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AAS 05-474

ASYNCHRONOUS OPTIMAL MIXED P2P SATELLITE REFUELING STRATEGIES

Atri Dutta? and Panagiotis Tsiotras?

Abstract In this paper, we study pure peer-to-peer (henceforth abbreviated as P2P) and mixed (combined single-spacecraft and P2P) satellite refueling in circular orbit constellations comprised of multiple satellites. We consider the optimization of two con?icting objectives in the refueling problem and show that the cost function we choose to determine the optimal refueling schedule re?ects a reasonable compromise between these two con?icting objectives. In addition, we show that equal time distribution between the forward and return legs for each pair of P2P maneuvers does not necessarily lead to the optimum cost. Based on this idea, we propose a strategy for reducing the cost of P2P maneuvers. This strategy is applied to pure P2P refueling scenarios as well as to mixed refueling scenarios. Furthermore, for the case of a mixed scenario, we propose an asynchronous P2P strategy that also leads to more e?cient refueling.

INTRODUCTION

It has long been recognized that servicing and refueling spacecraft in orbit has the potential to revolutionize spacecraft operations by extending the useful lifetime of the spacecraft, by reducing launching and insurance cost, and by increasing operational ?exibility and robustness.1–4 Several studies have been conducted over the past decade investigating the relative merit of satellite refueling when compared to satellite replacement.1, 5, 6 Crucial technologies that enable replenishment of satellites with propellant have already been tested or are in the process of being evaluated.7–12 Most of the previous studies in the literature have assumed that a single spacecraft alone undertakes the task of refueling the whole constellation. That is, a single service spacecraft plays the role of the sole supplier of fuel.1, 13, 14 Recently, an alternative scenario for distributing fuel amongst a large number of satellites has been proposed.15–17 In this scenario, no single spacecraft is in charge of the complete refueling process. Instead, all satellites share the responsibility of refueling each other on an equal footing. We call this the peer-to-peer (P2P) refueling strategy.16, 17

Graduate Student, D. Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150. Email: gtg048j@mail.gatech.edu. ? Professor, D. Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150. Tel: (404) 894-9526, Fax: (404) 894-2760, Email: p.tsiotras@ae.gatech.edu.

?

1

A P2P refueling strategy is, by de?nition, a distributed method for replenishing a constellation of spacecraft with fuel/propellant. Consequently, it o?ers a great degree of robustness and protection against failures. For instance, with a P2P strategy a failure of a single spacecraft will have almost no impact on the refueling of the rest of the constellation. On the contrary, a failure of the service vehicle in a single-spacecraft scenario will result in the failure of the whole mission. Although a stand-alone P2P scenario may seem unconventional at ?rst glance, it arises naturally as an essential component of a mixed refueling strategy. By mixed refueling strategy we mean a strategy which involves at least two stages. During the ?rst stage a single spacecraft refuels only a certain fraction (perhaps half) of the satellites. During the second stage the satellites that received fuel during the ?rst stage act as go-betweens, and distribute the fuel to the rest of the constellation in a P2P manner. That is, a P2P refueling strategy can be implemented as the ?nal distribution phase of a single-vehicle refueling strategy. In Refs. 18,19 it has been shown that a mixed refueling strategy is more fuel-e?cient than a single-spacecraft strategy, especially for a large number of satellites in the constellation and for short refueling periods. As a matter of fact, it is not di?cult to come up with cases for which the singlespacecraft scenario is infeasible (due to the time constraint), while a mixed refueling strategy is still possible. Pure P2P refueling for circular spacecraft constellations was originally proposed in Ref. 20 as a means to equalize fuel. In that work two P2P cases were analyzed. In the ?rst case the rendezvous costs were negligible when compared to the total amount of fuel exchange taking place. This situation arises only when the satellites are very closely spaced or when the time for refueling is su?ciently large.16 The optimal matching in this case is very simple, i.e., it is a symmetric matching§ . For the majority of cases encountered in practice however the cost incurred during the transfers is signi?cant and cannot be neglected in the optimization process. In order to achieve fuel equalization in this case, an optimization problem was formulated in Refs. 17, 19, where the absolute value of the deviation of each satellite’s fuel from the initial average fuel in the constellation is penalized. Ideally, one would like to minimize the deviation of each satellite’s fuel from the ?nal average fuel in the constellation. However, without any additional constraints, the later approach may lead to solutions where the satellites perform wasteful maneuvering just to equalize fuel. This undesirable situation does not occur in the formulation used in Ref. 19. By minimizing the deviation from the mean fuel before refueling takes place (as opposed to the mean fuel after refueling takes place) we eliminate this possibility. However, this is a rather heuristic way of addressing the objective of a P2P strategy, which is to both equally distribute fuel in the constellation and to ensure as little fuel expenditure as possible in the process. Fuel equalization and minimum fuel expenditure are two con?icting objectives. Fuel equalization requires transfer of fuel from one satellite to another and hence consumption of fuel because of the required orbital maneuvers. Minimizing total fuel consumption on the other hand, implies few and long transfers. In fact, if the fuel equalization requirement is missing, the optimal solution to the fuel maximization problem is simple: do nothing. No satellites are involved in refueling rendezvous. If, on the other hand, the requirement for fuel minimization

In a symmetric matching the satellite with the most amount of fuel pairs up with the satellite with the least amount of fuel, the satellite with the second most amount of fuel pairs up with the satellite with the second least amount of fuel, etc.

§

2

is missing, the opposite occurs: all eligible satellites are involved in refueling rendezvous. In Refs. 16, 17 and 19 the satisfaction of the previous two objectives was addressed via the introduction of a rather arti?cial cost function that minimizes satellite fuel deviation from the mean fuel in the constellation before refueling takes place. A correct formulation of the problem should involve an explicit incorporation of the two previous con?icting objectives. It is one of the objectives of this paper to ?ll this gap. In the ?rst part of the current paper we re-formulate the P2P refueling problem as a minimization problem of a cost function that is a convex combination of the previous two con?icting objectives. The cost function introduced this way is parameterized by a single nonnegative scalar 0 ≤ α ≤ 1 that plays the role of the relative weight of the two elementary optimization objectives. The choice of α thus becomes a design parameter to be tuned for best performance. This is a more direct method for formulating the P2P refueling problem than the one used in Refs. 16, 17, 19. Nonetheless, we show that the cost in Refs. 16, 17, 19 corresponds to the cost used herein for a proper choice of the parameter α. This analysis justi?es the methodology followed in Refs. 16, 17, 19. In the second, and major, part of the paper we revisit the P2P refueling problem, with the goal of further improving the transfer costs. Speci?cally, we relax two of the assumptions made in Refs. 17, 19 while calculating the fuel burnt for the orbital transfers during each fuel transaction. One of the assumptions for the P2P refueling problem studied in Ref. 19 is that when there is a fuel exchange between two satellites in a constellation, the time for the forward journey equals the time for the return journey for all satellite pairs. In the current paper, we will allow for unequal time sharing between the forward and return journeys, and we show that equal time sharing does not lead to optimal fuel consumption. We use this fact to formulate an algorithm that considerably reduces the cost of P2P maneuvers. This algorithm is also applied to a mixed refueling scenario in order to make it a more competitive option to the single-spacecraft refueling scenario. It is also shown that allowing asynchronous P2P maneuvers in such a mixed scenario further brings down the refueling cost. With the help of numerical examples, we demonstrate the improvements over Ref. 19 and we also show how the incorporation of the extensions proposed in this paper make the mixed refueling scenario a far better option than a single spacecraft strategy, particularly when the number of satellites is large.

THE P2P PROBLEM FORMULATION

The Constellation Graph

Given a collection of n ≥ 3 satellites C = {s1 , . . . , sn } with unequal amounts of fuel, the satellites with fuel greater than the average amount of fuel are termed fuel-su?cient satellites, whereas the satellites with fuel less than or equal the average amount of fuel in the constellation are termed the fuel-de?cient satellites. We use Cs to denote the set of all fuel-su?cient satellites, and Cd to denote the set of all fuel-de?cient satellites. Clearly, C = Cs ∪ Cd . It is assumed that all satellites are in the same circular orbit, but they do not have to be evenly distributed along the orbit. By a fuel/refuel transaction herein we assume a sequence of events that involves: (i) a satellite ?ring its thrusters so as to change its orbit and rendezvous with another satellite in 3

the constellation, (ii) exchange of fuel between the two satellites, and (iii) return of the ?rst satellite to its original slot. It will be assumed that during a refueling transaction, only one satellite, called the seller, can give fuel to another satellite. The latter is called the buyer. The set of seller satellites will be denoted by S and the set of buyer satellites will be denoted by B. Depending on the amount of fuel between the two, either of these two satellites can initiate a fuel transaction, i.e., perform a rendezvous with the other satellite, exchange fuel and return to its original orbital slot. The former satellite is said to be the active satellite and the latter satellite is said to be the passive satellite. The set of active satellites will be denoted by A and the set of passive satellites will be denoted by P. Note that, in general, S ∪ B ? C since not all satellites may be involved in fuel transactions. Similarly, A ∪ P ? C for the same reason. Also note that it is not necessarily true that S = A or that B = P, although this typically will be the case. For instance, it may happen that a satellite, say si , initiating a fuel transaction receives fuel (i.e., si ∈ A ∩ B) or that a passive satellite is the seller (si ∈ P ∩ S). Furthermore, it is not necessarily true that a fuel su?cient satellite will be active (i.e., Cs ? A). However, a fuel de?cient satellite is always a buyer, that is Cd ? B. Given now the set C we may construct a graph G having as nodes (or vertices) the satellites of C. We call G the constellation graph. Associated with G is a set of vertices V = {s1 , s2 , . . . , sn } and a set of edges L = { i, j : si , sj ∈ V} connecting the nodes of G. Without loss of generality, we enumerate the vertices such that i ? si for all 1 ≤ i ≤ n. This allows us in the sequel to refer to “vertex” si instead of i without the danger of confusion. We will make no distinction between the edge j, i and the edge i, j . That is, G is a undirected graph. This point needs some clari?cation. Since the propellant required for satellite si to rendezvous with satellite sj is not equal to the propellant required for satellite sj to rendezvous with satellite si , G is, in principal, a directed graph. By assigning the minimum fuel required between the two transfers si → sj and sj → si to the edge i, j we obtain an undirected graph. This is elaborated upon in the sequel. In the graph G, an edge between two vertices exists if a fuel transaction between the corresponding satellites is permissible. The number of elements of a subset of set X will be denoted by |X |. Clearly, |V| = n and for a complete graph |L| = n(n ? 1)/2. The set of vertices connected to vertex si is called the set of neighbors of si , and it is denoted by Ni . The edge neighborhood of si is de?ned by Qi = { i, j ∈ L : sj ∈ Ni }. Note that if si has no neighbors then no edges are connected to this vertex and Qi = ?. For example, we may impose that certain satellites are not involved in any fuel transactions due to operational constraints. By removing all satellites which are known a priori that cannot be involved in fuel transactions due to operational restrictions we get the core constellation graph G c . For simplicity, in the sequel we assume that G = G c . It should be kept in mind however that the following developments hold verbatim if we replace G with G c . To each edge i, j ∈ L we will assign a (positive) weight that re?ects the cost associated with a fuel transaction between the satellites connected by this edge. By an assignment or matching over the graph G we mean a partition of V into two sets Va and Vb , such that |Va | = |Vb | along with a subset M ? L and a one-to-one mapping σ : Va → Vb such that M = { i, j : si ∈ Va , sj ∈ Vb , and sj = σ(si )}. Given the positive weights on each edge, we seek the matching that maximizes the sum of the weights of all edges involved in this matching.

4

In the next section we show how the problem of ?nding the optimal pairings of satellites can be reduced to a problem of computing the maximum weighted matching in the constellation graph.

Construction of the Constellation Graph

Let, for convenience, I denote the index set of the vertices in the (core) constellation graph. That is, i ∈ I for si ∈ G. Let fi? and fi+ denote the fuel contained in each satellite before and after a fuel transaction, respectively. The average amount of fuel in the constellation ? ? before and after all fuel transactions will be denoted by f ? and f + , respectively. That is, ? ? = (1/n) + . Let pj denote the fuel burnt by satellite s ∈ A ? ? f i i∈I fi , and similarly for f i in order to rendezvous with satellite sj ∈ P and return to its original orbital slot. Notice that, in general, pj = pi . Also note that in a fuel transaction between si and sj either one j i can be the active satellite, provided that it has enough amount of fuel to rendezvous with the inactive satellite and return to its original orbital slot. Hence, the fuel cost assigned to a single rendezvous between satellites si , sj ∈ G is given by ? ?pj , if si can be active, but sj cannot, ? i ? ? i ?p , if sj can be active, but si cannot, j (1) pij = ?min{pj , pi }, if either si or sj can be active, ? i j ? ? ?∞, if neither si nor sj can be active. The objective is to minimize the square deviation of the fuel distributed among all satellites in the constellation. Therefore, the cost function to be maximized is given by Ja = ?

i∈I

? |fi+ ? f ? |2 .

(2)

The contribution of all matched vertices of G in Eq. (2) is easily computed as ?

i∈I i,j ∈Qi

? |fi+ ? f ? |2 xij ,

(3)

where xij is a binary variable associated with each edge as follows xij = 1 if i, j ∈ M, 0 otherwise. (4)

In order to ensure that each satellite is involved in at most one fuel transaction with another satellite we impose the inequality xij ≤ 1,

i,j ∈Qi

i ∈ I.

(5)

If satellite si is not involved in a fuel transaction, then fi+ = fi? . As a result, xij = 0 for all i, j ∈ Qi and the corresponding edges are not part of the optimal matching. As a matter of fact, we have that xij = 0 for all i, j ∈ L\M. 5

The contribution to Ja from all unmatched vertices is ?

i∈I

1?

i,j ∈Qi

? xij |fi? ? f ? |2 = ?

i∈I

? |fi? ? f ? |2 +

i∈I i,j ∈Qi

? |fi? ? f ? |2 xij .

(6)

? The term i∈I |fi? ? f ? |2 in the previous expression is constant, and thus it has no e?ect on the optimization process and it can be neglected. From Eqs. (3) and (7), and summing up the contributions from all satellites, we ?nally have Ja =

i∈I i,j ∈Qi

? ? |fi? ? f ? |2 ? |fi+ ? f ? |2 xij .

(7)

Recalling that each edge i, j ∈ L has contributions from two vertices i, j ∈ I of the graph, and rewriting the summation in Eq. (7) as a summation over all edges in the constellation graph, the objective function to be maximized is given by Ja =

i,j ∈L ? + ? ? ? ? |fi? ? f ? |2 ? |fi+ ? f ? |2 + |fj ? f ? |2 ? |fj ? f ? |2 xij

(8)

Letting πij denote the coe?cient of xij in the previous sum, the problem becomes one of maximizing πij xij . (9) Ja =

i,j ∈L

subject to (4) and (5). Since the objective of the refueling process is to equalize the fuel among all satellites in the constellation, we impose the constraint that after each fuel transaction between any pair of satellites, the two satellites end up with the same amount of fuel. In other words, we impose + the condition that fi+ = fj for all i ∈ I at the end of the refueling process. Noting that the di?erence between the total fuel in the satellites before and after refueling can be related to the total fuel burnt during the rendezvous,19 one obtains 1 + ? fi+ = fj = (fi? + fj ? pij ). 2 Using (10), the weight of each edge in the constellation graph becomes 1 ? ? ? ? ? πij = |fi? ? f ? |2 + |fj ? f ? |2 ? |fi? + fj ? 2f ? ? pij |2 . 2 (11) (10)

Given these weights on the edges of the constellation graph, we seek a matching M that will maximize the sum of the weights of all edges in M. This is a standard maximum weight matching problem in graph theory.21 The solution to this problem provides the pairs of satellites involved in the optimal distribution of fuel using a P2P refueling scheme.

AN ALTERNATIVE COST MINIMIZATION FORMULATION

As already mentioned, the two objectives to be satis?ed during a P2P refueling scenario are: (i) minimization of the fuel deviation among all satellites in the constellation, and (ii) minimization of the fuel expenditure during the orbital rendezvous transfers. These two objectives 6

are con?icting in nature. For instance, we can ful?l only the ?rst objective by performing continuous orbital transfers until all satellites have the same amount of fuel (perhaps even null). On the other hand, we can satisfy the second objective by not performing any orbital transfers at all. The cost function in Eq. (2) was introduced rather heuristically so that implicitly takes into account both of these objectives. In this section we show that this rationale is valid. We do this by introducing an optimization criterion Jb that incorporates explicitly the previous two con?icting objectives, and by unraveling the relationship of the cost Jb with the cost Ja in Eq. (2). Since we seek to minimize the fuel deviation among all satellites in the constellation at the end of the refueling process, we introduce the following cost function to be maximized J1 = ?

i∈I

? |fi+ ? f + |2 .

(12)

Since we also want to minimize the cost incurred during the orbital maneuvers required for the fuel transfers, we also introduce the following cost to be maximized J2 = ?

ν,? ∈M

p2 . ν?

(13)

Given J1 and J2 , we assign a relative weight between these two costs, and we combine them into a single cost function to be maximized, as follows Jb = αJ1 + (1 ? α) J2 , where 0 ≤ α ≤ 1 takes care of the relative importance assigned to the two objectives. The contribution to (12) from the satellites participating in fuel transactions is ?

i∈I i,j ∈Qi

(14)

? |fi+ ? f + |2 xij .

(15)

The contribution to J1 from the satellites not participating in fuel transactions is ?

i∈I

1?

i,j ∈Qi

? xij |fi? ? f + |2 .

(16)

Combining the contributions from the participating (matched) and nonparticipating (unmatched) satellites into (12), one obtains J1 = ?

i∈I i,j ∈Qi

? |fi+ ? f + |2 xij ?

i∈I

? |fi? ? f + |2 +

i∈I i,j ∈Qi

? |fi? ? f + |2 xij .

(17)

The average fuel available in the constellation before and after refueling are related by 1 ? ? f+ = f? ? n Using Eq. (18), we may rewrite Eq. (17) as J1 =

i∈I i,j ∈Qi

pν? .

ν,? ∈M

(18)

2 ? ? |fi? ? f ? |2 ? |fi+ ? f ? |2 + (fi? ? fi+ ) n 7

pν? xij ?

ν,? ∈M i∈I

? |fi? ? f + |2 . (19)

A simple calculation yields ? |fi? ? f + |2 =

i∈I i∈I

? |fi? ? f ? |2 + p2 + ν?

ν,? ∈M

2 ? ?? (f ? f ) n i pν?

pν?

ν,? ∈M

1 + n Note also that

pmk .

m,k ∈M\ ν,?

ν,? ∈M

? (fi? ? f ? ) = 0

i∈I ? i∈I |fi

? Moreover, the term ? f ? |2 is constant for a given constellation, and plays no role in the optimization process. Excluding this constant term, we have ? |fi? ? f + |2 =

i∈I

1 n

p2 + ν?

ν,? ∈M ν,? ∈M

pν?

m,k ∈M\ ν,?

pmk .

Hence the cost function to be maximized can be written as 2 ? ? |fi? ? f ? |2 ? |fi+ ? f ? |2 + (fi? ? fi+ ) Jb = α n ? α n

i∈I i,j ∈Qi

pν? xij

ν,? ∈M

p2 + ν?

ν,? ∈M ν,? ∈M

pν?

m,k ∈M\ ν,?

pmk ? (1 ? α)

ν,? ∈M

p2 . ν?

(20)

Writing the above summation as a summation over the edges and using Eq. (10), it follows that the criterion to be maximized takes the form 1 ? ? ? ? ? |fi? ? f ? |2 + |fj ? f ? |2 ? |fi? + fj ? pij ? 2f ? |2 xij Jb = α 2 α + n

i,j ∈L

pij

i,j ∈L m,k ∈L\ i,j

pmk xmk xij ? (1 ? α ?

α ) n

p2 xν? . ν?

ν,? ∈L

(21)

This expression consists of both linear and quadratic terms in the decision variables xij . This makes the problem a quadratic binary programming problem. One way to solve this problem is by introducing new variables in lieu of the quadratic terms. This also introduces new constraints involving the new and old variables. Formulating these as linear constraints, the problem can be converted to a linear binary programming problem for which e?cient algorithms exist. To this end, consider the quadratic term xij xmk where xij and xmk are binary variables. Note that two edges that are part of the matching cannot share the same vertex, that is, if i, j, m ∈ I, and xim = 1, then xij = 0 for all i, j ∈ L, j = m. Thus, we may only consider quadratic terms of the form xij xmk , i, j , m, k ∈ L and i, j, k, m ∈ I, all distinct. Let now I be a set of indices (of cardinality |L|) generated as follows q = n × i + j, for all i, j ∈ L, i, j ∈ I. (22)

Conversely, given q ∈ I the corresponding indices i and j are obtained via integer division by n using (22). We can therefore establish a one-to-one correspondence between elements of I and L, and we write q ? i, j to denote this correspondence. 8

Considering now distinct indices i, j, m, k ∈ I, and p, q ∈ I such that p ? i, j and q ? m, k we introduce new variables de?ned by xpq = xij xmk . These new variables are also binary since xpq = 1, when xij = 1 and xmk = 1, 0, otherwise. (24) (23)

The restrictions in Eq. (24) can be imposed on the new variables by introducing the following three linear constraints (25) xpq ≤ xij , xpq ≤ xmk , ?xpq + xij + xmk ≤ 1. (26) (27)

The ?rst two constraints ensure that whenever xij = 0 or xmk = 0, we have xpq = 0. The last of the previous three constraints ensures that xpq = 1 when xij = 1 and xmk = 1. Hence, the problem of minimizing the two objectives absorbed in Eq. (14) is equivalent to the following linear binary integer programming problem Jb =

i,j ∈L

1 ? ? ? ? ? α |fi? ? f ? |2 + |fj ? f ? |2 ? |fi? + fj ? 2f ? ? pij |2 xij 2 α ) n p2 xij + ij

i,j ∈L

?(1 ? α ?

2α n

pij pmk xpq ,

i,j ∈L m,k ∈L\ i,j

(28)

subject to the constraints given by Eqs. (25), (26), (27), and Eqs. (4)-(5). The parameter α in Eq. (14) weighs the relative importance for the ful?lment of the two performance objectives we have set for a P2P refueling scenario. If α = 0, no fuel equalization is desirable (Jb = J2 ), and we only minimize the rendezvous costs. Obviously, in such a case the optimal solution involves no satellite pairings: all satellites remain at their initial orbital slots and the matching set M is empty. Equivalently, |M| = 0. As we increase the value of α, fuel equalization becomes increasingly important and after a certain value of α = α > 0 at least one pair of satellites performs a fuel transaction. The matching set M is ? non empty, and consequently |M| > 0. For α = 1 fuel equalization is the only optimization objective (Jb = J1 ), which is achieved with a (perhaps) unacceptably large number of fuel transactions. A compromise between the performance objectives J1 and J2 is achieved via an intermediate value of α. To investigate the e?ect of α in the optimal number of satellite pairings, and compare with the original “two-in-one” cost Ja , several numerical examples have been conducted.

Numerical Example

In this section we investigate numerically the relationship between the solutions obtained via the two costs (2) and (14). Speci?cally, we show that solutions obtained via (2) correspond 9

to solutions obtained via (14) for a range of values of α that achieve a balanced compromise between the original con?icting optimization objectives J1 and J2 . Figure 1 shows a typical variation of Jb with α for the two constellations C4 and C8 in Table 1. The plots are piecewise linear, with each linear portion corresponding to a particular set of pairings of the satellites in the constellation.

Plot of Jb vs α

1000 900

800

700

600

Jb

500

400

300

200

Constellation C4 Constellation C

100

8

0

0

0.1

0.2

0.3

0.4

0.5

α

0.6

0.7

0.8

0.9

1

Figure 1: Typical variation of Jb with respect to α. Typical variations of the values of the two objective functions J1 and J2 are shown in Figures 2 and 3 for the constellations C4 and C8 , respectively. Each point on the curve in these plots is optimal, corresponding to a particular choice of α. The range of values of α for which the same pairings of satellites occur as with the optimization of Ja is also shown on these plots. Note that for this range of α the pairings of satellites are the same, hence the values of J1 and J2 are also the same. For this range of values of α we have a reasonable compromise between the two performance speci?cations J1 and J2 . Moreover, from these plots it is concluded that the use of the simpler cost Ja in lieu of Jb is justi?ed, as the former results in solutions which are identical to those obtained via Jb for values of α that provide a balance between the objectives J1 and J2 . The case for using Ja instead of Jb is made stronger in light of the fact that the calculation of the optimal matching using the cost Jb is computationally more intensive than using the cost Ja , owing to the larger number of decision variables and the associated constraints; see (23)-(27). As a result, in practice one can con?dently bypass the optimization of Jb and deal only with the optimization of Ja when computing the optimal satellite pairings in a P2P scenario. We will make use of this observation in all our subsequent computations from now on.

10

Plot of J2 vs J 1

7000

α = 0.99 ? 1.00

6000

5000

4000

α = 0.51 ? 0.98 These values of α give the same result as obtained by maximizing cost function J a

J2

3000

2000

α = 0.20 ? 0.50

1000

α = 0.09 ? 0.19

0 1000 2000 3000

α = 0.08

4000 5000

α = 0.00 ? 0.07

6000 7000 8000

0

J1

Figure 2: Variation of J2 with respect to J1 (constellation C4 ).

PURE P2P REFUELING STRATEGIES

It is well known22, 23 that coasting can signi?cantly reduce the fuel expenditure during a rendezvous. Therefore, during each transfer, initial or ?nal coasting intervals play an important role in the overall optimal rendezvous cost. Figure 4 shows a typical variation of the rendezvous cost between two satellites (in terms of non-dimensionalized ?V ) with respect to the transfer time. In this ?gure the initial separation angle between the satellites is 60 deg and both satellites are in the same circular orbit. The dotted line shows the cost if coasting is not allowed, while the solid line shows the cost when initial coasting is allowed. In the latter case, the active satellite stays for some time in its original orbit and the actual transfer occurs over a smaller time period. Therefore, by allowing a coasting period during an orbital transfer we can reduce the overall cost. The idea of allowing coasting intervals is utilized in this section to propose a strategy for reducing the overall P2P rendezvous cost. As it is evident from Figure 4 the optimal cost when coasting is included is a non-increasing function of time. That is, the inequality ?V (tf 1 ) ≤ ?V (tf 2 ), for tf 1 ≥ tf 2 (29)

holds for any two transfer times tf 1 and tf 2 . Note that this monotonicity of ?V versus the transfer time does not hold if there are no coasting intervals. In our previous investigation of P2P refueling strategies20 it was assumed that given the total amount of time to complete each fuel transaction, the time was equally divided between the forward and return orbital transfers for each fuel transaction. Here we relax this restriction. In particular, we show that by allowing unequal transfer times between the forward and return journeys for each fuel transaction, one can reduce the transfer cost. To see why this is true, let us consider a single refueling maneuver between two satellites si and sj , and let si ∈ A be the active satellite, and sj ∈ P be the passive satellite. Note 11

Plot of J 2 vs J 1

8000

α = 1.00

7000

6000

5000

α = 0.93 ? 0.99 α = 0.80 ? 0.92

4000

α = 0.73 ? 0.79 These values of α give the same result as obtained by the optimizing cost function Ja α = 0.69 ? 0.72 α = 0.25 ? 0.68 α = 0.00 ? 0.05

J2

3000

2000

1000

α = 0.18 ? 0.24 α = 0.06 ? 0.17

0 1000 2000 3000 4000 5000 6000 7000

0

J1

Figure 3: Variation of J2 with respect to J1 (constellation C8 ).

that either of the two satellites can be the seller or the buyer during the fuel transaction. The amount of fuel spent by si to rendezvous with sj is given by24 pf i = (msi + fi? )(1 ? e??Vij /c0i ), (30)

where msi is the mass of the permanent structure of satellite si , fi? is the initial fuel of satellite ? si , fj is the initial fuel of satellite sj , and ?Vij is the velocity increase required to transfer from the orbit of satellite si to the orbit of satellite sj . The parameter c0i is de?ned by c0i = g0 Ispi , where g0 is the acceleration due to gravity at the Earth’s surface, and Ispi is the speci?c thrust of satellite si . The amount of fuel consumed by satellite si to return back to its original position after a fuel exchange has taken place? is given by

? pri = (2msi + fi? + fj ? pf i )

(1 ? e??Vji /c0i ) , (1 + e??Vji /c0i )

(31)

where ?Vji is the optimum rendezvous cost for the return journey. Note that, in general ?Vji = ?Vij . Using the previous equations the total fuel used by satellite si during the two transfers is given by (32) pij = pf i + pri . Now let us denote by tij the total time allowed to complete both legs of the fuel transaction between satellites si and sj . Moreover, let tf denote the time for the forward journey and tr ij ij denote the time for the return journey, so that tij = tf + tr . ij ij

?

(33)

It is assumed that during the exchange of fuel the seller satellite gives enough fuel to the buyer satellite so that both have the same amount of fuel at the end of the fuel transaction.19

12

Plot of ? V vs tf for r1=r2=1 θ0 = 60 deg

14 without coasting with coasting 12

10

8

?V

6 4 2 0 0

1

2

3

4

5

6

Time of flight t

f

Figure 4: Variation of rendezvous cost with transfer time. Transfer from and to a circular orbit with an initial separation angle of 60 deg.

In case of an equal partition of the total time between the forward and return transfers, we have tf = tr = tij /2. In the sequel we use the superscript I, to denote quantities associated ij ij with such an equal time partition transfer. For simplicity, we assume a coasting period for the forward leg, and we will use the superscript II to denote the quantities associated with a transfer with unequal time partition of tij such that the forward and return legs are completed within the time intervals tf = tij /2 ? tij and tr = tij /2 + tij , where tij denotes the optimal ij ij coasting time for the forward leg. Similarly, we will use the superscript III to denote the quantities associated with a transfer with unequal time partition of tij such that the forward and return legs are completed within the time intervals tf = tij /2 + tij and tr = tij /2 ? tij , ij ij where tij denotes the optimal coasting time for the return leg. Let us concentrate on the case where coasting is part of the forward leg. Note that since coasting periods do not have any e?ect on the cost, one obtains,

I II ?Vij = ?Vij ,

which implies, according to (30) that

pI i = pIIi . f f

(34)

For the return ?ight, and since tij /2 + tij ≥ tij /2 we have, via (29), that

I II ?Vji ≥ ?Vji ,

which implies that e??Vji /c0i ≤ e??Vji /c0i . Using this inequality, it follows that 1?e??Vji /c0i ≥ II I II 1 ? e??Vji /c0i , and also 1 + e??Vji /c0i ≤ 1 + e??Vji /c0i . These two inequalities together yield 1 ? e??Vji /c0i 1 + e??Vji /c0i

I I

I

II

I

≥ 13

1 ? e??Vji /c0i 1 + e??Vji /c0i

II

II

(35)

which, via (31), yields

pI ≥ pII . ri ri pI ≥ pII . ij ij

(36) (37)

From equation (34) and inequality (36), the identity (32) yields

A similar analysis holds when a coasting period of length t is part of the return leg, in which case one can show that pI ≥ pIII . (38) ij ij We have therefore shown the following proposition. Proposition 1. For each fuel transaction between two satellites in the same circular orbit, and a given total time for the transaction to take place, an equal time allocation between the forward and return legs of the two associated rendezvous transfers is suboptimal. We will next utilize this idea to devise a coast time allocation (CTA) algorithm for reducing the fuel coast during each fuel transaction.

Coast Time Allocation Algorithm

The main idea behind the formulation of a fuel-reducing strategy is to allow for unequal time distribution between the forward and the return legs for each fuel transaction. To this end, we consider the following three cases: ? Case-I: tf = tr = tij /2 ij ij ? Case-II: tf = tij /2 ? tij and tr = tij /2 + tij ij ij ? Case-III: tf = tij /2 + tij and tr = tij /2 ? tij ij ij Assume a fuel transaction between satellites si ∈ A and sj ∈ P and let pjI , pjII and pjIII i i i denote the fuel spent for satellite si to rendezvous with sj and return back to its original position, for each of the previous three cases, respectively. The optimal time sharing is the one that satis?es pj? = min{pjI , pjII , pjIII }. (39) i i i i The corresponding time allocation is then given by ? ?(tij /2, tij /2), if pj? = pjI , ? i i (tf , tr ) = (tij /2 ? tij , tij /2 + tij ), if pj? = pjII , ij ij i i ? ? (tij /2 + tij , tij /2 ? tij ), if pj? = pjIII . i i We can similarly compute the cost of a single fuel transaction for the case si ∈ P and sj ∈ A. Finally, the optimum fuel consumption between any two satellites si , sj ∈ G is given by ? ?pj? , if si can be active, but sj cannot be active, ? i ? ? ?pi? , if sj can be active, but si cannot be active, j p? = ij ?min{pj? , pi? }, if either si or sj can be active, ? j i ? ? ?∞, if neither si nor sj can be active. 14

We demonstrate these results next via a numerical example. Let us consider a single fuel transaction between two identical satellites in the same circular orbit. We assume that the mass of permanent structure for satellites is ms = 60 units, and the characteristic constant of the rocket engine is c0 = 2943 units. The initial fuel of the active satellite is 100 units and of the passive satellite is 10 units. The allowed time to conduct the fuel transaction is chosen to be 12 units. Figure 5 shows a comparison between the three cases as a function of the separation angle between the two satellites. For all separation angles, an equal time allocation for the forward and return legs of a fuel transaction (Case I) always results in more or equal fuel expenditure than an unequal time allocation (Cases II or III).

Figure 5: E?ect of CTA algorithm to a single P2P maneuver. The e?ect of the CTA algorithm when refueling a constellation using a pure P2P strategy is evaluated by the introduction of the following ?gure of merit G= (

i,j ∈M

pij ?

? i,j ∈M pij ) ? i,j ∈M pij

× 100 %,

(40)

where M is the matching edge set for the optimal time allocation, and M is the matching edge set for the refueling strategy under evaluation. We call G the net percentage gain of the refueling. Several circular constellations with a varied number satellites of physical characteristics have been studied, and the ?nal fuel distribution and rendezvous costs associated with both pure P2P and mixed refueling strategies have been computed. The CTA algorithm has been applied to a complete P2P refueling scenario for the constellations given in Table 1. In this table, the initial fuel for each satellite is shown, along with the total allowed time T for the forward and return trips. Note that for all numerical results below one unit of time corresponds to one period of the circular orbit of the constellation. 15

The corresponding gains are shown in Figure 6. The results in Figure 6 indicate considerable

Figure 6: E?ect of CTA algorithm to an entire constellation; see also Table 1.

amount of fuel savings if the CTA algorithm is adopted. Note that in most cases, the application of the CTA algorithm has no e?ect on the satellite pairings. However, for constellation C6 , it was also found that the entire set of optimal pairings of satellites change when the algorithm is applied. This shows that the CTA algorithm can altogether a?ect the scheduling of the refueling process in order to reduce the cost.

MIXED REFUELING STRATEGIES

So far we have discussed pure P2P refueling strategies for the purpose of equalizing fuel among all satellites in the constellation. Although signi?cant unequal fuel distribution between identical satellites in the same orbit are rather unlikely (except in case of failures), and hence pure P2P strategies seem to be exceptional, nonetheless they arise naturally as a second stage of mixed refueling strategies. This has been demonstrated in Refs. 18, 19, where it was shown that a mixed strategy will typically outperform a single-spacecraft refueling strategy, as the number of satellites in the constellation increases. Let us consider a constellation in a circular orbit with an even number of satellites si , i ∈ I = {1, 2, ..., 2n}. For the sake of simplicity, we may assume that all satellites are initially depleted of fuel, that is, si ∈ Cd for all i ∈ I. Given a maximum refueling period, say T , we wish to refuel all of the satellites from a service vehicle s0 , such that after time T they all end up with approximately the same amount of fuel. In the process, we also want to minimize the total fuel expenditure during the ensuing orbital maneuvers. Equivalently, we want to 16

Table 1: Sample Constellations. Label C1 C2 C3 C4 C5 C6 C7 C8 Description 14 satellites, same structure and speci?c thrust fi (0? ): 38.8, 36, 35.2, 32.8, 29.6, 27.6, 26.8, 17.6, 14, 8, 6.8, 6.4, 5.6, 0.4 T = 12 6 satellites di?erent structure and speci?c thrust fi (0? ): 5, 45, 86, 31, 12, 90 T = 16 18 satellites, same structure and speci?c thrust fi (0? ): 62, 50, 40, 98, 70, 25, 88, 20, 72, 30, 82, 54, 42, 66, 35, 10, 90, 45. T =8 8 satellites, same structure and speci?c thrust fi (0? ): 85, 30, 95, 20, 65, 40, 75, 10 T = 12 20 satellites, same structure and speci?c thrust fi (0? ): 65, 70, 72, 65, 92, 44, 32, 16, 15, 28, 56, 88, 90, 92, 86, 30, 25, 36, 52, 60. T = 10 7 satellites, di?erent structure and speci?c thrust fi (0? ): 25, 40, 70, 82, 12, 95, 42 T =8 9 satellites, di?erent structure and speci?c thrust fi (0? ): 85, 30, 50, 95, 20, 65, 40, 75, 10 T = 12 10 satellites, di?erent structure and speci?c thrust fi (0? ): 25, 40, 50, 70, 82, 45, 12, 95, 30, 42 T =8

maximize the total amount of fuel that can be delivered to the constellation. We have two alternatives for solving this problem. The ?rst alternative is for s0 to refuel (perhaps sequentially15 ) all other satellites in the constellation. This scenario is shown in Figure 7. The second alternative is a mixed refueling strategy, consisting of two stages. During the ?rst stage, the service vehicle s0 delivers fuel to half the satellites in the constellation. During the second stage, these satellites share their fuel with the remaining satellites in P2P fashion. This alternative refueling scenario is shown in Figure 8. Let I1 denote the index set of the satellites refueled during the ?rst stage by the service vehicle s0 in a mixed strategy, and let I2 = I\I1 denote the remaining satellites which are to be refueled during the second stage. Without loss of generality we may assume that I1 = {1, 2, . . . , n} and I2 = {n + 1, n + 2, . . . , 2n}. Let also T (1) denote the time allotted for the ?rst stage and T (2) = T ? T (1) the time allotted for the second (P2P) stage in a mixed strategy. During T (1) the service vehicle s0 delivers fuel sequentially to the n satellites si (i ∈ I1 ) in (1) an optimal fashion. The optimal time distribution for these transfers, denoted by ti,i+1 (i =

17

1, . . . , n ? 1) then satis?es T

(1) (1) n?1

=

i=1

ti,i+1 ,

(1)

(41)

where the optimal values ti,i+1 are calculated by solving a binary integer programming problem.20 Note that the CTA algorithm can be implemented during this second stage to reduce the cost of the P2P maneuvers as was elaborated in the previous section. In Ref. 19 we showed that a mixed strategy will, in general, outperform a single-spacecraft strategy, especially as the number of satellites in the constellation increases. In Ref. 19 we assumed only synchronous implementation for the P2P second stage, that is, all P2P maneuvers during the second stage of the mixed refueling scenario, occur simultaneously and they all take time T (2) to be completed. However, we can further improve on the fuel savings incurred during the second stage by allowing asynchronous P2P maneuvers, as described next.

?V10,11 ?V11,12 S0 ?V12,0 S12 S0 ?V01 S1 S6 ?V12 S2 S3 ?V23 ?V34 ?V45 S5 S4 ?V56 ?V67 S7 ?V9,10 ?V89

S11 S10

S9 S8

?V78

Figure 7: Single-spacecraft refueling scenario.

Asynchronous P2P Refueling

In a synchronous P2P scenario all the satellite rendezvous take place simultaneously. In a mixed refueling strategy, this implies that all fuel de?cient satellites (at the end of the ?rst stage) are refueled within the time T (2) . Note, however that the time T (2) is binding only for satellite sn (the last satellite to be visited by s0 during the ?rst stage of a mixed strategy). All n?1 (1) other satellites si (i = 1, . . . , n ? 1) have available T (2) + k=i tk,k+1 time units to perform their fuel transactions. Thus, the time available for si to complete the P2P maneuver with its matching satellite sj is given by tij =

(2)

T (2) + T (2) ,

n?1 (1) k=i tk,k+1 ,

if i ∈ I1 \{n}, if i = n.

(42)

18

S11 S10 S12 S0 ?V01 S1

S9 S8

S7 S6

?V12

S2 S3 ?V23 ?V34 ?V45 S5 S4 S0 ?V56

Figure 8: Mixed refueling scenario.

We refer to this strategy as asynchronous P2P refueling, since not all satellite pairs complete (2) their corresponding fuel transactions within the same time period. Since tij ≥ T (2) for all satellite pairs, and referring again to Eq. (29), it is clear that each rendezvous between two satellites will require less fuel than a synchronous implementation. Consequently, the overall fuel consumption for the whole constellation will also be reduced by using an asynchronous P2P implementation. This is demonstrated next via numerical examples.

NUMERICAL EXAMPLES

We next apply the CTA algorithm along with an asynchronous (mixed) P2P refueling strategy to sample constellations. With the help of numerical examples we show how these improvements for a mixed refueling strategy make the latter a competitive alternative to a refueling strategy using a single service vehicle or to mixed synchronous P2P strategies. To this end, we assume a circular orbit constellation with an even number of satellites. The service spacecraft, denoted by s0 , starts with an initial amount of fuel f0 (0? ) = 500 units. We assume that s0 is initially at a higher circular orbit than the constellation orbit. It is required to return to the same orbit after completing the refueling process with f0 (T + ) = 10 units of fuel, where T = 20 is the maximum allowed time for completing the whole refueling process. Hence, the total amount of fuel to be delivered to the satellites in the constellation including the fuel to be used during the corresponding orbital transfers is 490 units. The mass of the permanent structure for each satellite is msi = 60 units and the characteristic constant of the engine is c0i = 2943 units for all satellites. In the ?rst example, we consider a constellation with six satellites evenly distributed in the circular orbit. The service vehicle s0 visits all these six satellites and distributes the fuel equally among all satellites in the constellation. There are ?ve rendezvous segments, and the maximum time of transfer allowed for each rendezvous segment is 6 time units. The

19

optimal time distribution for each of these ?ve rendezvous segments, and the corresponding fuel expenditure are given in Table 2. Table 2: Optimal Fuel Consumption With A Single Service Vehicle. Six Satellite Constellation. Segment i = 1, j = 2 i = 2, j = 3 i = 3, j = 4 i = 4, j = 5 i = 5, j = 6 tij 4.1607 4.1607 4.1607 4.1607 3.3570 ?Vij 0.1676 0.1676 0.1676 0.1676 0.2204 Fuel Expense 30.6311 24.8345 19.4244 14.3751 12.5754

At the end of this process, each of the six satellites ends up with an equal amount of fuel fi+ = 56.31 (i = 1, 2, ..., 6). The total amount of fuel used during all these transfers is thus 490 ? 6 × 56.31 = 152.14 units. Note that these values do not include the fuel consumption for the initial (?V = 44.2619) and ?nal (?V = 6.0094) transfers of s0 to and from the constellation orbit, which are constant and thus not part of the optimization process. Table 3: Optimal Fuel Consumption During the First Stage of a Mixed Refueling Strategy. Six Satellite Constellation. Segment i = 1, j = 2 i = 2, j = 3 tij 4.8279 3.8421

(1)

?Vij 0.1444 0.1826

Fuel Expense 22.0481 16.3783

Table 4: Optimal Fuel Consumption During the Second Stage of a Mixed Refueling Strategy. Six Satellite Constellation. Pairs (s1 , s6 ) (s2 , s4 ) (s3 , s6 ) T 20.00 15.17 11.33 T (1) /T (2) 10.17/9.83 7.85/7.32 6.00/5.33 Fuel Expense 9.0299 23.4042 31.3522

The optimum solution for a mixed refueling strategy yields that the ?rst step, during which s0 delivers fuel to satellites s1 , s2 and s3 requires two rendezvous segments with total time T (1) = 8.67 time units. The optimum time distribution and the corresponding fuel consumption for this step are given in Table 3. The three satellites refueled by s0 have 133.76 units of fuel each before performing the P2P maneuvers with the remaining satellites s4 , s5 and s6 . The available time and the corresponding fuel expenditures for the P2P maneuvers are given in Table 4. The ?nal fuel content of each satellite at the end of the refueling process are f1 (T + ) = f6 (T + ) = 62.37, f2 (T + ) = f4 (T + ) = 55.18 and f3 (T + ) = f5 (T + ) = 51.21. The average amount of fuel in the constellation then is equal to 56.25 units. The total amount of fuel burnt is 490 ? 6 × 56.25 = 152.50 units, which is 0.24% more than the amount of fuel burnt if the satellites are refueled by a single spacecraft. A single-spacecraft refueling strategy is marginally better than a mixed refueling strategy in this case. 20

For the second example we consider a constellation with twelve satellites evenly distributed in a circular orbit. The total time allowed for refueling is again T = 20 time units. There are eleven rendezvous segments with a single-spacecraft refueling strategy. The optimal time distribution for each of the ?ve rendezvous segments and the corresponding fuel consumption are given in Table 5. At the end of this process, each of the six satellites end up with an equal amount of fuel fi+ = 17.31. The total amount of fuel used during all the transfers is thus 490 ? 12 × 17.31 = 282.28 units. For the mixed strategy, there are ?ve rendezvous segments during the ?rst stage, which are all completed within T (1) = 9.59 units. The optimal time distribution for each of the ?ve rendezvous segments and the corresponding fuel consumption are given in Table 6. The six satellites refueled by s0 have fuel 55.53 units each before performing the P2P refueling part. The time available for the P2P maneuvers and the corresponding fuel consumption are given in Table 7. The ?nal fuel content of the satellites are f1 (T + ) = f10 (T + ) = 23.50, f2 (T + ) = f11 (T + ) = 23.04, f3 (T + ) = f12 (T + ) = 22.45, f4 (T + ) = f7 (T + ) = 21.74, f5 (T + ) = f8 (T + ) = 20.71, f6 (T + ) = f9 (T + ) = 19.35. The average amount of fuel in the constellation is 21.80 units. The total amount of fuel burnt using the mixed refueling strategy is 490 ? 12 × 21.80 = 228.4 units, which is about 19% less than the amount of fuel burnt if the satellites are refueled by a single spacecraft. Clearly, the mixed scenario outperforms the single service vehicle option in this case. Table 5: Optimal Fuel Consumption for Refueling with a Single Service Vehicle. Twelve Satellite Constellation. Segment i = 1, j = 2 i = 2, j = 3 i = 3, j = 4 i = 4, j = 5 i = 5, j = 6 i = 6, j = 7 i = 7, j = 8 i = 8, j = 9 i = 9, j = 10 i = 10, j = 11 i = 11, j = 12 tij 1.9084 1.9084 1.9084 1.9084 1.9084 1.9084 1.9084 1.9084 1.9084 1.9084 0.9163 ?Vij 0.1821 0.1821 0.1821 0.1821 0.1821 0.1821 0.1821 0.1821 0.1821 0.1821 0.3805 Fuel Expense 35.9746 32.1287 28.5604 25.2497 22.1779 19.3278 16.6834 14.2299 11.9535 9.8414 15.8334

Table 6: Optimal Fuel Consumption for First Step of Mixed Refueling Strategy. Twelve Satellite Constellation. Segment i = 1, j = 2 i = 2, j = 3 i = 3, j = 4 i = 4, j = 5 i = 5, j = 6 tij 1.9174 1.9174 1.9174 1.9174 1.9174

(1)

?Vij 0.1822 0.1822 0.1822 0.1822 0.1822

Fuel Expense 33.2517 26.8369 20.8556 15.3643 10.2419

21

Table 7: Optimal Fuel Consumption for Second Step of Mixed Refueling Strategy. Twelve Satellite Constellation. Pairs (s1 , s10 ) (s2 , s11 ) (s3 , s12 ) (s4 , s7 ) (s5 , s8 ) (s6 , s9 ) T 20.00 18.08 16.17 14.25 12.33 10.41 T (1) /T (2) 10.25/9.75 9.33/8.75 8.43/7.74 8.00/6.25 5.75/6.58 4.74/5.67 Fuel Expense 8.5335 9.4564 10.6270 12.0585 14.1137 16.8236

CONCLUSIONS

In this paper, we have studied peer-to-peer (P2P) satellite refueling scenarios in circular orbit constellations. P2P refueling strategies have been proposed recently as a viable, competitive alternative to single-satellite refueling. Although pure P2P strategies are rather unlikely for constellations with similar satellites, P2P refueling arises naturally as a second stage in mixed refueling strategies. For such mixed strategies we show via numerical examples that an unequal time distribution of the forward and return trips for each satellite pair, along with an asynchronous implementation of the P2P rendezvous sequence, result in more e?cient refueling than previous synchronous P2P/mixed or single-spacecraft refueling implementations.

ACKNOWLEDGEMENT

This work has been supported by AFOSR award no. FA9550-04-1-0135.

REFERENCES

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