Low Earth Orbit (LEO) dense constellations of small satellites have become an attractive solution for Internet of Things (IoT) applications in 5G . The constellation is composed of hundreds of spacecrafts 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  . 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 stations, gateways or end-devices through the Ground-to-Satellite Link (GSL), which is used for Telemetry and Telecommand (TMTC) data and device data. LEO satellites move at speeds km/h relative to the ground terminals. Therefore, the GSL 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 latency. Examples of current applications exploiting the ISL are TMTC, which sends telecommands to multiple LEO satellites, or retrieval of surveillance data from the constellation. In addition, the use of the ISL and the constellation as a relay network is an attractive solution to 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 .
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  provide a thorough compilation of the latest research efforts in the area of inter-satellite communications, organized in physical, data and network layer.  describes the main use cases and elements of a LEO constellation for IoT, including the use of the ISL. In , a power budget analysis for CubeSats that includes the ISL is conducted. 
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 . For unmanned aerial vehicles (UAVs),  investigates the assignment problem in a Flying Ad-Hoc Network composed of drones, formulating a dynamic matching game that uses the current trajectory of the drones. The matching problem addressed in this paper is to find the inter-plane connections that minimize the total cost of a LEO constellation of Cubesats at each time instant. The model includes power adaptation and uses the power consumption as exemplary cost, but any other QoS-oriented KPI can be optimized instead. Differently than , we address a planned network and solve the combinatorial problem by considering the predictability of the spacecrafts positions.
Specifically, we take a network-wise approach and propose two novel algorithms to solve the inter-plane matching problem with orbital planes and for up to two simultaneous ISL per satellite. The Hungarian algorithm  is known to find the optimal pairing between nodes in a graph, which corresponds to the case with only one ISL, but its computational cost is high. Conversely, our algorithms provide a near-optimal, and oftentimes the optimal, solution with a sharp reduction in the computational complexity.
Ii System model
The constellation is composed of satellites distributed in circular orbital planes. Planes are composed of evenly distributed satellites, and each plane is defined by the altitude , the inclination and the orbital period . Each of the satellites in the constellation is assigned an index that serves as a unique identifier. is the set of satellites in the same orbital plane as . The function gives the plane of a satellite. If the number of satellites per plane is the same for all planes (), then
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., close to ). There are two classical topologies: the Walker star or polar , and the Walker or Rosette  . Without loss of generality, the results of this study are obtained for a Walker constellation like the one shown in Fig. 1; the specific parameters for the geometry are given in Section V.
Ii-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.
Ii-C Link budget and power adaptation
For the sake of notation simplicity, we skip the time dependence in the following. At any given time, the received SNR at satellite from satellite is written as
where is the transmission power; and are the transmit and receive antenna gains, respectively; is Boltzmann’s constant; is the system noise temperature; is the data rate in the radio link; and is the free-space propagation path loss between satellites and . The latter is given as
where is the line-of-sight distance (or slant range) between satellites and , is the transmission frequency, and is the light speed.
The slant range between neighboring satellites and in orbital plane is given by
where is the radius of the Earth.
The slant range between satellites and in orbital planes , respectively, is given by
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 (1), let denote the orbital period of plane and denote the orbital angle of satellite in plane at time . 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, , with the x-axis aligned toward the equinox. After some calculations the equatorial coordinates are written
Then, the euclidean distance is calculated to obtain (1). ∎
The achievable ISL data rate is constrained by the usual link budget parameters that include, among others: modulation and coding schemes, link distance, operating 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 in the ISL. For this, we assume that the link budget parameters listed above remain constant, except for , for which two different power levels are defined: low power and high power . It is straightforward to extend our study to the case with any number of power levels. The high power level is set to be the minimum power to achieve a theoretical channel capacity with a bandwidth and within a distance , where is a design parameter. Mathematically,
The latter, in combination with the constellation geometry, determines the link opportunities, defined as the time a pair of satellites are within the communication range.
The maximum distance at which two satellites and can communicate at the low power level at a minimum rate is a design parameter, given by the hardware and energy constraints of the spatial mission. Throughout this study, the transmission power used for an inter-plane ISL is selected as
The impact on performance of different values for is explored in Section V.
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 for the ISL link ) 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 can be selected. Then, the matching problem can be solved once every by taking samples (snapshots) of the constellation. Hence, is denoted as the sampling period.
Finding an optimal value for is out of the scope of this paper. Instead, we focus on the solution of the matching problem (i.e., on minimizing the cost of the inter-plane ISL) for a constellation at each time instant , for a sufficiently short to consider a static geometry of the constellation during this period.
Inter-plane antennas are placed in the and 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 orbital planes, this is the matching problem in a -partite graph described in the following.
Let be a graph with set of vertices and set of edges . Graph is -partite if can be divided into disjoint subsets ; hence, , . Each subset represents one orbital plane, so and each edge in has endpoints , given and . 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 is denoted as .
At any given , is a function of the the power level required for communication at the edge between satellites and (i.e., from planes and , respectively). We write
where if the transmission buffer is not empty, i.e., and otherwise.
Let be the symmetric matrix with all costs . Matrix is formed by block matrices , which contain the costs for all and . Naturally, if ; hence, we set in these cases, which gives
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 be an assignment on . The assignment that results in the minimum cost among all the possible assignments is optimal.
The matching problem is mathematically written
where indicates a match (i.e., ) and indicates no match between spacecrafts.
Iii-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 without taking into consideration past decisions. Therefore, this solution minimizes the immediate cost at each independently.
The Hungarian algorithm  gives the optimal solution in polynomial time (worst case is ) for the independent experiments matching problem in (12). Nevertheless, the asymptotic complexity of the Hungarian algorithm is restrictive in constellations where the number of satellites is large. In these cases, we propose the use of 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 , the weights are updated, and the strategy is to recursively add edges to the set of assignments by finding the edge with the smallest weight. Then, the rows and columns with the indices of the new pair are deleted from .
It has been verified by simulations with a variety of constellation geometries that this near-optimal algorithm provides a closely similar solution and, oftentimes, the exact same solution as the Hungarian algorithm. The computational complexity of this near-optimal algorithm is compared to that of the Hungarian algorithm in Section V in terms of execution time.
Iii-B Markovian matching: maximization of the contact time
The weights change with the movement of the constellation, which is predictable. Given a sufficiently short , the movement of the constellation from to is smooth and relatively slow. Therefore, it is likely that most of the pairs assigned at are still viable and near-optimal choices at . Consequently, the slow and predictable time evolution of the geometry between these time instants can be exploited to increase the contact time 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 be the stochastic process with time index and state space that defines the inter-plane matching between satellites and at time index . That is, and vice versa. We denote as the event of a match between and at time index ; hence, is analogous to for experiment .
Algorithm 1 is extended to maintain existing satellite pairs for as long as . For this, we define
hence, these pairs are eliminated from . Only then, new satellite pairs are created according to Algorithm 1. Therefore,
is 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 problem with two transceivers
In this section, we extend the algorithm and the formulation for the independent experiments matching introduced in Section 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 , its orbital plane defines two possibilities for the relative position with respect to satellite . That is, is either to the side or to the side of . Since the inter-plane antennas are placed in opposite sides of the satellite, at most one ISL can be maintained at each side and . Building on this, the matching problem becomes
Algorithm 3 solves the problem in equation (14), using the same principles introduced for the case with one transceiver. The extension to two transceivers is possible by allowing one matching at each and ; these are indicated by , where . Satellite is removed from only when a matching is made at each side.
This section presents the most relevant results derived from our analysis. A Walker constellation, such as the one illustrated 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 satellites are distributed in orbital planes deployed at heights km with for . The inclination of plane is . The intra-plane distance at the highest orbital plane is used for power adaptation. Hence, and . The sampling period is set to s, which is much shorter than the minimum orbital period s given for the lowest orbital plane at km; hence, it is safe to consider the constellation is static throughout a single sampling period. The buffer of the satellites is never empty; hence, is constant and all pairs in the coverage volume are considered for the matching.
Simulators for the studied algorithms have been developed in MATLAB 2018b. Simulations were run on a PC with MS Windows 10 ( bit), an Intel Core i5-6200U processor, GHz, and GB RAM. No other processes with a relevant CPU usage were run during the execution of our code. At each simulation, a whole constellation period is considered, which is the least common multiple of the orbital periods.
Given the uniformity of the constellation, the differences in terms of contact time between the near-optimal algorithm for independent experiments and the Markovian algorithm are insignificant, with a slight improvement by using the Markovian solution. The differences in total cost (i.e., network energy consumption) are also negligible. Where the algorithms differ is in the execution time per matching, as plotted in Fig. 2 for the case with one transceiver, with and ; the execution time for the optimal Hungarian algorithm is also included. As observed, the reduction in execution time is significant when using the Markovian: the Markovian solution is executed, on average, and faster than the independent experiments matching and the Hungarian algorithm, respectively. Other uniform geometries have been simulated with similar conclusions. Such a sharp reduction in the execution time is highly relevant for the network operation, since the constellation must:(1) solve the matching problem; (2) perform handover; and (3) transmit data within .
Fig. 3 shows the relative average power consumption when applying Algorithm 3 to problem (14) in a constellation of orbital planes with for all . The average power, averaged over time and over the number of pairs (established inter-plane ISLs), is presented as a function of for different threshold values of . The value of the latter depends on the distribution of the inter-plane distances for each geometry. For example, the blue line at the bottom of Fig. 3 corresponds to the case where is sufficient to establish of all possible inter-plane connections within . As expected, the average power increases if the threshold is lower, and decreases as the number of planes increases, due to the lower inter-plane distances. Moreover, the low power links are prioritized, making the average power to remain very close to the lower bound for most of the scenarios. The variations observed across the x-axis of Fig. 3 are mainly due to the change of parity of , which affects the number of potential ISLs.
Fig. 4 shows the average number of satellite pairs established with and with one and two transceivers. As expected, the number of pairs increases with: (1) the number of planes; (2) the number of satellites per plane; and (3) 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.
We have addressed the inter-plane ISL in a LEO constellation of satellites using unicast communication. The constellation 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 near-optimal 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 proposed algorithms sharply reduce the computational complexity of the matching when compared to the Hungarian algorithm.
This paper provides basic results for understanding the limits in the inter-plane ISL connectivity. When all satellites have non-empty queues, the network-wise solution provided by our algorithms can be autonomously computed in each satellite. For the general case of variable load, a practical implementation requires a distributed protocol that takes into account the stochastic behaviour of the satellites’ buffer. This is left for future work.
-  3GPP, “Study on Scenarios and Requirements for Next Generation Access Technologies,” 3rd Generation Partnership Project (3GPP), Technical Recommendation (TR) 38.913, June 2018, version 15.0.0.
-  J. G. Walker, “Satellite constellations,” Journal of the British Interplanetary Society, vol. 37, p. 559:571, Nov 1984.
-  ——, “Circular orbit patterns providing whole earth coverage,” Journal of the British Interplanetary Society, vol. 24, pp. 369–384, 07 1971.
-  R. Radhakrishnan et al., “Survey of inter-satellite communication for small satellite systems: Physical layer to network layer view,” IEEE Communications Surveys Tutorials, vol. 18, no. 4, pp. 2442–2473, Fourthquarter 2016.
-  Z. Qu, G. Zhang, H. Cao, and J. Xie, “LEO satellite constellation for Internet of Things,” IEEE Access, vol. 5, pp. 18 391–18 401, 2017.
-  O. Popescu, “Power budgets for cubesat radios to support ground communications and inter-satellite links,” IEEE Access, vol. 5, pp. 12 618–12 625, 2017.
-  W. G. Cowley and M. P. Lavenant, “Analysis of uncoordinated ISL networks,” in GLOBECOM 2009 - 2009 IEEE Global Telecommunications Conference, Nov 2009, pp. 1–6.
-  M. Conforti, G. Cornuejols, and G. Zambelli, Integer Programming. Springer Publishing Company, Incorporated, 2014.
-  D. Liu et al., “Opportunistic mobility utilization in flying ad-hoc networks: A dynamic matching approach,” IEEE Communications Letters, vol. 23, no. 4, pp. 728–731, apr 2019.
-  J. Munkres, “Algorithms for the assignment and transportation problems,” Journal of the Society of Industrial and Applied Mathematics, vol. 5, no. 1, pp. 32–38, March 1957.
-  A. H. Ballard, “Rosette constellations of earth satellites,” IEEE Transactions on Aerospace and Electronic Systems, vol. AES-16, no. 5, pp. 656–673, Sept 1980.