TEST CASES

The GA is implemented with the equations shown in the previous section. The specific GA used was developed by Smith, Goldberg, and Earickson [4]. It is a simple genetic algorithm written in the C programming language, and is run on a UNIX operating system. All of these cases in this chapter use populations of 200 and 50 generations. Further details of this particular GA code can be found in [4].

In order to verify the validity of using a GA to solve a problem of this type, several test cases are run which have known solutions. These test cases are the Hohmann transfer [1] and the two cases defined in Lawden [2] - aligned elliptical orbits and non-aligned elliptical orbits of the same size. All three cases are described in more detail below.

Hohmann Transfer

The first test case that is evaluated using the GA is the Hohmann transfer. This transfer is composed of a circular (e = 0) initial and final orbit. The specific parameters of the initial and final ellipses are

where DU refers to the canonical distance unit. Since the GA must be given the set of possible values to try for the parameters, a range for each parameter is specified. For this case, the possible values allowed for the parameters, e, p, and ω, are 0 to 0.99, 0 to 10 DU, and 0 to 360°, respectively. The resulting transfer orbit is shown in Figure 7.5. The initial, transfer and final orbits are denoted by an i, tr, and , respectively. As can be seen in Figure 7.5, the transfer ellipse is not exactly tangent to the initial and final ellipses as the Hohmann transfer requires. This illustrates the pseudo-random, “brute force” optimization associated with the GA search. Although this solution is not the known best, it is remarkably accurate for being found with such relative ease.

If a more accurate solution is desired for a problem such as this, the GA can be modified slightly to facilitate this. If the range of the best solution can be identified to some “ball park” range, the maximum and minimum values for a parameter can be modified to be this narrower range. By narrowing the range of a parameter, the search space is decreased to include only the most useful parameter values. If this is done successfully, the GA will have an easier task of determining the optimum values because it will have fewer “useless” values to process.

Figure 7.5  Result for the initial computation of the transfer orbit for circular initial and final orbits.

The narrowing of the parameter range is done for each of the test cases to check the results found using the GA. For the Hohmann transfer case, the ranges of e and p are narrowed to 0.2 to 0.5 and 1 to 2 DU, respectively. The range of ω is reduced drastically to 0 to 0.1 radians. This large reduction in range is justified because ω for the Hohmann transfer case carries no meaning, due to the fact that the initial and final orbits are circular. After these parameter ranges are narrowed, the resulting transfer orbit is shown in Figure 7.6.

Figure 7.6  Transfer orbit result for circular initial and final orbits, after narrowing parameters.

This figure displays, almost exactly, the Hohmann transfer solution. Between the solution shown in Figure 7.5 and the one shown in Figure 7.6, the V requirements are reduced by 34% as a result of reducing the list of possible parameters to be used by the GA.

Transfer between Aligned Ellipses

The second case is one described in Lawden [2]. For this test case, the parameters specifying the initial and final ellipses are

For the first implementation of the GA, the parameter ranges are 0 to 0.99 for eccentricity, 0 to 10 DU for the semi-latus rectum, and 0 to 2π for the orientation angle. After the initial run of the genetic algorithm, the optimum orbit is shown in Figure 7.7. Then, the parameter ranges are narrowed to 0.5 to 0.99 for e, 1 to 2 DU for p, and 0 to 0.1 radians for ω, in order to obtain a more accurate solution. Figure 7.8 displays the transfer ellipse obtained from the reduction of the parameter ranges.

Figure 7.7  Initial solution for aligned orbits.

Figure 7.8  Transfer orbit solution for aligned elliptical orbits, after the parameter ranges have been reduced.

For this case, the optimum transfer as described in Lawden is found. By narrowing the parameter ranges, the required is reduced by 37%. This reduction of ΔV is expected because Figure 7.8 shows a transfer orbit that is more tangential to the initial and final orbits and is more elliptical in nature.

Non-Aligned, Identical Elliptical Orbits

The third test case is to find the transfer orbit between two non-aligned elliptical orbits of the same size. The parameters specifying the initial and final ellipses of this case are

After the first pass through the genetic algorithm, with the initial parameter ranges once again set to 0 to 0.99 for eccentricity, 0 to 10 DU for the semi-latus rectum, and 0 to 2π for the orientation angle, the transfer orbit solution is shown in Figure 7.9. After the parameter ranges are narrowed to 0.4 to 0.99 for e, 1 to 3 DU for p, and 0 to 1.047 radians for ο, the solution is shown in Figure 7.10.

Figure 7.9  Initial solution for the case of non-aligned elliptical orbits of the same size.

Figure 7.10  Solution for the transfer between two non-aligned elliptical orbits of the same size, after the parameter ranges have been reduced.

Here, once again, the resulting transfer orbit is very nearly tangential to both the initial and final orbits. The narrowing of the possible parameters to be used by the GA, reduced the required ΔV by 8%.

Based on the results displayed for the three test cases, the GA successfully identified the known solutions for each of the case and did this with relative ease. Now, the stage is set to find the optimum for cases in which the solution has not previously been proven.