrusters. Each different class of thruster has distinct advantages and disadvantages. Large thrusters have the advantage of applying more force (or thrust) to the spacecraft, but also have higher initial mass and consume greater amounts of fuel. Medium thrusters apply moderate amounts of force, while using moderate amounts of fuel. Small thrusters use lower amounts of fuel but apply less amount of force as a result. A final thruster cannot be decided upon until the exact use of the thruster is defined. Along with different size thrusters, there are also several different types of thrusters that vary by propulsion methods and fuel types. Different thruster types include monopropellant rockets, bipropel-lant rockets, resistojet, pulsed plasma, and ion thrusters. The specifications to choose different propulsion methods (or thrusters) can be a paper by itself. The reason for choosing each type of thruster will be covered briefly. Several meth-ods of propulsion that are very efficient, but require large onboard mass, were eliminated from the study. The excluded methods include those such as the ion thruster, Hall Effect thruster, and plasma thrusters. Monopropellant rockets use a single propellant (chemical) to produce thrust and are simple compared to the bipropellant rockets that require more than one chemical (an oxider and fuel) to produce the chemical reaction. The monopropellant rocket has less mass which will be an issue in the solar sail mission. The resistojet rocket produces thrust by heating a fluid and releasing the heated exhaust through a nozzle. The large thruster being considered is a monopropellant thruster, while the medium and small thrusters being considered are resistojet thrusters.
Figure 2.5: 1 N Monopropellant Thruster Model No. CHT-1(12)
The medium thruster is a Nitrous Oxide Resistojet 125mN Thruster(13). The mass flow rate of the medium thruster is 0.1 g/s at nominal thrust (125 mN). The small thruster is a Water Resistojet 45mN Thruster. The mass flow rate of the Water Resistojet is 0.03 g/s operating at the nominal thrust value (45 mN)(14). Both resistojets are produced by Surrey Satellite Technology Ltd and have been designed for station keeping and minor orbit adjustment which meet the requirements for the control system.
2.3
Final Design Decision
Current methods of construction allow for all sail film materials previously dis-cussed to be produced without special order. Any of these materials, due to the limited size of the sail, can be used as the sail film on the Solar Storm Mission. However, using the lower film density (thinner film) on larger sails can cause problems. The higher density sail film material would be more advantageously on a mission requiring extremely large sails, such as missions to other planets or beyond the solar system. The final materials chosen for the mission are shown below:
Table 2.3: Final Material Properties
Materials
Thruster A 1 N Monopropellant Thruster Model No CHT-1
Thruster B Surrey Satellite Technology Ltd 125mN Nitrous Oxide Resistojet Thruster C Surrey Satellite Technology Ltd 45mN Water Resistojet
Sail Film Metallized 0.9µm Mylar Sail Boom Bonded Kapton and Mylar
Tether SpectraT M(Polyethylene Polymer) 2 mm Diameter Tether
Figure 2.7: Sail-Loading Plot
Figure 2.7 shows the sail-loading based on the distance from the sail to the Sun. Based on payload ranges of 50 - 75 kgs used in the earlier analysis, the distance from the sail to the Sun was set at 0.95 AU. This distance required a sail loading of 9.8768 g/m2. If the overall sail-loading and material properties are known, the payload mass can be found from the following equation.
σL2 =ρsailf ilmL2+ρboomLb+mpayload (2.29)
Equation 2.29expresses the overall sail-loading as a function of the sail film area density, the sail boom length density, and the payload. The payload is a mass that is set by mission design and will be a constant value. The boom length is a constant that is a multiple of the length of the sail. The booms are on the diagonals of the sail and this is results in a boom length of √2
the length of the sail and mass of the payload.
(σ−ρsailf ilm)L2−ρboom 4
√
2L−mpayload = 0 (2.30) Based on an analysis of values for the mission sail length, a length of 100 meters was found to be the best length that resulted in a reasonable payload. A length of 100 meters results in a payload of 73.111 kg. The initial mass budget must be updated to reflect this new payload.
Table 2.4: Final Mass Budget Payload Mass
Thruster Nozzle and Fuel Lines 8 kg
Tank Computers, Regulators, and Sensors 10 kg
Ballast Mass 2 kg
Motion Sensors/Communication Instrumentation 5 kg Power Mechanisms (Solar, Thrusters etc) 10 kg
Thermal Control/Coating 1 kg
Payload Casing 5 kg
Additional Systems 5 kg
Total Mass 46 kg
Payload Mass Without Thruster Components 26 kg Tether Mass
Tether Length 0-8000 m
Tether Diameter 0.002 m
Tether Density 970 kg/m3
Tether Mass/Length 0.00304 kg/m
Tether Mass 0-24.378 kg
Payload, Tether, and Thruster Mass 72.9181 kg
Fuel and Fuel Tank Mass 2.733 kg
Total Mass Excluding Booms and Sail Film 73.111 kg
to be conservative, excess ballast mass was added. The ballast mass allows for an increase in mass for additional onboard components. The mass budget provides a realistic sail loading value that can then be used to find the distance the sail will be located from the Sun.
Table 2.5: Overall System Mass Mass (In grams) Percent of Mass
Boom Mass 5,657 5.73
Thruster Mass 20,000 20.25 Sail Mass 20,000 20.25 Tether Mass 24,378 24.68 Payload Mass 26,000 26.32
Fuel Mass 2,733 2.77
Total Mass 98,768 100
Table 2.5 gives a view of the overall mass of the solar sailcraft. The sailcraft is broken down into mass from the tether, payload, thruster components, fuel, sail film and the sail boom. The payload represents the highest percentage of the overall mass (26.32), which is to be expected with all instrumentation and materials included in this section. An interesting note is the sail’s percentage of the overall mass (25.98) considering the length of the sail and boom setup. The sailcraft mass breakdown shows the limiting factor for any solar sail mission will be the percentage of overall mass the sail represents. The sail mass will be higher for a mission that travels a greater distance. As the sail mass increases the amount of payload will have to be reduced to get desirable results. At the extreme (or limit) some missions would be impossible, since the entire mission mass would be comprised of the sail mass.
2.4
Check of Point Mass Assumption
desired angular velocity for the sailcraft. This point mass assumption will now be checked to make sure that it will not have an effect on the angular velocity of the sailcraft. The angular velocity of the system will be checked by summing up the forces acting on each system as a function of their actual location. In the earlier calculations it was assumed that each part of the sail was located at 0.95 AU from the Sun, even though a tether (up to two thousand meter length) is extended from the sail to the payload. To solve for the forces acting on the sail, the sailcraft will be broken down into three different sections (the sail component, the payload component, and the tether component). The sail component and payload component are located at a given distance from the Sun, so calculating the forces due to each component are relatively easy.
Fsailmass =
µsunmsail
r2 sail/sun
− µearthmsail
r2
sail/earth
(2.31)
Fpayloadmass=
µsunmpayload
r2
payload/sun
− µearthmpayload
r2
payload/earth
(2.32)
As stated earlier, the mission design assumes that the Earth travels in a circular orbit around the Sun. Applying the circular orbit assumption to Equations 2.31 and 2.32 yields:
Fsailmass=
µsunmsail
rsail/sun2 −
µearthmsail
(1−rsail/sun)2 (2.33)
rpl =rsail/sun−
p
ltetherltether−lhalf saillhalf sail (2.34)
Fpayloadmass=
µsunmpayload
r2 pl
− µearthmpayload
(1−rpl)2
(2.35)
term j refers to the section of tether being analyzed at the particular time. For the tethers that are in the x-direction:
xtetherj =ltetherjsinθ (2.36)
ytetherj =ltetherjcosθ (2.37)
xvectorj =x1+xtetherj (2.38)
yvectorj =y1+ynewj (2.39)
For the tethers that are in the y-direction:
xtether2j =ltetherjsinθ1 (2.40)
ytether2j =ltetherjcosθ1 (2.41)
xvector2j =x1+xtether2j (2.42)
yvector2j =y1+ynew2j (2.43)
In equations 2.38 and 2.39the terms x1 and y1 are the distances to the sail (0.95 AU) from the Sun.
rtetherj =
p
xvectorjxvectorj+yvectorjyvectorj (2.44)
rtether2j =
p
Ftotaltetherx=
X
Ftethermassx = 2000m
X
ltether=0
µsunmδmass
r2 tetherj
− µearthmδmass
(1(d)−rtetherj)2
(2.46)
Ftotaltethery =
X
Ftethermassy = 2000m
X
ltether=0
µsunmδmass
r2 tether2j
− µearthmδmass
(1(d)−rtether2j)2
(2.47)
Ftethermass = 2Ftethermassx+ 2Ftethermassy (2.48)
To find the total force acting on the sail, all of the forces acting on the system, which are tether force, sail force, and payload force, will need to be summed.
Fsystemmass=Fsailmass+Fpayloadmass+Ftethermass (2.49)
Relating the system force to the overall normal force produced the following equation used to calculate the angular velocity of the sailcraft.
ωs/c=
Fsailmass+Fpayloadmass+Ftethermass
ms/crsail/sun
(2.50)
Error= ωs/c−ω
ω 100 (2.51)
Applying the forces calculated for the tether, sail, and payload yields an error of 0.0047 percent for the point mass assumption. A second method to ensure that the point mass assumption will not effect the angular velocity of the sailcraft, requires calculating the force for the sail as if all of the mass was located at 0.95 AU and then calculating the force of the sail as if all of the mass was located at the placement of the payload (0.95 AU minus length of the tether).
Fmassatsail =
µsunms/c
r2 sail/sun
− µearthms/c
(1−rsail/sun)2
(2.52)
Fmassatpayload =
µsunms/c
r2 pl
− µearthms/c
(1−rpl)2
Chapter 3
Inertia Calculations
3.1
Orientation
This chapter develops the tools used to find the moments of inertia and basic orientation information for the sailcraft. The coordinate configuration of the sailcraft will be defined by frames B and S located on the sailcraft. Reference frame S has the origin located at the center of mass of the spacecraft and has three unit vectors attached in the defined direction.
Figure 3.1: Orientation of the Sailcraft Model with respect to Frame C
of each vector with respect to the overall mission, each vector’s orientation will be discussed. Vector s1 points towards the Sun. Vectors s2 and s3 are both located in the plane of the sail. Vectors2 is located along the orbital direction and vector
s3 is perpendicular to the orbital plane.
Figure 3.2: Relationship between Frames C and B
Frame B is a body-fixed reference frame that is not aligned with frame C. Frame B can be related to frame C by different three angles: θ1 (roll),θ2 (pitch), and θ3 (yaw). A visual description of the rotation each angle undergoes can be seen in figure 3.2.
3.2
Torque Calculation
The torque acting on the sailcraft is due to the thrust exerted by the thrusters. Details of the thrusters’ orientations will be explained so the thrust can be directly applied to the torque equations. The torque for a system can be found by using the relationship(15):
The cross product can be represented by the following matrix:
i j k
τ = rx ry rz
Fx Fy Fz
(3.2)
Solving the cross product for the torque from the matrix equation gives the final torque as a function of force vectors and direction vectors as shown below:
τ = (Fzry−Fyrz)i+ (Fxrz −Fzrx)j+ (Fyrx−Fxry)k (3.3)
Figure 3.3: Sail Lengths
it can be assumed to be approximately zero (rx = 0.9x10−6m << ry = 70.71m). The torque that occurs in the different axes (1, 2, and 3) can be simplified by applying the different force vectors. The torque equation will be broken down into torque components in each separate direction. Prior to reducing the torque equation, the assumption is made that the distance from the forces to the center of mass in the x direction is zero. In the roll axis, the torque is dependent on the force in the y and/or z direction, which will allow placement of the thruster in either axis.
τ1 = (Fzry −Fyrz)i (3.4)
The torques in the pitch and yaw axis are dependent only on the force acting in the x direction and the moment arm to the thruster.
τ2 = (Fxrz)j (3.5)
τ3 = (−Fxry)k (3.6)
The thrusters will be placed on both ends of the booms in the y and z directions, and the torque will change signs based on which thruster is operating. Each force vector for the thrusters will operate in the same direction; however, the moment arms will change signs, with respect to the center of mass of the sail, based on the thruster that is operating. Thus, the signs of the torques will be positive and the orientation of the force vector will reflect the sign change.
τ3 = (Fxry)k (3.7)
3.2.1
Thruster Placement
specific mission, the thrusters were placed in the x-y-z directions, which required extra mass. To make the sailcraft stable, more mass will need to be placed at the end of the boom in the y direction than in the z direction (this will be confirmed mathematically later). The additional mass at the end of the boom will come from torque one (equation 3.4) having the moment arm only in the y direction.
Figure 3.4: Torque One Setup
Figure 3.5: Overall Torque Setup
Figure 3.5 gives a visual description of thruster placement on the sailcraft. The notation that will be used throughout the rest of the paper on the thruster loca-tion is given in the following table.
Table 3.1: Thruster Definition Torque 1
+Fz +ry Thruster 1 +Fz −ry Thruster 2
Torque 2
+Fx +rz Thruster 3 +Fx −rz Thruster 4
Torque 3
−Fx +ry Thruster 5
−Fx −ry Thruster 6
3.3
Moment of Inertia Background
moments of inertia for the sailcraft will be calculated and then summed to find the total moment of inertia. Breaking the sailcraft into smaller components will not result in a different total moment of inertia from an analysis of the sailcraft as a single component, but will serve to simplify the integrations and mathematical analysis. The solar sailcraft will be broken down into six different components: sail film, sail boom, thruster, fuel tank, tethers, and payload.
The moment of inertia terms can be found by integrating the moment arm per-pendicular to the axis of interest with the change in mass. The equation can be shown as:
I =
Z
m
r2 dm (3.8)
Applying the moment of inertia terms to the components of the sailcraft in the x, y, and z axes will result in the following inertia matrix.
Ixx −Ixy −Ixz
I = −Ixy Iyy −Iyz
−Ixz −Iyz Izz
(3.9)
For a rigid body, the moment of inertia (Ixx,Iyy, and Izz) and products of inertia (Ixy,Ixz, and Iyz) about the x, y, and z axes are(16):
Ixx =
Z
m
(y2+z2)dm (3.10)
Iyy =
Z
m
(x2+z2)dm (3.11)
Izz =
Z
m
Ixy =
Z
m
(xy) dm (3.13)
Ixz =
Z
m
(xz)dm (3.14)
Iyz =
Z
m
(yz) dm (3.15)
The moment of inertia equations can be reduced in the case of the sailcraft, be-cause the density (ρ) can be assumed to be constant in each separate component. The mass element is equal to the density times the volume element.
I =ρ
Z
m
(r2) dV (3.16)
Assuming constant density in all of the different components of the sailcraft reduces the moment and product of inertia equations to:
Ixx =ρ
Z x Z y Z z
(y2+z2) dxdydz (3.17)
Iyy =ρ
Z x Z y Z z
(x2+z2) dxdydz (3.18)
Izz =ρ
Z x Z y Z z
(x2+y2) dxdydz (3.19)
Ixy =ρ
Z x Z y Z z
(xy)dxdydz (3.20)
Ixz =ρ
Z x Z y Z z
Iyz =ρ Z x Z y Z z
(yz) dxdydz (3.22)
The equations above can be applied to each component of the sailcraft individ-ually. Each component’s moment of inertia equations will differ by the limits of integration in each direction.
3.4
Component Description
The sailcraft will maneuvers (change in orientation) about the center of mass of the sail film and sail boom configuration. Therefore, an analysis of the sailcraft’s motion will be about the center of mass of the sail configuration. The sail film is a thin membrane supported by the sail booms located along the diagonals of the sails. The sail film will be constructed from metallized 0.9 micrometer Mylar, which has an area density of 2g/m2. Total area of the sail film is 100 meters x 100 meters with a thickness of 0.9 micrometers. The sail is not a square about the axes s2 and s3; therefore, the limits of integration cannot be achieved by integrating all of the variables separately. The solution will require an equation to relate the y and z directions (e2 and e3 vectors). In the analysis of the sail film, a new variable (a), equal to half the diagonal length of the sail, will be introduced. Using a new dimension (the diagonal length of the sail) the relationship between the y and z limits can be determined, but will require the sail film to be reduced to two separate integrations. The moment of inertia for the sail will be integrated using the limits based on the slope of the lines in the y-z plane. The sail film moment of inertia is solved by applying the limits seen in figure 3.6.
Figure 3.6: Sail Film Sectioning
Figure 3.7: Sail Boom 1 Sectioning
Figure 3.8: Sail Boom 2 Sectioning
to the method used for the sail film moments of inertia). Sail booms 1 and 2 can be seen in figures 3.7 and 3.8.
Figure 3.9: System Setup With Tether
The size of the tethers is 0.002 meters in diameter which has a density of 970
kg/m3. The tether density is for a common tether with a 0.002 meter diameter and no special construction is required to produce the tethers. Tethers can be represented by a slender rod assumption, since the length of the tether is much larger than the radius (0.001 meters) of the tether. The slender rod assumption is only valid about the center of the rod or about the end of the rod. The parallel-axis theorem is used to calculate the moment of inertia for an object which has a center of mass not located at the origin of interest (the tether inertia terms are not located at the center of mass of the sail film and boom configuration). The parallel-axis theorem states that(17):
Ixx = (Ix0x0)G+m(y2
G+z 2
G) (3.23)
Iyy = (Iy0y0)G+m(x2
G+z 2
G) (3.24)
Izz = (Iz0z0)G+m(x2
G+y 2
G) (3.25)
Ixy = (Ix0y0)G+m(x2
Gy 2
G) (3.26)
Ixz = (Iz0z0)G+m(x2
Gz 2
G) (3.27)
Iyz= (Iz0z0)G+m(y2
Gz 2
G) (3.28)
The prime moments and products of inertia terms (Ix0x0, Iy0y0, Iz0z0, Ix0y0, Ix0z0,
and Iy0z0) are the inertia terms about the end or center of the body. The prime
Figure 3.10: 2-D Payload Model
The payload is a sphere that encases computers and additional instrumentation for the desired mission. The payload is located at the end of the tethers attached to the sail booms. The payload can be seen as a 2-D model in figure 3.10. The mass of the payload is 26 kilograms and represents a large part of the overall mass of the sailcraft. The radius of the sphere is 2.5 meters.
The thrusters are not located about the center of the sailcraft and, to account for the offset, the parallel axis theorem will be used to find the total moment of inertia. Since the moment of inertia of the thrusters is very small compared with the moment of inertia from the parallel axis theorem for the placement of the thrusters, it is assumed that the Hydrazine thruster size closely resembles the size of the resistojets. The resistojets are rectangular and the dimensions are provided by the manufacture (13).
The fuel tanks and instrumentation for the thrusters will be located at the center of the sailcraft. The location of the instrumentation at the center of the sailcraft allows for direct substitution into the equations for the moments of inertia and does not require analysis using the parallel axis theorem. The fuel tanks are spherical and it is assumed that the computer, actuators, and other instrumenta-tion are just masses added to the fuel tanks. This assumpinstrumenta-tion was made because the sizes of the components weren’t listed and their effect on the overall moment of inertia of the sailcraft will be negligible.
3.5
Moment of Inertia Calculations
3.5.1
Sail Film Inertia
Applying the limits of integration discussed previously yields the following mo-ment and product of inertia equations for the sail film:
Ixx =ρf ilm
Z t/2 −t/2 dx Z a 0 Z −z+a z−a
(y2+z2)dydz+
Z 0
−a
Z z+a
−z−a
(y2+z2)dydz
(3.29)
Ixy =ρf ilm
Z t/2 −t/2 Z a 0 Z −z+a z−a
(xy)dydzdx+
Z t/2 −t/2 Z 0 −a Z z+a −z−a
(xy)dydzdx
!
(3.30)
Ixz =ρf ilm
Z t/2 −t/2 Z a 0 Z −z+a z−a
(xz)dydzdx+
Z t/2 −t/2 Z 0 −a Z z+a −z−a
(xz)dydzdx
!
Iyy =ρf ilm Z t/2 −t/2 dx Z a 0 Z −z+a z−a
(x2+z2)dydz+
Z 0
−a
Z z+a
−z−a
(x2+z2)dydz
(3.32)
Iyz =ρf ilm
Z t/2 −t/2 Z a 0 Z −z+a z−a
(yz)dydzdx+
Z t/2 −t/2 Z 0 −a Z z+a −z−a
(yz)dydzdx
!
(3.33)
Izz =ρf ilm
Z t/2 −t/2 dx Z a 0 Z −z+a z−a
(x2+y2)dydz+
Z 0
−a
Z z+a
−z−a
(x2+y2)dydz
(3.34)
Integrating the moment and product of inertia for the sail film results in the following inertia matrix:
2
3ρf ilmta
4 0 0
If ilm = 0 16(ρf ilmt3a2 + 2ρf ilmta4) 0
0 0 16(ρf ilmt3a2+ 2ρf ilmta4)
(3.35) For detailed calculation of the terms in the moment of inertia matrix consult Appendix B.
3.5.2
Sail Boom Inertia
z-direction are(18):
Ixx = Iyy =
ml2
12 (3.36)
Izz = 0 (3.37)
Ixy = Ixz = Iyz = 0 (3.38)
For calculations of the moments of inertia for the sail boom, sailboom1 will refer to the boom located in the y-direction and sailboom2 will be the sail boom located in the z-direction. Applying the slender rod assumption yields:
mbooml2boom
12 0 0
Isailboom1 = 0 0 0
0 0 mbooml2boom
12 (3.39)
mbooml2boom
12 0 0
Isailboom2 = 0
mbooml2boom
12 0
0 0 0
(3.40)
Summing up the moments of inertia for the sail boom yields:
mboomlboom2
6 0 0
Isailboom= 0
mboomlboom2
12 0
0 0 mbooml2boom
12
(3.41)
3.5.3
Tether Inertia
Figure 3.11: Tether 1 Configuration
The orientation for tether 1 can be seen in figure 3.11. Tether 1 decreases in the y direction with an increase in the distance from the sail (increase in the x direction). The tether extends off of the sail boom located in the positive y direction (70 meters), decreases in x direction of the sail (towards the Sun), and meets at the payload (0 meters in the y direction).
Figure 3.12: Tether 2 Configuration
Figure 3.13: Tether 3 Configuration
The orientation for tether 3 can be seen in figure 3.13. Tether 3 decreases in the z direction with an increase in the distance from the sail (x direction). The tether extends off of the sail boom located in the positive z direction (70 meters), decreases in the x direction of the sail (towards the Sun), and meets at the payload (0 meters in the z direction).
Figure 3.14: Tether 4 Configuration
Figure 3.15: Tether Angle
The tethers are all offset by the same angle since the sail is a square sail. The only difference is the direction (y or z direction) of the offset and/or the sign of the angle. Figure 3.15 shows the setup for the tether angle. To find the angle where all the tethers are located (θ), the following equations are applicable:
θ=sin−1
70
ltether
=tan−1 p 70 l2
tether−702
!
(3.42)
Before applying the parallel-axis theorem to the tethers, the moments of inertia for the offset tethers have to be calculated. The slender rod assumption is valid because the length is many orders of magnitude larger than the diameter of the rod. The moment of inertia matrix for a tether with the length in the x direction is(18):
0 0 0
Itether = 0
mtetherl2tether
12 0
0 0 mtetherl2tether
12
(3.43)
an elementary rotation matrix can be applied because the tethers use only single rotations. In more complex cases there would be many different combinations of rotations that could be used. The rotation matrix equation for the tethers can be defined as:
Itether1 = [Cγ1] [Itether] (3.44)
Itether2 = [Cγ2] [Itether] (3.45)
Itether3 = [Cγ3] [Itether] (3.46)
Itether4 = [Cγ4] [Itether] (3.47)
The rotation matrix terms are the C alpha terms in the moment of inertia equa-tions for each tether. Transformation matrices about one plane (x-y or x-z) are standard equations. For each tether the common transformation matrices are(19):
cos(γ1) sin(γ1) 0
Cγ1 = −sin(γ1) cos(γ1) 0
0 0 1
(3.48)
cos(γ4) 0 −sin(γ4)
Cγ4 = 0 1 0
sin(γ4) 0 cos(γ4)
(3.49)
the matrices.
cos(γ2) −sin(γ2) 0
Cγ2 = sin(γ2) cos(γ2) 0
0 0 1
(3.50)
cos(γ3) 0 sin(γ3)
Cγ3 = 0 1 0
−sin(γ3) 0 cos(γ3)
(3.51)
The resulting moment and product of inertia about the offset origin for each tether is:
0 mtetherl2tether
12 sin(γ1) 0
Itether10 = 0 mtetherl
2
tether
12 cos(γ1) 0
0 0 mtetherl2tether
12 (3.52)
0 −mtetherltether2
12 sin(γ2) 0
Itether20 = 0 mtetherl
2
tether
12 cos(γ2) 0
0 0 mtetherl2tether
12 (3.53)
0 0 mtetherltether2
12 sin(γ3)
Itether30 = 0 mtetherl
2
tether
12 0
0 0 mtetherl2tether
12 cos(γ3)
(3.54)
0 0 −mtetherl2tether
12 sin(γ4)
Itether40 = 0 mtetherl
2
tether
12 0
0 0 mtetherl2tether
12 cos(γ4)
(3.55)
axis theorem displaced distance. Tethers one and two will not be offset in the z direction, have the same distance in the x direction, and have opposite signs in the y direction. The distances will be defined as follows for the parallel-axis theorem matrices for the first two tethers:
xG1 = l11
yG1 = l12
zG1 = 0
The resulting in the parallel-axis matrices for tethers one and two are:
l212 l11l12 0
Itether1pa =mtether l11l12 l211 0
0 0 l2
11+l212
(3.56) l2
12 l11(−l12) 0
Itether2pa =mtether l11(−l12) l211 0 0 0 l211+l122
(3.57)
Tethers three and four were also grouped because they have the same relationship with the distance from the center of mass of the tether to the center of mass of the sail film and boom configuration(parallel axis theorem offset). Tethers three and four are not offset in the y direction, have the same offset in the x direction, and opposite offsets in z direction. The distances for the parallel-axis theorem for tethers three and four are:
xG1 = l21
yG1 = 0
zG1 = l22
The resulting in the parallel-axis matrices for tethers three and four are:
l222 0 l21l22
Itether3pa =mtether 0 l212 +l222 0
l21l22 0 l212
l2
22 0 l21(−l22)
Itether4pa =mtether 0 l212 +l222 0
l21(−l22) 0 l212
(3.59)
Combining the parallel-axis matrices and the transformed matrices gives the over-all moment of inertia term for each tether. The overover-all moment of inertia terms become:
1212 l2tether
12 sin(γ1) +l11l12 0
Itether1total =mtether l11l12 l2
tether
12 cos(γ1) +l 2
11 0
0 0 l2tether
12 +l 2 11+l122
(3.60) 12 12 − l2 tether
12 sin(γ2)−l11l12 0
Itether2total =mtether −l11l12 l2
tether
12 cos(γ2) +l 2
11 0
0 0 l2tether
12 +l 2 11+l212
(3.61) l2 22 0 l2 tether
12 sin(γ3) +l21l22
Itether3total =mtether 0
l2tether 12 +l
2
21+l222 0
l21l22 0
l2
tether
12 cos(γ3) +l 2 21 (3.62)
l222 0 −l2tether
12 sin(γ4)−l21l22
Itether4total =mtether 0 l2
tether
12 +l 2
21+l222 0
−l21l22 0
l2
tether
12 cos(γ4) +l 2 21 (3.63)
3.5.4
Payload Inertia
thru 3.28 for more detail about the parallel axis theorem).
Isphere =Isphere0 +Ispherepa (3.64)
The moment of inertia for a sphere is a common problem, similar to the slender rod, solved by a known equation. The moments of inertia for a sphere with the center of mass located at the origin are(18):
Ix0x0 =Iy0y0 =Iz0z0 = 2mr
2 5
Ix0y0 =Ix0z0 =Iy0z0 = 0
The payload inertia matrix about the center of mass of the sphere (x’, y’, and z’ axes) becomes: 2r2 sphere
5 0 0
Isphere0 =msphere 0
2r2
sphere
5 0
0 0 2r
2 sphere 5 (3.65)
Lengths from the center of mass of the sail film and boom configuration to the center of mass of the sphere have to be found in order to apply the parallel axis theorem to the sphere. The center of mass of the sphere is located at the center of mass of the sail film and boom configuration in the y and z directions. Therefore, the distances yGsphere and zGsphere are zero. The center of the mass of the two obejects is different in the x direction by the length of the tether in the x direction (lsail/sphere). The sphere inertia matrix, using the parallel-axis theorem, becomes:
0 0 0
Ispherepa =msphere 0 l2sail/sphere 0 0 0 l2sail/sphere
The resulting total moment of inertia for the payload is: 2r2 sphere
5 0 0
Isphere =msphere 0
2r2
sphere
5 +l 2
sail/sphere 0
0 0 2r
2
sphere
5 +l 2 sail/sphere (3.67)
3.5.5
Thruster Inertia
The thrusters are treated as a rectangular box and therefore are easy to integrate to get the moments of inertia. The process is similar to that of the solar sail film integration except the parallel-axis theorem will be used. The thruster shape is symmetric about the center and results in the products of inertia to all equal zero. The values for ly and lz are equal. Integrals for the center of the thrusters can be seen below:
Ixx = ρt
Z lx/2 −lx/2 dx Z lz/2 −lz/2 Z ly/2 −ly/2
(y2 +z2)dydz
!
=ρt
ly4lx
6 (3.68)
Iyy = ρt
Z ly/2 −ly/2 dy Z lz/2 −lz/2 Z lx/2 −lx/2
(x2+z2)dxdz
!
=ρt
lx3ly2+ly4lx
12 (3.69)
Izz = ρt
Z lz/2 −lz/2 dz Z ly/2 −ly/2 Z lx/2 −lx/2
(x2+y2)dxdy
!
=ρt
lx3ly2+ly4lx
12 (3.70)
The resulting moment of inertia for each separate thruster is:
ρtly 4lx
6 0 0
Ithruster = 0 ρtlx
3ly2+ly4lx
12 0
0 0 ρtlx
3ly2+ly4lx 12
(3.72)
Each thruster is offset from the position of interest (center of the sailcraft). In the x-direction the distance that the thruster is offset is from the center of the sail by the sum of the thickness of the thruster and the sail thickness (because the sail dynamics were all referenced to this point). The thrusters will be mounted on top of the sail film at the ends of the sail booms. The resulting total offset in the x-direction is equal to half of the sail film thickness plus the half of the thickness of the thruster in the x-direction. In the following calculations the distancelt will be used to denote the distance from the center of the sail to the thruster. The analysis for the thrusters will be done by combinations of thrusters. The first configuration will be for thrusters one and five because the thrusters are placed at the same point in the y direction and will have the same distance in the parallel axis theorem. The other configurations are thrusters two and six, thruster three, and thruster four. The distances for the thruster one and five configuration are:
xGt1 = tsail2+lx
yGt1 =lt m
zGt1 = 0
The moment of inertia due to the parallel-axis theorem is:
(lt)2 lttsail2+lx 0
Ithrust1pa =mth2 lttsail2+lx (tsail2+lx)2 0 0 0 (tsail+lx
2 ) 2+ (l
t)2
(3.73)
The distances for the thruster two and six configuration are:
yGt2 =−lt
zGt2 = 0
The moment of inertia due to the parallel-axis theorem is:
(−lt)2 −lttsail2+lx 0
Ithrust2pa =mth2 −lttsail2+lx (tsail2+lx)2 0 0 0 (tsail+lx
2 )
2+ (−l t)2
(3.74)
The distances for thruster three are:
xGm3 = tsail2+lx
yGm3 = 0
zGm3 =lt m
The moment of inertia due to the parallel-axis theorem is:
(lt)2 0 lttsail2+lx
Ithrust3pa =mth 0 (tsail2+lx)2+ (lt)2 0
lttsail2+lx 0 (tsail2+lx)2
(3.75)
The distances for thruster four are:
xGm4 = tsail2+lx
yGm4 = 0
zGm4 =−lt
The moment of inertia due to the parallel-axis theorem is:
(−lt)2 0 −lttsail2+lx
Ithrust4pa =mth 0 (tsail2+lx)2+ (−lt)2 0
−lttsail2+lx 0 (tsail2+lx)2
The total inertia for all of the thrusters can be calculated by adding the parallel-axis theorem to the inertia for each separate thruster. To simplify the equations the moment of inertia from the parallel-axis theorem for thrusters one, two, five, and six are summed. Also, the moment of inertia of thrusters three and four will be summed.
l2t 0 0
Ithrustpa12 = 2mth2 0 (tsail2+lx)2 0 0 0 (tsail+lx
2 ) 2+l2
t (3.77) l2
t 0 0
Ithrustpa34 = 2mth 0 (tsail2+lx)2+l2t 0 0 0 (tsail+lx
2 ) 2
(3.78)
Since the moment of inertia about the origin for each thruster is the same, the total moment of inertia would be equal to:
Ithrustertotal = 6Ithruster+Ithrustpa12+Ithrustpa34 (3.79)
3.5.6
Fuel Tank Inertia
The fuel tank is located at the center of the mass of the sail film and boom configuration. The moment of inertia for the fuel tank can be calculated by applying the common moment of inertia for a sphere (using equation 3.65).
2r2
tank
5 0 0
If ueltank =mtank 0
2r2tank 5 0 0 0 2rtank2
5
Chapter 4
System Dynamics
4.1
Euler’s Moment Equations
Principal moments of inertia occur when the object of inertest is oriented so that the product of inertia terms (I12, I13, I21, I23, I31, and I32) are zero. Analysis of the sailcraft’s moment of inertia (calculated in Chapter 3 Section 3) shows the moment of inertia terms for the sailcraft are principal moment of inertia. The principal moments of inertia for the sail will be denoted by I1, I2, and I3 with the subscript referring to the axis of interest. The motion of the spacecraft is done by using Euler’s equations of motion. Euler’s equations are the equations of motion for a rigid body described by the systems angular velocity and angular acceleration. The expanded Euler’s equations are(20):
I11α1−I12(α2−ω1ω3)−I13(α3+ω1ω2)−(I22−I33)ω2ω3−I23(ω2
2−ω32)=MGx (4.1)
I22α2−I23(α3−ω1ω2)−I12(α2+ω2ω3)−(I33−I11)ω3ω1−I13(ω2
3−ω12)=MGy (4.2)
I33α3−I13(α1−ω2ω3)−I23(α2+ω1ω3)−(I11−I22)ω1ω2−I12(ω2
1−ω22)=MGz (4.3)
equations for the sailcraft will reduce to become:
I1ω˙1−(I2−I3)ω2ω3 =τ1 (4.4)
I2ω˙2−(I3−I1)ω3ω1 =τ2 (4.5)
I3ω˙3−(I1−I2)ω1ω2 =τ3 (4.6)
4.2
Orbital Dynamics
Figure 4.1: Orbital Dynamics
body frame is S [s~1, ~s2, ~s3] with the origin located at the center of mass of the sail. The vectors directions are defined ass~1 directed towards the Sun,s~2located along the orbital direction, and vector s~3 directed perpendicular to the orbital plane. Body fixed reference frame B [b~1, ~b2, ~b3] can be related to principal body frame by the direction cosine matrix. Several different combinations of rotation sequences can be used to describe the motion of a system. This study uses a common no-tation known at the NASA standard airplane orienno-tation (1-2-3 rono-tation). The orientation about these axis and rotations can be represented as a transformation matrix b[c]s . The transformation matrix is obtained by multiplying the matrix for each separate rotation.
b[c]s= [R
θ1]x[Rθ2]y[Rθ3]z
c(θ2)c(θ3) c(θ2)s(θ3) −s(θ2)
= s(θ1)s(θ2)c(θ3)−c(θ1)s(θ3) s(θ1)s(θ2)s(θ3) +c(θ1)c(θ3) s(θ1)c(θ2)
c(θ1)s(θ2)c(θ3) +s(θ1)s(θ3) c(θ1)s(θ2)s(θ3)−s(θ1)c(θ3) c(θ1)c(θ2)
(4.7) In the rotation matrix sin is represented by s and cosine is represented by c for short hand notation. The matrix multiplication solution for the transformation was completed using MapleT M and can be seen in Appendix C. The relationship from the body fixed reference frame B to the principal body frame S can now be written as:
cθ2cθ3 cθ2sθ3 −sθ2
[~r]s= sθ1sθ2cθ3−cθ1sθ3 sθ1sθ2sθ3 +cθ1cθ3 sθ1cθ2
cθ1sθ2cθ3 +sθ1sθ3 cθ1sθ2sθ3−sθ1cθ3 cθ1cθ2
[~r]b (4.8)
The angular velocity of the sailcraft about the principal body frame with respect to the Sun’s frame (vector O) is the angular velocity due to the orbital motion.
O
~
