A threedimensional matrix method is proposed in this paper for Global Navigation Satellite System (GNSS) constellation Intrasatellite Link (ISL) topological design. The rows and columns of proposed matrix contain the information of both constellation orbit planes and satellites in each orbit plane. The third dimension of proposed matrix represents the time sequences during constellation movement. The proposed method has virtues of better advantage of conceptual clarity and computational efficiency, meanwhile, some properties of ISLs in the constellation can be proved easily. At the second part of this paper, a link assignment and optimal routing problem is proposed using threedimensional topology matrixes, which aimed to minimize the relay hops during the process of data uploading to whole constellation network. Moreover, some practical constrains as antenna beam coverage and relative velocity are considered and analyzed in detail. Finally, some numerical simulations are provided, and the results demonstrated the promising performance of proposed topological method in reduction of computation burden, clear ISL conception, and so forth; the efficiency of provided optimal ISL routing problem is also proved.
With the rapid progress in GNSS technology, it is now feasible and necessary to build ISL network to increase system functions as communications and range measurements. One of the most important aspects in satellite constellation ISL analysis is the topological design.
In previous researches, Werner [
Chang et al. [
Suzuki et al. [
As we can see from the analysis above, previous constellation ISL designs are basically based on intuitive graphical analysis for a taskspecific mission. Some of the ISL topology analyses are using matrix form, however, those of which are mostly graphical based and without any matrix associated properties analysis. In this paper, the authors proposed a threedimensional matrix topology analysis method, wherein the rows and columns of the matrix represent the orbital plane and satellite information, and the third dimension represents the time information. By theoretical analysis and numerical simulations, the results show that the proposed method has advantages of computational efficiency and clarity of ISL conceptual.
At the second stage of this paper, an optimal link assignment and routing problem is provided using proposed matrix topology method. The readers will find more details in Section
Assuming that a typical global navigation constellation consists of
Demonstration of 3orbit plane ISL topological.
The rows of the matrix have
Based on the definition of the matrix elements above, the following conclusions can be drawn:
the diagonal elements of matrix denote each satellite itself, as the blue region in Figure
according to the characteristics of the global constellation ISL, the topological matrix can be blocked to some regions, as shown in Figure
Demonstration of topological block matrix.
Thus, when the
Demonstration of 3orbit plane ISL topological.
Demonstration of 3dimensional ISL topological matrix.
In the subsequent content of this paper, for the sake of matrix operations efficiency, some minor changes could be adopted for ISL topology matrix. Supposing the ISL matrix is
Topological Method two (TM2):
Topological Method three (TM3):
It can be proved that those three methods above hold the equivalent meaning (which will be proved in Section
For a typical Walker
According to the definition of ISL topology matrix, under any epoch
Then it indicates that a particular pair of satellites ISL disconnected or redundant, by the timing accumulation of the ISL topology matrix, the upper triangular elements would be fully filled with ones.
ISLs between different orbit planes are established in pairs.
For a typical Walker
Note that all the satellites in a Walker constellation are geometrical symmetry, and any satellite in the constellation can geometrically be representative of all others. Let us take a look at the first satellite in first orbit plane (
Also, the phase difference of the
Obviously, when
The hop number for agile scanning beam analysis is divided into two different types:
For the orbit plane number
For the intraplane ISLs, each orbital plane has satellites of
coorbit plane with odd number satellites: each orbit plane has one satellite bye, traversing the hop count
coorbit plane with even number satellites: each orbit plane get the full pairing, traversing the hop count
According to Conclusion 1 and Figure
Also, the intraplane matrix area of Figure
Considering the geometric relationships of interplane area of Figure
Equation (
According to the orbit planes
An alternate way to prove it is as follows: each orbital plane has
This also proved that the formula (
According to the geometric relationship of coorbit areas in Figure
Equation (
The ISL topology matrix
Since each of its elements are all ones, so
Then the subunit matrixes with same column sequence can be obtained. For the coorbit ISL submatrix
The result of the multiplication between any two submatrices
Proposition 2 can be proved similar to proposition 1, which is not provided here.
The matrix
This section presents the analysis of optimal coorbit/different orbit ISL hops and the total ISL numbers under different satellite constellation conditions. It is assumed that each satellite has the capability of establishing only one agile scanning beam ISL during simulation. The selected constellations include practical engineering constellations and the theoretical ones as in Table
Analysis of ISL hops and link number.
Constellations Walker

Different orbit plane  Coorbit plane  Hops/cycle  ISLs/cycle  Remark  

ISL numbers  Hops (multiplication of two items)  Hops 
ISL numbers  
Pairs of ISLs/epoch  ISLs/difforbit*  Combinations of difforbit  ISL types**  ISLs/coorbit

Orbit plane number  
Walker9/3/2  1  3  3  3  3  1  3  12  36  
Walker18/6/2  3  3  3  5  3  1  6  18  153  
Walker24/3/2  1  8  8  3  7  4  3  31  276  Compass 
Walker24/6/1  3  4  4  5  3  2  6  23  276  GPS 
Walker25/5/1  2  5  5  5  5  2  5  30  300  
Walker27/9/0  4  3  3  9  3  1  9  30  351  
Walker72/6/1  3  12  12  5  11  6  6  71  2556  
Walker77/7/1  3  11  11  7  11  5  7  88  2926  Iridium 
Walker66/6/1  3  11  11  5  11  5  6  66  2145  Iridium 
Walker120/10/1  5  12  12  9  11  6  10  119  7140  NeLS 
Walker288/12/1  6  24  24  11  23  12  12  287  459504  Teledesic 
**“ISL types” denotes the ISL connection types for different orbit during one traversal cycle.
Here we take the Walker77/7/1 constellation as an example to demonstrate the computation method of total ISL hops and the meaning of each item in Table
It will be inevitable that in the event of ISLs failure during GNSS operation, supposing that one satellite failure occurred, the row and column
Demonstration of one ISL failure.
First of all, here we give a brief introduction of some properties of the topology matrix and linear space: Let
It can be proved that one group of bases in space
Obviously, the Topological Method one is a spanned upper triangular matrix linear space on the base vectors
In case of ISL failure, as shown in Figure
Here we assumed that the mapping
Then we have
The question now is how to calculate the coordinates
Let
This matrix is the linear transformation on the base
The analysis of ISLs redundancy situation and restructure is similar as above, which is not provided here.
Topological Methods one/two/three (TM 1/2/3) are equal.
In linear space
For any vector
For any
Clearly, mapping
Commonly used computation efficient evaluation algorithm on modern computers is flop (floating octal point); one flop is a floatingpoint operation. Matrix flops calculation can be realized by adding the basic operation loops in one algorithm. In the cases of matrix multiplication and addition, the basic algorithm is
In this paper, the matrix topology representation is in the form of upper triangular matrix; here we give the calculation efficiency analysis.
For any upper triangular matrix
Then, calculations of
That is, the amount of calculation for upper triangular matrix multiplication is only 1/6 of the general dense matrix, which clearly has higher computational efficiency.
Currently, ISL intersatellite communication has been widely used in new generation of global navigation satellite system, and operational data can be exchanged through ISL multihops [
However, unlike the traditional network system, the ISL nodes in the network are moving constantly, and the ISLs onoff switch frequently with the change of relative position between satellites. The satellite in the network needs to establish a new link when the original one disconnects to accomplish the data transmission mission. Therefore, finding an optimal ISL routing is a crucial task in the GNSS ISL network design, and it directly affects the ISL network performance.
Previous research results in ISL routing problem include the following. Harathi et al. [
Here we provide a tailored optimal link assignment and routing problem for a GNSS ISL system below.
An optimal problem includes two parts: optimal object and constraints.
Previous research results about optimal ISL routing problems have been focused on the object of short path for endtoend communications, including the critical up/downlink between space segment (satellites) and earth segment (mobile users and gateway stations), as well as the routing protocol model [
Some constraints should be considered in an optimal ISL routing problem such as linking onoff switch conditions, linking propagation distance, and linking time. Those of constraints directly affect the ISL network performance. Generally, ISLs establishment between satellites in constellation relies on two factors:
Finding
min (relay hops)
such as
The constraint models in the optimal ISL routing problem are introduced in detail hereafter, and the overall optimization process will finally be provided.
This section provides the analysis of the antenna beam coverage constraints. First, definition of elevation angle and related physical quantities are given; then, ISL visibility criterion between couple satellites is introduced.
Figure
Geometry of ISL elevation angle.
According to the geometrical relationship in Figure
Generally, the elevation angle is defined in interval of
Since the circular orbit approximation of GNSS can be adopted, then we have
Then the ISL antenna beam coverage condition of two satellites is
The analysis of ISL antenna coverage above is only of the schematic formulas, which cannot be used directly in practice. Here we provide the analytical analysis of specific visible satellites’ mean anomaly under the conditions of ISL antenna beam coverage.
Due to the relative motion between satellites in different orbital plane in constellation, the specific time of ISL visibility depends on the constellation configuration and time
Figure
The ISL antenna beam coverage analysis.
The shaded spatial region of Figure
Establish Cartesian Coordinates which are centered at the Earth, as in Figure
The coordinates of any point in orbit plane 2 are
The geocentric angle
The mean anomaly of point
Accordingly, the boundary of ISL antenna beam coverage has four points
Thus, when the mean anomaly
Similarly, the visible arcs of orbit plane 2 from satellite
Based on the analysis above, for a typical Walker
In summary, the procedures of calculating visible satellites in orbit plane
calculate mean anomaly of
calculate the visible arcs of orbit plane
calculate the mean anomaly
collect all the visible satellites according to the analysis of visible arcs and mean anomaly of each satellite.
Assume that satellite
Therefore, the second condition for ISL establish between two satellites, is
Based on the analysis of object and constrains for optimal routing problem. The overall optimization process can now be expressed as in Figure
Flow diagram of optimal ISL routing problem.
According to the introduction of optimal ISL routing problem in Section
Suppose we use Walker24/3/2 constellation in simulation, with orbital inclination of 55 deg, orbital altitude of 21528.8 km, intersection period of 12 h 53 min, and regression cycle of 13 laps/7 days. This constellation achieved availability of 65.18% and 94.52% with conditions of PDOP ≤ 2 and positioning error less than 10 m, respectively, after statistics analysis. Moreover, the distribution difference of constellation’s largest PDOP around the world is small, and the average positioning accuracy is about 6 m, which is more practical for engineering use. Control segment is selected as Beijing Station with coordinates of latitude 39°54′27′′ north, longitude 116°23′17′′ east, and 5 deg elevation angle constraint when observing zenith.
The ISL antenna beam range is assumed to be
The initial epoch of simulation is set to April 8, 2014, 141500 (GMT +08:00). Simulation duration is set to one regression period of 7 days.
The actual configuration of ISL antennas in each satellite is a practical engineering problem during the process of optimal ISL link assessment. Different optimal results will be obtained with different types of antenna configurations. Therefore, this section provides two types of simulation scenarios: one without ISL antenna configuration constraints and one with.
Figures
Constellation states at initial epoch.
Data uploading to visible satellite from ground.
Satellites that have received the uploading data.
Path searching for possible ISL establishment.
Satellites that have received data by relay process.
Thereafter, entering the phase of data relay, as shown in Figure
A variety of assess criterion could be adopted for the optimal ISL routing problem. Here we use the percentage of special time compared with total simulation time which achieved the uploading data from ground transmitted to whole constellation using just ONE relay. By using numerical statistics after simulation, the ISL routing can achieve 100% of the time that realize uploading data traversal using just ONE relay, under the condition of no ISL antenna configuration constraint; clearly, it is a pretty good result impersonality.
The simulation scenario in Section
This section assumes that each satellite is configured with two agile scanning ISL antenna beams. In each epoch, those two beams select the geometrical optimal paths and establish ISL links. Other simulation scenarios and parameters are identical to the above section. The illustrations of topological matrixes are not provided here for conciseness.
Figures
Initial epoch.
Data upload.
Data received (1st).
Path searching.
Data relaying (1st).
Data received (2nd).
Generally, two fundamental demands should be satisfied when using ISL routing algorithm: ISL ranging measurements and communications between satellites. The former emphasize on the number of established ISLs. The more the ISLs measurements acquired, the more the accuracy results that will be got. The latter emphasize on the quality of established ISLs, such as propagation delay or the carriertonoise ratio. Therefore, how to efficiently use of the limited onboard antenna beams that take into account both demands is the difficulty of the optimal routing algorithm.
The link assignment and routing algorithms used in previous two simulation scenarios are based on the realtime constellation status and dynamic calculations. The ISL antennas have to capture and track the calculated couple satellite with speed and accuracy at each epoch, which is almost impossible to achieve in engineering. According to the cyclical nature of the Walker constellation, routing assignment calculated in advance and stored in each satellite cannot cope with the situations as satellite failures and system abnormalities. Therefore, seeking for the combination of consistent antenna beams with agility ones is a more feasible solution in practical.
The consistent ISL antenna beams are the links between two satellites, in the same or different orbit planes, that can be seen from each other with no block during the entire constellation regression cycle. Those links can be analytically calculated in advance. Those ISL beams are commonly realized by using reflector antenna in engineering. With proper configuration of ISL antennas on satellite platform and the usage of high reliability shaft, uninterrupted tracking can be achieved with established linking satellite and intersatellite ranging and data transfer can be performed.
The consistent ISL beams have merits of high precision ranging and high rate of data transfer speed, and so forth, which are the preferred solutions for practical use. On the other hand, usage of ISL agile scanning beams has to consider the time sequence design in advance, which provides poor support to realtime intersatellite data relay.
Figure
Data relaying index under different conditions.
As we can see through the simulation data in Figure
option
option
option
option
option
According to the numerical statistics after simulation, the relaying indexes using those five options above are 93.1%; 98.1%; 81.2%; 73.1%; 82.3%. As we can see from the statistical data, the best relaying index can be obtained using option
In this paper, a threedimensional matrix topology representation method is proposed. This method compared with the previous ones, despite the increasing in matrix dimensions, possesses the merits of physical concept clarity, upper triangular matrix with sparse elements, smaller amount of computation, and the easier proof of ISL network properties. Moreover, an optimal link assignment and routing problem is provided, which considers the minimization of data relay times during the whole constellation operation. Some engineering problems as ISL antenna beam coverage and relative velocity constrains are also analyzed. A variety of scenarios are simulated for proposed threedimensional topology matrix and routing problem. The results demonstrated that the proposed method has advantage of conceptual clarity and computational efficiency; moreover, the effectiveness of provided optimal ISL routing problem is also proved.
The development of ISL technology in future will be focused on the following aspects: supporting to the autonomous navigation and constellation system operation; supporting to the system differential and integrity monitoring; and critical data transmitting to the whole network under critical conditions, as well as the rapid restructuring of ISL network during satellite failure or redundant. The application of threedimensional matrix, presented in this paper, to those areas will be provided in authors’ consequent researches.
The authors declare that there is no conflict of interests regarding the publication of this paper.