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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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<TITLE>Industrial Applications of Genetic Algorithms:Optimized Non-Coplanar Orbital Transfers Using Genetic Algorithms</TITLE>
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<H2><A NAME="Heading1"></A><FONT COLOR="#000077">Chapter 8<BR>Optimized Non-Coplanar Orbital Transfers Using Genetic Algorithms
</FONT></H2>
<P><I>Dedra C. Moore</I></P>
<P>KBM Enterprises, Inc.<BR>Huntsville, AL 35814<BR>e-mail: dmoore@kbm-inc.com</P>
<P><FONT SIZE="+1"><B>ABSTRACT</B></FONT></P>
<P>The problem of optimizing the transfer orbit between an initial orbit and a desired final orbit is not easily solved. The governing equations are difficult to solve in closed form. Additionally, the problem is even more difficult when the initial and final orbits are not coplanar. This report examines the use of a genetic algorithm (GA) to find the transfer orbit and plane change that minimizes the total velocity change (ΔV<SUB><SMALL>TOT</SMALL></SUB>) needed. Cases where the initial and final orbits are both circular and elliptical are investigated. Accuracy of the GA is verified by both coplanar and non-coplanar Hohmann transfers between circular orbits, for which analytical solutions exist.</P>
<P><FONT SIZE="+1"><B>LIST OF SYMBOLS</B></FONT></P>
<TABLE WIDTH="100%">
<TR>
<TD WIDTH="10%"><IMG SRC="images/08-01i.jpg"><TD WIDTH="90%">Total velocity change
<TR>
<TD><IMG SRC="images/08-02i.jpg"><TD>Velocity change
<TR>
<TD><IMG SRC="images/08-03i.jpg"><TD>Eccentricity of transfer orbit
<TR>
<TD><IMG SRC="images/08-04i.jpg"><TD>Eccentricity
<TR>
<TD><IMG SRC="images/08-05i.jpg"><TD>Argument of perigee
<TR>
<TD><IMG SRC="images/08-06i.jpg"><TD>Semi-major axis
<TR>
<TD><IMG SRC="images/08-07i.jpg"><TD>Inclination measured from the equatorial plane
<TR>
<TD><IMG SRC="images/08-08i.jpg"><TD>Longitude of the ascending node
<TR>
<TD><IMG SRC="images/08-09i.jpg"><TD>Initial plane change done in conjunction with the first velocity change
<TR>
<TD><IMG SRC="images/08-10i.jpg"><TD>True anomaly angle
<TR>
<TD><IMG SRC="images/08-11i.jpg"><TD>Perigee radius
<TR>
<TD><IMG SRC="images/08-12i.jpg"><TD>Velocity at perigee
</TABLE>
<P><FONT SIZE="+1"><B>INTRODUCTION AND BACKGROUND INFORMATION</B></FONT></P>
<P>In many instances, objects orbiting the earth are not initially in their desired orbits. For example, satellites placed in a low earth orbit by the space shuttle may require a higher altitude to provide maximum efficiency. Satellites also often drift over time from their original paths due to orbital perturbations. In both cases, an orbital transfer must take place to correct the problem. For many years, scientists who work in the field of orbital mechanics have been interested in finding the minimum velocity change to get from one orbit to another. Generally, this concern stems from the need to limit the weight and amount of fuel needed on board. However, in closed form, the governing equations which optimize the transfer are difficult to solve. Typically, assumptions have been made to elucidate the solution process.
</P>
<P>The best known and simplest transfers was developed by Hohmann [1]. He examined coplanar circular orbits and determined that the optimum transfer orbit was an elliptical orbit whose perigee point lay on the initial orbit and apogee point lay on the destination orbit. The maneuver is executed by applying a tangential ΔV at the departure point on the initial orbit and another tangential ΔV to circularize into the final orbit (Figure 8.1). This procedure can be solved analytically. Hohmann also developed a procedure for finding an optimum transfer for circular non-coplanar orbits. This maneuver is similar to the coplanar transfer in that the transfer orbit perigee and apogee points lie on the initial and final orbits, and velocity changes are performed at these locations. However, a plane change must also be incorporated into one or both velocity changes. It is the optimization of the velocity change that makes this problem difficult to solve analytically. The governing equation cannot be written in terms of the plane change angle; it must be solved iteratively.</P>
<P><A NAME="Fig1"></A><A HREF="javascript:displayWindow('images/08-01.jpg',250,273)"><IMG SRC="images/08-01t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-01.jpg',250,273)"><FONT COLOR="#000077"><B>Figure 8.1</B></FONT></A> Classic Hohmann transfer.</P>
<P>In 1962, Bender [2] published a paper which expanded Hohmann’s work to analyze elliptical orbits. Coplanar orbits were still assumed, however his work included the possibility of elliptic initial and final orbits. This study was restricted to two classes of transfer orbits — 180° transfer and cotangential transfer. The 180° transfer was defined as a transfer where the arrival and departure points are separated by 180 degrees and not necessarily at the apses of either orbit (Figure 8.2).
</P>
<P><A NAME="Fig2"></A><A HREF="javascript:displayWindow('images/08-02.jpg',250,173)"><IMG SRC="images/08-02t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-02.jpg',250,173)"><FONT COLOR="#000077"><B>Figure 8.2</B></FONT></A> 180° transfer case.</P>
<P>For the cotangential transfer case, the initial and final orbits were assumed to be coplanar (Figure 8.3). For each of these orbit types, once the departure point is defined, the transfer orbit is fixed. Therefore, each case was optimized by iteratively varying the location of the first ΔV or departure point.
</P>
<P><A NAME="Fig3"></A><A HREF="javascript:displayWindow('images/08-03.jpg',350,188)"><IMG SRC="images/08-03t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-03.jpg',350,188)"><FONT COLOR="#000077"><B>Figure 8.3</B></FONT></A> Cotangential transfer case.</P>
<P>Lawden [3] studied coplanar cases similar to the work of Bender, but also examined two cases with elliptic initial and final orbits. The first case involved elliptic orbits aligned along their major axis. He concluded that the optimum transfer is tangential at its apses to the initial and final orbits, and its axis is aligned with the other two axes. In this situation, the optimum transfer orbit is defined based on the apses location of the two given orbits and requires no iteration. The second case examined by Lawden involved initial and final orbits of identical size, but with skewed major axes (Figure 8.4). This is analogous to the cotangential case examined by Bender, where the transfer orbit is tangential to both given orbits and optimized by iteratively picking a departure point.
</P>
<P><A NAME="Fig4"></A><A HREF="javascript:displayWindow('images/08-04.jpg',350,268)"><IMG SRC="images/08-04t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-04.jpg',350,268)"><FONT COLOR="#000077"><B>Figure 8.4</B></FONT></A> Transfer between equal ellipses.</P>
<P>In 1966, Baker [4] examined the possibility of executing three-dimensional orbit transfers with a bi-elliptic transfer. This maneuver involved three velocity changes. The first and third were performed at the departure of the initial orbit and the arrival at the final orbit. The second is a velocity change at the apogee point on the initial transfer orbit onto another intermediate transfer orbit. Baker’s study was limited to both circular initial and final orbits and examined optimally, splitting the plane change required between the three ΔV’s. The governing equation which defines the total velocity change needed is written in terms of each of the three plane changes and cannot be solved for either angle. Here again, an iterative method is required to find the three angles.
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