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RESULTS
As with the test cases, each of these cases are run once to find a near optimum solution. For each of the initial runs of the genetic algorithm, the parameter ranges will remain 0 to 0.99 for eccentricity, 0 to 10 DU for semi-latus rectum, and 0 to 2π radians for the orientation angle. Then, the parameter ranges are narrowed to yield an answer that is closer to an exact solution.
Case 1
The first case to be attempted is a case with low eccentricity values. The orbit parameters used are
The initial implementation of the GA yields the solution illustrated in Figure 7.11. After narrowing the parameter ranges to 0.2 to 0.8 for e, 1 to 3 DU for p, and the range for ω remains 0 to 2π, the solution is slightly improved to the results shown in Figure 7.12.
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Figure 7.11 Initial solution for Case 1.
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Figure 7.12 Solution for Case 1 after the parameter ranges are narrowed.
In this case, reducing the ranges of parameters improved the required ΔV only by 4%. However, both figures show very similar transfer orbits that are both nearly tangent to the initial and final orbits, thus implying that both the initial and final solutions are near optimal.
Case 2
The second case uses one ellipse that is highly elliptical and one that is slightly more circular. The parameters of these ellipses are
The resulting transfer ellipse from the initial implementation of the GA is shown in Figure 7.13. After the parameters space is narrowed to 0.5 to 0.99 for e, 1 to 2 DU for p, and 0 to 1.047 radians for ω, the improved transfer ellipse is shown in Figure 7.14.
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Figure 7.13 Initial solution for Case 2.
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Figure 7.14 Solution for Case 2, after the parameter ranges have been narrowed.
For Case 2, reducing the parameter list results in a 22% reduction in the ΔV requirement. The reduction in ΔV is expected since the transfer ellipse is larger and more eccentric, thus allowing for a smoother velocity transition from the transfer to the final ellipse.
Case 3
The final orbits to be considered are both highly elliptical and are separated by almost 90°. The parameters for the initial and final ellipses are
Figure 7.15 displays the initial solution for the transfer ellipse. After refining the range of the parameters to 0.5 to 0.99 for e, 3 to 6 DU for p, and 0.578 to 2.047 radians for ω, the result improves, as shown in Figure 7.16.
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Figure 7.15 Initial transfer ellipse solution for Case 3.
After reduction of the allowable parameter ranges, the resulting transfer ellipse is larger, more eccentric, and requires 5% less ΔV than the case shown in Figure 7.15. However, the transfer orbit is not as tangential to the final ellipse as the cases previously shown. This case of two highly elliptical orbits separated by approximately 90° illustrates more of a challenge for the genetic algorithm. In a case such as this, many of the transfer orbits contained in a population that tangentially intersect the final orbit are likely to not intersect the initial orbit at all, rendering it useless. However, it should be noted that the solution in this case does come somewhat near to a tangential intersection, and this result was found with relative ease.
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Figure 7.16 Solution for the transfer ellipse for Case 3, after the parameter ranges have been narrowed.
CONCLUSIONS
From the results of the three test cases, it can be seen that the GA is getting near-exact solutions for these known cases. Limiting the ranges for the parameters did prove, in every case, to lower the amount of fuel required to make the orbit transfer. Therefore, if the user could provides the genetic algorithm with ranges of the parameters that are more likely to produce a good transfer orbit, better performance will result.
In addition, the results of each of the cases agree very closely with the intuitive solution for the transfer ellipse because all of the resulting transfer orbits shown are nearly tangent to both the initial and final ellipse, where Case 3 may prove to deviate the farthest from what is intuitively expected. In Case 3, the difficulty for the GA to find a solution that is more tangential to the final orbit may be due to the fact that the orbits are separated by almost 90° and are highly elliptical. Overall, the accuracy of the results and the ease with which they were found do nothing but further verify the use of a GA for such a problem.
For all of the cases, the GA proves to be a desirable method in which to find transfer-orbit solutions, because this type of solution method requires no derivative information for the complicated equations that describe this type of two-impulse orbital transfer. In order to find a solution, the GA only requires the straightforward equations that define the ΔV needed to transfer through some arbitrary transfer orbit, instead of complicated gradient information.
The successful performance of the genetic algorithm for the application demonstrated here, suggests that the minimum fuel for orbital transfer problem may be explored further using the GA. It may be applied in more realistic cases, such as multi-impulsive maneuvers and orbital transfers between non-coplanar orbits.
REFERENCES
APPENDIX A
Properties of an Elliptical Orbit
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Figure 7.A1 Various properties of an elliptical orbit.
 | = | semimajor axis  |
 | = | semiminor axis |
 | = | eccentricity  |
 | = | ea |
 | = | true anomoly |
 | = | apoapsis radius  |
 | = | periapsis radius  |
 | = | semilatus rectum  |
 | = | flight-path angle |
 | = | prime focus — location of the central attracting body |
 | = | empty focus |
 | = | satellite velocity |
 | = | normal satellite velocity |
 | = | radial satellite velocity |
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