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Inter-plane satellite matching in dense LEO constellations

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Inter-plane satellite matching in dense LEO constellations

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Dense constellations of Low Earth Orbit (LEO) small satellites are envisioned to extend the coverage of IoT applications via 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 that arises when trying to use the inter-plane ISL for unicast transmissions, with the aim of minimizing the total cost. The problem with any number of orbital planes and up to two transceivers is addressed, and we provide a near-optimal solution that is shown to perform very close to the optimal one. We also propose a Markovian algorithm to maintain the on-going connections as long as possible. This algorithm greatly reduces the switching and the computational complexity up to 176x with respect to optimal solutions 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.
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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
rErE
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) = i1
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(ab) 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|RBlog2(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 Ysides 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 MZ+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 iVaand jVb(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 iVaand jVb. 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 AEbe 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, jarg min
i,j
W
4: xij= 1 (associate satellite ito satellite j)
5: Delete rows and columns with indices iand 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 (s1) are the matching indicators at s1,
1: xij (s) = 0 for all i, j
2: for xij(s1) = 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, jarg min
i,j
W
11: xij(s)=1(associate satellite ito satellite j)
12: Delete rows and columns with indices iand 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)
s1= 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 Yside 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, jarg min
i,j
W
3: Find dfor jwith respect to iand vice versa
4: if xd
ij== 0 && xd
ji== 0 then
5: xd
ij= 1 and xd
ji= 1
6: wij =
7: if PkxY+
ik+xY
ik== 2 then
8: Delete the row and column with index i
9: end if
10: if PkxY+
jk+xY
jk== 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(m1)/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.
107106105104103102
0
0.2
0.4
0.6
0.8
1
Execution time [s]
CDF
Independent experiments Hungarian algorithm
Markovian
(a)
107106105104103102
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
107s. 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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