Astrodynamics and Mission Geometry:
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
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
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
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
List of Abbreviations
|AAR |Area access rate |
|ADCS |Attitude determination and control system |
|ARC |Atlantic research corporation |
|CDM |Chief decision maker |
|GEO |Geosynchronous Earth orbit |
|GLONASS |Global navigation satellite system |
|GN&C |Guidance navigation and control |
|GPS |Global positioning system |
|IAA |Instantaneous access area |
|INU |Inertial navigation unit |
|LEO |Low Earth orbit |
|MOE |Measure of effectiveness |
|MTB |Magnetic torquer bars |
|QUEST |Quaternion estimation |
|TDRS |Tracking and data relay satellite |
|TOF |Time of flight |
|ADCS |Attitude determination and control system |
|GLONASS |Global navigation satellite system |
|GPS |Global positioning system |
|QUEST |Quaternion estimation |
|TDRS |Tracking and data relay satellite |
|TOF |Time of flight |
List of Symbols
|h |Angular momentum of momentum wheel |
|[pic] |Angular momentum vector |
|[pic] |Angular velocity vector |
|[pic] |Applied torque |
|A |Area |
|[pic] |Argument of perigee |
|(V |Change in velocity |
|j |Current density |
|ρ |Density of fluid |
|Tm |Disturbance torque |
|Cd |Drag coefficient |
|e |Eccentricity |
|( |Efficiency |
|[pic] |Euler parameter or quaternion |
|u |Fluid velocity |
|F |Force |
|µ |Gravitational constant |
|Tg |Gravity gradient disturbance torque |
|i |Inclination |
|( |Launch angle |
|D |Magnetic dipole |
|[pic] |Magnetic dipole vector |
|[pic] |Magnetic field vector |
|m |Mass |
|[pic] |Mass flow rate |
|[pic] |Mass moment of inertia tensor |
|n |Mean motion |
|[pic] |Optimal eigenvalue |
|( |Payload ratio |
|( |Phase Angle |
|p |Pressure |
|[pic] |QUEST algorithm measurement weight |
|r |Radius |
|ra |Radius of apogee |
|rp |Radius of perigee |
|γ |Ratio of specific heats |
|( |Right ascension of the ascending node |
|Rbi |Rotation matrix from inertial to body-fixed frame |
|a |Semi-major axis |
|Tsp |Solar radiation torque |
|[pic] |Specific heat |
|Isp |Specific impulse |
|( |Specific mass |
|ε |Structural coefficient |
|t |Time |
|[pic] |True anomaly |
|G |Universal gravitational constant |
|[pic] |Value of the error function |
|V |Volume |
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.
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.
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 necessary information to operators on the ground that would then command the spacecraft to adjust its orbit.
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.
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 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.
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.
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:
[pic] (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:
[pic] (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.
[pic]
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
• [pic]right ascension of the ascending node: the angle from the vernal equinox to the ascending node
• [pic]argument of perigee: the angle from the ascending node to the eccentricity vector measured in the direction of the satellite’s motion
• [pic]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
[pic]
Figure 2.2: Orbital Elements8
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.
[pic] (2.3)
The velocity at arrival point on orbit 2 is found using Eq. 2.4.
[pic] (2.4)
where (T is the gravitational constant of the focus of the orbits.
The semi-major axis of transfer orbit is as follows,
[pic] (2.5)
The launch angle at arrival point on orbit 2 can be found using Eq. 2.6.
[pic] (2.6)
The velocity change to go from transfer orbit 1 to orbit 2 at RT_final is found using Eq. 2.7.
[pic] (2.7)
The mean angular motion of orbit 1, 2 and transfer orbit is described as follows,
[pic] (2.8)
[pic] (2.9)
[pic] (2.10)
The time of flight on transfer orbit from orbit 1 to orbit 2 is described as follows,
[pic] (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.[pic] is the phase angle of planet 1 at t = 0 sec.
[pic] (2.12)
The phase angle of planet 2 is described as follows,
[pic] (2.13)
The phase angle of planet 1 at t1 is described as follows,
[pic] (2.14)
[pic] is the phase angle of planet 2 at t1.
[pic] (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 [pic]and the radius at burn out [pic](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, [pic], eccentricity, [pic], velocity at radius R: [pic]). [pic] is the hyperbolic excess velocity from planet 1.
[pic] (2.16)
The eccentricity of the hyperbolic orbit from planet 1 is,
[pic] (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.
[pic] (2.18)
The velocity of the satellite in the orbit around planet 1 at the burn out position is defined by Eq. 2.19.
[pic] (2.19)
The velocity change to enter the hyperbolic escape orbit from the orbit around the planet is described as,
[pic] (2.20)
The total velocity change to go from the parking orbit around planet 1 to transfer orbit is described as,
[pic] (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 [pic]and the radius at burn in [pic](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 semi-major axis, [pic], eccentricity, [pic], velocity at radius R: [pic]). [pic] is the hyperbolic excess velocity from planet 2.
[pic] (2.22)
The eccentricity of capture orbit can be found using Eq. 2.23.
[pic] (2.23)
where[pic]is the gravitational constant of planet 2.
The burn in velocity to enter from the capture orbit to the parking orbit is,
[pic] (2.24)
The velocity of the satellite in the orbit around planet 2 at the burn in position is,
[pic] (2.25)
The velocity change to enter the parking orbit from the capture orbit is,
[pic] (2.26)
The total velocity change from the transfer orbit to the parking orbit around planet 2 can be found using,
[pic] (2.27)
This is the general procedure used in Mission Analysis.
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 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 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 current attitude and then calculate and implement the necessary actions to exert the appropriate torque to achieve the desired attitude.
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 measurement vectors. One of the vectors is assumed to be exact1. If [pic] is the exact vector and [pic] is the second vector, the triads are set up in Eq (2.28-2.30).
|[pic] |(2.28a) |
|[pic] |(2.28b) |
|[pic] |(2.29a) |
|[pic] |(2.29b) |
|[pic] |(2.30a) |
|[pic] |(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.
|[pic] |(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
|[pic] |(2.32) |
The subscript k denotes one of the N measurements, and [pic] 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 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.
|[pic] |(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:
|[pic] |(2.34) |
Equation (2.34) relates the torque applied,[pic], to the time derivative of the angular momentum vector, [pic].3 From Eq. (2.34) it is possible to see how the time derivative of the angular velocity vector, [pic], varies with the applied torque. The angular velocity vector, as shown in Eq. (2.35), directly affects attitude.1
|[pic] |(2.35) |
This equation relates the angular velocity vector, [pic], to the time derivative of the quaternion, [pic].
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, [pic], of the momentum wheel affects the change in the angular velocity of the spacecraft, [pic], where [pic] is the spacecraft's moment of inertia.3
|[pic] |(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, [pic], is defined by Eq. (2.37). 3
|[pic] |(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.
|[pic] |(2.38) |
Equation (2.38) shows the simple equation governing applied torques from thrusters, where [pic]is the applied force and [pic]is the moment arm.
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,
[pic] (2.39)
where Fx = component of force in x direction
[pic] = mass rate of flow (positive out)
ρ = density of fluid
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.
[pic] (2.40)
The symbols [pic] , [pic] and, [pic] are the exit velocity, exit pressure and, ambient pressure respectively. The exit area of the nozzle is[pic]. From this equation we can define an equivalent velocity
[pic] (2.41)
such that [pic]. The main performance measure of a chemical rocket is the specific impulse defined as
[pic] (2.42)
The constant [pic] is the gravitational acceleration at earth’s surface. In space we can assume constant [pic] and the absence of drag or gravity. Using Equation 2.39 and these assumptions we determine
[pic] (2.43)
where [pic] is the initial mass and [pic] is the burnout mass. Furthermore we can solve for the payload ratio defined as [pic] where [pic] is the payload mass
[pic] (2.44)
where ε is the structural coefficient [pic] where [pic] is the structural mass and [pic] is the propellant mass. The exit velocity can be determined from thermo-chemistry and is in general
[pic] (2.45)
where [pic], [pic], [pic] 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
[pic] (2.46)
[pic][pic] (2.47)
Here [pic]is the mass of the power plant and P is the power output. Isp is also defined for electric rockets as
[pic] (2.48)
The mass flow rate is assumed constant over a time [pic] (the burning time). Then by summing the masses and using Equations 2.46, 2.47 and, 2.43 we can obtain
[pic] (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.
[pic]
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.
[pic] (2.50)
[pic] (2.51)
In Equations 2.50 and 2.51 [pic] is the permitivity of free space and [pic], 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
[pic] (2.52)
To maximize the thrust, the[pic]and [pic] 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 [pic] is more difficult than [pic] 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 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),
|[pic] |(1) |
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.
|[pic] |(2) |
The velocity of the payload spacecraft, vps, can be calculated using Eq. (3). The symbol ( represents the angular velocity of the system.
|[pic] |(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.
|[pic] |(4) |
These equations can be used as a first approximation of the added velocity a payload spacecraft could receive using a momentum exchange tether.
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.
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.
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. 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.
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 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.
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.
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.
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.
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. Non-autonomous 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 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 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. |Output Torque (N-m) |Peak Power (W) |Power Holding Max. Speed |Type of Wheel |
| | | |(W) | |
|HR12, HR14, HR16 |0.1-0.2 |105 |22 |Reaction |
|HR0610 |0.075 |80 |15 |Reaction |
|HR2010, HR4510 |0.1 |115 |17 |Reaction |
|HR2020 |0.113 |175 |35 |Reaction |
|HR2030 |0.21 |190 |20 |Reaction |
|HR4820 |0.14 |165 |20 |Reaction |
|HR04 |0.028 |- |6 |Reaction |
|HM4520 |0.135 |150 |35 |Momentum |
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 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. |Dipole Moment (A-m2) |Power (W) |Mass (kg) |
|MT-30-2-CGS |30 |3.6 |1.135 |
|MT-80-1 |80 |3 |4.12 |
|MT-140-2 |145 |1.9 |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 |Power Consumption (W) |Mass (kg) |
|Star Tracker |2" |10 |3.1 |
|Horizon Tracker |10" |10 |2.5 |
|Sun Tracker | | | |
|SD-1 |10' |0.25 |0.2 |
|SD-2 |1" |5 |0.6 |
|SD-3 |1' |5 |0.7 |
|Gyroscope DNG-091 |0.02'/impulse |- |- |
|Accelerometer MA-1 |0.005 m/s/impulse |- |- |
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.
Table 3.4: Billingsley Magnetics space-rated magnetometers13
|Model Number |Accuracy |Power Consumption (W) |Mass (kg) |
|TFM1005 |(0.5% of full scale |0.56 |0.2 |
|TFM100G2 |(0.75% of full scale |1.02 |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 Range |7-28 bar |5-30 bar |3.5-28 bar |5-31 bar |5-31 bar |
|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 Impulse Bit |11 mN-s |90 mN-s |312 mN-s |891 mN-s |8910 mN-s |
|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 Diameter |2.5 cm |2.5 cm |3.8 cm |8.4 cm |14.8 cm |
|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 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 |Resistojet |Arcjet |Pulsed Plasma Thruster (PPT) |
|Supplier |Primex |Primex, TRW |IRS/ITT |Primex |Primex |TRW, Primex, |JHU/APL |Primex, TSNIMASH, |
| | | | | | |CTA | |NASA |
|Specific |296 |299 |480 |502 |> 580 |800 |847 |1200 |
|Impulse, | | | | | | | | |
|(sec) | | | | | | | | |
|Input Power, |0.5 |0.9 |0.85 |1.8 |2.17 |26* |< 0.03 |< 0.02 |
|(kW) | | | | | | | | |
|Thrust/ |743 |905 |135 |138 |113 |-- |20.8 |16.1 |
|Power, | | | | | | | | |
|(mN/kW) | | | | | | | | |
|Specific |1.6 |1 |3.5 |3.1 |2.5 |-- |195 |85 |
|Mass, (kg/kW)| | | | | | | | |
|Propellant |N2H4 |N2H4 |NH3 |N2H4 |N2H4 |NH3 |Teflon |Teflon |
Table 3.6b: Characteristics of selected electric propulsion flight systems (cont)20
|Concept |Hall Effect Thruster (HET) |Ion Thruster (IT) |
|Supplier |IST, Loral, |TSNIMASH, NASA |SPI, KeRC |HAC |MELCO, |MMS |HAC, NASA |DASA |
| |Fakel | | | |Toshiba | | | |
|Specific |1600 |1638 |2042 |2585 |2906 |3250 |3280 |3400 |
|Impulse, | | | | | | | | |
|(sec) | | | | | | | | |
|Input Power, |1.5 |1.4* |4.5 |0.5 |0.74 |0.6 |2.5 |0.6 |
|(kW) | | | | | | | | |
|Thrust/ |55 |-- |54.3 |35.6 |37.3 |30 |41 |25.6 |
|Power, | | | | | | | | |
|(mN/kW) | | | | | | | | |
|Specific |7 |-- |6 |23.6 |22 |25 |9.1 |23.7 |
|Mass, (kg/kW)| | | | | | | | |
|Propellant |Xenon |Xenon |Xenon |Xenon |Xenon |Xenon |Xenon |Xenon |
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.
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 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.
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 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.
| |Hohmann transfer orbit |Hyperbolic transfer orbit |
|Semi-major axis, a |1.262 AU |1.89*108 km |1 AU |1.49*108 km |
|Transfer orbit energy, En |-0.396 (AU/TU) 2 | |0.5 (AU/TU) 2 | |
|Velocity at transfer orbit perigee,|1.099 AU/TU |32.7km/s |1.73 AU/TU |51.6 km/s |
|V1 | | | | |
|Velocity at transfer orbit arrival,|0.721 AU/TU |21.5 km/s |1.52 AU/TU |45.3 km/s |
|V2 | | | | |
|Velocity change from Earth orbit to|0.0989 AU/TU |2.95 km/s |0.732 AU/TU |21.8 km/s |
|transfer orbit, (V1 | | | | |
|Velocity change from transfer orbit|0.0889 AU/TU |2.65 km/s |1.062 AU/TU |31.6 km/s |
|to Mars orbit, (V2 | | | | |
|Escape orbit velocity from the | |11.2 km/s | |24.4 km/s |
|parking orbit around Earth, Vbo | | | | |
|Escape orbit velocity from the | |5.25 km/s | |31.9 km/s |
|parking orbit around Mars Vbi | | | | |
|Velocity change to go from parking | |3.57 km/s | |16.7 km/s |
|orbit around Earth to transfer | | | | |
|orbit, (Vbo | | | | |
|Velocity change to go from transfer| |-2.05 km/s | |-28.7 km/s |
|orbit around to parking orbit | | | | |
|around Mars, (Vbi | | | | |
|Absolute value of total velocity | |5.61 km/s | |45.4 km/s |
|change required, (V | | | | |
|Time of Flight, TOF | |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 3σ accuracies of close to 50 m. If an autonomous solution is desired, GPS can produce 3σ 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, 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 |60 kg m2 |
|Iz |90 kg m2 |
|m/(CDA) |100 kg/m2 |
|A |4 m2 |
|cps – cg |0.4 m |
|cac – cm |0.2 m |
|CD |2.2 |
|q |0.6 |
|D |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 z-axis, (, is assumed to be 45o to maximize the value of the gravity gradient disturbance torque.
|[pic] |(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.
|[pic] |(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.
|[pic] |(4.4) |
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.
|[pic] |(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 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.
|[pic] |(4.6) |
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).
|[pic] |(4.7) |
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 [pic], 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.
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,
|[pic] |(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,
|[pic] |(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 |Value |
|e |1.6022×10-19 C |
|me |0.511 MeV |
| |9.1094×10-31 kg |
|k |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 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,
|[pic]. |(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 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.
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 |Value |
|rc |6778 km |
|L |100 km |
|( |0.1 rad/s |
|mps |200 kg |
|mtlf |2000 kg |
The results of the calculations are:
[pic]
The payload spacecraft receives a 10.11 km/s increase in its velocity due to the momentum exchange tether launch.
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.
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.
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 ΔV 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 ΔV 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 ΔV 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 propellant costs. If the time is more important that the cost, a type I transfer orbit is used to minimize the time of flight.
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.
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 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.
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 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 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.
References
1. Astro-Iki Hompage. 10/29/01.
2. Atlantic Research Corporation: Space Products 10/29/01
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.
5. Electrical Resistivity of Materials. 11/15/01 .
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 In-House Training Series. Germantown, MD April 1999.
8. Hall, Christopher. Spacecraft Attitude Determination and Control: Class Notes. 1/15/01. Blacksburg, VA
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.”
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.
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 .
17. Tethers Unlimited: home page 10/29/01
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.
20. Wertz, James R. and Larson, Wiley J., editors Space Mission Analysis and Design. Microcosm Press, El Segundo, California, 1999.
-----------------------
Perigee
Apogee
ra
rp
V
r
Launch
• Specify initial orbit
ADCS
• Position for ADCS models
• Station keeping maneuvers
Mission ops
• Orbit adjustments
• Orbit transfers
Power
• GN&C load
• Solar array positioning
Structure
• Acceleration constraints on DV
Communications
• GS Pass times
Propulsion
• Specify (V
• Thruster firing time
GN&C
ADCS
Thermal
• Thermal maneuvers
Propulsion
• ADCS thruster size
• Thruster pointing
Communications
• Antenna pointing
Structure
• Center of mass
• Moments of inertia
• Sensor placement
• Actuator placement
Power
• ADCS load
Power
• Solar array pointing
Mission ops
• Sensor pointing
• Special maneuvers
GN&C
• Position for ADCS models
• Station keeping maneuvers
Thermal
• Generates heat
ADCS
• ADCS thruster size
• Thruster pointing
Mission Ops
• Orbit adjustments
• Orbit transfers
Power
• Propulsion load
• Electric thrusters?
• Electrodynamic tether
Structure
• Withstand loads
• Hold propellant
Communications
• Plume interference
GN&C
• Specify (V
• Thruster firing time
Propulsion
Thermal
• Generates heat
ADCS
• ADCS thruster size
• Thruster pointing
Communications
• Plume interference
GN&C
• Specify (V
• Thruster firing time
Power
• Propulsion load
• Electric thrusters?
• Electrodynamic tether
Structure
• Withstand loads
• Hold propellant
Mission Ops
• Orbit adjustments
• Orbit transfers
Propulsion
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