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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:Using a Genetic Algorithm to Determine the Optimum Two-Impulse Transfer Between Coplanar, Elliptical Orbits</TITLE>
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<H2><A NAME="Heading1"></A><FONT COLOR="#000077">Chapter 7<BR>Using a Genetic Algorithm to Determine the Optimum Two-Impulse Transfer Between Coplanar, Elliptical Orbits
</FONT></H2>
<P><I>Angela K. Reichert</I></P>
<P>Department of Aerospace Engineering Sciences<BR>University of Colorado<BR>Campus Box 431<BR>Boulder, CO 80303-0431<BR>e-mail: reichera@orbit.colorado.edu</P>
<P><FONT SIZE="+1"><B>ABSTRACT</B></FONT></P>
<P>Many years ago, when the idea of space exploration was beginning to become a reality for mankind, scientists working in the field of orbital mechanics began studying the motions and maneuvering requirements for an orbiting spacecraft. Because fuel is an expensive commodity, the minimization of the amount required to maneuver a spacecraft from one orbit to another demanded high priority. The amount of fuel needed to perform an orbit transfer is directly related to the spacecraft’s velocity change, or Δ<I>V</I>, as it maneuvers. Therefore, in order to minimize fuel usage, the Δ<I>V</I> must be minimized. Because this problem of determining the minimum Δ<I>V</I> cannot be easily solved in closed form, simplifications were usually made concerning the initial and final orbit properties. With sufficient simplifications, it became possible to solve the governing equations in order to obtain a solution.</P>
<P>In this chapter, a different approach to the minimum Δ<I>V</I> problem is proposed for the case of coplanar, elliptical orbits. Instead of trying to solve the governing equations directly, a genetic algorithm (GA) is utilized to find the optimum transfer trajectory required for minimum Δ<I>V</I>. With the geometry of the initial and final elliptical orbits given, the GA is employed as a “brute force” method of solving the problem, by first identifying a population of pseudo-random transfer orbits, specified by the three parameters required to define an ellipse in a given plane: the eccentricity, the semi-latus rectum, and the orientation angle of the ellipse from a given reference point. For each possible transfer orbit, the minimum total Δ<I>V</I> is calculated, provided that a total of two Δ<I>V</I> impulses are applied — the first to change the orbit from the initial to the transfer orbit, and the second to change from the transfer to the final orbit. In this chapter, the deciding factor in determining the optimum transfer orbit is the value of the total Δ<I>V</I> required to maneuver from the initial to the final orbit. Therefore, the equations for computing Δ<I>V</I> define the fitness function (the function to be optimized) by the GA.</P>
<P>In order to prove the validity of the solution found using a GA for a general case, the transfer orbits for three known test cases are calculated using the GA search method. These cases include (1) the transfer between coplanar circular orbits, or Hohmann transfer, (2) the transfer between coplanar, aligned ellipses, and (3) the transfer between identical, non-aligned elliptical orbits. Once the above cases have been verified using a GA, other general cases are solved. The GA used is a simple genetic algorithm written in the C programming language and is run on a UNIX operating system.</P>
<P><FONT SIZE="+1"><B>INTRODUCTION</B></FONT></P>
<P>Several types of transfers between coplanar, non-aligned elliptical orbits have been solved in the past. One of the first and most significant cases of optimal two-impulse transfer was studied in 1925 by Hohmann in the particular case of coplanar, circular orbits. This problem produces an intuitive solution of a transfer ellipse that is tangent to the initial and final orbits at its periapsis and apoapsis. This particular orbit transfer defines what is refered to as the Hohmann transfer [1].
</P>
<P>Since the discovery of the Hohmann transfer, many have investigated the case of optimum transfer (minimum total <IMG SRC="images/07-01i.jpg">) between elliptical orbits. However, the general equation for this problem is not easily solved for an optimum solution. Instead, the problem was usually simplified to yield an attainable solution. One who has derived these equations fully is Lawden [2]. After deriving them, he simplified the problem into two specific cases. The first is the case in which two elliptical orbits are aligned along their major axis and share a common focus. (Note that because all of the transfer ellipses to be considered in this chapter will always be orbiting the same central body, every ellipse to be discussed shares a common focus.) The solution to this case leads to a trajectory that is similar to the Hohmann transfer, where the transfer ellipse is tangent to the initial and final ellipses at periapsis and apoapsis. A second case is one in which the initial and final elliptical orbits are identical in size, but are non-aligned. This configuration results in the transfer ellipse being tangent to both the initial and final ellipse.</P>
<P>The Hohmann transfer and Lawden’s two cases are well defined and are accepted as being correct for coplanar orbits. Because they are straightforward and easily implemented, they will be used as test cases for the GA solution method described in this chapter.</P>
<P>The approach taken in this chapter to define the minimum two-impulse <IMG SRC="images/07-02i.jpg"> between coplanar, non-aligned elliptical orbits is done in a general manner. Since this problem is to be solved using a genetic algorithm, the search remains as general as the programmer wishes. Using a GA simply requires the fitness function and the parameters that are to be varied in order to find the best fitness function value. In the case of non-aligned, coplanar elliptical orbits, the variable parameters to be used are eccentricity (<I>e</I>), semi-latus rectum (<I>p</I>), and the orientation angle of the transfer ellipse (ω). Various properties of elliptical orbits are defined in Appendix A. The GA fitness function is related to the total ΔV; required to make a transfer between the initial and final orbits, using a transfer ellipse defined by the parameters <I>e</I>, <I>p</I> and ω.</P><P><BR></P>
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