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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:Using a Genetic Algorithm to Determine the Optimum Two-Impulse Transfer Between Coplanar, Elliptical Orbits</TITLE>
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<P><BR></P>
<P><FONT SIZE="+1"><B>RESULTS</B></FONT></P>
<P>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.
</P>
<P><FONT SIZE="+1"><B><I>Case 1</I></B></FONT></P>
<P>The first case to be attempted is a case with low eccentricity values. The orbit parameters used are
</P>
<P ALIGN="CENTER"><IMG SRC="images/07-16d.jpg"></P>
<P>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 <I>e</I>, 1 to 3 DU for <I>p</I>, and the range for ω remains 0 to 2π, the solution is slightly improved to the results shown in Figure 7.12.</P>
<P><A NAME="Fig11"></A><A HREF="javascript:displayWindow('images/07-11.jpg',400,341)"><IMG SRC="images/07-11t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-11.jpg',400,341)"><FONT COLOR="#000077"><B>Figure 7.11</B></FONT></A> Initial solution for Case 1.</P>
<P><A NAME="Fig12"></A><A HREF="javascript:displayWindow('images/07-12.jpg',400,361)"><IMG SRC="images/07-12t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-12.jpg',400,361)"><FONT COLOR="#000077"><B>Figure 7.12</B></FONT></A> Solution for Case 1 after the parameter ranges are narrowed.</P>
<P>In this case, reducing the ranges of parameters improved the required Δ<I>V</I> 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.</P>
<P><FONT SIZE="+1"><B><I>Case 2</I></B></FONT></P>
<P>The second case uses one ellipse that is highly elliptical and one that is slightly more circular. The parameters of these ellipses are
</P>
<P ALIGN="CENTER"><IMG SRC="images/07-17d.jpg"></P>
<P>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 <I>e</I>, 1 to 2 DU for <I>p</I>, and 0 to 1.047 radians for ω, the improved transfer ellipse is shown in Figure 7.14.</P>
<P><A NAME="Fig13"></A><A HREF="javascript:displayWindow('images/07-13.jpg',392,509)"><IMG SRC="images/07-13t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-13.jpg',392,509)"><FONT COLOR="#000077"><B>Figure 7.13</B></FONT></A> Initial solution for Case 2.</P>
<P><A NAME="Fig14"></A><A HREF="javascript:displayWindow('images/07-14.jpg',350,423)"><IMG SRC="images/07-14t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-14.jpg',350,423)"><FONT COLOR="#000077"><B>Figure 7.14</B></FONT></A> Solution for Case 2, after the parameter ranges have been narrowed.</P>
<P>For Case 2, reducing the parameter list results in a 22% reduction in the Δ<I>V</I> requirement. The reduction in Δ<I>V</I> 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.</P>
<P><FONT SIZE="+1"><B><I>Case 3</I></B></FONT></P>
<P>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
</P>
<P ALIGN="CENTER"><IMG SRC="images/07-18d.jpg"></P>
<P>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 <I>e</I>, 3 to 6 DU for <I>p</I>, and 0.578 to 2.047 radians for ω, the result improves, as shown in Figure 7.16.</P>
<P><A NAME="Fig15"></A><A HREF="javascript:displayWindow('images/07-15.jpg',340,322)"><IMG SRC="images/07-15t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-15.jpg',340,322)"><FONT COLOR="#000077"><B>Figure 7.15</B></FONT></A> Initial transfer ellipse solution for Case 3.</P>
<P>After reduction of the allowable parameter ranges, the resulting transfer ellipse is larger, more eccentric, and requires 5% less Δ<I>V</I> 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.</P>
<P><A NAME="Fig16"></A><A HREF="javascript:displayWindow('images/07-16.jpg',350,352)"><IMG SRC="images/07-16t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-16.jpg',350,352)"><FONT COLOR="#000077"><B>Figure 7.16</B></FONT></A> Solution for the transfer ellipse for Case 3, after the parameter ranges have been narrowed.</P>
<P><FONT SIZE="+1"><B>CONCLUSIONS</B></FONT></P>
<P>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.
</P>
<P>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.</P>
<P>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 Δ<I>V</I> needed to transfer through some arbitrary transfer orbit, instead of complicated gradient information.</P>
<P>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.</P>
<P><FONT SIZE="+1"><B>REFERENCES</B></FONT></P>
<DL>
<DD><B>1</B> Hohmann, W. (1960). <I>Die Ereichbarkeit der Himmelskorper</I> (The Attainability of Heavenly Bodies), NASA, Technical Translations F-44.
<DD><B>2</B> Lawden, D. S. (1962). Impulsive transfer between elliptical orbits. <I>Optimization Techniques</I>, edited by G. Leitmann, Academic, New York, Chapter 11.
<DD><B>3</B> Bate, R. R., Mueller, D. D., White, J, E. (1971). <I>Fundamentals of Astrodynamics</I>. New York, NY: Dover Publications, Inc.
<DD><B>4</B> Smith, R. E., Goldberg, D. E., Earickson, J. A. (1991). SGA-C: A C-language implementation of a simple genetic algorithm. <I>TCGA Report No. 91002</I>.
</DL>
<P><FONT SIZE="+1"><B>APPENDIX A</B></FONT></P>
<P><U>Properties of an Elliptical Orbit</U></P>
<P><A NAME="Fig17"></A><A HREF="javascript:displayWindow('images/07-17.jpg',600,344)"><IMG SRC="images/07-17t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-17.jpg',600,344)"><FONT COLOR="#000077"><B>Figure 7.A1</B></FONT></A> Various properties of an elliptical orbit.</P>
<TABLE WIDTH="100%">
<TR>
<TD WIDTH="5%"><IMG SRC="images/07-23i.jpg"><TD WIDTH="10%">=
<TD WIDTH="85%">semimajor axis <IMG SRC="images/07-24i.jpg"><TR>
<TD><IMG SRC="images/07-25i.jpg"><TD>=
<TD>semiminor axis
<TR>
<TD><IMG SRC="images/07-26i.jpg"><TD>=
<TD>eccentricity <IMG SRC="images/07-27i.jpg"><TR>
<TD><IMG SRC="images/07-28i.jpg"><TD>=
<TD>ea
<TR>
<TD><IMG SRC="images/07-29i.jpg"><TD>=
<TD>true anomoly
<TR>
<TD><IMG SRC="images/07-30i.jpg"><TD>=
<TD>apoapsis radius <IMG SRC="images/07-31i.jpg"><TR>
<TD><IMG SRC="images/07-32i.jpg"><TD>=
<TD>periapsis radius <IMG SRC="images/07-33i.jpg"><TR>
<TD><IMG SRC="images/07-34i.jpg"><TD>=
<TD>semilatus rectum <IMG SRC="images/07-35i.jpg"><TR>
<TD><IMG SRC="images/07-36i.jpg"><TD>=
<TD>flight-path angle
<TR>
<TD><IMG SRC="images/07-37i.jpg"><TD>=
<TD>prime focus — location of the central attracting body
<TR>
<TD><IMG SRC="images/07-38i.jpg"><TD>=
<TD>empty focus
<TR>
<TD><IMG SRC="images/07-39i.jpg"><TD>=
<TD>satellite velocity
<TR>
<TD><IMG SRC="images/07-40i.jpg"><TD>=
<TD>normal satellite velocity
<TR>
<TD><IMG SRC="images/07-41i.jpg"><TD>=
<TD>radial satellite velocity
</TABLE>
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