FLIGHT DYNAMICS
AOE 4065
Virginia Tech, Blacksburg, VA
16 November 2001
Team Members:
Rodolf Biro de Bona
Doug Firestone
Ben MacKay
Kevin Pisterman
Matthew VanDyke
Table of Contents
Table of Contents ................................................................................................................ ii
List of Figures .................................................................................................................... iv
List of Tables ...................................................................................................................... v
List of Abbreviations ......................................................................................................... vi
List of Symbols ................................................................................................................. vii
Chapter 1: Introduction ...................................................................................................... 1
1.1 Astrodynamics ......................................................................................................... 1
1.2 Mission geometry .................................................................................................... 1
1.3 Mission analysis ....................................................................................................... 2
1.4 Guidance, navigation, and control ........................................................................... 2
1.5 Propulsion ................................................................................................................ 3
1.6 Attitude determination and control .......................................................................... 3
1.7 Overview .................................................................................................................. 5
Chapter 2: Subsystem Modeling ......................................................................................... 6
2.1 Astrodynamics ......................................................................................................... 6
2.1.1 Satellite equations of motions ........................................................................... 6
2.1.2 Orbital elements ................................................................................................ 8
2.2 Mission analysis ..................................................................................................... 10
2.2.1 Characteristics of the orbits............................................................................. 10
2.2.2 Phase angle and time required for an interplanetary transfer.......................... 11
2.2.3 Escape orbit at planet 1 ................................................................................... 12
2.2.4 Capture orbit at planet 2 .................................................................................. 13
2.3 Guidance navigation and control ........................................................................... 14
2.3.1 Non-autonomous systems ............................................................................... 14
2.3.2 Autonomous systems ...................................................................................... 15
2.4 Attitude determination and control ........................................................................ 15
2.4.1 Attitude determination ..................................................................................... 16
2.4.2 Attitude control ................................................................................................ 18
2.5 Propulsion .............................................................................................................. 20
2.5.1 Chemical rockets .............................................................................................. 20
2.5.2 Electrical rockets .............................................................................................. 22
2.5.3 Electrodynamic tether ...................................................................................... 24
2.5.4 Momentum exchange tether ............................................................................ 24
2.6 Subsystem interactions .......................................................................................... 26
2.6.1 Guidance navigation and control interactions ................................................. 26
2.6.2 Attitude determination and control subsystem interactions ............................ 28
2.6.3 Propulsion subsystem interactions .................................................................. 29
2.7 Summary ................................................................................................................ 30
Chapter 3: Subsystem Examples ....................................................................................... 31
3.1 Astrodynamics ........................................................................................................ 31
3.2 Mission analysis ...................................................................................................... 32
3.3 Guidance navigation and control ............................................................................ 33
ii
3.4 Attitude determination and control ......................................................................... 35
3.5 Propulsion ............................................................................................................... 37
3.6 Summary ................................................................................................................. 39
Chapter 4: Application Examples ..................................................................................... 40
4.1 Astrodynamics ........................................................................................................ 40
4.2 Mission analysis ...................................................................................................... 42
4.3 Guidance Navigation and Control .......................................................................... 44
4.4 Attitude Determination and Control ....................................................................... 45
4.5 Propulsion ............................................................................................................... 48
4.5.1 Conventional propulsion ................................................................................. 48
4.5.2 Electrodynamic tether propulsion ................................................................... 49
4.5.3 Momentum exchange tether ............................................................................ 52
4.6 Summary ................................................................................................................. 53
Chapter 5: Conclusion....................................................................................................... 54
5.1 Mission analysis and astrodyanmics ...................................................................... 54
5.2 Guidance navigation and control ........................................................................... 55
5.3 Attitude dynamics and control ............................................................................... 55
5.4 Propulsion .............................................................................................................. 56
5.5 Summary ................................................................................................................ 57
iii
List of Figures
Figure 2.1: Geometry of an Ellipse and Orbital Parameters ............................................... 8
Figure 2.2: Orbital Elements8 ............................................................................................. 9
Figure 2.3: Potential Distribution for a Planar Triode6 ..................................................... 23
Figure 2.4: GN&C Subsystem Interactions ...................................................................... 27
Figure 2.5: ADCS Subsystem Interactions ....................................................................... 29
Figure 2.6: Propulsion Subsystem Interactions ................................................................ 30
Figure 4.1: Measures of effectiveness vs. altitude for IAA, AAR, and ∆V/year. ............. 42
iv
List of Tables
Table 3.1: Honeywell reaction and momentum wheels10 ................................................. 35
Table 3.2: Microcosm magnetic torque bars13 .................................................................. 36
Table 3.3: Attitude determination sensors provided by Astro-Iki Corporation1 ............... 36
Table 3.4: Billingsley Magnetics space-rated magnetometers13 ....................................... 37
Table 3.5: Atlantic Research Corporation’s MONARC line of hydrazine thrusters2 ....... 37
Table 3.6a: Characteristics of selected electric propulsion flight systems20 .................... 38
Table 4.1: Characteristics of a Hohmann transfer orbit and a hyperbolic transfer orbit. . 43
Table 4.3: Physical characteristics of the satellite15.......................................................... 45
Table 4.4: Electrodynamic tether calculation constants ................................................... 50
Table 4.5: Values for momentum exchange tether launch numerical example ................ 52
v
List of Abbreviations
AAR
ADCS
ARC
CDM
GEO
GLONASS
GN&C
GPS
IAA
INU
LEO
MOE
MTB
QUEST
TDRS
TOF
ADCS
GLONASS
GPS
QUEST
TDRS
TOF
Area access rate
Attitude determination and control system
Atlantic research corporation
Chief decision maker
Geosynchronous Earth orbit
Global navigation satellite system
Guidance navigation and control
Global positioning system
Instantaneous access area
Inertial navigation unit
Low Earth orbit
Measure of effectiveness
Magnetic torquer bars
Quaternion estimation
Tracking and data relay satellite
Time of flight
Attitude determination and control system
Global navigation satellite system
Global positioning system
Quaternion estimation
Tracking and data relay satellite
Time of flight
vi
List of Symbols
h

L


N
A

V
j
ρ
Tm
Cd
e

q
u
F
µ
Tg
i

D

M

B
m

m
I
n
optimal


p
wk
r
ra
rp
γ

Rbi
a
Tsp
Cp
Angular momentum of momentum wheel
Angular momentum vector
Angular velocity vector
Applied torque
Area
Argument of perigee
Change in velocity
Current density
Density of fluid
Disturbance torque
Drag coefficient
Eccentricity
Efficiency
Euler parameter or quaternion
Fluid velocity
Force
Gravitational constant
Gravity gradient disturbance torque
Inclination
Launch angle
Magnetic dipole
Magnetic dipole vector
Magnetic field vector
Mass
Mass flow rate
Mass moment of inertia tensor
Mean motion
Optimal eigenvalue
Payload ratio
Phase Angle
Pressure
QUEST algorithm measurement weight
Radius
Radius of apogee
Radius of perigee
Ratio of specific heats
Right ascension of the ascending node
Rotation matrix from inertial to body-fixed frame
Semi-major axis
Solar radiation torque
Specific heat
vii
Isp

ε
t

G
J
V
Specific impulse
Specific mass
Structural coefficient
Time
True anomaly
Universal gravitational constant
Value of the error function
Volume
viii
Chapter 1: Introduction
This report details the specifics of the subsystems that allow such a spacecraft to
maneuver accurately in space. Topics included in this subsystem are: Astrodynamics,
Mission Geometry, Mission Analysis, Guidance and Navigation, Propulsion, and Attitude
Determination and Control. In each case, both the hardware and the mathematical
models are discussed.
1.1 Astrodynamics
Astrodynamics is the study of a satellite’s orbit. Astrodynamicists are responsible
for the satellite’s path through space. Their primary objective is to identify the parameters
of the orbit that best meets the requirements of the mission. Typically, there is a separate
initial parking orbit, transfer orbit, and final mission orbit. Orbit propagator software is
used to estimate spacecraft’s future orbit from the present orbital elements. The mission
orbit significantly influences every element of the mission and provides many options for
trades in mission architectures.
1.2 Mission geometry
Mission geometry is the study of a satellite’s trajectory using graphical
representations of orbits, coverage and access (time and area). A coordinate system has to
be defined in order to begin any formal problem in space mission geometry. In general,
any coordinate system can be used, but choosing the most effective one will reduce the
prospect for errors.
1
1.3 Mission analysis
Mission analysis that is closely linked to astrodynamic consists in finding how a
satellite gets into a specified orbit. The velocity changes, the position of the planets( if an
interplanetary transfer is done), the time at which the velocity changes are applied,… are
determined by the mission analysis.
1.4 Guidance, navigation, and control
Guidance and navigation deals with determining the location of the spacecraft in
orbit and the direction of its motion. Determining the position and velocity vectors of a
spacecraft are critical in timing orbital maneuvers, ground station passes, and rendezvous
with other spacecraft. This data can be obtained in several different ways. Ground
stations track spacecraft by monitoring the satellite’s telemetry signals or with radar.
There are several on orbit tracking systems that use TDRS, GPS, and GLONASS to
determine position and velocity. GPS is rapidly becoming the most common choice for
orbit determination. These systems are discussed in detail in chapter 3. As computers
have advanced, the ability to have fully autonomous orbit determination has become
available. These systems use on-board sensors and software to determine the orbital
elements without the use of outside measurements. When changes to the spacecraft’s
flight path are necessary the GN&C system determines the necessary adjustments. In an
autonomous system the spacecraft would then be commanded to make the necessary
adjustments. For a ground controlled system the guidance system would supply the
2
necessary information to operators on the ground that would then command the
spacecraft to adjust its orbit.
1.5 Propulsion
The propulsion subsystem includes all components that impart a force on the
spacecraft. This includes small thrusters used for attitude control as well as large thrusters
used to insert a satellite into its initial orbit. Thrusters are also used to maintain the orbit
as it decays due to drag or other forces.
Propulsion takes many forms such as a solar sails and electrodynamic or
momentum exchange tethers. The most common form however is chemical propulsion.
Chemical propellants include monopropellant, bipropellant, cold gas, warm gas, or any
combination. With chemical systems, thrust is generated by expelling mass in the
direction opposite of the desired motion. Ion thrusters are closely related to chemical
thrusters. These engines use electrical rather than chemical energy to accelerate the
propellant. There are many variations on this theme. Nuclear power can also be used to
heat the propellant as well as energy beamed in by laser or microwave.
The electrodynamic tether uses electrical energy to push off the earth through the
magnetic field. The momentum exchange tether uses the orbital energy of one satellite to
propel another.
1.6 Attitude determination and control
The attitude determination and control subsystem gives the spacecraft the ability to
achieve a desired orientation in space. A successful spacecraft must be able to determine
3
and control its attitude. Attitude determination is the computation of a spacecraft's
orientation in space. The ability to assess its attitude allows the spacecraft to calculate
the torques required to reach its desired attitude. The attitude control system then exerts
the desired torques over the appropriate time period. This can be performed by a variety
of attitude control devices such as momentum wheels, magnetic torque coils, and
magnetic torque rods. Many of a spacecraft's subsystems rely on the accurate pointing of
the satellite. Thrusters must be aligned in the direction of the desired propulsive force.
Solar panels need to be abreast of the sun to provide power to the spacecraft.
Communication antennas require proper alignment to transfer information. There are few
subsystems that can operate successfully without having to correctly align the spacecraft's
attitude.
Attitude control is the process of rotating a spacecraft from one orientation in
space to another5. An uneven torque must be exerted on the spacecraft to induce an
angular displacement. The combination of the magnitude of the torque and the length of
duration determine the magnitude of the angular displacement. There are several
methods used for attitude control. Magnetic Torque devices use the Earth's
magnetosphere in combination with an induced magnetic field to exert a torque on the
spacecraft. Gas jets, such as hydrazine thrusters, can be used to exert a force away from
the spacecraft's center of mass, thus causing a torque. Varying the velocity of a
momentum wheel can change the rotation of a spacecraft about a single axis, and with
multiple wheels all three angular displacements can be affected.
4
1.7 Overview
In the chapters that follow, each of the aspects of this subsystem are further explored.
In chapter 2, the models that are used in designing and implementing these systems are
discussed. In chapter 3, hardware is discussed and subsystem examples are detailed in
chapter 4. Finally, chapter 5 summarizes the report.
5
Chapter 2: Subsystem Modeling
Mathematical models are needed to effectively design and analyze a subsystem.
Accurate subsystem modeling is necessary to size components as well as define control
laws. This chapter details the equations involved in analyzing the subsystem disciplines.
The interaction between spacecraft subsystems is also identified and discussed.
2.1 Astrodynamics
Satellite orbits are described by orbital elements calculated from position and
velocity vectors. First, Kepler’s three laws of planetary motion have to be defined to
understand how satellite orbits work:

First Law: The orbit of each planet is an ellipse, with the sun at a focus.

Second Law: The line joining the planet to the sun sweeps out equal areas in
equal times.

Third Law: The square of the period of a planet is proportional to the cube of its
mean distance from the sun.
2.1.1 Satellite equations of motions
Isaac Newton explained mathematically why the planets (and satellites) follow
elliptical orbits. Newton formulated the law of gravity by stating that any two bodies
attract one another with a force proportional to the product of their masses and inversely
proportional to the square of the distance between them. Equation (1) represents the
magnitude of the force caused by gravity:
6
F  GMm / r 2
(2.1)
where F is the magnitude of the force caused by gravity, G is the universal constant of
gravitation, M is the mass of the Earth, m is the mass of the satellite, r is the distance
from the center of the Earth to the satellite. The product GM is usually represented by the
letter µ, the Earth’s gravitational constant ( 398,600 km3s-2). Combining Newton’s
second law with his law of gravitation, the two-body equation of motion can be obtained:
..


r   r 3 r  0
(2.2)
Equation (2) illustrates the relative equation of motion of a satellite position vector as the
satellite orbits the Earth. Some assumptions are made while deriving Eq. (2): the gravity
force is the only force, the Earth is spherically symmetric, the Earth’s mass is much
greater than the satellite’s mass, and the Earth and the satellite are the only two bodies in
the system. Figure 1 represents an elliptical orbit around the Earth and its orbital
parameter.
7
V
Perigee
r
Apogee
ra
rp
r: position vector of the satellite relative to the Earth’s center
V: velocity vector of the satellite relative to Earth’s center
ra: radius of apogee
rp: radius of perigee
Figure 2.1: Geometry of an Ellipse and Orbital Parameters
2.1.2 Orbital elements
To completely describe the shape and orientation of an orbit, five independent
quantities are needed. These quantities, called orbital elements are defined below:

a, semimajor axis: describes the size of the ellipse

e, eccentricity: describes the shape of the ellipse

i, inclination: the angle between the angular momentum vector and the unit vector
in the k-direction

, right ascension of the ascending node: the angle from the vernal equinox to
the ascending node
8

 , argument of perigee: the angle from the ascending node to the eccentricity
vector measured in the direction of the satellite’s motion

 , true anomaly: the angle from the eccentricity vector to the satellite position
vector, measured in the direction of satellite motion
Figure 2.2 illustrates the five orbital parameters
Figure 2.2: Orbital Elements8
9
2.2 Mission analysis
Mission analysis determines the optimal transfer from one orbit to another. First,
the transfer orbit is determined. The following equations and methods are used in
mission analysis. Assume the initial and final orbital parameters are known.
The insertion point from orbit 1 to the transfer orbit is RT_initial. The launch angle at
this point is T_initial and the velocity of the satellite on the transfer orbit is VT_initial .The
insertion point from the transfer orbit to orbit 2 is RT_final. The velocities of the satellite
on the initial and final orbit are V1_initial and V2_final respectively. We assume that the two
orbits are co-planar.
2.2.1 Characteristics of the orbits
The velocity change to go from orbit 1 to transfer orbit at RT_initial is found using Eq. 2.3.
VT _ initial  V12_ initial  VT2_ initial  2V1 _ initial VT _ initial cos( T _ initial )  [km / s ]
(2.3)
The velocity at arrival point on orbit 2 is found using Eq. 2.4.
 1
1 
VT _ final  VT2_ initial  2 T 

 [km / s]
R

R
T
_
final
T
_
initial


(2.4)
where T is the gravitational constant of the focus of the orbits.
The semi-major axis of transfer orbit is as follows,
aT 

 [km]
 RT _2in itia l
2
T _ initial
V
The launch angle at arrival point on orbit 2 can be found using Eq. 2.6.
10
(2.5)
cos( T _ final ) 
RT _ initial VT _ initial cos( T _ initial )
RT _ final VT _ final
(2.6)
The velocity change to go from transfer orbit 1 to orbit 2 at RT_final is found using Eq. 2.7.
VT _ final  V22_ final  VT2_ final  2 V2 _ final VT _ final cos( T _ final )  [km / s]
(2.7)
The mean angular motion of orbit 1, 2 and transfer orbit is described as follows,
nT   T aT3  [ s 1 ]
(2.8)
n1   T a13  [ s 1 ]
(2.9)
n2   T a 23  [ s 1 ]
(2.10)
The time of flight on transfer orbit from orbit 1 to orbit 2 is described as follows,
TOF   nT  t1  t 0  [ s]
(2.11)
2.2.2 Phase angle and time required for an interplanetary transfer
The insertion from orbit 1 to the transfer orbit takes place at t = 0 sec.  1 (t 0 ) is the phase
angle of planet 1 at t = 0 sec.
 1 (t 0 )  [rad ]
(2.12)
The phase angle of planet 2 is described as follows,
 2 (t 0 )   1 (t 0 )   (1  2k )  [rad ](k  1, 2, 3, )
(2.13)
The phase angle of planet 1 at t1 is described as follows,
 1 (t1 )   1 (t 0 )  n1 (t1  t 0 )   1 (t 0 )  n1 TOF
(2.14)
 2 (t1) is the phase angle of planet 2 at t1.
 2 (t1 )   2 (t 0 )  n 2 (t1  t 0 )   2 (t 0 )  n 2 TOF  [rad ]
11
(2.15)
The characteristics of the transfer orbit in a heliocentric system are now defined.
Now we have to determine the escape trajectory from the first planet and the final orbit
around the second planet.
2.2.3 Escape orbit at planet 1
The system considered here is planet 1. The focus of the parking orbit is the center of
planet 1. We assume we know the launch angle at burn out  1 _ bo and the radius at burn
out R1 _ bo (position when the satellite goes to the transfer orbit).
The characteristics of the parking orbit around planet 1 are known: (e.g. its semi-major
axis, a1 _ park , eccentricity, e1 _ park , velocity at radius R: V1 _ park _ R ). VT _ initial is the
hyperbolic excess velocity from planet 1.
V1 _   VT _ initial  [km / s ]
(2.16)
The eccentricity of the hyperbolic orbit from planet 1 is,
e1 _ 
 R1 _ bo V12_ 


1


 cos 2 ( 1 _ bo )  sin 2 ( 1 _ bo )


(2.17)
where 1 is the gravitational constant of planet 1.
The burn out velocity to enter in the hyperbolic escape orbit is defined by Eq. 2.18.
V1 _ bo  V12_  
2 1
 [km / s]
Rbo
(2.18)
The velocity of the satellite in the orbit around planet 1 at the burn out position is defined
by Eq. 2.19.
 2
1 
V1 _ park _ bo  1 

 [km / s]
R

a
1
_
bo
1
_
park


12
(2.19)
The velocity change to enter the hyperbolic escape orbit from the orbit around the planet
is described as,
V1 _ bo  V1 _ bo  V1 _ park _ bo  [km / s ]
(2.20)
The total velocity change to go from the parking orbit around planet 1 to transfer orbit is
described as,
V1  VT _ initial  V1 _ bo  [km / s ]
(2.21)
2.2.4 Capture orbit at planet 2
The procedure to determine the capture orbit is similar to that for the escape orbit
of planet 1. The system considered is planet 2. The focus of the parking orbit is the
center of planet 2. We assume we know the launch angle at burn in  2 _ bi and the radius
at burn in R2 _ bi (position when the satellite goes from the transfer orbit to the parking
orbit). The characteristics of the parking orbit around planet 2 are known: (e.g. its semimajor axis, a 2 _ park , eccentricity, e2 _ park , velocity at radius R: V2 _ park _ R ). VT _ final is the
hyperbolic excess velocity from planet 2.
V2 _   VT _ final  [km / s]
(2.22)
The eccentricity of capture orbit can be found using Eq. 2.23.
e2 _ 
 R2 _ bi V22_ 


2


 cos 2 ( 2 _ bi )  sin 2 ( 2 _ bi )


where  2 is the gravitational constant of planet 2.
The burn in velocity to enter from the capture orbit to the parking orbit is,
13
(2.23)
V2 _ bi  V22_  
2 2
 [km / s]
Rbi
(2.24)
The velocity of the satellite in the orbit around planet 2 at the burn in position is,
 2
1 
V2 _ park _ bi   2 

 [km / s]
R

 2 _ bi a 2 _ park 
(2.25)
The velocity change to enter the parking orbit from the capture orbit is,
V2 _ bi  V2 _ bi  V2 _ park _ bi  [km / s]
(2.26)
The total velocity change from the transfer orbit to the parking orbit around planet 2 can
be found using,
V2  VT _ final  V2 _ bi  [km / s]
(2.27)
This is the general procedure used in Mission Analysis.
2.3 Guidance navigation and control
The purpose of the guidance system is to obtain the position and velocity of the
spacecraft. These vectors can then be used to determine the orbital elements. Different
systems accomplish this in different ways. Guidance systems can be separated into two
groups: non-autonomous and autonomous systems.
2.3.1 Non-autonomous systems
Non-autonomous systems determine the spacecraft’s orbit outside of the
spacecraft. The two main ways to do this are using ground station tracking or TDRS. A
ground station tracks a satellite using either radar or telemetry signals. In both cases the
14
ground station uses range and range rate data to determine the orbit. Determining an
accurate orbit this way requires several ground passes.
The TDRS system is a constellation of spacecraft used to track satellites in LEO
and relay data. TDRS provides range and range rate data as well as less accurate angular
measurements. Position data is much more accurate than angular data. TDRS can
achieve 3 accuracies of 50 meters.
2.3.2 Autonomous systems
Autonomous systems calculate the orbit on-board the spacecraft. Sensors are
used to measure range and orientation data then compared to a mathematical model to
determine the orbital elements. Using GPS is now the most common autonomous system
for LEO satellites. A GPS receiver uses signals from four GPS satellites. Each GPS
spacecraft sends a signal containing the GPS satellite ephemeris. This data can be used to
determine the position of the GPS satellite at a reference time. The signal also contains a
time stamp that displays the time the signal was sent. This time is compared to a clock
on the spacecraft that runs in GPS time. The length of time the signal takes to arrive at
the receiver multiplied by the speed of light calculates range and range data for each
spacecraft. Knowing the range and position of four GPS spacecraft, a LEO satellite can
solve for its position and time. GPS systems can obtain 3 accuracies of as little as 15
meters.
2.4 Attitude determination and control
The purpose of the attitude determination and control subsystem (ADCS) is to
orient a spacecraft in space. The system must be able to determine the spacecraft's
15
current attitude and then calculate and implement the necessary actions to exert the
appropriate torque to achieve the desired attitude.
2.4.1 Attitude determination
There are two degrees of attitude determination: single-axis attitude determination
and three-axis attitude determination. One body-fixed axis of the spacecraft is
determined in single-axis attitude determination. Only two independent numbers are
required for this type of determination. For example, a vector measured in the body-fixed
frame can be compared to the same vector in the inertial frame, and the rotation from one
frame to the next can be calculated. However, this leaves a single degree of freedom for
the spacecraft, a rotation about the determined axis. Such a determination scheme is
sometimes used in spin-stabilized spacecraft4. Three-axis attitude determination
completely defines a spacecraft's orientation in inertial space. The addition of a third
independent number is required for three-axis attitude determination. The number serves
to define the rotation about the axis determined in the single-axis determination method,
thereby fully determining the spacecraft's attitude4.
Attitude determination is often an over constrained problem. A spacecraft usually
has more attitude sensors than necessary to define its orientation. Two methods to solve
such an over constrained problem are deterministic and statistical determination.
The deterministic approach involves assuming some of the information is exact
and discarding the extra information. The triad algorithm is an example of the
deterministic approach. It is called the triad algorithm because two triads of orthonormal
vectors are constructed to solve the problem. The algorithm requires two independent
16

measurement vectors. One of the vectors is assumed to be exact1. If A is the exact

vector and B is the second vector, the triads are set up in Eq (2.28-2.30).


t1b  Ab
(2.28a)


t1i  Ai
(2.28b)



Abi  Bb
t 2b   
| Ab  Bb |
(2.29a)
 

Ai  Bi
t 2i   
| Ai  Bi |
(2.29b)

 
t3b  t1b  t 2b
(2.30a)
  
t3i  t1i  t 2i
(2.30b)
The b subscript denotes the body-fixed reference frame and the i subscript denotes the
inertial reference frame. The attitude matrix for the spacecraft is then calculated using
the two triads.

R bi  [t1b

t 2b
 
t3b ][t1i

t 2i

t3i ]T
(2.31)
Statistical attitude determination makes use of all the information, and therefore
should provide a better estimate of the spacecraft's attitude than calculated by the
deterministic method. The goal of attitude determination is to determine the attitude
matrix that provides the smallest value for the loss function.1
J ( R bi ) 

 2
1 N
bi
w
|
A

R
A
 k kb
ki |
2 k 1
(2.32)
The subscript k denotes one of the N measurements, and wk represents the relative
weights given to each measurement. A commonly used statistical attitude determination
algorithm is the QUEST algorithm. This algorithm uses an approximation of the optimal
17
eigenvalue found in the least square optimization to save computational resources. The
value of the loss function at the optimal eigenvalue is close to zero and can be discarded.
optimal   wk  J   wk
(2.33)
The eigenvector that corresponds to the optimal eigenvalue is the quaternion that
represents the optimal attitude estimate.1 A Kalman filter is another method used in
statistical attitude determination. It is a set of equations that provide a recursive solution
of the least squares method. Kalman filters are able to estimate past, present, and future
states. The most appealing aspect of Kalman filters is that the exact nature of the system
does not need to be known.2
2.4.2 Attitude control
The attitude dynamics of a spacecraft is governed by the following equation:



dL 
d
 N   L  I
dt
dt
(2.34)

Equation (2.34) relates the torque applied, N , to the time derivative of the angular

dL 3
momentum vector,
. From Eq. (2.34) it is possible to see how the time derivative of
dt

d
the angular velocity vector,
, varies with the applied torque. The angular velocity
dt
vector, as shown in Eq. (2.35), directly affects attitude.1

q  1  q  q 4 1
2   qT 

This equation relates the angular velocity vector,  , to the time derivative of the
quaternion, q .
18
(2.35)
The choice and design of an attitude control system is based on the required
pointing accuracy and slew rate, as well as the magnitude of disturbance torques. The
two broad categories of attitude control are passive and active attitude control.
Spin stabilization and gravity-gradient stabilization are two examples of passive
attitude control. A spacecraft with single-spin stabilization rotates about a single axis to
keep its angular momentum vector approximately constant. An asymmetric spacecraft
exposed to a gravity well will tend to align its minor axis normal to the gravity potential.
Gravity-gradient stabilization exploits this property. Earth-pointing spacecraft are often
gravity-gradient stabilized for this reason.
Spacecraft with more stringent pointing requirements use active attitude control
systems. In an active attitude control system, mechanisms are used to deliberately apply
a torque to the spacecraft. The mechanisms used to control attitude include momentum
wheels, magnetic torquers, and thrusters. Momentum wheel control systems alter the
attitude of a spacecraft by varying the speed at which the wheel spins. Equation (2.36)
shows how a change in the angular moment, h , of the momentum wheel affects the
change in the angular velocity of the spacecraft,  p , where I p is the spacecraft's
moment of inertia.3
 p  
h
Ip
(2.36)
Magnetic torquers use the interaction of the Earth's magnetic field with an induced
magnetic dipole to enact a torque on a spacecraft. The magnetic dipole is induced by

means of running a current through a coil of wire. The resulting applied torque, N , is
defined by Eq. (2.37). 3
19
  
N  M B
(2.37)
A prominent property, shown in Eq. (2.37), is that the applied torque is always
perpendicular to the magnetic field. 3 Thrusters are another mechanism used in attitude
control. They apply a force at a fixed distance from the center of mass of the spacecraft.
  
N  F r
(2.38)
Equation (2.38) shows the simple equation governing applied torques from thrusters,


where F is the applied force and r is the moment arm.
2.5
Propulsion
A spacecraft’s propulsion system usually accounts for most of the mass though
newer systems have begun to address this issue. The type of propulsion considered here
are chemical rockets, electrical rockets, and electrodynamic tethers. They are chosen
because they are the farthest along in development and seem the most promising as far as
general propulsion duties. Launch vehicle propulsion is not considered. Only in-space
propulsion requirements are considered.
2.5.1 Chemical rockets
The performance of a space propulsion system is generally modeled using the
momentum equation,
F
x
where

d
u x dV   u x dm
dt cv
cs
Fx = component of force in x direction
m = mass rate of flow (positive out)
ρ
= density of fluid
20
(2.39)
V = volume
ux = component of fluid velocity in x direction
and cv and cs denote integration over the control volume and control surface respectively.
Using Eq. (2.39) and accounting for pressure differences Eq. (2.40) is obtained.
T  m u e  ( pe  pa ) Ae
(2.40)
The symbols u e , p e and, p a are the exit velocity, exit pressure and, ambient pressure
respectively. The exit area of the nozzle is Ae . From this equation we can define an
equivalent velocity
 p  pa 
u eq  u e   e
 Ae
 m

(2.41)
such that T  m ueq . The main performance measure of a chemical rocket is the specific
impulse defined as
I sp 
ueq
(2.42)
ge
The constant g e is the gravitational acceleration at earth’s surface. In space we can
assume constant u eq and the absence of drag or gravity. Using Equation 2.39 and these
assumptions we determine
u  ueq ln
M0
Mb
(2.43)
where M 0 is the initial mass and M b is the burnout mass. Furthermore we can solve for
the payload ratio defined as  
ML
where M L is the payload mass
M0  ML
21
 
e
u
u eq
1 e
where ε is the structural coefficient  
1
(2.44)
u
u eq
Ms
where M s is the structural mass and
M p  Ms
M p is the propellant mass. The exit velocity can be determined from thermo-chemistry
and is in general
p 
u e  2C p T0 [1   e 
 p0 
 1

(2.45)
where T0 , p 0 , C p and, γ are the stagnation temperature, stagnation pressure, specific
heat at constant pressure, and the ratio of specific heats respectively. Equations 2.39
through 2.43 are taken from Ref.6.Equation 2.45 is from Ref. 7.
2.5.2 Electrical rockets
For electrical rockets, in particular electrostatic devices there are a few key
performance parameters. They are the specific mass of the power plant and the
efficiency, defined as

MP  Ms
P

m ue2
2P
(2.46)
(2.47)
Here M P is the mass of the power plant and P is the power output. Isp is also defined for
electric rockets as
I sp 
1
ge
2
q
Va
m
22
(2.48)
The mass flow rate is assumed constant over a time t b (the burning time). Then by
summing the masses and using Equations 2.46, 2.47 and, 2.43 we can obtain
ML
e
M0
 u
ue

u e2  u e2
1 

2t b  2t b

(2.49)
The most common way to maximize the current density in an electrostatic propulsion
device is to use and accel-decel system, a planar triode as described in Figure 2.3.
Figure 2.3: Potential Distribution for a Planar Triode6
This potential distribution gives the maximum current density ( j ) and thus thrust
per unit area. The maximum current density and thrust per unit area in this configuration
are given by Equations 2.50 and 2.51.
4
2q V  2
j  o
9
m L2
3
(2.50)
1
T 8  V   V a  2
 o 

 
A 9  L   V 
2
23
(2.51)
In Equations 2.50 and 2.51  o is the permitivity of free space and V , Va, and L are as
defined in Figure 2.3. A is the area of the thruster.
2.5.3 Electrodynamic tether
An electrodynamic tether is rather simple to model. The main equation is
  
F J B
(2.52)


To maximize the thrust, the J and B vectors should be perpendicular and the maximum
possible. The orbit should be at the altitude where the Earth’s magnetic field is greatest


without incurring too much of a drag penalty. Maximizing J is more difficult than B as
efficient high capacity plasma contactors must be used and there is no consensus on what
they consist of, though recent research has suggested a bare wire would provide the best
performance.
Equations 2.46 through 2.49 and Figure 2.3 are taken from Ref.6. Equations 2.50
through 2.52 are from Ref.7.
2.5.4 Momentum exchange tether
Momentum exchange tether propulsion is a propellant less method of changing a
spacecraft’s orbit. A momentum transfer between a payload spacecraft and a tether
launch facility achieves the change in velocity. Momentum is transferred to the payload
spacecraft to achieve a higher energy orbit, and momentum is transferred from the
payload spacecraft to decrease the energy of its orbit.
The momentum exchange is achieved through two methods: the exploitation of
the difference in the gravity potentials and spinning the system. The masses of the tether
launch facility and the payload spacecraft are assumed to be point masses. The system of
24
the two spacecraft is assumed to be in a circular orbit. Tethered spacecraft create a new
system where the center of mass is located between the two spacecraft. The distance of
the center of mass from the payload spacecraft, l2, can be determined from Eq. (1),
l2 
(1)
mtlf L
(mtlf  m ps )
where, mps is the mass of the payload spacecraft, mtlf is the mass of the tether launch
facility, and L is the length of the tether. The system is traveling at the speed of the
center of mass as shown in Eq. (2). The symbols , vcm, and rcm represent the
gravitational constant, the velocity of the system, and the radius of the system’s orbit.
v cm 

(2)
rcm
The velocity of the payload spacecraft, vps, can be calculated using Eq. (3). The symbol
 represents the angular velocity of the system.
v ps  rcm  l 2 

rcm
3
   l2
(3)
Equation 4 defines the added velocity received by the payload spacecraft, v, as the
difference between the velocity of the center of mass and the velocity of the payload
spacecraft.
v  v ps  v cm
(4)
These equations can be used as a first approximation of the added velocity a
payload spacecraft could receive using a momentum exchange tether.
25
2.6 Subsystem interactions
Defining how subsystems will react with the flight dynamics subsystems that are
discussed in this paper is an important issue. Knowing these interactions is essential for a
well-conceived design. Ensuring that all subsystems will work together in the design
phase is necessary before attempting to manufacture and integrate a spacecraft.
2.6.1 Guidance navigation and control interactions
The guidance, navigation and control system senses the spacecraft orbit and
determines the necessary alterations. To accomplish this task, information and
constraints from other subsystems must be obtained. In addition, the GN&C subsystem
supplies other parts of the spacecraft with orbital information that is required. The
interaction of GN&C with other subsystems is shown in Figure 2.4.
The GN&C subsystem has to moderate the accelerations that are
performed during the velocity changes in order to respect the structural constraint of the
subject. The mission operation subsystem tells the GN&C subsystem when the orbit
adjustment have to be done. For example, to execute an orbit transfer burn, the mission
operation subsystem sends the data to the GN&C subsystem that at time t, a transfer to a
specified orbit will be executed. The GN&C system then calculates the velocity change
V necessary to accomplish this maneuver.
26




Mission ops
Orbit adjustments
Orbit transfers
Power
GN&C load
Solar array
positioning

GN&C


Launch
Specify initial
orbit
Communications

GS Pass times
ADCS
Position for ADCS
models
Station keeping

maneuvers


Structure
Acceleration
constraints on
DV
Propulsion
Specify V
Thruster firing
time
Figure 2.4: GN&C Subsystem Interactions
The GN&C subsystem specifies the velocity change to the propulsion subsystem
and when they have to be applied in order to get to the correct orbit. The ADCS receives
the information of the GN&C in order to know what correction have to be done. If the
subject has to be at a given point which is specified by the GN&C, the ACDS may
correct a position error by referring to the given point. The GN&C determines the
position of the subject in space, which is used in the attitude determination models to
calculate the suns direction and the earth’s magnetic field. Determining orbital position
and therefore the direction to the sun, is also used to determine the optimal direction to
point solar arrays.
27
2.6.2 Attitude determination and control subsystem interactions
The ADCS is an important subsystem that greatly interacts with many other
subsystems. Figure 2.5 shows the dependencies and requirements of the ADCS to ensure
successful operation of a spacecraft.
The ADCS is reliant on the power subsystem to provide the appropriate amount
power to operate its components. The power subsystem must have a method of
collecting energy. For example, if solar panels are used then the ADCS is required to
orient the spacecraft so that the panels are exposed to the sun.
Some attitude sensors, such as magnetometers, earth horizon sensors, and sun
sensors, require the position of the spacecraft to execute the determination algorithms.
The position of the spacecraft is provided by the GN&C subsystem.
The ADCS must meet the slew rate requirements of the mission. If thrusters are
used for both ADCS and the propulsion subsystem, they must be sized so that they are
able to produce the required torques. Propulsion subsystem requires that the spacecraft is
correctly orientated to properly direct the applied V.
Properties of the structure of the spacecraft, including the location of the center of
mass and the mass moments of inertia, are required knowledge for control maneuvers that
need to be performed. The ADCS imposes some restraints on the structure subsystem. It
needs to have proper placement of attitude sensors and actuators to operate correctly and
efficiently.
Mission operations requires ADCS for the appropriate pointing of scientific
sensors and any other special maneuvers. The ADCS is required to orient the spacecraft
so that their antennas are positioned to allow for data transfer and communication.
28
Without this ability the communication subsystem would not be able to function. Some
sensors on spacecraft are sensitive to temperature. The ADCS can keep from exposing a
certain sensor to the sun and thereby protecting it from extreme temperature changes.


Mission ops
Sensor pointing
Special maneuvers
Power

ADCS load

Solar array pointing

Position for ADCS
models
Station keeping
maneuvers

Thermal

Power
ADCS
GN&C


Propulsion
Thermal maneuvers
Communications

Antenna pointing



Center of mass
Moments of inertia
Sensor placement
Actuator placement

Structure
ADCS thruster size
Thruster pointing
Figure 2.5: ADCS Subsystem Interactions
2.6.3 Propulsion subsystem interactions
The propulsion subsystem interacts mainly with seven other subsystems: mission
operations, thermal, communications, structure, GN&C, ADCS, and power. To change or
adjust the spacecraft’s orbit, the propulsion system modifies the V required for the new
orbit. The GN&C subsytem specifies the V required and when to apply it during the
orbit change. Because thrusters generate heat while burning propellant, the thermal
subsytem interacts strongly with propulsion. The structure subsystem has to withstand the
29
load induced by the propellant. During launch, the exhausted propellant makes some
interference between the ground station and the spacecraft, the communication system is
therefore affected by the propulsion system. Finally, the propulsion system needs enough
power to function properly.





Power
Mission Ops
Orbit adjustments
Orbit transfers
Propulsion load
Electric thrusters?
Electrodynamic
tether
Thermal

Propulsion



Generates heat
Communications
ADCS
ADCS thruster size
Thruster pointing




Plume interference
Structure
Withstand loads
Hold propellant
GN&C
Specify V
Thruster firing
time
Figure 2.6: Propulsion Subsystem Interactions
2.7 Summary
This chapter explores the mathematical models that are used in analyzing the
flight dynamics and propulsion of a spacecraft. These equations are used to design and
analyze the subsystem to meet the mission requirements. The analysis of the mission
requirements yields subsystem performance requirements. These requirements are then
used to select subsystem components and architecture. The next chapter gives examples
of various subsystem components that can be used to fulfill the performance
requirements.
30
Chapter 3: Subsystem Examples
This chapter details specific examples of subsystem hardware. Three of the
disciplines discussed in this paper have hardware components on the spacecraft. The
ADCS, GN&C, and propulsion subsystems all have hardware that is chosen and sized to
meet mission requirements. Astrodynamics and mission analysis are analytic disciplines
that have an influence on other subsystems, but have no specific hardware of their own.
3.1 Astrodynamics
The orbit selection process is complex, involving trades between a number of
different parameters. The orbit is generally selected to meet the largest number of mission
requirements at the least possible cost. Several different orbit designs may be credible
due to changes in mission requirements or simply improvements in mission definition.
The orbit selection process is divided into several steps. First, one needs to
establish the orbit types. The orbit-related requirements are then defined. These
requirements may include orbital limits, individual requirements such as the altitude
needed for specific observations, or a range of values constraining any of the orbit
parameters. Once the orbit parameters are defined, it is possible to select the mission orbit
by evaluating how they affect each of the mission requirements. As the orbit design
depends principally on altitude, the best way to begin is by assuming a circular orbit and
then conducting altitude and inclination trades. One or more alternatives can then be
selected from this range of potential altitudes and inclinations.
31
3.2 Mission analysis
There are several transfer orbits to consider when attempting to maximize
performance and minimize cost. The most common type of transfer orbit used is the
Hohmann transfer orbit. A Hohmann transfer orbit is between the perigee of an initial
orbit and the apogee of the final orbit. It is advantageous because the total velocity
change required to complete the orbit transfer is minimized. This minimization is due to
the fact that the flight path angles at the initial and final orbit is of zero degree. So, from
Eq. (2.3) one can see that the velocity change V is minimum when the flight path angle
is zero. If the velocity changes are minimized, the cost of the mission is minimized as
well.
There are two different types of transfer orbits for interplanetary transfers called
type I and type II. The type I orbit is when the interplanetary trajectory carries the
spacecraft less than 180 around the sun. A spacecraft in a type I orbit has a high
velocity with respect to the sun. Therefore, the spacecraft can reach its destination in a
short amount of time, but more fuel is used. The type II orbit is when the interplanetary
trajectory carries the spacecraft more than 180 degree around the sun. A spacecraft in a
type II orbit has a lower velocity with respect to the sun than that of a type I orbit.
Therefore, the spacecraft take more time to arrive to its destination but it costs less.
Some interplanetary missions are using the planets that are on the trajectories of
the spacecraft in order to increase the energy of the spacecraft by an exchange of
momentum between the spacecraft and the planet. This kind of transfer orbit is called a
gravity assist trajectory.
32
For example, the Mars Pathfinder that was sent to Mars on December 4th 1996
used a type I orbit. The time of flight of this mission was of 212 days. And for another
Mars mission, Mars Global Surveyor that was sent on November 7th 1996, the orbit used
was a type II orbit. The time of flight of the mission was of 309 days.
The Voyager 2 mission that is going to the outer planets was launched on August
20th 1977 and it used a type II Hohmann transfer orbit. The spacecraft flew by Jupiter
and Saturn. As it flew by the gaseous planets, a momentum exchange between the
spacecraft and the planet occurred such that the spacecraft was able to leave the sphere of
influence of the planets with a higher velocity than when it arrived to the planet.
3.3 Guidance navigation and control
Examples of GN&C systems are categorized by the degree of autonomy. Nonautonomous orbit determination is accomplished either on the ground or using TDRS.
Both systems use range and range rate data to determine information about the spacecraft
orbit. This data can be obtained using radar or telemetry signals from the orbiting
satellite. The data is then used by orbit propagating software to estimate the orbit of the
spacecraft, and its present and future location. Determining orbital position in this
manner is operationally expensive and requires either the use of existing tracking stations
or building a new ground station.
There are several different options for on-board autonomous orbit determination.
The most popular and proven technology available is GPS. A spacecraft with a single
GPS receiver can determine orbital position accurate to 50 m. Adding a second antenna
and GPS receiver can increase accuracy and allows for the failure of one receiver. By
using differential GPS and software, the accuracy of commercial systems can reach 15
33
meters. There are several companies that make GPS units for commercial and military
use. Honeywell, Motorola, Rockwell, and Magellen all make GPS receivers as well as
other space hardware.
Inertial navigation units (INU) sense changes in acceleration and angular rates.
An INU uses devices such as ring laser gyros and accelerometers to determine these
accelerations and rates. In general there are sets of three orthogonal accelerometers and
three gyros. These components produce relative acceleration and angular rate
information. This data is used to determine the relative motion of the spacecraft.
Companies that build INUs include Honeywell, Litton, and Lockheed Martin.
Honeywell manufactures a complete GN&C system called the space integrated
GPS/INS (SIGI). This device uses a combination of GPS receivers and INUs for
autonomous orbit determination. With selective availability turned on, the unit can
calculate position with an accuracy of better than 50 m and velocity better than 0.3 m/s.
The SIGI can operate at an altitude of up to 600 miles, withstand accelerations less than
10 g’s and survive a temperature range from –55 to 60 C. The SIGI requires 35-45
Watts and weighs approximately 20 lbs. Honeywell advertises the unit as capable of
providing autonomous guidance for target tracking, range tracking, and rendezvous and
docking maneuvers.
Orbit control is also either accomplished through ground commands or
autonomously on-board the spacecraft. Ground controlled orbital maneuvers consist of a
series of commands that are sent to the satellite, which then executes the maneuver. The
calculations for these burns are done on the ground before the commands are sent.
Autonomous orbit control is accomplished using a computer on the spacecraft. Readings
34
from an orbit determination component such as SIGI are used to calculate any necessary
maneuvers. The computer then autonomously generates the same type of commands that
would be sent from the ground and the spacecraft executes the appropriate burns. In the
past this kind of automation was not possible due to the limited capability of computers.
With the advances made in computer systems, fully autonomous GN&C systems are
possible. In addition, by removing the human and machine resources necessary to
control a spacecraft from the ground, an autonomous GN&C system saves money.
Assuming a spacecraft has a sufficiently powerful computer, autonomous control is
largely a software issue.
3.4 Attitude determination and control
Honeywell offers a number of models of reaction wheels and one momentum wheel
model. Table 3.1 shows the vital statistics of some of the reaction/momentum wheel models that
Honeywell offers. The information provided includes the available output torque, peak power
requirement, and the power required for the wheel to maintain its maximum speed.
Table 3.1: Honeywell reaction and momentum wheels10
Model No.
Peak
Power (W)
105
Power Holding
Max. Speed (W)
22
Type of Wheel
HR12, HR14, HR16
Output Torque
(N-m)
0.1-0.2
HR0610
HR2010, HR4510
HR2020
HR2030
HR4820
HR04
HM4520
0.075
0.1
0.113
0.21
0.14
0.028
0.135
80
115
175
190
165
150
15
17
35
20
20
6
35
Reaction
Reaction
Reaction
Reaction
Reaction
Reaction
Momentum
Reaction
Magnetic torque bars are usually built to a specification provided by the space system
engineers, so that it is optimized for the particular mission. A company that produces custom
35
MTBs is Microcosm. Table 3.2 displays the vital statistics of some of the MTBs that Microcosm
has produced for prior missions.
Table 3.2: Microcosm magnetic torque bars13
Model No.
Power (W)
Mass (kg)
MT-30-2-CGS
Dipole Moment
(A-m2)
30
3.6
1.135
MT-80-1
MT-140-2
80
145
3
1.9
4.12
5.3
The Astro-Iki Corporation produces systems and instruments for spacecraft attitude
determination. The optico-electronic instruments offered include star, horizon, and sun trackers.
This company also manufactures gyroscopes and accelerometers that may be used with an
integrator for attitude determination.1 Table 3.3 is a summary of the attitude sensors offered by
the Astro-Iki Corporation.
Table 3.3: Attitude determination sensors provided by Astro-Iki Corporation1
Sensor Type
Accuracy
Mass (kg)
2"
10"
Power
Consumption (W)
10
10
Star Tracker
Horizon Tracker
Sun Tracker
SD-1
SD-2
SD-3
Gyroscope DNG-091
Accelerometer MA-1
10'
1"
1'
0.02'/impulse
0.005 m/s/impulse
0.25
5
5
-
0.2
0.6
0.7
-
3.1
2.5
Magnetometers are a lightweight, inexpensive spacecraft attitude sensor. Billingsley
Magnetics produces two magnetometers designed for space systems.13 The magnetometer model
specifications are summarized in Table 3.4.
36
Table 3.4: Billingsley Magnetics space-rated magnetometers13
Model Number
Accuracy
TFM1005
TFM100G2
0.5% of full scale
0.75% of full scale
Power
Consumption (W)
0.56
1.02
Mass (kg)
0.2
0.117
3.5 Propulsion
Atlantic Research Corporation specializes in all types of rocket propulsion and
has a long history of producing highly reliable engines. They also have a wide product
range, from very small attitude control thrusters to upper stage and apogee kick motors.
Table 3.5 is a listing of ARC’s line of monopropellant hydrazine thrusters. The
largest of which is capable of providing 445 N of thrust, while the smallest has a
minimum impulse bit of just 11 mN-s. ARC also has a good selection of bi-propellant
systems as well as the Agena 2000 upper stage. The original Agena system was the most
reliable US developed upper stage ever.2
Table 3.5: Atlantic Research Corporation’s MONARC line of hydrazine thrusters2
MODEL
MONARC-1 MONARC-5 MONARC-22 MONARC-90 MONARC-445
Thrust
1N
5N
22N
90N
445N
Specific Impulse
232 s
230 s
235 s
235 s
235 s
Inlet Pressure
7-28 bar
5-30 bar
3.5-28 bar
5-31 bar
5-31 bar
Range
Max. Impulse
111,250 N-s
311,500 N-s 501,000 N-s 3,500,00 N-s 5,600,00 N-s
Minimum
11 mN-s
90 mN-s
312 mN-s
891 mN-s
8910 mN-s
Impulse Bit
Weight
0.33 kg
0.37 kg
0.24 kg
1.0 kg
1.6 kg
Engine Length
13.3 cm
20.3 cm
20.3 cm
30.4 cm
40.6 cm
Nozzle Exit
2.5 cm
2.5 cm
3.8 cm
8.4 cm
14.8 cm
Diameter
Status
Flight
Flight
Flight
Flight
Flight
Qualified
Qualified
Qualified
Qualified
Qualified
Tethers Unlimited is a tether production company. This organization is
advertising a complete electrodynamic tether propulsion system for microsatellites called
37
μPET. Tethers Unlimited also market the Terminator Tether. This tether uses
electrodynamic effects to deorbit spent upper stages. By far the most important product is
the HoyTether which is predicted to have a lifetime on the order of 100 years even in a
debris rich environment.17
For other electric propulsion Tables 3.6a and 3.6b summarizes the characteristics
of some common systems in use. Unfortunately with the present state of the art in electric
propulsion one must choose between extreme efficiency in propellant usage and high
thrust levels.
Table 3.6a: Characteristics of selected electric propulsion flight systems20
Concept
Supplier
Specific
Impulse,
(sec)
Input
Power,
(kW)
Thrust/
Power,
(mN/kW)
Specific
Mass,
(kg/kW)
Propellant
Resistojet
Arcjet
Pulsed Plasma Thruster
(PPT)
TRW,
Primex,
Primex, JHU/APL TSNIMASH,
CTA
NASA
Primex
Primex,
TRW
IRS/ITT
Primex
Primex
296
299
480
502
> 580
800
847
1200
0.5
0.9
0.85
1.8
2.17
26*
< 0.03
< 0.02
743
905
135
138
113
--
20.8
16.1
1.6
1
3.5
3.1
2.5
--
195
85
N2H4
N2H4
NH3
N2H4
N2H4
NH3
Teflon
Teflon
38
Table 3.6b: Characteristics of selected electric propulsion flight systems (cont)20
Concept
Hall Effect Thruster (HET)
Supplier
IST,
TSNIMASH, SPI,
Loral,
NASA
KeRC
Fakel
HAC
MELCO,
Toshiba
MMS
HAC,
NASA
DASA
Specific
Impulse,
(sec)
1600
1638
2042
2585
2906
3250
3280
3400
1.5
1.4*
4.5
0.5
0.74
0.6
2.5
0.6
55
--
54.3
35.6
37.3
30
41
25.6
7
--
6
23.6
22
25
9.1
23.7
Xenon
Xenon
Xenon
Xenon
Xenon
Xenon
Xenon
Input
Power,
(kW)
Thrust/
Power,
(mN/kW)
Specific
Mass,
(kg/kW)
Propellant Xenon
Ion Thruster (IT)
3.6 Summary
Chapter 3 introduces some examples of the components used in the subsystems
we have been discussing. These examples were chosen based on their reputation and
flight heritage, as well as potential performance. The next chapter discusses some
example applications using the modeling solutions developed in chapter 2 and the
components discussed here.
39
Chapter 4: Application Examples
This chapter illustrates the use of the methods developed in chapter 2 to solve a
specific problem in each discipline using the components described in chapter 3. These
applications show how to choose orbits and size hardware for specific design
requirements.
4.1 Astrodynamics
The orbit design micro-project14 can be used to illustrate an orbit determination
method. The goal of this project was to determine the three most efficient orbital paths
for a satellite of unknown design given three different sets of weights for each measure of
effectiveness (MOE). For this example, only one set of weights will be used. The
satellite’s purpose is to gather sensor data in the tropical and temperate zones of the earth,
and communicate that data to the ground station located in Blacksburg, Virginia, which is
located at a latitude of 37.23274° N. The space-to-ground communication system
requires a minimum elevation of 10º. The scope of this project is limited to choosing a
combination of altitude from 500 km to 15000 km and inclination from 0º to 45º which
will provide us with the optimal orbit. We must first select an altitude at which the set of
weights, as dictated by the CDM, will provide for the maximum total Value. Once this
altitude is chosen, we proceed with the task of picking an orbital inclination that will best
fulfill the requirements of sensing the tropics and temperate zones while also providing
for contact with the ground station. A spreadsheet was used to model all the possible
alternatives for orbits14. For each altitude between 500 and 15000 km, the earth-center
angle λ was calculated. From this angle, the latitude of the ground station, and the
40
inclination of the satellite, it is possible to eliminate some options due to the spacecraft
never being in view of the ground station. For the remaining options, the Instantaneous
Access Area (IAA) is calculated. Then the IAA is normalized over the maximum
theoretical IAA, which for a satellite at an infinite altitude is half the surface area of the
earth. After calculating the IAA for each inclination, the Area Access Rate (AAR) is
found based on the satellites altitude and period for a circular orbit and then normalized.
Finally, the ΔV needed to maintain orbit annually is found for each option. Once each
MOE has been determined over the range of altitudes, the calculation of the net Value
may be completed.
Value = w1 MOE1 + w2 MOE2 + w3 MOE3
(4.1)
where MOEi is the IAA, AAR, and ΔV respectively and each wi is the weight assigned to
that MOE by the CDM.
Figure 4.1 shows the normalized MOE’s with respect to altitude. The ΔV plot
shows that the atmosphere only affects the spacecraft at low altitudes. Otherwise, it has
no effect at all. That means the final design should fly at an altitude of at least 1000 km
based on this MOE alone.
The instantaneous access area plot is somewhat linear, increasing in value as it
increases in altitude. Based on this factor, the satellite should orbit as high as possible.
However, the area access rate, increases until about 2500 km, and then starts to decrease.
Therefore, above 2500 km as the IAA increases the AAR will decrease.
41
Figure 4.1: Measures of effectiveness vs. altitude for IAA, AAR, and ∆V/year.
4.2 Mission analysis
To have a better representation of the discussion of section 4.1, we compute the
characteristics of a transfer orbit from Earth to Mars using a Hohmann transfer and using
a hyperbolic transfer orbit of eccentricity e = 2. The spacecraft is originally on a circular
orbit around the Earth of altitude 400 km and the spacecraft has to get to a circular orbit
around Mars of altitude 800 km.
From Table 4.1, we can see that the velocity change required in order to go from
Earth parking orbit to Mars parking orbit is much lower for the Hohmann transfer orbit
than for the hyperbolic transfer orbit. Therefore, the cost needed for a Hohmann transfer
orbit is less than that for a hyperbolic transfer orbit. However, the time of flight required
42
to go from one parking orbit to another is much higher for the Hohmann transfer orbit
than for the hyperbolic transfer orbit. This result is logical because a spacecraft in a
hyperbolic transfer orbit has more energy and so, more velocity than a spacecraft in a
Hohmann transfer orbit. As shown in chapter 3, the transfer orbit generally used is the
Hohmann transfer. This orbit it used because the Hohmann transfer costs less than other
sorts of transfer orbits. However, depending on the urgency of a mission, another
transfer orbit could be used.
Table 4.1: Characteristics of a Hohmann transfer orbit and a hyperbolic transfer orbit.
Semi-major axis, a
Transfer orbit energy, En
Velocity at transfer orbit
perigee, V1
Velocity at transfer orbit
arrival, V2
Velocity change from
Earth orbit to transfer
orbit, V1
Velocity change from
transfer orbit to Mars orbit,
V2
Escape orbit velocity from
the parking orbit around
Earth, Vbo
Escape orbit velocity from
the parking orbit around
Mars Vbi
Hohmann transfer orbit
1.262 AU
1.89*108 km
-0.396 (AU/TU) 2
Hyperbolic transfer orbit
1 AU
1.49*108 km
0.5 (AU/TU) 2
1.099 AU/TU
32.7km/s
1.73 AU/TU
51.6 km/s
0.721 AU/TU
21.5 km/s
1.52 AU/TU
45.3 km/s
0.0989 AU/TU
2.95 km/s
0.732 AU/TU
21.8 km/s
0.0889 AU/TU
2.65 km/s
1.062 AU/TU
31.6 km/s
Velocity change to go from
parking orbit around Earth
to transfer orbit, Vbo
Velocity change to go from
transfer orbit around to
parking orbit around Mars,
Vbi
Absolute value of total
velocity change required,
V
Time of Flight, TOF
43
11.2 km/s
24.4 km/s
5.25 km/s
31.9 km/s
3.57 km/s
16.7 km/s
-2.05 km/s
-28.7 km/s
5.61 km/s
45.4 km/s
259 days
49.3 days
4.3 Guidance Navigation and Control
To demonstrate the process of choosing a GN&C system for a spacecraft, three
different mission types will be examined. First, a LEO imaging spacecraft that has strict
position accuracy requirements (better than 100 m). Second, a LEO communications
satellite that needs only a 3 km accuracy requirement. Finally, a GEO communications
spacecraft will be examined.
The first case involves a system that has strict position accuracy requirements.
This requirement eliminates ground tracking as a possible solution. There are two proven
technologies that can calculate orbital positions with better than 100 m accuracy. If a
non-autonomous solution is desired, TDRS can achieve 3 accuracies of close to 50 m.
If an autonomous solution is desired, GPS can produce 3 accuracies of as little as 15 m.
If the success of the mission relies heavily on orbit determination, then GPS would give
the best possible solution.
The second case also involves a LEO spacecraft, however a high degree of
position accuracy is not necessary. In this case ground tracking may be an option. If
existing ground station architecture is used, the major costs of ground tracking are
operationally based. However, as the cost of GPS decreases, ground tracking will
become a less favorable option.
In the final example, a GEO spacecraft is examined. Being at such a high altitude
places many constraints on the choice of GN&C architecture. In practice, GPS cannot be
used at such an altitude. There have been studies done on using the “spillover” of signals
from the opposite side of the Earth, but this technique is not proven. In general, ground
tracking will be the best choice. Ground tracking provides only 50 km accuracy at GEO,
44
however 50 km is a small error relative to the height of the orbit and requirements of the
mission.
4.4 Attitude Determination and Control
A proposed space mission is slated to provide global Earth coverage using a single
satellite. The satellite will be stationed in a 600-km sun-synchronous orbit. The ground
station is located in Blacksburg, Virginia at 37.23274o latitude and 80.42841o longitude.
The communications subsystem of the satellite requires a minimum elevation angle of
10o. Table 4.3 contains the other pertinent data describing the satellite's physical
characteristics.
Table 4.3: Physical characteristics of the satellite15
Characteristic
Value
Ix
80 kg m2
Iy
Iz
m/(CDA)
A
cps – cg
cac – cm
CD
q
D
60 kg m2
90 kg m2
100 kg/m2
4 m2
0.4 m
0.2 m
2.2
0.6
1 A m2
The worst-case disturbance torque is calculated from the given information. The actuators
are sized so that they allow the satellite to maintain a given attitude in the presence of the
disturbance torques. The attitude sensors are chosen to meet the constraints of the system
including the accuracy.
This example is limited to the effects of the Earth's gravity gradient, solar radiation,
magnetic field, and aerodynamic forces on the satellite. The maximum deviation of the
45
z-axis, , is assumed to be 45o to maximize the value of the gravity gradient disturbance
torque.
Tg 
3
I z  I y sin 2 
2R3
(4.2)
Equation 4.2 gives a gravity gradient torque of 5.28  10-5 N-m for this mission. The
magnitude of the torque due to solar radiation is given by Eq. (4.3). The value of the
solar radiation torque is calculated to be 1.17  10-5 N-m.
F

T   s A 1  q  cosi  c  cg 
sp  c s
 ps



(4.3)
The angle on incidence, i, is assumed to be 0o to maximize the value of cos(i) to
maximize Tsp. The symbol c represents the speed of light which is equal to 3.0  108 m/s.
Equation 4.4 is used to calculate the magnitude of the disturbance torque due to the
Earth's magnetic field.
Tm 
(4.4)
2 DM
R2
A value of 4.69  10 –5 N-m is calculated for Tm. The disturbance torque due to
aerodynamic forces is calculated using Eq. (4.5). The atmospheric density, , is found to
be 4.89  10-13 kg/m3. The velocity used is the circular orbital velocity at a radius equal
to R.


Tm  CD AV 2 c pa  cg 
(4.5)
The sizing of the attitude actuators is shown below using reaction wheels,
momentum wheels, and magnetic torque bars as examples. The gravity gradient torque is
the maximum disturbance torque, TD, for this actuator. A 10% margin is used for this
calculation. The necessary torque, given the 10% safety margin, is 5.8110-5 N-m. This
46
magnitude of torque is well below the capabilities of most reaction wheels. If a reaction
wheel is chosen to control the disturbance torques, it would be specified based upon
storage or slew requirements instead of disturbance rejection.
The angular momentum change, h, which a momentum wheel can enact on a
satellite, is defined by Eq. (4.6). The orbital period, P, for a 600 km orbit is 5800
seconds. A pointing accuracy, a, of 1 is assumed for this mission. Momentum wheels
can produce angular momentum changes of 0.4 to 400 N-m-s.
h
(4.6)
TP
4 a
The value for this mission is 4.39 N-m-s, and is at the low end of the available range.
Therefore, a momentum wheel is sufficient to counteract the maximum disturbance
torques.
The Earth's magnetic field can be used to control a spacecraft's attitude. A
spacecraft’s MTBs are used in conjunction with the Earth's magnetic field to apply torque
to a spacecraft in LEO. They are constrained to LEO orbit because the magnetic field
weakens with increasing altitude. A drawback of MTBs is that the moment created is
always perpendicular to the local magnetic field, thus limiting the versatility of this type
of attitude control actuator. The MTBs need to generate a magnetic dipole of 1.126 A-m2
to counteract the worst-case torque on this spacecraft. This torque requires a weak
actuator. An MTB of 4 to 10 A-m2 is sufficient to counteract the disturbance torques on
the satellite based on calculations done using Eq. (4.7).
D
(4.7)
T
B
47
Several different types of sensors may be used to determine the orientation of the
spacecraft including sun sensors, cameras, star trackers, and magnetometers. Normally,
the attitude determination sensors chosen for a particular spacecraft meet certain accuracy
constraints set by the mission. Sun sensors detect the intensity of sunlight and can
determine at what angle the sun is relative to the spacecraft. Sun sensors typically have
an accuracy of 0.005 to 3 degrees and use approximately 0 to 3 Watts of power. Cameras
take different images that are used to determine a spacecraft’s orientation. Star trackers
determine a spacecraft’s orientation relative to the stars with an accuracy of 0.0003 to
0.01 degrees and use 5 to 20 Watts of power.
Magnetometers are sensors that measure the magnitude and direction of the local
magnetic field. Attitude determination is accomplished by comparing the measured
magnetic field vector to an inertial magnetic field vector. The inertial magnetic field
vector is calculated using a mathematical model of the magnetic field. Magnetometers
are considered to give the most coarse attitude measurements.
4.5 Propulsion
4.5.1 Conventional propulsion
This example investigates the sizing of a propulsion system to transfer a 2000 kg
payload from LEO to GEO. Two options are considered, a low thrust electrical rocket
and a high thrust conventional chemical rocket.
For a low thrust transfer orbit from LEO to GEO the total ΔV needed is 4.59 km/s.
This velocity change was calculated assuming the LEO to have 400 km altitude and the
GEO to have 5.6 Earth radii altitude and using the equation18 v 
48
 
RLEO
1

RLEO 
RGEO

,


for a low thrust transfer. For a 2000 kg payload the total impulse needed is
approximately 9180 kg-km/s. Performing the transfer using a 1 N Xenon ion thruster
would take 106.25 days. Assuming an Isp of 3400 s and an efficiency of 0.90 the thruster
from DASA would add 1948 kg of propellant and power plant. It would also have to
generate 39 kW. Lower thrust levels can lower this number but would increase the
transfer time proportionally.
For a chemical rocket a Hohmann transfer is used to move from LEO to GEO,
which gives a total ΔV of 3.85 km/s. If we assume a structural coefficient of 0.1 and an
Isp of 235 s, like the MONARC series, the payload ratio is only 0.108. Using these values,
the spacecraft will have a mass of 18,472 kg before leaving LEO. Using the MONARC
445 will require multiple engines to actually achieve a Hohmann orbit with only two
burns, so the structural coefficient is probably optimistic. However, the time from LEO to
GEO is only 5.28 hours.
4.5.2 Electrodynamic tether propulsion
The thrust achievable by electrodynamic propulsion is current limited. The
current is limited by the electron gathering capabilities of the anode. Currently the best
prospect for high current tether operation is the “bare wire” anode. It is simply a length of
conducting tether without insulation. This configuration places it in the optimum orbital
motion limited (OML) regime. The current that can be collected is approximately as
follows6,
I OML  J th  2Rc Lc 
49
4eVa
kTe
(1)
where Jth is the thermal current density, Rc and Lc are the geometry of the bare portion of
the tether, e, is the electronic charge, Va, is the anodic potential and k and Te are Boltzman
constant and the temperature of the electron gas, respectively. The thermal current
density is defined as,
J th  eN  kTe 2 me
(2)
where N∞, and me are the number density in the plasma and mass, of the electron,
respectively. There are restrictions to the geometry in order to achieve the OML regime6.
The geomagnetic field and ionospheric conditions can be found from web pages
of NASA's National Space Science Data Center16. Some other useful constants are listed
in the following table12.
Table 4.4: Electrodynamic tether calculation constants
Constant
e
me
k
Value
1.6022×10-19 C
0.511 MeV
9.1094×10-31 kg
8.6174×10-5 eV/K
1.3807×10-23 J/K
Using the previous example for transferring a 2000 kg payload from LEO to GEO the
total ΔV necessary is again 4.59 km/s for a low thrust transfer. This only approximate
since the relation used assumes a spiral, circular orbit. It is assumed that instead of a
circular orbit the final orbit is elliptical with the thruster only operating around periapsis.
This is of course because the earth’s magnetic field drops off as 1/r2. The calculation of
the time for transfer will be much more involved and require a numerical analysis. Still
we can determine the approximate input power levels needed. This time we’ll use a 10 N
thrust, 50 km long, 3 mm radius electrodynamic tether for propulsion. The 10 km closest
50
to the earth are assumed to be bare. Also, rather than integrating over the length of the
tether, it is assumed that 44 km are conducting the full current. The final assumption is a
400 km altitude during the thrusting portion of the orbit, which is equatorial. The
component of the magnetic field normal to a gravity gradient stabilized tether is 24167.5
nT. The electron number density and temperature for the same part of the same orbit are
752281 1/cm3 and 938 K respectively.
From the magnetic field, we can solve for the necessary current using Eq. (2.52).
The current necessary for 10 N of thrust is 9.404 A. Using the OML equation for current
the anode bias is solved for. Va equals 4.8 V assuming the entire collecting length is used.
This is a measure of how efficient a plasma contactor the bare wire system is. The next
step is to calculate the voltage induced along the length of the tether by the orbital motion
through the geomagnetic field. Faraday’s law along with the definition of voltage or
potential gives,
  
VF  v  B  L .
(3)
The voltage will vary with the orbital speed. For this scenario the speed varies from 7.67
km/s at the beginning of the transfer to 10.85 km/s at the end. These correspond to
voltages of 9,268 V for the former to 13,110 V for the latter. Another significant voltage
drop will come from the resistance of the tether. Using relatively pure aluminum with
resistivity ρ equal to 2.65×10-8 Ω∙m we can calculate the resistance of the tether5. The
resistance for the tether is 46.86 Ω. Assuming the entire current passes through the entire
length the voltage drop associated with this resistance is 441 V. Another resistance in the
system is that of the plasma return path, which is neglected in this example. So the
average power needed can now be calculated. The average power was found by
51
averaging the orbital velocity induced voltage drops and then adding the rest to obtain an
average power of 109.4 kW. This probably an optimistic estimate as the system was
design assuming periapsis was on the daylight side of Earth.
4.5.3
Momentum exchange tether
The following is an example of the calculations involved in a momentum
exchange tether boost. Table 4.5 contains the given data required for the calculations.
Table 4.5: Values for momentum exchange tether launch numerical example
Variable
rc
L

mps
mtlf
Value
6778 km
100 km
0.1 rad/s
200 kg
2000 kg
The results of the calculations are:
l2 
mtlf L
(mtlf  m ps )
v cm 

rcm


2000kg  100km
 90.91km
(2000kg  200kg)
3.986  10 5 km3 / s 2
 7.669km / s
6778km
v ps  6778km  100km
3.986  10 5 km3 / s 2
6778km3
 0.1rad / s  100km  17.78km / s
v  v ps  v cm  17.78km / s  7.669km / s  10.11km / s
The payload spacecraft receives a 10.11 km/s increase in its velocity due to the
momentum exchange tether launch.
52
4.6 Summary
This chapter 4 presents applications of the methods and example components
presented in previous chapters. These applications are the type of analysis that is
performed when choosing spacecraft components and operations to fulfill flight dynamics
requirements. The next chapter will present some conclusions about this subsystem in
general.
53
Chapter 5: Conclusion
The information in this paper details the important aspects of the flight dynamics
disciplines for a spacecraft. In this chapter, each of the subsystems are summarized and
recommendations are made to help design an optimal flight dynamics system.
5.1 Mission analysis and astrodyanmics
Mission analysis and astrodynamics involve the orbits in which a spacecraft
travels. In order for a spacecraft to go from one orbit to another, it has to get into a
transfer orbit. Four types of transfer orbits are available. The hyperbolic transfer orbit
has the lowest time of flight, but requires a higher change of velocity. A type I transfer
orbit is an elliptic orbit that carries the spacecraft less than 180 around the sun. Type I
orbits require a smaller velocity change than a hyperbolic orbit. A type II transfer orbit
carries the spacecraft more than 180 around the sun. Type II orbits require a smaller V
than a type I transfer orbit. However, the time of flight is longer than the type I orbit.
The Hohmann transfer orbit is an elliptic orbit that starts at the periapsis of the initial
orbit and of the transfer orbit and finishes at the apoapsis of the final orbit and of the
transfer orbit. A Hohmann transfer requires the least amount of V of any transfer orbit.
Previous space missions such as Voyager 2 and Mars Global surveyor have used
type II transfer orbits because, if the change of velocity required to get into the transfer
orbit is small, the cost of the mission decreases. The Mars pathfinder mission used a type
I transfer orbit that required a V of velocity than the type II transfer orbit but the time of
flight was less. The choice for the transfer orbit is therefore dependant on the kind of
mission. If time is not important, a type II transfer orbit is preferred to minimize
54
propellant costs. If the time is more important that the cost, a type I transfer orbit is used
to minimize the time of flight.
5.2 Guidance navigation and control
The GN&C subsystem determines the orbit of the spacecraft and generates
commands to maintain or alter that orbit. This can be accomplished through the use of
autonomous or non-autonomous systems. The major advantage of using an autonomous
system is that it removes additional architecture and operations costs required with
ground-controlled systems. In addition, autonomous systems such as GPS can provide
position data accurate to 15 m. This makes GPS far superior to ground tracking position
determination.
When choosing a guidance system for a spacecraft it is important to identify the
mission requirements and resources. A spacecraft that requires high position accuracy
will benefit from an autonomous GPS system. However, a GEO satellite will not have
access to the GPS constellation, making ground tracking a better alternative. Choosing a
control system depends on the capabilities of a satellite’s computer system. Many
modern spacecraft have sufficient computing power to integrate a fully autonomous orbit
control system. However, for a particular mission, greater control by the ground may be
desired in which case fully autonomous control may not be warranted. Balancing
requirements with mission resources is the most important aspect of choosing a GN&C
system.
5.3 Attitude dynamics and control
The ADCS provides a spacecraft with the ability to orient itself in space. Attitude
determination and attitude control are required for successful space systems. Attitude
55
determination is completed through the use of sensors (magnetometers, cameras, star
trackers, rate gyros, sun and horizon sensors, etc.) that supply data to an algorithm that in
turn determines the attitude. A control algorithm calculates the difference in the desired
and current attitude and computes the necessary torque. Attitude actuators (momentum
and reactions wheels, magnetic torquers, thrusters, etc.) are then commanded to apply the
proper torque on the spacecraft to reach the desired attitude.
An important issue not discussed in this report is the effect of flexible
appendages, solar panels or a tether for example, on the rotational dynamics of the
spacecraft. This issue should be studied to ensure that these effects could be determined.
Also, further study into the implementation of more advanced and efficient algorithms for
attitude determination may be required if greater accuracy is desired. This report only
dealt with the sizing of the components for an attitude control system and not in their
other dynamic characteristics, these should be looked into further.
The ADCS interacts greatly with several other subsystems, including power,
communications, propulsion, and guidance navigation and control. The design of an
ADCS is therefore greatly dependent on the properties of the other subsystems. It is
important to ensure the interactions are carefully studied and any problems resolved. The
type of ADCS, such as spin stabilized, gravity gradient stabilized, or three-axis
stabilization, should be the first decision made. The components can then be determined
based on the requirement enforce by the other subsystems of the spacecraft.
5.4 Propulsion
For propulsion the tradeoff is generally between efficiency and thrust. An
electrostatic rocket will use very little propellant but can provide only a small amount of
56
thrust. A chemical rocket can produce large thrusts but uses exponentially more
propellant. Another way of expressing this is that chemical systems are energy limited
where as electrodynamic methods are power limited. Chemical rockets are fully mature
and no significant advances can be expected from them. Electrical rockets on the other
hand should increase in performance with power generation technology. Within electrical
rocket technology itself there are also promising avenues for improvement.
Electrodynamic tethers are very promising for their minimal to no propellant usage. The
downside is they can only be used near a body with a magnetic field of sufficient
strength.
5.5 Summary
The previous chapters detail information about flight dynamics theory. It helps the
reader understand how a spacecraft orients itself and maneuvers accurately in space. The
modeling chapter demonstrates how to analyze and design the aspects of the flight
dynamics system. Examples in this paper illustrate the important differences between
alternative components and the methods used to choose and size them. These models and
examples are useful in designing the flight dynamics subsystems for a space mission.
57
References
1. Astro-Iki Hompage. 10/29/01. http://wildcat.iki.rssi.ru/ASTRO_1_E.htm
2. Atlantic Research Corporation: Space Products 10/29/01
http://www.atlanticresearchcorp.com/docs/space.shtml
3. Bate, Roger R., Mueller, Donald D. and White, Jerry E. Fundamentals of
Astrodynamics, Dover Publications, Inc., New York, 1971.
4. Billingsley Magnetics. "Feature and Specification Comparison: TFM100S
v.TFM100G2." 10/29/01. http://www.magnetometer.com/attmag.htm
5. Electrical Resistivity of Materials. 11/15/01
http://www.ai.mit.edu/people/tk/tks/rcon.html.
6. Estes, Robert D, et al. Bare T ethers for Electrodynamic Spacecraft Propulsion.
Journal of Spacecraft and Rockets. Vol. 37 No. 2, March-April 2000.
7. Haas, Lin. Orbit Determination Part 1. Orbital Sciences Corporation ACS InHouse Training Series. Germantown, MD April 1999.
8. Hall, Christopher. Spacecraft Attitude Determination and Control: Class Notes.
1/15/01. Blacksburg, VA
http://www.aoe.vt.edu/~chall/courses/aoe4984/page4.html
9. Hill, Pillip G. and Peterson, Carl R. Mechanics and Thermodynamics of
Propulsion. Addison-Wesley Publishing Company, Inc., New York, 1992.
10. Honeywell Corporation. “Commercial Electromagnetic Products: Specification
Sheets.”
http://content.honeywell.com/Space/CEM/RWApages/RWAproduct_spec_sheets.
htm
11. J. R. Wertz, editor. Spacecraft Attitude Determination and Control. D. Reidel,
Dordrecht, Holland, 1978.
12. Liboff, Richrad L., Introductory Quantum Mechanics: Third Edition. 1998,
Addison-Wesley. Reading, Mass.
13. Microcosm. “Low-Cost Spacecraft Magnetic Torquers.” 10/28/01.
http://www.smad.com/analysis/torquers.html
14. Micro-project: DIPSTICS Orbit Design. AOE 4065 Blacksburg, VA Fall 2001.
15. Mini Project: ALL-STARS ADCS Design. AOE 4065 Blacksburg, VA. VA
Tech Fall 2001.
16. Space Physics models at National Space Science Data Center. 11/15/01
http://nssdc.gsfc.nasa.gov/space/model/models_home.html .
17. Tethers Unlimited: home page 10/29/01
http://www.atlanticresearchcorp.com/docs/space.shtml
18. Wang, Joseph. Spacecraft Propulsion: Class Notes. Blacksburg, VA 1/15/01.
19. Welch, G. and Bishop, G. An Introduction to the Kalman Filter. 2/8/01.
http://www.cs.unc.edu/~welch/kalman/kalman_filter/kalman.html
20. Wertz, James R. and Larson, Wiley J., editors Space Mission Analysis and
Design. Microcosm Press, El Segundo, California, 1999.
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