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14th U.S. National Congress of Theoretical and Applied Blacksburg, VA, June 23–28, 2002

SPACECRAFT RELATIVE GEOMETRY DESCRIPTION THROUGH ORBIT ELEMENT DIFFERENCES

Hanspeter Schaub∗ ORION International Technologies, Albuquerque, NM 87106

The relative orbit geometry of a formation can be elegantly described in terms of a set of orbit element diﬀerences relative to a common chief orbit. For the non-perturbed orbit these orbit element diﬀerences remain constant if the anomaly diﬀerence is expressed in terms of mean anomalies. A general method is presented to estimate the linearized relative orbit motion for both circular and elliptic chief reference . The relative orbit is described purely through relative orbit element diﬀerences, not through the classical method of using Cartesian initial conditions. Analytical solutions of the relative motion are provided in terms of the angle. By sweeping this angle from 0 to 2π, it is trivial to obtain estimates of the along-track, out-of-plane and orbit radial dimensions which are dictated by a particular choice of orbit element diﬀerences. The main assumption being made in the linearization is that the relative orbit radius is small compared to the relative orbit radius. The resulting linearized relative motion solution can be used for both formation ﬂying control applications or to assist in selecting the orbit element diﬀerences that yield the desired relative orbit geometry.

Introduction ther, as the orbits decay and the orbit elements such as the The relative orbits within a spacecraft formation are typi- semi-major axis change, the orbit element diﬀerences which cally prescribed through sets of Cartesian coordinate initial determine the relative motion will only vary slowly. conditions in the rotating Hill reference frame.1–4 The six Using orbit element diﬀerences in the control of a space- Cartesian initial conditions are the invariant parameters of craft formation has advantages too. At any instant it is the relative orbit. However, to solve for the relative orbit possible to map the current inertial position and motion that will result from such initial conditions, the dif- vectors into an equivalent set of orbit elements. By diﬀer- ferential equations of motion must be solved. Except for encing the orbit elements of the deputy to those of the unperturbed circular reference (or chief) orbit special the chief satellite, we are able to compare these diﬀerences case, solving the diﬀerential equations for the resulting rel- to the prescribed orbit element diﬀerences and determine ative motion analytically is a very challenging task that has any relative orbit errors. Note that no diﬀerential equations been tackled in various papers. Melton develops in Refer- have to be solved to determine any relative orbit tracking ence 5 a state transition matrix that can be used to predict errors. Several relative orbit control strategies have recently the relative motion for chief orbits with small eccentricities. been suggested that feed back orbit errors in terms of orbit Tschauner and Hempel have solved the relative equations of element diﬀerences.13–17 On the other hand, using Cartesian motion directly for the general case of having an elliptic chief coordinates to describe the desired relative orbit requires orbit.6 However, their solution is not explicit and requires solving the nonlinear diﬀerential relative equations of mo- the computation of an integral. Kechichian develops in Ref- tion of the desired relative orbit and diﬀerencing these states erence 1 the analytical solution to the relative orbit motion with the current position and states, unless the con- under the inﬂuence of both the J2 and J3 zonal harmonics trol tolerances allow any of the linearized relative motion assuming that the eccentricity is a very small parameter. solutions to be used. Using the orbit element diﬀerence de- Unfortunately these methods yield relatively complex solu- scription we are not required to perform any linearization or tions and the six Cartesian relative motion initial conditions solve any nonlinear diﬀerential equations to ﬁnd the current do not easily reveal the nature of the resulting relative orbit. desired deputy satellite position. It is not intuitive to the relative orbit design process how to In References 16 and 18, a linearized mapping is presented adjust the initial Cartesian conditions to obtain a relative between a particular set of orbit element diﬀerences and orbit of the desired shape and size. More recently, Broucke the Cartesian position and velocity coordinates in the ro- has presented in Reference 7 an analytical solution to the tating Hill reference frame. This work was then expanded linearized relative equations of motion for eccentric chief or- in Reference 19 to provide the state transition matrix for bits. His solution uses both time and true anomaly and ﬁnds the relative orbit motion using orbit elements. This pa- the current Cartesian coordinates of a deputy satelite given per expands on this theory and develops direct relationships the initial Cartesian coordinates. between the orbit element diﬀerences and the resulting rel- An alternate set of six invariant parameters to describe ative orbit geometry for both circular and eccentric chief the relative orbit is to use orbit element diﬀerences relative orbits. In particular, the relative orbit along-track, out-of- 8–12 to the chief orbit. In References 8 and 9 the anomaly plane and orbit radial dimensions are estimated for speciﬁc diﬀerence is prescribed in terms of the mean anomalies, sets of orbit element diﬀerences. Contrary to previous work not true or . The reason being that for in this area,11, 12 the presented results apply to both circular elliptic , a diﬀerence between two and elliptic chief orbits. But more generally, analytic solu- remains constant under the assumption of classical tions are provided for the linearized relative orbit motion, Keplerian two-body motion. Prescribing the relative orbit where the linearizing assumption is that the relative orbit geometry through sets of relative orbit element diﬀerences radius is small compared to the inertial orbit radius. While has the major advantage that these relative orbit coordi- Tschauner and Hempel provide the complex solution of the nates are constants of the non-perturbed orbit motion. Even true diﬀerential equations of motion, the linearized ana- if perturbations are considered, they typically have a sim- lytical solution here is obtained from geometric arguments ilar inﬂuence on each of the satellites in the formation if without solving any diﬀerential equations. The resulting these satellites are of equal build and type. For example, linearized analytic relative orbit solution is useful when de- atmospheric will cause the orbits to decay. However, signing a relative orbit that must meet scientiﬁc mission all satellites in the formation will experience nearly identical requirements. If the relative orbit must have a certain along- amounts of drag. As such, the relative motion between the track behavior, then this solution directly shows how to satellites is only minorly aﬀected by this . Fur- adjust the relative orbit element diﬀerences to achieve the desired motion. Further, the general solution for elliptic ∗Aerospace Research Engineer, AIAA and AAS Member. chief orbits is specialized for the small eccentricity and near

1 2 SCHAUB: RELATIVE ORBIT GEOMETRY case. With the small eccentricity case, linear Thus, given both the relative position vector ρ and the chief terms in the eccentricity e are retained, while higher order position vector r, we are able to determine the inertial mo- terms of e are dropped. The near-circular analysis drops any tion of a deputy satellite. terms containing e. The linearized relationship between the Instead of using Cartesian coordinates to describe the rel- orbit element diﬀerence description of the relative orbit and ative position of a deputy to the chief, we can also use orbit the classical Cartesian coordinate description is usefull in element diﬀerences.8, 9 Let the vector e be deﬁned through both designing and controlling relative orbits in terms of the orbit elements orbit element diﬀerences. T e = (a, θ, i, q1, q2, Ω) (5) Relative Orbit Deﬁnitions where a is the semi-major axis, θ = ω+f is the true latitude To following nomenclature is adopted to describe the angle, i is the orbit inclination angle, Ω is the argument of satellites within a formation. The satellite about which the ascending node and q are deﬁned as all other satellites are orbiting is referred to as the chief i satellite. The remaining satellites, referred to as the deputy q = e cos ω (6) satellites, are to ﬂy in formation with the chief. Note that 1 it is not necessary that the chief position actually be occu- q2 = e sin ω (7) pied by a physical satellite. Sometimes this chief position is simply used as an orbiting reference point about which The parameter e is the eccentricity, ω is the argument of the deputy satellites orbit. To express how the relative perigee, and f be the true anomaly. Let the relative or- orbit geometry is seen by the chief, we introduce the Hill bit be described through the orbit element diﬀerence vector coordinate frame O.20 Its origin is at the osculating chief δe. Whereas all six elements of the relative orbit state X satellite position and its orientation is given by the vec- vector are time varying, all the orbit element diﬀerences, tor triad {oˆ , oˆ , oˆ } shown in Figures 1. The unit vector except for the true anomaly diﬀerence, are constant for a r θ h non-perturbed Keplerian orbit. This has many advantages oˆr is in the orbit radius direction, while oˆh is parallel to the orbit momentum vector in the orbit normal direction. when measuring the relative orbit error motion and applying it to a control law. In References 18 and 16 a convenient The unit vector oˆθ then completes the right-handed coordi- nate system. Let r be the chief orbit radius and h be the direct mapping is presented which translates between the chief vector. Unless noted otherwise, Cartesian states X and the orbit element diﬀerences δe. any non-diﬀerenced states or orbit elements are assumed to In deriving this mapping, it is assumed that the relative be those of the chief. Diﬀerenced states are assumed to be orbit radius ρ is small in comparison to the inertial chief diﬀerences between the deputy and chief satellite. Mathe- orbit radius r. While Reference 16 shows both the forward matically, these O frame orientation vectors are expressed and backward mapping between these relative orbit coor- as dinates, only the mapping from orbit element diﬀerences to Cartesian Hill frame coordinates is used in the following r oˆr = (1a) development. The relative position vector components are r given in terms of orbit elements through: oˆθ = oˆh × oˆr (1b) r Vr r h x ≈ δa + r δθ − (2aq1 + r cos θ)δq1 oˆh = (1c) a V p h t (8a) r − (2aq + r sin θ)δq with h = r × r˙. Note that if the inertial chief orbit is p 2 2 ˆ circular, then oθ is parallel to the satellite velocity vector. y ≈ r(δθ + cos i δΩ) (8b) z ≈ r(sin θ δi − cos θ sin i δΩ) (8c)

t i 2 2 rb The parameter p = a(1−e ) = aη is the semi-latus rectum, O √ e 2 v with η = 1 − e being another convenient measure of the ti oˆh Deputy la e Satellite orbit eccentricity. The chief radial and transverse velocity R oˆθ ρ z components Vr and Vt are deﬁned as x h rc y V =r ˙ = (q sin θ − q cos θ) (9) r p 1 2 Chief Satellite oˆr h V = rθ˙ = (1 + q cos θ + q sin θ) (10) t p 1 2 Deputy The orbit radius r is deﬁned in terms of the orbit elements Inertial Orbit used in Eq. (5) as Chief Inertial Orbit a(1 − q2 − q2) aη2 Fig. 1 Illustration of a General Type of Spacecraft For- r = 1 2 = (11) mation with Out-Of-Plane Relative Motion 1 + q1 cos θ + q2 sin θ 1 + e cos f

The relative orbit position vector ρ and velocity vector Note that the ratio Vr/Vt in Eq. (8a) can be rewritten as ρ˙ of a deputy satellite relative to the chief is expressed in Cartesian O frame components as Vr q1 sin θ − q2 cos θ e sin f = = (12) Vt 1 + q1 cos θ + q2 sin θ 1 + e cos f ρ = (x, y, z)T (2) T Alternate mappings between orbit element diﬀerences and ρ˙ = (x, ˙ y,˙ z˙) (3) Cartesian relative orbit coordinates are found in Refer- The relative position and velocity vectors are compactly ences 10 and 21. written as General Elliptic Orbits ρ X = (4) Note that Eq. 8 provides us a direct linear mapping ρ˙ between orbit element diﬀerences δe and the Hill frame SCHAUB: RELATIVE ORBIT GEOMETRY 3

Cartesian coordinates ρ. The only linearizing assumption express the relative position coordinates (x, y, z) in terms of that was made is that the relative orbit radius ρ is small the orbit element diﬀerences in Eq. (20) through compared to the inertial chief orbit radius r. However, when describing a relative orbit through orbit element diﬀer- r ae sin f x(f) ≈ δa + δM − a cos fδe (22a) ences, it is not convenient to describe the anomaly diﬀerence a η through δθ or δf. For elliptic chief orbits, the diﬀerence in r 2 true anomaly between two orbits will vary with time. To y(f) ≈ (1 + e cos f) δM + rδω η3 avoid this issue, the desired anomaly diﬀerence between two (22b) r sin f orbits is expressed in terms of a mean anomaly diﬀerence + (2 + e cos f)δe + r cos iδΩ δM. This anomaly diﬀerence will remain constant, assum- η2 ing unperturbed Keplerian motion, even if the chief orbit is z(f) ≈ r(sin θδi − cos θ sin iδΩ) (22c) elliptic. To express the mean anomaly diﬀerences in terms of other anomaly diﬀerences, we make use of the mean anomaly Note that with this linearized mapping the diﬀerence in the deﬁnition argument of perigee δω does not appear in the x(f) expres- sion. Further, these equations are valid for both circular M = E − e sin E (13) and elliptic chief orbits. Only the δM and δe terms con- tribute periodic terms to the radial x solution. Due to the where E is the eccentric anomaly. Taking its ﬁrst variation dependence of r on the true anomaly f, all orbit element dif- we express diﬀerences in mean anomaly in terms of diﬀer- ference terms in the along-track y motion contribute both ences in eccentric anomaly and diﬀerences in eccentricity. static oﬀsets as well as periodic terms. For the out-of-plane z motion both the δi and δΩ terms control the out-of-plane ∂M ∂M δM = δE + δe oscillations. By dividing the dimensional (x, y, z) expres- ∂E ∂e (14) sions in Eq. (22) by the chief orbit radius r and making use = (1 − e cos E)δE − sin Eδe of Eq. (11), we obtain the non-dimensional relative orbit coordinates (u, v, w). Using the mapping between eccentric anomaly E and true anomaly f δa e sin f u(f) ≈ + (1 + e cos f) δM a η3 r (23a) f 1 + e E (1 + e cos f) tan = tan (15) − cos fδe 2 1 − e 2 η2 δM and taken its ﬁrst variation, diﬀerences in E are then ex- v(f) ≈ (1 + e cos f)2 + δω η3 pressed as diﬀerences in f and e through (23b) sin f + (2 + e cos f)δe + cos iδΩ η sin f δe η2 δE = δf − (16) 1 + e cos f 1 + e cos f η w(f) ≈ sin θδi − cos θ sin iδΩ (23c)

Substituting Eq. (16) into Eq. (14) and making use of the Since (y, z)  r, the non-dimensional coordinates (v, w) orbit identities are the angular deputy satellite relative orbit position with respect to the chief orbit radius axis. η2 (1 − e cos E) = (17) However, the present form of Eq. (23) is not convenient (1 + e cos f) to determine the overall non-dimensional shape of the rela- η sin f tive orbit. Reason is that there are several sin() and cos() sin E = (18) (1 + e cos f) functions being added here. Using the identities

p  A  the desired relationship between diﬀerences in true and A sin t+B cos t= A2 +B2 cos t − tan−1 mean anomalies is found. B    (24) 2 p B (1 + e cos f) sin f = − A2 +B2 sin t − tan−1 δf = δM + (2 + e cos f)δe (19) −A η3 η2

Let us redeﬁne the orbit element diﬀerence vector δe to as well as standard trigonometric identities, we are able to consist of rewrite the linearized non-dimensional relative orbit motion as T δe = (δa,δM,δi,δω,δe,δΩ) (20) s δa 1 e2δM 2 u(f) ≈ + + δe2 cos(f − f ) Note that all these orbit element diﬀerences are constants a η2 η2 u for Keplerian two-body motion. Further, while using q1 and s (25a) 2 2 q2 instead of e and ω allows us to avoid singularity issues eδe e e δM − + + δe2 cos(2f − f ) for near-circular orbits, for the following relative orbit ge- 2η2 2η2 η2 u ometry discussion such singularities do not appear. In fact, describing the relative orbit path using δe and δω instead of  e2  δM  v(f) ≈ 1 + + δω + cos iδΩ δq1 and δq2 yields a simpler and more elegant result. Using 2 η3 Eqs. (6) and (7), the diﬀerences in the parameters are s expressed as 2 e2δM 2 + + δe2 cos(f − f ) η2 η2 v (25b) δq1 = cos ωδe − e sin ωδω (21a) s e e2δM 2 δq2 = sin ωδe + e cos ωδω (21b) + + δe2 cos(2f − f ) 2η2 η2 v After substituting Eqs. (19) and (21) into the linear map- p 2 2 2 ping in Eq. (8) and simplifying the result, we are able to w(f) ≈ δi + sin iδΩ cos (θ − θw) (25c) 4 SCHAUB: RELATIVE ORBIT GEOMETRY with the phase angles fu, fv and θw being deﬁned as Deputy Orbit hc Plane  eδM  f = tan−1 (26a) u −ηδe Chief Orbit Plane   hd −1 ηδe π δ f = tan = f − (26b) w v eδM u 2   −1 δi θw = tan (26c) − sin i δΩ δw At these phase angles, the trigonometric terms will reach either their minimum or maximum value. Note that 180 de- Ω i+δi grees can be added or subtracted from these angles to yield i the second extrema point of the trigonometric functions. To further reduce the expression in Eq. (25), let us introduce the small states δu and δw: δΩ s e2δM 2 δ = + δe2 (27a) u η2 p δ = δi2 + sin2 iδΩ2 (27b) Fig. 2 Illustration of Orbit Plane Orientation Diﬀerence w between Chief and Deputy Satellites

Using these δu and δw deﬁnitions as well as Eq. (26b), the linearized relative orbit motion is described through of the chief orbit and the angular momentum vector of the deputy orbit. To prove that δw is indeed this angle, let us δa eδe u(f) ≈ − make use of the spherical law of cosines for angles. Using a 2η2 the spherical trigonometric law of cosines, we are able to (28a) 22 δ  e  relate the angles δΩ, i, δi and δw through: + u cos(f − f ) + cos(2f − f ) η2 u 2 u 2  e  δM  cos δw = cos i cos(i+δi) + sin i sin(i+δi) cos δΩ (29) v(f) ≈ 1 + + δω + cos iδΩ 2 η3 (28b) δu  e  Assuming that δΩ, δi and δw are small angles, we approxi- − 2 sin(f − fu) + sin(2f − fu) 2 η2 2 mate sin x ≈ x and cos x ≈ 1 − x /2 to solve for δw.

w(f) ≈ δw cos (θ − θw) (28c) p δ = δi2 + sin2 iδΩ2 (30) Note that the cos(2f) and sin(2f) terms are multiplied by w the eccentricity e. Only if the chief orbit is very eccentric will these terms have a signiﬁcant contribution to the over- Using the angle δw, the out-of-plane motion w(f) in all relative orbit dimension. For the more typical case of Eq. (28c) is written in the compact form shown.12, 23 having a chief orbit with a small eccentricity e, these terms only provide small perturbations to the dominant sin(f) and cos(f) terms. Using Eq. (28), it is trivial to determine the maximum radial, along-track and out-of-plane dimension of Chief Orbits with Small Eccentricity a relative orbit provided that the relative orbit geometry is prescribed through the set of orbit element diﬀerences In this section we assume that the chief orbit eccentricity {δa, δe, δi, δΩ, δω, δM}. Note that this linearized relative e is a small quantity. In particular, we assume that e is orbit motion is valid for both circular and elliptic chief ref- small but greater than ρ/r, while powers of e are smaller erence orbits. The only linearizing assumption made so far is than ρ/r. In this case we only retain terms which are linear that the relative orbit radius is small compared to the in e and drop higher order terms of e. The orbit radius r is centric inertial orbit radius. However, note that we are only now approximated as estimating the non-dimensional relative orbit shape. To ob- tain the true radial, along-track and out-of-plane motions, aη2 we need to mulitpy (u, v, w) by the chief orbit radius r. r = ≈ a(1 − e cos f) (31) Since r is time dependent for an elliptic chief orbit, the 1 + e cos f points of maximum angular separation between deputy and chief satellites may not correspond to the point of maximum while η2 ≈ 1. The linearized dimensional relative orbit mo- physical . To plot the dimensional linearized rela- tion in Eq. (22) is written for the small eccentricity case tive orbit motion, we use Eq. (22) instead. However, due to as: the ratio’s of sin() and cos() terms, it is not trivial to obtain the maximum physical dimensions of the relative orbit. Let us take a closer look at the out-of-plane motion. The true latitude angle θ , at which the maximum angular ae sin f w x(f) ≈ (1 − e cos f)δa + δM out-of-plane motion will occur, is given by Eq. (26c). As η (32a) expected, if only a δΩ is prescribed, then the maximum w − a cos fδe motion occurs during the equator crossing at θ = 0 or 180 a degrees. If only a δi is prescribed, then the maximum w y(f) ≈ (1 + e cos f)δM + a(1 − e cos f)δω motion occurs at θ = ±90 degrees. η The maximum angular out-of-plane motion is given by the + a sin f(2 − e cos f)δe (32b) angle δw as shown in Figure 2. This angle δw is the tilt angle + a(1 − e cos f) cos iδΩ of the deputy orbit plane relative to the chief orbit plane. As such, it is the angle between the angular momentum vector z(f) ≈ a(1 − e cos f)(sin θδi − cos θ sin iδΩ) (32c) SCHAUB: RELATIVE ORBIT GEOMETRY 5

Making use of the trigonometric identity in Eq. (24), the then a bounded relative motion will occur. Assuming this (x, y, z) motion is written as constraint is satisﬁed, then the diﬀerential CW equations can now be solved for an analytical solution of the relative x(f) ≈ δa + aδx cos(f − fx) (33a) orbit motion.  δM  y(f) ≈ a + δω + cos iδΩ x(t) = A0 cos(nt + α) (38a) η (33b) ae y(t) = −2A0 sin(nt + α) + yoff (38b) − aδy sin(f − fy) − sin(2f)δe 2 z(t) = B0 cos(nt + β) (38c) ae z(f) ≈ aδ cos(θ − θ ) − δ cos(2f − f ) z z z z The integration constants A , B , α, β and y are deter- 2 (33c) 0 0 off ae mined through the relative orbit initial conditions. These − (sin ωδi − cos ω sin iδΩ) 2 equations have been extensively used to generate relative orbits if the chief orbit is circular. Let us now compare the with the small states δx, δy and δz deﬁned as predicted (x, y, z) motion in terms of the true anomaly in s Eq. (33) to the CW solution in Eq. (38) if the chief orbit e2δM 2  δa 2 is assumed to be near-circular (i.e. e < ρ/r). In this case δ = + δe + (34a) x η2 a terms containing the eccentricity e are dropped, as com- pared to the small eccentricity case studied earlier where s  δM 2 only higher order terms of e were dropped. Assuming that δ = 4δe2 + e2 − δω − cos iδΩ (34b) y η all δe components are small (i.e. the relative orbit radius is assumed to be small compared to the inertial orbit radius), p 2 2 2 δz = δi + sin iδΩ (34c) and letting e → 0, we ﬁnd that r → a and η → 1. Fur- ther, note that fx and fy approach 0. Using Eq. (33) the and the phase angles fx, fy, θz and fz deﬁned as relative orbit motion (x(f), y(f), z(f)) is expressed for the near-circular chief orbit special case as ! −1 eδM fx = tan (35a) x(f) ≈ δa − a cos fδe (39a) −η δe + δa  a y(f) ≈ a(δω + δM + cos iδΩ) + 2a sin fδe (39b)   δM   e η − δω − cos iδΩ p 2 2 2 f = tan−1 (35b) z(f) ≈ a δi + sin iδΩ cos (θ − θz) (39c) y  −2δe  Note that the maximum width of the oscillatory along-track   −1 δi motion y is given by 2aδe. This result has been previously θ = tan (35c) z − sin iδΩ presented in References 11 and 12. Comparing Eqs. (38)   and (39) and noting that nt = f for this case, we are able −1 cos ωδi + sin ω sin iδΩ fz = tan (35d) to establish a direct relationship between the CW constants sin ωδi − cos ω sin iδΩ and the orbit element diﬀerences. Note that the orbital radial motion x(f) for the small eccen- A = −aδe (40a) tricity case is identical to the general orbit radial coordinate 0 p 2 2 2 in Eq. (22a) if δa is zero. The semi-major axis diﬀerence B0 = a δi + sin iδΩ (40b) must be zero for bounded relative motion if no perturba- α = 0 (40c) tions are present. With perturbations present, δa may be non-zero and the orbit radial coordinate will then be diﬀer- β = ω − θz (40d) ent between the linearizing approximations. The estimated yoff = a(δω + δM + cos iδΩ) (40e) along-track motion y(f) and out-of-plane motion z(f) will always be numerically diﬀerent between the generally ellip- Recall that Eqs. (38) require that the bounded relative mo- tic case and the small eccentricity case. tion constraint is satisﬁed. Thus the δa term is set to zero The dimensional form of the relative orbit motion in when comparing the two forms of the relation orbit motion Eq. (32) is convenient to determine the amplitudes of the expression. sinusoidal motion in either the along-track, orbit radial or out-of-plane motion. Note that since e is considered small, the double-orbit frequency terms sin(2f) are only a minor Prescribing Bounded Relative Motion perturbutation to the dominant orbit frequency sinusoidal Conditions terms. Eqs. (22), (28), (33) or (39) show how the orbit element diﬀerences and the Hill frame Cartesian coordinates must Near-Circular Chief Orbit be related for any chief true anomaly angle f. Since this If the chief orbit is circular or near-circular, then the lin- mapping must hold at any instance of time, however, these earized relative equations of motion are given through the linearized equations also approximate a solution for the rel- famous Clohessy-Wiltshire or CW equations.24 These are ative orbit motion ρ(t). To map between time and the sometimes also referred to as Hill’s equations.20 true anomaly we must solve Kepler’s equation. However, to be able to describe the relative orbit geometry in terms x¨ − 2ny˙ − 3n2x = 0 (36a) of the Hill frame Cartesian coordinates, the forms presented in Eqs. (22), (28), (33) or (39) in terms of the true anomaly y¨ + 2nx˙ = 0 (36b) f are preferred. The reason for this is that by sweeping z¨ + n2z = 0 (36c) f through a complete revolution, the (x, y, z) coordinates found through these equations will yield the linearized rel- Further, these diﬀerential equations are only valid if the ative orbit approximation that results due to a prescribed relative orbit radius is small compared to the planet centric set of constant orbit element diﬀerences. Note that no dif- orbit radius. If the relative orbit initial conditions satisfy ferential equations are solved here to determine the relative the constraint orbit motion, and that the dominant relative orbit radial (x-direction), along-track (y-direction) and out-of-plane mo- y˙0 + 2nx0 = 0 (37) tion (z-direction) can be trivially extracted. 6 SCHAUB: RELATIVE ORBIT GEOMETRY

However, note that Eqs. (22), (28), (33) or (39) do not the constraint is written in its ﬁnal form as25 explicitly contain any secular terms. For the classical two- body orbital motion, the only condition on two inertial or- y˙0 −n(2 + e) = p (52) bits to have a closed relative orbit is that their orbit x0 (1 + e)(1 − e)3 must be equal. In terms of orbit elements this corresponds to demanding that Let us linearize this constraint about a small eccentricity. In this case terms which are linear in e are retained and higher δa = 0 (41) order terms in e are dropped. The bounded relative motion This constraint is valid for both circular and elliptical chief constraint on the initial Cartesian coordinates is then given orbits. Also, note that this constraint is the precise require- by ment of the Keplerian motion for bounded relative orbit y˙ + (2 + 3e)nx = 0 (53) paths; no linearizations have been made here. For Keple- 0 0 rian two-body motion, all the orbit element diﬀerences will The ﬁnd the initial Cartesian coordinates constraint for naturally remain constant except for the mean anomaly dif- bounded relative motion at the chief orbit apoapses, we set ference. If δa is not zero between two orbits, then these r(t0) = ra = a(1 + e) and follow the same steps. The re- orbits will drift apart due to having diﬀerent orbit periods. sulting constraint for chief orbits with a general eccentricity In this case δM will not remain a constant, but grow larger is with time. The linearizations in Eqs. (22), (28), (33) or (39) can still be used to predict the relative orbit motion, y˙0 −n(2 − e) but only until the relative orbit radius ρ is no longer small = p (54) x0 (1 − e)(1 + e)3 compared to the inerital chief orbit radius r. Next we would like to see what conditions the δa = 0 while the constraint for chief orbits with a small eccentricty constraint imposes on the initial Hill frame Cartesian coordi- is given by nates X(t0) = X0. Let us use the inverse mapping between orbit element diﬀerences δe and the Hill frame Cartesian y˙0 + (2 − 3e)nx0 = 0 (55) coordinates X found in Reference 16 to express δa as Note that if the chief orbit is circular and e = 0, then this δa = 0 = 2α(2 + 3κ1 + 2κ2)x(t) + 2αν(1 − 2κ1 + κ2)y(t) bounded relative motion constraint reduces to the familiar 2α2νp 2a form of + x˙(t) + (1 + 2κ1 + κ2)y ˙(t) (42) Vt Vt y˙0 + 2nx0 = 0 (56) where the parameters which is classically obtained by solving the Clohessy- α = a/r (43) Wiltshire diﬀerential equations. The more general bounded ν = Vr/Vt (44) relative orbit constraint in Eq. (52) is valid for eccentric  p  chief orbits. However, its form requires that t0 be deﬁned κ1 = α − 1 (45) to be at the orbit perigee point. r If we include the J2 perturbation, then the various orbit 2 p κ2 = αν (46) elements will experience short period, long period and sec- r ular drift.26, 27 Mapping the instanteneous are used to simplify the expression. This general constraint elements to mean elements, only the ascending node, argu- can be further simpliﬁed by expressing it at the initial time, ment of perigee and mean anomaly will experience secular where t0 is deﬁned as the time where the true anomaly f drifts. It was shown in References 8 and 9 that the following is equal to zero and the satellite is at the orbit periapses. two mean orbit element constraints will make all orbits drift Note that the orbit radius is now given by on average at the same angular rates θ˙ and Ω:˙ r(t ) = r = a(1 − e) (47) η 0 p δη = − tan i δi (57a) 4 Further, the V is given by r J δa = 2 (4 + 3η) 1 + 5 cos2 i r δη (57b) h 5 eq V (t ) =r ˙(t ) = (q sin ω − q cos ω) = 0 (48) 2η r 0 0 p 1 2 The scalar parameter req is the equatorial Earth radius. Thus, using Eqs. (44) and (46) we ﬁnd that ν = 0 and Note that all orbit elements and orbit element diﬀerences κ2 = 0. The bounded relative orbit constraint equation is in Eq. (57) are assumed to be mean states, not osculat- now written speciﬁcally at the initial time as ing states.26 Whereas a Keplerian motion requires δa to    be zero to obtain a repeating, closed relative orbit, if the a a p J2 induced orbit drift is also considered than δa must be 0 = 2 2 + 3 − 1 x0 rp rp rp non-zero if either a change in eccentricity or change in incli- a  a  p  nation is required. However, note that with this constraint + 2 1 + 2 − 1 y˙ (49) ˙ V (t ) r r 0 the mean δθ will be zero, but the mean δω and δM will t 0 p p have secular drifts. The linearized relative motion predic- tions in Eqs. (22), (28), (33) or (39) should be treated as Since Vt(t0) = rpθ˙p and making use of Eq. (47), this con- straint is further reduced to the simpler form the mean relative motion where the short and long period components have been removed. Over one orbit the aﬀect 1 of the J2 induced drift will be very small. As such, we can (2 + e)x0 + (1 + e)y ˙0 = 0 (50) θ˙p ignore the small drifts in the orbit element diﬀerences and simply hold them constant to evaluate the shape and geome- Let n = pµ/a3 be the chief mean angular motion. Express- try of relative orbit. To see the long term eﬀect of the orbit ing the true latitude rate θ˙ at perigee as elements drift (i.e. study multiple orbit revolutions), the orbit element diﬀerences δω and δM must be considered to √ s be slowly time varying.9 Other orbit parameter constraints ˙ h µp 1 + e θp = = = n (51) could be imposed as well, as long as these constraint will r2 a2(1 − e)2 (1 − e)3 p yield a bounded relative orbit. SCHAUB: RELATIVE ORBIT GEOMETRY 7

10 case 4 case 3

5 0 rack [km] Along-T 10 case 1,2,3 20 case 1,2 case 4 10 0

5

Out-of-Plane [km]

0

-5 Out-of-Plane [km] Out-of-Plane -5 Along-T 5 rack [km] -10 7.5 5 10 Orbit Radial0 [km] -10 12.5 Orbit Radial [km] -5 -2 2 0 a) Relative Orbits in Hill Frame for e = 0.03 b) Relative Orbits in Hill Frame for e = 0.13

case 3

case 2 case 3

f f

o o o case 2 180 0 180 0o 0.02 0.04 0.06km 0.2 0.4 0.6km

c) RMS Relative Orbit Error in kilometers vs. Chief True d) RMS Relative Orbit Error in kilometers vs. Chief True Anomaly Angle f for e = 0.03 Anomaly Angle f for e = 0.13

Fig. 3 Comparison of the Linearized Relative Orbit Solutions for Cases 1–4 with e = 0.03 and e = 0.13.

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