# Path Formation Time in the Noise-Limited Fractionated Spacecraft Network with FDMA

Authors: Tingting Yan; Shengbo Hu; Jinrong Mo
Journal: International Journal of Aerospace Engineering
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1, 2]. FSN is a kind of ad hoc network which needs to accomplish cluster flight on orbit, and it is composed of multiple module nodes which can be homogeneous or heterogeneous. The module nodes are required to maintain bounded relative distances between tens of kilometers and hundreds of kilometers and to keep loose geometry for the entire mission lifetime. This leads to greater dynamic and random characteristics between node connectivity [3–5]. However, FSN needs a higher demand for signal dissemination between the nodes: even if the whole nodes cannot be completely connected in each time, FSN can realize the information distribution in the full network finally. Therefore, the study on message dissemination delay, that is, the research on the characteristic of path formation time, is quite significant. Undoubtedly, it is also worthwhile to research the design and performance optimization of FSN.Recently, percolation theory has been used to study connectivity and message dissemination delay in static multihop wireless networks [6–10]. Percolation theory studies the existence of phase transitions in random graphs. For instance, by constructing a weighted graph model and adopting first passage percolation, the first-arrival path and time delay in weighted graph is met in [11], but their model did not consider multiple access interference. Node connection depends not only on Euclidean distance between nodes but also on the location of other nodes in interference environment [12]. For the dynamic network topology caused by Medium Access Control (MAC) planning and load change in [13], the authors research the dynamic connection between nodes within Additive Links Online Hawaii Area (ALOHA) access control, and they derive the distributional properties of message dissemination delay.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 [14, 15].For this problem, percolation theory is a natural tool, where percolation refers to the formation of the infinite component in a graph [16–18]. Also, first passage percolation is used to describe the paths reachable in a graph [19–21]. Many scholars have studied path formation time using first passage percolation. For instance, taking information distribution time or path formation time as first passage percolation is simulated in [22, 23]. Path formation time is also studied by using the subadditive ergodic theorems in [24, 25]. But few references are available in path formation time for FSN with a loose geometric structure.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 Section2. Secondly, using the principles of orbital dynamics, we analyze the orbital element constraints to percolation in noise-limited FSN and the condition of ensuring cluster flight of fractionated spacecraft and FSN percolating in Section 3. Then using first passage percolation in Section 4, we study the relationship between nodes first docking time and spatial initial distribution and the relationship between first separating time and spatial initial distribution in FSN. Section 5 presents simulation results. Finally, Section 6 concludes the paper.


## 2. System Models

### 2.1. Earth-Centered Inertial Coordinate System

The Earth-centered inertial (ECI) coordinate system is defined in the following standard manner [2]: the fundamental plane is the equatorial plane, x axis towards the vernal equinox, z axis points towards the geographic north pole, and y=z×x. The vector of classical orbital elements in the ECI coordinate, which describes natural orbits of fractionated spacecraft, is defined as
(1)α≜a,e,β,ω,f,ΩT,where a is the semimajor axis, e is the eccentricity, β is the inclination, ω is the argument of perigee, f is the true anomaly, and Ω is the right ascension of the ascending node (RAAN).If the vectorr=x,y,zT denotes the position of any satellites in FSN in the ECI coordinate, v=dr/dt is the velocity, and η=x2+y2 is equatorial projection of the position vector, as well as the maximal and minimal equatorial projections of the position vector of satellite at time t, given by ηmax=maxtηt and ηmin=mintηt.


### 2.2. The Evolution of the Dynamic Topology Graph in FSN

We assume that FDMA with subcarrier BPSK modulation is used as the MAC scheme for FSN. Let vectorsrik and rjk denote position of the node i and node j at time k in the ECI coordinate, respectively. The node si with transmission power P sends the signal to node sj, and the node sj also receives interfering signals from other nodes with same transmission power P. So, when the signal-to-interference-plus-noise ratio (SINR) in the node sj satisfies (2), the node sj can successfully receive the signal from node si. SINR is given as [26]
(2)SINR=Plri−rjN0+∑w≠i,jγPlrw−rjSBPSKfw−fj≥Γ,where SBPSKfw−fj can be approximated as [26]
(3)SBPSKfw−fj≈12πTffw−fj2,and Γ is the threshold value of SINR, lri−rj is the path loss factor with respect to channel, and γ is interference cancellation ratio 0<γ≤1 which depends on multiple access modes, modulation mode, and other wireless communication technologies. N0 is the power of additive white Gaussian noise (AWGN). SBPSKfw−fj is the power spectrum in BPSK modulation, Tf is the symbol duration, fi is subcarrier frequency of node i.Based on (2), it is denoted that the construction of network topology depends entirely on the path loss factor lri−rj, if P, N0, and Γ are known.Therefore, letV be the set of nodes of FSN, Vtk denote the set of transmitting nodes at time k, and Vrk denote the receiving set of nodes, as well as Vtk∪Vrk=V. If the threshold Γ is less than SINR between transmitting node si∈Vtk and receiving node sj∈Vrk, there is a bidirectional connection path to form an edge set denoted by Ek. So it is seen that SINR directly affects the construction of network topology.Hence, the dynamic topology graph can be defined as the following.Definition 1.
Letsi∈Vtk and sj∈Vrk at time k. If the edge set between nodes is denoted by Ek, the dynamic topology graph is gk=GV,Ek.
Based on orbit dynamics theory, the orbital hyperperiod can be divided intoT0,T1,T2,…,TT times for fractionated spacecraft [27, 28]. So there are T time slots in an orbital period. If the edge set is denoted by Ekσ in time slot σk=Tk−1,Tk k=1,2,…,T, the topology graph denoted by Gk=V,Ekσ can be supposed to remain static in the time slot. The orbital hyperperiod is H=TT−T0 [27]. Then one defines the evolving graph of the dynamic topology graph for FSN as follows.Definition 2.
If the orbital hyperperiod can be divided intoT0,T1,T2,…,TT times for fractionated spacecraft, there are T time slots. Then the evolving graph of dynamic topology for FSN in the orbital hyperperiod is defined as
(4)G1,T=GV,E1σ∪⋯∪GV,ETσ=V,⋃k=T1Ekσ.
Certainly, the evolving graph of dynamic topology in time slotl can be denoted by G1,l according to Definition 2.


### 2.3. Constraint to Complete Percolating in Noise-Limited FSN

For convenience, we considerγ=1 and ignore the noise contribution. So the system is noise limited. And SINR in (2) can be changed into the signal-to-interference ratio (SIR), that is,
(5)SIR=Plri−rj∑w≠i,jγPlrw−rjSBPSKfw−fj.Consideringlri−rj=ri−rj−2 in free space, we can rewrite (5) as follows:
(6)SIR=rik−rjk−2∑w≠i,jrwk−rjk−2SBPSKfw−fj.Equation (6) indicates that the topology of the noise-limited FSN completely depends on spatial distribution of nodes.Additionally, fractionated spacecraft are required to maintain bounded relative distance for the entire mission lifetime, which is denoted asm≤rik−rjk≤M at any time k. So let SBPSKminfw−fj=Smin, minrik−rjk2=m2, and Q be the number of interfering nodes. Equation (6) can be changed as follows:
(7)SIR=rik−rjk−2∑w≠i,jrwk−rjk−2SBPSKfw−fj≥rik−rjk‐2QSminm−2.For any timek, if rik−rjk2≤m2/ΓQSmin, that is, SIR≥Γ, the node i can connect with the node j.Usingmaxrik−rjk2=M2 and (7), it is also seen that if m/M≥QΓSmin, the FSN is complete percolated at any time. So we can obtain the constraint to complete percolating in the noise-limited FSN, and the constraint can be given as follows:
(8)mM≥QΓSmin.Letλ=QSmin, (8) can be simplified as
(9)mM≥Qλ.Furthermore, due to the high-speed mobility of FSN, even if (9) is not satisfied at some time, that is, node i cannot connect with node j, node i will send the signal to node j next one or few time slots successfully by using store and forward function of fractionated spacecraft. The time slots are just node connection time, that is, path formation time.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 edgeei in the graph is nonnegative random variables tei of independent distribution, the path connection time between node i and node j is given as
(10)Tp=∑ttei.So, the path formation time is defined as(11)Ti,j=infTp:apathfromitoj.


### 2.4. Node Mobility Model

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 withM−m/4 radius in Figure 1.Figure 1
Distribution area of node positions of FSN.So the mobility modelModelt of nodes for FSN can be defined as the following.Definition 3.
In ECI coordinates, if the position set ofn nodes of FSN is R0=r10,r20,…,rn0 at initial time T0, the position set is Rk=r1k,r2k,…,rnk, and the positions are uniformly distributed within sphere Bri0,ai=1,2,…,n at time k, where ri0 and a=M−m/4 are the center and radius of the sphere, respectively. Moreover, positions among all nodes are mutually independent and independent of all previous locations.
In addition, in order to study dynamic connectivity, it is necessary to define the two important concepts about time.Definition 4.
In ECI coordinates, givenR0 at initial time T0 and Mt of FSN, the first docking time of two nodes i and j can be defined as follows:
(12)Tdockingi,j≜inft≥T0:rit−rjt2≤m2λΓ.
Conversely, Definition5 is given as the following.Definition 5.
In ECI coordinates, givenR0 at initial time T0 and Mt of FSN, the first separating time of two nodes i and j can be defined as follows:
(13)Tseparatingi,j≜inft≥T0:rit−rjt2>m2λΓ.
Obviously, ifrit−rjt2≤m2/λΓ, Tdockingi,j=0 and Tseparatingi,j>0; if rit−rjt2>m2/λΓ, Tdockingi,j>0 and Tseparatingi,j=0.
So, one can also define the concepts about the FSN graph using the two definitions above.Definition 6.
GivenR0 at initial time T0 and Mt of FSN, if and only if rit−rjt2≤m2/λΓ, there exists an instant connectivity graph Gs and a path between node i and node j, and if and only if ETdockingi,j<∞, there exists a long-time connection graph G1,t and a path between the two nodes.


## 2.1. Earth-Centered Inertial Coordinate System

The Earth-centered inertial (ECI) coordinate system is defined in the following standard manner [2]: the fundamental plane is the equatorial plane, x axis towards the vernal equinox, z axis points towards the geographic north pole, and y=z×x. The vector of classical orbital elements in the ECI coordinate, which describes natural orbits of fractionated spacecraft, is defined as
(1)α≜a,e,β,ω,f,ΩT,where a is the semimajor axis, e is the eccentricity, β is the inclination, ω is the argument of perigee, f is the true anomaly, and Ω is the right ascension of the ascending node (RAAN).If the vectorr=x,y,zT denotes the position of any satellites in FSN in the ECI coordinate, v=dr/dt is the velocity, and η=x2+y2 is equatorial projection of the position vector, as well as the maximal and minimal equatorial projections of the position vector of satellite at time t, given by ηmax=maxtηt and ηmin=mintηt.


## 2.2. The Evolution of the Dynamic Topology Graph in FSN

We assume that FDMA with subcarrier BPSK modulation is used as the MAC scheme for FSN. Let vectorsrik and rjk denote position of the node i and node j at time k in the ECI coordinate, respectively. The node si with transmission power P sends the signal to node sj, and the node sj also receives interfering signals from other nodes with same transmission power P. So, when the signal-to-interference-plus-noise ratio (SINR) in the node sj satisfies (2), the node sj can successfully receive the signal from node si. SINR is given as [26]
(2)SINR=Plri−rjN0+∑w≠i,jγPlrw−rjSBPSKfw−fj≥Γ,where SBPSKfw−fj can be approximated as [26]
(3)SBPSKfw−fj≈12πTffw−fj2,and Γ is the threshold value of SINR, lri−rj is the path loss factor with respect to channel, and γ is interference cancellation ratio 0<γ≤1 which depends on multiple access modes, modulation mode, and other wireless communication technologies. N0 is the power of additive white Gaussian noise (AWGN). SBPSKfw−fj is the power spectrum in BPSK modulation, Tf is the symbol duration, fi is subcarrier frequency of node i.Based on (2), it is denoted that the construction of network topology depends entirely on the path loss factor lri−rj, if P, N0, and Γ are known.Therefore, letV be the set of nodes of FSN, Vtk denote the set of transmitting nodes at time k, and Vrk denote the receiving set of nodes, as well as Vtk∪Vrk=V. If the threshold Γ is less than SINR between transmitting node si∈Vtk and receiving node sj∈Vrk, there is a bidirectional connection path to form an edge set denoted by Ek. So it is seen that SINR directly affects the construction of network topology.Hence, the dynamic topology graph can be defined as the following.Definition 1.
Letsi∈Vtk and sj∈Vrk at time k. If the edge set between nodes is denoted by Ek, the dynamic topology graph is gk=GV,Ek.
Based on orbit dynamics theory, the orbital hyperperiod can be divided intoT0,T1,T2,…,TT times for fractionated spacecraft [27, 28]. So there are T time slots in an orbital period. If the edge set is denoted by Ekσ in time slot σk=Tk−1,Tk k=1,2,…,T, the topology graph denoted by Gk=V,Ekσ can be supposed to remain static in the time slot. The orbital hyperperiod is H=TT−T0 [27]. Then one defines the evolving graph of the dynamic topology graph for FSN as follows.Definition 2.
If the orbital hyperperiod can be divided intoT0,T1,T2,…,TT times for fractionated spacecraft, there are T time slots. Then the evolving graph of dynamic topology for FSN in the orbital hyperperiod is defined as
(4)G1,T=GV,E1σ∪⋯∪GV,ETσ=V,⋃k=T1Ekσ.
Certainly, the evolving graph of dynamic topology in time slotl can be denoted by G1,l according to Definition 2.


## 2.3. Constraint to Complete Percolating in Noise-Limited FSN

For convenience, we considerγ=1 and ignore the noise contribution. So the system is noise limited. And SINR in (2) can be changed into the signal-to-interference ratio (SIR), that is,
(5)SIR=Plri−rj∑w≠i,jγPlrw−rjSBPSKfw−fj.Consideringlri−rj=ri−rj−2 in free space, we can rewrite (5) as follows:
(6)SIR=rik−rjk−2∑w≠i,jrwk−rjk−2SBPSKfw−fj.Equation (6) indicates that the topology of the noise-limited FSN completely depends on spatial distribution of nodes.Additionally, fractionated spacecraft are required to maintain bounded relative distance for the entire mission lifetime, which is denoted asm≤rik−rjk≤M at any time k. So let SBPSKminfw−fj=Smin, minrik−rjk2=m2, and Q be the number of interfering nodes. Equation (6) can be changed as follows:
(7)SIR=rik−rjk−2∑w≠i,jrwk−rjk−2SBPSKfw−fj≥rik−rjk‐2QSminm−2.For any timek, if rik−rjk2≤m2/ΓQSmin, that is, SIR≥Γ, the node i can connect with the node j.Usingmaxrik−rjk2=M2 and (7), it is also seen that if m/M≥QΓSmin, the FSN is complete percolated at any time. So we can obtain the constraint to complete percolating in the noise-limited FSN, and the constraint can be given as follows:
(8)mM≥QΓSmin.Letλ=QSmin, (8) can be simplified as
(9)mM≥Qλ.Furthermore, due to the high-speed mobility of FSN, even if (9) is not satisfied at some time, that is, node i cannot connect with node j, node i will send the signal to node j next one or few time slots successfully by using store and forward function of fractionated spacecraft. The time slots are just node connection time, that is, path formation time.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 edgeei in the graph is nonnegative random variables tei of independent distribution, the path connection time between node i and node j is given as
(10)Tp=∑ttei.So, the path formation time is defined as(11)Ti,j=infTp:apathfromitoj.


## 2.4. Node Mobility Model

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 withM−m/4 radius in Figure 1.Figure 1
Distribution area of node positions of FSN.So the mobility modelModelt of nodes for FSN can be defined as the following.Definition 3.
In ECI coordinates, if the position set ofn nodes of FSN is R0=r10,r20,…,rn0 at initial time T0, the position set is Rk=r1k,r2k,…,rnk, and the positions are uniformly distributed within sphere Bri0,ai=1,2,…,n at time k, where ri0 and a=M−m/4 are the center and radius of the sphere, respectively. Moreover, positions among all nodes are mutually independent and independent of all previous locations.
In addition, in order to study dynamic connectivity, it is necessary to define the two important concepts about time.Definition 4.
In ECI coordinates, givenR0 at initial time T0 and Mt of FSN, the first docking time of two nodes i and j can be defined as follows:
(12)Tdockingi,j≜inft≥T0:rit−rjt2≤m2λΓ.
Conversely, Definition5 is given as the following.Definition 5.
In ECI coordinates, givenR0 at initial time T0 and Mt of FSN, the first separating time of two nodes i and j can be defined as follows:
(13)Tseparatingi,j≜inft≥T0:rit−rjt2>m2λΓ.
Obviously, ifrit−rjt2≤m2/λΓ, Tdockingi,j=0 and Tseparatingi,j>0; if rit−rjt2>m2/λΓ, Tdockingi,j>0 and Tseparatingi,j=0.
So, one can also define the concepts about the FSN graph using the two definitions above.Definition 6.
GivenR0 at initial time T0 and Mt of FSN, if and only if rit−rjt2≤m2/λΓ, there exists an instant connectivity graph Gs and a path between node i and node j, and if and only if ETdockingi,j<∞, there exists a long-time connection graph G1,t and a path between the two nodes.


## 3. Noise-Limited FSN Percolating

Without controlling, two initially close modules—achief and a deputy—rapidly separate due to differential accelerations. It is thus necessary to identify orbits on which the modules remain within some prespecified relative distance for the entire mission lifetime [2, 5].For obtaining the initial constraint to fractionated spacecraft with respect to keeping bounded relative distance, the lemma is given as follows [2, 5].Lemma 1 (see [2]).
Consider two modules, the chief moduleC and the deputy module D, having identical constant ballistic coefficients, that is,
(14)CDCSCmC=CDDSDmD=const,where S is the cross-sectional area, CD is the drag coefficient defined with respect to the cross-sectional area, and mC and mD are mass of satellites C and D, respectively. In the following development, the subscripts ⋅C and ⋅D denote quantities corresponding to the satellites C and D, respectively. If only the atmospheric drag and the zonal harmonics with gravitational potential are considered, satellite D requires initial conditions to be satisfied as follow:
(15)αDt0=αCt0+∫t0t0+Δtα̇Adt+05×1ΔΩ,where 05×1 denotes a five-dimensional-zeros vector and differential RAAN satisfies ΔΩ=ΩDt0−ΩCt0+Δt in Figure 2.
Then, the distance between the satellites C and D is required to be satisfied as follows:(16)2ηminsinΔΩ2−Δtvmax≤rDt−rCt≤2ηmaxsinΔΩ2+Δtvmax,where Δt∈R is any given interval time.Figure 2
The geometry of two identical orbits with a different RAAN,ΔΩ, shown in the inertial reference frame.According to Lemma1, the following theorem can be proved.Theorem 1.
The chief moduleC and the deputy module D of FSN have identical constant ballistic coefficients. If the SIR between the two modules is satisfied (7), ∣Δt∣ and ∣ΔΩ∣ are satisfied as
(17)2arcsinm2ηmin≤ΔΩ≤2arcsinM+m2ηmax+2ηmin,Δt≤2ηminsinΔΩ/2−mvmax,(18)2arcsinM+m2ηmax+2ηmin≤ΔΩ≤2arcsinM2ηmax,Δt≤M−2ηmaxsinΔΩ/2vmax,noise-limited FSN percolates; that is, the nodes can be connected.Proof 1.
By Lemma1, if modules are required to remain bounded relative distance, the following conditions must be met:
(19)2ηmaxsinΔΩ2+Δtvmax≤M,(20)2ηminsinΔΩ2−Δtvmax≥m.
Using (19) and (20), it can be obtained the feasible region with respect to sinΔΩ/2 and ∣Δt∣. And the feasible region includes two cases as follow.Case 1.
WhenM/ηmax≥m/ηmin, the feasible region is shown in Figure 3.Case 2.
WhenM/ηmax<m/ηmin, the feasible region is shown in Figure 4.Figure 3
The feasible region in Case 1.Figure 4
The feasible region in Case 2.By analyzing Figure4, it is obviously unreasonable that the feasible region exists Δt<0 in Figure 4, that is, Case 2 shown in Figure 4 is false. So it is observed in Figure 3 that Δt and ΔΩ should be satisfied as
(21)2arcsinm2ηmin≤ΔΩ≤2arcsinM+m2ηmax+2ηmin,Δt≤2ηminsinΔΩ/2−mvmax,2arcsinM+m2ηmax+2ηmin≤ΔΩ≤2arcsinM2ηmax,Δt≤M−2ηmaxsinΔΩ/2vmax.So, if the SIR between the two modules satisfies (9), ∣Δt∣ and ∣ΔΩ∣ satisfy (17) or (18), respectively, the noise-limited FSN percolates. q.e.d.


## 4. First Docking Time, First Separating Time, and Initial Spatial Distribution

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.Theorem 2.
For any two nodesi and j in FSN, the initial position ri0 and rj0 and mobility model Mt are given. If and only if m/λΓ≤ri0−rj0≤M−m/2+m/λΓ, one has 0<ETdockingi,j<∞.Proof 2.
For convenience, one gives a proof in 2-dimension because of the symmetry of a sphere. So, the circular area atr with radius a is denoted by Ar,a.Remark 1.
Nodesi and j are located in Ari0,M−m/4 and Arj0,M−m/4, respectively, in Figure 1. And if ri0−rj0≥M−m/2+m/λΓ, then rik−rjk≥m/λΓ for all time k. According to (6), Tdockingi,j=∞ and ETdockingi,j=∞.
Now, supposem/λΓ≤ri0−rj0≤M−m/2+m/λΓ, three cases are considered.Case 1.
Letrdocking0 denote the midpoint between ri0 and rj0 and draw a circle centered at rdocking0 with radius m/2λΓ. If the circle is completely contained in the overlapping area of Ari0,M−m/4 and Arj0,M−m/4 as shown in Figure 5, then we have m/λΓ≤ri0−rj0≤M−m/2−m/λΓ. Obviously, the nodes i and j can appear inArdocking0,m/2λΓ.
When nodesi and j simultaneously appear in Ardocking0,m/2λΓ, the two nodes are able to communicate with each other directly. Hence, one has ETdockingi,j≤ET1i,j, where T1i,j is the first time that the two nodes appear in the circle and T1i,j follows geometric distribution with parameter p1. Parameter p1 can be calculated as follows:
(22)p1=prrik,rjk∈Armeet0,m2λΓ=Ardocking0,m/2λΓArdocking0,m/2λΓAri0,M−m/4Arj0,M−m/4=4πm2λΓM−m22π−2∂+sin2∂,
where⋅ is two-dimensional Lebesgue measure, that is, the area, ∂=arccos2ri0−rj0/M−m is the angle between the straight line which pass through points ri0 and E and another straight line which pass through points ri0 and rj0 at point ri0, and E is the upper intersection point of the cycles ri0 and rj0 in Figure 5.
So, one has(23)ETdockingi,j≤ET1i,j=λΓM−m22π−2∂+sin2∂4πm2.Case 2.
Similar to Case1, if the circle of radius m/2λΓ is incompletely contained in the overlapping area of the circles ri0,M−m/4 and rj0,M−m/4, then M−m/2−m/λΓ≤, as shown in Figure 6.
When nodesi and j simultaneously appear in the shaded area ABCD, the two nodes are able to communicate with each other directly. Hence, we have ETdockingi,j≤ET2i,j, where T2i,j is the first time that the two nodes appear in the shaded area ABCD and ET2i,j can be given as follows:
(24)ETdockingi,j≤ET2i,j=2π−2∂+sin2∂π+2θ1−sin2θ1+2θ2−sin2θ22m/M−mλΓ2,
whereθ1=arcsin4Ay/M−m, θ2=arcsin2λΓAy/m, and Ay=m2/4λΓ−ri0−rj0/4+4m2−M−m2λΓ/16ri0−rj0λΓ2 in Figure 6.Case 3.
Applying the same method as Case2, T3i,j is the first time that the two nodes simultaneously appear in the two shaded areas in Figure 7.
Whenri0−rj0≤M−m/2+m/λΓ, ET3i,j can be given as follows:
(25)ETdockingi,j≤ET3i,j=2π2θ1−sin2θ1+2θ2−sin2θ22m/M−mλΓ2.
In summary, whenm/λΓ≤ri0−rj0≤M−m/2+m/λΓ, there exists 0<ETdockingi,j<∞. q.e.d.Figure 5
Case 1.Figure 6
Case 2.Figure 7
Case 3.Similar to Theorem2, we can prove Theorem 3 as follows.Theorem 3.
In ECI coordinates, the initial positionri0 and rj0 and mobility model Mt are given. If and only if ri0−rj0≤m/λΓ for any nodes in Mt, 0<ETseparatingi,j<∞.


## 5. Numerical Results and Analysis

### 5.1. Orbital Elements

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 thatm=1 km and M=100 km. For this case, the modules have the following parameters: the drag coefficient is 2.2, the cross-sectional area is 0.09 m2, and the mass is 15 kg if the near circular reference orbital elements of A are given as follow:
(26)αA=7500km,0.1,35°,0°,0°,0°T.Using STK (Satellite Tool Kit), we haveVmax=8.061 km/s, ηmax=8231.725 km, and ηmin=5523.656 km.Using (16), when 0.010°≤ΔΩ≤0.421°, we have Δt≤1370.46sinΔΩ/2−0.124. In accordance with (17), when 0.421°≤ΔΩ≤0.696°, we have Δt∣≤12.405−2042.358sinΔΩ/2.Hence, ifΔΩ of B, C, D, and E are chosen as −0.023°, 0.057°, 0.086°, and 0.126°, respectively, all near circular orbital elements are listed in Table 1.Table 1
All near circular orbital elements.
Parameter
B
C
D
E
ΔΩ (deg)
−0.023
0.057
0.086
0.126
Δt
0.1
−0.2
−0.5
1.2
at0 (km)
7499.9996
7500.001
7500.0015
7500.0022
et0
0.09960
0.10100
0.10150
0.10219
βt0 (deg)
34.9771
35.0573
35.0860
35.1260
ωt0 (deg)
359.9771
0.05725
0.08581
0.12635
ft0 (deg)
359.9827
0.04591
0.05747
0.19439
Ωt0 (deg)
359.9771
0.0573
0.0860
0.1260According to Table1, all orbital periods can be calculated and are approximated as 6464 seconds using STK; so we believe the orbital hyperperiods of the FSN are also 6464 s. In addition, we can also calculate all relative distances shown in Figure 8 in 244 days by STK. It is observed that the relative distance between any two modules always remains below 100 km and above 1 km. It is shown that the orbital elements are effective, because of satisfying the condition of bounded relative distance, and this outcome verifies Theorem 1 reasonable.Figure 8
The relative distance between any modules within 244 days.


### 5.2. Construction of the Adjacency Matrix

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 [29–31].According to Definition2, we can model the topology of the FSN in slot σkk=1,2,…,T for an orbital hyperperiod by the 5×5 adjacency matrix denoted as Ak,
(27)Ak=0a12ka13ka14ka15ka21k0a23ka24ka25ka31ka32k0a34ka35ka41ka42ka43k0a44ka51ka52ka53ka54k0,where aijk=1i,j=1,…,5,i≠j, if the SIR is greater than Γ in slot σk; otherwise, aijk=0.LetTf=1/2000 s, fw−fj=2.5/Tfw−j=1, and σk=60sk=1,2,…,108. The orbital hyperperiod of the FSN is divided into 108 time slots, and each time slot is 60 s. So, we can construct the adjacency matrix of the FSN using STK and MATLAB, and other parameters used are listed in Table 2.Table 2
Parameters for numerical analysis.
Parameter
Value
Parameter
Value
Γ
12 dB
f1
5 kHz
P
0.15 W
f2
10 kHz
Q
3
f3
15 kHz
Smin
1/(5π)2
f4
20 kHz
—
—
f5
25 kHzBased 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:(28)0110010100110000000100010.And in slot 94 for the 5th orbital hyperperiod, the adjacency matrix is also given as follows:(29)0110010110110000100100010.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.


### 5.3. Path Formation Time and the Path Length

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 Section5.2, we calculate the path formation time and path length within 26 orbital hyperperiods using STK and MATLAB. In addition, we consider three types of path formation from the source node to the destination node, that is, E is the destination node, while A, B, and C are source nodes, respectively. First, we construct the adjacency matrices of the FSN within 26 orbital hyperperiods using previous results. Second, we use Dijkstra’s algorithm [32, 33] to find the shortest path. Dijkstra’s algorithm is an algorithm for finding the shortest paths between nodes in a graph. Using the algorithm, if the last visited node is not a destination node in some time slot, then this node is viewed as a source node to find the shortest path again. Until the last visited node is a destination node, this algorithm is over. In this process, the number of time slots to find the shortest paths is just the path formation time. Also, we can calculate the minimum number of hops, that is, hop length between two nodes, using the algorithm.Based on above analysis, we give the simulation results shown in Figures9–12. Figures 9, 10, and 11 are the simulation results of the 1st hyperperiod, the 5th hyperperiod, and the 12th hyperperiod, respectively. And Figure 12 is the simulation results of the 25th hyperperiod.Figure 9
(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)
(b)Figure 10
(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)
(b)Figure 11
(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)
(b)Figure 12
(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.
(a)
(b)It is observed that the path formation time and hop length nearly scale linearly with path length within each orbital hyperperiod shown in Figures9, 10, 11, and 12. Moreover, although a part of relationships within 26 orbital periods are shown, the path formation time and hop length change periodically. Maybe due to the influence of orbital perturbation, there exists a slight difference in each orbital hyperperiod.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.


### 5.4. Connection Time Constant between Two Module Nodes

The connection time constant depends on the ratio between path formation time and path length. Just as the results in Section5.3, path formation time nearly scales linearly with path length. For convenience, we analyze the relationships between connection time constant and path formation time. Under the same simulation environment as described in Section 5.2, Figure 13 shows the relationships between connection time constant and path formation time of three paths, that is, A to E, B to E, and C to E, within 26 orbital hyperperiods.Figure 13
The connection time constant as a function of path formation time for different source-destination.In Figure13, when path formation time is smaller than 50 slots, connection time constant increases with path formation time. When path formation time is larger than 50 slots, the increasing trend of time constant is gradually slow. Finally, the trend is nearly constant. Obviously, the time constant gradually tends to a stable value with path formation time increasing, although the path formation time and hop length change periodically. This result strongly supports percolation theory further.


## 5.1. Orbital Elements

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 thatm=1 km and M=100 km. For this case, the modules have the following parameters: the drag coefficient is 2.2, the cross-sectional area is 0.09 m2, and the mass is 15 kg if the near circular reference orbital elements of A are given as follow:
(26)αA=7500km,0.1,35°,0°,0°,0°T.Using STK (Satellite Tool Kit), we haveVmax=8.061 km/s, ηmax=8231.725 km, and ηmin=5523.656 km.Using (16), when 0.010°≤ΔΩ≤0.421°, we have Δt≤1370.46sinΔΩ/2−0.124. In accordance with (17), when 0.421°≤ΔΩ≤0.696°, we have Δt∣≤12.405−2042.358sinΔΩ/2.Hence, ifΔΩ of B, C, D, and E are chosen as −0.023°, 0.057°, 0.086°, and 0.126°, respectively, all near circular orbital elements are listed in Table 1.Table 1
All near circular orbital elements.
Parameter
B
C
D
E
ΔΩ (deg)
−0.023
0.057
0.086
0.126
Δt
0.1
−0.2
−0.5
1.2
at0 (km)
7499.9996
7500.001
7500.0015
7500.0022
et0
0.09960
0.10100
0.10150
0.10219
βt0 (deg)
34.9771
35.0573
35.0860
35.1260
ωt0 (deg)
359.9771
0.05725
0.08581
0.12635
ft0 (deg)
359.9827
0.04591
0.05747
0.19439
Ωt0 (deg)
359.9771
0.0573
0.0860
0.1260According to Table1, all orbital periods can be calculated and are approximated as 6464 seconds using STK; so we believe the orbital hyperperiods of the FSN are also 6464 s. In addition, we can also calculate all relative distances shown in Figure 8 in 244 days by STK. It is observed that the relative distance between any two modules always remains below 100 km and above 1 km. It is shown that the orbital elements are effective, because of satisfying the condition of bounded relative distance, and this outcome verifies Theorem 1 reasonable.Figure 8
The relative distance between any modules within 244 days.


## 5.2. Construction of the Adjacency Matrix

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 [29–31].According to Definition2, we can model the topology of the FSN in slot σkk=1,2,…,T for an orbital hyperperiod by the 5×5 adjacency matrix denoted as Ak,
(27)Ak=0a12ka13ka14ka15ka21k0a23ka24ka25ka31ka32k0a34ka35ka41ka42ka43k0a44ka51ka52ka53ka54k0,where aijk=1i,j=1,…,5,i≠j, if the SIR is greater than Γ in slot σk; otherwise, aijk=0.LetTf=1/2000 s, fw−fj=2.5/Tfw−j=1, and σk=60sk=1,2,…,108. The orbital hyperperiod of the FSN is divided into 108 time slots, and each time slot is 60 s. So, we can construct the adjacency matrix of the FSN using STK and MATLAB, and other parameters used are listed in Table 2.Table 2
Parameters for numerical analysis.
Parameter
Value
Parameter
Value
Γ
12 dB
f1
5 kHz
P
0.15 W
f2
10 kHz
Q
3
f3
15 kHz
Smin
1/(5π)2
f4
20 kHz
—
—
f5
25 kHzBased 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:(28)0110010100110000000100010.And in slot 94 for the 5th orbital hyperperiod, the adjacency matrix is also given as follows:(29)0110010110110000100100010.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.


## 5.3. Path Formation Time and the Path Length

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 Section5.2, we calculate the path formation time and path length within 26 orbital hyperperiods using STK and MATLAB. In addition, we consider three types of path formation from the source node to the destination node, that is, E is the destination node, while A, B, and C are source nodes, respectively. First, we construct the adjacency matrices of the FSN within 26 orbital hyperperiods using previous results. Second, we use Dijkstra’s algorithm [32, 33] to find the shortest path. Dijkstra’s algorithm is an algorithm for finding the shortest paths between nodes in a graph. Using the algorithm, if the last visited node is not a destination node in some time slot, then this node is viewed as a source node to find the shortest path again. Until the last visited node is a destination node, this algorithm is over. In this process, the number of time slots to find the shortest paths is just the path formation time. Also, we can calculate the minimum number of hops, that is, hop length between two nodes, using the algorithm.Based on above analysis, we give the simulation results shown in Figures9–12. Figures 9, 10, and 11 are the simulation results of the 1st hyperperiod, the 5th hyperperiod, and the 12th hyperperiod, respectively. And Figure 12 is the simulation results of the 25th hyperperiod.Figure 9
(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)
(b)Figure 10
(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)
(b)Figure 11
(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)
(b)Figure 12
(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.
(a)
(b)It is observed that the path formation time and hop length nearly scale linearly with path length within each orbital hyperperiod shown in Figures9, 10, 11, and 12. Moreover, although a part of relationships within 26 orbital periods are shown, the path formation time and hop length change periodically. Maybe due to the influence of orbital perturbation, there exists a slight difference in each orbital hyperperiod.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.


## 5.4. Connection Time Constant between Two Module Nodes

The connection time constant depends on the ratio between path formation time and path length. Just as the results in Section5.3, path formation time nearly scales linearly with path length. For convenience, we analyze the relationships between connection time constant and path formation time. Under the same simulation environment as described in Section 5.2, Figure 13 shows the relationships between connection time constant and path formation time of three paths, that is, A to E, B to E, and C to E, within 26 orbital hyperperiods.Figure 13
The connection time constant as a function of path formation time for different source-destination.In Figure13, when path formation time is smaller than 50 slots, connection time constant increases with path formation time. When path formation time is larger than 50 slots, the increasing trend of time constant is gradually slow. Finally, the trend is nearly constant. Obviously, the time constant gradually tends to a stable value with path formation time increasing, although the path formation time and hop length change periodically. This result strongly supports percolation theory further.


## 6. Conclusion

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:(i)
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.
(ii)
The initial spatial distribution, that is, the initial orbital elements, directly affects the first docking time and the first separating time.
(iii)
The path formation time and hop length nearly scale linearly with path length within each orbital hyperperiod, and this relationship changes periodically.
(iv)
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.



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*Source: 9124132-2018-10-02.xml*