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<META name=vstitle content="Industrial Applications of Genetic Algorithms">
<META name=vsauthor content="Charles Karr; L. Michael Freeman">
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<META name=vspubdate content="12/01/98">
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<TITLE>Industrial Applications of Genetic Algorithms:Optimized Non-Coplanar Orbital Transfers Using Genetic Algorithms</TITLE>
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<P>Once the string is defined, an initial population of strings is randomly generated. The fitness of each string in the population is found by evaluating Equation 8.13. Then, reproduction using tournament selection, two-point crossover, and mutation are used to generate subsequent generations and search for acceptable values for e, ω and Ω of the transfer orbit and α<SUB><SMALL>1</SMALL></SUB> which minimize Equation 8.13.</P>
<P>Tournament selection is executed by randomly picking two strings from the current population, and comparing their fitness values. The string with the lower fitness value is automatically put into the mating pool for the new population. Two-point crossover is accomplished by randomly picking two strings from the mating pool, then randomly picking two locations in the string length, based on a probability of crossover of 0.9, crossing the strings at the first location, and then crossing the strings at the second location. A mutation operator with a probability of 0.001 is used to introduce new genetic material into the population. If this operator is enacted on a bit in the string, it will be flipped. In other words, a “1” in a bit position is replaced with a “0,” and a “0” with a “1” [9]. Once a new population of strings is generated, the fitness values of each string are found. In this investigation, this procedure was used to generate ten generations of populations.</P>
<P><FONT SIZE="+1"><B>RESULTS</B></FONT></P>
<P>The ability of the GA to locate optimum orbital transfers can be tested by comparing its results to some well-known results. If the initial and final orbits are circular then the genetic algorithm should produce a transfer orbit equal to or near a Hohmann transfer, which is the minimum energy transfer between two coplanar circular orbits (Figure 8.12). Because the analytical solution to this situation is known, it was used to find a value for the weighting factor “k” in Equation 8.13 which produced a transfer near the known optimum.
</P>
<P><A NAME="Fig12"></A><A HREF="javascript:displayWindow('images/08-12.jpg',250,268)"><IMG SRC="images/08-12t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-12.jpg',250,268)"><FONT COLOR="#000077"><B>Figure 8.12</B></FONT></A> Hohmann transfer between two circular orbits.</P>
<P>For the first test case, two coplanar circular orbits were used. The initial orbit had a semi-major axis of 7,000 km and the final orbit had a semi-major axis of 7,500 km. When the genetic algorithm was run and compared to the known analytical solution, the following results were obtained (Table 8.3):
</P>
<TABLE WIDTH="100%" BORDER><CAPTION ALIGN=LEFT><B>Table 8.3</B> Comparison of GA to analytical solution for transfer orbit for co-planar circular initial and final orbits. (GA population size=80, number of generations=10, probability of crossover=0.9, probability of mutation=0.001).
<TR>
<TH WIDTH="30%">
<TH WIDTH="35%"><U>Analytical</U>
<TH WIDTH="35%"><U>GA</U>
<TR>
<TD>Eccentricity
<TD ALIGN="CENTER">0.03448
<TD ALIGN="CENTER">0.03519
<TR>
<TD>α<SUB><SMALL>1</SMALL></SUB>
<TD ALIGN="CENTER">0 deg
<TD ALIGN="CENTER">0 deg
<TR>
<TD>ω<SUB><SMALL>t</SMALL></SUB>
<TD ALIGN="CENTER">0 deg
<TD ALIGN="CENTER">23.6 deg
<TR>
<TD>Ω<SUB><SMALL>t</SMALL></SUB>
<TD ALIGN="CENTER">0 deg
<TD ALIGN="CENTER">243 deg
<TR>
<TD>ΔV<SUB><SMALL>TOT</SMALL></SUB>
<TD ALIGN="CENTER">0.2558 km/sec
<TD ALIGN="CENTER">0.2661 km/sec
</TABLE>
<P>The transfer orbit resulting from the GA is slightly more elliptical than the optimum Hohmann transfer orbit (Figure 8.13). However, genetic algorithms do not promise to locate optimum solutions, only near optimum solutions. If this is considered, the GA performed quite well in this situation, as indicated in the fact that the ΔV it determined was only 4.02% larger than the optimum value.
</P>
<P><A NAME="Fig13"></A><A HREF="javascript:displayWindow('images/08-13.jpg',450,224)"><IMG SRC="images/08-13t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-13.jpg',450,224)"><FONT COLOR="#000077"><B>Figure 8.13</B></FONT></A> (a) Optimum transfer orbit (Hohmann transfer), (b) GA produced transfer orbit.</P>
<P>The next test case is a non-coplanar Hohmann transfer. If the inclination between the initial and final orbits is set to zero, the total ΔV needed can found using Equations 8.7, 8.9 and 8.12. For transfers between circular orbits where the inclination between them is not zero, the accepted solution method is to solve Equation 8.15 which is derived by applying the law of cosines at the time of ΔV<SUB><SMALL>1</SMALL></SUB> and ΔV<SUB><SMALL>2</SMALL></SUB>, iteratively for the optimum α<SUB><SMALL>1</SMALL></SUB>.</P>
<P ALIGN="CENTER"><IMG SRC="images/08-15d.jpg"></P>
<P>If the same circular orbits are used again, however, with an initial inclination of 5 deg and a final inclination of 10 deg, the following comparison results are obtained (Table 8.4):
</P>
<TABLE WIDTH="100%" BORDER><CAPTION ALIGN=LEFT><B>Table 8.4</B> Comparison of GA to analytical solution for transfer orbit for non-planar circular initial and final orbits. (GA population size=80, number of generations=10, probability of crossover=0.9, probability of mutation=0.001.)
<TR>
<TH WIDTH="30%">
<TH WIDTH="35%"><U>Analytical</U>
<TH WIDTH="35%"><U>GA</U>
<TR>
<TD>Eccentricity
<TD ALIGN="CENTER">0.03448
<TD ALIGN="CENTER">0.03499
<TR>
<TD>α<SUB><SMALL>1</SMALL></SUB>
<TD ALIGN="CENTER">0.4869 deg
<TD ALIGN="CENTER">0.4890 deg
<TR>
<TD><IMG SRC="images/08-27i.jpg"><TD ALIGN="CENTER">1.349 km/sec
<TD ALIGN="CENTER">1.351 km/sec
</TABLE>
<P>The transfer orbit produced by the GA is slightly more elliptical than the analytical optimum (Figure 8.14). However, the ΔV is only 0.148% greater than the optimum value.
</P>
<P><A NAME="Fig14"></A><A HREF="javascript:displayWindow('images/08-14.jpg',500,240)"><IMG SRC="images/08-14t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-14.jpg',500,240)"><FONT COLOR="#000077"><B>Figure 8.14</B></FONT></A> (a) Optimum transfer orbit, (b) GA produced transfer orbit for non-coplanar circular initial and final orbits.</P>
<P>The GA was further tested with the non-coplanar Hohmann case by applying it to a range of final orbit inclinations. Circular initial and final orbits were used with respective semi-major axes of 7,500 km and 8,000 km. The final orbit inclination was varied between 0° and 90°. The total ΔV’s obtained from the GA produced orbits were then plotted with the analytical solutions (Figure 8.15). The GA produced results were comparable to the analytical for each inclination.
</P>
<P><A NAME="Fig15"></A><A HREF="javascript:displayWindow('images/08-15.jpg',500,282)"><IMG SRC="images/08-15t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-15.jpg',500,282)"><FONT COLOR="#000077"><B>Figure 8.15</B></FONT></A> Total ΔV produced by numerical and GA methods versus final orbit inclination for a non-coplanar Hohmann transfer (GA population size = 150, number of generations = 10, probability of crossover = 0.9, probability of mutation = 0.001).</P>
<P>Once of the GA was validated by comparison to analytical solutions, more complex problems were addressed; problems which do not have known analytical solutions. The following cases were used to apply the GA to non-coplanar elliptical orbits:
</P><P><BR></P>
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