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Inter-plane satellite matching in dense LEO

constellations

Beatriz Soret, Israel Leyva-Mayorga, and Petar Popovski

Department of Electronic Systems, Aalborg University, 9220, Aalborg, Denmark

Email:{bsa, ilm, petarp}@es.aau.dk

Abstract—Dense constellations of Low Earth Orbit (LEO)

small satellites are envisioned to make extensive use of the inter-

satellite link (ISL). Within the same orbital plane, the inter-

satellite distances are preserved and the links are rather stable.

In contrast, the relative motion between planes makes the inter-

plane ISL challenging. In a dense set-up, each spacecraft has

several satellites in its coverage volume, but the time duration

of each of these links is small and the maximum number of

active connections is limited by the hardware. We analyze the

matching problem of connecting satellites using the inter-plane

ISL for unicast transmissions. We present and evaluate the

performance of two solutions to the matching problem with

any number of orbital planes and up to two transceivers: a

heuristic solution with the aim of minimizing the total cost; and a

Markovian solution to maintain the on-going connections as long

as possible. The Markovian algorithm reduces the time needed

to solve the matching up to 1000×and 10×with respect to

the optimal solution and to the heuristic solution, respectively,

without compromising the total cost. Our model includes power

adaptation and optimizes the network energy consumption as the

exemplary cost in the evaluations, but any other QoS-oriented

KPI can be used instead.

I. INTRODUCTION

Low Earth Orbit (LEO) dense constellations of small satel-

lites, based on the CubeSat architecture, have become an

attractive solution for Internet of Things (IoT) applications in

5G [1]. The constellation is composed of hundreds of space-

crafts plus several ground stations, working all together as a

relay communication network. The space segment is organized

in several orbital planes that can be deployed at different

inclinations and altitudes [2] [3]. The satellites are connected

to each other via the Inter-Satellite Links (ISL), a two-way

connection. The ISL can be intra-plane ISL, connecting with

the satellite in front and the satellite behind in the same plane;

and inter-plane ISL, connecting satellites from different orbital

planes. In addition, the satellites are connected to ground sta-

tions, gateways or end-devices through the Ground-to-Satellite

Link (GSL). LEO satellites move at speeds >25 000 km/h

relative to the ground terminals. Therefore, the GSL is only

available for a few minutes before handover to another satellite

occurs.

The use of the ISL unleashes the true potential of a LEO

constellation, ensuring continuous connectivity, and reducing

the number of required ground stations and the end-to-end la-

tency. One example of application is to use the constellation as

a relay network, which can dramatically increase the coverage

of machine-type communication (MTC) and IoT deployments

in rural or remote areas, where the cellular and other relaying

networks are out of range [1].

Inter-satellite distances are usually preserved within a plane.

However, inter-satellite distances between different planes are

time-variant: longest when satellites are over the Equator, and

shortest over the polar region boundaries. Moreover, the orbital

periods are different if the planes are deployed at different

altitudes, or if these contain a different number of satellites,

which results in aperiodic topologies. In a dense set-up, each

spacecraft has several inter-plane satellites in its coverage

volume, which leads to a matching problem of who should

communicate to whom.

Although less investigated than the GSL, several works

have addressed the communication challenges of the ISL. The

authors in [4] provide a thorough compilation of the latest

research efforts in the area of inter-satellite communications,

organized in physical, data and network layer. [5] describes the

main use cases and elements of a LEO constellation for IoT,

including the use of the ISL. In [6], a power budget analysis

for CubeSats that includes the ISL is conducted. [7] addresses

the communication among a group of independent satellites in

an unstructured constellation, treating the spacecraft positions

as random variables.

Matching problems are among the most important problems

in network optimization [8]. For unmanned aerial vehicles

(UAVs), [9] investigates the assignment problem in a Flying

Ad-Hoc Network composed of drones, formulating a dynamic

matching game that uses the trajectory of the drones. In this

paper, we address the matching problem of ﬁnding the inter-

plane ISL connections that minimize the total cost of the

constellation at each time instant. The model includes power

adaptation and the power consumption is the exemplary cost,

but any other QoS-oriented Key Performance Indicator (KPI)

can be optimized instead. Differently than [7], we address

a planned network and solve the combinatorial problem by

considering the predictability of the spacecrafts positions.

Speciﬁcally, we aim to solve the inter-plane matching prob-

lem with Morbital planes and for up to two simultaneous

ISLs per satellite. The Hungarian algorithm [10] is known to

ﬁnd the optimal pairing in bipartite graphs, which corresponds

to the case with only M= 2 and one ISL. Furthermore, its

computational cost is high. Conversely, we take a network-

wise approach and propose two novel algorithms that provide

a near-optimal solution to the matching problem with any

Mand up to two simultaneous ISLs per satellite without

arXiv:1905.08410v2 [cs.IT] 7 Aug 2019

−rE

0

rE−rE

0

rE

−rE

0

rE

Fig. 1: Walker δconstellation with N= 200 satellites in M=

5orbital planes with altitudes starting at 1000 km; the Earth

radius is rE= 6371 km.

compromising the total cost. In addition, the computational

complexity of our two algorithms is much lower than that of

the Hungarian algorithm.

The rest of the paper is organized as follows. In Section II,

we describe the system model. Sections III and IV address

the problem when the CubeSat is equipped with one and two

transceivers, respectively. Section V presents the performance

results and Section VI the conclusions.

II. SY ST EM M OD EL

A. Geometry

The constellation is composed of Nsatellites distributed

in Mcircular orbital planes. Planes m= 1,2, . . . , M are

composed of Nmevenly distributed satellites, and each orbital

plane is deﬁned by the altitude hm, the inclination mand the

orbital period Tm. Each of the N=PmNmsatellites in

the constellation is assigned an index i∈ {1,2, . . . , N}that

serves as a unique identiﬁer. P(i)is the set of satellites in the

same orbital plane as i. The function p(·)gives the plane of a

satellite. If the number of satellites per plane is the same for

all planes (N1=N2=...), then

p(i) = i−1

N1+ 1 (1)

Orbits with a low inclination are called equatorial or near

equatorial orbits, and polar orbits are those passing above or

nearly above both poles on each revolution (i.e., mclose to

π/2). There are two classical topologies: the Walker star or

polar [2], and the Walker δor Rosette [3] [11]. Without loss of

generality, the results of this study are obtained for a Walker

δconstellation like the one shown in Fig. 1, whose speciﬁc

parameters are given in Section V.

B. Antennae placement

The attitude determination and control subsystem of

CubeSats is often speciﬁed to be 3-axis, stabilized with the

yaw axis (x-axis) pointing towards the zenith, the z-axis (pitch)

aligned to the orbit angular momentum (i.e., perpendicular to

the orbit plane), and the y-axis (roll) aligned to the satellite

velocity vector.

Although a set of coordinated small satellites have similar

functionality as a big satellite, there are practical constraints

in the design of each CubeSat in terms of energy, weight and

processing. Some of these constraints are related to the cube

structure itself. For instance, the position of the antennas is

rarely free due to the satellite geometry and the placement of

other subsystems like thrusters, payload, and heat shielding.

Furthermore, even when the inter-plane ISL is implemented,

a practical mission will typically prioritize the stability of the

GSL and the intra-plane ISL. Under these premises, the GSL

antennas will be pointing towards the Earth’s center, in the yaw

axis, with a dedicated modem. The intra-plane ISL antennas

are deployed in both sides of the roll axis, and two intra-plane

transceivers are required to ensure two-way communication

within an orbital plane. The pitch axis is then left for the inter-

plane ISLs antennas and, depending on weight restrictions, one

or two transceivers can be placed for this connectivity type.

Both cases, one and two modems, are considered in this paper.

C. Link budget and power adaptation

For the sake of notation simplicity, we skip the time

dependence tin the following. At any given time, the received

SNR at satellite jfrom satellite i6=jis written as

SNR(i, j) = PtGtGr

kTsRLp(i, j )(2)

where Ptis the transmission power; Gtand Grare the transmit

and receive antenna gains, respectively; kis Boltzmann’s

constant; Tsis the system noise temperature; Ris the data

rate in the radio link; and Lp(i, j)is the free-space propagation

path loss between satellites iand j. The latter is given as

Lp(i, j) = 4πl(i, j)f

c2

(3)

where l(i, j)is the line-of-sight distance (or slant range)

between satellites iand j,fis the transmission frequency,

and cis the light speed.

Proposition 1. The slant range between neighboring satellites

iand jin orbital plane a=p(i) = p(j)is given by

lintra(a) = min {l(i, j)|a=p(i) = p(j)}

= 2(rE+ha) cos π

Na

tan π

Na

(4)

where rEis the radius of the Earth.

The slant range between satellites iand jin orbital planes

a=p(i)6=b=p(j), respectively, is given by

l(i, j) = (ha+rE)2+ (hb+rE)2

−2(ha+rE)(hb+rE) cos θa,i cos θb,j

−2(ha+rE)(hb+rE) cos(a−b) sin θa,i sin θb,j ]1/2.

(5)

Proof. Equation (4) is derived from a circular orbit with evenly

distributed satellites, by calculating the distance between two

points in a circle. To calculate the distance between spacecrafts

in different orbital planes, as in (5), let Tadenote the orbital

period of plane aand θa,i(t) = (2πt/Ta) + (2πi/Na)denote

the orbital angle of satellite iin plane aat time t. Notice

the notation emphasis here regarding the time dependence.

The equatorial coordinates of the satellite are ﬁrst expressed

in terms of the ecliptic coordinates in the ecliptic plane,

(xecl,yecl ,zecl), with the x-axis aligned toward the equinox.

After some calculations the equatorial coordinates are written

xa,i(t) = (ha+rE) cos θa,i (t)(6a)

ya,i(t) = (ha+rE) cos isin θa,i (t)(6b)

za,i(t) = (ha+rE) sin isin θa,i (t)(6c)

Then, the euclidean distance is calculated to obtain (5).

The achievable ISL data rate is constrained by the usual link

budget parameters including, among others: modulation and

coding schemes, link distance, frequency, RF power, antenna

gains, noise temperature, and equipment losses. We study

a scenario in which the spacecrafts aim to transmit at a

minimum data rate Rin the ISL. We assume that the link

budget parameters listed above remain constant, except for

Pt. Mathematically, the minimum Ptneeded to achieve Ris

min

Pt

{Pt|R≤Blog2(1 + SNR(i, j))}

=2R/B −1kTsR

GtGr4πl(i, j )f

c2

.(7)

The power adaptation policy together with the constella-

tion geometry determines the link opportunities, or contact

times, deﬁned as the time a pair of satellites are within the

communication range. Although not considered in this paper,

the contact time is also inﬂuenced by the directivity and the

radiation pattern of the antennae (i.e., if both transmitting and

receiving satellite are pointing to each other), but we assume

omnidirectional antennae throughout the paper and leave the

antenna characterization for future work.

As power adaptation policy, we deﬁne two levels: low power

P`and high power Ph. The maximum distances at which two

satellites iand jcan communicate at the low power and high

power level at a minimum rate Rare design parameters, given

by the hardware and energy constraints of the spatial mission.

Through this study, P`is set to be the minimum power to

achieve a theoretical channel capacity that is higher than R

with a bandwidth Band within a distance l`=η·lintra(m),

where ηis the design parameter. Phis deﬁned analogously

but within a distance lh= 2η·lintra(m) = 2l`. The inter-plane

ISL transmission power is then selected as

Pt=(P`,if l(i, j)≤l`

Ph,if l`< l(i, j)≤lh.(8)

III. MATCHING PROBLEM WITH ONE TRANSCEIVER

In this section we deﬁne the inter-plane ISL matching

problem with one transceiver and propose two approaches to

solve it. Then, the matching problem and our approaches are

extended to the case with two transceivers in Section IV.

The satellite network can be represented as a time-varying

graph in which the inter-plane ISL link opportunities are

short. The dynamics is such that satellites may perform early

handover (i.e., before reaching the maximum distance lh) to

increase their link budget and, hence, transmit at a higher

data rate and/or at a lower power when compared to the

previous inter-plane link. Therefore, the actual link duration

(or contact time) between two satellites may be shorter than the

link opportunity. However, handover has an inherent signaling,

delay, and processing overhead. Consequently, excessively

frequent handovers must be avoided, for which a minimum

period between handovers Tho can be selected. Then, the

matching problem can be solved once every Tho by taking

samples (snapshots) of the constellation. Hence, Tho is denoted

as the sampling period. Finding an optimal value for Tho is out

of the scope of this paper. Instead, we set a sufﬁciently short

Tho to consider a static geometry of the constellation during

this period.

Inter-plane antennas are placed in the Y+and Y−sides of

the spacecraft, so the coverage volume with one transceiver

is assumed to be the same as the implementation with two

transceivers, but only one simultaneous inter-plane connection

is possible. For M∈Z+orbital planes, this is the matching

problem in a M-partite graph described in the following.

Let G(V, E )be a graph with set of vertices Vand set

of edges E. Graph G(V, E )is M-partite if Vcan be di-

vided into Mdisjoint subsets Vm; hence, V=SM

m=1 Vm,

TM

m=1 Vm=∅. Each subset Vmrepresents one orbital plane,

so |Vm|=Nmand each edge in Ehas endpoints (i, j), given

{i, j} ∈ Vand j /∈ P(i). These endpoints are usually called

agents and tasks. Any agent can be assigned to perform any

task, incurring some cost that may vary depending on the

agent-task assignment. The cost of using edge (i, j)∈Eis

denoted as wij .

At any given s,wij is a function of the the power level

Pij required for communication at the edge (i, j)between

satellites i∈Vaand j∈Vb(i.e., from planes aand b,

respectively). We write

wij =Pij

(Q > 0) (9)

where (Q > 0) = 1 if the transmission buffer is not empty,

i.e., Q > 0and (Q > 0) = 0 otherwise.

Let Wbe the symmetric matrix with all costs wij . Matrix

Wis formed by block matrices Wa,b, which contain the costs

wij for all i∈Vaand j∈Vb. Naturally, (i, j)/∈Eif j∈ P(i);

hence, we set wij =∞in these cases, which gives

W=

∞W1,2· · · W1,M

W2,1∞ · · · W2,M

.

.

..

.

.....

.

.

WM,1WM,2· · · ∞

The goal is to assign exactly one agent to each task and

exactly one task to each agent in such a way that the total

cost of the assignment is minimized. Let A⊆Ebe an

Algorithm 1: Heuristic algorithm for independent experiments

matching with a single transceiver.

Input: Wis matrix of costs

1: xij = 0 for all i, j

2: while ∃min W < ∞do

3: i∗, j∗←− arg min

i,j

W

4: xi∗j∗= 1 (associate satellite i∗to satellite j∗)

5: Delete rows and columns with indices i∗and j∗

6: end while

assignment on G(V, E ). The assignment Athat results in the

minimum cost among all the possible assignments is optimal.

The matching problem is mathematically written

min

N

X

i=1

N

X

j=1

j /∈P (i)

wij xij

subject to

N

X

j=1

j /∈P (i)

xij = 1 ∀i

xij ∈ {0,1}

(10)

where xij = 1 indicates a match (i.e., (i, j )∈A) and xij = 0

indicates no match between spacecrafts.

A. Independent experiments matching

This is the classical static approach, in which the underlying

graph is assumed to be time-invariant. Hence, the matching

problem is solved at each time instant twithout taking into

consideration past decisions. Therefore, this solution mini-

mizes the immediate cost at each tindependently.

The basic Hungarian algorithm [10] gives the optimal solu-

tion in polynomial time (O(N3)) for the independent experi-

ments matching problem in (10) for M= 2. Nevertheless, the

asymptotic complexity of the Hungarian algorithm is restric-

tive in constellations where the number of satellites is large

and extensions are needed for the cases where M > 2. In these

cases, we propose the use of the heuristic Algorithm 1, which

greatly reduces the computational complexity with respect to

the Hungarian algorithm and provides a near-optimal solution

for the independent experiments matching. Its operation is

summarized as follows. At each time instant t, the weights

wij are updated, and the strategy is to recursively add edges

to the set of assignments Aby ﬁnding the edge (i, j)with the

smallest weight. Then, the rows and columns with the indices

of the new pair are deleted from W.

B. Markovian matching: maximization of the contact time

The weights wij change with the movement of the constel-

lation, which is predictable. Given a sufﬁciently short Tho, the

movement of the constellation from tto t+Tho is smooth

and relatively slow. Therefore, it is likely that most of the

pairs assigned at tare still near-optimal choices at t+Tho.

Consequently, the slow and predictable time evolution of the

Algorithm 2: Markovian algorithm for a single transceiver

Input: Wis the matrix of costs at time index s

Input: xij (s−1) are the matching indicators at s−1,

1: xij (s) = 0 for all i, j

2: for xij(s−1) = 1 do

3: if wij <∞then

4: {The pair of satellites is still reachable}

5: xij (s)=1(maintain association)

6: Delete rows and columns with indices iand j

7: end if

8: end for

9: while ∃min W < ∞do

10: Find i∗, j∗←− arg min

i,j

W

11: xi∗j∗(s)=1(associate satellite i∗to satellite j∗)

12: Delete rows and columns with indices i∗and j∗

13: end while

geometry between these time instants can be exploited to in-

crease the contact time (and reduce the handovers) and reduce

the computational complexity of the matching algorithm. For

this, we formulate a dynamic assignment problem in which

the previous state of the system is considered.

Let nX(i,j)

sosbe the stochastic process with time index

sand state space {0,1}that deﬁnes the inter-plane match-

ing between satellites iand jat time index s. That is,

Pr hX(i,j)

s= 1i= Pr [(i, j)∈A]and vice versa. We denote

xij (s)as the event of a match between iand jat time index

s; hence, xij (s)is analogous to xij for experiment s.

Algorithm 1 is extended to maintain existing satellite pairs

for as long as l(i, j)≤lh. For this, we deﬁne

Pr hX(i,j)

s= 1 |X(i,j)

s−1= 1, wij <∞i= 1; (11)

hence, these pairs are eliminated from W. Only then, new

satellite pairs are created according to Algorithm 1. Therefore,

nX(i,j)

sosis a discrete-time Markov chain. Algorithm 2

summarizes the Markovian approach, whose computational

complexity is expected to be considerably lower than that of

the independent experiments matching.

IV. MATCHING PROB LE M WI TH TWO TRANSCEIVERS

In this section, we extend the algorithm and the formulation

for the independent experiments matching introduced in Sec-

tion III-A to the case in which each satellite is equipped with

two inter-plane transceivers. The extension of the following

to the Markovian approach described in Section III-B is

straightforward.

For each satellite i, its orbital plane deﬁnes two possibilities

for the relative position with respect to satellite j /∈ P(i). That

is, jis either to the Y+side or to the Y−side of i. Since

the inter-plane antennas are placed in opposite sides of the

Algorithm 3: Algorithm for two-transceivers

Input: Wis the matrix of costs

1: while ∃min W < ∞do

2: Find i∗, j∗←− arg min

i,j

W

3: Find dfor jwith respect to iand vice versa

4: if xd

i∗j∗== 0 && xd

j∗i∗== 0 then

5: xd

i∗j∗= 1 and xd

j∗i∗= 1

6: wij =∞

7: if PkxY+

i∗k+xY−

i∗k== 2 then

8: Delete the row and column with index i∗

9: end if

10: if PkxY+

j∗k+xY−

j∗k== 2 then

11: Delete the row and column with index j∗

12: end if

13: end if

14: end while

satellite, at most one ISL can be maintained at each side Y+

and Y−. Building on this, the matching problem becomes

min

N

X

i=1

N

X

j=1

j /∈P (i)

wij xY+

ij +wij xY−

ij

subject to

N

X

j=1

j /∈P (i)

xY+

ij = 1 ∀i

N

X

j=1

j /∈P (i)

xY−

ij = 1 ∀i

xd

ij ∈ {0,1}, d ∈ {Y+, Y −}

(12)

Algorithm 3 solves the problem in equation (12), using the

same principles introduced for the case with one transceiver.

The extension to two transceivers is possible by allowing

one matching at each Y+and Y−(indicated by xd

ij , where

d∈ {Y+, Y −}). Satellite iis removed from Wonly when a

matching is made at each side.

V. RE SU LTS

This section presents the most relevant results derived from

our analysis. A Walker δconstellation, such as the one illus-

trated in Fig. 1, and the model from Section II are considered.

The default parameters involved in the geometry, link budget

and power adaptation are as follows. A total of Nsatellites

are distributed in Morbital planes deployed at heights hm=

900 + 100mkm with Nm=N/M for m= 1,2, . . . , M.

The inclination of plane mis m= 2π(m−1)/M. The

intra-plane distance at the highest orbital plane is used for

power adaptation. Unless otherwise speciﬁed, η= 1, such that

l`=lintra(m)and lh= 2lintra (m). The buffer of the satellites

is never empty; hence, (Q > 0) = 1 is constant and all pairs

in the coverage volume are considered for the matching.

10−710−610−510−410−310−2

0

0.2

0.4

0.6

0.8

1

Execution time [s]

CDF

Independent experiments Hungarian algorithm

Markovian

(a)

10−710−610−510−410−310−2

0

0.2

0.4

0.6

0.8

1

Execution time [s]

CDF

(b)

Fig. 2: Comparison of the execution time per matching with (a)

M= 2 and (b) M= 5 using the Hungarian algorithm (only

for M= 2), the independent experiments, and the Markovian

approaches for a single transceiver; Nm= 40 for all m.

Simulators for the studied algorithms have been developed

in Python 3. Simulations were run on a PC with Ubuntu

18.04.2 LTS (64 bit), an Intel Core i7-7820HQ CPU, 2.9GHz,

and 16 GB RAM. The clock precision of this platform is

10−7s. No other processes with a relevant CPU usage were

run during the execution of our code.

The constellation is simulated for at least ﬁve orbital periods

of the lowest orbital plane (h= 1000 km) and, at least, 10 000

matching problems are solved. Given the orbital period for the

lowest orbital plane at h1= 1000 km is T1= 6298 s, the

sampling period is Tho = 5T1/10 000 = 3.41 s; hence, it is

safe to consider the constellation is static throughout a single

sampling period.

Given the uniformity of the constellation, the differences

in terms of contact time between the heuristic algorithm for

the independent experiments and the Markovian algorithm

are insigniﬁcant, with a slight improvement by using the

Markovian one. The differences in total cost (i.e., network

energy consumption) are also minor. Where the algorithms

differ is in the execution time per matching, as plotted in

Fig. 2 for the case with one transceiver and Nm= 40. In

Fig. 2 (a), for M= 2, the execution time of the optimal

Hungarian algorithm is also included. For M= 5, plotted

in Fig. 2 (b), only the heuristic (independent experiments)

and the Markovian solution are compared. The Markovian

algorithm reduces the time needed to solve the matching up to

1000×and 10×with respect to the optimal solution, and to

the heuristic solution, respectively. Other uniform geometries

have been simulated with similar conclusions.

Fig. 3 shows the relative average power consumption

5 6 7 8

5

10

15

1

Number of orbital planes M

Relative average inter-plane ISL

transmission power Pt/Pintra

Independent experiments η= 1 Markovian η= 1

Independent experiments η= 2 Markovian η= 2

Fig. 3: Relative average inter-plane ISL transmission power

Pt/Pintra per satellite pair versus number of planes with the

independent experiments and Markovian approaches; single

transceiver.

¯

P=¯

Pt/P`when applying Algorithm 3 to problem (12) in a

constellation of M= 4,5,...,8orbital planes with Nm= 40

for all m. The power is averaged over time and over the

number of pairs (established inter-plane ISLs). The cases

with η= 1 and η= 2 are plotted for the independent

experiments and the Markovian algorithms. As expected, the

relative average power consumption is higher when η= 2, and

consequently more pairs are candidates to establish a com-

munication link. The Markovian solution results in a higher

relative power as compared to the independent experiments,

showing the price to pay in energy consumption for reducing

the matching complexity and increasing the link duration,

therefore reducing the handover signaling.

Fig. 4 shows the average number of satellite pairs estab-

lished with Nm={40,60}and with one and two transceivers.

As expected, the number of pairs increases with the M,Nm,

and the number of transceivers. Nevertheless, the number of

pairs with two transceivers is less than twice the number of

pairs with one transceiver; this is the upper bound introduced

by the geometry of the constellation. Therefore, the throughput

in the inter-plane ISL of a constellation is not doubled by

adding a second transceiver.

VI. CONCLUSIONS

We have addressed the inter-plane ISL in a LEO constella-

tion of satellites using unicast communication. The constella-

tion is modelled like a dynamic graph, in which vertices are

satellites and edges are the communication links. The case

in which the CubeSat is equipped with a single transceiver

for this connectivity type is ﬁrst studied, with a heuristic

algorithm and a Markovian solution, the latter for maximizing

the link duration. Then, the case with two transceivers is

analyzed, considering the relative position of the planes. The

cost of assigning a pair of spacecrafts is abstracted in our

model, although the examples illustrate the minimization of

the network energy consumption under a power adaptation

scheme. The simulation results show that the Markovian

5678

0

25

50

75

100

125

150

Number of orbital planes M

Mean number of pairs

1transceiver, Nm= 40 1 transceiver, Nm= 60

2transceivers, Nm= 40 2 transceivers, Nm= 60

Fig. 4: Mean number of satellite pairs versus number of planes

and satellites per plane with one and two transceivers.

solution sharply reduces the computational complexity of

the matching when compared to the baseline algorithm. The

algorithms are periodically executed with a sampling period

sufﬁciently small, so it is straightforward to address changes

in the topology due to malfunctioning spacecraft.

This paper provides benchmark results for understanding the

limits in the inter-plane ISL connectivity. The network-wise

solution provided by our algorithms can be the basis to deﬁne

practical implementations of distributed protocols that can be

autonomously computed in each satellite. Another extension

is the characterization of directional antennas and the pointing

in the inter-plane ISL.

ACKNOWLEDGMENT

This work has been in part supported by the European

Research Council (Horizon 2020 ERC Consolidator Grant Nr.

648382 WILLOW).

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