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Connectivity and path formation time are very important for the design and optimization in fractionated spacecraft network. Taking frequency division multiple access (FDMA) with subcarrier binary phase-shift keying (BPSK) modulation as an example, this paper focuses on the issues of constraint to orbital elements and path formation time for the noise-limited fractionated spacecraft network percolating. First, based on the proposed evolution of the dynamic topology graph in the fractionated spacecraft network, we prove the constraint condition of orbital elements for noise-limited fractionated spacecraft network percolating, and the definition of path formation time is provided and the mobility model is established. Next, we study the relationship between first docking time and spatial initial distribution and the relationship between first separating time and spatial initial distribution. These relationships provide an important basis for the orbit design in the fractionated spacecraft network. Finally, the numerical results show that the network topology for fractionated spacecraft is time-varying and dynamic. The path formation time and hop length scale linearly with path length within each orbital hyperperiod and change periodically. Besides, the time constant gradually tends to a stable value with path formation time increasing, that is, path length. These results powerfully support percolation theory further under the environment of the noise-limited fractionated spacecraft network.

Since fractionated spacecraft network (FSN) has the advantages of fast response, strong robustness, flexibility, low cost, and long lifetime, this innovative structure has been considered as the next generation of distributed space system [

Recently, percolation theory has been used to study connectivity and message dissemination delay in static multihop wireless networks [

In static wireless networks, in order to guarantee that any node in the network can receive the message from the source node, it is required that the network is fully connected. In mobile wireless networks, however, the situation is quite different from that for static networks. In a moving environment, connectivity should not be assessed at a fixed time instant. Rather, two nodes can exchange a message and thus share a link. Because different nodes can share a link at different times, the connection is time-dependent. Ignoring the propagation delay, all the nodes on the connected branch of the source node will receive the signal from the source node. However, as time goes on and nodes move, the nodes carrying a message will transmit the signal to the new nodes whenever the transmitted nodes share a link with the new nodes. Of course, it will naturally be put forward such a problem, namely, how long it takes for all nodes to receive the message. This is the so-called path formation time or node connection time [

For this problem, percolation theory is a natural tool, where percolation refers to the formation of the infinite component in a graph [

In this paper, we focus on some important issues of node connection and path formation time in FSN. Firstly, the system model is established. We mainly discuss the evolving graph of network topology associated with FSN, the path formation time, the node mobility model, and constraint to complete percolating in the noise-limited network in Section

The Earth-centered inertial (ECI) coordinate system is defined in the following standard manner [

If the vector

We assume that FDMA with subcarrier BPSK modulation is used as the MAC scheme for FSN. Let vectors

Based on (

Therefore, let

Hence, the dynamic topology graph can be defined as the following.

Let

Based on orbit dynamics theory, the orbital hyperperiod can be divided into

If the orbital hyperperiod can be divided into

Certainly, the evolving graph of dynamic topology in time slot

For convenience, we consider

Considering

Equation (

Additionally, fractionated spacecraft are required to maintain bounded relative distance for the entire mission lifetime, which is denoted as

For any time

Using

Let

Furthermore, due to the high-speed mobility of FSN, even if (

The path formation time is similar to first passage percolation in percolation theory. Roughly speaking, first passage percolation can be described as path formation time or node connection time. So if connection time of each edge

So, the path formation time is defined as

It is needed to construct a mobility model of nodes, because of high-speed mobility of FSN.

For convenience, we use a twin-satellites model to study the mobility model of nodes. Because fractionated spacecraft are required to maintain bounded relative distance for the entire mission lifetime, the node positions can be viewed as a uniform distribution within a sphere with

Distribution area of node positions of FSN.

So the mobility model

In ECI coordinates, if the position set of

In addition, in order to study dynamic connectivity, it is necessary to define the two important concepts about time.

In ECI coordinates, given

Conversely, Definition

In ECI coordinates, given

Obviously, if

So, one can also define the concepts about the FSN graph using the two definitions above.

Given

Without controlling, two initially close modules—a

For obtaining the initial constraint to fractionated spacecraft with respect to keeping bounded relative distance, the lemma is given as follows [

Consider two modules, the chief module

Then, the distance between the satellites C and D is required to be satisfied as follows:

The geometry of two identical orbits with a different RAAN,

According to Lemma

The chief module

By Lemma

Using (

When

When

The feasible region in Case 1.

The feasible region in Case 2.

By analyzing Figure

So, if the SIR between the two modules satisfies (

In aerospace engineering, the spacecraft performance is directly affected by the initial orbital elements, which determine the initial spatial distribution and affect the connectivity of FSN. Therefore, it is very important to study the relationship of the first docking time and the first separating time with the initial spatial distribution.

First, the following theorem is given.

For any two nodes

For convenience, one gives a proof in 2-dimension because of the symmetry of a sphere. So, the circular area at

Nodes

Now, suppose

Let

When nodes

where

So, one has

Similar to Case

When nodes

where

Applying the same method as Case

When

In summary, when

Case 1.

Case 2.

Case 3.

Similar to Theorem

In ECI coordinates, the initial position

For simplicity, it is supposed that the FSN is composed of 5 same duplex module nodes, which are marked as A, B, C, D, and E, respectively. It is also assumed that ^{2}, and the mass is 15 kg if the near circular reference orbital elements of A are given as follow:

Using STK (Satellite Tool Kit), we have

Using (

Hence, if

All near circular orbital elements.

Parameter | B | C | D | E |
---|---|---|---|---|

−0.023 | 0.057 | 0.086 | 0.126 | |

0.1 | −0.2 | −0.5 | 1.2 | |

7499.9996 | 7500.001 | 7500.0015 | 7500.0022 | |

0.09960 | 0.10100 | 0.10150 | 0.10219 | |

34.9771 | 35.0573 | 35.0860 | 35.1260 | |

359.9771 | 0.05725 | 0.08581 | 0.12635 | |

359.9827 | 0.04591 | 0.05747 | 0.19439 | |

359.9771 | 0.0573 | 0.0860 | 0.1260 |

According to Table

The relative distance between any modules within 244 days.

The topology in wireless networks can change dynamically depending on the link connectivity between each node pair. The network topology is modeled by the adjacency matrix [

According to Definition

Let

Parameters for numerical analysis.

Parameter | Value | Parameter | Value |
---|---|---|---|

12 dB | 5 kHz | ||

0.15 W | 10 kHz | ||

3 | 15 kHz | ||

1/(5^{2} |
20 kHz | ||

— | — | 25 kHz |

Based on above analysis, we construct adjacency matrices within 26 orbital hyperperiods. For example, in slot 3 for the 1st orbital hyperperiod, the adjacency matrix is given as follows:

And in slot 94 for the 5th orbital hyperperiod, the adjacency matrix is also given as follows:

Constructing the adjacency matrix within 26 orbital hyperperiods, we can also observe that the adjacency matrix of the FSN is a sparse matrix in which most of the elements are zero in most time slots of orbit hyperperiod.

In order to investigate the relationship between the path formation time and path length, we use shortest path algorithms to find shortest path length and calculate path formation time from the source node to the destination node in STK and MATLAB.

Under the same simulation environment as described in Section

Based on above analysis, we give the simulation results shown in Figures

(a) The relationship between path formation time and path length. (b) The relationship between hop length and path length for different source-destination in the 1st orbit hyperperiod.

(a) The relationship between path formation time and path length. (b) The relationship between hop length and path length for different source-destination in the 5th orbit hyperperiod.

(a) The relationship between path formation time and path length. (b) The relationship between hop length and path length for different source-destination in the 12th orbit hyperperiod.

(a) The relationship between path formation time and path length. (b) The relationship between hop length and path length for different source-destination in the 25th orbit hyperperiod.

It is observed that the path formation time and hop length nearly scale linearly with path length within each orbital hyperperiod shown in Figures

It is also observed that the formation times and hop lengths of three paths exist slight difference. Its reasons are the adjacency matrix is a sparse matrix and there exists a few edges in most slots of hyperperiod. When Dijkstra’s algorithm is used to find the shortest path, a part of shortest paths are same in the three paths.

The connection time constant depends on the ratio between path formation time and path length. Just as the results in Section

The connection time constant as a function of path formation time for different source-destination.

In Figure

Connectivity in FSN is time-varying and dynamic. Recent research on connectivity is mainly focused on static graph models. So, we have introduced an evolving graph of network topology associated with FSN, path formation time. We have presented constraint to orbital elements for the noise-limited FSN percolating, taking FDMA with subcarrier BPSK modulation as an example. According to first passage percolation in percolation theory, we defined the path formation time. Then we have investigated relationship between first docking time and spatial initial distribution and the relationship between first separating time and spatial initial distribution. Also, we have analyzed the relationship between the path formation time and path length using Dijkstra’s algorithm. While the approach discussed is useful, the simulation part is just an example, and the data included are not necessarily representing a real case. Finally, some insights on how to design and optimize in noise-limited FSN, for example, how to choose the hop length for efficient routing in FSN, can be also concluded as follows:

Due to the mobility, network topology is time-varying for fractionated spacecraft, and the connection is dynamic. Satisfying the constraints presented in this paper, the cluster flight and FSN percolating can be maintained in long-term bounded relative distances.

The initial spatial distribution, that is, the initial orbital elements, directly affects the first docking time and the first separating time.

The path formation time and hop length nearly scale linearly with path length within each orbital hyperperiod, and this relationship changes periodically.

The time constant increases with path formation time. When path formation time is larger, time constant gradually tends to a stable value, despite of the path formation time and hop length changing periodically.

The data generated or analyzed during this study are included within the article, but the programs analyzed during the current study are not publicly available as they are part of the simulation programs in use.

The authors declare that there is no conflict of interest regarding the publication of this paper.

The authors wish to thank the reviewers for their valuable comments, corrections, and suggestions, which led to an improved version of the original paper. The authors thank Professor Yang Zhen from National Space Science Center, CAS, China. This research is a project partially supported by the National Natural Science Foundation of China (Grant no. 61561009).

^{2}—a simulation study