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Title Optimum Low Thrust Elliptic Orbit Transfer using Numerical Averaging
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Author Tarzi, Zahi
Publication Date 2012
Peer reviewed|Thesis/dissertation
eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA
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Optimum Low Thrust Elliptic Orbit Transfer
Using Numerical Averaging
A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy
in Aerospace Engineering
by
Zahi Bassem Tarzi
2012
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ABSTRACT OF THE DISSERTATION
Optimum Low Thrust Elliptic Orbit Transfer
Using Numerical Averaging
by
Zahi Bassem Tarzi
Doctor of Philosophy in Aerospace Engineering
University of California, Los Angeles
Professor Jason Speyer, Chair
Low-thrust electric propulsion is increasingly being used for spacecraft missions primarily due to its high propellant efficiency. Since analytical solutions for general low-thrust transfers are not available, a simple and fast method for low-thrust trajectory optimization is of great value for preliminary mission planning. However, few low-thrust trajectory tools are appropriate for preliminary mission design studies. The method presented in this paper provides quick and accurate solutions for a wide range of transfers by using numerical orbital averaging to improve solution convergence and include orbital perturbations. Thus allowing preliminary trajectories to
be obtained for transfers which involve many revolutions about the primary body. This method
considers minimum fuel transfers using first order averaging to obtain the fuel optimum rates of
change of the equinoctial orbital elements in terms of each other and the Lagrange multipliers.
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Constraints on thrust and power, as well as minimum periapsis, are implemented and the equations are averaged numerically using a Gaussian quadrature. The use of numerical averaging allows for more complex orbital perturbations to be added without great difficulty.
Orbital perturbations due to solar radiation pressure, atmospheric drag, a non-spherical central body, and third body gravitational effects have been included. These perturbations have not ben considered by previous methods using analytical averaging. Thrust limitations due to shadowing have also been considered in this study. To allow for faster convergence of a wider range of problems, the solution to a transfer which minimizes the square of the thrust magnitude is used as a preliminary guess for the minimum fuel problem. Thus, this method can be quickly applied to many different types of transfers which may include various perturbations. Results from this model are shown to provide a reduction in propellant mass required over previous minimum fuel solutions. Minimum time transfers are also solved and compared to minimum fuel.
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The dissertation of Zahi Bassem Tarzi is approved.
Richard Wirz
Christopher Russell
Lewis Mingori
Todd Ely
Jason Speyer, Committee Chair
University of California, Los Angeles
2012
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Dedication
To my Mom and Dad
Who experience more frustration when I struggle
And more joy when I succeed
Than I could ever manage to feel myself
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Table of Contents
1. Introduction ...... 1 2. Reference Frames and Orbital Elements ...... 4 2.1 Position and Velocity ...... 4 2.2 Keplerian Orbital Elements ...... 7 2.3 Equinoctial Elements and Frame ...... 7 2.4 Spacecraft Fixed Frame ...... 11 3. Optimal Control Problem Formulation ...... 14 3.1 Equations of Motion ...... 14 3.2 Maximizing Delivered Mass ...... 15 3.2.3 Transitioning Between Cases ...... 17 3.3 Minimizing Transfer Time ...... 18 4. Averaging and Multiple Time Scales ...... 20 4.1 Small Variable ...... 20 4.2 Averaging Method...... 21 4.3 Multiple Time Scale Formulation ...... 25 4.4 Numerical Averaging ...... 30 4.5 Reacquiring Periodic Performance Data ...... 32 4.6 Mean Element Transformation...... 33 5. Optimal Control Problem Solution ...... 37 5.1 Maximizing Delivered Mass ...... 37 5.2 Minimizing Time...... 39 6. Limits, Penalty Functions, and Shadowing ...... 42 6.1 Thrust and Power Limits ...... 42 6.2 Periapsis Penalty Function ...... 43 6.3 Shadowing Effects...... 44 7. Perturbations ...... 47 7.1 Solar Radiation Pressure ...... 47 7.2 Atmospheric Drag ...... 48 7.3 Zonal Harmonics ...... 50 7.4 Tesserel/Sectorial Harmonics ...... 52 7.4.1 Slow Sidereal Rate ...... 55 7.4.2 Fast Sidereal Rate ...... 55 7.4.3 Resonant Sidereal Rate ...... 56 7.5 Multi-Body Effects ...... 57 7.5.1 Long Perturbing Body Period ...... 58 7.5.2 Short Perturbing Body Period ...... 58 8. Example Transfers ...... 60 8.1 Case A: Highly Elliptic Orbit Transfer ...... 60 8.2 Case B: GTO to GEO Transfer (ARIANE 4) ...... 67 8.3 Case C: GTO to GEO Transfer (Taurus) ...... 71 8.4 Case D: Elliptic Low Earth Orbit Raising ...... 75 8.5 Case E: Mercury Orbiter ...... 79 8.6 Case F: Molniya De-Orbiting ...... 84 vi
9. Conclusion ...... 89 10. Future Work ...... 90 A. Appendix ...... 92 A.1 The M Matrix ...... 92 A.2 Partials of the Hamiltonian ...... 93 A.3 Partials of the Perturbing Accelerations...... 98 A.3.1 Solar Radiation Pressure ...... 99 A.3.2 Atmospheric Drag ...... 100 A.3.3 Zonal Harmonics ...... 102 A.3.4 Tesserel Harmonics ...... 104 A.3.5 Multi-Body Effects ...... 108 A.4 Atmospheric Density Model ...... 109 A.5 Earth Non-Spherical Gravity Coefficients ...... 111 Bibliography ...... 118
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List of Figures
Figure 2.1 Body-Centered Inertial Equatorial Frame ...... 4 Figure 2.2 Heliocentric Inertial Frame (Placeholder) ...... 5 Figure 2.3 Body Fixed Equatorial Frame ...... 6 Figure 2.4 Equinoctial Frame ...... 9 Figure 2.5 Radial Transverse Normal Frame ...... 11 Figure 7.1 Frame Rotation Geometry (Placeholder)...... 12 Figure 8.1 Case A - Equinoctial Element Time History ...... 61 Figure 8.2 Case A - Lagrange Multiplier Time History ...... 61 Figure 8.3 Case A - Periapisis Time History ...... 62 Figure 8.4 Case A - Keplerian Element and Spacecraft Mass Time Histories ...... 64 Figure 8.5 Case A - a) Average Thrust b) Periodic Thrust ...... 65 Figure 8.6 Case A - Thrust Direction Angles ...... 66 Figure 8.7 Case A - Orbit Transformation with Thrust Vectors ...... 66 Figure 8.8 Case B - Keplerian Element and Spacecraft Mass Time Histories ...... 68 Figure 8.9 Case B - a) Average Thrust b) Periodic Thrust ...... 69 Figure 8.10 Case B - Thrust Direction Angles ...... 70 Figure 8.11 Case B - Min Fuel Keplerian Element and Spacecraft Mass ...... 71 Figure 8.12 Case B - Orbit Transformation ...... 71 Figure 8.13 Case C - Keplerian Element and Spacecraft Mass Time Histories ...... 73 Figure 8.14 Case C - a) Average Thrust b) Periodic Thrust ...... 74 Figure 8.15 Case C - Thrust Direction Angles ...... 75 Figure 8.16 Case C - Orbit Transformation ...... 75 Figure 8.17 Case D - Keplerian Element and Spacecraft Mass Time Histories ...... 77 Figure 8.18 Case D - a) Average Thrust b) Periodic Thrust ...... 78 Figure 8.19 Case D – Thrust Direction Angles...... 79 Figure 8.20 Case E - Keplerian Element and Spacecraft Mass Time Histories ...... 81 Figure 8.21 Case E – a) Average Thrust b) Periodic Thrust ...... 82 Figure 8.22 Case E – Thrust Direction Angles ...... 83 Figure 8.23 Case E - Orbit Transformation ...... 83 Figure 8.24 Case F - Keplerian Element and Spacecraft Mass Time Histories ...... 85 Figure 8.25 Case F – a) Average Thrust b) Periodic Thrust ...... 86 Figure 8.26 Case F – Thrust Direction Angles ...... 87 Figure 8.27 Case F - Orbit Transformation ...... 88
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List of Tables
Table 8.1 Case A – Comparison of Results, ISP = 3000s, Tmax = 0.50N ...... 63 Table 8.2 Case B – Comparison of Results, ISP = 2000s, Tmax = 0.35N ...... 68 Table 8.3 Case C – Comparison of Results, ISP = 3300s, Pmax = 5kW ...... 72 Table 8.4 Case D – Comparison of Results, ISP = 3000s, Tmax = 0.01N ...... 76 Table 8.5 Case E – Comparison of Results, ISP = 3000s, Tmax = 0.15N ...... 80 Table 8.6 Case F – Comparison of Results, ISP = 3000s, Tmax = 0.001N ...... 85 Table A.1 Earth Atmospheric Density ...... 110 Table A.2 Earth Zonal Gravity Coefficients ...... 111 Table A.3 Earth Sectoral and Tesserel Gravity Coefficients ...... 117
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Acknowledgements
I would like to thank my adviser, Jason Speyer, for all of his support and guidance, which has directed me through the many the challenges presented during this work. I would also like to express my appreciation to Richard Wirz for all of his help in encouraging me to present and publish this work, and to work towards increasing its applications. Additionally, I extend my thanks to Todd Ely for helping me work out the numerical averaging method and non-spherical gravity perturbations used in this work.
I would like to express my gratitude to everyone I have worked with at the Jet Propulsion
Laboratory for their support and flexibility which allowed me to carry out this research while working part-time. I would especially like to thank Kobie Boykins for allowing my work schedule to be so adaptable and driving me to graduate.
My thanks also go to fellow student Javier Fernandez for his support and assistance during our joint time graduate experience at UCLA, and his help in refining this paper.
Finally, I would like to express my gratitude to my family and friends who have supported and encouraged me throughout this work.
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VITA
2006 B.S., Aerospace Engineering University of California, Los Angeles
2005-2006 Reliability Engineer (Part-Time) Boeing Satellite Systems El Segundo, California
2007 M.S., Aerospace Engineering University of California, Los Angeles
2006-2012 Mechanisms Engineer (Part-Time) Jet Propulsion Laboratory Pasadena, California
2007-2012 Teaching Assistant Mechanical and Aerospace Engineering University of California, Los Angeles
PUBLICATIONS AND PRESENTATIONS
Tarzi, Z., Speyer, J., and Wirz, R., “Optimum Low-Thrust Transfer between Arbitrary Elliptic Orbits using Asymptotic Expansions and Averaging” Paper IEPC-2009-222, 31st International Electric Propulsion Conference, Ann Arbor, Michigan, September 2009
Tarzi, Z., Speyer, J., and Wirz, R., “Optimum Low-Thrust Transfer for Power Limited Spacecraft Using Numerical Averaging and Continuously Variable Thrust” Paper AAS 11-234, AIAA/AAS Astrodynamics Specialist Conference, New Orleans, Louisiana, February 2011
Berry, D., Guinn, J., Tarzi, Z., and Demcak, S., “Automated Spacecraft Conjunction Assessment at Mars and the Moon”, AIAA SpaceOps 2012 Conference, Stockholm, Sweden, June 2012
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1. Introduction
With the growing use of electric propulsion systems for earth-orbiting and interplanetary spacecraft, it is
increasingly important to be able to design optimal trajectories for these missions. Existing low-thrust optimization
methods are too complex and difficult to use for quickly generating preliminary mission trajectories. Except for
simple circular spiral transfers, there are no medium and low fidelity tools available for use in preliminary mission
design studies. In particular, current preliminary design models generally do not consider orbital perturbations other
than J2 effects.
The study of optimal transfers using an electric propulsion system for rendezvous between two neighboring
elliptic orbits (orbits whose elements differ by a small amount) was solved analytically by Edelbaum [4]. He then
extended the formulation to include transfer between coplanar and coaxial orbits assuming many revolutions [5].
Marec outlined an extension of this to the general case of transfer between arbitrary elliptic orbits and analytically solved the problem for a few specific types of transfers [12-13]. Both Edelbaum and Marec used the Keplerian orbital elements which contain singularities for key orbits (i.e., polar, circular). Ketchichian has suggested using the non-singular equinoctial orbital elements to solve the minimum time, constant thrust problem [9]. Sacket, Malchow, and Edelbaum have developed the SESPOT program which uses averaging and the equinoctial elements to solve the minimum time, constant thrust transfer problem and includes the analytically averaged J2 effects and shadowing
[16]. The SESPOT program also includes a penalty on areas of high magnetic flux around the Earth. Kluever and
Oleson have used a more robust direct optimization method to obtain better convergence for the same problem [10].
Their results compare well to the SESPOT program. Geffroy and Epenoy obtained results for both minimum time and fuel problems using bang-bang thrust with constraints on thrust direction [7]. A continuation strategy was used which consists of starting at the minimum time solution and solving a succession of slightly varied problems which culminate at the minimum fuel problem. Haberkorn, Martinon, and Gergaud used a similar homotopic technique to move from the minimum quadratic thrust to minimum fuel problem [8].
The objective of this work is to provide a simple and fast method for obtaining preliminary solutions for a wide range of low-thrust orbit transfers. Although previous low-thrust averaging techniques have provided useful solutions, they typically require longer run-times and accurate initial guesses to converge. In addition, they cannot
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account for some orbital perturbations which are important to certain low-thrust trajectories. The combined minimum quadratic thrust, minimum fuel problem allows approximations to the minimum fuel solution to be obtained very quickly and provides a starting point to the minimum fuel problem which increases the likelihood and speed of solution convergence. In addition, the orbital averaging in this work is carried out numerically whereas previous methods have used analytical averaging. The use of numerical averaging allows for the consideration of complex orbital perturbations not previously taken into account in such problems. This method includes the effects of solar radiation pressure, atmospheric drag, third bodies, higher order non-spherical gravity terms, and a periapsis penalty function, which have not been considered by previous averaging methods. Thus, this method can be quickly applied to many different types of transfers which may include various perturbations. Furthermore, the method used in this paper results in a reduction in propellant mass required over the minimum fuel solution presented in [8].
In this work, a first order averaging method is used to obtain optimum trajectories which involve many revolutions about a central body. The optimal control problem is written as a combination of the minimum quadratic thrust and minimum fuel mass problems. The optimum rates of change of the equinoctial orbital elements are derived for both cases: minimum fuel and minimum thrust squared. The resulting equations are averaged numerically using a Gaussian quadrature, which allows for the straightforward addition of complex orbital perturbations. The two-point boundary value problem is then solved numerically using a shooting method. Due to the non-linear nature of the problem, convergence to an optimum solution is highly dependent on the initial guess.
However, a simple guess method is developed in this work to allow solution convergence when minimizing the square of thrust for a wide variety of transfers in a matter of minutes. If further accuracy is desired, the minimum fuel problem can be solved with the quadratic result used as an initial guess. A smooth transition between the two problems is made possible through the use of a sliding weighted average of the optimum thrust values from both solutions. This allows for quicker and more likely convergence for the ill-behaved optimum fuel problem. A shooting method is also used to solve minimum time problem.
A periapsis penalty function is implemented as a method of avoiding orbits with low periapsis which may dip into the atmosphere or below the surface of the central body. Such a penalty was used by Petropoulos in conjunction with his Q-Law algorithm for low-thrust orbit transfers [16]. The effects of shadowing on spacecraft using solar electric propulsion are also included. Such spacecraft are assumed to be unable to thrust during portions
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of their orbits which are shadowed. Limits on the available thrust due to thruster physical properties and available
propulsion power are also utilized.
Orbital perturbations due to solar radiation pressure, atmospheric drag, a non-spherical central body, and third
body gravitational effects are accounted for. Although forces due the pressure of solar radiation are typically very
small, they can cause significant orbital perturbations over long periods of time or if the spacecraft is considerably
close to the Sun. Atmospheric drag can cause large orbital alterations for spacecraft in low orbits, and has even been
used to assist in orbit transfers (Aerobraking). Significantly large perturbations can be caused by the second order
gravity perturbations induced by the oblateness of an orbited body (J2). Higher order effects due to a non-spherical
body are generally very small for large bodies such as the Earth, but can become significant for orbits in periodic
resonance with the rotation of the orbited body or for smaller bodies such as asteroids. Substantial orbital
perturbations can also be imparted from gravity due to celestial bodies other than the main orbited body; especially
for spacecraft in large orbits. For example, geostationary satellites are subject to significant gravitational forces
from the Lunar and Solar bodies.
The results of five orbit transfer cases are studied which demonstrate the various optimization methods, perturbations, limits and penalties implemented. The first case examines a transfer between two highly elliptical
Earth orbits. Two Geosynchronous Transfer Orbit (GTO) to Geostationary Orbit (GTO) transfers are examined and one is compared to the results from [7] and [10] which solve a minimum time, constant thrust problem; the other is compared to the minimum fuel solution given in [10]. An orbit transfer involving a Mercury orbiter is used to
demonstrate the gravitational effects of a third body (the Sun) on the transfer. Orbit raising for an eccentric low
earth orbiting satellite shows the effects of drag on Low Earth Orbit (LEO) transfers. Orbit lowering for a spacecraft
in a Molniya orbit is used to demonstrate the effects of resonances induced by the tesserel and sectoral non-spherical
gravity effects. The advantages of using minimum fuel solutions are shown in the reduction of the fuel mass in
comparison to the same transfer with slightly more time than the minimum time case.
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2. Reference Frames and Orbital Elements
In this section, reference frames and orbital elements are introduced which will be used throughout the work.
2.1 Position and Velocity
The state of a spacecraft in orbit about a body can be completely defined by the six components which make up
its position and velocity vectors. These vectors are typically given in a Body-Centered Inertial Equatorial (BCIE)
frame as shown in Figure 2.1. The BCIE frame is defined by the unit vectors xˆ , yˆ and zˆ such that xˆ and yˆ lie in the equatorial plane and zˆ is normal to it in the direction of the geographic north pole. The xˆ axis points
towards in the vernal equinox direction (towards the Sun at the Northern Hemisphere Vernal Equinox), with ˆ y placed to create a right-handed coordinate system. Although this frame is not truly inertial as the equinox and plane of the equator change slightly over time, the inertial approximation is valid for the relatively short time intervals considered here.
Figure 2.1 Body-Centered Inertial Equatorial Frame
Thus, the spacecraft position and velocity would be given by: 4
r=rrrxyz x+ˆˆ y + zˆ (2.1.1) v= r=++rrrvxyz x+ˆˆ y z=ˆˆ x x+ ˆˆ v y y v z z (2.1.2)
In this work, the BCIE frame is used as the base coordinate system for any orbited body except the Sun.
Position and velocity vectors for objects orbiting the Sun are given in the Heliocentric Inertial Ecliptic (HIE) frame shown in Figure 2.2. The HIE frame is defined by the unit vectors xˆ H , yˆ H and zˆH such that xˆ H and yˆ H lie in the ecliptic plane and zˆH is normal to it aligned with the angular momentum vector of the Earth’s orbit. The xˆ H
axis points towards the vernal equinox, with yˆ H placed to create a right-handed coordinate system.
Figure 2.2 Heliocentric Inertial Frame
Thus, the position and velocity of a body orbiting the Sun would be given by:
r= rrrx x+ˆˆ H y y H+ zH zˆ (2.1.3) v= r=++rrrvx x+ˆˆ H y y H zH z=ˆˆ x x+ ˆˆ H v y y H v zH z (2.1.4)
In order to account for certain orbital perubtation effects, we must also consider a Body-Fixed Equatorial (BFE) frame as shown in Figure 2.3. The BFE frame is defined by the unit vectors xˆ BFE , yˆ BFE and zˆBFE such that xˆ BFE and yˆ BFE lie in the equatorial plane and zˆBFE is normal to it in the direction of the geographic north pole. The
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ˆ axis points towards the prime meridian, with ˆ placed to create a right-handed coordinate system. The xBFE yBFE angle from the vernal equinox to the prime meridian is known as the sidereal angle, ϑ . Thus, the transformation
from the BCIE to BFE frame is simply a rotation of ϑ about the z axis:
rBFE =Rz (ϑ) rBCIE (2.1.5) where
cos(ϑϑ) sin( ) 0 ϑ= − ϑϑ Rz ( ) sin( ) cos( ) 0 (2.1.6) 0 01
Since this frame is non-inertial, the non-spherical gravity effects are calculated in the BFE frame, but
transformed into an inertial frame before integration.
Figure 2.3 Body Fixed Equatorial Frame
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2.2 Keplerian Orbital Elements
The classical Kelperian orbital elements consist of five variables which describe the size, shape and location of an orbit, and one which describes the position of an object within the orbit. This separation between the slow-
varying (or constant) and fast-varying components allows for the use of orbital averaging over the fast element. The six elements are as follows:
a – semi-major axis
e – eccentricity
i – inclination
Ω – longitude of the ascending node
M – mean anomaly
The quickly varying mean anomaly which describes position within the orbit is related to the eccentric anomaly (E)
and true anomaly (ν) as follows:
M= Ee − sin E (2.2.1)
1/2 ν 1+ eE tan = tan (2.2.2) 21− e 2
The mean anomaly is used as one of the six main elements since it varies linearly with time making it easier to convert between time and angle. However, the eccentric and true anomalies provide more even spacing over the ellipse sample for eccentric orbits. They are all equal to one another for circular orbits.
2.3 Equinoctial Elements and Frame
The equinoctial orbital element set is preferred to the Keplerian for orbit transfer problems because the variational equations for the equinoctial elements are non-singular for all elliptic orbits. The six equinoctial elements are related to the Keplerian elements as follows:
aa= (2.3.1) he=sin(ω +Ω ) (2.3.2)
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ke=cos(ω +Ω ) (2.3.3)
pi= tan( / 2)sin Ω (2.3.4)
qi= tan( / 2)cosΩ (2.3.5)
LM= +ω +Ω (2.3.6)
The relationship between eccentric anomaly (E) and eccentric longitude (F) is given by:
FE= +ω +Ω (2.3.7)
The above set of elements is the direct equinoctial set. There is a singularity in the direct equinoctial set which occurs for equatorial retrograde orbits (i =180 °) . This can be solved by using a retrograde factor to switch to a slightly modified equinoctial set for retrograde orbits. However, only the direct set is used in this work since equatorial retrograde orbits are very rare.
The mean and eccentric longitudes are related via the following expression:
LFk=−+sin Fh cos F (2.3.8)
The inverse relationship between the Keplerian and equinoctial element sets is given as follows:
aa= (2.3.9)
1/2 ehk=( 22 + ) (2.3.10)
−−11 ω =tan(hk /) − tan( pq / ) (2.3.11)
−1 221/2 i= 2 tan ( pq+ ) (2.3.12)
−1 Ω=tan( pq / ) (2.3.13)
ML= −ω −Ω (2.3.14) ˆ ˆ The equinoctial frame is defined by the unit vectors f , gˆ and wˆ such that f and gˆ lie in the orbital plane
ˆ and wˆ is normal to it aligned with the angular momentum vector. The direction of f is obtained by a clockwise rotation of Ω from the ascending node. This frame is shown in Figure 2.4.
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Figure 2.4 Equinoctial Frame
The position and velocity vectors are given in the Equinoctial Frame by:
r= XY f+ˆ gˆ (2.3.15) v= r = XY f+ˆ gˆ (2.3.16) where the components are written in terms of the equinoctial elements and the eccentric longitude, F.
2 X=−+− a(1 hββ) cos F hk sin F k (2.3.17)
2 Y= a hkββcos F +−( 1 k) sin F − h (2.3.18) 21− 2 X= a nr hkββcos F−−( 1 h) sin F (2.3.19) 21− 2 Y=−− a nr(1 kββ) cos F hk sin F (2.3.20) with β, and n defined as:
β ≡+1/( 1 G) (2.3.21)
1/2 na≡ (µ / 3 ) (2.3.22) where μ is the specific gravitational constant of the orbited body and G is defined as:
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1/2 G≡−(1 hk22 −) (2.3.23)
The magnitude of the radius vector is given by:
1/2 22 r ==+=−−r( X Y) a(1 k cos Fh sin F) (2.3.24)
In order to transform from the equinoctial frame into either the BCIE or HIE coordinate systems, the equinoctial
frame basis vectors must be found in the other frame. In both cases, the vectors are given as follows:
1−+pq22 ˆ 1 f= 2 pq (2.3.25) K −2 p 2 pq 1 22 g=ˆ 1+−pq (2.3.26) K 2q 2 p 1 w=ˆ −2q (2.3.27) K 22 1−+pq where
K≡+1 pq22 + (2.3.28)
These vectors can then be used in Eqs. (2.3.15) and (2.3.16) to convert to either BCIE or HIE frame depending on what body the spacecraft is orbiting. In order to convert from BCIE/HIE frame to the equinoctial frame, the inverse relationships can be used:
T ˆ ˆˆ (2.3.29) r=feq g w rBCIE/ HIE ˆ ˆˆ (2.3.30) rBCIE/ HIE = f g wreq
A transformation between the equinoctial frame and the BFE frame and be obtained by applying Eq. (2.1.5) to the above BCIE equations:
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T T ˆ ˆˆ ϑ (2.3.31) r=feq g w Rz ( ) rBCIE/ HIE ϑ ˆ ˆˆ (2.3.32) rBFE =Rz ( ) f g w req
2.4 Spacecraft Fixed Frame
Another transformation that will be used is that from Equinoctial frame to the spacecraft body centered Radial
Transverse Normal (RTN) frame shown in Figure 2.5. The RTN frame is defined by the unit vectors rˆ , tˆ and nˆ such that rˆ and tˆ lie in the orbital plane and nˆ is normal to it aligned with the angular momentum vector (n=wˆˆ) . The direction of rˆ is equal to that of the position vector such that r=r/ˆ r .
Figure 2.5 Radial Transverse Normal Frame
The rotation matrix required to transform from the equinoctial frame to the RTN frame is as follows:
XY0 1 R= − YX0 (2.3.33) eq/ RTN r 00r
In order to invert the transformation, the transpose of Req/ RTN is used.
The RTN frame can also be located in terms of Spacecraft Spherical Coordinates: given by radius, geocentric latitude, and geographic longitude. The geocentric latitude is the angle from the equator to the spacecraft position
11 vector, and the geographic longitude is the angle from the prime meridian to the spacecraft position vector. These angles are shown in Figure 2.6.
Figure 2.6 Spacecraft Polar Coordinates
The geocentric latitude can be written in terms of the BFE/BCIE position vector components as follows:
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−−11 zBCIE zBFE φ = tan 1/2 = tan 1/2 (2.3.34) 2++ 2 22 ( xBCIE y BCIE ) ( xyBFE BFE )
Similarly, the geographic longitude is written in terms of BFE coordinates below:
− y ψ = tan 1 BFE (2.3.35) xBFE
These coordinates are useful when considering non-spherical gravity orbital perturbations.
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3. Optimal Control Problem Formulation
In this section, the optimal control problem is written as a combination of the minimum quadratic thrust and
minimum fuel mass problems. The optimum rates of change of the equinoctial orbital elements are derived for both
cases: minimum fuel and minimum thrust squared. A smooth transition between the two cases is outlined, and the
minimum time problem is also formulated.
3.1 Equations of Motion
The spacecraft thrust acceleration vector u can be expressed in the equinoctial or RTN frames with magnitude
given by u.
ˆ ˆ u=uf f + u g gˆˆˆ + u w w= uuu rtn r ++ t n ˆ (3.1.1)
1/2 1/2 222 222 u==u ( uuuf ++ gw) =( uuu rt ++ n) (3.1.2)
The thrust components in the RTN frame can be expressed in terms of the thrust magnitude, yaw angle (γ ) and
pitch angle (ψ ) as shown in Eq. (3.1.3). A conversion between the two frames is given in the previous chapter.
uurtn= cosψγ sin uu= cos ψ cos γuu= sin ψ (3.1.3)
The six state equations which completely define the motion of the spacecraft in a strong inverse-square force-
field have previously been written using the equinoctial orbital elements [9] and are represented by Eq. (3.1.4).
x = Mu + b (3.1.4)
T b= [00000 n] na≡ (/)µ 3 1/2 (3.1.5)
where the components of the M matrix are as given in the Appendix and are functions of the first five equinoctial
states and the eccentric longitude, F. Expanding Eq. (3.1.4) to give the individual rates of change of the elements
and including the mass rate of change as the seventh state results in the following:
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x1 = da/ dt = M11 ufg + M 12 u (3.1.6)
x2 ==++ dh/ dt M21 uf M 22 u gw M 23 u (3.1.7)
x3 ==++ dk/ dt M31 uf M 32 u gw M 33 u (3.1.8)
x4 = dp/ dt = M43 uw (3.1.9)
x5 = dq/ dt = M53 uw (3.1.10)
x6 = dL/ dt =+++ n M61 uf M 62 u gw M 63 u (3.1.11)
x7 = dm// dt = T ve (3.1.12)
where the thrust magnitude, T, and effective exhaust velocity, ve , are given by
T= mu (3.1.13)
ve= Ig SP e (3.1.14)
3.2 Maximizing Delivered Mass
The final dry spacecraft mass is given by integrating Eq. (3.1.12) over the transfer time.
t f f =0 − m m∫ ( T/ ve ) dt (3.2.1) 0
Thus, for a fixed transfer time, t f , maximizing the delivered mass is equivalent to minimizing the
instantaneous thrust magnitude over the transfer. However, the optimization problem of minimizing the thrust
magnitude is ill-behaved with poor convergence characteristics. Very similar results can be obtained by minimizing
the square of the thrust, which places a greater penalty on larger thrust values. This problem is much less sensitive
to initial guesses and more likely to converge to a solution. Both payoffs are used along with a weighting factor, α.
2 t f T J=αα m f +−(1 ) ∫ − dt (3.2.2) t0 2
The problem is first solved for α = 0 since it has better convergence characteristics. If more accurate results are
desired, the solution to the quadratic thrust (α = 0) problem is used as a starting point to solve the minimum fuel (α =
15
1) problem. The payoff, J, is to be maximized subject to the system dynamics given by Eqs. (3.1.6)-(3.1.12). The variational Hamiltonian can be written as
7 dJ dxi H = + ∑λi (3.2.3) dti=1 dt
This work is only concerned with the orbit transfer problem without rendezvous. Thus, the final value of the
mean longitude which describes position within the orbit is not specified. The Lagrange multiplier associated with
the mean longitude is set equal to zero to allow it to be uncontrolled.
λ6 = 0 (3.2.4)
Therefore, λ6 and is not used from this point forward. The Hamiltonian is then given by:
HTMMMmTMMMm=fg[ 11λλλ 1 ++ 21 2 31 3 ] // +[ 12 λλ 1 ++ 21 2 32 λ 3 ] (3.2.5) 2 +TMMMMmTvTwe[ 23λλλλλ 2 + 33 3 + 43 4 + 53 5 ] / −7 / −−( 1 α) /2
The problem is more easily solved by splitting the thrust into two independent direction components (TTfg, ) and the magnitude, T, so that the Hamiltonian is rewritten as:
HTMMMmTMMMm=fg[ 11λλλ 1 ++ 21 2 31 3 ] // +[ 12 λλ 1 ++ 21 2 32 λ 3 ] 1/2 (3.2.6) 222 +(TTT −−fg) [ M23λλλλλ 2 + M 33 3 + M 43 4 + M 53 5 ] // m −7 Tve
The optimality conditions for the control variables are then given by Eqs. (3.2.7)-(3.2.9), which have been
solved for the thrust components:
∂HT/0 ∂=ff ⇒T = TM[ 11λ 1 + M 21 λ 2 + M 31 λτ 3 ] / (3.2.7)
∂HT/0 ∂=gg ⇒T = TM[ 11λ 1 + M 21 λ 2 + M 31 λτ 3 ] / (3.2.8)
2=++ 222 ⇒ =λλλλτ + + + TTTTfgw TTw 2 M 23 3 M 33 4 M 43 5 M 53 / (3.2.9)
where τ is defined as
16
221/2 [MMMλλλ++] +[ MMM λλ ++ λ] 11 1 21 2 31 3 12 1 21 2 32 3 τ ≡ 2 (3.2.10) +λλλλ +++ 2MMMM 23 3 33 4 43 5 53
∂HT/ ∂=τλ / m −7 /1 ve −−( α) T = 0 (3.2.11) τλ//mv− T = 7 e (3.2.12) (1−α )
For the special case when α = 1, the value of the thrust magnitude is determined by picking the value that will maximize the Hamiltonian. Since the Hamiltonian is a linear function of the thrust magnitude, the partial derivative of the Hamiltonian with respect to the thrust magnitude can be used as a switching function.
∂HT/// ∂=τλ m −7 ve (3.2.13)
When ∂HT/0 ∂>, the maximum value of thrust available will maximize the Hamiltonian. When
∂HT/0 ∂<, the minimum value of thrust magnitude, zero, is used to maximize the Hamiltonian. The case when
∂HT/0 ∂=for a finite time is disregarded; the assumption is that there is no singular arc.
The rates of change of the remaining Lagrange multipliers are given as follows:
dλi ∂H =−=it1,...5,7 λ7 (f ) = 1 (3.2.14) dt∂ xi
where the final values of the remaining multipliers must be guessed (the initial and final values of the states are
known). The partial derivatives of the Hamiltonian with respect to the states are given in the Appendix.
3.2.3 Transitioning Between Cases
For some cases, the solution to the minimum thrust squared problem will not provide a close enough
approximation to ensure the convergence of the minimum fuel problem. In such instances, the value of α can be
varied incrementally from 0 to 1 so that the previous solution remains close enough to ensure convergence.
However, instead of maximizing the payoff for intermediate values of α, a weighted average of the optimal thrust
17
profiles from both α = 0 and α = 1cases is used. This leads to a smoother transition between the two thrust profiles.
The transitional optimal thrust is then given by:
TTff=αα,1αα= +−(1 ) Tf,0= (3.2.15)
TTgg=αα,1αα= +−(1 ) Tg,0= (3.2.16)
TTww=αα,1αα= +−(1 ) Tw,0= (3.2.17)
The rates of change of the states and Lagrange multipliers are then calculated using these weighted average
thrust values.
3.3 Minimizing Transfer Time
The problem of minimizing the transfer time. The payoff, J, is to be maximized subject to the system dynamics
given by Eqs. (3.1.6)-(3.1.12).
Jt= − f (3.3.1)
The variational Hamiltonian can be written as
55 dJ dxiidx H = +∑∑λλii =−+1 (3.3.2) dtii=11 dt = dt
For this case, the Lagrange multiplier associated with mass, λ7 , is equal to zero at the final time since the mass
is not being driven to a specific boundary condition. The Lagrange multiplier associated with the mean longitude,
λ7 , is also equal to zero since only orbit transfer is considered as explained in the previous section. The partial
derivative of the Hamiltonian with respect to the thrust components is the same in this case as for the minimum fuel
problem. Thus, the optimum thrust components for the minimum time problem are equal to those for the minimum
fuel problem and are given by Eqs. (3.2.7)-(3.2.9). However, in this case the thrust magnitude is chosen so as to
minimize the transfer time. The value of the thrust magnitude is determined by picking the value that will maximize
the Hamiltonian. Since the Hamiltonian is a linear function of the thrust magnitude, the partial derivative of the
Hamiltonian with respect to the thrust magnitude can be used as a switching function.
18
∂HT// ∂=τ m (3.3.3)
The variable, τ , is always positive so that ∂HT/0 ∂> at all times so that the maximum value of thrust available will always maximize the Hamiltonian. Thus, the minimum time problem leads to the use of constant maximum thrust over the course of the transfer.
The rates of change of the remaining Lagrange multipliers are given as follows:
dλ ∂H i =−=i 1,...5,7 (3.3.4) dt∂ xi where the initial values ust be guessed (the initial and final values of the states are known). The partial derivatives of the Hamiltonian with respect to the states are given in the Appendix.
19
4. Averaging and Multiple Time Scales
In this section, a small variable associated with the magnitude of the thrust is introduced which is then used to
justify an orbital averaging method for the dynamics equations. The averaging method is shown to be equivalent to
a multiple time scales formulation. The dynamic equations are then averaged numerically using a Gaussian
quadrature, which allows for the straightforward addition of complex orbital perturbations. A method for the
reacquisition of the periodic thrust profile is suggested, and the transformation between mean and oscillating orbital
elements is presented.
4.1 Small Variable
A small variable, ε , related to the magnitude of the thrust acceleration can be extracted from the dynamic equations and is defined as follows in order to have units of a time rate of change (s−1 ) .
1/2 amax ε ≡ umax (4.1.1) µ
The natural small variable associated with the spacecraft mass rate of change is different than that associated
with the orbital elements and Lagrange multipliers. It can be written as the max thrust acceleration magnitude over
the exhaust velocity which is typically very large. Thus, the mass small variable, εm , can be written in terms of ε
since it is typically much smaller (εεm << ) .
1/2 umax εµ ε m ≡= (4.1.2) vee vamax
Rewriting the rates of change of the states and Lagrange multipliers in terms of ε results in the following:
x == da/ dtε M** u += M ** u ελ g x,, F (4.1.3) 1 11f 12 gx1 ( 1...5,7 1...5,7 )
x == dh/ dtε M** u ++ M ** u M ** u =ελ g x,, F (4.1.4) 2 21f 22 g 23 wx2 ( 1...5,7 1...5,7 )
x == dk/ dtε M** u ++ M ** u M ** u =ελ g x,, F (4.1.5) 3 31f 32 g 33 wx3 ( 1...5,7 1...5,7 )
20
x = dp/ dt =ε M** u = ελ g x,, F (4.1.6) 4 43 wx4 ( 1...5,7 1...5,7 )
x = dq/ dt =ε M** u = ελ g x,, F (4.1.7) 5 43 wx5 ( 1...5,7 1...5,7 )
x = dL/ dt =+++ nε M** u M ** u M ** u =+ nελ g x,, F (4.1.8) 6 61f 62 gw 63 x6 ( 1...5,7 1...5,7 )
** 1/2 x7 = dm/ dt =−=−εεµme u m u m// v( amax ) = ε g x( x1...5,7,, λ 1...5,7 F ) (4.1.9) 7
where
* u≡ uu/ max (4.1.10)
1/2 1/2 ** Ma= ( max / µ ) ( a) M (4.1.11)
* a≡ aa/ max (4.1.12) Similarly
dλxi/ dt=∂∂= H / x ελ gλ ( x1...5,7 ,, 1...5,7 F ) (4.1.13) ii
Thus, all of the states and Lagrange multipliers except the mean anomaly are slowly varying, and the mean longitude is a fast variable.
4.2 Averaging Method
This formulation is very similar to Cefola [3] which follows Nayfeh's Generalized Method of Averaging [13]
with the exception that the natural rate of the fast variable is a function of the slow variables. Nayfeh assumes it is a
constant. Sanders [17] has shown that there exists a near-identity transformation from osculating to mean states
( xx→→,λλ) which eliminates the fast variable by averaging and produces a system of equations that
approximate the original system to first order O()ε on a time scale of order O(1 /ε ) . A near-identity transformation consists of a family of transformations which depend on ε and are reduced to the identity when
ε = 0 . The transformation takes the following form:
[1] 2 [2] xxxxFxxFiii=+ε i( 1...5,7,, λε 1...5,7 ) +i( 1...5,7, λ 1...5,7 ,) += ... 1,..7 (4.2.1) 21
[1] 2 [2] FFFxFFxF=++ε( 1...5,7,, λε 1...5,7 ) ( 1...5,7, λ 1...5,7 ,) + ... (4.2.2)
[1] 2 [2] λii=+ λ ελ i( xFxFi1...5,7,, λ 1...5,7 ) + ε λi( 1...5,7, λ 1...5,7 ,) += ... 1,..7 (4.2.3)
for some xj[]jj,λ [] ( = 1,2,...) which are 2π periodic in F . The goal is to choose x[]jj,λ [] such that the original
equations (4.1.3)-(4.1.9) and (4.1.13) are transformed into the averaged equations:
[1] 2 [2] dxix/ dt=ε G( x1...5,7, λε 1...5,7 ) +Gx( x1...5,7, λ 1...5,7 ) += ...i 1,..5,7 (4.2.4) ii
[1] 2 [2] dx6 / dt= n++ε Gxx( x1...5,7, λε 1...5,7 ) G( x1...5,7, λ 1...5,7 ) + ... (4.2.5) 66
[1] 2 [2] dλεi / dt= Gλλ( x1...5,7, λε 1...5,7 ) +G( x1...5,7, λ 1...5,7 ) += ...i 1,..5,7 (4.2.6) ii
j for some unknown Gj[] ( =1,2,...) induced by the transformation; where
[1] 2 [2] nn=+++εε nx(11 ) n ( x ) ... (4.2.7)
1/2 3 nx= (µ / 1 ) (4.2.8)
[]j In order to find the Gjx ( =1,2,...) induced by this transformation, Eq. (4.2.1) is substituted into (4.1.3)-
(4.1.9) with the following result:
[1] dxii dx dx i = +ε( xFgxFi1...5,7, λ 1...5,7 ,) += ... ελx ( 1...5,7, 1...5,7 ,) = 1,..5,7 (4.2.9) dt dt dt i
The time derivatives of the higher order states can be written in terms of the average mean longitude as
follows:
dx[1]∂∂ x [1] dL dx[2] x [2] dL ii= , i= i , .... (4.2.10) dt∂∂ L dt dt L dt
Substituting Eqs. (4.2.4) and (4.2.10) into Eq. (4.2.9) leads to the following:
22
[1] [1] ∂xi εGxx ( 1...5,7, λε 1...5,7 ) ++ ... n( x1...5,7, λ 1...5,7 , F) += ... i ∂L (4.2.11) [1] [1] [1] εgxx ( 1...5,7+ ε x 1...5,7 +..., λ 1...5,7 + ελ 1...5,7 ++...,FFε + ...) i
The original equations of motion g can be expanded in ε as follows: ( xi )
[1] [1] [1] gxx ( 1...5,7+ε x 1...5,7 +..., λ 1...5,7 + ελ 1...5,7 ++...,FFε + ...) i (4.2.12) =++gg[0]εε [1] 2 g [2] +... xxii x i
with the zeroth order form given by:
[0] gxx= gx( 1...5,7,,λ 1...5,7 F) (4.2.13) ii
Equating equal coefficients of ε in Eq.(4.2.11) results in equations of the form
[1] [1] ∂xi GxnxFgxFxx( 1...5,7,λ 1...5,7 ) +=( 1...5,7,, λλ 1...5,7 ) ( 1...5,7 ,, 1...5,7 ) (4.2.14) ii∂L
[2] [2] ∂xi [1] Gxxx( 1...5,7,λλ 1...5,7 ) += n( x1...5,7,, 1...5,7 Fg) (4.2.15) ii∂L
[]j []jj∂xi [− 1] Gxxx( 1...5,7,λλ 1...5,7 ) += n( x1...5,7,, 1...5,7 Fg) (4.2.16) ii∂L
The function g is assumed to be the sum of short periodic terms and long secular terms which do not xi contain the fast variable, the eccentric longitude. As stated earlier, the higher order states ( xx[1], [2] ,...) are chosen so
that the functions G[]j contain only secular terms (are not functions of the longitude). Thus, the higher order states xi
[j− 1] []j will contain all of the periodic components of g , leaving the remaining secular terms to G . The simplest xi xi
way of separating out the periodic components from the secular components is to integrate with respect to the
periodic variable, F , over one period and divide by the period (take the average). Since we are solving for a rate of
23
change with respect to time, the average must be taken over the mean longitude, L , which naturally varies linearly with time. The first order function is then given by:
1 L +π G[1] x ,λλ= g x,, F dL (4.2.17) xxii( 1...5,7 1...5,7 ) ∫ ( 1...5,7 1...5,7 ) 2π L −π
In order to evaluate the integral, the relationship between the average eccentric and mean longitudes must be
obtained. If the expanded variables are substituted into Eq. (2.3.8) relating the two longitudes and equal coefficients
of ε are equated, the following relationship between the average longitudes is found:
LFx=−+32sin Fx cos F (4.2.18)
Differentiating with respect to the average eccentric longitude and remembering that the average states are not
a function of the eccentric longitude results in the following:
dL =−−1x cos Fx sin F (4.2.19) dF 32
This relationship between the differential anomalies can be used to evaluate the integral in Eq. (4.2.17).
F +π dL G[1] x ,λλ= g x,, F dF (4.2.20) xxii( 1...5,7 1...5,7 ) ∫ ( 1...5,7 1...5,7 ) F −π dF As long as the integral is taken over one period, the integration limits can be arbitrary. Centering the integral about the mean eccentric longitude allows for more accurate numerical evaluation later. Identical relationships exist for the Lagrange multipliers. The solution can be found for the mean longitude using a similar method with the addition of the fast varying term, n .
[1] [1] ∂x6 GxnxFgxFnxx( 1...5,7,λ 1...5,7 ) +=+( 1...5,7,, λλ 1...5,7 ) ( 1...5,7 ,, 1...5,7 ) (4.2.21) 66∂L 1 F +π dL G[1] x ,λλ= g x,, F dF+ n (4.2.22) xx66( 1...5,7 1...5,7 ) ∫ ( 1...5,7 1...5,7 ) 2π F −π dF
Thus, the problem can be reduced to the following first order system:
24
xii=+= xO(ε ) i 1,..7 (4.2.23)
λλii=+=Oi( ε ) 1,..5,7 (4.2.24) with rates given by:
ε F +π dL dx/ dt= g x,,λε F dF+= O(2 ) i 1,..5,7 (4.2.25) ix∫ i ( 1...5,7 1...5,7 ) 2π F −π dF
ε E+π dL dx/ dt= g x,,λε F dF++ n O()2 (4.2.26) 6 ∫ x6 ( 1...5,7 1...5,7 ) 2π E−π dF
F +π ε dL 2 dλ/ dt= gλ x,, λε F dF+= O( ) i 1,..5,7 (4.2.27) i ∫ i ( 1...5,7 1...5,7 ) 2π F −π dF
These equations are the basis of the first-order averaging methods that have been widely applied to orbit
perturbation and low thrust problems and are valid for time scales of order 1/ε . Higher order terms can be found
by extension of this process.
4.3 Multiple Time Scale Formulation
The first-order equations obtained through averaging can also be derived through a multiple time-scale method
[17]. Considering the same problem given by Eqs. (4.1.3)-(4.1.13) we introduce a second time scale τ such that
τε= t (4.3.1)
The states and Lagrange multipliers can be expanded in ε as follows:
[0] [1] 2 [2] xxii=++εε x i x i i =1,..7 (4.3.2)
FF=++[0]εε F [1] 2 F [2] (4.3.3)
[0] [1] 2 [2] λii=++ λ ελ i ε λ i +...i = 1,..5,7 (4.3.4)
With the two time scales t and τ used as independent variables, the differential operator becomes:
d ∂∂ = + ε (4.3.5) dt∂∂ t τ 25
Applying this to Eq. (4.3.2) for the expanded states results in:
∂∂∂xxx[0] [0] [1] ∂ x[1] ∂ x[2] iii+εεε + +22 i + ε i +=... ∂∂ttττ ∂ ∂ ∂ t [0] [1] [0] [1] [0] [1] εgxx ( 1...5,7+ ε x 1...5,7 +..., λ 1...5,7 + ελ 1...5,7 +..., FF ++ε ...) (4.3.6) i i =1,..5,7 ∂∂∂xxx[0] [0] [1] ∂ x[1] ∂ x[2] 666+εεε + +22 6 + ε 6 +=... ∂∂ttττ ∂ ∂ ∂ t [0] [1] [0] [1] [0] [1] εgxx ( 1...5,7+ ε x 1...5,7 +..., λ 1...5,7 + ελ 1...5,7 +..., FF ++ε ...) (4.3.7) 6 [0] [1] +++nnε ... where
nn=++[0]εε n [1] 2 n [2] +... (4.3.8)
3 1/2 nx[0] = µ / [0] (4.3.9) ( 1 )
The equations of motion g can be expanded in ε as follows: xi
[0] [1] [0] [1] [0] [1] gxx ( 1...5,7+ε x 1...5,7 +..., λ 1...5,7 + ελ 1...5,7 +..., FF +ε +=...) i (4.3.10) gg[0]++εε [1] 2 g [2] +... i = 1,..7 xxii x i with the zeroth order form given by:
[0] [0] [0] [0] gxx= gx( 1...5,7,,λ 1...5,7 F) (4.3.11) ii
Equating equal coefficients of ε from Eq. (4.3.6) results in equations of the form:
∂x[0] i = 0 (4.3.12) ∂t ∂∂xx[1] [0] ii=−+g[0] (4.3.13) ∂∂t τ xi ∂∂xx[1] [1] ii=−+g[1] (4.3.14) ∂∂t τ xi 26
which can be solved successively. Applying the same process to Eq. (4.3.7) results in equations of the form:
∂x[0] 6 = n[0] (4.3.15) ∂t ∂∂xx[1] [0] 66=−++gn[0] [1] (4.3.16) ∂∂t τ x6 ∂∂xx[2] [1] 66=−++gn[1] [2] (4.3.17) ∂∂t τ x6
Integrating Eq. (4.3.12) and Eq. (4.3.15) with respect to time leads to:
xB[0]= [0] (τ ) (4.3.18) ixi
[0] [0] [0] x= B(τ ) + nt (4.3.19) 6 x6
where B[0] (τ ) and B[0] (τ ) have not yet been determined. Integrating Eq. (4.3.13) and Eq. (4.3.16) with respect xi x6 to a dummy variable, s, associated with time, t, results in:
t [0] dBx x[1] =−++i g[0] ds B[1] (τ ) (4.3.20) i∫ τ xxii 0 d
t [0] dBx x[1] =−6 +g[0] ds ++ n [1] B [1] (τ ) (4.3.21) 6 ∫ τ xx66 0 d
In order to avoid terms in the expansion that become unbounded with time, t, the non-secularity condition is
applied:
T [0] dBx −+i g[0] ds =0 B[1] (τ ) bounded (4.3.22) ∫ τ xxii 0 d
T [0] dBx −+6 g[0] ds =0 n[1] , B [1] (τ ) bounded (4.3.23) ∫ τ xx66 0 d
27
where T is the orbital period. This condition prevents the emergence of "false secular terms" which grow on the time scale 1/ε and destroy the asymptotic validity of the solution [17]. Since the integration is over an orbital period, a new dummy variable, , associated with the zeroth order mean longitude can be introduced. The SL[0]
relationship between and s is given as follows: SL[0]
[0] dS [0] ∂M M = = n[0] (4.3.24) ds∂ t
The variable n[0] is constant as so can be removed from the integrals and eliminated from the equations.
Applying the change of integration variables and rearranging the terms results in the following relationships:
dF [0] 2π xi 1 [0] [0] [0] = g x,,λ S[0] dS [0] (4.3.25) ∫ xi ( 1...5,7 1...5,7 FL) dτπ2 0
dF [0] 2π x6 1 [0] [0] [0] = g x,,λ S[0] dS [0] (4.3.26) ∫ x6 ( 1...5,7 1...5,7 FL) dτπ2 0
In order to evaluate the integral, the relationship between the zeroth order eccentric and mean longitudes must
be obtained (since the equations of motion are known in terms of eccentric longitude). If the expanded variables are
substituted into Eq. (2.3.8) and equal coefficients of ε are equated, the following relationship between the zeroth order longitudes is found:
[0] [0] [0] [0] [0] [0] LFx=−+32sin Fx cos F (4.3.27)
Differentiating with respect to the zeroth order eccentric longitude and noting that the zeroth order eccentricity is not a function of the longitude results in the following:
dL[0] =−−1x[0] cos Fx [0] [0] sin F [0] (4.3.28) dF [0] 32
Thus, a new integration variable associated with the zeroth order eccentric longitude can be used with relationship to given by: SL[0] 28
[0] dS [0] dL L = =−−[0] [0] (4.3.29) [0] (SF[0] ) 1 x32 cos Sx FF[0] sin S[0] dSF[0] dF
The limits of the integral are not important as long as it is taken over one period. However, if the evaluation of the integral is done numerically, more accurate results will be obtained if it is centered about the zeroth order (or average) eccentric longitude. Thus, Eqs. (4.3.25) and (4.3.26) can be rewritten as:
[0] F[0]+π dB dS [0] xi 1 [0] [0] [0] L = gx ( x1...5,7,,λ 1...5,7 S[0] ) dS [0] (4.3.30) τπ∫ i FF d 2 F[0]−π dSF[0]
[0] F[0]+π dB dS [0] x6 1 [0] [0] [0] L = gx ( x1...5,7,,λ 1...5,7 S[0] ) dS [0] (4.3.31) τπ∫ 6 FF d 2 F[0]−π dSF[0]
A similar method can be applied to the remaining slow variables. Thus the problem can be reduced to the following first order system:
[0] xxii= + O(ε ) (4.3.32)
[0] λλii= + O( ε) (4.3.33) with the rates given by the application of Eq. (4.3.5):
[0] F[0]+π dxi ε [0] [0] dSL[0] 2 = gx ( x1...5,7,,λε 1...5,7 S[0] ) dS[0] + O( ) (4.3.34) π ∫ i FF dt 2 F[0]−π dSF[0]
[0] F[0]+π dx6 ε [0] [0] dSL[0] [0] 2 = gx ( x1...5,7,,λε 1...5,7 S[0] ) dS[0] ++ n O( ) (4.3.35) π ∫ 6 FF dt 2 F[0]−π dSF[0]
[0] F[0]+π dλi ε [0] [0] dSL[0] 2 = gλ ( x1...5,7,,λε 1...5,7 S[0] ) dS[0] + O( ) (4.3.36) π ∫ i FF dt 2 F[0]−π dSF[0]
29
These equations are identical to those obtained using the generalized method of averaging and are valid for time
scales of order 1/ε .
4.4 Numerical Averaging
Due to the complexity of the system dynamics (especially when perturbations are included) an analytical
average cannot always be obtained in close form, so it is calculated numerically. An nth order Gaussian quadrature is used to solve Eqs. (4.2.25)-(4.2.26) and obtain the averaged state equations. This operation is also conducted on the equations for the rates of change of the Lagrange multipliers, Eq.(4.2.27). A Gaussian quadrature obtains the best numerical approximation of a definite integral by picking optimal values of the integration variable (in this case the eccentric longitude, F) at which to evaluate the function being integrated [1]. The optimal values of F for an nth order Gaussian quadrature are the roots of the orthogonal polynomial for the same interval and weighting function.
The weighting function is given by:
2 wj = 2 jn=1,2.... (4.4.1) − 2' (1Fj) PF nj ()
th where Pn is given by the n order Legendre Polynomial and Fj are the n roots of this orthogonal polynomial. The integral is conventionally defined over the interval [−1,1] so that the approximation takes the form
1 n ≅= ∫ fFdF( ) ∑ wfFjj( ) j1,2.... n (4.4.2) −1 j=1
for some generic function, f. An integral over a more general interval of [ab, ] must be transformed into an integral over [−1,1] before applying the quadrature:
b ba−n ba −+ ab ≅ += (4.4.3) ∫ f( F) dF∑ wjj f F j1,2.... n a 2j=1 22
30
This can be applied to approximate the first-order average rates of change of the states and Lagrange multipliers
as defined in Eqs. (4.2.25)-(4.2.27):
dx 1 F +π dL i = g[0] x,,λ F dF ∫ xi ( 1...5,7 1...5,7 ) dt 2π −π dF F (4.4.4) 1 n dL ≅ w g[0] x,,λπ F dF ∑ jxi ( 1...5,7 1...5,7 j) 2 j=1 dF
F +π dλi 1 [0] dL = gλ x,,λ F dF ∫ i ( 1...5,7 1...5,7 ) dt 2π −π dF F (4.4.5) 1 n dL ≅ w g[0] x,,λπ F dF ∑ jjλi ( 1...5,7 1...5,7 ) 2 j=1 dF
th where Fj are given by the roots of the n order Legendre polynomial. Once the averaged equations of motion are obtained, a Runge-Kutta-Fehlberg (RK45) method is used to integrate the states and Lagrange multipliers forward in time.
xti(+∆ t ) = xt ii () +∆ x−RK45 (4.4.6)
λi(tt+∆ ) = λλ ii () t +∆ −RK45 (4.4.7)
where the initial state variables are known (and assumed to be equal to the mean value) and the initial Lagrange multipliers must be guessed. A shooting method is used to solve the two-point boundary value problem of arriving at the specified final orbit.
When solving the maximum delivered mass problem (α = 1), the switching times are found by linear numerical integration over an orbital period. An iterative stepping method is used around the switching times to ensure that the switch timing is highly accurate (∂HT/ ∂≤ 10−16 ) . Once the switching times are known, an nth order Gaussian
quadrature is used to estimate the change in elements and Lagrange multipliers over each interval of uninterrupted
thrusting. The total change over the orbit is then equal to the sum of all of the changes over each interval. This
approach ensures a high degree of accuracy in the timing of the thrust switching. The use of averaging over less
than one orbital period remains valid due to the use of numerical integration via a Gaussian quadrature:
31
F Off dx 1 m k dL i = g[0] x,,λ F dF ∑ ∫ xi ( 1...5,7 1...5,7 ) dt 2π k =1 On dF Fk λ (4.4.8) mnOff On x1...5,7,, 1...5,7 1 FF− [0] dL kk Off On Off On ≅ wjx g dF ∑∑i FFkk−+ FF kk 22π kj=11= F + dF 22j
On Off where FFk and k represent the average eccentric anomaly when the thrusters are turned on and off, respectively,
for the kth thrusting interval; and m represents the number of thrusting intervals in one period. A similar expression can be written for the rates of change of the Lagrange multipliers.
4.5 Reacquiring Periodic Performance Data
If the number of time steps used is such that there will be at least one step per revolution about the central body
during the transfer, then an estimate of the periodic changes in thrust can be obtained. This is done by storing values
calculated for the eccentric longitudes, Fi , used during the Gaussian quadrature. The eccentric longitudes are sorted in order of increasing value over the interval [−ππ, ] . The thrust acceleration magnitude is then calculated according to Eq. (3.1.2). In order to view the periodic changes over the length of the transfer, the eccentric longitudes corresponding to the thrust magnitudes must be converted to values of time. A relationship between time and eccentric longitude can be found by taking the time derivative of Eq. (2.3.8)and setting it equal to Eq. (3.1.11).
dL d =−+(F ksin F h cos F) =+++ n Mu Mu Mu (4.5.1) dt dt 61f 62 gw 63
The assumption of low thrust leads to the approximation that n>> Mu61f + Mu 62 gw + Mu 63 in Eq. (4.5.1), which allows for simplification to:
dt =−−(1k cos Fh sin F) / n (4.5.2) dF
The state variables are assumed to remain constant and equal to their mean values over one orbit. Thus, integrating
Eq. (4.5.2) leads to the following relationship between time and eccentric longitude: 32
0 tt= + F ++−π hksin Fh + cos F / n (4.5.3)
where t 0 is the initial time. The values of eccentric longitude used for the numerical averaging procedure are taken
relative to the start of each orbital period. Thus, t 0 in Eq. (4.5.3) represents the time at the beginning of each orbital
period. The approximate time to complete one period is given by 2/π n , where na= (/µ 3 ) 1/2 . From this, the approximate time at the start of each orbital revolution is given as follows, with j denoting the value for the current orbit.
00 ttjj=−1 +=2π / n j2,3,.... (4.5.4)
0 For the first orbit: t1 = 0 . This method can be used to generate an approximate time history of periodic thrust and
power over the transfer. Unlike the state variables which change very little over one period, the thrust acceleration
magnitude can undergo large variations over one orbit. It is important to study the periodic values to determine
factors which drive mission design such as maximum and minimum thrust and power values and when they occur.
The thrust acceleration components can also be used to analyze periodic changes in thrust direction.
4.6 Mean Element Transformation
The true periodic behavior of the elements and thrust is also desirable. This can be obtained by applying the
solution to the averaged problem (initial Lagrange multipliers) to the unaveraged problem. However, in order to do
this the initial mean elements and Lagrange multipliers must be transformed into their osculating counterparts. This can be done using the mean element transformation described in [6], assuming the initial osculating mean longitude is given. The orbital elements, Lagrange multipliers, and mean and eccentric longitudes can be separated into the average and periodic components via the following transformation:
[P 1] 2 xxii=++εε x i O( ) i=1,...5 (4.6.1)
[P 1] 2 λii=++ λ ελ i Oi( ε ) =1,...5 (4.6.2)
[P 1] 2 LL=++εε L O( ) (4.6.3)
33
[P 1] 2 FF=++εε F O( ) (4.6.4)
with rates would be given by:
2 xii=+=εε xO( ) i1,...5 (4.6.5)
2 λii=+= ελOi( ε ) 1,...5 (4.6.6)
2 Ln =++εε LO( ) (4.6.7) FL=−−/( 1 x32 cos Fx sin F) (4.6.8)
[P 1] [ PP 1] [ 1] [ P 1] where xii,,,λ LF represent the first order periodic components of the orbital elements and Lagrange multipliers. Substituting the first order approximation of Eqs. (4.6.1)-(4.6.2) into Eqs. (4.1.3)-(4.1.7) and (4.1.13) results in the following:
dx[P 1] i [P 1] [ PP 1] [ 1] xix+=ε ε gx( 1,...5 + ε x 1,...5,, λ 1,...5 + ελ 1,...5 FF + ε ) (4.6.9) dt i
[P 1] dλi [P 1] [ PP 1] [ 1] λi += ε εgxλ ( 1,...5 + ε x 1,...5,, λ 1,...5 ++ ελ 1,...5 FF ε ) (4.6.10) dt i
where the time derivative of the first order periodic components of the orbital elements and Lagrange multipliers can
be written as:
[PP 1] [ 1] [ P 1] [ P 1] dxii dx ∂ x i dx i =+=+L xn O(ε ) (4.6.11) dt dL∂ xi dL
[PP 1] [ 1] [ P 1] [ P 1] ddλλii ∂ λ i d λ i =+=+Lλε nO( ) (4.6.12) dt dL∂ xi dL
Substituting Eqs. (4.6.11)-(4.6.12) into (4.6.9)-(4.6.10) and collecting first order terms leads to:
∂x[P 1] i xnix+= gx( 1,...5,,λ 1,...5 F) (4.6.13) ∂L i
[P 1] ∂λi λλi +=n gxλ ( 1,...5,, 1,...5 F) (4.6.14) ∂L i 34
Recall that the average rates of change of the states and Lagrange multipliers are given by Eqs.(4.2.25)-(4.2.26)
reproduced below:
ε F +π dL x = gx,,λ F= gx, λε , F dFO+ ()2 ixii( 1,...5 1,...5 ) ∫ x( 1...5,7 1...5,7 ) 2π F −π dF (4.6.15) i =1,..5,7
F +π ε dL 2 λλ= gxλλ,, F= gx,λ , F dFO+ () ε i ii( 1,...5 1,...5 ) ∫ ( 1...5,7 1...5,7 ) 2π F −π dF (4.6.16) i =1,..5,7
where the function g is evaluated at the current values of the average states, Lagrange multipliers and eccentric
longitude. Thus, Eqs. (4.6.13)-(4.6.14) can be written as:
[P 1] ∂xi n= gxxx1,...5,,λλ 1,...5 FFF( ,∆−) gx( 1,...5 ,, 1,...5 F) (4.6.17) ii( ) ∂L
[P 1] ∂λi n= gxλλ1,...5,,λλ 1,...5 FFF( ,∆−) gx( 1,...5 ,, 1,...5 F) (4.6.18) ii( ) ∂L
where the short periodic variable is now ∆F and is evaluated on the interval [−ππ, ] around the mean value F and
∂(FF + ∆) = ∂∆ F. Assuming the functions g can be expanded in a Fourier series as functions of the eccentric longitude variation ∆F , the right hand side of Eqs. (4.6.17)-(4.6.18) can be approximated as follows:
∞ m jm∆ F gxFFFgxFgxFex1,...5,,λ 1,...5 ( ,∆−) xx( 1,...5 ,, λλ 1,...5 ) = ( 1,...5,, 1,...5 ) (4.6.19) i( ) ii∑ m=−∞ m≠0
∞ m jm∆ F gxλ1,...5,,λ 1,...5 FFF( ,∆−) gx λλ( 1,...5 ,, λλ 1,...5 F) = gx( 1,...5,, 1,...5 Fe) (4.6.20) i( ) ii∑ m=−∞ m≠0
where the zero-frequency term is simply the average value, hence the Fourier series has a zero mean value.
Substituting Eqs. (4.6.19)-(4.6.20) into Eqs. (4.6.17)-(4.6.18) and integrating yields the following solution for the first order periodic components of the states and Lagrange multipliers:
35
∞ gxm ,,λ F [P 1] 1 xi ( 1,...5 1,...5 ) jm∆ F xei = ∑ (4.6.21) n m=−∞ jm m≠0
∞ gxm ,,λ F [P 1] 1 λi ( 1,...5 1,...5 ) jm∆ F λi = ∑ e (4.6.22) n m=−∞ jm m≠0
where the constant of integration is also zero by virtue of defining the average of the transformation to be zero. The first order transformation from average to periodic elements and Lagrange multipliers at the current mean longitude is found by setting ∆L equal to zero:
∞ gxm ,,λ F 1 xi ( 1,...5 1,...5 ) 2 xxii=++∑ O(ε ) (4.6.23) n m=−∞ jm m≠0
∞ gxm ,,λ F 1 λi ( 1,...5 1,...5 ) 2 λλii=++∑ O(ε) (4.6.24) n m=−∞ jm m≠0
The first order transformation from periodic to average elements and Lagrange multipliers is given by rearranging the previous equations:
∞ gxm ,,λ F 1 xi ( 1,...5 1,...5 ) 2 xxii=−+∑ O(ε ) (4.6.25) n m=−∞ jm m≠0
∞ gxm ,,λ F 1 λi ( 1,...5 1,...5 ) 2 λλii=−+∑ O(ε) (4.6.26) n m=−∞ jm m≠0
These values are then used to propagate the un-averaged problem forward to the final time and compare to the
averaged solution.
36
5. Optimal Control Problem Solution
In this section, the two-point boundary value problem is solved numerically using a shooting method for both the minimum fuel and minimum time problems.
5.1 Maximizing Delivered Mass
In order to solve the averaged two-point boundary value problem, the transition matrix, G, of the average
Hamiltonian system must be found for the final time specified.
δ x δ x0 = (5.1.1) [G]0 δλ δλ
Because the averaging is done numerically, the transition matrix cannot be found analytically, but must be solved for numerically. To do this, the initial values of the Lagrange multipliers are guessed and the equations are integrated forward to the specified final time as outlined in the previous section. Then, a small change is made to one of the Lagrange multipliers and the equations are integrated forward again to a new set of final conditions. This is done in succession to each Lagrange multiplier in both directions such that 10 additional integrations are made.
00 ff λi, new−=−⇒(1 ελ) i xnew−−,λ new i =1,...,5 (5.1.2)
00 ff λi, new+=+⇒(1 ελ) i xnew++,λ new i =1,...,5 (5.1.3)
In this way, the transition matrix shown below can be calculated numerically. ΦΦ 11 12 G = (5.1.4) ΦΦ 21 22
The only component of the transition matrix that is calculated is Φ since it is the only one necessary to 12
calculate the desired change in Lagrange multipliers.
Φ f 0 12−i = δδx / λ (5.1.5)
37
The value of Φ12 is computed for changes in both directions, and the average of those two values is used.
Φ 1 ff 00 ff 00 th =(xx − )/(λλ − ) +− (xx )/(λλ − ) (5.1.6) 12,i column 2 new−− i i,, new new ++ i i new
The change in initial Lagrange multipliers necessary to arrive at the desired final state variables is given by
solving part of Eq. (5.1.1):
01− ff ∆=λ Φ12 ε ( xxdesired −) (5.1.7)
In order to maintain the linearity assumed in Eq. (5.1.7), the changes made must be very small. Thus, the
calculated change is multiplied by a small number, ε , chosen large enough to affect meaningful change but small enough to maintain linearity. This can be accomplished by checking that the predicted change in x for a given
change in λ predicted by Eq. (5.1.1) is approximately equal to what is observed. Once the change in the initial
Lagrange multipliers is determined, then the new initial Lagrange multipliers are given by:
000 λλλnew = +∆ (5.1.8)
The equations are integrated forward again with these new initial conditions and the entire process is repeated
until the final states are reached within the desired accuracy. In practice, this process is carried out in reverse, with the final Lagrange multipliers being guessed and the integration carried out from final to initial time. This is done because the value of the Lagrange multiplier associated with the spacecraft mass is known at the final time.
Carrying out a backwards integration allows for one less Lagrange multiplier to be varied.
It is necessary to ensure that no conjugate points are passed through or the inverse in Eq. (5.1.7) will not exist.
It has been shown that many conjugate points exist for this problem [19], so the shooting method outlined here will only converge to an optimal solution if the initial guess is very good and the extremal path does not pass through a conjugate point. For the minimum thrust squared problem, this issue can be avoided for many transfers by guessing very small initial values for all of the Lagrange multipliers except for the one corresponding to semi-major axis.
Then, as long as the initial Lagrange multiplier for the semi-major axis is guessed in the vicinity of the solution, the method will tend to converge for the unperturbed problem. When adding perturbations or thrust limits it is best to start with the unperturbed/unlimited problem. Once it has converged to a solution, the final values of the initial 38
Lagrange multipliers are used as a guess to the same problem with other effects added. When a final solution has
been obtained for the minimum thrust squared problem, the Lagrange multipliers at the final time can be used as an
initial guess to the minimum fuel problem. Alternatively, the minimum time solution can be used as a starting point
to the minimum fuel problem. However, convergence behavior tends to be better if starting with the minimum
thrust squared solution. The error in desired boundary values is calculated as the sum of the final (or initial) errors
in the states scaled by their desired values follows:
f ff ∆=xScaled ∑ ( xDesired − xx Actual) / Actual (5.1.9)
−5 It has been observed that the fuel mass varies by less than 0.03% after converging with ∆≤xScaled 10 , so this value is what was used as sufficiently converged for the example cases. However, convergence is more difficult when shadowing is included, so an error of 10-3 is allowed (fuel mass varies by less than 0.3%).
5.2 Minimizing Time
For the minimum time problem, the same averaged two-point boundary value problem must be solved with the terminal time, t f , no longer specified, but determined implicitly by the terminal conditions. The new transition matrix, G, of the average Hamiltonian system at the final time is given by:
δδxxf 0 f 0 δδλλ= [G] (5.2.1) f dΩ dt
where
5 dxi Ω=∑ λi (5.2.2) i=1 dt tt= f Because the averaging is done numerically, the transition matrix cannot be found analytically, but must be solved for numerically. To do this, the initial values of the Lagrange multipliers and the final time are guessed and the equations are integrated forward to the guessed final time. Then, a small change is made to one of the Lagrange multipliers or final time and the equations are integrated forward again to a new set of final conditions. This is done 39
in succession to each Lagrange multiplier and the final time in both directions such that 12 additional integrations
are made.
00 ff λi−− new =−⇒(1 ελ) i xnew−,λ new − i =1,...,5 (5.2.3)
00ff λi−+ new =+⇒(1 ελ) i xnew+,λ new + i =1,...,5 (5.2.4)
f f ff ttnew−=−⇒(1,ε ) xnew−−λ new (5.2.5)
f f ff ttnew−=−⇒(1,ε ) xnew++λ new (5.2.6)
In this way, the transition matrix shown below can be calculated numerically. ΦΦΦ 11 12 13
ΦΦΦ = (5.2.7) G 21 22 23 ΦΦΦ 31 32 33
The only components of the transition matrix needed to calculate the desired change in Lagrange multipliers and final time are shown in the following equation:
δδx f ΦΦ λ 0 = 12 13 f (5.2.8) dΩ ΦΦ32 33 dt
The values of these components are computed for changes to the Lagrange multipliers and final time in both directions, and the average of those two values is used.
Φ 1 12 =νν −λλ00 − +−νν λλ00 − (5.2.9) ( new−) //( i i,, new −+) ( new) ( i i new +) Φ th 2 32 i column
Φ 1 13 =νν −ff − +−νν ff − (5.2.10) ( new−−) //(tt i new ) ( new ++) (ttnew ) Φ33 2
where
x f ν = (5.2.11) Ω
40
The change in initial Lagrange multipliers necessary to arrive at the desired final state variables is given by
solving Eq.(5.2.8):
−1 ∆λ 0 ΦΦ = 12 13 ε νν− f ( desired ) (5.2.12) ∆t ΦΦ32 33
In order to maintain the linearity assumed in Eq. (5.2.8), the changes made must be very small. Thus, the
calculated change is multiplied by a small number, ε , chosen large enough to affect meaningful change but small enough to maintain linearity. This can be accomplished by checking that the predicted change in x and Ω for a given change in λ and t f predicted by Eq. (5.2.12) is approximately equal to what is observed. Once the change in the initial Lagrange multipliers and final time is determined, then the new initial Lagrange multipliers and final time are given by:
000 λλλnew = +∆ (5.2.13)
ff f tttnew = +∆ (5.2.14)
41
6. Limits, Penalty Functions, and Shadowing
In this section, limits on the available thrust due to thruster physical properties and available propulsion power are established. A periapsis penalty function is implemented as a method of avoiding orbits with low periapsis
which may dip into the atmosphere or below the surface of the central body. The effect of shadowing on spacecraft
using solar electric propulsion is also examined.
6.1 Thrust and Power Limits
For the minimum thrust squared case (α = 0), it is useful to specify the maximum power available for propulsion in order better represent the limitations of real spacecraft. Maximum and minimum limits on thrust also reflect physical limitations of real thrusters. Eq. (6.1.1) shows that the power used for propulsion is a scalar multiple of the thrust; the exhaust velocity and thruster efficiency, η , are assumed to be constant.
Ttv() Pt()= e (6.1.1) 2η
Thus, any limit placed on power is essentially just a limit on the thrust magnitude. Limits can be placed on the thrust magnitude as follows:
Tmin ≤≤ TTmax (6.1.2)
The thrust limits can be imposed on either the periodic or average thrust values. Periodic thrust limits allow the
placement of a physical limit on the absolute thrust available at any given time. It should be used if the spacecraft is
to be operated using the variable thrust solution provided by the minimum thrust squared optimization. However,
placing a limit on the average thrust results in a solution that more closely resembles that given by the minimum fuel
optimization. Results using both methods are presented in the examples section.
The minimum fuel problem requires a maximum thrust (or power) value to be specified. The thrust magnitude
then alternates between maximum and zero depending on when it is most efficient to thrust.
42
The minimum time problem also required a maximum thrust or power to be specified. The thruster then operates at the maximum value for the entire transfer (unless shadowed).
6.2 Periapsis Penalty Function
The payoff can be multiplied by a penalty function in order to avoid trajectories which dip below a minimum periapsis [15]. This is useful in order to avoid trajectories which pass too closely to the surface of an orbited body or dip into its atmosphere. The new payoff is given as below, with the penalty function defined in Eq. (6.2.2).
2 t f f T J=αα m +−(1 ) ∫ − +R dt (6.2.1) t0 2
RW= expρ 1− rr (6.2.2) R ( ppmin )
where WR is a switching variable equal to one or zero (penalty function on or off) and ρ is a scaling factor. The periapsis is given by the following relationship:
1/2 22 ra=[11 −= ea] − hk + (6.2.3) p ( )
The updated Hamiltonian is then:
6 222 dxi H=−−− R(1 αλ)( TTTfgw + +) /2 +∑ i (6.2.4) i=1 dt
Since R is a function of the states, the equations used for the rates of change of the Lagrange multipliers must be
updated. The following additions are made to the right hand side of Eqs. (3.2.14):
dλ1,R ∂R = (6.2.5) dt∂ a
dλ2,R ∂R = (6.2.6) dt∂ h
43
dλ3,R ∂R = (6.2.7) dt∂ k
where the partial derivatives of the penalty function are given by:
∂ − ρ ∂r R WR p = expρ ( 1− rrpp) (6.2.8) ∂∂xr x min ipmin i
∂ 1/2 rp 22 =−+1 (hk) (6.2.9) ∂a
∂rp −ha = 1/2 (6.2.10) ∂h (hk22+ )
∂rp −ka = 1/2 (6.2.11) ∂k (hk22+ )
6.3 Shadowing Effects
For missions utilizing Solar Electric Propulsion (SEP), thrusting is assumed to not take place during periods
when the spacecraft is in shadow. This can be accounted for by changing the limits of the integrals of Eqs. (4.4.4)-
(4.4.5) to be rewritten as
FF+ S dx 1 EN dL i = g[0] x,,λ F dF (6.3.1) ∫ xi ( 1...5,7 1...5,7 ) dt 2π S dF FF− EX
S FF+ EX dλi 1 [0] dL = gλ x,,λ F dF (6.3.2) ∫ i ( 1...5,7 1...5,7 ) dt 2π S dF FF− EN
SS where FFEXand EN represent the eccentric longitudes at the exit and entrance of the shadowed portion of the orbit
(no distinction is made between penumbra and umbra). The following assumptions are made in the calculation of the shadow entry and exit angles: the shadow is cylindrical, the Sun’s direction is fixed over one spacecraft
44
revolution, and the thruster is turned on immediately upon exit of the shadow. Recall from Chapter 2 that the
position vector of the spacecraft relative to the central body in the spacecraft equinoctial frame is given by:
r=XY fgˆ + ˆ (6.3.3)
The position vector from the central body to the sun can also be found in the spacecraft equinoctial frame as
shown in Chapter 2 and can be written as:
ˆ rSS=XYZ f ++ S gwˆˆ S (6.3.4)
with its unit vector given by:
ˆ rS ˆˆ ˆˆ rS== XYZ SS f ++ gwˆˆ S (6.3.5) rS
ˆ If rCB designates the mean radius of the central body, then the cosine of the angle between r and rS is given by:
1/2 2 2 ˆ r − rCB rrS ⋅ ( ) = − (6.3.6) rr
or
1/2 ˆˆ+=−− 2 2 X XSS YY( r rCB ) (6.3.7)
Squaring and rearranging the terms in (6.3.7), the above equation results in:
ˆ2 2 ˆ 22 ˆˆ 2 S≡−(1 XS ) X +−( 12 YS ) Y − XYSS XY − rCB = 0 (6.3.8)
This is the shadow equation which must be satisfied by the entry and exit angles. Further manipulation results
in a quartic equation in cos F :
43 2 SA≡0cos FA + 1 cos FA + 2 cos FA + 34 cos FA += 0 (6.3.9) 45 where
22 Ahh01= + 4 (6.3.10)
A11245=22 hh + hh (6.3.11)
2 22 A2=+ h 22 hh 31 −+ h 4 h 5 (6.3.12)
A0=22 hh 32 + hh 45 (6.3.13)
22 Ahh035= + (6.3.14) and
22 22 h1= d 1( b 1 −+ b 2) d 2( b 2 −− b 3) d 3( bb 1 2 − bb 23) (6.3.15)
h2=−−+22 dkb 11 dhb 2 2 dkb 3( 2 + hb ) (6.3.16)
22 22 22 hdbkdbhdbbhkra3= 12( ++) 23( +−) 323( +−) CB / (6.3.17)
2 h4=22 bb 12 d 1 + bbd 23 2 −+ d 3( b 2 bb 13) (6.3.18)
h5=−−22 kbd 21 hbd 32 + dkb 3( 3 + hb 2) (6.3.19)
ˆ 2 dX1 =1 − S (6.3.20) ˆ 2 dY2 =1 − S (6.3.21)
d3 = 2 XYSS (6.3.22)
2 bh1 =1 − β (6.3.23)
b2 = hkβ (6.3.24)
2 bk2 =1 − β (6.3.25)
ˆ Spurious roots can be eliminated by the criteria that rrS ⋅<0 . The entry and exit angles can be distinguished by noting that ∂SF/0 ∂
46
7. Perturbations
In this section, orbital perturbations due to solar radiation pressure, atmospheric drag, a non-spherical central
body, and third body gravitational effects are adjoined to the optimum transfer problem.
7.1 Solar Radiation Pressure
The acceleration acting on a spacecraft due to solar radiation pressure is given by:
pASR (1+ β ) S rS / C -Sun uSR = (7.1.1) mt()rS / C -Sun
where
p – Solar radiation pressure at orbited body SR
β – Optical reflection constant −≤11β ≤ ( )
AS – Spacecraft projected area facing the sun rS / C -Sun – Spacecraft position relative to the sun
The solar radiation pressure and spacecraft area facing the sun are assumed to be constant. The position of the
spacecraft relative to the sun is assumed to be approximately equal to the position of the central body relative to the
sun.
rrS / C -Sun CB-Sun (7.1.2)
The first order average rates of change of the states due to solar radiation pressure are given by:
FF+ S 1 EN dL = (7.1.3) x SR ∫ MuSR,eq dF 2π S dF FF− EX
where uSR,eq is the acceleration due to solar radiation pressure expressed in the spacecraft equinoctial frame. In order to obtain uSR,eq it is necessary to find the position vector of the spacecraft relative to the sun in the spacecraft 47
SS equinoctial frame using the transformations given in Chapter 2. The variables FFEXand EN represent the eccentric longitudes at the exit and entrance of the shadowed portion of the orbit as calculated in Section 6.3. If the orbit is unshadowed, then the integral is taken over a full period.
The equations for the Lagrange multipliers must also be updated to account for these additions to the state dynamics. The average rates of change of the Lagrange multipliers due to solar radiation pressure are given by:
FF++SSFF EN EN ∂ 11∂x TTdL ∂M uSR,eq dL =−SR λλ =−+ (7.1.4) λSR ∫∫dF uMSR,eq dF 22ππSS∂xdF ∂∂xxdF FF−−EX FFEX where the partial derivatives are given in the Appendix. The integrals for the average rates of the states and
Lagrange multipliers are calculated using Gaussian quadrature as explained in Chapter 4. The average rates are then added to the right hand side of Eqs. (4.4.4)-(4.4.5).
7.2 Atmospheric Drag
For orbits which extend into the atmosphere of the central body, the force due to atmospheric drag will affect the orbit. The acceleration acting on a spacecraft due to atmospheric drag can be modeled as follows:
1 ρCA u= − DD vv (7.2.1) AD 2mt () S// C−− ATM S C ATM where
CD – Drag coefficient AD – Spacecraft projected area in the vS/ C− ATM direction
ρ – Atmospheric density vS/ C− ATM – Spacecraft velocity relative to the Atmosphere
The drag coefficient and projected area are assumed to be constant over the transfer. The density is a function of the spacecraft altitude (ha ) given by:
a hr=S/ C − r CB (7.2.2)
48
where rCB is the mean radius of the central body. The density is interpolated from tabular data using the following
equation:
aa hh0 − ρρ= 0 exp (7.2.3) H S where
ρ0 – Reference density
a h0 – Reference altitude
H S – Scale Height For purposes of calculating vS/ C− ATM , the atmosphere is assumed to rotate uniformly with the same angular velocity as the body. This assumption is adequate for obtaining first-order results.
vS/ C− ATM = vv − ATM (7.2.4) vrATM =ϑ × (7.2.5) where ϑ is the angular velocity vector of the spacecraft central body (sidereal rate). The perturbation due to
atmospheric drag can be accounted for by simply adding the following terms to the right hand side of Eqs. (3.1.6)-
(3.1.11):
FF+ D 1 EX dL = (7.2.6) x AD ∫ MuAD,eq dF 2π D dF FF− EN
where uAD,eq is the acceleration due to atmospheric drag expressed in the spacecraft equinoctial frame. In order to obtain uAD,eq it is necessary to find the velocity vector of the spacecraft relative to the atmosphere in the spacecraft
DD equinoctial frame using the transformations given in Chapter 2. The variables FFEXand EN represent the eccentric
longitudes at the exit and entrance of the portion of the orbit when the spacecraft is within the atmosphere. If the
radial distance to the top of the atmosphere ratm is known, then the atmospheric entry and exit angles are given by:
49
D −1 1/− raatm FEN = −cos (7.2.7) e
D −1 1/− raatm FEX = cos (7.2.8) e
If the orbit lies fully within the atmosphere, then the integral is taken over a complete period.
The equations for the Lagrange multipliers must also be updated to account for these additions to the state dynamics. The average rates of change of the Lagrange multipliers due atmospheric drag are given by:
FF++SSFF EN EN ∂ 11∂x TTdL ∂M uSR,eq dL =−SR λλ =−+ (7.2.9) λSR ∫∫dF uMSR,eq dF 22ππSS∂xdF ∂∂xxdF FF−−EX FFEX where the partial derivatives are given in the Appendix. The integrals for the average rates of the states and
Lagrange multipliers are calculated using Gaussian quadrature as explained in Chapter 4. The average rates are then added to the right hand side of Eqs. (4.4.4)-(4.4.5):
7.3 Zonal Harmonics
The non-spherical axial symmetric gravity acceleration perturbations (zonal harmonics) can significantly affect a spacecraft's orbit. The effects can be expressed as the gradient of the following potential:
µ ∞ r n = ∇ CB φ (7.3.1) UZ ∑ n JPnn0 (sin ) rrn=2 where
Pn0 – Associated Legendre function of order 0 and degree n
φ – Geocentric latitude
Jn – Un-normalized zonal harmonic coefficient of order n
n – Degree of geopotential field
The geocentric latitude can be written in terms of the BCIE or BFE position vector components as:
50
−1 z φ = tan 1/2 (7.3.2) 22 ( xy+ )
From Vallardo [17], using matrix differentiation in the BCIE frame, we can directly determine the partial derivatives of the perturbing potential function. We differentiate the Legendre function in spherical coordinates, but find the acceleration in Cartesian coordinates, so we have to apply the chain rule:
TT ∂∂UUTT∂∂r φ uT, BCIE = + (7.3.3) ∂∂r rr ∂∂φ where the partial derivatives of the potential with respect to the spherical coordinates is given by:
∂U ∞ µr n T = − + CB φ (7.3.4) ∑(n 1) n+2 JPnn0 (sin ) ∂rrn=2 ∂U ∞ µr n T = CB φ (7.3.5) ∑ n+1 Jn Pn1 (sin ) ∂φ n=2 r and the partial derivatives of the spherical coordinates with respect to the BFE position vector are given by:
T ∂r r = (7.3.6) ∂r r
T ∂∂φ 1 r zz = −+ 1/2 2 (7.3.7) ∂∂rr( xy22+ ) r
Substituting Eqs. (7.3.6)-(7.3.7) into Eq. (7.3.3) results in the following accelerations in the BFE frame:
∂∂ 1 UUTTz uxTx, = − 1/2 (7.3.8) rr∂∂22 2 φ rx( + y) ∂∂ 1 UUTTz uyTy, = − 1/2 (7.3.9) rr∂∂22 2 φ rx( + y)
1/2 22 z ∂∂UU( xy+ ) u =TT + (7.3.10) Tz, rr∂∂ r2 φ 51
These accelerations can be transformed into the equinoctial frame using the relation given in Chapter 2.
T ˆ ˆˆ (7.3.11) uT,, eq =f gw uT BCIE
The first order average rates of change of the states due to zonal harmonics are then given by:
F +π 1 dL = (7.3.12) xT ∫ MuZ,eq dF 2π F −π dF
where the limits of the integration always encompass a full spacecraft orbital period.
The equations for the Lagrange multipliers must also be updated to account for these additions to the state
dynamics. The average rates of change of the Lagrange multipliers due to zonal harmonics are given by:
FF++ππ∂ 11∂x TTdL ∂M uZ,eq dL =−Z λλ =−+ (7.3.13) λZ ∫∫dF uMZ,eq dF 22ππFF−−ππ∂xdF ∂∂ xxdF
where the partial derivatives are given in the Appendix. The integrals for the average rates of the states and
Lagrange multipliers are calculated using Gaussian quadrature as explained in Chapter 4. The average rates are then
added to the right hand side of Eqs. (4.4.4)-(4.4.5).
7.4 Tesserel/Sectorial Harmonics
The non-spherical, non-symmetric gravity acceleration perturbations (tesseral/sectorial harmonics) are usually
very small for near-spherical bodies such as Earth. However, resonant effects can have a significant impact over
long periods of time for certain orbits. In addition, these effects are magnified in smaller, less uniform bodies.
These effects can be expressed as the gradient of the following potential:
µ ∞ n r n =∇+CB φψ ψ (7.4.1) UT ∑∑ n Pnm (sin)( Cnm cos( mS )nm sin( m )) rrnm=21 = where
Pnm – Associated Legendre function of order m and degree n
52
φ – Geocentric latitude
ψ – Geographic longitude
– Geopotenetial Constant Coefficients CSnm, nm
m – Maximum order of geopotential field (mn≤ )
n – Maximum degree of geopotential field
The geocentric latitude was written in terms of the BFE/BCIE position vector components in Eq. (7.3.2) and the
geographic latitude can be calculated using Eq. (7.4.2):
− y ψ = tan 1 BFE (7.4.2) xBFE These angles were shown in Figure 2.6. The gravitational potential is expressed in terms of spherical coordinates, but the acceleration is desired in equinoctial coordinates. Since the geographic attitude must be expressed in terms of the BFE frame position components, we first find the acceleration in the BFE frame by
applying the chain rule as in the previous section:
TT T ∂∂∂UUUTT∂∂∂r φψ T uT, BFE =++ (7.4.3) ∂∂r rr ∂∂φψ ∂ ∂ r
where the partial derivatives of the potential with respect to the spherical coordinates is given by:
∂U ∞ n µr n T = − + CB φψ+ ψ (7.4.4) ∑∑(nP1) n+2 nm (sin)(Cnm cos( mS )nm sin( m )) ∂rrn=21m =
µr n ∞ n CB φ− φφ ∂UT n+1 (PPnm+1 (sin) m tannm ( sin )) = ∑∑r (7.4.5) ∂φ nm=2 =1 ×+(Cnmncos( mSψψ )m sin( m ))
∂U ∞ n µr n T =CB φ−+ ψψ (7.4.6) ∑∑ n+1 Pnm (sin)mC( nm sin( m ) Snm cos( m )) ∂ψ nm=21 = r
and the partial derivatives of the spherical coordinates with respect to the BFE position vector are given by:
T ∂r rBFE = (7.4.7) ∂rBFE r 53
T ∂∂φ 1 r z z = −+BFE 1/2 2 (7.4.8) ∂∂rr22 r BFE ( xyBFE+ BFE ) BFE
∂φ 1 ∂∂yx = BFE − BFE 22xyBFE BFE (7.4.9) ∂+∂∂rBFExy BFE BFE rrBFE BFE
Substituting Eqs. (7.4.4)-(7.4.9) into Eq. (7.4.3) results in the following accelerations in the BFE frame:
∂∂ ∂ 1 UT z UT yUBFE T uxT, x =−−1/2 BFE 1/2 (7.4.10) rr∂∂222++φψ22 ∂ rxy( BFE BFE ) ( xyBFE BFE ) ∂∂ ∂ 1 UT z UT xUBFE T uyT, y =−+1/2 BFE 1/2 (7.4.11) rr∂∂222++φψ22 ∂ rxy( BFE BFE ) ( xyBFE BFE )
1/2 22 z ∂∂UU( xyBFE+ BFE ) u =TT + (7.4.12) Tz, rr∂∂ r2 φ
These accelerations can be transformed into the equinoctial frame using the relations given in Chapter 2:
T T ˆ ˆˆ ϑ (7.4.13) uT,, eq =f gw Rz( ) u T BFE
The accelerations induced by the tesseral and sectoral harmonics are dependent on the central body's orientation
relative to the spacecraft, expressed via the sidereal angle, θ . The sidereal rate θ introduces a new time scale into the problem that may be fast or slow. If the sidereal rate is O(ε ) then it is slow and can be assumed to be constant over the spacecraft's orbital period. It can be shown that faster sidereal rates only introduce periodic effects into the state dynamics and thus can be ignored in the average case. However, if a rational ratio can be formed from the sidereal and mean longitude rates, then a resonance effect will introduce secular terms into the equations of motion.
The three cases are examined in the following sections.
54
7.4.1 Slow Sidereal Rate
If the sidereal rate is O(ε ) then it is considered to be one of the slow variables and is assumed to be constant over the period of averaging. The first order average rates of change of the states due to tesserel/sectoral effects would then be given by:
F +π 1 dL = (7.4.14) xT ∫ MuT,eq dF 2π F −π dF where the limits of the integration always encompass a full spacecraft orbital period.
The equations for the Lagrange multipliers must also be updated to account for these additions to the state dynamics. The average rates of change of the Lagrange multipliers due to tesserel/sectoral effects are given by:
FF++ππ∂ 11∂x TTdL ∂M uT,eq dL =−T λλ =−+ (7.4.15) λT ∫∫dF uMT,eq dF 22ππFF−−ππ∂xdF ∂∂ xxdF where the partial derivatives are given in the Appendix. The integrals for the average rates of the states and
Lagrange multipliers are calculated using Gaussian quadrature as explained in Chapter 4. The average rates are then added to the right hand side of Eqs. (4.4.4)-(4.4.5).
7.4.2 Fast Sidereal Rate
For faster sidereal rates when the mean longitude and sidereal angle rates are not resonant, it is necessary to use a double average over the mean longitude and sidereal angle to obtain accurate average results. The first order average rates of change of the states due to tesserel/sectoral effects would then be given by:
ππF + 1 dL x = Mu dFdθ (7.4.16) T 2 ∫∫ T,eq 4π −−ππF dF
Since the sidereal angle is only present in the gravitational potential, the integral of the potential over a sidereal period can be examined in order to find an analytical average:
55
π µ ∞ n r n Udθφ= ∇ CB P sin ∫ T ∑∑ n nm ( ) −π rrnm=21 = (7.4.17) π αθ−+ αθθ − ∫ (Cnm cos( mm ) Snm sin( mmd )) −π
The above integral is identically equal to zero. Thus, for fast sidereal rates, the tesserel/sectoral effects do not
contribute any secular accelerations to the spacecraft and can be ignored when integrating the averaged equations of
motion. There is an exception when the spacecraft mean longitude and body sidereal rates are in resonance as
discussed in the following section.
7.4.3 Resonant Sidereal Rate
If a rational ratio can be formed from the sidereal and mean longitude rates, then a resonance effect will
introduce secular terms into the equations of motion. In a resonant region, there exists a rational ratio of two
integers such that
LQ = → QP: Resonance (7.4.18) θ P
The resonances will typically be more significant for smaller values of P and Q. Due to this relationship
between the sidereal and mean longitude rates, the average integral can be taken over the full period with the values
of the appropriate sidereal angle calculated for each value of mean longitude in the quadrature analyzed. The first
order average rates of change of the states due to tesserel/sectoral effects would then be given by:
F +π 1 dL = (7.4.19) xT ∫ MuT,eq dF 2π F −π dF
The average rates of change of the Lagrange multipliers due to tesserel/sectoral effects are given by:
FF++ππ∂ 11∂x TTdL ∂M uT,eq dL =−T λλ =−+ (7.4.20) λT ∫∫dF uMT,eq dF 22ππFF−−ππ∂xdF ∂∂ xxdF
During orbit transfer, the rate of change of the mean longitude will vary as the semi-major axis changes. Thus, it may be important to be able to ascertain when a region of resonance has been entered and exited. For example, 56
the resonance averaging structure could only be used when the ratio Q/P is rational as explained earlier to within
some percentage. Then if 1% is chosen, a spacecraft would be considered in the 2:1 resonance region as long as
1.98≤≤QP / 2.02 . In all other regions, unless the sidereal rate is slow as discussed previously, the tesserel/sectoral effects are neglected since they do not produce any secular terms as shown in the previous section.
However, the size of the resonance region can change for each problem so that defining such a percentage is difficult. Alternatively, the sidereal rate can be treated as in a resonant state for the entire transfer. In areas of non- resonance, the integral in (7.4.19) should calculate out to be zero if it is a fast variable. This method is more computationally expensive, but easier to use when unsure of the extent of the region of resonance.
7.5 Multi-Body Effects
The acceleration caused by celestial bodies other than the central body can be accounted for in the following
manner:
r− rr u = µ CB-PB − CB-PB (7.5.1) PB PB 33 rCB-PB − rrCB-PB
where
µPB – Specific gravitational constant of perturbing body rCB-PB – Position vector from central body to perturbing body
If there is more than one perturbing body, then the total acceleration is obtained by summing the accelerations
due to each perturbing body:
N uu= (7.5.2) PB∑ PBi i=1
where N is the number of perturbing bodies. As shown in Eq. (7.5.1), the acceleration depends on the position of the
perturbing body. In most cases, the perturbing body's period is large enough so that its position can be assumed to
be constant over the spacecraft orbital period. However, if the period is sufficiently small, a second average over the
perturbing body's orbital period may be necessary. The two cases are treated separately in the following sections.
57
7.5.1 Long Perturbing Body Period
If the orbital period of the perturbing body is sufficiently large such that the mean longitude rate is O(ε ) , it is assumed to be constant over the period of averaging. The first order average rates of change of the states due to the perturbing body is then be given by:
F +π 1 dL = (7.5.3) x PB ∫ MuPB,eq dF 2π F −π dF
where the limits of the integration always encompass a full spacecraft orbital period.
The equations for the Lagrange multipliers must also be updated to account for these additions to the state
dynamics. The average rates of change of the Lagrange multipliers due to the perturbing body are given by:
FF++ππ∂ 11∂x TTdL ∂M uPB,eq dL =−PB λλ =−+ (7.5.4) λPB ∫∫dF uMPB,eq dF 22ππFF−−ππ∂xdF ∂∂ xxdF
where the partial derivatives are given in the Appendix. The integrals for the average rates of the states and
Lagrange multipliers are calculated using Gaussian quadrature as explained in Chapter 4. The average rates are then
added to the right hand side of Eqs. (4.4.4)-(4.4.5).
7.5.2 Short Perturbing Body Period
For perturbing bodies with periods not sufficiently long enough to be consider the mean longitude constant
over one spacecraft orbit, the value can be updated for each point in the Gaussian quadrature from which the
numerical average rate are calculated based. As the period grows shorter, higher and higher order Guassians must
be used in order to maintain the accuracy of the solution. At some point it becomes more effective to perform a
second average over the period of the perturbing body. In this case, the first order average rates of change of the
states due to the perturbing body would be given by:
F −π F +π 1 PB dL x = Mu dFdF (7.5.5) PB 4π 2 ∫∫ PB,eq dF PB FFPB −−ππ
58
Calculating this double average numerically is computationally expensive and there are few interesting cases for which it is required. Thus, this case has not been implemented in this work.
59
8. Example Transfers
In this section, the results of five orbit transfer cases are studied which demonstrate the various optimization
methods, perturbations, limits and penalties implemented.
8.1 Case A: Highly Elliptic Orbit Transfer
An example minimum quadratic thrust transfer between two elliptic orbits was solved for the following
conditions for a specified time of 90 days (the values of λ 0 were guessed according to the method described in the shooting method section):
0 T x =[12956km −− 0.2614 0.4026 0.0092 0.1761]
f T x = [15000km 0.5416− 0.0955 0.3546 0.7605]
f −−−−−−7 1 19 19 19 19 T λ = 2.2X 10km 10 10 10 10
aa0 12956km f 15000km ee0 0.48 f 0.55 µ = 398574km32 / s 0 = °°f = ωω210 2 75 0 ge = 9.81 ms / f ii20°°80 0 f ΩΩ3°°35
0 m=500 kg η = 0.65ISP = 3000 s
The problem was first solved without any perturbations, shadowing, penalties, or thrust limits imposed for the
α = 0 case. A 10th order Gaussian quadrature was used for the numerical averaging, and a variable integration step size of 6 orbital periods for the RK45 integration. The time histories of the equinoctial elements are shown in Figure
8.1 for both the initial guess and final converged solution. The elements stay fairly constant for the initial guess trajectory since the initial Lagrange multiplier guesses were so small. Figure 8.2 shows a similar effect for the
Lagrange multiplier time histories. It took 22 steps and 127 seconds to converge to the solution within the desired accuracy (on a dual core 2.86 GHZ processor), specified according to the below condition. The error in the final
60
-5 states is represented by the variable ∆xScaled and was specified to be less than 10 . The value of ε used for this case was 0.5.
f ff −5 ∆xScaled =∑ ( xDesired − xx Actual) / Actual ≤×1 10 (8.1.1)
Figure 8.1 Case A - Equinoctial Element Time History
Figure 8.2 Case A - Lagrange Multiplier Time History
Once the minimum quadratic thrust problem has been solved, the solved values of the Lagrange multipliers at the final time can be used as an initial guess to the minimum fuel problem. A smooth transition was not necessary in 61 this case. It took 20 steps and 428 seconds (on a dual core 2.86 GHZ processor) to converge to the solution to the same accuracy and using the same small variable as the minimum quadratic thrust case. Accurately finding the switching times for the minimum fuel solution is computationally expensive, thus causing the optimal convergence process to take longer even though fewer steps were used. The maximum thrust specified was 0.5N which corresponds to 11.32kW of power delivered to the thruster or thrusters.
For this problem, the solution passes very close to the surface of the Earth unless the periapsis penalty function is used. A comparison of the periapsis time histories for the minimum fuel problem with the penalty function on
(with the values specified below) and off is shown in Figure 8.3.
−6 W =1 ρ =1.1X 10 r= 6578.1 km Rpmin
Figure 8.3 Case A - Periapisis Time History
The time histories of the Keplerian elements and spacecraft mass are shown in Figure 8.4 when the effects of solar radiation pressure are considered with the parameters specified below.
2 −62 A=10 m β = 0 pSR = 4.57 x 10 Nm /
62
The longitude of ascending node is increased most efficiently at high eccentricity, so the eccentricity initially
grows while the longitude of ascending node grows rapidly at maximum eccentricity and then levels off. The
inclination and eccentricity are most easily changed at large semi-major axis values, so the semi-major axis also initially increases past the final value while the inclination grows rapidly when the semi-major axis is at its peak value. The eccentricity also decreases rapidly at the maximum semi-major axis. The argument of perigee is most efficiently changed at low eccentricities, so it drops quickly at the minimum eccentricity value. The mass change is directly related to the average thrust: mass decreases more rapidly during period of higher thrusting and vice versa.
The propellant mass required for the transfers are as follows: minimum time with solar radiation pressure and penalty function – 105.9 kg, minimum fuel with solar radiation and penalty function – 91.7 kg, minimum fuel without solar radiation and penalty function – 91.0 kg, minimum thrust squared with solar radiation and penalty function - 97.4 kg. A comparison of these cases is shown in Table 8.1. The minimum fuel solution results in a
13.4% decrease in fuel mass and a 24.1% increase in transfer time over the minimum time solution. As expected, the minimum quadratic thrust solution results in greater propellant mass used, but still provides a relatively close estimate. The effects of solar radiation pressure and the penalty function increase the required propellant mass slightly, which is also expected. The fuel mass values remained consistent to 10-4 kg when tested using Gaussian quadratures of orders 10 to 100.
Transfer Fuel Fuel Mass ΔV Revs Time Mass Ratio (days) (kg) (mP/ m0) (km/s) Min Time 72.5 105.9 0.211 7.04 244 Min Fuel: On 90.0 91.7 0.183 5.43 306 Min Fuel: Off 90.0 91.0 0.182 5.41 325 Min T2 90.0 97.4 0.195 6.19 297 Table 8.1 Case A – Comparison of Results, ISP = 3000s, Tmax = 0.50N
63
Figure 8.4 Case A - Keplerian Elements and Spacecraft Mass Time Histories
The periodic and average thrust values are plotted in Figure 8.5. The average thrust is limited to less than 0.5N for the minimum quadratic thrust problem. The peak thrusting occurs at the times when the values of semi-major axis and eccentricity are at their extremes. Changing certain orbital elements is done most efficiently at these points as discussed earlier. The periodic thrust values for the minimum fuel problem can be seen to vary discretely between the maximum value and zero for similar reasons: it is more efficient to change the orbital elements during certain portion of the orbital period. Thruster on/off periods on the order of a few hours can be seen in plot. This type of quick switching is operationally undesirable, and constraints on thruster on/off time may be a useful future addition to eliminate such events. Such constraints would increase the fuel mass required for this transfer, as thrusting would no longer take place only during the most efficient portions of the orbit.
64
a)
b)
Figure 8.5 Case A - a) Average Thrust b) Periodic Thrust
Figure 8.6 shows the changes in thrust direction angles (pitch and yaw as shown in Figure 2.4) over time. The thrust angles tend to vary drastically over each orbital period. This is typically not operationally desirable as it implies large attitude maneuvering. A constraint on thrust angles could reduce this effect, however it would also
65
cause more fuel to be used. Figure 8.7 shows the orbit transformation over time with one orbit plotted approximately every 10 revolutions from the initial orbit to the final orbit.
Figure 8.6 Case A - Thrust Direction Angles
Figure 8.7 Case A - Orbit Transformation
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8.2 Case B: GTO to GEO Transfer (ARIANE 4)
This example is an attempt to reproduce the case of a transfer from a Geosynchronous transfer orbit to a
Geostationary orbit used by Geffory and Epenoy. [7] The transfer is solved without any orbital perturbations or
shadowing effects present. The initial and final values of the Keplerian elements are shown below; the initial orbit
is meant to simulate the GTO of an ARIANE 4 launch vehicle. The spacecraft is assumed to have an initial mass of
2000kg and the thruster is assumed to have an ISP of 2000s and maximum thrust of 0.35N.
aa0 24505.9km f 42165km ee0 0.725 f 0 µ = 398600.47km32 / s 0 = ° f = ωω0 Undefined 2 0 f ge = 9.8665 ms / ii70°° 0 f ΩΩ0° Undefined
0 m= 2000 kg TMAX = 0.35 N ISP= 2000 s
The minimum time solution is given by [7] as taking 137.5 days, 190 revolutions and 212 kg of fuel mass. The method of this paper gives a minimum time solution of 136.3 days, 190 revolutions, 211kg of fuel mass; so the results are in close agreement.
For a transfer lasting 150 days, the minimum fuel solution given in [7] takes 216 revolutions and 192kg of fuel.
The method used in this paper provides a minimum fuel solution which takes 218 revolutions and uses 187.6 kg of
fuel. This gives an 11.1% decrease in fuel mass for a 10.1% increase in time compared to the minimum time
solution. It is a 2.3% decrease in fuel mass from the solution given in [7].
If the square of thrust is minimized with a limit placed on the periodic thrust value, the fuel mass used is 194.3
kg. If the limit is instead placed on the average thrust, the fuel mass used is 187.3 kg. However, it should be
reiterated that the periodic thrust is allowed to take on values greater than the maximum thrust allowed by the
thruster so that this case is not physically possible. It is used because it tends to provide a closer estimate to the
minimum fuel solution.
The minimum fuel solution was solved using Gaussian quadratures of varying orders to study the effective error
on fuel mass used with the following results: 10th order – 187.61 kg, 32nd order – 187.58 kg, 64th order – 187.58 kg,
100th order - 187.58 kg. Based on this, a 64th order quadrature was chosen for this problem.
67
Transfer Fuel Fuel Mass ΔV Revs Time Mass Ratio (days) (kg) (mP/ m0) (km/s) Reference 7 Min Time 137.5 212 0.106 * 190 Min Fuel 150 192 0.096 * 216 Method of this Paper Min Time 136.3 211 0.106 2.18 190 Min Fuel 150 187.6 0.094 1.71 218 Min T2: Avg 150 187.3 0.097 1.85 218 Min T2: Per 150 194.3 0.094 1.73 218 Table 8.2 Case B – Comparison of Results, ISP = 2000s, Tmax = 0.35N
Figure 8.8 shows the time history of the semi-major axis, eccentricity, inclination, and spacecraft mass over the transfer for the minimum fuel and both minimum thrust squared cases. They are very similar for all three cases.
Figure 8.9 shows the average thrust profile for the three cases and the min time case, and the periodic thrust profile for the minimum fuel case. The average thrust profiles are all similar, with dips during the middle of the transfer when it is least efficient to thrust. Thus, using a minimum thrust square solution gives a good preliminary result for a minimum fuel transfer which could be obtained very quickly.
Figure 8.8 Case B - Keplerian Elements and Spacecraft Mass Time Histories 68
a)
b)
Figure 8.9 Case B - a) Average Thrust b) Periodic Thrust
Figure 8.10 shows the changes in thrust direction angles (pitch and yaw as shown in Figure 2.4) over time. It is very similar to what is shown in [7].
69
Figure 8.10 Case B - Thrust Direction Angles
The final converged values for the initial Lagrange multipliers can be used to propagate the un-averaged
trajectory forward to the final time. Figure 8.11 compares the un-averaged and averaged states for this case, and the results are very similar. The un-averaged elements oscillate about the average values since the periodic effects have not been removed via orbital averaging. The fuel mass used for the un-averaged problem is 186.4 kg, very similar to the averaged case. Figure 8.11 shows the orbit transformation over time with one orbit plotted approximately every
10 revolutions from the initial orbit to the final orbit. The transfer is shown from two views: the left highlighting the plane change, and the right the change in the size and shape of the orbit
70
Figure 8.11 Case B - Min Fuel Keplerian Element and Spacecraft Mass
Figure 8.12 Case B - Orbit Transformation
8.3 Case C: GTO to GEO Transfer (Taurus)
This example will compare the minimum fuel solution to the minimum time solution for a transfer from a
Geosynchronous transfer orbit to a Geostationary orbit used by Kluever and Oleson. [10]. Although [10] solved a minimum time, constant thrust problem, the results using minimum fuel, variable thrust optimization are similar.
Shadowing and J2 effects are accounted for with the start date of the transfer set to be Jan 1, 2000. The initial and final values of the Keplerian elements are shown below; the initial orbit is meant to simulate the GTO of a Taurus launch vehicle. The maximum power available is assumed to be a constant 5kW and the ion engine specifications 71
are given below. The transfer time was set to 75 days, 8 days longer than the minimum time solution given in [10] of 67 days.
aa0 24364.3km f 42164km ee0 0.731 f 0 µ = 398600km32 / s 0 = ° f = ωω0 Undefined 2 0 f ge = 9.81 ms / ii27°° 0 0 f ΩΩ99° Undefined
0 m=450 kg η = 0.65 ISP= 3300 s
The minimum time solution is given by [10] as taking 66.6 days. The method of this paper gives a solution of
64.6 days, so the results are in close agreement.
For a transfer lasting 75 days, the minimum fuel solution given by the method used in this paper takes 107
revolutions and uses 30.56 kg of fuel. If the square of thrust is minimized with a limit placed on the periodic thrust
value, the fuel mass used is 31.94 kg. If the limit is instead placed on the average thrust, the fuel mass used is 32.31
kg. The minimum fuel solution uses 12.4% less propellant mass and takes 16% more time than the minimum time
solution. A comparison of these cases is shown in Table 8.3.
Transfer Fuel Fuel Mass ΔV Revs Time Mass Ratio (days) (kg) (mP/ m0) (km/s) Reference 10 Min Time 66.6 * * * * Method of this Paper Min Time 64.9 34.9 0.078 2.58 88 Min Fuel 75.0 30.6 0.068 1.98 107 Min T2: Avg 75.0 31.9 0.071 2.17 107 Min T2: Per 75.0 32.3 0.072 2.21 107 Table 8.3 Case C – Comparison of Results, ISP = 3300s, Pmax = 5kW
The time histories of the Keplerian elements and spacecraft mass are shown in Figure 8.13 for this transfer and
compared to minimum time results from [10]. The results are very similar to the reference cases as expected.
72
Figure 8.13 Case C - Keplerian Element and Spacecraft Mass Time Histories
Figure 8.14 shows the average periodic thruster power used over the transfer. The average power is seen to dip during the middle of the transfer when it is less efficient to thrust. The initial dip in the average power plot is caused by the large shadowed portion of the orbit when the spacecraft is in a smaller orbit. This effect then disappears as the orbit grows larger. This is also the reason that the average power for the minimum time solution is not constant.
73
a)
b)
Figure 8.14 Case C - a) Average Thrust b) Periodic Thrust
The changes in thrust direction angles (pitch and yaw as shown in in Figure 2.4) over time are displayed in
Figure 8.15. Figure 8.16 shows the orbit transformation over time with about one orbit plotted for each of the approximately 107 revolutions completed. The transfer is shown from two views: the left highlighting the plane change, and the right the change in the size and shape of the orbit.
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Figure 8.15 Case C - Thrust Direction Angles
Figure 8.16 Case C - Orbit Transformation
8.4 Case D: Elliptic Low Earth Orbit Raising
This case evaluates the use of electric propulsion for the long term orbit raising of a spacecraft in an elliptic
Low Earth Orbit (LEO), where atmospheric drag can affect the orbit. The initial orbit is slightly elliptical with a minimum periapsis altitude of 192 km. The goal is to increase the semi-major axis by 20 km over 90 days, bringing the minimum periapsis altitude up to approximately 210 km. The atmospheric density model used is explained in 75
Section A.4 of the Appendix. The spacecraft projected area and drag coefficient are assumed to be 20 m2 and 2, respectively. The initial mass is assumed to be 2000 kg, and the maximum thrust allowed is 0.01 N. The following transfer is then considered:
aa0 7300 f 7320 ee0 0.10 f 0.10 µ = 398574km32 / s 0 = °°f = ωω149 149 2 0 f ge = 9.81 ms / ii51.4°°51.4 0 f ΩΩ322°°322
0 = η = = m2000 kg 0.50 ISP3000 s
For a transfer lasting 90 days, the minimum fuel solution given by the method used in this paper takes 1250 revolutions and uses 1.28 kg of fuel. If the square of thrust is minimized the fuel mass used is 1.46 kg. If drag effects are not considered, the minimum fuel solution only uses 0.68 kg of fuel. This is expected, since atmospheric drag tends to reduce the semi-major axis, whereas the thruster is working to increase it. The minimum time solution takes 43.5 days and uses approximately the same amount of fuel mass as the minimum fuel problem. This is due to spending less time in the higher drag region, which works to reduce the orbit size. Thus, there is no real advantage to using the longer minimum fuel transfer to raise the orbit. However, a minimum fuel solution would be beneficial for station keeping in low elliptic orbits, as the thruster would only fire when it would be most efficient to counteract the drag forces and maintain constant orbital elements.
Table 8.4 Case D – Comparison of Results, ISP = 3000s, Tmax = 0.01N
76
The time histories of the Keplerian elements and spacecraft mass are shown in Figure 8.17 for this transfer. The
plot shows an approximately linear increase in semi-major axis when drag is not considered, with almost no change in the other elements. When the drag effects are introduced, slightly larger changes in the other elements are seen, though they are still kept very small. The semi-major axis initially increasing more sharply to reduce the amount of time spent at the lowest altitudes where drag is highest.
Figure 8.17 Case D - Keplerian Element and Spacecraft Mass Time Histories
Figure 8.18 shows the average and periodic thrust magnitude used over the transfer. When drag is not considered, the average thrust is nearly constant. However, with drag effects included, the maximum thrust is used in the beginning of the transfer in order to quickly increase the semi-major axis and reduce orbit degradation caused by higher densities at low altitudes. The thrust value then slowly reduces over the rest of the transfer.
77
a)
b)
Figure 8.18 Case D - a) Average Thrust b) Periodic Thrust
The changes in thrust direction angles (pitch and yaw as shown in Figure 2.4) over time are displayed in Figure
8.19. As expected, the thrust yaw angle varies little over any given period since the drag perturbation is always opposed to the velocity vector and the orbit is nearly circular. The pitch angle also changes little since the drag force does not affect the non-planar elements significantly. 78
Figure 8.19 Case D – Thrust Direction Angles
8.5 Case E: Mercury Orbiter
This example examines the effects of the Sun's gravity and solar radiation pressure on changing the orientation of a Mercury orbiter similar to Messenger. The Messenger spacecraft orbits Mercury in a highly elliptical orbit since many of its instruments require it to be close to the planetary surface, but cannot stay there long due to thermal considerations. Thus, it only gets a close look at the northern hemisphere of the planet. This example examines a low-thrust transfer to change the argument of periapsis so that the Mercury orbiter can examine the Southern hemisphere of the planet as well. The initial orbit is very similar to that of Messenger, and the only desired change is a decrease in argument of periapsis of 180 degrees. The initial mass is assumed to be 500 kg, and the maximum thrust allowed is 0.15N. The effects of solar radiation pressure and the gravity of the Sun are accounted for. The transfer time was set to 88 days, or approximately the orbital period of Mercury about the Sun and the transfer was started when Mercury's mean anomaly was equal to 179º.
79
aa0 10039.7 f 10039.7 ee0 0.7371 f 0.7371 µ = 22030.6km32 / s 0 = °°f = ωω60 240 2 0 f ge = 9.81 ms / ii82.5°° 82.5 0 f ΩΩ90°° 90
0 m=500 kg η = 0.50 ISP= 3000 s
The minimum time solution with the effects of solar radiation pressure and the Sun’s gravity included results in
a transfer which takes 58.4 days and uses 25.71 kg of fuel. For a transfer lasting 88 days, the minimum fuel solution
with perturbations included uses 20.56 kg of fuel. This gives a 57.8% decrease in fuel mass for a 50.7% increase in
time compared to the minimum time solution. If the perturbations are not included, then the fuel mass used goes
down to 20.40 kg. The values are nearly identical since the perturbations re very small. Minimizing the square of
the thrust with perturbations included results in 23.36 kg of fuel used. The results are summarized in Table 8.5.
Transfer Fuel Fuel Mass ΔV Revs Time Mass Ratio (days) (kg) (mP/ m0) (km/s) Min Time: On 58.4 25.71 0.0514 1.55 94 Min Fuel: On 88.0 20.56 0.0411 0.99 131 Min Fuel: Off 88.0 20.40 0.0408 0.97 130 Min T2: On 88.0 23.36 0.0467 1.27 137 Table 8.5 Case E – Comparison of Results, ISP = 3000s, Tmax = 0.15N
The time histories of the Keplerian elements and spacecraft mass are shown in Figure 8.20 for this transfer. The
eccentricity is reduced to near zero where it is almost free to change the argument of periapsis, and then it returns to
the initial value. The semi-major axis increases since it is more efficient to change eccentricity at larger values of semi-major axis (lower orbital velocities), then decreases back to its original value. Thus, the greatest changes in eccentricity are seen to occur at the same time as the peak value of semi-major axis. The inclination and longitude of ascending node change very little (almost zero if no perturbations are included), so that only the planar elements are changed.
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Figure 8.20 Case E - Keplerian Element and Spacecraft Mass Time Histories
Figure 8.22 shows the average periodic thruster power used over the transfer. In all cases (except minimum time), the average thrust is seen to increase in the middle of the transfer when it is most efficient to thrust in order to change the eccentricity and argument of periapsis as discussed above.
81
a)
b)
Figure 8.21 Case E – a) Average Thrust b) Periodic Thrust
The changes in thrust direction angles (pitch and yaw as shown in Figure 2.4) over time are displayed in Figure
8.22. The figure shows that the thrust direction stays largely in plane since there are only small perturbations to the
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orbital plane caused by solar radiation pressure and the gravitational effects of the Sun. Most of the thrusting takes
place in the plane of the orbit to change the semi-major axis, eccentricity, and argument of periapsis.
Figure 8.22 Case E – Thrust Direction Angles
Figure 8.23 shows the orbit transformation over time with about one orbit plotted for every 10 revolutions made. The transfer is shown from two views: the left highlighting the lack of non-planar changes, and the right the
change in the size and shape of the orbit.
Figure 8.23 Case E - Orbit Transformation 83
8.6 Case F: Molniya De-Orbiting
This example examines the non-spherical gravity harmonic resonance which occurs for satellites in the 12 hour highly eccentric Molniya orbits (QP:= 2:1) , which are used to obtain high-altitude coverage of the northern hemisphere. Although these effects are somewhat small for Earth, they can be much larger and more significant for smaller, less spherical bodies such as asteroids and small moons. However, detailed gravitational coefficients are not readily available for such bodies. The objective is to reduce the semi-major axis by 150 km in order to reduce the periapsis and begin de-orbiting of the spacecraft. The optimization is carried out in reverse, starting at the final orbit and ending at the initial orbit; so all of the elements other than semi-major axis are free at the initial time, and correspond to a Molniya orbit at the final time. The initial value of semi-major axis corresponds to that of a
Molniya orbit, while the final value is reduced by 150 km. Gravity harmonics up to 20th order are used with coefficients given in the Section A.5, and the end of the transfer is assumed to coincide with a sidereal angle of 0º.
The initial mass is assumed to be 2000 kg, and the maximum thrust allowed is 0.001N. The transfer time is set to
300 days.
aa0 26610 f 26460 ee0 free f 0.7220 µ = 398574km32 / s 0 = f = ° ωωfree 270 2 0 f ge = 9.81 ms / iifree 63.4° 0 f ΩΩfree 10°
0 m= 2000 kg η = 0.50 ISP= 3000 s
For a transfer lasting 300 days, the minimum fuel solution with the full non-spherical gravity effects included uses 0.64 kg of fuel. If only the zonal harmonics are included, then the fuel mass used goes up to 0.86 kg. This mass usage is nearly identical to that used if no gravity perturbations are included. This is expected, since the zonal harmonics do not affect semi-major axis, the only element being driven to a specified boundary condition. In contrast, the spacecraft can use less fuel to change its semi-major axis by taking advantage of the resonances created
84
by the tesserel/sectoral gravity effects in this region. Minimizing the square of the thrust with perturbations included
results in 0.55 kg of fuel used. The results are summarized in Table 8.6.
Transfer Fuel Fuel Mass ΔV Revs Time Mass Ratio (days) (kg) (mP/ m0) (km/s) Min Fuel: Full 300 0.64 3.2E-4 0.0108 603 Min Fuel: Zonals 300 0.86 4.3E-4 0.0128 603 Min Fuel: Spherical 300 0.86 4.3E-4 0.0128 603 Min T2: Full 300 0.55 2.8E-4 0.0101 603 Table 8.6 Case F – Comparison of Results, ISP = 3000s, Tmax = 0.001N
The time histories of the Keplerian elements and spacecraft mass are shown in Figure 8.24 for this transfer. The
semi-major axis decreases linearly when the tesseral/sectroal gravity harmonics are not included, and follows a
similar path with some oscillations when they are taken into account. The rest of the elements, except the longitude
of ascending node, vary little over the transfer and also show oscillatory behavior when the tesseral/sectoral
harmonics are present. The longitude of ascending node decreases by about 40 degrees over the transfer mainly due to the lower order zonal gravity effects (J2, J3).
Figure 8.24 Case F - Keplerian Element and Spacecraft Mass Time Histories 85
Figure 8.25 shows the average and periodic thrust used over the transfer. In all cases (except minimum time), the average thrust is seen to decrease very slightly over the transfer. This is most likely due to the decreasing semi- major axis, as it is more difficult (takes more fuel) to change the semi-major axis when it is smaller (orbital velocity is higher).
a)
b)
Figure 8.25 Case F – a) Average Thrust b) Periodic Thrust
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The changes in thrust direction angles (pitch and yaw as shown in Figure 2.4) over time are displayed in Figure
8.26. The figure shows that the thrust direction stays completely in plane since the only in-plane thrusting influences the semi-major axis. The yaw angle oscillates between 130º and 230º in order to thrust in the opposite direction to the velocity vector over the eccentric orbit, thus reducing the semi-major axis.
Figure 8.26 Case F – Thrust Direction Angles
Figure 8.27 shows the orbit transformation over time with about one orbit plotted for every 30 revolutions made. The change in semi-major axis reduction is difficult to see since it is very small, but the large change in the longitude of the ascending node caused by the lower order zonal harmonics (J2, J3) is easily noticeable.
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Figure 8.27 Case F - Orbit Transformation
88
9. Conclusion
The method laid out in this work allows optimum low-thrust trajectories to be quickly obtained for transfers involving many revolutions about a central body. The fuel optimum solution can be obtained using the minimum quadratic thrust conditions as an initial guess. A method for the smooth transition between the minimum quadratic thrust solution and the fuel optimum solution has been applied to allow for enhanced convergence of the ill-behaved minimum fuel problem. Minimum time solutions have also been calculated and compared to the fuel optimum solutions. Limits on thrust and power have been successfully implemented into the control problem. The periapsis penalty has been shown to be useful in avoiding transfers which may dip below operationally hazardous altitudes, such as into an orbited body's atmosphere or surface. The effects of shadowing on spacecraft using Solar Electric
Propulsion (SEP) have been applied and shown to be especially important in low altitude orbits. Perturbations due to solar radiation pressure, atmospheric drag, a non-spherical central body, and third-body gravitational effects have been accounted for. Various examples have shown that these perturbations can be important in certain types of transfers. Example cases have also shown to compare well to previous solutions. Although previous low-thrust averaging techniques have provided useful solutions, they typically require longer run-times and accurate initial guesses to converge. The use of non-singular equinoctial elements and numerical averaging methods leads to better conditioned behavior of the numerical shooting method (the solution is more likely to converge). The numerical averaging has also allowed for the straightforward addition of orbital perturbations which have not been considered by previous averaging methods. The method presented here has been implemented into a simple-to-use MATLAB
Graphical User Interface (GUI) and is available by contacting the author. Run-times are typically on the order of minutes on a standard laptop computer.
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10. Future Work
One area for further improvement involves using a direct gradient optimization method in place of a shooting
method. Such a method relies on making an initial guess for the control variable time history instead of the initial
Lagrange multipliers. The control variable history is then directly updated based on satisfying the two point
boundary value problem. Such a method is typically less sensitive to accurate initial guesses and has better
convergence characteristics than the shooting method. However, it is difficult to implement a direct gradient
method for the averaged problem. Since the Lagrange multipliers change very little over one orbit, the average
values can be used to determine the period thrust behavior which drives the changes to the mean orbital elements.
However, the periodic thrust characteristics cannot be found from the average thrust values. Thus, it is difficult to implement a direct gradient method for the averaged problem. A technique which combines the two methods may be necessary to achieve the greater convergence characteristics of the direct gradient method.
Another region for future development is a penalty function on time spent in certain regions of space. This would be useful for limiting the time spent in areas with high radiation (i.e. Van Allen belts) or a high concentration of debris (i.e planetary rings, space junk littered orbits). Specifying these hazardous regions is difficult as they may vary greatly with time. If the time spent within the undesirable region over a particular orbit can be calculated, then the ratio of this time over the period of the orbit can be used as a penalty factor. Such a ratio could be multiplied by a specified penalty scaling factor in order to control the magnitude of the penalty. The penalty function would then be similar to that used for the periapsis, but with the suggested period ratio used in the exponential instead of the periapsis ratio. Such a penalty factor was not implemented due to the difficulty in specifying hazardous regions of space.
An additional useful feature would be a limit on the spacecraft slew rate or thrust direction angles in order to reduce the amount of fuel necessary for attitude maneuvers required for the orbit transfer. However, it is difficult to determine such limits. A more useful approach may be to model the fuel used for attitude control and attempt to minimize the overall fuel mass used. Such a model would add a large amount of complexity. One way to try to account for excessive attitude adjustments is to place a limit of the magnitude of the thrust direction angles, pitch and yaw. In many of the examples, the pitch and yaw angles travel from positive to negative 180 degrees over the
90
course of each orbit. Such "rapid" oscillations may not be desirable. In such a case, limits could be placed on the
max direction magnitudes to reduce attitude oscillation while increasing fuel mass required for the transfer.
Another, slightly more complex route would be to place limits on the slew rate of the spacecraft, implemented via
limits on the rates of change of the thrust direction angles. This would not correspond directly to slew rate for
spacecraft with gimbaled thrusters, but it could help. While this would add some complexity to the problem, it would also allow for more realistic thrust profiles.
The operational feasibility of the optimum trajectories could be improved by constraining thruster on/off times.
There are typically warm-up and cool-down periods associated with turning electric thrusters on and off which may use fuel. Thus, short intervals of on or off times are undesirable and difficult to implement operationally. A limit on the length of each on/off interval of thrusting for the minimum fuel problem could help avoid this issue, but may degrade convergence of the optimization.
The addition of rendezvous to the orbit transfer problem has been considered by previous authors solving the minimum time problem. For orbit transfers in which averaging is useful, many revolutions are made about the central body. Thus, the fuel required for rendezvous is very small compared to that required for the orbit transfer.
However, small changes to the location of the spacecraft within its orbit can play an important when effects such as shadowing or perturbations are considered. The rendezvous problem could be solved by keeping the mean longitude and its associated Lagrange multiplier in the problem. The addition of another specified boundary value and
Lagrange multiplier would aggravate convergence of the solution, especially since the added state (the anomaly) would be a fast-varying element. In order to calculate the rate of change of the added Lagrange multiplier, new partial derivatives of the Hamiltonian with respect to the mean longitude need to be found. These are typically more complex than for the slow-varying elements, especially in the case of orbital perturbations.
The additions discussed here would increase the fidelity of the optimal low thrust orbit transfer method put forth in this work. However, there were significant issues with applying them, though most come down to large increases in the complexity and difficulty in use of the method for small increases in solution fidelity. If some of these issues can be worked out, then the increased fidelity may be worth the added complexity.
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A. Appendix
A.1 The M Matrix
The components of the M matrix are defined as follows:
2 MX= (A.1.1) 11 an2 2 MY= (A.1.2) 12 an2 GX∂ X = − β Mh21 2 (A.1.3) na∂ k n
GY∂ Y = − β Mh22 2 (A.1.4) na∂ k n k M=( qY − pX ) (A.1.5) 23 Gna2 GX∂ X = − β Mh31 2 (A.1.6) na∂ h n
GY∂ Y = − β Mh32 2 (A.1.7) na∂ h n −h M=( qY − pX ) (A.1.8) 33 Gna2 KY M = (A.1.9) 43 2Gna2 KX M = (A.1.10) 53 2Gna2
MM11 12 0 MMM21 22 23 M = MMM (A.1.11) 31 32 33 00M 43 00M 53
92 where
K=++1 pq22 (A.1.12)
A.2 Partials of the Hamiltonian
The rates of change of the Lagrange multipliers are given by:
dλ ∂H ∂∂MM ∂∂ MM∂M 1 =−=−λλ11 +12 −21 +22 + 23 12uufg uuu fgw dt∂ a ∂∂a a ∂∂∂a a a ∂∂∂MMM ∂ M ∂ M −λ31 ++32 33 −λλ43 −53 (A.2.1) 3uuufgw45 u w u w ∂∂∂aaa ∂ a ∂ a 1/2 ∂∂∂MMM3 µ −λλ61 ++62 63 − 66uuufgw 5 ∂∂∂aaa2 a dλ ∂H ∂∂MM ∂∂ MM∂M 2 =−=−λλ11 +12 −21 +22 + 23 12uufg uuu fgw dt∂ h ∂∂h h ∂∂∂h h h ∂∂∂MMM ∂ M ∂ M −λ31 ++32 33 −λλ43 −53 3uuufgw45 u w u w (A.2.2) ∂∂∂hhh ∂ h ∂ h ∂∂∂MMM −λ 61 ++62 63 6 uuufgw ∂∂∂hhh dMλ ∂H ∂∂MM ∂∂ MM∂ 3 =−=−λλ11 +12 −21 +22 + 23 12uufg uuu fgw dt∂ k ∂∂k k ∂∂∂k k k ∂∂∂MMM ∂ M ∂ M −λ31 ++32 33 −λλ43 −53 (A.2.3) 3uuufgw45 u w u w ∂∂∂kkk ∂ k ∂ k ∂∂∂MMM −λ 61 ++62 63 6 uuufgw ∂∂∂kkk dλ ∂H ∂∂∂∂MMMM 4 =−=−λλλλuuuu23 −33 −43 − 53 (A.2.4) dtppppp∂2345wwww ∂∂∂∂ dλ ∂H ∂∂∂∂ MMMM 5 =−=−λλλλuuuu23 −33 −43 − 53 (A.2.5) ∂∂∂∂∂∂tqqqqq2345wwww dλ ∂H 7 =−=λλMTMTm + //22 + MTMTMTm + + ∂∂tm1 11fg 12 2 21f 22 gw 23 +λ ++ 2 (A.2.6) 3MT 31f MT 32 gw MT 33 / m 22 ++λλ4MTm 43 ww//5 MTm 53
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where the partial derivatives of the M matrix with respect to a are:
∂M 42∂X 11 =X + (A.2.7) ∂∂a na22 na 2 a ∂M 42∂Y 12 =Y + (A.2.8) ∂∂a na22 na 2 a ∂M G1 ∂∂ X2 Xhββ hX∂ 21 =− +− − 2 X (A.2.9) ∂a na2 a∂ k ∂∂ a k na n∂ a
∂M G1 ∂∂ Y2 Yhββ hY ∂ 22 =− +−− 2 Y (A.2.10) ∂a na2 a∂ k ∂∂ a k na n∂ a ∂M k1 ∂∂YX 23 = − −+ − (A.2.11) 2 (qY pX) q p ∂a na G2 a ∂∂a a
∂M G1 ∂∂ X2 Xhββ k∂ X 31 =− +− − 2 X (A.2.12) ∂a na2 a∂ h ∂∂ a h na n∂ a
∂M G1 ∂∂ Y2 Ykββ k ∂ Y 32 =− +−− 2 Y (A.2.13) ∂a na2 a∂ h ∂∂ a h na n∂ a ∂M −h1 ∂∂YX 33 = − −+ − (A.2.14) 2 (qY pX) q p ∂a na G2 a ∂∂a a ∂M KY1 ∂ 43 = −+Y (A.2.15) ∂∂a22 na2 G a a ∂M KX1 ∂ 53 = −+X (A.2.16) ∂∂a22 na2 G a a
The partial derivatives of M with respect to h are:
∂M 2 ∂X 11 = (A.2.17) ∂∂h na2 h ∂M 2 ∂Y 12 = (A.2.18) ∂∂h na2 h ∂M −∂h Xhβ G∂2 X X h23ββ h∂ X 21 = − + −+β − (A.2.19) 22X ∂h na G ∂ k n na∂∂ h k n 1 −β n ∂ h 94
∂M −∂h Yhβ G∂2 Y Y h23ββ h∂ Y 22 = − + −+β − (A.2.20) 22Y ∂h na G ∂ k n na∂∂ h k n 1 −β n ∂ h ∂M hkG k∂∂ Y X 23 =[qY −+ pX] q − p (A.2.21) ∂h na22na G∂∂ h h
∂M h∂∂ Xkβ G23 XhkXkββ ∂ X 31 = −− + − (A.2.22) 2X 22 ∂h na G ∂ h n na ∂− h1 β n n ∂ h
∂M h∂∂ Ykβ G23 YhkYkββ ∂ Y 32 = −− − − (A.2.23) 2Y 22 ∂h na G ∂ h n na ∂− h1 β n n ∂ h
∂M −1 ∂∂YXh2 ( qY− pX ) 33 = −+ − − 2 (qY pX) k q p 23 (A.2.24) ∂h na G ∂∂h h na G ∂M K∂ Y hY 43 = + (A.2.25) ∂∂h2 na22 G h G ∂M K∂ X hX 53 = + (A.2.26) ∂∂h2 na22 G h G
The partial derivatives of M with respect to k are:
∂M 2 ∂X 11 = (A.2.27) ∂∂k na2 k ∂M 2 ∂Y 12 = (A.2.28) ∂∂k na2 k ∂M −∂k X hβ G∂23 X hkββ h∂ X 21 = −+ − − (A.2.29) 2XX22 ∂k na G ∂ k n na ∂− k n(1 β ) n ∂ k
∂M −∂k Y hβ G∂23 Y hkββ h∂ Y 22 = −+ − − (A.2.30) 2YY22 ∂k na G ∂ k n na ∂− k n(1 β ) n ∂ k
∂M qY− pX 1 ∂∂YX k2 ( qY− pX ) 23 = + −+ 22kq p 2 (A.2.31) ∂k na G na G∂∂ k k G
∂M k∂∂ Xkβ G2 X k23ββ Xk ∂ X 31 = − − ++β − (A.2.32) 22X ∂k na G ∂ h n na∂∂ k h 1 −β n n ∂ k
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∂M k∂∂ Ykβ G2 Y k23ββ Yk ∂ Y 32 = − − −+β − (A.2.33) 22Y ∂k na G ∂ h n na∂∂ k h 1 −β n n ∂ k
∂M −∂∂hYXhk( qY− pX ) 33 =qp −− (A.2.34) ∂k na2 G ∂∂ h h na23 G ∂M K∂ Y kY 43 = + (A.2.35) ∂∂k2 na22 G k G ∂M K∂ X kX 53 = + (A.2.36) ∂∂k2 na22 G k G
The only non-zero partial derivatives of M with respect to p are:
∂M −kX 23 = (A.2.37) ∂p na2 G ∂M hX 33 = (A.2.38) ∂p na2 G ∂M pY 43 = (A.2.39) ∂p na2 G ∂M pX 53 = (A.2.40) ∂k na2 G
The only non-zero partial derivatives of M with respect to q are:
∂M kY 23 = (A.2.41) ∂q na2 G ∂M −hY 33 = (A.2.42) ∂k na2 G ∂M qY 43 = (A.2.43) ∂p na2 G ∂M qX 53 = (A.2.44) ∂k na2 G
The partial derivatives of X are:
∂X kY = (A.2.45) ∂a na2 G 96
∂X kY = (A.2.46) ∂h na2 G ∂2 X kY = (A.2.47) ∂∂h k na2 G ∂2 X kY = (A.2.48) ∂h22 na G ∂X kY = (A.2.49) ∂k na2 G ∂2 X kY = (A.2.50) ∂∂k h na2 G ∂2 X kY = (A.2.51) ∂k22 na G
The partial derivatives of Y are:
∂X kY = (A.2.52) ∂a na2 G ∂X kY = (A.2.53) ∂h na2 G ∂2 X kY = (A.2.54) ∂∂h k na2 G ∂2 X kY = (A.2.55) ∂h22 na G ∂X kY = (A.2.56) ∂k na2 G ∂2 X kY = (A.2.57) ∂∂k h na2 G ∂2 X kY = (A.2.58) ∂k22 na G
The partial derivatives of X are:
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∂X kY = (A.2.59) ∂a na2 G ∂X kY = (A.2.60) ∂h na2 G ∂X kY = (A.2.61) ∂k na2 G
The partial derivatives of Y are:
∂X kY = (A.2.62) ∂a na2 G ∂X kY = (A.2.63) ∂h na2 G ∂X kY = (A.2.64) ∂k na2 G
A.3 Partials of the Perturbing Accelerations
The partial derivatives of the equinoctial direction vectors in the BCIE frame are:
−2 p ∂ffˆˆ12p = 2q − (A.3.1) ∂pK K −2 2q ∂ggˆˆ12 p = 2 p − (A.3.2) ∂pK K 0 2 ∂wwˆˆ12p = 0 − (A.3.3) ∂pK K −2 p
98
2q ∂ffˆˆ12 q = 2 p − (A.3.4) ∂qK K 0 2 p ∂ggˆˆ12q = −−2q (A.3.5) ∂pK K 2 0 ∂wwˆˆ12q = −−2 (A.3.6) ∂pK K −2q
A.3.1 Solar Radiation Pressure
The perturbing acceleration due to solar radiation pressure is written in the spacecraft equinoctial frame below:
pASR (1+ β ) S rCB-Sun,eq uSR,eq = (A.3.7) mt()rCB-Sun where the position vector of the central body relative to the sun has been substituted for that of the spacecraft with respect to the sun (as explained in Section 7.1). The partial derivatives of this acceleration with respect to the states are as follows:
∂∂∂uuuSR,eq SR,eq SR,eq = = = 0 (A.3.8) ∂∂∂akk ∂∂urSR,eq pASR (1+ β ) S CB-Sun,eq = (A.3.9) ∂∂p mt()rCB-Sun p ∂∂urSR,eq pASR (1+ β ) S CB-Sun,eq = (A.3.10) ∂∂q mt()rCB-Sun q ∂u pA(1+ β ) SR,eq = − SR S 2 (A.3.11) ∂m mt()rCB-Sun
99
If rCB-Sun,BCIE represents the position vector of the central body with respect to the sun in the central body
BCIE frame, then the following equation gives the position vector of the central body with respect to the sun in the spacecraft equinoctial frame.
T ˆ ˆˆ (A.3.12) rCB-Sun,eq =f gw rCB-Sun,BCIE
with the partial derivatives with respect to the states given by:
ˆ T ∂rCB-Sun,eq ∂∂∂f gwˆˆ = rCB-Sun,BCIE (A.3.13) ∂p ∂∂∂ ppp
ˆ T ∂rCB-Sun,eq ∂∂∂f gwˆˆ = rCB-Sun,BCIE (A.3.14) ∂q ∂∂∂ qqq
A.3.2 Atmospheric Drag
The perturbing acceleration due to atmospheric drag is written in the spacecraft equinoctial frame below:
1 ρCA u= − DD vv (A.3.15) AD, eq 2mt () S/ C−− ATM S /, C ATM eq
The partial derivatives of the acceleration with respect to the states are given by:
∂ρ ∂ v + ρ S/ C− ATM vvS/ C−− ATM S /, C ATM eq vS/, C− ATM eq ∂uAD, eq 1 CA ∂∂xx1,...5 1,...5 = − DD (A.3.16) ∂x1,...5 2 mt () ∂vS/, C− ATM eq +ρ vS/ C− ATM ∂x1,...5 ∂uAD, eq 1 ρCA = DDvv (A.3.17) ∂m2 mt2 () S/ C−− ATM S /, C ATM eq
where the atmospheric density is written as a function of the spacecraft altitude as follows:
aa hh0 − ρρ= 0 exp (A.3.18) H S
100
with partial derivative relative to the states given by:
aa a ∂∂ρ 1 hh0 − h = − ρ0 exp (A.3.19) ∂∂xxHHSS
where:
∂∂hra ==−−(1k cos Fh sin F) (A.3.20) ∂∂aa ∂∂hra = = −aFsin (A.3.21) ∂∂hh ∂∂hra = = −aFcos (A.3.22) ∂∂kk
The velocity vector of the spacecraft relative to the atmosphere in the equinoctial frame and its partial derivative with respect to the states are given by:
vSC/,− ATMeq= vv eq − ATMeq, (A.3.23) ∂vSCATMeq/,− ∂∂ vv eq ATMeq, = − (A.3.24) ∂x ∂∂ xx
The partial derivative of the velocities of the spacecraft and atmosphere in the equinoctial frame are as follows:
∂∂X x ∂veq =∂∂Y x (A.3.25) ∂ x 0 T ˆ T ∂vATM, eq ∂∂∂f gwˆˆ ∂vATM,BCIE = + ˆ ˆˆ (A.3.26) vATM,BCIE f gw ∂x ∂∂∂ xxx ∂x
where ∂∂X x and ∂∂Y x were given in Section A.2. The velocity of the atmosphere in the BCIE frame and its
partial derivative with respect to the states can be written as:
vrATM, BCIE=ϑ BCIE × BCIE (A.3.27)
101
∂v ∂r ATM, BCIE =ϑ × BCIE (A.3.28) ∂∂xxBCIE
with the BCIE position vector and partial derivative as follows:
ˆ rBCIE =XY fg + ˆ (A.3.29) ∂ ∂ ∂ ∂∂ˆ ˆ rBCIE XYˆ fg =++f g+ˆ XY (A.3.30) ∂x ∂ x ∂ x ∂∂ xx
The partial derivative of the magnitude of the velocity vector of the spacecraft relative to the atmosphere is given by:
∂ vS/, C− ATM eq vvSC/,−− ATMeq∂ SC /, ATMeq = ⋅ (A.3.31) ∂∂xxvS/, C− ATM eq
A.3.3 Zonal Harmonics
The perturbing acceleration due to zonal effects in the equinoctial frame was given in Chapter 7 and its partial
derivative with respect to the states can be written as:
T ˆ T ∂uZ, eq ∂∂∂f gwˆˆ ∂uZ, BCIE + ˆ ˆˆ (A.3.32) =uZ, BCIE f gw ∂x ∂∂∂ xxx ∂x
where the partial derivative of the acceleration in the BCIE frame with respect to the states is given by:
102
11∂∂r∂∂UU 2zr∂ ∂ U −+ZT + Z 2 1/2 r∂∂xx rr ∂ ∂ r 32+ 2 ∂∂xφ rx( y) ∂u 1 ∂z∂∂UU z∂∂ xy Zx, ZZ =−1/2 ++3/2 xy x ∂xrx22( ++ y 2) ∂∂xφφrx22( y 2) ∂xx ∂ ∂ (A.3.33) z ∂ ∂U Z − 1/2 22 2 ∂∂x φ rx( + y) ∂∂∂ 1 UUZZz x +− 1/2 rr∂∂22 2 φ ∂x rx( + y) 11∂∂r∂∂UU 2zr∂ ∂ U −+ZZ + Z 2 1/2 r∂∂xx rr ∂ ∂ r 32+ 2 ∂∂xφ rx( y) ∂uZy, 1 ∂z∂∂UU z∂∂ xy ZZ =−1/2 ++3/2 xy y ∂xrx22( ++ y 2) ∂∂xφφrx22( y 2) ∂xx ∂ ∂ (A.3.34) z ∂ ∂U Z − 1/2 22 2 ∂∂x φ rx( + y) ∂∂∂ 1 UUTTz y +− 1/2 rr∂∂22 2 φ ∂x rx( + y)
1/2 22+ ∂uZz, 1 ∂z∂∂∂UUU zr ∂∂ z 2( xy) ∂r ∂ U =−+ZZ Z − Z ∂xxr ∂∂ rr23 ∂∂ x rr ∂ x ∂ r r ∂∂xφ 1/2 (A.3.35) 22 1 ∂∂xy∂∂UU(xy+ ) ∂ ++ZZ 1/2 xy 2 rx22( + y 2) ∂xx ∂∂φφr ∂∂x where:
∂∂∂UU22 ∂∂r ∂ Uφ ZZ= + Z (A.3.36) ∂x ∂rr ∂2 ∂ xx ∂∂ rφ ∂ ∂∂∂UU22 ∂∂r ∂ Uφ ZZ= + Z (A.3.37) ∂x ∂φφ ∂∂r ∂ xx ∂ φ2 ∂
103
The second order partial derivatives of the gravity potential with respect to the spherical coordinates are then given by:
∂2U ∞ n µr n Z = + + CB φ (A.3.38) 23∑∑((nn1) 2) n+ JPnn0 (sin ) ∂rrnm=21 = ∂2U ∞ n µr n Z = + CB φ (A.3.39) ∑∑(n 1) n+2 JPnn1 (sni ) ∂r∂φ nm=21 = r ∂∂22UU ZZ= (A.3.40) ∂∂φφrr ∂∂
∂2U ∞ n µr n Z = CB φ (A.3.41) 21∑∑ n+ JPnn1 (sin ) ∂φ nm=21 = r
The partial derivative of the radius with respect to the states was given in Section 8.1 and the partial of the geographic longitude is given as follows:
∂φ 11∂z z∂∂ xy = −+xy (A.3.42) 1/2 2 1/2 3/2 ∂x 22 22∂x22∂∂xx 1/++( zx( y) ) ( xy++) ( xy)
Finally, the partial derivatives of the BCIE Cartesian elements can be written as:
T ˆ T ∂r ∂∂∂f gwˆˆ ∂req BCIE + ˆ ˆˆ (A.3.43) =req f gw ∂x ∂∂∂ xxx ∂x
A.3.4 Tesserel Harmonics
The perturbing acceleration due to Tesserel/Sectoral effects in the equinoctial frame was given in Chapter 7 and its partial derivative with respect to the states can be written as:
T ˆ T ∂uT, eq ∂∂∂f gwˆˆ TT ∂uT, BFE ϑϑ+ ˆ ˆˆ (A.3.44) =Rz( ) u T, BFE f gw Rz ( ) ∂x ∂∂∂ xxx ∂x
104
where the partial derivative of the acceleration in the BFE frame with respect to the states is given by:
11∂∂rr∂∂UU 2 z∂∂ U −+T T +BFE T 2 1/2 r∂∂xx rr ∂ ∂ r 32 2 ∂∂xφ rx( BFE+ y BFE ) ∂ ∂ 1 z UT − 1/2 22 2 ∂∂x φ rx( BFE+ y BFE ) ∂uTx, = x ∂x z∂∂ xy∂UT ++3/2 xyBFE BFE 22 2 ∂xx ∂∂φ rx( BFE+ y BFE )
z ∂ ∂UT − 1/2 22+ 2 ∂∂x φ rx( BFE y BFE ) ∂∂∂ 1 UzT BFE UT x +− 1/2 rr∂ 22+ 2 ∂∂φ x rx( BFE y BFE ) 1 ∂y ∂U T (A.3.45) − 1/2 22∂∂x ψ ( xyBFE+ BFE )
y∂∂ xy∂UT ++3/2 xyBFE BFE 22∂xx ∂∂ψ ( xyBFE+ BFE )
y ∂ ∂UT − 1/2 22∂∂x ψ ( xyBFE+ BFE )
105
11∂∂rr∂∂UU 2 z∂∂ U −+T T +BFE T 2 1/2 r∂∂xx rr ∂ ∂ r 32 2 ∂∂xφ rx( BFE+ y BFE ) ∂ ∂ 1 z UT − 1/2 22 2 ∂∂x φ rxBFE+ y BFE ∂uTy, ( ) = y ∂x z∂∂ xy∂UT ++3/2 xyBFE BFE 22 2 ∂xx ∂∂φ rx( BFE+ y BFE )
z ∂ ∂UT − 1/2 22+ 2 ∂∂x φ rx( BFE y BFE ) ∂∂∂ 1 UzT BFE UT y +− 1/2 rr∂ 22+ 2 ∂∂φ x rx( BFE y BFE ) 1 ∂x ∂U T (A.3.46) + 1/2 22∂∂x ψ ( xyBFE+ BFE )
x∂∂ xy∂UT −+3/2 xyBFE BFE 22∂xx ∂∂ψ ( xyBFE+ BFE )
x ∂ ∂UT + 1/2 22∂∂x ψ ( xyBFE+ BFE )
∂uTz, 1 ∂∂zr∂∂Uz Uz ∂ ∂ U =−+T BFE T BFE T ∂xxr ∂∂ rr2 ∂∂ x r r ∂ x ∂ r 1/2 22 2( xyBFE+ BFE ) ∂r ∂U − T r3 ∂∂x φ (A.3.47) 1 ∂∂xy∂UT ++1/2 xyBFE BFE 22 2 ∂xx ∂∂φ rx( BFE+ y BFE ) 1/2 22 ( xyBFE+ BFE ) ∂ ∂U + T r 2 ∂∂x φ where:
∂∂∂UU22 ∂∂r ∂ Uφψ ∂ 2 U ∂ TT=++ T T (A.3.48) ∂x ∂rr ∂2 ∂ xx ∂∂ rφψ ∂ ∂∂ r ∂ x
106
∂∂∂UU22 ∂∂r ∂ Uφψ ∂ 2 U ∂ TT=++ T T (A.3.49) ∂x ∂φ ∂∂ φr ∂ xx ∂ φ2 ∂ ∂∂ φψ ∂ x ∂∂∂UU222 ∂∂∂r ∂ Uφψ ∂ U TT=++ T T (A.3.50) ∂x ∂ψ ∂∂∂ ψr xxx ∂∂ ψφ ∂ ∂ ψ2 ∂
The second order partial derivatives of the gravity potential with respect to the spherical coordinates are then given by:
∂2U ∞ n µr n T = + + CB φψ+ ψ (A.3.51) 23∑∑((n 1)nP 2) n+ nm (sin)(Cnm cos( mS )nm sin( m )) ∂rrnm=21 =
2 ∞ n n ∂ U µr (Pnm+1 (sinφ) − m tan φφPnm ( sin )) T = + CB (A.3.52) ∑∑(n 1) n+2 ∂∂rrφ = = nm21 ×+(Cnm cos( mSψψ )nm sin() m) ∂2U ∞ n µr n T =+ CB φ−+ ψψ (A.3.53) ∑∑(nP1) n+2 nm (sin)mC( nm sin( m ) Snm cos(m )) ∂rr∂ψ nm=21 = ∂∂22UU TT= (A.3.54) ∂∂φφrr ∂∂
PP(sinφφ) − m tan( sin φ) nm++21nm 2 ∞ n n ∂ U µr −−mmtanφ(PPnm+1 ( sin φ) tan φφnm ( sin )) T = CB (A.3.55) 21∑∑ n+ 2 ∂φ = = r nm21 −msecφφPnm ( sin ) ×+(Cnm cos( mSψψ )nm sin( m ))
2 ∞ n n ∂ U µr (PPnm+1 (sinφ) − m tan φφnm ( sin )) T = CB (A.3.56) ∑∑ n+1 ∂∂φψ = = r nm21 ×−( Cnm sin(mSψψ ) + nm cos( m ))
∂∂22UU TT= (A.3.57) ∂ψψ ∂rr ∂∂ ∂∂22UU TT= (A.3.58) ∂ψφ ∂ ∂∂ φψ
∂2U ∞ n µr n T =−+CB φ2 ψψ (A.3.59) 21∑∑ n+ Pnm (sin)mC( nm cos( m ) Snm sin( m )) ∂ψ nm=21 = r
107
The partial derivative of the radius with respect to the states was given in Section 8.1 and the partial of the geographic longitude is given as follows:
1 ∂z − 1/2 22+ ∂x ∂φ 1 ( xyBFE BFE ) = (A.3.60) 1/2 2 ∂x 22 z ∂∂xy 1/++zxy BFE + ( BFE( BFE BFE ) ) 3/2 xyBFE BFE 22+ ∂∂xx ( xyBFE BFE )
The partial derivative of the geographic latitude is given by:
∂ψ 11∂∂yxy = − BFE (A.3.61) ∂ 2 ∂∂2 x1/+ ( yxBFE BFE ) xxBFE xxBFE
Finally, the partial derivatives of the BFE Cartesian elements can be written as:
T ˆ T ∂r ∂∂∂f gwˆˆ ∂req BFE + ˆ ˆˆ (A.3.62) =req f gw ∂x ∂∂∂ xxx ∂x
A.3.5 Multi-Body Effects
The perturbing acceleration due to solar radiation pressure is written in the spacecraft equinoctial frame below:
− rCB-PB,eq rreq CB-PB,eq uPB,eq = µPB 33− (A.3.63) − rCB-PB,eq rreq CB-PB,eq
The partial derivatives of this acceleration with respect to the states is given as follows:
108
∂∂rrCB-PB,eq eq 1 − 3 ∂∂xxrr− CB-PB,eq eq ∂−u ∂−rrCB-PB,eq eq rr PB,eq =µ −3 CB-PB,eq eq (A.3.64) PB 4 ∂∂xxrr− CB-PB,eq eq ∂ ∂rrCB-PB,eq 1 CB-PB,eq rCB-PB,eq −+3 ∂∂xx34 rrCB-PB,eq CB-PB,eq
If rCB-PB,BCIE represents the position vector from the central body to the perturbing body in the central body
BCIE frame, then the following equation gives the position vector from the central body to the perturbing body in the spacecraft equinoctial frame:
T ˆ ˆˆ (A.3.65) rCB-PB,eq =f gw rCB-PB,BCIE
with the partial derivatives with respect to the states given by:
ˆ T ∂rCB-PB,eq ∂∂∂f gwˆˆ = rCB-PB,BCIE (A.3.66) ∂p ∂∂∂ ppp
ˆ T ∂rCB-PB,eq ∂∂∂f gwˆˆ = rCB-PB,BCIE (A.3.67) ∂q ∂∂∂ qqq
and the partial derivative of the magnitude can be found similarly.
A.4 Atmospheric Density Model The atmospheric density model of the Earth used for Case D is taken from Reference [20] and reproduced in
Table A.1 below:
109
Altitude Min Density Max Density Altitude Min Density Max Density (km) (kg/m3 ) (kg/m3 ) (km) (kg/m3 ) (kg/m3 ) 100 4.974E-07 4.974E-07 420 1.588E-12 5.684E-12 120 2.490E-08 2.490E-08 440 1.091E-12 4.355E-12 130 8.377E-09 8.710E-09 460 7.701E-13 3.362E-12 140 3.899E-09 4.059E-09 480 5.474E-13 2.612E-12 150 2.122E-09 2.215E-09 500 3.916E-13 2.042E-12 160 1.263E-09 1.344E-09 520 2.819E-13 1.605E-12 170 8.008E-10 8.758E-10 540 2.042E-13 1.267E-12 180 5.283E-10 6.010E-10 560 1.488E-13 1.005E-12 190 3.617E-10 4.297E-10 580 1.092E-13 7.997E-13 200 2.557E-10 3.162E-10 600 8.070E-13 6.390E-13 210 1.839E-10 2.396E-10 620 6.012E-14 5.123E-13 220 1.341E-10 1.853E-10 640 4.519E-14 4.121E-13 230 9.949E-11 1.455E-10 660 3.430E-14 3.325E-13 240 7.488E-11 1.157E-10 680 2.620E-14 2.691E-13 250 5.709E-11 9.308E-11 700 2.043E-14 2.185E-13 260 4.403E-11 7.555E-11 720 1.607E-14 1.779E-13 280 2.697E-11 5.095E-11 760 1.036E-14 1.190E-13 290 2.139E-11 4.226E-11 780 8.496E-15 9.776E-14 300 1.708E-11 3.526E-11 800 7.069E-15 8.059E-14 320 1.099E-11 2.511E-11 840 4.680E-15 5.741E-14 340 7.214E-12 1.819E-11 880 3.200E-15 4.210E-14 360 4.824E-12 1.337E-11 920 2.210E-15 3.130E-14 380 3.274E-12 9.955E-12 960 1.560E-15 2.360E-14 400 2.249E-12 7.492E-12 1000 1.150E-15 1.810E-14
Table A.1 Earth Atmospheric Density
An exponential interpolation was used to find the density (ρ) at any given altitude (h: 100-1000 km) in solar minimum and maximum conditions. The density is assumed to be zero above 1000 km, altitudes of less than 100 km are not considered.
(hhH1 − )/ ρρ= 1e (A.3.68) where
hh− H = 12 (A.3.69) ln(ρρ21 / )
110
This is done for both solar maximum and solar minimum conditions; then a weighted average is carried out according to the actual solar flux condition:
ρ=−+(1 ηFlux) ρ Min ηρ Flux Max (A.3.70)
where ηFlux = 0 corresponds to minimum solar flux, and ηFlux =1to maximum solar flux. Case D is assumed to
occur during a solar minimum period, ηFlux = 0 .
A.5 Earth Non-Spherical Gravity Coefficients
The un-normalized zonal gravity coefficients for Earth are taken from Reference [18] and reproduced below to
20th order:
n Jn 2 4.84169389054810E-04 3 -9.57185084154389E-07 4 -5.39991435260740E-07 5 -6.87159810011790E-08 6 1.49940119731249E-07 7 -9.05046302291320E-08 8 -4.94813341047799E-08 9 -2.80231882033290E-08 10 -5.33200105265499E-08 11 5.07724962976520E-08 12 -3.64369678942379E-08 13 -4.17302524097659E-08 14 2.26756302235759E-08 15 -2.19383226933879E-09 16 4.71289814233569E-09 17 -1.91872007314310E-08 18 -6.09762019207260E-09 19 3.30340851762670E-09 20 -2.15566597029229E-08
Table A.2 Earth Zonal Gravity Coefficients 111
The un-normalized sectorial and tesseral gravity coefficients for Earth are taken from Reference [18] and reproduced below to 20th order:
n m Cnm Snm 2 1 -2.04583381847449988e-10 1.39681953795510009e-09 2 2 2.43932330011909994e-06 -1.40026620038669993e-06 3 1 2.03047526560640003e-06 2.48174169030309976e-07 3 2 9.04800669750679979e-07 -6.19004414271030050e-07 3 3 7.21289242476499998e-07 1.41435564340519991e-06 4 1 -5.36175837894340016e-07 -4.73568022874759985e-07 4 2 3.50511599310869996e-07 6.62439448491860019e-07 4 3 9.90855035417340077e-07 -2.00975294423420010e-07 4 4 -1.88467504745159995e-07 3.08822282787560013e-07 5 1 -6.29040513804809977e-08 -9.43732633569280060e-08 5 2 6.52103930803030047e-07 -3.23348384507880010e-07 5 3 -4.51879654499089980e-07 -2.15001408012229996e-07 5 4 -2.95336339969189978e-07 4.98178346139759973e-08 5 5 1.74793678615289994e-07 -6.69374448511330027e-07 6 1 -7.59035347130909983e-08 2.65169698204040001e-08 6 2 4.86714774108890018e-08 -3.73790636327859976e-07 6 3 5.72352511408300006e-08 8.93559083171720000e-09 6 4 -8.60243060207619966e-08 -4.71424927789280018e-07 6 5 -2.67170445353750025e-07 -5.36488705937620020e-07 6 6 9.46678581176680060e-09 -2.37405638786950001e-07 7 1 2.80888981450810025e-07 9.51196440627809965e-08 7 2 3.30415607465350017e-07 9.29852397872480005e-08 7 3 2.50451662631419986e-07 -2.17147482110030010e-07 7 4 -2.74989380595529996e-07 -1.24067044114940011e-07 7 5 1.66013173285799999e-09 1.79338880393250011e-08 7 6 -3.58807758799609996e-07 1.51794547356989991e-07 7 7 1.50628842190990003e-09 2.41162592779400003e-08 8 1 2.31599797277340015e-08 5.88966651248619973e-08 8 2 8.00151521912720048e-08 6.52788350208980056e-08 8 3 -1.93788337749689995e-08 -8.59774731751730026e-08 8 4 -2.44367441684680017e-07 6.98081695736609943e-08
112
8 5 -2.56953810206029996e-08 8.91955034832059959e-08 8 6 -6.59624718394710010e-08 3.08944796734990016e-07 8 7 6.72614398495850048e-08 7.48756708730800001e-08 8 8 -1.24038735032040009e-07 1.20553306867689996e-07 9 1 1.42150218128849999e-07 2.13994106247449993e-08 9 2 2.14133083108420005e-08 -3.16937570995879973e-08 9 3 -1.60617775556249998e-07 -7.42880657277480046e-08 9 4 -9.37444947818120001e-09 1.99034264838489990e-08 9 5 -1.63104761535860006e-08 -5.40419232294500026e-08 9 6 6.27822143512300017e-08 2.22951781384670004e-07 9 7 -1.17977052318579999e-07 -9.69284348571229945e-08 9 8 1.88140030800960012e-07 -3.00068454705350017e-09 9 9 -4.75567140556599992e-08 9.68800583876869964e-08 10 1 8.37648432514100020e-08 -1.31091512846909992e-07 10 2 -9.39862026605370018e-08 -5.12798227487940026e-08 10 3 -7.01664817883609972e-09 -1.54145220563250002e-07 10 4 -8.44726355863289952e-08 -7.90319737950219981e-08 10 5 -4.92920348548950010e-08 -5.06117179099050027e-08 10 6 -3.75851950343959976e-08 -7.97695742669969963e-08 10 7 8.26013221779260035e-09 -3.04728133397479990e-09 10 8 4.05961164818980001e-08 -9.17186963760239960e-08 10 9 1.25382723795859987e-07 -3.79543677891850023e-08 10 10 1.00424569864040005e-07 -2.38595956665140001e-08 11 1 1.56032527408559992e-08 -2.71190625962799997e-08 11 2 2.01136100235820015e-08 -9.90084672152959978e-08 11 3 -3.05812020910249986e-08 -1.48843476111799995e-07 11 4 -3.79526410394709980e-08 -6.37697596921109989e-08 11 5 3.74212841430079996e-08 4.95900853900059973e-08 11 6 -1.55896380201770003e-09 3.42744197854409971e-08 11 7 4.65796029356270015e-09 -8.98194914568629956e-08 11 8 -6.30309872473060039e-09 2.45442225150679992e-08 11 9 -3.10692602092850018e-08 4.20658696345280002e-08 11 10 -5.22526930351960019e-08 -1.84237317816489987e-08 11 11 4.62408057364670013e-08 -6.96685420430340048e-08 12 1 -5.35876692081920033e-08 -4.31635871002140025e-08 12 2 1.42673370959969996e-08 3.10930835560089992e-08 12 3 3.96159512836029988e-08 2.50634092008970002e-08
113
12 4 -6.77366702836919940e-08 3.84058906059060015e-09 12 5 3.08791067349680023e-08 7.59012793304670080e-09 12 6 3.13340086473739992e-09 3.89747843170290008e-08 12 7 -1.90449416343199990e-08 3.57328054423330000e-08 12 8 -2.58897571157559999e-08 1.69363269941199985e-08 12 9 4.19194899634250019e-08 2.49642488754230009e-08 12 10 -6.20100705381310013e-09 3.09469248714420029e-08 12 11 1.13619537434260007e-08 -6.39105782949500014e-09 12 12 -2.43029216732679983e-09 -1.11042488945409997e-08 13 1 -5.14415553340140015e-08 3.86985687758529993e-08 13 2 5.53089943394350009e-08 -6.26962679384670008e-08 13 3 -2.15547683779850007e-08 9.76810828228700017e-08 13 4 -3.65443469982859992e-09 -1.17478080179210007e-08 13 5 5.83729395923200018e-08 6.72279048557479972e-08 13 6 -3.50446388120099986e-08 -6.27495330096750017e-09 13 7 3.01986079573209980e-09 -7.32392030420839983e-09 13 8 -1.00536020531250006e-08 -9.86429220629199931e-09 13 9 2.47729957871640014e-08 4.58859926770249997e-08 13 10 4.11036621578139974e-08 -3.68387486209449997e-08 13 11 -4.45216615349650001e-08 -4.83362936830670036e-09 13 12 -3.13060301562770019e-08 8.79393757220940006e-08 13 13 -6.12140382689340066e-08 6.81481678107430046e-08 14 1 -1.87709982167690017e-08 2.88608371451520000e-08; 14 2 -3.59203236000659997e-08 -4.05362954404810012e-09 14 3 3.65099838114599973e-08 1.96946216072540006e-08 14 4 1.60337316755770010e-09 -2.26651359545060012e-08 14 5 2.93102647109429997e-08 -1.67890253324039986e-08 14 6 -1.90676038887109999e-08 2.46025081737260016e-09 14 7 3.76320783676020032e-08 -3.93362272325419970e-09 14 8 -3.49422636417469990e-08 -1.54445821167739997e-08 14 9 3.19517813524400003e-08 2.84631547271799997e-08 14 10 3.88016070368050004e-08 -1.29430651638440001e-09 14 11 1.56472473594289984e-08 -3.90464327962610014e-08 14 12 8.46583759965109921e-09 -3.11237574603869972e-08 14 13 3.22366804070620016e-08 4.51530398138999994e-08 14 14 -5.18704795525289993e-08 -4.81344218328860011e-09 15 1 9.42994886893369949e-09 1.04832100925509998e-08
114
15 2 -2.05279487432490015e-08 -3.03005930223270015e-08 15 3 5.34206362316779988e-08 1.76611280699239987e-08 15 4 -4.01738701190320027e-08 6.81601512714589996e-09 15 5 1.22428149267349999e-08 7.62250510465490023e-09 15 6 3.28485131212280027e-08 -3.64711011629550028e-08 15 7 5.96498682904089971e-08 5.07497586312400014e-09 15 8 -3.20884645780170020e-08 2.21702666037359992e-08 15 9 1.32980490877230004e-08 3.79925545028240013e-08 15 10 1.02597199391729997e-08 1.46872480939710006e-08 15 11 -1.30849842226460005e-09 1.85195031633929998e-08 15 12 -3.24124742904200011e-08 1.56062051106059991e-08 15 13 -2.83631857753099990e-08 -4.58013989085019980e-09 15 14 5.21252435688719992e-09 -2.44210664932160011e-08 15 15 -1.90312941357980008e-08 -4.69152249642509984e-09 16 1 2.61840631135990015e-08 3.33377215875599987e-08 16 2 -2.45087790522699998e-08 2.80358111949140003e-08 16 3 -3.39181410054569982e-08 -2.13382306452860007e-08 16 4 4.08537057083779998e-08 4.79847689378779998e-08 16 5 -1.21222848667180001e-08 -3.44466503218689994e-09 16 6 1.38701147093969997e-08 -3.55992360699520000e-08 16 7 -8.06411234896849968e-09 -8.65067283256839938e-09 16 8 -2.12036221540929992e-08 5.40353293208479983e-09 16 9 -2.24178389260349995e-08 -3.96685595843799991e-08 16 10 -1.18023467338590006e-08 1.15352510170639992e-08 16 11 1.91128532330080005e-08 -3.20483476186190011e-09 16 12 1.95670095727400004e-08 6.71888839819530000e-09 16 13 1.37736285347249996e-08 1.05230674497279996e-09 16 14 -1.93432141011899989e-08 -3.86513514132810028e-08 16 15 -1.44027831875449996e-08 -3.27433356227189972e-08 16 16 -3.83010663453479996e-08 2.95949781342570019e-09 17 1 -2.53645890124030014e-08 -3.17048491735240032e-08 17 2 -2.00998976254329998e-08 6.81168738089760014e-09 17 3 6.31161504355159963e-09 5.08357000835809973e-09 17 4 6.47721223237970021e-09 2.53347007929110014e-08 17 5 -1.62215194878090008e-08 8.03104948761439984e-09 17 6 -1.17313493297189994e-08 -2.94424895003409990e-08 17 7 2.49788910164179998e-08 -4.38593277790830006e-09
115
17 8 3.89995266808730016e-08 3.63367389830130016e-09 17 9 3.48242766124059996e-09 -2.76407631154300002e-08 17 10 -3.80175782770059999e-09 1.83753713251019984e-08 17 11 -1.60186798217950013e-08 1.10878775058280007e-08 17 12 2.87145782035479999e-08 2.04620070284580013e-08 17 13 1.65104794069319994e-08 2.01367491972439994e-08 17 14 -1.42718374688830004e-08 1.15669222878590006e-08 17 15 5.53622241280290018e-09 5.24128739839249959e-09 17 16 -3.05099267701230003e-08 3.68718363520690004e-09 17 17 -3.47014840997759975e-08 -1.98726190068019985e-08 18 1 7.20136680561089982e-09 -3.92994410802759974e-08 18 2 1.47260987462909997e-08 1.08362744751619996e-08 18 3 -5.04738512528959963e-09 -5.84530219835769984e-09 18 4 5.46274076624539967e-08 -7.91173183317689987e-10 18 5 5.97770333096470011e-09 2.61402090546590013e-08 18 6 1.35738191697540002e-08 -1.32436832662799998e-08 18 7 6.79395646178379961e-09 7.46911259352460028e-09 18 8 3.04998787777440004e-08 4.34478595287109986e-09 18 9 -1.95698881351369995e-08 3.61278690091630028e-08 18 10 5.21493810354730036e-09 -4.24280133530450020e-09 18 11 -6.88937720027069980e-09 2.11550793460429982e-09 18 12 -2.97450068288230008e-08 -1.65660143879329988e-08 18 13 -6.25357294939640016e-09 -3.49359649152400018e-08 18 14 -8.29104162455619945e-09 -1.28285852726959992e-08 18 15 -4.04571867023889981e-08 -2.02750517003450000e-08 18 16 1.01552590150960003e-08 6.50554738959369994e-09 18 17 3.48779882525350018e-09 4.37974535704149993e-09 18 18 2.99166710907890008e-09 -1.08385545018600006e-08 19 1 -8.97152581600269925e-09 1.19377067894970007e-09 19 2 3.57386494001920025e-08 -2.37139623328140011e-09 19 3 -7.55791479000760019e-09 1.10491259433630000e-09 19 4 1.58077466687629985e-08 -8.13830452835910075e-09 19 5 1.03892253584950004e-08 2.74518897620719999e-08 19 6 -4.80183681859449972e-09 1.87554119382439996e-08 19 7 5.64177233070960008e-09 -8.74152037730210024e-09 19 8 2.98880347900319977e-08 -9.99465416610019970e-09 19 9 3.28680071944019990e-09 7.23634584032820026e-09
116
19 10 -3.38801674488169970e-08 -7.58394068018700062e-09 19 11 1.63020160399820010e-08 1.03548032978129999e-08 19 12 -2.44873449007329981e-09 9.45762201358140039e-09 19 13 -7.53394347200489937e-09 -2.85229718313379999e-08 19 14 -4.74359610228510013e-09 -1.28490435521779992e-08 19 15 -1.75890776811189998e-08 -1.40729486340649998e-08 19 16 -2.18001197119249998e-08 -7.05829953037430015e-09 19 17 2.88074919621179991e-08 -1.53431974603419990e-08 19 18 3.50640232205389968e-08 -9.70945569937920009e-09 19 19 -2.70653954978629986e-09 5.19928596083409991e-09 20 1 5.56666707465730038e-09 7.02867770838819981e-09 20 2 2.02897183380859989e-08 1.71853676970230012e-08 20 3 -4.75410572226800040e-09 3.89012688346970024e-08 20 4 4.26981367329409973e-09 -2.26280972522290015e-08 20 5 -1.01255023348610001e-08 -8.25947979092540065e-09 20 6 1.21595707584600003e-08 -4.36291907245539989e-09 20 7 -2.12251781862860009e-08 -7.05951911611569989e-10 20 8 5.09494448347359970e-09 2.15133663406369993e-09 20 9 1.71853683893989985e-08 -7.01296727516599983e-09 20 10 -3.22952345437520006e-08 -4.79148481816510004e-09 20 11 1.44473053315319992e-08 -1.91527802100489993e-08 20 12 -6.46447513436439973e-09 1.81305295533090005e-08 20 13 2.74148976104239991e-08 6.76428714561850033e-09 20 14 1.15328239415760000e-08 -1.43800119088810007e-08 20 15 -2.57883586533800006e-08 -8.72730065194060009e-10 20 16 -1.24319744422269997e-08 -3.41519213407269979e-10 20 17 4.50520493208190023e-09 -1.37281218085739997e-08 20 18 1.53599856444770005e-08 -8.88076820980340028e-10 20 19 -3.04401463589460016e-09 1.09257403007539996e-08 20 20 3.73469847236800008e-09 -1.27022696687589997e-08
Table A.3 Earth Sectoral and Tesserel Gravity Coefficients
117
Bibliography
[1] Acton, F. S., Numerical Methods That Work, Harper & Row, New York, 1970
[2] Bryson, A. E., and Ho, Y.C., Applied Optimal Control, Taylor and Francis Group, New York, 1975
[3] Cefola, R., Long, A.C., and Holloway, G. Jr., "The Long-Term Prediction of Artificial Satellite Orbits", AIAA 12th
Aerospace Sciences Meeting, AIAA 74-170, Washington, D.C., February 1974
[4] Edelbaum, T. N., “Optimum Low-Thrust Rendezvous and Station Keeping,” AIAA Journal, Vol. 2, No. 7, 1964, pp. 1196-
1201
[5] Edelabum, T. N., “Optimum Power-Limited Orbit Transfer in Strong Gravity Fields,” AIAA Journal, Vol. 3, No. 5, 1964, pp.
921–925.
[6] Ely, T.A., “Mean Element Propagations using Numerical Averaging,” AAS/AIAA Astrodynamics Specialist Conference, AAS
09-440, Pittsburgh, Pennsylvania, August 2009
[7] Geffory, S., and Epenoy, R. “Optimal Low-Thrust Transfers with Constraints – Generalization of Averaging Techniques,”
Acta Astonautica, Vol. 41, No. 3, 1997, pp. 133-149.
[8] Haberkorn, T., Martinon, P., and Gergaud, J., "Low Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach",
Journal of Guidance, Control, and Dynamics, Vol. 27, No. 6, 2004, pp. 1046-1060
[9] Kechichian, J.A., “Optimum Low-Thrust Rendezvous using Equinoctial Orbital Elemenets,” Acta Astonautica, Vol. 38, No.
1, 1996, pp. 1-14.
[10] Kluever, C.A., and Oleson, S.R., “Direct Approach for Computing Near-Optimal Low-Thrust Earth-Orbit Transfers,”
Journal of Spacecraft and Rockets, Vol. 35, No. 4, 1998, pp. 509-515
[11] Lutzky, D., and Uphoff, C., “Short-Periodic Variations and Second-Order Numerical Averaging,” AIAA 13th Aerospace
Sciences Meeting, Pasadena, California, 1975
[12] Marec, J. P., Optimal Space Trajectories, Elsevier Scientific Publishing Company, Amsterdam, 1979, Chaps. 6, 12.
[13] Marec, J. P., and Vinh, N.X., “General Study of Optimal Low Thrust Limited Power Transfers Between Arbitrary Elliptical
Orbits,” ESA TT-697, 1981.
[14] Nayfeh, A., Perturbation Methods, John Wiley & Sons, New York, 1973
[15] Petropoulos, A.E., “Low-Thrust Orbit Transfers Using Candidate Lyapunov Functions with a Mechanism for Coasting,”
AIAA/AAS Astrodynamics Specialist Conference, AIAA 2004-5089, Providence, Rhode Island, August 2004
[16] Sacket, L. L., Malchow, H.L., and Edelbaum, T.N., “Solar Electric Geocentric Transfer with Attitude Constraints:
Analysis,” NASA CR-134927, Aug. 1975. 118
[17] Sanders, J.A., Verhulst, F., and Murdock, J., Averaging Methods in Nonlinear Dynamical Systems, Second Edition,
Springer, New York, 2007
[18] Speyer, J. L., and Jacobson, D.H., Primer on optimal Control Theory, Society of Industrial and Applied Mathematics,
Philadelphia, 2010
[19] Tapley, B., Reis, J., Bettadpur, S., Chambers, D., Cheng, M., Condi, F. and Poole, S., “The GGM03 Mean Earth Gravity
Model from GRACE, Eos Trans. AGU, 88(52), Fall Meet Suppl., Abstract G42A-03, 2007
[20] Tarzi, Z. T., Speyer, J.L., and Wirz, R., “Optimum Low-Thrust Transfer between Arbitrary Elliptic Orbits using Asymptotic
Expansions and Averaging,” 31st International Electric Propulsion Conference, IEPC-2009-222, Ann Arbor, Michigan,
September 2009
[21] Tarzi, Z. T., Speyer, J.L., and Wirz, R., “Optimum Low-Thrust Elliptic Transfer for Power Limited Spacecraft Using
Numerical Averaging and Continuously Variable Thrust”, AAS/AIAA Astrodynamics Specialist Conference, AAS 11-234,
New Orleans, Louisiana, February 2011
[22] Vallado, D. A., Fundamentals of Astrodynamics and Applications, Microcosm Press, El Segundo, 2001
119