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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><BR></P>
<P>The second step is to use a coordinate transformation to obtain a representation of each orbit in a common reference frame. The common coordinate system is the “geocentric equatorial coordinate system.” This system has its origin at the earth’s center. The fundamental plane is the earth’s equatorial plane and the positive x-axis points in the vernal equinox direction. The z-axis points in the direction of the north pole (Figure 8.9). The transformation [7] between the two coordinate systems is done by use of a rotation matrix, R (Equation 8.8).
</P>
<P><A NAME="Fig9"></A><A HREF="javascript:displayWindow('images/08-09.jpg',297,267)"><IMG SRC="images/08-09t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-09.jpg',297,267)"><FONT COLOR="#000077"><B>Figure 8.9</B></FONT></A> Geocentric equatorial coordinate system.</P>
<P ALIGN="CENTER"><IMG SRC="images/08-08d.jpg"></P>
<P>The point of intersection occurs when each coordinate in the geocentric equatorial coordinate system of the transfer equals that of the final orbit. This system of nonlinear equations can be solved for the variables, vt and vf, at the point of intersection using a Taylor Series approach [8].
</P>
<P>Once the location of intersection has been found, the radius to the intersection can be determined using the conic equation (Equation 8.7). The velocity on each orbit at the intersection point can then be found using Equation 8.9.</P>
<P ALIGN="CENTER"><IMG SRC="images/08-09d.jpg"></P>
<P>The velocity change required to go from the transfer orbit to the final orbit at the intersection location can then be calculated by applying the law of cosines (Figure 8.10, Equation 8.10).
</P>
<P><A NAME="Fig10"></A><A HREF="javascript:displayWindow('images/08-10.jpg',200,144)"><IMG SRC="images/08-10t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-10.jpg',200,144)"><FONT COLOR="#000077"><B>Figure 8.10</B></FONT></A> Physical representation of the law of cosines at the second velocity change.</P>
<P ALIGN="CENTER"><IMG SRC="images/08-10d.jpg"></P>
<P>where
</P>
<DL>
<DD><IMG SRC="images/08-15i.jpg"> = velocity on the transfer orbit at intersection point<DD><IMG SRC="images/08-16i.jpg"> = velocity on the final orbit at intersection point</DL>
<P ALIGN="CENTER"><IMG SRC="images/08-11d.jpg"></P>
<P>where
</P>
<DL>
<DD><IMG SRC="images/08-17i.jpg"> = the inclination of the initial orbit measured from the equatorial plane<DD><IMG SRC="images/08-18i.jpg"> = the inclination of the final orbit measured from the equatorial plane</DL>
<P>Hence, the total ΔV is:
</P>
<P ALIGN="CENTER"><IMG SRC="images/08-12d.jpg"></P>
<P>It should be noted that for those transfer orbits which intersect the final ellipse twice, two values for ΔV<SUB><SMALL>TOT</SMALL></SUB> will exist. However, since the semi-major axes were assumed to be aligned, the magnitude of each ΔV<SUB><SMALL>TOT</SMALL></SUB> is equal.</P>
<P><FONT SIZE="+1"><B>GENETIC ALGORITHM PARTICULARS</B></FONT></P>
<P><FONT SIZE="+1"><B><I>Fitness Function</I></B></FONT></P>
<P>Since the goal is to determine a transfer orbit which minimizes the total ΔV needed to go from an initial orbit to a final orbit, the fitness value is considered to be the value of Equation 8.12 plus a penalty for those orbits which do not cross the final orbit (Equation 8.13).
</P>
<P ALIGN="CENTER"><IMG SRC="images/08-13d.jpg"></P>
<P>where
</P>
<DL>
<DD><IMG SRC="images/08-19i.jpg"> a weighting factor of 1500 (determined by numerical experimentation)<DD><IMG SRC="images/08-20i.jpg"> the radius of apogee for the final orbit<DD><IMG SRC="images/08-21i.jpg"> the distance the proposed transfer orbit misses the final orbit (Figure 8.11)</DL>
<P><A NAME="Fig11"></A><A HREF="javascript:displayWindow('images/08-11.jpg',250,216)"><IMG SRC="images/08-11t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-11.jpg',250,216)"><FONT COLOR="#000077"><B>Figure 8.11</B></FONT></A> Diagram of how penalty function is assessed when the transfer orbit does not intersect the final orbit.</P>
<P>The penalty is added, instead of setting the fitness values to a large number, to ensure that any good genetic material which might exist in these solutions has a chance to survive to the next generation.
</P>
<P><FONT SIZE="+1"><B><I>Coding Scheme</I></B></FONT></P>
<P>As described above, the goal of this project is to demonstrate the utility of using a genetic algorithm to find a transfer orbit to minimize the total velocity change needed to go from an initial orbit to a final orbit. The transfer orbit will be defined by the eccentricity (e), the argument of periapsis (ω), the longitude of the ascending node (Ω), and the initial plane change (α<SUB><SMALL>1</SMALL></SUB>). The length of the binary string which represents these four parameters is found by specifying the accuracy of each parameter and finding the necessary substring length, and then concatenating (Equation 8.14).</P>
<P ALIGN="CENTER"><IMG SRC="images/08-14d.jpg"></P>
<P>where
</P>
<DL>
<DD><IMG SRC="images/08-22i.jpg"> the accuracy of the parameter<DD><IMG SRC="images/08-23i.jpg"> for a binary representation<DD><IMG SRC="images/08-24i.jpg"> length of substring<DD><IMG SRC="images/08-25i.jpg"> maximum value the parameter can take<DD><IMG SRC="images/08-26i.jpg"> minimum value the parameter can take</DL>
<P>The following table shows the accuracy chosen for each parameter and the substring length needed based on solving Equation 8.14.
</P>
<TABLE WIDTH="100%" BORDER><CAPTION ALIGN=LEFT><B>Table 8.1</B> Substring length for each parameter based on chosen accuracy and maximum and minimum values.
<TR>
<TH WIDTH="20%">Parameter
<TH WIDTH="20%">Accuracy
<TH WIDTH="15%">U<SUB><SMALL>min</SMALL></SUB>
<TH WIDTH="20%">U<SUB><SMALL>max</SMALL></SUB>
<TH WIDTH="25%">Substring Length
<TR>
<TD ALIGN="CENTER">e
<TD ALIGN="CENTER">0.001
<TD ALIGN="CENTER">0
<TD ALIGN="CENTER">1
<TD ALIGN="CENTER">10
<TR>
<TD ALIGN="CENTER">ω
<TD ALIGN="CENTER">0.001
<TD ALIGN="CENTER">0
<TD ALIGN="CENTER">2π radians
<TD ALIGN="CENTER">13
<TR>
<TD ALIGN="CENTER">Ω
<TD ALIGN="CENTER">0.001
<TD ALIGN="CENTER">0
<TD ALIGN="CENTER">2π radians
<TD ALIGN="CENTER">13
</TABLE>
<P>The substring length for α<SUB><SMALL>1</SMALL></SUB> is not easily calculated. It must range between the inclination of the initial and final orbits which vary from problem to problem. Therefore, a set length of fifteen was chosen for this parameter. If the substring lengths of each of the four parameters are concatenated, the total string length is 51.</P>
<P>The string is then arranged with the eccentricity and plane change parameter representations adjacent to one another. The eccentricity is important in defining the shape of the transfer orbit, and the plane change angle is significant in determining the amount of plane change which must be transversed at each velocity change. They are placed adjacent to one another to reduce the likelihood of destroying good combinations of these two parameters by crossover. Therefore, the binary string is set up as follows:</P>
<TABLE WIDTH="100%" BORDER><CAPTION ALIGN=LEFT><B>Table 8.2</B> Position of each parameter in string.
<TR>
<TH WIDTH="24%">Parameter
<TH WIDTH="19%">E
<TH WIDTH="19%">α<SUB><SMALL>1</SMALL></SUB>
<TH WIDTH="19%">ω
<TH WIDTH="19">Ω
<TR>
<TD ALIGN="CENTER">Position in string
<TD ALIGN="CENTER">1-10
<TD ALIGN="CENTER">11-25
<TD ALIGN="CENTER">26-38
<TD ALIGN="CENTER">39-51
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