Relative Orbital Motion Dynamical Models for Orbits about Nonspherical Bodies
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Authors Burnett, Ethan Ryan
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/628098 Relative Orbital Motion Dynamical Models for Orbits about Nonspherical Bodies
by Ethan R. Burnett
Copyright c Ethan R. Burnett 2018
A Thesis Submitted to the Faculty of the Department of Aerospace and Mechanical Engineering In Partial Fulfillment of the Requirements For the Degree of Master of Science With a Major in Aerospace Engineering In the Graduate College The University of Arizona
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Acknowledgments
I would like to thank my advisor, Dr. Eric Butcher, for his encouragement and excellent teaching.
I greatly enjoyed working with my research colleagues: Mohammad Maadani, Bharani Malladi, David Yaylali, Jingwei Wang, Morad Nazari, and Arman Dabiri.
I thank my parents and my brothers for their steadfast support. 4
Vita
Education
Master of Science, Aerospace Engineering Emphasis: Dynamics and Control University of Arizona, Tucson, AZ May 2018
Bachelor of Science, Aerospace Engineering Minor, Physics University of Arizona, Tucson, AZ May 2016
Publications
Ethan Burnett and Eric Butcher, “Linearized Relative Orbital Motion Dynamics in a Ro- tating Second Degree and Order Gravity Field,” AAS 18-232, AAS/AIAA Astrodynamics Specialist Conference, Snowbird, UT, August 19 – 23, 2018 (submitted)
Ethan Burnett, Eric Butcher, and T. Alan Lovell, “Linearized Relative Orbital Motion Model About an Oblate Body Without Averaging,” AAS 18-218, AAS/AIAA Astrodynam- ics Specialist Conference, Snowbird, UT, August 19 – 23, 2018 (submitted)
Ethan Burnett and Andrew J. Sinclair, “Interpolation on the Unit Sphere in Laplace’s Method,” AAS 17-793, AIAA/AAS Astrodynamics Specialist Conference, Columbia River Gorge, Stevenson, WA, August 20 – 24, 2017
Kristofer Drozd, Ethan Burnett, Eric Sahr, Drew McNeely, Vittorio Franzese, and Natividad Ramos Mor´on,“Block-Like Explorer of a Near-Earth Body by achieving Orbital Proximity (BEEBOP),” AAS 17-846, AIAA/AAS Astrodynamics Specialist Conference, Columbia River Gorge, Stevenson, WA, August 20 – 24, 2017
Eric Butcher, Ethan Burnett, Jingwei Wang, and T. Alan Lovell, “A New Time-Explicit J2-Perturbed Nonlinear Relative Orbit Model with Perturbation Solutions,” AAS 17-758, AIAA/AAS Astrodynamics Specialist Conference, Columbia River Gorge, Stevenson, WA, August 20 – 24, 2017
Eric Butcher, Ethan Burnett, and T. Alan Lovell, “Comparison of Relative Orbital Motion Perturbation Solutions in Cartesian and Spherical Coordinates,” AAS 17-202, AIAA/AAS Spaceflight Mechanics Meeting, San Antonio, TX, February 5 – 9, 2017 5
Table of Contents
List of Figures ...... 7
List of Tables ...... 9
Abstract ...... 10
1. Introduction ...... 11
2. Orbital Motion and Relative Motion ...... 15 2.1. Classical Theory of Satellite Orbits ...... 15 2.1.1. Gravitation ...... 15 2.1.2. The Two-Body Problem and Elliptic Orbits ...... 16 2.1.3. Orbital Elements for Elliptic and Circular Orbits ...... 18 2.2. Analytical Relative Dynamics of Co-orbiting Bodies ...... 19 2.2.1. Relative Orbital Motion ...... 20 2.2.2. The Hill-Clohessy-Wiltshire Model ...... 21 2.2.3. Representing Model Error ...... 26 2.3. Orbital Perturbations ...... 28 2.3.1. Perturbed Orbital Motion ...... 28 2.3.2. Types of Perturbations ...... 29 2.3.3. Perturbed Relative Orbital Motion ...... 30
3. Nonspherical Body Gravitation and Orbital Mechanics .... 34 3.1. Gravitational Potential Expressed by Spherical Harmonics ...... 34 3.2. The Shape of the Earth ...... 37
3.3. Orbital Motion under the Influence of J2 ...... 37 3.4. Moons, Planetoids, and Asteroids ...... 38 6
Table of Contents—Continued
3.5. Orbital Motion about Small Asteroids ...... 39
4. A J2-Perturbed Relative Orbit Model ...... 43 4.1. Formulation ...... 43 4.1.1. Case of Vanishing Chief Eccentricity ...... 47 4.2. Model Analysis and Validation ...... 50
5. A Linearized Relative Orbit Model for Orbits in a Rotating Second Degree and Order Gravity Field ...... 55 5.1. Formulation ...... 55 5.1.1. Case of Vanishing Chief Eccentricity ...... 70 5.1.2. Extension to Eccentric Chief Orbits ...... 74 5.2. Model Analysis ...... 77 5.2.1. The Case of Equatorial Orbits ...... 77 5.2.2. A Special Case LTI Model ...... 78 5.2.3. Analysis of Unstable and Stable Eigenspaces ...... 82 5.3. Model Validation via Simulation ...... 87 5.3.1. Hypothetical Asteroid for Case Studies ...... 88 5.3.2. Variable Chief Inclination with Constant Γ ...... 88 5.3.3. Variable Γ with Constant Chief Inclination ...... 95 5.3.4. Variable Chief Semimajor Axis with Constant Body Rotation Rate ...... 99 5.3.5. Interesting Cases ...... 101
6. Conclusions and Suggestions for Future Work ...... 107
References ...... 109 7
List of Figures
Figure 2.1. Elliptical Orbit Geometry22 ...... 18 Figure 2.2. The 3-1-3 Euler Angles and the Orbit Plane23 ...... 19 Figure 2.3. Relative Motion Problem Geometry ...... 20 Figure 2.4. Relative Distance for Unperturbed and Perturbed Cases . . . . 32 Figure 2.5. ∆h(t) for Unperturbed and Perturbed Cases ...... 32
Figure 3.1. Primary Body Mass Distribution Geometry ...... 34 21 Figure 3.2. Nodal Regression due to J2 ...... 38 Figure 3.3. Asteroids and Comets Visited by Spacecraft24 ...... 39 Figure 3.4. Spacecraft Orbital Ejection due to SRP Forces ...... 42 Figure 3.5. Spacecraft Orbital Velocity Before and During Ejection . . . . . 42
Figure 4.1. Relative Orbit for 4 Orbits (LEO with J2)...... 51
Figure 4.2. Small Parameter Values for Two Orbits (LEO with J2)..... 52 Figure 4.3. Radial Error (LEO) ...... 53 Figure 4.4. Along-Track Error (LEO) ...... 53 Figure 4.5. Cross-Track Error (LEO) ...... 53 Figure 4.6. Error Norm (LEO) ...... 53 Figure 4.7. Maximum Error Norm over Two Orbits vs. Chief Orbit Radius 54
Figure 5.1. Problem Geometry for Orbiting a Rotating Ellipsoidal Body . . 58 Figure 5.2. Libration Points for Rotating Ellipsoidal Body ...... 79 Figure 5.3. Superposition of In-Plane Modes at Stable Libration Point . . . 84 Figure 5.4. Simulated In-Plane Motion at Stable Libration Point ...... 84 Figure 5.5. Stable and Unstable Manifolds (Unstable Libration Point) . . . 87 Figure 5.6. Average Modeling Error vs. Inclination, Γ = 2.4 ...... 90 8
List of Figures—Continued
Figure 5.7. Maximum Deviation of Relative Orbit Angular Momentum, Γ = 2.4...... 92 Figure 5.8. Average Modeling Error vs. Inclination, Γ = 1.2 ...... 93 Figure 5.9. Maximum Deviation of Relative Orbit Angular Momentum, Γ = 1.2...... 93 Figure 5.10. Average Modeling Error vs. Inclination, Γ = 0.8 ...... 94 Figure 5.11. Maximum Deviation of Relative Orbit Angular Momentum, Γ = 0.8...... 95 Figure 5.12. Average Modeling Error vs. Γ, i = 10◦ ...... 96 Figure 5.13. Maximum Deviation of Relative Orbit Angular Momentum, i = 10◦ 97 Figure 5.14. Average Modeling Error vs. Γ, i = 45◦ ...... 97 Figure 5.15. Maximum Deviation of Relative Orbit Angular Momentum, i = 45◦ 98 Figure 5.16. Average Modeling Error vs. Γ, i = 75◦ ...... 98 Figure 5.17. Maximum Deviation of Relative Orbit Angular Momentum, i = 75◦ 99 Figure 5.18. Average Modeling Error vs. Semimajor Axis, i = 75◦ ...... 100 Figure 5.19. Max Deviation of Relative Angular Momentum vs. Semimajor Axis...... 100 Figure 5.20. Relative Distance with Relative Orbit Resonance ...... 103 Figure 5.21. ∆h(t) with Relative Orbit Resonance ...... 103 Figure 5.22. Model Error (Relative Orbit Resonance Case) ...... 104 Figure 5.23. Relative Orbit (Resonance Case) ...... 104 Figure 5.24. Relative Orbit (Marginal Stability Case) ...... 106 Figure 5.25. Model Error (Marginal Stability Case) ...... 106 9
List of Tables
Table 2.1. Orbital Elements and Differences ...... 31
Table 3.1. Asteroid Physical Parameters ...... 40
Table 4.1. Orbital Elements for LEO Relative Orbit Scenario with J2 ... 51
Table 5.1. Unique O Matrix Terms in 5.31, Oij = Oji ...... 62
Table 5.2. Unique E Matrix Terms in 5.32, Eij = Eji ...... 62 Table 5.3. Unstable and Stable Eigenvalues for Example Case ...... 83 Table 5.4. Simulated Asteroid Data ...... 88 Table 5.5. Orbital Elements, Variable Chief Inclination ...... 88 Table 5.6. Orbital Elements, Variable Chief Semimajor Axis ...... 99 10
Abstract
RELATIVE ORBITAL MOTION DYNAMICAL MODELS FOR ORBITS ABOUT NONSPHERICAL BODIES
by
Ethan R. Burnett
Master of Science
University of Arizona
Tucson, Arizona, 2018
Dr. Eric A. Butcher, Chair
Relative orbital motion dynamical models are presented and discussed. Two types of models are primarily discussed in this work: a linear relative motion model account- ing for J2 (the gravitational parameter associated with the oblateness of the Earth), and a new linear relative motion model accounting for both nonzero second degree and order gravity terms C20 = −J2 and C22. The latter model, referred to as the “second-order model,” is useful for simulating and studying spacecraft relative motion in orbits about uniformly rotating asteroids. This model is derived in two alternate forms. The first makes use of averaging in the kinematics and the second avoids any use of averaging. Additional work is devoted to analyzing the stability of relative orbital motion in rotating second degree and order gravity fields. To facilitate this, a parameter called the relative orbit angular momentum is introduced. For commensurate angular rates of primary body rotation and orbital mean mo- tion, a special case linear time-invariant (LTI) model is obtained from the unaveraged second-order model. This is connected to the topic of libration points in the body frame of rotating gravitating triaxial ellipsoids, and shown to successfully predict the instability of libration points collinear with the long axis and the stability of libration points collinear with the short axis. Analytical and numerical results confirm the accuracy of all models discussed. 11
1. Introduction
The need to coordinate the relative orbital motion of multiple spacecraft is present in many modern scientific, commercial, and manned space missions, as well as in orbital space station construction. Numerous analytical challenges have been introduced by the small separation and even smaller margin of error necessary for space vehicle coordination in these scenarios. This has led to the burgeoning study of spacecraft relative motion, a demanding and increasingly relevant subject within the larger field of the analytical mechanics of space systems. In the short history of spaceflight, much of the research in spacecraft relative motion has been based on linearized dynamic models that neglect the perturbations that affect real orbital motion. The Hill-Clohessy-Wiltshire (HCW) model is the most famous and well-known example of such models.1 While useful for short-term applications like terminal guidance and rendezvous, this model is less useful for longer time spans, after the effect of the neglected orbital perturbations (namely higher-order gravitational terms and drag) have sufficiently accumulated. This makes it inherently undesirable for use in long-term formation flying missions. In recent work, researchers have proposed different methods and models for incor- porating the effect of perturbations on relative orbital motion. Much of this work has been focused on accounting for the perturbations due to the J2 zonal harmonic, which are considerable in low-Earth orbits. Two major approaches have been undertaken for incorporating the effects of J2: state transition matrices (STMs) and ODE models. The STMs and ODE models typically describe the relative motion of the spacecraft in the local-vertical-local-horizontal (LVLH) frame of the chief, but some also use the orbit element differences or relative orbit elements (ROEs). A comparison of these approaches was given in Johnson et al.2 Gim and Alfriend developed a well-known
3 state transition matrix (called the GA STM) that incorporates the J2 perturbation. 12
It maps orbit element differences and makes use of a spherical coordinate description of relative motion in the LVLH frame. Use of Brouwer-Lyddane theory4, 5 enables mapping between the mean and osculating orbital elements such that the STM can be expressed with either set. The STM is an analytical solution that explicitly de- pends on the orbit elements of the chief. This is advantageous because the model is valid for any eccentricity (a common challenge for linear ODE models), but the ex- plicit dependence on a computed or observed set of chief orbit elements is a potential disadvantage. The reliance on a continuous update of information about the chief orbit may not always be desirable or achievable in on-board computers. Several modifications of the GA STM have been made to include drag6 and all zonal harmonics.7 Riggi and D’Amico8 proposed a modified STM that enables sepa- ration of in-plane and out-of-plane motion using a new set of orbital elements. Finally, Biria and Russell10 also obtained a new STM using Vinti’s intermediary, which incor- porates both the J2 and J3 perturbations. The linearized restriction of the GA STM is retained by all of these additional works. One of the advantages of some ODE models over STM models is that they sometimes do not rely on a continuous update of information about the chief or- bit. Instead, they may only make use of the chief orbit elements at some initial time. Furthermore, ODE models, especially LTI models or LTV models of the form x˙ = A (t) x = A (t + T ) x can be analyzed with the many useful tools from linear systems theory and Floquet theory, providing the potential for additional dynamical insight. Such insights are often more difficult to extract from STM models. Both ODE models and STM models offer the potential for closed-form, time-explicit solutions, which have many uses ranging from terminal guidance to relative orbit determina- tion. However, it is often easier to include nonlinear and perturbation effects in ODE models than in STM models. Some researchers have developed nonlinear ODE models to account for larger sepa- ration or eccentric chief orbits. These can yield time-explicit solutions via straightfor- 13
ward perturbation expansions9 or other means. In 1963 London11 obtained second- order nonlinear equations in this manner to account for larger separation, and de Vries12 obtained linear time-varying equations to account for eccentric chief orbits. Both types of perturbations were included to second order by Anthony and Sasaki in 1965.13 In 2016, Butcher et al. developed third-order Cartesian relative motion perturbation solutions for slightly eccentric chief orbits,14 then extended this work to account for larger chief eccentricity in both Cartesian and spherical coordinate perturbation solutions.15 In 2002, Schweighart and Sedwick16, 17 obtained a linearized ODE model that at-
tempted to correct for the effects of the J2 perturbation. Their approach was to
time-average the gradient of the J2 potential to obtain constant coefficient linearized equations. They noted that such averaging resulted in a loss of some information about the perturbed relative motion, and they made efforts to correct for this. Their procedure introduced analytic corrections to an initially unperturbed chief orbit, in- stead of treating the kinematics of the perturbed LVLH frame in the more formal manner of describing the angular velocity of the frame in terms of the perturbed orbit element rates, outlined by many sources such as Prussing and Conway,18 and implemented by Casotto.19 However, Casotto’s implementation still did not result in a stand-alone model like that of Schweighart and Sedwick. There is thus a need for developing a relative motion model that incorporates the strengths of both of these
models. Furthermore, a linearized model which accounts for both C20 = −J2 and
C22 has not been previously developed, but could prove to be relevant for spacecraft rendezvous/docking and formation flying in orbits around asteroids. Asteroids are increasingly mentioned as targets for exploration by space agencies and even private companies, due to their mystery to science, potential risk to human civilization, and opportunity for future resource extraction. This thesis is organized as follows: Chapters 2 and 3 introduce much of the nec- essary background material for the work in this thesis, but also contain some new 14
analytical arguments and concepts, and the results of two carefully written numer- ical simulations. Chapter 4 focuses on the derivation and testing of a linearized
20 J2-perturbed relative motion ODE model developed in Butcher et al. This model treats the kinematics more formally (an advantage of Casotto’s work) and the per- turbation solution for the linear ODEs provides a stand-alone model. Chapter 5 introduces a completely new linearized relative motion ODE model
which incorporates the effects of C20 and C22. This model assumes that the primary body is a triaxial ellipsoid, such that all nonzero higher-order gravitational coefficients
can be expressed as polynomials in terms of C20 and C22. The model derivation assumes that these higher-order gravity terms are small enough to ignore, and that the primary body is in stable rotation about its axis of maximum inertia. Further analytical work is done with this model, and it is tested extensively with numerical
simulations. Finally, zeroing the C22 terms in this linearized model also provides a new J2 = −C20 model that seems to offer improved performance over the model developed in Butcher et al. 15
2. Orbital Motion and Relative Motion
2.1 Classical Theory of Satellite Orbits
This section serves as a basic, non-exhaustive review of introductory orbital mechanics and the two-body problem. We begin with a discussion of gravitational forces and then derive the two-body problem and include a brief review of the geometry of the bound orbits that occur in the two-body problem. There are numerous texts that discuss this subject, but this discussion is adapted from Prussing and Conway’s Orbital Mechanics,18 Vallado’s Fundamentals of Astrodynamics and Applications,21 and Battin’s An Introduction to the Mathematics and Methods of Astrodynamics.22
2.1.1 Gravitation
From Newton’s law of gravitation, gravity is an attractive force between two bodies in proportional to the product of their masses and inversely proportional to the square of the distance that separates them:
Gm m F = 1 2 ˆr (2.1) r2 where G is the gravitational constant, and G = 6.67259 × 10−11 m3kg−1s−2. For a
system of n masses, for which rij = rj − ri is a vector from mass i to mass j, the
attraction felt by the ith mass is given below, where δij is the Kronecker delta.
n X mimj F = m ¨r = G (1 − δ ) r (2.2) i i i ij r3 ij j=1 ij It can be easily shown using this expression that the linear and angular momentum of this system is constant, along with the total mechanical energy. 16
2.1.2 The Two-Body Problem and Elliptic Orbits
Using equation 2.2 with n = 2:
Gm1 Gm2 ¨r2 − ¨r1 = 3 r21 − 3 r12 (2.3) r12 r12
Dropping the subscript so r12 = r and rearranging, we arrive at the two-body equa- tion: G (m + m ) µ ¨r = − 1 2 r = − r (2.4) r3 r3
In the case where m1 >> m2, µ ≈ Gm1. This standard gravitational parameter is catalogued for all large well-known celestial bodies in the solar system. Vector manipulations of equation 2.4 result in transformed forms that are per- fect differentials, which can be integrated directly. The constants of this integration are called the integrals of motion or orbital elements.22 There are many formula- tions of orbital elements, but in this thesis we will use only the most common sets which describe the geometry of bound orbits in the two-body problem. These will be introduced after the geometry of such bound orbits has been introduced. Crossing equation 2.4 with r, we obtain:
µ r × ¨r + r × r = 0 (2.5) r3
This is integrated to obtain: r × r˙ = h (2.6) where h is the angular momentum vector, which is a constant vector. Thus, the two-body orbital motion (satisfying conservation of angular momentum) lies in the plane normal to this vector. To solve equation 2.4, take the cross product with h:
−µ −µ r˙ rr˙ ¨r × h = r × h = r × (r × r˙) = µ − (2.7) r3 r3 r r2 17
Re-arranging equation 2.7: d r ¨r × h = µ (2.8) dt r This can be integrated to yield:
r r˙ × h = µ + e (2.9) r
The vector e is constant and normal to h, so it is fixed in the orbital plane. Dotting equation 2.9 with r yields:
r · (r˙ × h) = (r × r˙) · h = h2 = µ (r + r · e) = µr (1 + e cos f) (2.10)
where f is the angle between r and e. Rearranging and solving for r, we obtain:
h2 r = (2.11) µ (1 + e cos f)
Isolating p = h2/µ and defining this variable (with length dimensions) as the parame- ter, we note that it can also be defined in terms of the dimensionless quantity e and a new (for now, undefined) quantity a, which has dimensions of length: p = a (1 − e2). Substituting this into equation 2.11, we obtain an alternate form: r = p/ (1 + e cos f). This is a polar coordinate representation of the equation of the orbit. It can be shown that for e < 1, this describes an ellipse, given in Fig. 2.1 (reproduced from Battin22). Now we define a as the semimajor axis, and b in the figure is the semiminor axis, b = pa2 (1 − e2). Two important points on the orbit are the periapsis (f = 0) and the apoapsis (f = π). Note that for e = 0, we have a circular orbit and these points (along with f) are undefined. In this case, f must be replaced with an orbit angle measured from a point of interest in the orbit. This subject will be revisited during the discussion of orbital elements. Using the equations already defined in this section, we can obtain many of the common equations used to describe two-body orbital mechanics. A few more useful equations will be presented for reference, but without derivation. First, the period 18
Figure 2.1. Elliptical Orbit Geometry22 of an orbit T and mean motion n (the mean angular rate of the orbiting body) are defined: s a3 T = 2π (2.12) µ 2π r µ n = = (2.13) T a3 The mean motion n defines the mean anomaly M, which is a linearly increasing angular variable that will be useful for several expansions and equations used in the following chapters:
M = n (t − tp) (2.14) where tp is the time of periapsis passage.
2.1.3 Orbital Elements for Elliptic and Circular Orbits
It is necessary to describe the orientation of the elliptical or circular orbit with respect to an (assumed) inertially fixed coordinate system. This is done with a 3-1-3 Euler angle rotation through the angles Ω (called the right ascension of the ascending node, RAAN), i (called the inclination), and ω (called the argument of periapsis).23 The rotation from an inertially fixed coordinate system to the inertially fixed orbit plane (using these angles) is depicted in Fig. 2.2 (reproduced from Schaub and Junkins23). 19
Figure 2.2. The 3-1-3 Euler Angles and the Orbit Plane23
Note that the argument of latitude θ = ω + f is also defined in the figure. This is the angle between the ascending node and the current location in the orbit of the satellite. This angle must be used for inclined circular orbits, since the argument of periapsis ω is undefined. For equatorial circular orbits, the angular position of the ˆ orbiting satellite must be measured from the inertially fixed reference direction ix. Since a 3D orbit is a 3 degree-of-freedom problem, we need six quantities to define the state of a satellite in an elliptical orbit. The most common orbit element set to be used in this thesis is [a, e, i, ω, Ω, f], and all of these quantities have already been defined in this chapter. Note that for circular orbits, this set is undefined, and the state can be described using fewer than 6 quantities. For example, for a circular inclined orbit, we would use the orbit element set [a, i, Ω, θ].
2.2 Analytical Relative Dynamics of Co-orbiting Bodies
Often, we are interested in the simultaneous behavior of multiple orbiting spacecraft. This is especially true if the spacecraft are in similar orbits (either due to coincidence or to fulfill a certain mission). In this section, we will describe the motion of a 20
secondary spacecraft with respect to a primary spacecraft, in a coordinate system centered on the primary. This relative orbital motion description can be very useful for formation flying and rendezvous and docking of multiple spacecraft.
2.2.1 Relative Orbital Motion
In the spacecraft relative motion problem, the motion of the secondary (deputy) spacecraft is described in a rotating frame centered on the primary (chief) spacecraft. This frame is called the Hill frame, or equivalently the Local Vertical-Local Horizontal (LVLH) or R-T-N frame. The problem geometry is depicted in Fig. 2.3 below.
Figure 2.3. Relative Motion Problem Geometry
The positions of the chief and deputy spacecraft with respect to the primary body
are given by rc and rd, respectively. The vector from the chief to the deputy is
∆r = rd − rc, and it is resolved in its LVLH components ∆r = xeˆr + yeˆt + zeˆn. The
unit vectors eˆr, eˆt, eˆn are defined:
rc rc × r˙ c eˆr = , eˆn = , eˆt = eˆn × eˆr (2.15) rc krc × r˙ ck 21
The relative motion dynamics are obtained by first treating the kinetics of the problem, by subtracting the accelerations acting on the chief from those acting on the deputy:
∆¨r = a (rd) − a (rc) (2.16)
µ where a (r) = − r3 r if we are only considering two-body gravitational dynamics. Treating the kinematics involves resolving the inertial acceleration of the relative
position vector into the Hill frame, which rotates with angular velocity ωH . This is done by applying the transport theorem twice: H d2∆r H d∆r ∆¨r = + ω˙ × ∆r + 2ω × + ω × (ω × ∆r) (2.17) dt2 H H dt H H Pure two-body dynamics are not encountered in reality; there are always perturb- ing accelerations acting on any orbiting satellite. The nature of these perturbations will determine the exact forms of a (r) and ωH . In Earth orbits, these perturbations are generally small and two-body dynamics will approximate the behavior of orbiting satellites very well for short time spans.
2.2.2 The Hill-Clohessy-Wiltshire Model
The Hill-Clohessy-Wiltshire model approximates the relative motion of satellites in similar orbits in the two-body problem. While George Hill also obtained linearized relative motion equations to describe the Moon’s orbit in a rotating Earth-centric frame, the best known introduction of this model is from 1960, in which it was used to design a preliminary terminal guidance system for satellite rendezvous.1 This simple but useful model (which is restricted for use with small separations, small perturbing accelerations, and a circular chief orbit) will now be derived. We begin by treating the kinetics of the problem, by subtracting the two-body accelerations acting on the chief from those acting on the deputy: µ µ ¨ ∆r = − 3 rd + 3 rc (2.18) rd rc 22
Noting rd = rc + ∆r, we expand the equation:
µ µ ¨ ∆r = − 3 (rc + ∆r) + 3 rc (2.19) 2 r ((rc + ∆r) · (rc + ∆r)) c Factoring the denominator of the deputy acceleration term, and keeping only terms linear in ∆r: − 3 µ (∆r · r ) 2 µ ¨ c ∆r ≈ − 3 1 + 2 2 (rc + ∆r) + 3 rc (2.20) rc rc rc Note that linearization renders the equality in equation 2.19 as an approximation in equation 2.20, but the approximation notation will be dropped. Using the first term in a binomial expansion ((1 + )k ≈ 1 + k) to linearize the exponential term, and factoring: µ (∆r · r ) ¨ c ∆r = − 3 1 − 3 2 (rc + ∆r) − rc (2.21) rc rc Expanding the expression in parentheses and keeping only terms linear in ∆r, drop-
ping the subscript notation so rc = r:
µ (∆r · r) ∆¨r = − ∆r − 3 r (2.22) r3 r2
Noting ˆr = r/r and expressing this in dyadic notation:
µ ∆¨r = − I − 3ˆrˆr · ∆r (2.23) r3
where I is the unit dyadic, and ˆrˆr is also a dyadic. For readers unfamiliar with dyads, the pair ab is called a dyad or direct product with antecedent a and consequent b. A dyad maps any vector v into a vector parallel to a according to the definition ab · v = a (b · v).27
We resolve ∆r into its LVLH components as ∆r = xeˆr + yeˆt + zeˆn, and noting
ˆr = eˆr, we obtain the linear terms due to the difference in acceleration felt by the deputy and chief: µ ∆¨r = (2xeˆ − yeˆ − zeˆ ) (2.24) r3 r t n 23
For a circular chief orbit, r = a, so we can rewrite this using the definition of the mean motion:
2 2 2 ∆¨r = 2n xeˆr − n yeˆt − n zeˆn (2.25)
We must resolve the inertial acceleration of the relative position vector into the LVLH
frame as well, which rotates with angular velocity ωH = neˆn. This is done by applying the transport theorem twice: H d2∆r H d∆r ∆¨r = + ω˙ × ∆r + 2ω × + ω × (ω × ∆r) (2.26) dt2 H H dt H H
If the chief spacecraft is in a circular orbit, ωH = neˆn is constant: H d2∆r H d∆r ∆¨r = + 2ω × + ω × (ω × ∆r) (2.27) dt2 H dt H H Resolving this in LVLH components and simplifying: