FEFF methodology for spacecraft formations reconfiguration in the vicinity of libration points

Acta Astronautica 67 (2010) 810–817

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FEFF methodology for spacecraft formations reconfiguration in the vicinity of libration points$ Laura Garcia-Taberner a,, Josep J. Masdemont b a b

 Departament d’Informatica i Matema tica Aplicada, Universitat de Girona, Escola Polite cnica Superior, 17071 Girona, Spain IEEC & Departament de Matema tica Aplicada I, Universitat Polite cnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

a r t i c l e i n f o

abstract

Article history: Received 29 January 2010 Received in revised form 30 April 2010 Accepted 7 May 2010 Available online 2 June 2010

Formation flight of spacecraft clusters needs of reliable and accurate techniques to design the proximity maneuvers required in situations like the deployment or the reconfiguration of the formation. This work is a contribution to this set of techniques considering a finite element approach. The FEFF (finite elements for formation flight) formulation provides a systematic way to compute such maneuvers and, at the same time, incorporates optimal control techniques which can include collision avoidance or many other types of constraints. In this paper we perform basic FEFF computations in a libration point regime of the Sun–Earth system using the linearized equations about a nonlinear nominal base orbit. The nominal trajectories obtained for this model are then corrected considering a full nonlinear model (JPL-ephemeris). Also random errors have been added to the nominal reconfiguration maneuvers in order to simulate small thrust deviations. The extended FEFF methodology proves to be robust both in the computation of the nominal reconfiguration maneuvers and in the correction of the perturbations. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Astrodynamics Formation flying

1. Introduction Since some years ago different concepts of spacecraft formations have been considered for scientific missions. Main examples are the allocation of several satellites in a baseline of few hundred meters forming a virtual telescope or pairs of satellites performing the roles of mirror and focus for space observations. At the same time, missions about the libration points have had remarkable interest since the launch of ISEE3 in 1978 with the purpose of doing solar studies from a halo orbit about the Sun  Earth+ Moon L1 point. The special geometric and dynamical properties of the neighborhood of the Lagrangian points L1 and L2 provide strategic sites

$

This paper was presented during the 60th IAC in Daejeon.

 Corresponding author.

E-mail addresses: [email protected] (L. Garcia-Taberner), [email protected] (J.J. Masdemont). 0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.05.007

to put the spacecraft [1,2]. Since L1 is located between the Sun and the Earth, it has been considered for missions to observe the Sun, such as ISEE3, SOHO or Genesis, while L2 keeps the Sun and the Earth at the same direction and it is a good place for deep space observations like in the ACE mission or in formation flying concepts such as Darwin or TPF [3–5]. On the other hand, formation flight for spacecraft demands a lot in terms of technology hardware requirements and also in terms of mission design. Astrodynamic problems like station keeping maintaining relative positions during observational periods, or the deployment and reconfiguration of the formation have to be addressed and solved in an efficient way. Currently, the existing literature on formation flight trajectory design about the collinear libration points focuses mainly on rough estimates of the mission cost, like in the works of Beichman, Go´mez, Lo, Masdemont, Museth and Romans [5,6], or on the control strategies for the formation. With respect to this, Folta, Hartman,

L. Garcia-Taberner, J.J. Masdemont / Acta Astronautica 67 (2010) 810–817

Howell, Marchand [7,8] consider the formation control of the MAXIM mission about L2 and more control techniques can be found in Farrar et al. [9] and references therein. In another line, Elosegui, Go´mez, Marcote, Masdemont, Mondelo, Perea and Sa´nchez have studies concerning the transfer of the formation from the Earth to L2 with suitable geometries and control procedures (see [10–12]). Also reconfigurations and deployments of spacecraft formations have been mostly considered for missions about the Earth. Representative techniques of proximity maneuvering have been studied by McInnes [13,14] by means of Lyapounov functions and by Hadaegh, Beard, Wang and McLain [15,16] considering rotations of the formations or using a sequence of simple maneuvers. In general, these techniques have to be adapted someway to the particular reconfiguration problem under consideration. They require some tuning of the parameters or to select an appropriate sequence of motions in a set of possible permutations. Most likely the reconfiguration of a formation will need to be done many times in the lifetime of the mission. Using the procedures of our work we can address such potential complex maneuvers in a fairly general way. The deployment of the formation, to point it towards different goals, or to change the pattern for different purposes, are some examples of maneuvers that can be computed with the FEFF methodology. The basic part of FEFF consists in computing and obtaining nominal reconfiguration maneuvers in a linearized model about a nonlinear libration point orbit of the restricted three body problem (RTBP). This methodology has been developed by the authors in a recent work [17,18] and will be summarized in the next section. It essentially consists of splitting the time interval of the reconfiguration maneuver in a set of smaller intervals (mesh), where the nominal Dv’s required to accomplish it are computed using a finite element approach. Moreover, computations are carried out in such a way that they minimize a functional related to the fuel expenditure of the spacecraft, and of course, avoiding collision risks. This paper focus on representative examples where the previously truncated nonlinearities are now included with the purpose of quantifying the magnitude and influence of the deviations with respect to the nominal values obtained using the basic FEFF. To this aim we consider full JPL ephemeris and also we include some random errors in the execution of the maneuvers. The nominal maneuvers resulting from the basic FEFF demand in this case of small corrections which are compared with them.

2. Basics of the FEFF methodology An accurate description of the basic FEFF methodology that we use can be found in [17,18] and here we give just a summary for completeness. One of the main purposes of FEFF is to compute reconfigurations of a set of N spacecraft about a nominal halo orbit (or any libration point orbit), in a fixed time T. In the examples we consider in this paper, these nominal reconfiguration trajectories are computed using a linearized

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model about a nominal halo orbit of 120 000 km of z-amplitude about L2 in the Sun–Earth system. Let us consider the linearized equations for the motion about the selected halo orbit of the RTBP. These equations have the form _ XðtÞ ¼ AðtÞXðtÞ,

ð1Þ

where A(t) is a 6  6 matrix and X refers to the state of the satellite. The origin of the reference frame for the X coordinates is the nominal point on the base halo orbit at time t being the orientation of the coordinate axis parallel to the one of the RTBP. Halo orbits are periodic orbits and so A(t) is also a periodic matrix. However, the matrix A(t) also has some properties related to the characteristics of libration point orbits. For a fixed value of t it has six eigenvalues: two of them are real with opposite sign (the ones which give the hyperbolic part to the halo orbit) and the other four are pure imaginary numbers and conjugated in pairs (the ones which are related with the rotations about the orbit), as can be seen in [19]. In the case of choosing other libration orbits this is not exactly in this way, but the characteristics of hyperbolicity and rotation are maintained. For the reconfiguration maneuvers of the spacecraft some controls will be needed. Let us consider a control U i ðtÞ applied to the i-th spacecraft of the formation. Then the equations of motion for this spacecraft are of the form X_i ðtÞ ¼ AðtÞXi ðtÞ þ U i ðtÞ, where the control U i ðtÞ only changes the acceleration components, i.e. it is of the form, U i ðtÞ ¼ ð0,0,0,u xi ðtÞ,u yi ðtÞ,u zi ðtÞÞt . Then including the initial and final states of the spacecraft in the reconfiguration problem we obtain the equations 8 _ > > < Xi ðtÞ ¼ AðtÞXi ðtÞ þ U i ðtÞ, Xi ð0Þ ¼ X0i , > > : Xi ðTÞ ¼ XT ,

ð2Þ

i

where X0i and XTi stand for the initial and final states of the i-th spacecraft of the formation. The goal of FEFF is to find optimal controls, U 1 , . . . ,U N , subjected to a given set of constraints, including the fundamental ones of collision avoidance. Using the properties of the halo orbits, or of any libration point orbit, inherited by the matrix A(t) as discussed previously, we can perform a change of variables and to simplify our problem by uncoupling the equations of motion (see [18]). For each spacecraft (the index i is now dropped for clarity) Eq. (2) cast into six equations, each one of the form 8 € _ > < xðtÞ þ lðtÞxðtÞ þ tðtÞxðtÞ ¼ uðtÞ, xð0Þ ¼ x0 , xðTÞ ¼ xT , > : xð0Þ _ _ ¼ vT , ¼ v0 , xðTÞ

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where now x refers to a variable that is a function of the state of the spacecraft. As a matter of fact, we have to mention that this process of uncoupling variables is not strictly necessary. It is suitable for periodic or quasiperiodic libration point orbits to improve the computational cost, and probably it can be suitable also when considering formation flight about the Earth, but in general terms, one could skip this step. Our basic objective now is to find the set of six controls u(t) for each spacecraft which perform the required reconfiguration using minimum fuel consumption and avoiding collisions. The procedure FEFF that we implement to obtain these controls is based on the finite element methodology (see [18,20] for references related to this important computational technique). The domain for the variables (one-dimensional in our case) is the time interval [0,T] considered for the reconfiguration. We split this time interval into M elements which are subintervals of the domain. The elements can be of different lengths, depending on the nature of the reconfiguration problem or on the numerical accuracy that one wants to attain for the trajectories. Each element joins to the neighbor ones by means of a node, a distinguished point at each end of the subinterval. The nodes are the places where the maneuvers (delta-v) will be applied, and so, the objective of the problem is essentially to determine the maneuver that needs to be performed at each node. More precisely, the finite element formulation is carried out for each variable x in an element Ok (from time tk to tk + 1) obtaining equations of the form 0 1 ! ! k k K1,2 K1,1 Dukk þ Qkk xk @ A ¼ , k k xk þ 1 Dukk þ 1 þQkkþ 1 K2,1 K2,2 where the values Kki,j are known (computed from lðtÞ and tðtÞ evaluated inside Ok ) and the magnitudes Qkk and Qkk + 1 correspond to boundary terms in Ok (which disappear when assembling the full set of equations). The nodal variables xk, xk + 1 of x(t) giving the trajectory, as well as the controls Dukk and Dukk þ 1 , are the unknowns of our reconfiguration problem. Finally, assembling the elements of the mesh by means of the ordinary procedure of the finite element method, we end up with a set of 6N systems of linear equations of +1 the form (we denote Ki =Ki2,2 + Ki1,1 ) 0 1 0 1 1 K1,2 0 K0 x1 B 1 CB 2 B K2,1 CB x 2 C K1 K1,2 C B CB B& CB ^ C C & & CB B C B C M3 M2 C B CB KM3 K1,2 K2,1 x @ A M2 @ A M2 xM1 0 K2,1 KM2 0 1 0 1 0 x0 K2,1 Du1 B C B B C B Du2 C 0 C B C B B C B ^ C C, ^ þB C¼B C B C B B C @ DuM2 C 0 A @ A M2 xT K1,2 DuM1 which relate the reconfiguration trajectories of the spacecraft, determined by the M nodal xj values for each

satellite and component, with the nodal maneuvers Duj applied. The procedure FEFF then continues to reduce our reconfiguration problem to an optimization problem with constraints. The function chosen to be minimized has to be directly related to the fuel consumption, while for the constraints, in the examples of this paper, we only take into account collision avoidance. Since the fuel expenditure of spacecraft is directly related to the norm of delta-v, the functional we want to minimize is essentially

J1 ¼

Mi N X X

ri,k JDvi,k J,

ð3Þ

i¼1k¼0

where J J denote the Euclidean norm and ri,k are weight parameters that can be used, for instance, to penalize the fuel consumption of selected spacecraft with the purpose of balancing fuel resources (here for clarity we consider that ri,k multiplies the modulus of the delta-v, but in a similar way we can impose a weight on each component). Of course the functional of Eq. (3) is ill conditioned to compute derivatives when delta-v values are small (and our objective is precisely to find the delta-v as small as possible). In order to avoid this problem, we start minimizing a functional without ill conditioning problems and also somewhat related to fuel consumption,

J2 ¼

Mi N X X

ri,k JDvi,k J2 :

ð4Þ

i¼1k¼0

Using the optimal value of this functional, the mesh is adapted in order to obtain an optimal solution for (3) controlling and avoiding the ill condition (see more details in [18,21] with respect to the procedures FEFFDV, FEFF-A and FEFF-DV2 related specifically with the minimization of functionals (3) and (4)). One of the key advantages of the FEFF methodology is that we can include easily many constraints into the problem. As it is general in optimization problems, these constraints can be equality constraints, this is c(x)= 0, or inequality constraints, cðxÞ 40. Of course our most important set of constraints is collision avoidance, but in the same way, we can add other constraints to obtain for instance trajectories with thrust restrictions in magnitude or time, or geometrical constraints like confinements and exclusion zones. As a final remark, the methodology to avoid collision between spacecraft consists of assuring a minimum distance between them during the reconfiguration process. To accomplish this objective, we surround each spacecraft with an imaginary sphere of radius half the security distance. The constraint we impose is that in all the reconfiguration time the spheres do not intersect, accepting only a tangency point between them. The finite element implementation allows to check this constraint during the computational process in a suitable and convenient element basis.

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3. Dealing with nonlinearities and perturbations The FEFF methodology previously sketched provide us with reconfiguration trajectories computed by means of linearized equations about a nonlinear libration point orbit. The purpose of this section is to check up to which extent the model used in FEFF is good enough to obtain nominal trajectories when further nonlinearities are considered. Of course, in principle this is something directly related to the size of the formation. For small diameter formations the effects of the nonlinearities of the model will be almost negligible while they will become much more apparent when the diameter is big. In this section more than to evaluating and quantifying this fact in general situations, and to decide under what conditions direct outputs from FEFF are good enough and when they are not, we present a methodology that could be used to deal with nonlinearities, and in particular, we focus on the behavior and results for an ordinary example. The main idea is to consider the reconfiguration process of the formation given by FEFF as the nominal one and then to add some small maneuvers between the nodes to force the spacecraft to go through the nominal nodal states. Since we work with small formations when compared in size to the halo orbit, the linearized equations provide us with a good approximation for the nonlinear model. The current objective is to give a way to measure how the truncated nonlinear terms, as well as other perturbations, affect the nominal reconfiguration trajectory, and the corrections that should be applied to the nominal maneuvers (corrective maneuvers) in order to reach the mission goal. The study of the influence of these new nonlinearities could be done in two steps: the first one, taking into account the full RTBP equations, and the second one using JPL-ephemeris. Since in both scenarios the order of magnitude for the corrections are of the same size here we only will discuss examples in the JPL case. We refer the reader to [18] for more details. The corrective maneuvers will be computed using a strategy similar to [6]. Our nominal trajectory, the one that the spacecraft should follow, is the trajectory obtained with the FEFF methodology which minimizes (3) and consists of some nodal states that the spacecraft must encounter, plus the nominal maneuvers to be applied at the nodes in order to follow them. When we consider a nominal trajectory in the full RTBP or the JPLephemeris model (for instance by means of giving the initial condition plus the set of maneuvers at the nodal times), or when we include an execution error in the maneuvers, we obtain a new trajectory differing from the nominal one. Let us call it the true trajectory. The idea is that in each element, the difference between nominal and true trajectories will be corrected including some small corrective maneuvers (see Fig. 1). The correction of the trajectory that we implement uses a fixed number of corrective maneuvers inside the 0 1 n element Ok : Dv^ k , Dv^ k , . . . , Dv^ k , which will be applied at some given times t^ 0 , t^ 1 , . . . , t^ n . Eventually, these maneuvers should be applied as soon as possible in order to avoid the inherent exponential grow of errors with respect to time, and will be computed in such a way that

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state after the nodal maneuver corrective maneuvers True path nΔt

Deviation of nominal state at node k

(1 − n )Δt

Node k+1 nodal maneuver Nominal path Node k Fig. 1. Scheme of corrective maneuvers inside the element Ok (from nodes k to k +1).

the state of the spacecraft at node k+ 1 be the corresponding one of the nominal path. In the case of considering two corrective maneuvers, Dv^ 1k and Dv^ 2k , they are obtained by means of the equation " !! !# 0 0 fð1aÞDt faDt xk þ þ ¼ xk þ 1 , Dv^ 1k Dv^ 2k where xk and xk + 1 are the initial and final states in Ok , a 2 ½0,1=n, Dt ¼ tk þ 1 tk and ft is the time-t flow of the equations of motion. In case of doing more corrective maneuvers the controller is constructed in a similar way. We note that aDt is the time between consecutive corrective maneuvers when they are evenly applied in time. Furthermore, corrective maneuvers are chosen to minimize the functional, n X

j

2j JDv^ k J2 ,

j¼0

where the weights 2  j grant someway that the corrective delta-v decay in magnitude at each step approximately by a factor of two. 4. Fundamental case examples In this section we present few fundamental case examples to see how the technique applies. The first one is a parallel transfer example and the second one consists in swapping the positions of two spacecraft. 4.1. The parallel transfer case A very common case of reconfiguration maneuvers corresponds to satellite trajectories without collision risk. For instance when the formation needs to experience a parallel shift, or even in cases of proximity maneuvering with collision hazard there could be some satellites of the formation without such a risk. In this case FEFF converges towards a bang-bang controlled trajectory for the satellites. In Table 1 we give the result corresponding to few examples of a pure translation in the x direction (200 or 400 m in 8 or 24 h). In this table DvL represents the nominal cost given by FEFF while Dv^ LJmax and Dv^ LJ correspond to the maximum

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Table 1 Corrective maneuvers for some cases of parallel shifts. Case

DvL

n

a

Dv^ L Jmax

%

Dv^ L J 3

3

200 m, 8 h

0.69 0.69 0.69 0.69

3 4 5 6

0.167 0.125 0.100 0.083

3.3 2.5 2.2 2.2

 10  10  3  10  3  10  3

6.7 6.1 5.8 5.6

 10  10  3  10  3  10  3

0.96 0.88 0.84 0.80

200 m, 24 h

0.23 0.23 0.23 0.23

3 4 5 6

0.167 0.125 0.100 0.083

3.6 2.7 2.2 2.2

 10  3  10  3  10  3  10  3

7.5 6.7 6.4 6.1

 10  3  10  3  10  3  10  3

3.25 2.89 2.77 2.65

2.8 2.8 2.8 2.8

3 4 5 6

0.167 0.125 0.100 0.083

5.3 3.9 3.1 2.7

 10  3  10  3  10  3  10  3

9.7 9.2 8.1 7.5

 10  3  10  3  10  3  10  3

0.35 0.33 0.29 0.27

400 m, 8 h

Model equations considered correspond to JPL ephemeris and all delta-v are given in cm/s.

and total value of the corrective maneuvers when the solution is corrected using JPL ephemeris. We also present the percentage amount of corrective maneuvers with respect to the nominal DvL . Only tiny differences are found, and so, corrections essentially will have only to be taken into account for execution errors, in a similar way for the ones considered in the case of swapping satellites. 4.2. Swapping positions of two spacecraft Let us consider two spacecraft in a neighborhood of a halo orbit of z-amplitude 120 000 km about L1 in the Sun– Earth system. One of the spacecraft is located 100 m far from the orbit, in the positive direction of the X coordinate of the local frame which is parallel to the one of the RTBP. The second spacecraft is located symmetrically to the first one with respect to the orbit, i.e., at the point (  100,0,0) in local coordinates centered in the halo orbit nominal position. In a first step we consider the direct application of FEFF to obtain low thrust trajectories for this reconfiguration example while in a second step we see how the previously truncated nonlinearities of the model affect the nominal solution. Finally we also consider random execution errors in the nominal maneuvers. We see how these errors are cancelled someway by the corrective methodology and the cost associated with such corrections. The reconfiguration we consider is a swap in positions of these two spacecraft in 8 h time and with a security distance of 20 m. To attain the new desired position, the optimal trajectory for each spacecraft is essentially a bang-bang control. But if both spacecraft follow a bangbang trajectory at the same time the result ends up with a collision at 4 h of the reconfiguration process. An intuitive reasoning also determines that the optimal path to be followed by the satellites should be two arcs joined at the end points (corresponding to the initial/final positions of the spacecraft) where the central parts are enough apart to keep the exclusion spheres without intersection. This

solution is optimal, even when considering transfers that include time shifts in the departure of the satellites inside the set of feasible reconfigurations, and the FEFF methodology converges towards it. We have computed the nominal reconfiguration using different number of initial nodes in the mesh. First of all we start with 10 elements and then we use the FEFF solution with these 10 elements as initial seed to obtain the solution with 20 elements and so on. The required controls in terms of delta-v using 10, 20, 50 and 100 elements are shown in Fig. 2. As one could expect, we see that when the number of elements is increased the deltav expenditure of the reconfiguration process tends towards a low thrust profile. 4.3. The impact of further nonlinearities Let us now take the initial distance between the spacecraft as a parameter of the reconfiguration. In the previous computation we considered to swap two spacecraft that initially were at a distance of 200 m. Now we consider the same reconfiguration maneuver using the equations of motion given by full JPL ephemeris. With the initial distance we also change the security distance. Just for illustrative purposes of the methodology, the security distance has been taken here as 10% of the initial distance between the satellites in all the cases. In each case we compute the nominal delta-v control profile required using the FEFF methodology based on the linearized equations about the nonlinear halo orbit of the RTBP ðDvL Þ, and some magnitudes measuring the corrective maneuvers like: the maximum of the corrective maneuvers ðDv^ max Þ, the total amount of corrective ^ and the percentage of them with respect maneuvers ðDvÞ to the nominal DvL . On each of the elements we put four corrective maneuvers. Numerical results are shown in Table 2 while Fig. 3 represents the percentage amount of corrective maneuvers depending on the initial spacecraft distance. As a result of this simulation we see that FEFF provide good nominal reconfiguration maneuvers still when the diameter of the formation is much bigger than for proximity maneuvering should be required. In many of the applications of formation flight less than 1% of the nominal maneuvers will need to be corrected. 4.4. The impact of thrust errors Let us now consider the case where the FEFF nominal maneuvers are executed with some random errors in the magnitude of the delta-v execution. To simulate such a fact we scale each maneuver by a factor which depends on the percentage of the error (a parameter p that we can choose) and a random variable Z, which follows a normal distribution, N(0,1) (i.e., at each node we apply a scaled delta-v: Dv ¼ Dvð1 þ ZpÞ, where Dv is the nominal maneuver obtained at the output of the basic FEFF methodology). Considering again the equations of motion given by JPL ephemeris, we perform corrective maneuvers to account for two errors at the same time: the truncation error due

L. Garcia-Taberner, J.J. Masdemont / Acta Astronautica 67 (2010) 810–817

815

20 elements

10 elements

sat 1 sat 2

2

Δ a (mm/s )

2

Δ a (mm/s )

sat 1 sat 2

0.001

0.001

0

0 0

4

8

0

Time (hours)

4 100 elements

50 elements

sat 1 sat 2

2

2

Δ a (mm/s )

sat 1 sat 2

Δ a (mm/s )

8

Time (hours)

0.001

0.001

0

0 0

4 Time (hours)

8

0

4 Time (hours)

8

Fig. 2. Delta-v divided by the element length for the case example of swapping two spacecraft, using 10, 20, 50 and 100 elements. The results tend to a low-thrust profile.

4

Dist. (m) 200 350 500 750 1000 2000 5000 7500 10 000 15 000 20 000

D vL

Dv^ max

Dv^

% Dv^

3.78 6.64 9.50 14.32 19.05 39.13 94.37 142.18 188.70 282.87 378.50

3.5  10  4 4.2  10  4 5.7  10  4 1.2  10  4 2.1  10  3 6.5  10  3 1.9  10  2 6.8  10  2 9.9  10  2 3.9  10  1 1.2  100

0.021 0.039 0.063 0.133 0.202 0.557 1.673 2.820 3.829 7.657 13.970

0.55 0.59 0.66 0.93 1.03 1.42 1.77 1.98 2.03 2.71 3.69

The equations of motion correspond to full JPL ephemeris and all delta-v are given in cm/s.

to the nonlinear equations and the thruster error. In each element, we insert n small corrective delta-v to attain the nodal position and velocity given by the FEFF methodology at the end of the element. In Table 3, we present typical results we obtain for a low thrust case. In this particular example we maintain n=4 and change p, the percentage of error of the thrust. For each

Percentage of error

Table 2 Corrective maneuvers to swap the positions of two spacecraft in 8 h depending on their initial distance.

2

0 0

10000 Distance (m)

20000

Fig. 3. Percentage amount of corrective maneuvers with respect to the nominal ones depending on the initial distance between satellites. Case example of swapping two spacecraft using the model equations given by JPL ephemeris.

value of the parameters (p and n), we compute the mean for

Dv^ L Jmax (the maximum delta-v of corrections), Dv^ E (the total delta-v of corrections) and % in Dv^ E (the percentage amount of corrective delta-v with respect to the nominal

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Table 3 Corrective maneuvers (in cm/s) for the execution error in the model equations given by JPL ephemeris.

DvL

%p

n

Dv^ L Jmax

0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63

0 2 4 6 8 10 12 14 16 18 20

4 4 4 4 4 4 4 4 4 4 4

8.7 2.5 3.2 6.5 9.4 1.3 2.0 2.5 2.8 3.1 3.6

 10  5  10  4  10  4  10  4  10  4  10  3  10  3  10  3  10  3  10  3  10  5

Dv^ E

% Dv^ E

0.007 0.011 0.019 0.029 0.034 0.042 0.056 0.064 0.070 0.075 0.085

1.06 1.80 3.08 4.62 5.41 6.64 8.84 10.16 11.13 11.92 13.42

Percent of corrections

14

7

the equations of motion about a nonlinear libration point orbit of the RTBP model (although any other model could be used). In top of this methodology we have implemented a correction procedure to account for the truncated nonlinearities and for execution errors. A main result is that for most of the formations involving close encounters (say confined in a sphere of 10 km of diameter) FEFF provide accurate and optimal nominal reconfiguration maneuvers that would need of very small corrections in real scenarios. Also the whole methodology is flexible enough to be tuned for different constraints like confinements inside a region or avoidance of certain regions, time constraints in the execution of maneuvers, etc. Although the computations and case examples of this paper have been carried out in a libration point regime, the methodology is general enough to be applied in other scenarios like for missions about the Earth. Future research will be directed in this sense. Also in providing an accurate description of the cost associated with general maneuvering and confinement of formations in the libration point regime. For instance, taking into account the most important parameters defining the process, like the formation diameter, location and orientation as well as the size of collision avoidance spheres.

Acknowledgements 0 0

10

20

Percent of engine error Fig. 4. Percentage amount of corrective maneuvers with respect to the total nominal delta-v for the reconfiguration, plotted as a function of the error parameter p. The plot is obtained using a test bench of 25 reconfigurations with n= 4 and considering 500 simulations for each scenario.

delta-v obtained in the initial computations) with 500 simulations. We have not continued after 20% of execution error since the corrections would be considered very big and the accuracy of the thruster too poor. However, we have performed computations up to this big execution error to see the robustness of the methodology. In Fig. 4 we show the results of a simulation using 25 different cases of reconfigurations and doing 500 simulations for each one of the examples. Again we fix n = 4 and we study how the percentage of error grows depending on the parameter p. With the hypothesis taken into account, essentially we see a linear behavior with respect to this variable, but still corrections with respect to the nominal trajectory provided by FEFF should be very small for high performance thrusters. 5. Conclusions In this paper we present a systematic methodology to compute reconfigurations of spacecraft formations specially adapted for proximity maneuvering. The basic FEFF methodology is discussed and used to compute the nominal reconfiguration process. FEFF is based on the linearization of

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