Optimisation process

The optimisation process is carried out in three steps, as shown in Figure 2.3; firstly the input parameter are specified by the user, then the driver script pre- pares the environment for the execution of the GA and starts the optimization. The GA parameters specified in the driver script are of paramount importance to the final solution; the final solution, and the time required to find it, strongly depends on how the generations are evolved, on the size of the population and on the GA operators parameters (crossover frequency, mutation probability, use of elitism, to name a few). Given the limited computing resources available for this study the population size and the maximum number of generations have been chosen to be relatively small. The number of sites and the number of satellites utilized has been also kept as small as possible. Examples of the FC optimizations are provided in the following.

2.1 Optimisation of Flower Constellations

Figure 2.3: Optimisation process diagram.

Examples

The examples concerns regions in the U.S.A. simply because Matlab provides political boundaries of the states within the U.S.A. and thus the regions are easier to specify, but there is no loss of generality for the method.

Example 1

This example provides a solution to the observation of Alaska, an oil rich region. The GA has been set to stop as soon as it found a 100% coverage solution. Matlab output is provided below:

Region selected: Alaska Ns = 8, Nd = 1, Fd = 16, T_min = 2 h, T_max = 8 h a_min = 8059.0 [Km], a_max = 20307.4 [Km] (Np_min, Np_max) = (3, 11) Sensor FOV: 30 [deg] N. of points: 6

2.1 Optimisation of Flower Constellations

Output: Optimization terminated: minimum fitness limit reached. Optimization time: 79.4 [s] fval = 0.0 Np = 3, Nd = 1, Ns = 8, Fn = 7, Fd = 16, Fh = 0 inc = 61.27 [deg], w = 0.00 [deg], e = 0.488 a = 20270.4 [Km], e = 0.49, T = 0.332 [days]

Average coverage: 100.00 % Average gap: 0.00 %

2.1 Optimisation of Flower Constellations

Figure 2.5: Three-dimensional relative path in ECEF coordinates for Example 1. Example 2

In this case the region of interest is Texas, using 23 points to approximate the re- gion. The GA finds an optimal solution using 6 satellites, that achieve continuous coverage of the region.

N. of points: 23 Selected region: Texas Ns = 6, Nd = 1, Fd = 18, T_min = 2.0 [h], T_max = 15.0 [h] a_min = 8059.0 [Km], a_max = 30878.4 [Km] --> (Np_min, Np_max) = (2, 11) Sensor FOV = 30.0 [deg]

Output: Optimization terminated: minimum fitness limit reached. Optimization time: 310.2 [s] fval = 0.0 Np = 2, Nd = 1, Ns = 6, Fn = 5, Fd = 18, Fh = 0 inc = 27.36 [deg], w = 0.00 [deg], e = 0.750 a = 26561.7 [Km], e = 0.75, T = 0.499 [days]

Average coverage: 100.00 % Average gap: 0.00 %

2.1 Optimisation of Flower Constellations

Figure 2.6: Solution for continuous coverage of Texas with 6 satellites. Example 3

In this example only 6 satellites are allowed, in MEO orbit. In this example we are trying to achieve continuous coverage with a lower orbit. The solution found is not 100% coverage but still very close, as it is show in Figure 2.8. Four points on the Florida region are considered. An plot that shows how the GA converges toward better solutions is shown in Figure 2.9.

N. of points: 4 Input parameters: Selected region: Florida Simulation start date: 10/1/2006 12.0 Ns = 6, Nd = 1, Fd = 18, Tmin = 2.0 [h], Tmax=6.5 [h] a{\_}min = 8059.0 [Km], a{\_}max = 17682.2 [Km] -->(Np_min, Np_max) = (4, 11) Sensor FOV = 30.0 [deg]

Output: Optimization terminated: maximum number of generations exceeded. Optimization time: 2012.9 [s] fval = 27117.2 Np = 4, Nd = 1, Ns = 6, Fn = 1, Fd = 18, Fh = 0, inc = 45.03 [deg], w = 3.83 [deg], e =0.074 a = 16732.9 [Km], e = 0.07, T = 0.249 [days]

2.1 Optimisation of Flower Constellations

Average gap: 8.25 %

Figure 2.7: Ground track plot of best solution found in Example 3.

Figure 2.8: Percentage of visible sites as function of time for the solution found in Example 3.

2.1 Optimisation of Flower Constellations

Figure 2.9: Best and mean population fitness during GA optimization for example 3.

Example 4

Another run on Florida, but the desired orbits should have a period between 3 and 4 hours, thus more satellites are needed to achieve satisfactory coverage. N. of points: 4 Optimization terminated: maximum number of

generations exceeded. Optimization time: 701.5 [s]

Input parameters: Selected region: Florida Simulation start date: 10/1/2006 12.0 Ns = 12, Nd = 2, Fd = 48, T_min = 3.0 [h],

T_max=4.0 [h] a_min = 10560.3 [Km], a_max = 12792.9 [Km] --> (Np_min, Np_max) = (12, 15) Sensor FOV = 30.0 [deg]

Output: fval = 89717.5 Np = 13, Nd = 2, Ns = 12, Fn = 1, Fd = 48, Fh = 0, inc = 34.29 [deg], w = 0.00 [deg], e = 0.304 a = 12106.0 [Km], e = 0.30, T = 0.153 [days]

Average coverage: 86.37 % Average gap: 13.41 %

2.1 Optimisation of Flower Constellations

Figure 2.10: Ground track plot of best solution found in Example 4.

Figure 2.11: Ground track plot of best solution found in Example 4. Examples 1-4 are shown to validate the feasibility of the approach and the correctness of the Matlab code. The examples show expected trends: if higher orbits are allowed the preferred solutions tend towards highly eccentric orbits

2.1 Optimisation of Flower Constellations

Figure 2.12: Ground track plot of best solution found in Example 3. with high apogees above the regions of interest (example 1 and 2), thus achieving continuous coverage with a limited number of satellites. On the other end, if the height-range of allowed orbit is reduced, more satellites are needed to achieve the continuous coverage requirement, as expected.

In Figure2.9,2.12 the convergence history of the Genetic algorithm is shown. Blue dots represent the average fitness in the population, whereas black dots rep- resent the history of the best solution. The plot shows that the average converges toward the best found solution, and there is also an improvement of the best individuals with the progression of generations. Some important GA parameters are shown and described in Table 2.3; for further details refer to Matlab help documents on the GA optimisation toolbox.

BIBLIOGRAPHY

Parameter Description Value

Population Size Number of individuals in the population 30 Crossover Fraction

Fraction of the population in the next gen- eration that is created using the cross over operator.

90% Generations Maximum number of generations before

halt. 30

Stall time limit

Maximum time allowed during runtime without an improvement in the best indi- vidual.

10 minutes Fitness Limit If this value is reached, no further optimi-

sation is possible. 0

Table 2.3: GA parameters

References

[1] J. R. Wertz, Mission Geometry; Orbit and Constellation Design and Man- agement, Microcosm Press, 2001. 2.1

Chapter 3

Applications and Benefit

Analysis

FCs have been designed in the past for Earth’s global navigation, giving much better navigation performance (in terms of GDOP, ADOP, and coverage) than the existing Global Positioning System (GPS), Global Navigation Satellite System (GLONASS), and GALILEO constellations, using the same number of spacecraft or achieving the same performance using fewer satellites [1] [2].

3.1

Telemedicine

In this Section, we deal with telemedicine applications which are a particular type of telecommunication service which can exploit real-time and/or store-and- forward applications. For this type of service we can identify a specific number of locations involved in providing and accessing to the service. There are sev- eral telemedicine providers which exploit existing satellite systems for the provi- sion of this telecommunication service. Most of the sites interested in accessing telemedicine services are located in rural areas and, hence, satellite systems are the most suitable choice for the platform service. However, none of these satel- lite systems are specifically designed for telemedicine services. Furthermore they do not provide a direct connection between the service suppliers and the service customers. In this work we design a specific FC for the provision of telemedicine services with the following features:

3.1 Telemedicine

• near continuous coverage of a list of targets interested in providing and accessing the service;

• direct connection of service suppliers and service customers via satellites and Inter Satellite Links (ISLs);

• maximum Round Trip Time (RTT) of 200 ms.