Topology design algorithm for optical inter-satellite links in future navigation satellite networks

Abstract

At present, deployment of optical inter-satellite link (OISL) payloads on future navigation satellites are planned for each global navigation satellite system (GNSS). The communication mechanism of an OISL is quite different from that of a radio link, so the existing topology optimization algorithms for radio links is difficult to adapt to the constraints of OISLs. We propose a topological optimization algorithm called multi-objective discrete binary particle swarm optimization (MODBPSO) to optimize the topology of OISLs for future navigation satellites. The emphasis is on minimizing the position dilution of precision (PDOP), point-to-point link delay and terminal idle rate costs. In our simulation test, the multiple complex constraints of OISL, such as maximum link distance, earth occlusion, terminal viewing field, number of terminals and fixed links, are fully considered. The simulation results show that in static topology optimization, MODBPSO can obtain 1–7 network topologies with lower PDOP, link delay, and link idleness rate than the multi-objective simulated annealing (MOSA) simultaneously in a single simulation test with different auxiliary parameters. Besides, it is convenient to select the appropriate solution from a non-dominated solution set for different priority requirements in dynamic topology optimization. In the simulation, we get a solution that can obtain lower average PDOP and link delay, but the terminal idle rate is relatively higher due to the influence of the link switching optimization strategy.

Appendix 1

Main flow of the MODBPSO algorithm.

For a certain epoch of the orbit period of satellites, the MODBPSO algorithm is as follows.

• 1. The adjacency matrix of the network (denoted as ${\mathbf{L}}$) is initialized by setting all elements to zero and then setting specific elements to one according to the fixed link constraint.

• 2. Initialize the visibility constraint symmetrical matrix ${\mathbf{R}}_{{\mathbf{V}}}$ according to the visibility constraints (1), (4) and (5). If the i-th satellite and j-th satellite are visible, then ${\mathbf{R}}_{{\mathbf{V}}} (i,j)$ is set to 1; otherwise, it is set to 0.

• 3. Transform ${\mathbf{R}}_{{\mathbf{V}}}$ into a one-dimensional vector $\mathop{x}\limits^{\rightharpoonup}$, and then, a one-dimensional index vector $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{x_{in} }}$ is established, which collects the indexes of all 1 bits in $\mathop{x}\limits^{\rightharpoonup}$ except for the corresponding bits of the fixed links.

• 4. One element $x_{in}^{j}$ in $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{x_{in} }}$ is randomly selected. According to the index represented by $x_{in}^{j}$, the corresponding element in network adjacency matrix ${\mathbf{L}}$ (denoted as ${\mathbf{L}}(j_{1} ,j_{2} )$) is temporarily set to 1, and then, whether it meets the azimuth and elevation angle constraints of the two related satellites is checked. If it meets these requirements, then the loading factor of the terminals on each satellite is increased by 0 or 1/s according to the geometric relationship between viewing field of the terminals (see Fig. 6). If the loading factor exceeds 1, the ${\mathbf{L}}(j_{1} ,j_{2} )$ is corrected to 0; otherwise, it keep 1. Finally, $x_{in}^{j}$ is removed from $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{x_{in} }}$. The above step is repeated until all terminal loading factors are 1 or $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{x_{in} }}$ is empty. This process is called random selection.

• 5. Step (4) is repeated m times to generate m particle position vectors, which represent m initial feasible solutions.

• 6. The corresponding cost is calculated for the m initial feasible particles according to formulas (10), and the non-dominated particles are selected to form the initial non-dominated particle set.

• 7. While the number of cycles does not reach the maximum

•  (1) The global optimal particle is selected from the non-dominated particle set by the roulette method.

•  (2) According to the current position $\overrightarrow {{p_{i} }} (t)$ and velocity $\overrightarrow {{v_{i} }} (t)$ of each particle and the current optimal position $\overrightarrow {{p_{g} }} (t)$ and velocity $\overrightarrow {{v_{g} }} (t)$ of global optimal particles, the current velocity $\overrightarrow {{v_{i} }} (t + 1)$ of each particle is calculated by formulas listed below. Here we denote $v_{ij}^{c}$ as the probability of a change in the j-th bit of the i-th particle, $v_{ij}^{1}$ is the probability of the j-th bit of the i-th particle changing to one, and $v_{ij}^{0}$ is the probability that it changes to zero. Then, we take $c_{1}$ and $c_{2}$ as two learning factors, $w$ as the inertia weight, $r_{1}$ and $r_{2}$ as two random values between 0 and 1. The $p_{i}^{j}$ and $p_{g}^{j}$ are j-bit of $\overrightarrow {{p_{i} }} (t)$ and $\overrightarrow {{p_{g} }} (t)$, respectively.

  $$v_{ij}^{^{\prime}} (t + 1) = sig(v_{ij} (t + 1)) = \frac{1}{{1 + e^{{ - v_{ij} (t + 1)}} }},$$

  (15)
  $$v_{ij}^{c} = \left\{ {\begin{array}{*{20}c} {v_{ij}^{1} = wv_{ij}^{1} + d_{ij,1}^{1} + d_{ij,2}^{1} ,\,{\text{if}}\,x_{ij} = 0} \\ {v_{ij}^{0} = wv_{ij}^{0} + d_{ij,1}^{0} + d_{ij,2}^{0} ,\,{\text{if}}\,x_{ij} = 1} \\ \end{array} } \right.,$$

  (16)
  $$\left\{ {\begin{array}{*{20}c} {If\,p_{i}^{j} = 1\,{\text{then}}\,d_{ij,1}^{1} \, = \,c_{1} r_{1} \,{\text{and}}\,d_{ij,1}^{0} \, = \, - c_{1} r_{1} } \\ {If\,p_{i}^{j} = 0\,{\text{then}}\,d_{ij,1}^{0} \, = \, - c_{1} r_{1} \,{\text{and}}\,d_{ij,1}^{1} \, = \,c_{1} r_{1} } \\ {If\,p_{g}^{j} = 1\,{\text{then}}\,d_{ij,2}^{0} \, = \,c_{2} r_{2} \,{\text{and}}\,d_{ij,2}^{1} \, = \, - c_{2} r_{2} } \\ {If\,p_{g}^{j} = 0\,{\text{then}}\,d_{ij,2}^{0} \, = \, - c_{2} r_{2} \,{\text{and}}\,d_{ij,2}^{1} \, = \,c_{2} r_{2} } \\ \end{array} } \right.,$$

  (17)

•  (3) The current optimal position of each particle is calculated by formula shown below, where $x_{ij}$ is the j-th bit of position vector $\overrightarrow {{x_{i} }}$. It does not necessarily meet the constraint conditions, so it is saved as a temporary optimal particle, which is recorded as $\overrightarrow {{p_{i}^{^{\prime}} }} (t + 1)$ $i \in \{ 1,2,...,M\}$. where the symbol $\varepsilon_{ij}$ is random in the range of (0,1).

  $$x_{ij} (t + 1) = \left\{ {\begin{array}{*{20}c} {1\,{\text{if}}\,\varepsilon_{ij} < sig(v_{ij} (t + 1))} \\ {0\,{\text{otherwise}}} \\ \end{array} } \right.\,\,\,\,j \in \left\{ {1,2,...,d} \right\},$$

  (18)

•  (4) A fixed link constraint is added to $\overrightarrow {{p_{i}^{^{\prime}} }} (t + 1)$, the j-th component $p_{ij}^{^{\prime}} (t + 1)$ is set to 1 if it corresponds to a fixed link, and the j^’-th component $p_{{ij^{^{\prime}} }}^{^{\prime}} (t + 1)$ is set to 0 if it does not meet the visibility constraint according to ${\mathbf{R}}_{{\mathbf{V}}}$.

•  (5) Similar to the random selection mechanism in step 4), the other 1 component in vector $\overrightarrow {{p_{i}^{^{\prime}} }} (t + 1)$ except for the fixed link components are randomly selected, and feasible particles satisfying the constraints are obtained. This step is repeated n times to obtain feasible particle sets, denoted as $\overrightarrow {{p_{i}^{k} }} (t + 1)\,k \in \{ 1,2,...,n\}$.

•  (6) Among the n feasible particles, the particle with the shortest Hamming distance from the temporary optimal particle $\overrightarrow {{p_{i}^{^{\prime}} }} (t + 1)$ is chosen as the actual position of the current particle $\overrightarrow {{p_{i} }} (t + 1)$.

•    (7) $\overrightarrow {{p_{i} }} (t + 1)$ is transformed into adjacency matrix ${\mathbf{L}}_{{\mathbf{i}}}$, and the cost of the particle is calculated through formulas (10).

•    (8) The nondominated particles are selected from the particle swarm through formulas (11). If the number of nondominated particles exceeds the capacity of the repository, then the particles in the grid area with the highest density in the repository are deleted until the capacity is met, and the grid size of the repository is adjusted at the same time. This process is referred to as research (Coello et al. 2004).

•    (9) The learning factors $c_{1}$ and $c_{2}$ and inertia weight $w$ are updated according to recursion as listed below, where $w_{d}$, $c_{d1}$ and $c_{d2}$ are constants used to adjust the change rate.

  $$\left\{ {\begin{array}{*{20}c} {w(t + 1) = w(t) \cdot w_{d} } \\ {c_{1} (t + 1) = c_{1} (t) \cdot c_{d1} } \\ {c_{2} (t + 1) = c_{1} (t) \cdot c_{d2} } \\ \end{array} } \right.,$$

  (19)

•    (10) The cycle number is increased.

• End while

• 8. The final Pareto optimal solution and its cost and ending algorithm are obtained.