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Application of the fundamental equation to celestial mechanics and astrodynamics
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Application of the fundamental equation to celestial mechanics and astrodynamics
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APPLICATION OF THE FUNDAMENTAL EQUATION TO
CELESTIAL MECHANICS AND ASTRODYNAMICS
by
Darren David Garber
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ASTRONAUTICAL ENGINEERING)
May 2012
Copyright 2012 Darren David Garber
ii
DEDICATION
This dissertation is dedicated to:
My beautiful and patient wife, Carlie, who has provided me with nothing but
love, constant support, perspective and the ability to focus on this endeavor for us
My wonderful children Samuel and Elizabeth, and now having the time and
ability to see them achieve all of their hopes and dreams
My family for always believing in me and instilling in me the belief that dreams
can come true
And lastly to my friends for always listening to me and challenging me, especially
when I was completely wrong
iii
ACKNOWLEDGEMENTS
I would like to express my sincerest thanks to my advisor Firdaus E. Udwadia for
his time, guidance and mentorship. It has truly been an honor to work with him and I look
forward to future research efforts and investigations with him.
iv
TABLE OF CONTENTS
DEDICATION ii
ACKNOWLEDGEMENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
ABBREVIATIONS viii
ABSTRACT ix
CHAPTER 1: INTRODUCTION 1
CHAPTER 2: PROBLEM STATEMENT 2
CHAPTER 3: CURVATURE AND THE
FUNDAMENTAL EQUATION 4
CHAPTER 4: APPLICATION TO CELESTIAL MECHANICS
AND ASTRODYNAMICS 17
CHAPTER 5: INTEGRATED MISSION EXAMPLES 32
CHAPTER 6: SUMMARY 47
REFERENCES 48
v
LIST OF TABLES
Table 1: Standard Orbital Elements ..................................................................... 1
Table 2: Curves and Curvature........................................................................... 14
Table 3: Precessing Orbit Initial Conditions and Equations of Motion ............... 18
Table 4: Comparison of Previous Approaches with Curvature Technique .......... 22
Table 5: MoonSail Initial Conditions and Equations of Motion .......................... 25
Table 6: Generalized Transfer Equation Example .............................................. 29
Table 7: ED Tether Initial Conditions ................................................................ 34
Table 8: RPO Initial Conditions ......................................................................... 41
Table 9: Cluster Initial Conditions ..................................................................... 45
vi
LIST OF FIGURES
Figure 1: From Conic Sections to Perturbed Motion ....................................................... 2
Figure 2: Addressing Non-Keplerian Motion .................................................................. 4
Figure 3: Precessing Orbit Simulation ........................................................................... 17
Figure 4: Newton’s Letter to Hooke on Precessing Orbits ............................................. 18
Figure 5: Control Force for Precessing Orbits Example ................................................ 19
Figure 6: Control Force Components for Precessing Orbits ........................................... 19
Figure 7: Outward Spiral From GEO to the Moon at 1 Year ......................................... 22
Figure 8: MoonSail Approach and Insertion.................................................................. 23
Figure 9: MoonSail Approach, Insertion & 48 Hours In Lunar Orbit ............................. 24
Figure 10: MoonSail Control Force Magnitude and Lunar Altitude ............................... 26
Figure 11: MoonSail Control Force Components .......................................................... 26
Figure 12: Hohmann Transfer Schematic ...................................................................... 28
Figure 13: Curvature Plot LEO to GEO ........................................................................ 30
Figure 14: Radius of Curvature Plot LEO to GEO ........................................................ 31
Figure 15: ED Tether Curvature Profile ........................................................................ 32
Figure 16: ED Tether Orbital Geometry ........................................................................ 33
Figure 17: ED Tether Control Force Components ......................................................... 35
Figure 18: Horseshoe Trajectory Overview ................................................................... 36
Figure 19: Horseshoe Curvature Profile ........................................................................ 37
Figure 20: Horseshoe Control Force Components ......................................................... 38
Figure 21: Resonant 2:1 Orbit ....................................................................................... 38
vii
Figure 22: Resonant 2:1 Curvature Profile .................................................................... 39
Figure 23: Resonant 2:1 Control Force Components ..................................................... 39
Figure 24: Hill Frame and RPO Profile ......................................................................... 40
Figure 25: RPO Curvature Profile ................................................................................. 41
Figure 26: RPO Control Force Components .................................................................. 42
Figure 27: GEO Sparse Array ....................................................................................... 43
Figure 28: Cluster Geometry & Orientation .................................................................. 44
Figure 29: Cluster Formation Geometry ........................................................................ 45
Figure 30: Cluster Control Force Components & Magnitude ........................................ 46
viii
ABBREVIATIONS
ED Electro-Dynamic
GEO Geosynchronous Orbit
GPS Global Positioning System
GTE Generalized Transfer Equation
LCO Latitude Controlled Orbit
LEO Low Earth Orbit
UAV Unmanned Aerial Vehicle
ix
ABSTRACT
This paper proposes a new general approach for describing, generating and
controlling the trajectory of an object by combining recent advances in analytical
dynamics with the underlying theorems and concepts from differential geometry. By
using the geometric construct of curvature to define an object’s motion and applying the
fundamental equation of constrained dynamics, the resulting solutions are both explicit
and exact for the minimum acceleration necessary to maintain the specified trajectory.
The equations detailing the control force required to follow the selected trajectory can be
expressed in closed form, regardless if the object is in Keplerian free-flight about a single
central body or following a non-Keplerian trajectory in a highly disturbed environment.
Examples are provided in both the inertial and non-inertial frames to demonstrate the
utility of this combined approach for solving common problems in both celestial
mechanics and astrodynamics. The practical aspects of exploiting curvature for maneuver
and mission planning is also investigated resulting in the formulation of the Generalized
Transfer Equation which extends the method of patched conics to include any curve.
CHAPTER 1: INTRODUCTION
All orbital motion is inherently non-Keplerian. The inclusion of any disturbance
either natural or artificial on an orbiting object results in a deviation from the constancy
of the Keplerian elements used to describe the motion. As the magnitude of the
perturbation increases, the utility and intuitiveness of simple conic sections as defined by
eccentricity decreases. Numerous techniques and ancillary parameters have been
developed to account for natural and artificial perturbations. Examples of these methods
include Gauss’ Variational Equations, descri bing the orbit with respect to equinoctial
elements and applying the tenants of gauge theory. In each of these cases, the additional
terms are specific to a given perturbation, reference frame or simplifying assumption. The
myriad of unique elements and special reference frames that exist illustrates that this is
not a solved problem for either celestial mechanics or astrodynamics [1]. Table 1 lists a
representative set of orbital elements in common use.
Table 1: Standard Orbital Elements
In this paper, a general method is presented to fully and succinctly detail the
perturbed motion of an object through the curvature of its trajectory and application of
the fundamental equation of analytical dynamics. This new approach provides the
common capabilities to define, describe and generate perturbed orbits for celestial
1
2
mechanics, while enabling the precision control of a maneuvering spacecraft necessitated
by astrodynamics.
This paper is organized into five sections. Section II presents the elementary
trajectory problem for perturbed motion that is unsatisfied by existing techniques. Section
III provides an overview of curvature and the fundamental equation of analytical
dynamics. In Section IV, an overview of examples with simulation results are detailed
and their relevancy for celestial mechanics and astrodynamics is summarized. Section V
details missions specifically enabled by this integrated approach utilizing both curvature
and the fundamental equation. Section VI concludes with a summary of the overall effort.
CHAPTER 2: PROBLEM STATEMENT
The problem examined by this paper is how to accurately and succinctly define,
generate, and control both Keplerian and non-Keplerian motion. The difference between
Keplerian and non-Keplerian motion is the duration over which the perturbing
acceleration is applied. Instantaneous accelerations are accommodated by two body
motion. Applying
an instantaneous
acceleration to an
object following a
Keplerian orbit,
results in the orbit
changing to a different conic section as a function of the magnitude and direction of the
Figure 1: From Conic Sections to Perturbed Motion
3
perturbing force. Exploiting this ability to transition from one conic-based orbit to
another under an instantaneous impulsive force is the basis of the trajectory modeling
technique known as patched conics. The patched conics technique allows for complex
trajectories to be approximated through a collection of arcs utilizing circles, ellipses,
parabolas and hyperbolas as segments of the overall trajectory.
As the duration of the perturbation increases, the trajectory rapidly deviates from
being modeled by a single conic section. Figure 1 illustrates this transition from two body
motion to non-Keplerian trajectories based upon the magnitude, direction and duration of
the perturbing acceleration. As depicted, the perturbing acceleration can have three gross
effects on the orbit. Retarding forces such as drag will result in the orbit spiraling inward.
Net accelerations in line with the velocity vector will generate an outward spiral, while
orthogonal accelerations will induce torques on the orbit plane causing either regression
or precession. For long duration, small magnitude perturbations, Gauss’ variatio nal
equations provide an approximate solution to the trajectory through time varying
osculating Keplerian parameters. But as the magnitude of the continuous acceleration
grows, the physical significance and utility of the time varying parameters also wanes.
Applying gauge theory to account for perturbations has produced simplified equations for
unique disturbances, but at the expense of osculation and the introduction of additional
parameters and convective terms (Φ) without physical significance [9].
The spiral orbit serves as a clear example for illustrating the short-comings of the
current techniques ability to model even the simplest non-Keplerian trajectory is
4
illustrated in Figure 2. The spiral trajectory is commonplace in celestial mechanics and
the hallmark of modern low thrust propulsion systems. Through the investigation of
spiral orbits, insight was gained to describe the motion of an object under acceleration. In
the same manner that conic sections provide the foundation for Keplerian motion, this
paper details how the geometric construct of curvature provides the physical parameters
required to fully describe non-Keplerian motion.
Figure 2: Addressing Non-Keplerian Motion: Generating osculating ellipses at each point (on left), invoking gauge
freedom (middle) and utilizing curvature to completely describe the curve (on right)
CHAPTER 3: CURVATURE AND THE FUNDAMENTAL EQUATION
Newton is credited with the first detailed definition and analysis of curvature [3]. In
the Principia, curvature is employed to provide an elegant solution to Kepler’s Second
Law. Despite its prevalence and demonstrated utility in the Principia, curvature has been
neglected by celestial mechanics and astrodynamics [18]. In contrast, curvature is utilized
by nearly every engineering discipline from road construction to genetics.
5
The reason for this neglect is understandable given the fact that two-body
assumptions have provided adequate solutions for over 300 years. The inertia behind this
eccentricity centric approach to orbital motion, slowly changed with the advent of
astrodynamics and modern low-thrust propulsion systems. Concurrently, advances in
observational astronomy and astrophysics have forced celestial mechanics to come to
terms with the non-Keplerian motion of natural bodies. In both cases, the exponential
increase in computing power allowed both astrodynamics and celestial mechanics to
forego a re-examination of how orbits are defined, for the more expedient method of
brute force numerically integrating the equations of motion. This computational crutch
has propagated the belief that only simple two body trajectories with at most finite
impulsive maneuvers can be solved via hand calculations, relegating all other solutions to
numerical integration. With curvature, additional trajectories can now be defined and
solved for with pencil and paper, providing both accurate solutions along with insight
into the overall mechanics of the system.
Curvature is a scalar mathematical construct which describes the rate of change of a
curve. Newton referred to curvature as the “crookedness” of a line [18]. All curves have
curvature. Circles have constant curvature, while straight lines are defined to have
identically zero curvature. Geometrically, given a point on a curve the value of curvature
at that point is equal to the reciprocal of the radius from the largest circle tangent to the
same point [8]. The osculating circle’s radius is also defined as the radius of curvature
and is inversely proportional to the rate of change of the curve. Curvature increases as the
rate of change of the curve increases and decreases as the rate of change decreases. To
6
define complex curves, curvature is customarily defined as a function with respect to arc
length, angle or time. These simple parameterized functions provide the ability to
generate any planar curve. Any unique curve can be generated if the initial coordinates
and the curvature are given [8]. Extending this geometric theory to orbital motion,
provides the ability to define the trajectory of an object with only its initial state and a
time history of its curvature.
There are multiple expressions for curvature. The most useful form for trajectory
generation and analysis is the vector formulation. The vector formulation clearly shows
how curvature describes the rate of change of the curve as a function of the velocity and
acceleration. Equation 1 defines velocity as a scalar component in the direction of the
velocity unit vector T.
(1)
Differentiating velocity with respect to time results in the total acceleration as
shown in Equation 2.
(2)
Equation 3 provides the relationship between the time rate of change of the velocity
unit vector T and the normal component of the acceleration as scaled by the velocity and
curvature.
(3)
7
Substituting equation 3 into equation 2, produces an expression for the acceleration
with respect to a component parallel to the motion of the object and a normal component
accounting for centripetal acceleration of the object via curvature.
(4)
Crossing the acceleration expression in equation 4 with the velocity relationship in
equation 1 provides the vector formulation of curvature.
(5)
By definition the cross product of the velocity unit vector with the normal vector
results in the binormal vector as shown in equation 6.
(6)
Substituting equation 6 into equation 5 produces the vector form which relates
curvature to velocity and acceleration as detailed in equation 7.
(7)
For this effort, it is assumed that all motion is planar, so the following relationships
hold:
(8)
8
For orbital motion restricted to a plane, the angular momentum vector is parallel to
the binormal vector, since the acceleration unit vector is equal to the negated unit position
vector as shown below:
(9)
(10)
(11)
Therefore in the planar case, the relationship between curvature, velocity,
acceleration and the angular momentum unit vector is given by equation 12.
(12)
This vector formulation clearly illustrates that the curvature directly relates the rate of
change of the orbit to the object’s velocity and acceleration about the angular momentum
vector. In its vector form, counter-clockwise rotation about the angular momentum vector
is considered positive curvature, while clockwise motion is negative curvature. Since
curvature is a scalar value, the vector formulation is usually simplified and the magnitude
is taken of both sides to remove the orthogonal unit vector to both velocity and
acceleration.
(13)
This seemingly innocuous simplification needlessly complicates the analysis of an
orbiting body by introducing a square root into the cross product between velocity and
9
acceleration. This standard scalar representation along with curvature being coupled to
several non-inertial frames (e.g. Frenet-Serret frame) has precluded previous attempts to
utilize curvature for orbital analysis [12], [19]. These earlier attempts tried to recast all of
celestial mechanics and astrodynamics into the frame of the object, similar to how
curvature is used to plan the route for UAVs and the motion of other autonomous
vehicles [24]. The difficulty in this approach is that unlike a UAV, where the primary
acceleration is the thrust vector coupled to the body frame, an orbiting object’s dominant
acceleration is gravity and is best expressed in the inertial frame.
This new approach utilizes equation 12 and since the frame is not specified, curvature
can be expressed and has physical significance in any reference frame. This frame
independence is the most significant break from eccentricity based orbital solutions,
where eccentricity (along with the other constant orbital elements) is only defined in the
inertial frame [1]. Eccentricity’s inertial dependence is determined by equation 14, where
the magnitude of the eccentricity vector is the constant value of eccentricity for the
Keplerian orbit.
(14)
Specific examples of this unique characteristic of curvature are reviewed in Section IV.
Even with frame independence and the ability to uniquely define the trajectory via a
scalar function, the overall utility of curvature over eccentricity is negligible. The
enabling feature of this curvature based approach is leveraging the recently proven
10
mathematics of the Fundamental Equation of Analytical Dynamics to generate, control
and correct the orbital motion. Again without this machinery, curvature is in many
respects no better than the other myriad of special parameters and techniques.
The utility of the Kalaba-Udwadia fundamental equation for trajectory analysis has
been exploited to investigate many challenging problems relating to orbital motion. From
tethered vehicle dynamics to the orbital control of a vehicle about an oblate planet,
numerous papers have utilized the fundamental equation to express the seemingly
impossibly complex motion in a simple and straight forward way [4], [5], [13], [14], [15],
[20], [21], [22], [23]. The resulting trajectory solutions based upon the Kalaba-Udwadia
equations have resulted in unique solutions for very specific problems. Examples of these
individual solutions include maintaining a perfectly circular orbit with a constant
inclination about an oblate planet in inertial space or the control law necessary for
maintaining a constant relative distance and orientation from a orbiting object in the
relative Hill’s frame [4] [ 14]. These rigid solutions further exemplify the continuing
problem described in Section I where celestial mechanics and astrodynamics continue to
perpetuate special techniques, parameters and reference frames without investigating a
common solution.
The fundamental equation is based upon Gauss’ principle and provides an analytical
method to solve constrained motion [24]. The resulting equations represent a closed form
solution for the exact force required to achieve the constraints [25]. Unlike similar
approaches which use Lagrangian multipliers, the technique developed by Kalaba and
11
Udwadia does not differentiate between holonomic and non-holonomic constraints, while
yielding a unique vector that globally minimizes the Gaussian via Gauss’principle of least
constraint [4]. By accommodating the path independence of holonomic constraints and
the time varying path dependent nature of non-holonomic constraints, the utility of this
approach provides the necessary flexibility to account for any perturbation directly [27].
The Kalaba-Udwadia equations require no linearizations or approximations on the
constraints or the environment [4]. Additionally, the Kalaba-Udwadia equations do not
require any transformations or normal forms, since the control force is expressed in the
same coordinates as the uncontrolled system [26]. This critical aspect of Kalaba-
Udwadia equations further complement curvature’s frame independence by truly
enabling a new and more general approach for investigating non-Keplerian motion.
Applying the fundamental equation is exceptionally straight forward. Unlike other
approaches, the fundamental equation requires only simple matrix multiplications and
additions [27]. The simplicity of the fundamental equation is illustrated in equation 15
and details that the force of constraint F
c
is explicitly given by [24] [27]:
(15)
where M is the mass matrix, A is the constraint matrix, b is a vector of scalars, a is the
unconstrained acceleration of the object and the + superscript denotes the Moore-Penrose
pseudoinverse. The Moore-Penrose pseudoinverse provides a least squares solution to the
system of equations as defined in equation 15 & 16. If the mass is unchanging the
Kalaba-Udwadia equation simplifies to [24]:
12
(16)
(17)
2-D: Planar Motion
To demonstrate the utility of curvature used in conjunction with the fundamental
equation the spiral trajectory problem outlined in Section II will be exercised. First, place
a single object in a Cartesian inertial frame of reference about a central gravitating body.
The unconstrained motion for this object is the well known inverse square law of
universal gravitation [1]:
(18)
where,
. (19)
To this unconstrained motion a set of constraints are applied to bound the motion. The
vector form of curvature is utilized as the primary constraint. The A matrix then contains
the velocity of the object, while the scalar curvature terms are the components of the b
matrix.
(20)
(21)
13
Taking the Moore-Penrose Pseudo-Inverse of A and then simplifying the Kalaba-
Udwadia equation, yields the closed form, explicit and exact solution for the minimum
acceleration necessary to be applied to an object’s unconstrained motion to achieve the
trajectory defined by the constraints in the absence of noise [27]. Investigations regarding
the insensitivity to the fundamental equation to noise sources and errors in the initial
conditions are ongoing.
For the planar problem presented in Section II, all that is required to express the
object’s motion from a circular orbit to a spiral i s changing the curvature term as
illustrated below:
(22)
(23)
(24)
(25)
The curvature for conic sections and other common curves is detailed in Table 2 [30].
14
Table 2: Curves and Curvature
Curve Type Curvature
Circle
R > 0
Ellipse
a ≥ b; b ≠ 0
Parabola
a > 0
Hyperbola
Archimedean Spiral
c > 0
Integer values of n generate the following curves:
Lituus -2
Hyperbolic Spiral -1
Archimedes’ Spiral 1
Fermat’s S piral 2
Logarithmic Spiral
15
This simple expression for the control acceleration, illustrates curvature’s orbital
shape-defining characteristic for non-Keplerian motion that is similar to that captured by
eccentricity when dealing with Keplerian orbits. Again, unlike eccentricity, curvature is
not restricted to defining only conic sections, but any curve.
Using the fundamental equation and curvature in this way is akin to designing the
on-ramp to a freeway. The path is determined, but the speed and rate at which the vehicle
can move along the path is unspecified. In the special case of a circular orbit, the velocity
is constant, but the true utility of this new general approach is the ability to capture both
Keplerian and non-Keplerian motion with the same equations. Key to accommodating
Keplerian motion, is the ability to move about an ellipse (or any conic section) while
maintaining equal areas in equal time. Kepler’s 2
nd
Law is a direct expression of the
conservation of angular momentum, which is fundamentally captured and preserved by
the Kalaba-Udwadia equation.
For specific maneuver planning problems in astrodynamics, where the thrust of
the vehicle is known and an input to the overall system, the inclusion of a second
constraint to provide a clear and intuitive mechanism for explicitly manipulating the
magnitude of the angular momentum vector. Where the first constraint of curvature
defined the overall path of the object, the second constraint provides the means modify
the rate at which the object traverses the curve defined by the curvature constraint. Again
to ensure simple two-body free flight, setting the scalar α to zero supplies a redundant
constraint ensuring angular momentum is a constant as embodied in the primary
16
curvature constraint. Updating equations 20 and 21 with this new constraint is detailed
below.
(26)
(27)
3-D: Non-Planar Motion
To account for motion out of the plane, a closed form equation was developed to
produce non-planar trajectories as a function of curvature in three dimensions as detailed
in equations 28-30.
(28a)
(28b)
(29a)
(29b)
(30)
17
The fundamental equation combined with curvature provides a powerful new
mathematical tool for defining, generating and controlling orbital motion. For celestial
mechanics, given a set of observations of a perturbed body, the underlying net
acceleration acting on the body can be determined directly. As applied to astrodynamics,
the minimal acceleration necessary to achieve a selected trajectory is uniquely and
precisely determined and the resulting closed form equations are easily accommodated by
real-time on-board processing.
CHAPTER 4: APPLICATIONS TO CELESTIAL MECHANICS AND
ASTRODYNAMICS
The application of curvature and the fundamental equation to orbital motion
enables four capabilities:
description, generation,
control, and instruction.
First is the ability to describe
a trajectory completely,
regardless of how complex,
constrained or in a time-
varying reference frame. The utility provided by this new approach with respect to
describing a trajectory is the common problem of orbit determination for both celestial
mechanics and astrodynamics. In the case where observations exist of an object and its
motion has been characterized, these equations can be used to solve for the time history
of the curvature of the orbit. By solving for the curvature, the resulting function can be
Figure 3: Precessing Orbit Simulation
18
used as a template for perturbed objects. These curvature templates provide a clear means
for detailing the naturally occurring perturbed motion of shepherd moons, horseshoe
orbits and comets. The complementary problem is also enabled by these equations, where
the motion has been captured, but the magnitude and time varying nature of the
perturbation is still undefined. This has direct applicability to multi-body systems such as
star clusters or the prevalence of spiral motion in astronomical systems.
An example of the utility of these templates is illustrated
by the trajectory plot in Figure 3. Figure 3 depicts a celestial
object whose planar trajectory follows a hypotrochoid. This
repeating closed curve is exceedingly difficult to model with
existing techniques and was the subject of many
correspondences between Newton and Hooke as shown in Figure
4 [18]. Again by exercising the fundamental equation and
inserting the equation of curvature for the hypotrochoid shown in Figure 3 as detailed in
Table 3, the control force necessary to maintain this repeating orbit can be found exactly
and is shown in Figures 5 and 6.
Table 3: Precessing Orbit Initial Conditions and Equations of Motion
X (km) Y (km) Xdot (km/s) Ydot(km/s)
Initial Inertial State 32000 0 0 3.52
Unconstrained Equation of Motion
where μ = 3.986e5 km
3
/s
2
Hypotrochoid Curvature Constraint
Hypotrochoid Curvature Parameters
a = 20000
b = 8000
h = 20000
Constrained Equation of Motion
Figure 4: Newton’s
Letter to Hooke on
Precessing Orbits [15]
19
Figure 5: Control Force, Radius and Velocity Magnitude for Precessing Orbits Example (1/3 of a orbit)
Figure 6: Control Force Components for Precessing Orbits (1/3 orbit)
The generation and control of an object under acceleration, marks the transition
from celestial mechanics to navigation. This new general approach enables multiple
capabilities for improving orbital flight. The most important use is the ability to
accurately generate and control new non-impulsive propulsion systems [2]. Ion engines,
20
solar sails, and electrodynamic tethers all require significant modeling to generate
accurate predicted ephemerides for mission planning. Through this approach numerous
potential trajectories can be evaluated precisely and then scored with respect to the
necessary thrust required. Non-viable profiles which require more acceleration than what
the system is capable of are rejected. An example of the non-Keplerian nature of these
low thrust/continuous thrust propulsion systems is the eletrodynamic (ED) tether. This
propulsion system exploits the Lorentz force to impart an acceleration on the vehicle by
running a current along the tether while moving through the magnetosphere. The ED
tether’s orbit is not closed and the orbit’s increased precession rate causes a significant
problem for Keplerian representations and solutions of the ED tether’s orbit. The inability
to accurately describe and predict the ED tether’s trajectory for mission planning and
safety of flight was the impetus for this research, especially when operating in the high
drag environment of low earth orbit. This formulation provides a simple closed form
control solution for modulating the current across the tether where the only required
sensor inputs are state knowledge from a GPS receiver and accelerometer measurements
of the net acceleration generated by the system.
These same advanced propulsion systems also provide the means to generate new
and exotic orbits of exceptional utility and interest. From latitude controlled orbits (LCO)
that maintain a constant orbit along a line of latitude to super-stationary and sub-
stationary points above and below geosynchronous altitude, again the new approach
outlined in this paper provides the requisite mathematical machinery to model, assess and
execute this new class of non-Keplerian orbits. Again the curvature and torque
21
constraints are simple to generate either exotic orbit. For the LCO, the resulting
acceleration from the Kalaba-Udwadia equations precludes the velocity vector from
deviating from the line of latitude while maintaining the altitude of the satellite.
Similarly, the sub- and super-stationary points below and above geosynchonous altitude
utilize an approach that maintains the altitude while increasing the angular momentum of
the object to keep pace with an object at geosynchronous altitude. To match the orbital
rate of a target object above or below the non-Keplerian orbit, the resultant acceleration
either increases the centripetal acceleration (i.e. object is above GEO and must “catch
up”) to increase the rate or decreases the centripetal acceleration to slow down (i.e. object
is below GEO and must move “slower”).
Through curvature and the fundamental equation, trajectories can be developed
that are seamless from launch through insertion and operations. This approach’s ability to
be applied to any reference frame provides the ability to utilize rotating and other non-
inertial frames, without having to change the underlying mathematical approach.
Nowhere is this applicability more apparent than with detailing the trajectories necessary
to achieve specific relative motion, formation flying, rendezvous and proximity
operations requirements. This approach expands upon H. Cho’s use of the Hill’s frame
and rigidly defining the separation parameter to be maintained between the two vehicles
[5], [6]. Now the motion between the two or more vehicles can be as complex or
mundane by simply changing the curvature term. Table 4 details the previous unique A
and b matrices generated by Lam and Cho using the Kalaba-Udwadia equation versus the
general curvature based approach.
22
Table 4: Comparison of Previous Approaches with Curvature Technique
As shown in Table 4, the curvature approach provides a common means to solve
each problem where the only
update required is the
determination of the scalars in
the vector b. The ability to
develop closed form exact
control laws for these same
advanced propulsion systems
represents a clear advance over
existing techniques [25]. Not only are these control laws simpler and more robust, but the
proven aspect of the fundamental equation to provide the minimum necessary
acceleration to achieve the desired trajectory, represents a direct application for efficient
Previous Approach Curvature Based Approach
Case A B A B
Maintain a
circular orbit
at a fixed
inclination in
Inertial Space
[14]
Maintain a
constant
offset from a
target object
in the Hill
Frame [5]
NOTE: I
1,2,3
are complex algebraic expressions [14]; i is the orbital inclination and Ω is the
angle between the principle axis and the ascending node; R is the desired orbital radius
Figure 7: Outward Spiral From GEO to the Moon at 1 Year
23
fuel usage. This critical feature represents not just the potential for extended mission life,
but increased mission life cycle cost savings and revenue generation. Additionally, these
closed form control laws are mathematically simple requiring only multiplication and
addition of matrices allowing for real-time on-board processing [27]. An example
demonstrating the robustness of these closed loop control laws is the proposed 2.7 year
MoonSail mission from GEO to the Moon using a solar sail. The first year is depicted in
Figure 7 below with initial conditions in Table 5. Like all low thrust propulsion
techniques, the long flight times are significantly influenced by even the smallest
perturbations and mis-modeled control laws. The sensitivity to lunar perturbations is
further exacerbated by a simple control law that only orients the sail in a manner to
generate a net acceleration. The initial regular spiral motion outward from GEO is
quickly disturbed by the Moon’s gravity, resulting in a fly by and then ejection from the
Earth-Moon system altogether. To correctly and easily account for the Moon’s gravity
and apply the correct control law, the Kalaba-Udwadia equations are recast into the
rotating Earth-Moon frame. On top of this frame, the curvature for the spiral away from
GEO is overlaid, providing a template for the required motion from the Earth to the
Moon as illustrated in Figure 7. A MatLab simulation of the final approach and capture in
Figure 8: MoonSail Approach and Insertion
24
inertial space is depicted in Figures 9 and 10. The initial conditions and curvature
constraints for this simulation are listed in Table 5. For the final half orbit prior to
entering the Moon’s sphere of influence the vehicle’s curvature constraint transitions to
maintaining a circular orbit that will intersect with the final 5760 km lunar orbit. When
the vehicle crosses the Earth-Moon vector (for this simulation the X axis), the constraint
changes to a logarithmic spiral about the Moon, with an initial altitude set to 4000 km
above the lunar surface. At this altitude there is a 97 m/s velocity excess between the
approach velocity and desired lunar orbit. Figure 9 shows a close up of the capture and
subsequent 4 orbits. The resultant control force profile for the approach and capture is
plotted in Figures 10 and 11 with the velocity correction post insertion clearly visible.
Figure 9: MoonSail Approach, Insertion & 48 Hours In Lunar Orbit
25
Table 5: MoonSail Initial Conditions and Equations of Motion
In Figure 10, the red line is the resultant control force magnitude, where the
increasing acceleration over the 9 hours before insertion is due to maintaining the Earth
centered circular orbit within the sphere of influence of the Moon. Once the vehicle
reaches the insertion point 4000km above the Moon, the required acceleration profile,
quickly drops off with the exception of a small correction to remove the excess approach
velocity while maintaining the final slow logarithmic spiral inward towards the Moon (Y
component and X components in Figure 11 respectively).
X (km) Y (km) Xdot (km/s) Ydot(km/s)
MoonSail Initial GEO Inertial
State 42159.5 0 0 3.074829
MoonSail Approach Inertial
State 353615.8 -128706 0.35200384 0.96712261
Moon Inertial State at Insertion
382050 0 0 1.02142985
Unconstrained Equation of
Motion
where r is the distance to the Earth, =
3.986e5 km
3
/s
2
, r
M
is the distance to the Moon and
M
= 0.0123*
Initial GEO Departure Curvature
Constraint
Initial GEO Departure Curvature
Parameters
θ = 0
c = 42159.8km
b = 0.003
Approach Curvature Constraint
Approach Curvature Parameters R = 376310
Insertion Curvature Constraint
Insertion Curvature Parameters θ = 0
c = 7200km
b = 0.5
Constrained Equation of Motion
for Free-flight and Insertion
26
Further investigation
of the utility of curvature
and the fundamental
equation generated new and
unique equations for
expressing Keplerian and
non-Keplerian motion. The
various standard Keplerian
relationships provide a
simple and effective means to describe the motion and velocity of an object on a conic
trajectory. These same
equations provide a means
to maneuver and transfer
between two-body orbits.
The minimum energy
maneuvers provided by the
Hohmann transfer is the best
example of the patched
conic technique. Despite the
inherent utility in these equations, they are unable to calculate the velocity necessary to
transfer to the spiral trajectory described in Section II. Via curvature, the patched conic
technique is enhanced to account for non-conic orbits.
Figure 10: MoonSail Control Force Magnitude and Lunar Altitude
Figure11: MoonSail Control Force Components
27
The key to this approach is the usual scalar form for curvature as shown in
equation 13. Rewriting this equation to make use of the definition of the cross product,
results in a simple relationship for the instantaneous curvature of a point on a given
trajectory as a function of the magnitude of the acceleration as scaled by sine of the angle
between the acceleration and velocity vectors (ϕ) and inversely proportion to the square
of the current speed of the object.
(31a)
Given two coplanar orbits, the change in curvature necessary to transfer from one
orbit to another orbit can be found by applying equation 31a to both orbits at the point of
arrival and departure and then subtracting. Equation 31b details this change in curvature
between an initial and final orbit.
(31b)
To transition from one orbit to another, the orbits must intersect or a transfer orbit
must be created to connect the two orbits. Equation 31b is greatly simplified at these
intersections, since at the crossing point, the acceleration due to the environment (gravity,
drag, radiation pressure, etc.) is common and constant to both curves. This simplification
allows for the equation to be easily rearranged and the value of the destination orbit’s
velocity to be found directly as shown in equation 32.
(32)
28
Equation 32 is termed the General Transfer Equation (GTE) and augments the
method of patched conics, by providing a new
ability to compute and assess the velocity needed
to transfer between any two curves. By removing
the limitation of only being able to model orbits
as arcs of circles, ellipses, parabolas and
hyperbolas, now complex non-Keplerian
trajectories can be modeled for mission planning by a piecing together spirals and any
other curve through the geometric definition of curvature. Applying the GTE to the
standard problem of maneuvering between two co-planar concentric orbits of differing
radii, illustrates that from equation 31b, the Hohmann transfer can be derived and is
shown as a special case of the GTE. Figure 12 illustrates the standard two maneuver
Hohmann transfer problem with an initial orbit in low earth orbit (LEO) and a desired
final orbit in geosynchronous orbit (GEO) (NOTE: figure is not to scale). To minimize
energy, the Hohmann transfer generates a transfer ellipse with its perigee and apogee
tangent to the initial orbit and final orbit respectively. At these points with circular initial
and final orbits, the angle between the velocity and acceleration vectors for all three
orbits at the tangent points is equal to 90º. With the sine of the angle equaling 1, the GTE
further simplifies to equation 33 for the perigee and apogee transfer maneuvers.
(33)
Figure 12: Hohmann Transfer Schematic [1]
29
Using the GTE, the velocity required to transfer from one orbit to another can be directly
computed. Table 6 details using the GTE to compute the required velocity to transition
from a circular orbit at LEO to two potential transfer orbits. The first case generates the
velocity at perigee for a standard Hohmann transfer ellipse at perigee and the second case
details the velocity necessary to begin a low thrust transfer along a logarithmic spiral.
Table 6: Generalized Transfer Equation Example
CASE 1 CASE 2
Initiate Hohmann Transfer
Initiate Logarithmic Spiral
Transfer Units
Initial Radius 6578 6578 Km
Initial Velocity 7.78 7.78 km/s
Initial Acceleration 0.009211 0.009211 km/s^2
Initial Curvature Equation 1/R 1/R N/A
Initial Curvature Value 0.000152 0.000152 1/km
Transfer Curvature Equation
N/A
Transfer Curvature
Parameters
R = 6578 km
e = eccentricity = 0.73
θ = 0
r = 6578km
b = 0.1
Transfer Curvature Value 8.7879e-5 0.000149 1/km
Velocity on Transfer Orbit 10.24 7.86 km/s
Delta V Required 2.45 0.08 km/s
In the simple case of the Hohmann transfer, all of the equations are well known
and the GTE provides only a redundant method of computing the changes in velocity to
accomplish the transfer. The true utility of the GTE is for utilizing non-conic curves as
the final trajectory post-maneuver. With the GTE, the required velocity can always be
computed given the initial state and the desired final orbit’s curvature. Using the GTE,
the method of patched conics is now expanded to include any curve.
30
The final benefit derived from this new general approach is the ability to
graphically represent the orbit with respect to curvature as a function of time, as depicted
in Figures 13 and 14. By plotting curvature the overall mission profile can be examined
in a linear fashion providing insight into timing, magnitude and planning of the overall
trajectory. To demonstrate this characteristic, the two cases detailed in Table 6 are plotted
in Figures 13 and 14. The traditional Hohmann transfer from LEO to GEO is in blue with
a low thrust spiral transfer
between the same orbits
denoted in red. Both vehicles
start from the same 200km
altitude circular orbit at the
same time. In curvature
space, circular orbits are
horizontal lines, ellipses are
sinusoidal and spirals follow
a smooth curve. After half of a period the vehicle following the blue Hohmann transfer
line reaches apogee at GEO and the second maneuver occurs transitioning to the new
horizontal line equal to the inverse of the final geosynchronous radius. The corresponding
low thrust spiral between the two orbits is represented by the red curve and it clearly
illustrates the significant time it takes for the spiral transfer to achieve geosynchronous
altitude.
Figure 13: Curvature Plot LEO to GEO
31
Figures 13 and 14 both display the same trajectories, but the curvature is
represented directly and
inversely (i.e. radius of
curvature) in each
respectively. By plotting
the radius of curvature as
a function of time, the
subsequent result lends
itself to a more intuitive
means of understanding
the motion over plotting curvature itself. The reason for this is the radius of curvature
varies directly with the altitude of the object. For the cases depicted in Figure 14, the
orbit transfer is from LEO to GEO; from a lower orbit to a higher altitude with the
resulting plot showing this motion directly and not inverted as is the case which uses
curvature. This clear representation gained by using the radius of curvature increases the
utility of assessing a low thrust transfer from LEO to GEO against the standard Hohmann
transfer with respect to time and velocity change required. This improvement again
readily shows that increasing the acceleration on the low thrust spiral will result in the
line rotating upward and decreasing the time of flight during transfer. Displaying the
overall trajectory in this linear fashion provides a single view of the motion, providing a
useful means of examining and assessing complex mission plans via the GTE enhanced
patched conics method.
Figure 14: Radius of Curvature Plot LEO to GEO
32
CHAPTER 5: INTEGRATED MISSION EXAMPLES
Using the Generalized Transfer Equation in conjunction with radius of curvature
plots to determine initial conditions and then applying the fundamental equation with a
specified curvature profile as the constraint provides an integrated method for defining
and assessing complex trajectories. This overall capability provides a unique means for
expanding the overall mission tradespace for future missions employing continuous or
non-traditional propulsion systems or requiring specific geometries between vehicles.
Four specific examples are provided to illustrate the effectiveness of this integrated non-
Keplerian approach to relevant missions currently under investigation by various
agencies.
Further highlighting
the utility and uniqueness of
this approach is the ability
to operate in the frame of
reference that is most
convenient and intuitive for
the mission to be expressed
in. From inertial to non-
inertial rotating frames each case uses the same machinery and overall approach to
determine initial conditions, define the overall trajectory and determine the requisite
control force necessary to maintain the required motion. Case 1 details the initial
Figure 15: ED Tether Curvature Profile
33
checkout, activation and transfer of an Electro-Dynamic (ED) Tether from 700km to
1700km in inertial space. Case 2 explores two possible natural motion trajectories in the
barycentric frame for exploring cislunar space. Case 3 illustrates a robust rendezvous and
proximity operations mission against an uncooperative target. Case 4 determines the delta
V required by individual spacecraft to maintain formation and synthesize a sparse
aperture at GEO with a specific geometry.
Several organizations are investigating the utility of electro dynamic tethers to
achieve the goal of propellant-less orbital maneuvering by manipulating a current while
travelling through the magnetosphere. Current designs require a 3km long tether
supported by several hundred kilogram hosts and end mass elements capable of
generating and controlling 1 amp of current across the tether. The sheer size of this
vehicle requires its initial altitude to be
above a significant portion of the
atmosphere to preclude drag effects
from severely impacting the propulsion
system. For this example the ED tether
is inserted into an initial 45° inclined
circular orbit at 700km altitude. During
the first 48 hours the tether remains
stowed and the end mass remains
connected to the host and perturbations due to the geopotential and solar and lunar n-
body effects cause the orbit to regress in inertial space. Upon activation prior to the end
Figure 16: ED Tether Orbital Geometry
34
of the second day, the vehicle is to circularize and freeze the plane of the orbit and negate
any perturbations. Once the orbit has been confirmed to be frozen, the command is then
given to begin a 5 day controlled logarithmic spiral from 700 to 1700km. Over the course
of the 1000km change in altitude, the orbit plane must be maintained. Finally, at 1700km
the trajectory returns to a circle continuing the negation of all external perturbations. A
plot of the radius of curvature constraint and corresponding timeline is illustrated in
Figure 15.
Initial conditions are easily determined from Keplerian relationships and are listed
in Table 7 along with the curvature equations and parameters for insertion into the
fundamental equation. The resulting overall 3D trajectory illustrating the implementation
of the simple radius of curvature 2D plot from Figure 15 from 700 to 1700km is depicted
in Figure 16. The initial effect due to the perturbations torquing the orbit plane is clearly
shown in the inner dark blue bands at 700km. Similarly the transition from free-flight to
controlled motion is illustrated by the thin blue lines depicting the logarithmic transfer
Table 7: ED Tether Initial Conditions
35
and subsequent insertion at
1700km all the while
maintaining the final
mission orbit plane. The
control force components
necessary to achieve this
profile is shown in Figure
17. The small oscillations
up to 48 hours from insertion represent the forces necessary to maintain the 700 km
circular initial orbit. From 48 to 54 hours post launch, the tether is further exercised to
raise the orbit 1000km as highlighted in blue in Figure 17. Once the vehicle reaches
700km the tether than transitions again to maintanence mode and the final 1700km
circular orbit is kept.
Case 2 reflects the renewed enthusiasm in returning to the Moon by both the
government, commercial and private interests. The desire to minimize overall mission
cost has lead to a investigation of the utility of natural motion orbits in the Earth-Moon
barycentric rotating frame for the Planetary Society’s LightSail solar sail . In this common
frame from celestial mechanics, the Earth and Moon lie on the X-axis at an normalized
opposite distance from the origin equal to the corresponding mass fraction of the overall
system. The entire frame rotates about the Z axis at a rate equal to the period of the
Moon’s orbit. The equations of motion for the barycentric sys tem are detailed in
equations 34 and 35 [16].
Figure 17: ED Tether Control Force Components
36
(34)
(35)
where µ is the ratio of the GM of the Moon to the Earth (1/82.314) and Re and Rm are
the distances from the Earth and Moon to the spacecraft respectively [17].
For this mission two candidate orbits have been selected. The first is the aptly
named horseshoe mission which is capable of visiting three of the Earth-Moon Lagrange
Points within 300 days. The overall trajectory (in blue) and initial conditions in the
barycentric frame is shown in Figure 18. The horseshoe trajectory starts below L3 and
then moves toward’s the Moon’s sphere of influence before falling away at the reversal
point closest to the Moon and then heading to the edge of the Earth-Moon sphere of
Figure 18: Horseshoe Trajectory Overview
37
influence where the process is repeated again this time about L5 before returning to L3.
This horseshoe oscillation about the L4 and L5 libration points occurs naturally about
Saturn by its moons Epimetheus and Janus [17].
Again the GTE is used to compute the initial conditions, but unlike previous
examples the curvature constraint is not just a simple equation; it is now a profile
generated from the nominal free-flight trajectory. This curvature profile of the nominal
horseshoe trajectory is depicted in Figure 19 with the reversal between the Moon and L4
highlighted.
While in cislunar space,
the dominant perturbation is solar
radiation pressure which both aids
and retards the motion of the solar
sail and must be correctly
accounted for to achieve mission
success. This non-conservative
force is a function of the area and
mass of the vehicle, the reflectivity of the sail, the orientation of and to the Sun and the
incident solar flux at 1 AU. The proposed LightSail vehicle has a aluminized sail with an
area of 35 square meters, a mass of 10 kg and a reflectivity of 0.999. The solar flux at 1
AU provides a constant pressure of 9.5 µPa for the sail to exploit and accommodate [28].
The equations defining the effects of solar radiation pressure are listed below:
Figure 19: Horseshoe Curvature Profile
38
(36)
(37)
Including these
perturbations in
conjunction with the
desired curvature profile
via the fundamental
equation supplies the
required control force
necessary to maintain the free flight trajectory in the presence of solar radiation pressure.
The components of this required force are listed for the first half of the horseshoe
trajectory in Figure 20
along with the details at
the reversal.
The second
natural motion trajectory
is the 2:1 resonant orbit
which oscillates between
the Earth and Moon over
a 120-day period. This trajectory begins at L2 and then rapidly falls towards the Moon,
Figure 20: Horseshoe Control Force Components
Figure 21: Resonant 2:1 Orbit
39
orbits the Moon twice before transiting through L1 and then back towards the Earth
where it orbits the Earth four
times before returning to the
Moon. This dynamic
trajectory is depicted in blue
in Figure 21 along with the
corresponding barycentric
initial conditions.
The same analysis
and approach was applied
to the 2:1 resonant orbit as
was performed to the
horseshoe orbit. Again the
curvature profile
corresponding to the natural
motion was determined and
used as the constraint to the
fundamental equation with solar radiation pressure as the perturbing force. The resonant
curvature profile is depicted in Figure 22. The resulting control force component
accelerations are shown in Figure 23.
Figure 22: Resonant 2:1 Curvature Profile
Figure 23: Resonant 2:1 Control Force Components
40
By comparing the two acceleration profiles, the ability of this approach to
quantify the challenge and corresponding requirements necessary to levy on the LightSail
mission due to solar radiation pressure is unique and further highlights the utility of this
approach.
The third case
demonstrates the utility of this
approach in the common Hill
frame used to define and assess
rendezvous and proximity
operations about another co-
orbital vehicle. The Clohessy-
Wiltshire (CW) equations
describe the relative motion of a maneuvering satellite with respect to a target satellite in
the Hill frame
as equations 38-41 below [7].
(38)
(39)
(40)
(41)
Figure 24: Hill Frame and RPO Profile
41
The Hill frame
is centered on the
target satellite with
the x axis aligned
with the radial vector
outward from the
primary body and the
z axis aligned with the
orbital angular
momentum vector. The y axis is defined as the resulting cross product of x and z to
complete the triad. In this rotating frame the same equations can be utilized to easily
describe and specify the motion of the maneuvering satellite as it closes in on the target
satellite.
For this
example the
target satellite is
assumed to be in
a circular orbit at
GEO and the
maneuvering satellite has matched planes with the target satellite. The required approach
and ingress procedure requires the maneuvering satellite to close the distance from 100
to 5 kilometers over 4 days with two 24 circumnavigations of the target at 100 and 50
Figure 25: RPO Curvature Profile
Table 8: RPO Initial Conditions
42
km. For this example approach and final ingress are accomplished by the maneuvering
vehicle following a logarithmic spiral. Figure 24 depicts the overall motion of the
maneuvering satellite in the Hill frame.
For this example,
the maneuvering satellite
loiters at 100 km for 1 day
and then over the next 24
hour period spirals inward
to 50 km. At 50 km from
the target, the
maneuvering vehicle again
holds for 24 hours and
completes another circumnavigation of the target vehicle. Once the final
circumnavigation is completed the maneuvering vehicle moves inward on a tighter
logarithmic spiral trajectory to reach the final 5 km offset from the target. Of note in this
final ingress maneuver is the fact that the trajectory is purposely offset from the target
allowing for a greater diversity of observations from any onboard RPO sensors to
reconstruct and maintain the target’s orbit. Again this complex multi -day scenario can be
easily visualized in a curvature versus time plot as depicted in Figures 25 (radius of
curvature vs time).
Figure 26: RPO Control Force Components
43
The initial conditions and RPO profile and parameters are shown in Table 8. The
resulting control force necessary to execute this RPO sequence is shown in Figure 26.
Even in this planar case, the utility the ability to use the same techniques and
method’s previously employed in both the inertial and barycentric frames further
demonstrates the utility of this approach.
The fourth and final case illustrates the ability of this
overall approach to address formation flying while in a
significantly perturbed environment. Significant effort and
resources have been invested in ways to generate large
apertures in space from fractionated systems. Key to these
proposed systems is the ability for the dispersed elements to work in coordinated manner
to maintain both relative geometry within the cluster and absolute geometry with a
terrestrial or inertial target. For this example, a 3-ring concentric cluster of 15 free flying
satellites is developed spaced at 50m, 100m and 150m from a geosynchronous node as
depicted in Figure 27. The overall “wagon wheel” hub and spoke geometry must be
maintained for the system to act as a sparse aperture. Therefore, each free flying element
must maintain its relative distance and rate within each spoke while also maintaining the
overall shape and orientation of the aperture towards 30° N latitude as illustrated in
Figure 28.
Figure 27: GEO Sparse Array
44
Like the third example,
this system is also best described
in the rotating Hill frame, but
with the additional complexity
that the Z component and
motion is not 0. Additionally,
the dominant perturbations
acting on each element in the array are the Earth’s geopotential as modeled by the
dominant J
2
zonal harmonic and solar radiation pressure. The equations for each of these
perturbations are listed below.
(42)
(43)
(44)
(45)
(46)
where J
2
= 0.0010827, Re is the mean radius of the Earth, i is the orbit’s inclination, is
the argument of latitude, is the declination of the Sun, and µ is the GM of the Earth.
Figure 28: Cluster Geometry & Orientation
45
The resulting torques on each element and the overall formation are time varying
and unique. If left uncorrected the entire
aperture would disperse and preclude
the mission. The initial conditions for
each spoke as they cross the equatorial
plane is shown in Table 9. By exploiting
the frame independence of curvature and
the fundamental equation, the resulting
constraint is simply a circle (1/R).
Applying this single constraint results in the correct geometry and rate as shown
in Figure 29. As illustrated in the isometric picture, all three elements on the spoke are
rotating about the hub while remaining in a plane with the proper radial alignment. The
resulting control force per element is explicitly determined by the fundamental equation
and for comparison the unperturbed (two body only) and perturbed component and
magnitude are shown in Figure 30.
Table 9: Cluster Initial Conditions
Figure 29: Cluster Formation Geometry
46
As illustrated in Figure 30 even in the two-body only scenario, the control force
required is time varying in all three components. The inclusion of the perturbations
significanlty changes the magnitude and phase of the resulting control force. Analysis of
the control force neccesary at 50m increrments along the spoke is found to be nearly
linear with each subsequent ring outward from the hub scaling directly with the distance
from the hub. For even this small array and cluster, maintaining this formation requires
the control force to be continuously applied requiring approximately 1m/s of delta V per
day for the outer most element per day. Again the utility of this approach is evident in
providing a clear and unique means for quantifying the required propulsion necessary to
achieve this formation.
Figure 30: Cluster Control Force Components (50m top) and Magnitude (All bottom)
47
CHAPTER 6: SUMMARY
This paper provides an overview of a new method utilizing curvature and the
Kalaba-Udwadia fundamental equation for describing, generating and controlling a
trajectory. The utility of this approach was investigated for problems in both celestial
mechanics and astrodynamics. A Generalized Transfer Equation was developed to
supplement the standard Keplerian relationships for determining the velocity needed to
transfer between orbits. The General Transfer Equation increases the utility of the
patched conic approach by enabling the use of all curves as templates for potential
transfer orbits. By plotting curvature as a function of time, complex trajectories can easily
be visualized and provide direct support for mission planning and design. Simulation
results demonstrate the ability to generate and control highly perturbed orbits using
curvature as the primary constraint in both inertial and non-inertial frames.
48
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The Space Technology Library, El Segundo, 2001.
[29] M. Vasile, O. Schutze, et al., “Spiral trajectories in global optimisation of
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Abstract (if available)
Abstract
This paper proposes a new general approach for describing, generating and controlling the trajectory of an object by combining recent advances in analytical dynamics with the underlying theorems and concepts from differential geometry. By using the geometric construct of curvature to define an object’s motion and applying the fundamental equation of constrained dynamics, the resulting solutions are both explicit and exact for the minimum acceleration necessary to maintain the specified trajectory. The equations detailing the control force required to follow the selected trajectory can be expressed in closed form, regardless if the object is in Keplerian free-flight about a single central body or following a non-Keplerian trajectory in a highly disturbed environment. Examples are provided in both the inertial and non-inertial frames to demonstrate the utility of this combined approach for solving common problems in both celestial mechanics and astrodynamics. The practical aspects of exploiting curvature for maneuver and mission planning is also investigated resulting in the formulation of the Generalized Transfer Equation which extends the method of patched conics to include any curve.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Garber, Darren David
(author)
Core Title
Application of the fundamental equation to celestial mechanics and astrodynamics
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Astronautical Engineering
Publication Date
03/30/2012
Defense Date
02/29/2012
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
analytical dynamics,astrodynamics,curvature,fundamental equation,maneuver planning,non-keplerian motion,OAI-PMH Harvest,perturbations
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Gruntman, Michael (
committee chair
), Udwadia, Firdaus E. (
committee chair
), Effroimsky, Michael (
committee member
), Erwin, Daniel A. (
committee member
), Hintz, Gerald (
committee member
)
Creator Email
darren.garber@nxtrac.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-59
Unique identifier
UC11288110
Identifier
usctheses-c3-59 (legacy record id)
Legacy Identifier
etd-GarberDarr-561.pdf
Dmrecord
59
Document Type
Dissertation
Rights
Garber, Darren David
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
analytical dynamics
astrodynamics
curvature
fundamental equation
maneuver planning
non-keplerian motion
perturbations