SPACECRAFT FORMATION CONTROL USING ANALYTICAL INTEGRATION OF GAUSS’ VARIATIONAL EQUATIONS

Mohamed Khalil Ben Larbi, Enrico Stoll

Institute of Space Systems Technische Universitaet Braunschweig Hermann-Blenk-Str. 23, 38108 Braunschweig, Germany

ABSTRACT Apollo space program, which had the Lunar Excursion Mod- ule and the Command and Service Module being assembled This paper derives a control concept for far range Formation in orbit. The purpose is not to correct the Earth relative orbit Flight (FF) applications assuming circular reference orbits. itself during this maneuver, but rather to adjust and control The paper focuses on a general impulsive control concept for the relative orbit between two vehicles. The relative distance FF which is then extended to the more realistic case of non- is decreased to zero in a very slow and controlled manner dur- impulsive thrust maneuvers. The control concept uses a de- ing the docking maneuver [1]. scription of the FF in relative orbital elements (ROE) instead of the classical Cartesian description since the ROE provide a The modern-day focus of s/c formation flying has ex- direct access to key aspects of the relative motion and are par- tended to maintain a formation of various s/c. Several forma- ticularly suitable for relative orbit control purposes and col- tion flying missions are currently operating or in the deisgn lision avoidance analysis. Although Gauss’ variational equa- stage: synthetic aperture interferometers for Earth obser- tions have been first derived to offer a mathematical tool for vation (e.g. TanDEM-X/TerraSAR-X), dual s/c telescopes processing orbit perturbations, they are suitable for several (e.g. XEUS) and laser interferometer for the detection of different applications. If the perturbation acceleration is due gravitational waves (e.g. LISA). It became obvious that to a control thrust, Gauss’ variational equations show the ef- the formation flying concept overcomes significant techni- fect of such a control thrust on the keplerian orbital elements. cal challenges and even avoids financial limitations. Indeed, Integrating the Gauss’ variational equations offers a direct re- the distribution of sensors and payloads among several s/c lation between velocity increments in the local vertical lo- allows higher redundancy, flexibility, and new applications cal horizontal (LVLH) frame and the subsequent change of that would not be achievable with a single s/c [2]. Recently, keplerian orbital elements. For proximity operations, these formation flying with non-cooperative objects has emerged equations can be generalized from describing the motion of as a new focus, motivated through the increasing number of single spacecraft to the description of the relative motion of such objects especially in low Earth Orbit. The exploitation two spacecraft. This will be shown for impulsive and finite- of satellite-based resources has led to an ever increasing num- duration maneuvers. Based on that, an analytical tool to es- ber of non-cooperative objects such as defunct satellites and timate the error induced through impulsive planning is pre- rocket upper stages termed as space debris. A collisional sented. The resulting control schemes are simple and effective cascading effect has been postulated by Kessler in the late and thus, also suitable for on-board implementation. Simula- seventies [3] and seems more real than ever since 2009 when tions show that the proposed concept improves the timing of two intact artificial satellites, Iridium 33 (operational) and the thrust maneuver executions and thus reduces the residual Kosmos 2251 (out of service) collided distributing debris error of the formation control. across thousands of cubic kilometers. Using the the NASA long-term orbital debris projection model (LEGEND), Liou Index Terms— Gauss’ variational equations, relative or- showed that, besides already implemented mitigation mea- bital elements, impulsive thrust, continuous thrust sures, the annual active debris removal (ADR) of 5-10 prior- itized objects from orbit, is required in order to stabilize the LEO environment [4]. Those results have been backed by 1. INTRODUCTION several other studies [5] and meanwhile are widely accepted in the space debris community. The theory of spacecraft (s/c) formation flying has become the focus of considerably extensive research and development This evolution has led to a substantial research effort to effort during the last decades. Earlier design techniques ad- develop a theory that could simply and explicitly address dressed rendezvous and docking missions such as those of the the relative motion and collision avoidance issue in control design. Thus the development of the upcoming theories, 2. DYNAMICS OF RELATIVE MOTION such as the relative orbital elements (ROE) and Eccentric- ity/Inclination (E/I) vector separation, originally developed First of all we define some notations adopted in this paper. for collocation of geostationary satellites [6], were conducted. The motion of a single s/c orbiting the Earth is described in the Earth Centered Inertial (ECI) frame. The relative mo- The control of satellite formation is performed by the ac- tion of two s/c orbiting the Earth is described in the radial- tivation of on-board thrusters. Typically, impulsive control tangential-normal (RTN) frame (Fig. 1). (very short duration thrust) is preferred to continuous control The ∆(.) operator indicates arithmetic differences be- (finite-duration thrust). This is due to historical limitation on tween absolute Cartesian or Orbital parameters. The δ(.) propulsion technologies, typical payload requirements espe- operator indicates the relative Cartesian position and veloc- cially for scientific missions, and the simplicity of impulsive ity in the RTN Frame. It refers generally to a non-linear planning often allowing pure analytical maneuver design. Re- combination of the absolute Cartesian/orbital parameters. cent advances in propulsion and computer technologies sug- The s/c about which all other s/c motions are referenced gest a deeper study of continuous planning. This approach is called the Client and is denoted with subscript (• ). The is not only justified by the precision of continuous maneuvers 1 second s/c, referred to as Servicer, is to fly in formation with planning (the finite duration is explicitly addressed and no im- the Client and is denoted with the subscript (• ). Absolute pulsive assumptions are made) but also because of the typical 2 Cartesian and orbital parameters without subscript are to be advantages of low thrust propulsion systems, such as reduced understood as Client parameters. mass, limited required power, and variable exhaust velocity.

A vast amount of literature exists on formation recon- 2.1. Linearized Equations of Relative Motion figuration maneuvers with impulsive thrust mainly modeled with Clohessy-Wiltshire and Lawden’s equations of relative The Client position vector in the Earth-centered inertial frame motion. The Gauss’ variational equations (GVE) of motion (ECI) is noted r1. The relative orbit will be described in the however offer an ideal mathematical framework for design- rotating local orbital frame RTN in terms of the Cartesian co- T ing impulsive control laws [7]. These equations have been ordinate vector δr = x y z
. We introduce dimension- extensively used in the last decades for absolute orbit keeping of single s/c, but have only recently been exploited for forma- tion flying control in LEO as introduced by Schaub et. al [8] and subsequently by Vaddi et al. [9], Breger et al. [10] and D’amico [11]. The reason for such slow development is that GVE describe the effect of control acceleration on the time derivative of the Keplerian orbital elements which were nor- mally used to parametrize the motion of a single s/c but not the relative motion of a formation [10]. In this paper, we build upon the previously mentioned ref- erences and give the following original contributions. Firstly, a comprehensive literature survey of the relative motion parametrization and Gauss’ variational equations for relative Fig. 1. Illustration of a s/c formation in RTN frame motion is presented demonstrating the convenience of ROE and GVE-based maneuver planning as opposed to Cartesian less spatial coordinates and a dimensionless time τ via the parametrization. Secondly, Gauss’ variational equation for equations: relative motion using finite duration thrust are derived for δr T a specific set of ROE. Thirdly, an impulsive maneuver plan δρ = = x¯ y¯ z¯
and dτ = ndt , (1) based on[2] is extended to the general case of non-zero rel- r1 ative semi major axis and finally translated to the case of finite-duration thrust. The paper is organized as follows. δr
T In section 2, an overview of the theory of relative motion, δρ = = x¯ y¯ z¯ (2a) r including ROE, is presented. In section 3, an overview of 1 the GVE and their application in relative orbit control is dτ = ndt , (2b) presented and the integrated GVE are derived. Section 4 is with n the mean motion. The differentiation with respect to dedicated to the study of the effects of impulsive and finite the independent variable τ is written here as duration thrust on ROE. Subsequently the impulsive and fi- nite duration maneuver schemes are derived in section 5 and d () ()0 ≡ . (3) verified via numerical simulation. dτ The dimensionless state vector is then noted The amplitudes and phases of the in-plane and out-of-plane relative motion oscillations are T δρ δρ 0
δx = . (4) q q 2 2 2 2 c34 = c3 + c4 c56 = c5 + c6 (8) The Clohessy-Wiltshire (CW) equations of motion take a very c  c  elegant numerically advantageous form if written in a non- ϕ = arctan 4 θ = arctan 6 . (9) dimensional form [12]: c3 c5 We can easily see from (7) that the Servicer moves in an x¯00 − 2¯y0 − 3¯x = 0 (5a) y¯00 + 2¯x0 = 0 (5b) z¯00 +z ¯ = 0 . (5c)

Note that these equations of motion are valid only if :

• the Client orbit is circular,

• δρ  r1,

• the Client and Servicer s/c have a pure Keplerian mo- tion. Fig. 2. Illustration of the integration constants in the projected instantaneous (no drift) relative motion ellipse at τ = τ0. Further, the out-of-plane component z¯ in equation (5c) decou- ples from the radial and along track directions (in-plane). A elliptical-like pattern around the Client as illustrated in Fig. 1 more detailed view on how the equations are obtained can be and Fig. 2. Indeed: found in [1, 12]. • The projection of the relative path in the RT plane is an ellipse centered in (c1, c2). Because of the drift 2.2. Solution of the Linearized Equations of Motion 3 term c1 2 (τ − τ0) the ellipse is instantaneous and the y¯- value of the center is valid only at τ = 0. Bounded rel- The CW equations (5) are a set of three coupled ordinary ative motion is hence obtained for c = 4¯x +2¯y0 = 0. homogeneous second order equations with constant coeffi- 1 0 0 cients. Six independent constants are thus required to de- • The RN projection is an ellipse centered in (c1, 0) for termine a unique solution for a relative orbit. The general (ϕ − θ) ∈ {0, π} which gets tighter and dwindle to a homogenous solution can be written as the product of a state π line for (ϕ − θ) → 2 . In the case of bounded rela- transition matrix Φ(τ.τ0) with an integration constants vector tive motion (c1 = 0) the Client lies on this line which
T c c1 ··· c6 . A possible representation is: leads to collision risk if along-track position uncertain- ties exist and suggests choosing (ϕ − θ) ∈ {0, π} to δx (τ) = Φ (τ, τ0) c , with Φ (τ, τ0) = minimize this risk. This Presumption will be discussed in section 2.4.  1 0 − cos τ − sin τ 0 0  3 − 2 (τ − τ0) 1 2 sin τ −2 cos τ 0 0  2.3. Relative Orbital Elements  0 0 0 0 sin τ − cos τ   , In conventional analysis, the set of independent variables c  0 0 sin τ − cos τ 0 0    could be computed using the initial conditions consisting of  − 3 0 2 cos τ 2 sin τ 0 0   2  the position r and velocity v at some specific initial time 0 0 0 0 cos τ sin τ t , often taken at zero for convenience. However, any six (6) 0 independent constants can describe the solution, with the and c a vector containing a set of six independent integration physical nature of the problem usually dictating the choice. constants as described in [13]. To study the geometry of the Many authors worked on that issue searching combinations relative path we rewrite the first three rows of equation (6) in of Keplerian elements of the co-orbiting s/c to describe their amplitude-phase form relative motion ([14]). The motivation for this effort was the advantages of the Keplerian elements description, expe- x¯ = c1 − c34 cos (τ − ϕ) (7a) rienced for single s/c in the last decades, compared with the 3 y¯ = c2 − c1 (τ − τ0) + 2c34 sin (τ − ϕ) (7b) classical position-velocity description. Similar advantages 2 were expected and could be noticed. This new approach pro- z¯ = + c56 sin (τ − θ) . (7c) vides direct insight into the formation geometry and allows the straightforward adoption of variational equations such as the Gauss’ ones to study the effects of orbital perturbations on the relative motion. In this paper a set of non-singular orbital elements α =
T a u ex ey i Ω is used to describe the absolute or- bit of a s/c with u = M + ω, ex = e cos ω, and ey = e sin ω where a denotes the semi major axis, e the eccentricity, i the inclination, Ω the right ascension of the ascending node, ω the argument of periapsis, M the mean anomaly, and u the mean argument of latitude. Fig. 3. Illustration of the relative orbital elements in the We define the ROE introduced by D’Amico [11]: projected instantaneous (no drift) relative motion ellipse at

   −1  u = u0. δa (a2 − a1) a1  δλ  (u2 − u1) + (Ω2 − Ω1) cos i1     δex  ex − ex  are found to be zero, then it can immediately be concluded δα =   =  2 1  , δey  ey2 − ey1  that the amplitude of the out-of-plane motion is zero (c56 =     p 2 2 δix   i2 − i1  kδik = (∆i) + (∆Ω sin i1) ). δiy (Ω2 − Ω1) sin i1 Furthermore, we obtain a mapping tool between the rel- (10) ative state vector δx (τ) at a generic time τ and the initial where δλ denotes the relative mean longitude, δe and δi the ROE vector δα(τ0). Keeping in mind the equivalence be- relative eccentricity and inclination vectors. The ROE defined tween mean argument of latitude u and the independent vari- in (10) are all invariants of the unperturbed relative motion able τ, we write : with the exception of δλ, which evolves linearly with time. δλ˙ can be approximated to first order as δx(u) = Φ(u, u0)δα(u0) (14) Of practical use is mainly the inverse linear mapping from the ˙ 3 ∆a δλ = ∆u ˙ = n2 − n1 = − n1 (11) relative state vector to the initial ROE vector with Φ−1(u, u ) = 2 a1 0 The general linearized relative motion of the Servicer relative  4 0 0 0 2 0  to the Client is provided in terms of ROE by 6(u − u0) 1 0 −2 3(u − u0) 0     3 cos u 0 0 sin u 2 cos u 0  3   δα (t) = δα − (u(t) − u ) δα δ2 (12)  3 sin u 0 0 − cos u 2 sin u 0  j j0 0 10 j   2  0 0 sin u 0 0 cos u where j denotes the vector index (j = 1, ..., 6), the subscript 0 0 − cos u 0 0 sin u 2 (15) 0 indicates quantities at the initial time t0 and δj is the Kro- necker delta. Note that the only assumptions made here are It may be noted that this choice of the ROE maintain the de- coupling of the motion. In other words δa, δλ, and δe de- pure Keplerian motion and ∆u, ∆a << r1. These equations are hence valid for arbitrary eccentricities. scribe the in-plane motion δi describes the out-of-plane mo- tion. Moreover the usage of ROE increases the accuracy of the 2.4. Final Comments CW general solution because it retains higher order terms These ROE have the distinct advantage of matching exactly which are normally dropped using Cartesian description [1]. the integration constants of equation (6) [11]. It follows For example the first order Cartesian constraint for bounded 0 relative motion (c1 = 4¯x0 + 2¯y0 = 0) translated in ROE (c1, c2, c34, c56) = (δa, δλ, kδek , kδik) (13) yields δa = 0. This is in fact the only condition on two in- ertial orbits to have a closed relative orbit since their energies ϕ and θ are the arguments of the vectors δe and δi in polar are equal. The ROE constraint is thus universally valid (no coordinates as depicted in Fig. 3. linearization). That means that the vector c is not only an integration Because of the coupling between semi major axis and or- constant vector which could be geometrically interpreted in bital period, small uncertainties in the initial position and ve- the relative trajectory (Fig. 2) but also receives a geometri- locity result in a corresponding drift error and thus in a grow- cal meaning by means of Servicer’s and Client’s Keplerian ing along-track error [11]. Long-term predictions of the rel- elements. Statements about the relative orbit geometry can ative motion between Servicer and Client are therefore sen- directly be made based on the absolute Keplerian elements sitive to both orbit determination errors and maneuver execu- without solving any equation. For example if ∆i and ∆Ω tion errors. In order to minimize the collision risk of the two ∆v s/c in the presence of along-track position uncertainties, they impulse provides the relation between the maneuver M must be properly separated in RN directions. As expected and the subsequent change in orbital elements ∆α where t+ from the results of section 2.2 and shown in [6] this can be ∆v = R M γ dt and (· ) denotes the maneuver execution M t− p M achieved by a (anti-)parallel alignment of the δe and δi vec- M time tors ((ϕ − θ) ∈ {0, π}) . In this case RN separations never For proximity operations, these equations can be gener- vanish at the same time and provide a minimum safe separa- alized from describing the motion of single spacecraft to the tion between the s/c at all times. This principle is termed E/I description of the relative motion of two spacecraft. This ap- separation. proach is the most natural way to control relative orbital ele- Perturbations of the motion such as J and atmospheric 2 ments and have the main advantage that it allows us to trans- drag effects can be easily incorporated through the convenient late the aforementioned advantages of the ROE parametriza- orbital elements description. The only perturbation which af- tion into maneuver planning. fect the E/I separation is the earth oblateness [11]. It turned out that choosing δix = 0 avoids a secular motion of δλ and 3.1. Impulsive Thrust δi due to J2 and provides hence a more stable configuration. Therefore the passively safe and stable configuration given We can extend the result above for relative motion. Let τ the
T f through δαnom = δa δλ 0 ±kδek 0 ±kδik is dimensionless final time immediately after the maneuver and adopted as nominal configuration in this work. τi the dimensionless initial time. The alteration in relative or- Finally, all six relative state variables (position and ve- bital elements can be expressed a s a function of the absolute locity) are fast varying variables, meaning that they vary orbital elements at different dimensionless times throughout the orbit. Using ROE simplifies the relative or- + − + − bit computation because even within a perturbed orbit , e.g. ∆δα =f(α2(τM), α2(τM), α1(τM), α1(τM)) (17a) gravitational perturbations, ROE will only change slowly. We obtain the direct relation between velocity increments in Due to its curvilinear nature large rectilinear distances can be the RTN frame and the consequent change of ROE: captured by small ROE variations. This property is exploited 1 and illustrated by the use of GVE for the relative control as ∆δα(u) = G(u, u )∆v M M , with described in the next section. n2a2  0 2 0 

3. GAUSS’ VARIATIONAL EQUATIONS  −2 −3(u − uM ) 0     sin uM 2 cos uM 0  G(uM ) =   . (18) The Gauss’ variational equations derived in [7] describe in the − cos uM 2 sin uM 0    RTN frame the alteration of Keplerian Orbital elements due to  0 0 cos uM  a disturbance acceleration γ . If the perturbation acceleration 0 0 sin u p M is due to a control thrust, GVE show what effect such a control ∆δα denotes the alteration of the ROE, (·M ) the maneuver ex- thrust would have on the keplerian orbital elements. Based on ∆v ecution time and M . It is possible to summarize the factor the general GVE description in [7] it is possible to derive the 1/n a ∆v 2 2 and M to obtain the dimensionless velocity incre- GVE in terms of the non-singular orbital elements vector α ment ∆ρ0 . Taking into account the evolution of the ROE M and build the limit for e → 0 for a circular orbit. As result we described in (12) we deduce that the alteration of δλ evolves get after the maneuver linearly according to the following law

da γT ∆δλ = δλ − 3(u − u ) (19) =2a (16a) M M dt na This result has been already incorporated in equation (18), du γR sin u γN =n − 2 − (16b) so that G(u , u ) decribes the instantaneous change in ROE dt na tan i na M M de γ γ while G(u, uM ) takes into account the linear evolution of δλ. x =2 cos u T + sin u R (16c) dt na na de γ γ 3.2. Relation to The State Transition Matrix y =2 sin u T − cos u R (16d) dt na na It is worth noting that the inverse state transition matrix di γN −1 = cos u (16e) Φ (u, u0) in equation (15) and the variation matrix G(u, uM ) dt na in equation (18) are closely connected. We rewrite the map- dΩ sin u γ = N (16f) ping rule for convenience: dt sin i na 0 ∆δα(u) =G(u, u0)∆ρ (20a) 0 Given an impulsive acceleration γ at a generic time t0, p −1 the integration of the system of equations (16) over the δα(u0) =Φ (u, u0)δx(u) (20b) If we set δρ = 0, implying that both spacecrafts have same 4. ASSESSMENT OF ORBITAL MANEUVERS position but different velocities in the inverted linear motion model of equation (20b), we obtain the effect of an instanta- We assume that there are three different thrust directions: ra- neous velocity increment on the ROE vector [11]. Let Φ−1 in dial, along-track and normal. The fuel consumption FC due equations (15) be divided in four 3 × 3 blocks Φ−1(u, u ), it to a single impulse is proportional to the norm of the impulse ij 0 follows vector. One impulse in normal direction is required to recon- figure the out-of-plane motion, while two in RT-direction are h −1 −1 i 0 needed to reconfigure the in-plane motion. Based on the re- δα(u0) = Φ (u, u0)Φ (u, u0) δρ (u) (21) 12 22 sults of Vaddi etal.[9] and D’amico [11] we choose π as the Equations (21) and (18) are equivalent, the only difference optimal separation between two impulses. Based on equation (18), one can deduce that radial maneu- between them is the opposite sign in the term −3(u − uM ). This is due to the fact that the result is calculated for different vers are safer than along-track ones since they do not affect mean argument of latitudes: u refers to after the maneuver the relative semi major axis and thus do not induce an evolv- ing change (drift) in mean longitude. However, they are twice while u0 refers to the initial state. The equations above assume impulsive maneuver which as expensive as along-track and are not able to achieve com- would require a propulsion source of infinite Thrust. In prac- plete formation reconfiguration because the relative semi ma- tice thrusts have always a finite duration which can be approx- jor axis can only be changed using along-track maneuvers. In imated as follow: this paper far range formation are considered, in which an ap- proach via drift is desirable and thus exclusively along-track maneuvers are used for in-plane control. The insecurity is ∆tM = m2 ∆v /Fmax , (22) M compensated through appropriate separation in RN-direction. with Fmax the available thrust level. Considering the execu- tion as impulsive is a valid assumption for small maneuvers 4.1. Impulsive Thrust Maneuvers and thus adequate for maneuver planning. For large maneu- vers the finite duration of the thrust has to be considered ex- 4.1.1. Out-of-Plane Maneuvers plicitly. The desired variation in the inclination vector can be obtained using maneuvers in normal direction. The influence of impul- 3.3. Finite-Duration Thrust sive thrusts is given by equatio (18) where ∆vN is the nor- mal impulsive thrust (scalar value) and u the location of the Depending on the amplitude of the maneuver, the mass of the N thrust. The non-trivial solution is given by a single thrust lo- s/c and the available thrust level, it is possible that the com- cated at u or u + π. Depending on the actual position on puted maneuver duration (22) is too large to be considered as N1 N1 the orbit one can choose the closest solution to the actual po- impulsive. In this case, u cannot be considered constant dur- M sition. However, operational constraints may suggest splitting ing the maneuver anymore. Let γ be the available level of M the out-of-plane maneuver in two components located at uN1 acceleration in the RTN directions and [t1, t2] the maneuver and uN1 + π. This allows for example to correct maneuver execution time interval. We integrated (18) and obtain execution errors. Let p be the thrust distribution coefficient, 1 the double thrust solution is: ∆δα = 2 H(˜uM , uˆM )γ , with (23) n a M (   2 2 ∆v = pnak∆δik at u = arctan ∆δiy N1 N1 ∆δix

∆vN2 = (p − 1)nak∆δik at uN2 = uN1 + π, p ∈ [0, 1]

H(˜uM , uˆM ) = (26)   0 2ˆuM 0 2  −4ˆu − 3 (2ˆu ) 0  4.1.2. Out-of-Plane Delta-V Budget  M 2 M   2 sinu ˜ sinu ˆ 4 cosu ˜ sinu ˆ 0 ,  M M M M  Single and double thrust impulsive maneuvers have the same −2 cosu ˜ sinu ˆ 4 sinu ˜ sinu ˆ 0   M M M M  delta-v budget:  0 0 2 cosu ˜M sinu ˆM  0 0 2 sinu ˜ sinu ˆ M M FC = |∆v | + |∆v | = nak∆δik (27) (24) N1 N2 u(t ) + u(t ) u(t ) − u(t ) u˜ = 2 1 and uˆ = 2 1 (25) M 2 M 2 4.1.3. In-Plane Maneuvers

Note that uˆM is the halved dimensionless maneuver duration The desired variation of the relative eccentricity vector δe, and u˜M the mean argument of latitude at the middle point of relative semi major axis δa and relative mean longitude δλ the maneuver. can be obtained using maneuvers in along-track and/or radial direction. The influence of impulsive thrusts is given by equa- thrust. Analogously to the impulsive case, the solution for a tion (18) where ∆vR and ∆vT are respectively the radial and homogeneous distribution (p = 0.5) is given by along-track impulsive thrusts (algebraic values) , uR and uT  2  2 k∆δik   ∆δiy  the locations of the thrusts. ∆tN1 = arcsin n a at u˜N1 = arctan n 4|γN | ∆δix It is possible to control the relative eccentricity with a sin- 2  2 k∆δik  ∆tN2 = n arcsin n a 4|γ | at u˜N2 =u ˜N1 + π gle thrust. The side effect is a persistent variation of the mean N (31) semi major axis and thus, an increasing change of the mean where γ = ±|γ | longitude. Beside operational constraints suggesting the split- N N ting of maneuvers, a double-thrust solution can limit the in- stantaneous variation of semi major to between the thrusts. 4.2.2. Out-of-Plane Delta-V Budget That means that, for p = 0.5, there is no change of the semi We consider the simple (and not limiting the generality) case major axis but a change of the mean longitude at the end of of p = 0.5. The index refers to the number of maneuvers the maneuver. (   FC1 = |γN |∆tN ( 1 ∆δey (32) ∆vT 1 = p 2 nak∆δek at uT 1 = arctan ∆δe P x FC2 = k|γN |∆tNk = |γN | (∆tN1 + ∆tN1 ) 1 ∆vT 2 = (p − 1) 2 nak∆δek at uT 2 = uT 1 + π p ∈ [0, 1] (28) Lemma 1. The splitting of the out-of-plane maneuvers re- The pair of along-track maneuvers planned to settle the new duces the propellant consumption δe vector change temporarily the relative semi major axis δa x
1 Proof. We define the function h(x) = arcsin 2 − 2 arcsin (x). and thus cause a change of δλ. The caused drift between It can be shown that this function is monotonically decreasing ∆δλ = ± 3 pπk∆δek the maneuvers is 2 . Along-track maneu- for x ∈ [−1, 1] . It follows that vers can be exploited to correct additionally the semi major ( axis[2]. One solution for the control of δa and δe is given by h(0) = 0 ⇒ h(x) < 0∀x ∈ ]0, 1] (   h monotone decreasing ∆v = 1 na (+k∆δek + ∆δa) at u = arctan ∆δey T 1 4 T 1 ∆δex x 1 1 ⇒ arcsin < arcsin (x) ∆vT 2 = 4 na (−k∆δek + ∆δa) at uT 2 = uT 1 + π 2 2 (29) k∆δik The double thrust solution induces a non vanishing semi we set x = n2a ∈ ]0, 1] 2|γN | major axis difference that makes the spacecraft drift from  k∆δik 1  k∆δik each other. We have to take that into account in our control ⇒ arcsin n2a < arcsin n2a strategy and plan a second pair of maneuvers to stop the drift 4|γN | 2 2|γN | and acquire the desired mean longitude δλ alteration (along- 1 ⇒ ∆t < ∆t track separation). To stop (counteract) the variation of δλ we N1 2 N1 aim a (slightly non) zero at the end of the second pair of ma- ⇒ FC < F C neuvers which make the spacecraft maintain the target sepa- 2 1 ration (drift back). The caused drift between the maneuvers In a physical sense we can derive this result in the follow- 3 remains ∆δλ = ± 4 πk∆δek. ing way. The finite-duration thrust takes place along the arc

length uˆM The effectiveness of the maneuver is at maximum near the middle point u˜ . Splitting the maneuver provides 4.1.4. In-Plane Delta-V Budget M two arcs separated by π and thus increases the effectiveness Since the choice of p does not influence the total fuel con- of the maneuver since the length of the portion in vicinity of sumption we set p = 0.5 and obtain a homogeneous distribu- u˜M (two in our case) increases. tion. 1 Lemma . FC = |∆vT 1 | + |∆vT 2 | = nak∆δek (30) 2 The minimum of the (splitted) out-of-plane ma- 2 neuver cost is given by a homogenous distribution (p = 0.5)

4.2. Finite-Duration Thrust Maneuvers Proof. Let h(p) = arcsin (kp) + arcsin (k(1 − p)), where k is a positive constant and p a real variable in[0, 1]. 4.2.1. Out-of-Plane Maneuvers d h(p) = 0 ⇔ k[(1 − (kp)2)−1 − (1 − k2(1 − p)2)] = 0 The equation describing the influence of finite-duration dp thrusts on the out-of-plane motion is given by equation (23) ⇔ p2 = (1 − p)2 where ∆t is the duration of the thrust, γ is the normal N N ⇔ p = 0.5 thrust acceleration (scalar value) and uN the location of the d2 Since dp2 h(0.5) > 0, we deuce that h(p) has a global mini- • To stop a drift (i.e. set δa to zero) we need two pulses so mum at p = 0.5. that only δa is affected. Obviously, δλ is also affected from the maneuver but has a constant value after the In a physical sense, we can derive this result in the follow- execution. δex and δey change only between the two maneuvers and finally take on their initial values at the ing way. The length of the arc in vicinity of u˜M is maximized for homogenous distribution and thus the effectiveness of the end of the maneuvers. maneuver is maximized for p = 0.5 The formation reconfiguration and formation keeping are 4.2.3. In-Plane Maneuvers performed using the concept illustrated in Fig. 4 which is valid for the general case of δa 6= 0 and for the finite-duration The equation describing the influence of finite-duration thrust case. The desired change of the relative semi major thrusts on the in-plane motion is given by equation (23) axis is split in ∆δa = ∆δaI + ∆δaII. Two along-track ma- where ∆tR /∆tT is the duration of the thrust, γR /γT is the neuvers settle a change of the relative eccentricity vector ∆δe radial/along-track thrust acceleration (algebraic value) and and the intermediate change of the relative semi major axis uR /uT the location of the thrust. Analog to the impulsive ∆δaI. Due to the new settled semi major axis δλ will evolve case, the solution for a homogeneous distribution (p=0.5) is with a constant rate until a second pair of along-track ma- given by neuvers provoke the second semi major axis change ∆δaII. Commonly the second pair will stop the drift by setting δa  ! to zero. ∆δaI has to be chosen in a way so that the total drift  2 2 k∆δ~ek 1 an∆δa ∆tT1 = arcsin n a  1 an∆δa  +  n 8kγT k 4kγT k (inclusive between the maneuver pairs) corresponds to the de-  cos 2 n 4kγ k T ! sired change ∆δλ. For the out-of-plane motion a single pulse  2 2 k∆δ~ek 1 an∆δa split in two equivalent pulses will reconfigure δi. ∆tT2 = arcsin n a  1 an∆δa  −  n 8kγT k 4kγT k  cos 2 n 4kγ k T (33) π nT π   respectively at u˜ = arctan ∆δey , u˜ =u ˜ + π with T1 ∆δex T2 T 1 π

γT = ±kγT k u0 u (rad) uT1 uN1 uT2 uN2 uT3 uT4

4.2.4. In-Plane Delta-V Budget Δu ( FC = |γ |∆t Fig. 4. Exemplary maneuver sequence for complete forma- 1 T T (34) P tion reconfiguration FC2 = k|γT |∆Tk = |γT | (∆tT1 + ∆tT1 )

5. CONTROL OF RELATIVE MOTION

5.1. Control Scheme 5.1.1. Impulsive Scheme

The control scheme in this paper is based on the following Based on the results of the last section, we derived the 6 im- considerations pulsive thrusts (IT) as follows: • Three different thrust directions (radial, along-track 1 and normal) are available. This increases the fuel ∆vT1 = na (∆δaI + k∆δek) at uT1 = ϕ (35a) consumption in case skewed thrusts are needed but is 4 1 still a valid assumption since maneuvers are typically ∆vT2 = na (∆δaI − k∆δek) at uT2 = uT1 + π (35b) constraint to certain directions. 4 1 ∆vT3 = na∆δaII at uT3 = uT2 + nT • The reconfiguration of δe using two along track maneu- 4 vers is an optimal solution in that case [9, 11]. (35c) 1 ∆v = na∆δa at u = u + π (35d) • The two along-track thrusts can be used to simultane- T4 4 II T4 T3 ously reconfigure δe , δe and δa (and hence a non- 1 x y ∆v = + nak∆δik at u = θ (35e) vanishing variation of δλ). This is still an adequate sub- N1 2 N1 optimal solution for the in-plane reconfiguration since 1 ∆v = − nak∆δik at u = u + π (35f) generally δa remains small. N2 2 N2 N1 The intermediate change of the relative semi major axis can Table 1. Initial and final configurations be determined analytically ([15]):

aδa aδλ aδex δey aδix aδiy − 2∆δλ − πk∆δek − δa (u − u + nT + 2π) − ∆δa ∆δa = 3 2 0 T1 0 2 Initial(m) 29 −8000 0 0 0 0 I nT + π Final (m) 0 −6000 0 −500 0 250 (36)

5.1.2. Finite-Duration Scheme 5.2.1. Formation reconfiguration

Based on equation (23), we extended the solution in equa- We examine an approach phase including a complete for- tion (35) to the case of finite-duration thrusts (FDT) mation reconfiguration (FR) and keeping(FK) (same desired configuration) with a maneuver set interval ∆u correspond-   ing to 12h. It’s assumed that this frequency corresponds to n2ak∆δek an2∆δa I the radar measurement update. The implemented algorithm uˆT1 = arcsin   + (37a)  n2a∆δa  I 8|γT | 8|γT | cos compares the measured configuration (initial) to the desired 8|γT |   one (final) and computes a set of 6 maneuvers based on the 2 2 n ak∆δek an ∆δaI finite-duration scheme. This step is repeated after 12h to uˆT2 = arcsin   − (37b)  n2a∆δa  compute a maneuver set for formation keeping in case the I 8|γT | 8|γT | cos 8|γ | T desired configuration has not been reached or did not remain 2 an ∆δaII stable. uˆT3 = + (37c) 8|γT | 9 2 Error in a/ e an ∆δaII y uˆ = + (37d) Error in a/ e @900m T4 8 y 8|γT | Error in a/ i y Error in a/ i @500m   y 2 2 k∆δik 7 uˆN1 = arcsin n a (37e) n 4|γN |   6 2 2 k∆δik uˆN1 = arcsin n a (37f) n 4|γN | 5

4 Note that the middle point of each maneuver u˜M matches

exactly the pulse location uM of the IT above. Furthermore (%) Error Execution 3 ∆δaI involves solving a transcendental equation and can not be calculated analytically in case of finite-duration maneu- 2 vers. One possible approach is to approximate ∆δaI to the value computed via impulsive planning in equation (36) [16]. 1 The transcendental equation can be approximated to a poly- 0 nomial equation and solved via numerical iteration. First tests 0 100 200 300 400 500 600 700 800 900 1000 Target a/ e /a/ e (m) show that sufficiently accurate results are obtained after 3 to y y 6 iteration steps. This issue will not be treated in this paper. Fig. 5. Analytical assessment of formation error induced 5.2. Simulation through impulsive planning.

In this section the proposed control schemes are verified Fig. 6 shows the temporal evolution of the individual through a numerical integration of the nonlinear differen- ROE. The desired formation is achieved with high precision tial equations of motion using a Matlab-Simulink simula- within the first 12h and remains stable so that no maneuvers tion environment. An assessment of the achievable perfor- are needed for the formation reconfiguration. The error is mance via closed loop simulation is presented. The bound- 9.8m for aδλ and remains under 1m for the rest of the ROE. ary conditions were defined assuming unperturbed motion The temporal evolution of the elements coincides with the and a Servicer s/c with a mass of 870 kg and a maximal predicted behavior described in section 5.1 with the excep- thrust level of 0.4N. As described in section 2.4 a pas- tion of small variations in aδex and aδix which are supposed sively safe and stable configuration has been defined as to remain zero. This would be the case if the maneuvers
T δαnom = δa δλ 0 ±kδek 0 ±kδik . The ini- are executed in an impulsive manner. Since maneuvers are tial and targeted final ROE used in the simulation are listed in executed along an arc centered at uM a temporal change is Table 1. induced which sums up to zero at the end of the maneuver. 200 -4000

0 (m)

6 -6000

a (m) a

/

-200 Simulation / a Target value a -400 -8000 0 5 10 15 20 25 0 5 10 15 20 25 Time (h) Time (h)

5 0

0 -200

(m) (m)

x y

e e /

/ -5 -400

a a

-10 -600 0 5 10 15 20 25 0 5 10 15 20 25 Time (h) Time (h)

0.5 300

0 200

(m) (m)

x y

i i /

/ -0.5 100

a a

-1 0 0 5 10 15 20 25 0 5 10 15 20 25 Time (h) Time (h)

Fig. 6. Temporal evolution of the ROE during formation reconfiguration and formation keeping maneuvers

Further small discrepancies are explained as linearization 6. REFERENCES errors. [1] H. Schaub and J.L. Junkins, Analytical Mechanics of Space Systems, Second Edition, AIAA Education series, 5.2.2. Integrated GVE as Error Assessment Tool virginia, 2009.

It is possible to use the integrated GVE as precise (opposed [2] J.S. Ardaens and S. D’Amico, “Spaceborne autonomous to the standard GVE) analytical tool to propagate the effect relative control system for dual satellite formations,” of thrust maneuvers on ROE. We approximate the duration of Journal of Guidance, Control, and Dynamics, vol. 32, pp. 1859–1870, 2009. maneuvers ∆tM For IT using (22). Inserting this ∆tM in(23) yields the induced alteration in ROE. This provides an ana- lytical tool to estimate the error induced through impulsive [3] D. J. Kessler and B. G. Cour-Palais, “Collision fre- planning. Depending on mission profile, this allows us to quency of artificial satellites: The creation of a debris determine a threshold value (targeted alteration in ROE), for belt,” Journal of Geophysical Research, vol. 83, pp. which impulsive planning is sufficient to achieve the required 26372646, Januar 1978. precision. Fig. 5 depicts the analytical error estimation for aδey and aδiy. The estimated errors (1.6 % for aδey and 2 % [4] J.-C. Liou, N. Johnson, and N. Hill, “Controlling the for aδiy) coincide with simulation results. growth of future leo debris populations with active de- bris removal,” Acta Astronautica, vol. 66, pp. 648–653, 2010. [5] C. Kebschul, J. Radtke, and H. Krag, “Deriving a prior- ity list based on the environmental criticality,” Interna- tional Astronautical Congress 2014, Toronto, Canada, 2014. [6] M.C. Eckstein, C.K. Rajasingh, and P. Blumer, “Coloca- tion strategy and collision avoidancefor the geostation- ary satellites at 19 degrees west,” International Sympo- sium on Space Flight Dynamics, 1989. [7] R. H. Battin, An Antroduction to the Mathematics and Methods of Atsrodynamics, Revised Edition, AIAA Ed- ucation series, virginia, 1999. [8] H. Schaub and K.T. Alfriend, “Impulsive feedback con- trol to establish specific mean orbit elements of space- craft formations,” Journal of Guidance, Control and Dy- namics, vol. 24, pp. 739–745, July 2001. [9] S.S. Vaddi, K.T. Alfriend, S.R. Vadali, and P. Sengputa, “Formation establishment and reconfiguration using im- pulsive control,” Journalof Guidance, Control and Dy- namics, vol. 28, pp. 262–268, March 2005. [10] L.S. Breger and J. P. How, “Gve-based dynamics and control for formation flying spacecraft,” 2nd Interna- tional Symposium on Formation Flying Missions and Technologies, September 2004. [11] S. D’Amico, Autonomous Formation Flying in Low Earth Orbit, Ph.D. thesis, Technical University of Delft, March 2010. [12] Kyle T. Alfriend, R. Vadali Srinivas, Pini Gurfil, Jonathan P. How, and Louis S. Breger, Spacecraft For- mation Flying: Dynamics, control and navigation, El- sevier, Oxford, 2010. [13] D. L. Richardson and J. W. Mitchell, “A third-order an- alytical solution for relative motion with a circular ref- erence orbit,” AAS/AIAA Space Flight Mechanics Meet- ing, vol. 51, January 2002. [14] H. Schaub, Srinivas R. Vadali, J.L. Junkins, and K. T. Alfriend, “Spacecraft formation flying control using mean orbit elements,” AAS Journal of Astronautical Sci- ences, 2000. [15] M.K. Ben Larbi, “Control concept for deos far range formation flight,” Diploma Thesis - University of Stuttgart, 2011. [16] M.K. Ben Larbi and P. Bergner, “Concept for the control of relative orbital elements for non-impulsive thrust ma- neuver,” Embedded Guidance, Navigation and Control in Aerospace (EGNCA 2012), IFAC, 2012.