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<META name=vsisbn content="0849398010">
<META name=vstitle content="Industrial Applications of Genetic Algorithms">
<META name=vsauthor content="Charles Karr; L. Michael Freeman">
<META name=vsimprint content="CRC Press">
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<META name=vspubdate content="12/01/98">
<META name=vscategory content="Web and Software Development: Artificial Intelligence: Other">
<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>Once the intersection points between the initial and transfer ellipse are found, the points are converted from <I>x</I> and <I>y</I> rectangular coordinates to radius, <I>r</I>, and true anomaly, θ. Since the origin of the rectangular coordinate frame is at the focus, the radius of the intersection point is given by</P>
<P ALIGN="CENTER"><IMG SRC="images/07-05d.jpg"></P>
<P>Because the initial and transfer ellipses share a common focus, this radius will be the same for each ellipse. The procedure for finding true anomaly is slightly more involved. First, the direction of the satellite is assumed to be orbiting in a clockwise direction about the focus. From the geometry shown in Figure 7.2, it is seen that the true anomaly (θ, in radians) is given by the following equation:
</P>
<P ALIGN="CENTER"><IMG SRC="images/07-06d.jpg"></P>
<P>where α is a reference angle from an arbitrary x-axis and is given by
</P>
<P><A NAME="Fig2"></A><A HREF="javascript:displayWindow('images/07-02.jpg',320,245)"><IMG SRC="images/07-02t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-02.jpg',320,245)"><FONT COLOR="#000077"><B>Figure 7.2</B></FONT></A> Representation of the true anomaly for a given point of intersection.</P>
<P>Once an intersection point of the two elliptical orbits is found in terms of <I>r</I> and θ, the velocities at both the initial <IMG SRC="images/07-04i.jpg"> and transfer ellipse (<IMG SRC="images/07-05i.jpg">) are found at this point using</P>
<P ALIGN="CENTER"><IMG SRC="images/07-07d.jpg"></P>
<P>The flight path angle, <IMG SRC="images/07-06i.jpg">, for a given velocity is determined by</P>
<P ALIGN="CENTER"><IMG SRC="images/07-08d.jpg"></P>
<P>where the normal velocity is given by the equation,
</P>
<P ALIGN="CENTER"><IMG SRC="images/07-09d.jpg"></P>
<P>The flight-path angle is determined for both the initial and transfer orbits. Since clockwise orbital motion is assumed, γ will be positive for 0° < θ < 180 ° and negative for 180° < θ < 360°.
</P>
<P>The Δ<I>V</I> required for transfer at a particular intersection point is found using the law of cosines for the velocities at that point on the initial and transfer ellipses and the angle between these two velocities. This angle is represented in Figure 7.3. As can be seen, the angle between <IMG SRC="images/07-07i.jpg"> and <IMG SRC="images/07-08i.jpg"> is <IMG SRC="images/07-09i.jpg"> where <IMG SRC="images/07-10i.jpg"> and <IMG SRC="images/07-11i.jpg"> have opposite signs. The case where <IMG SRC="images/07-12i.jpg"> and <IMG SRC="images/07-13i.jpg"> have the same sign is shown in Figure 7.4. This figure illustrates that when <IMG SRC="images/07-14i.jpg"> and <IMG SRC="images/07-15i.jpg"> have the same sign, the angle between <IMG SRC="images/07-16i.jpg"> and <IMG SRC="images/07-17i.jpg"> is once again the difference of the two flight path angles. With this mind, the change in velocity required to maneuver from the initial orbit to the transfer orbit is</P>
<P ALIGN="CENTER"><IMG SRC="images/07-10d.jpg"></P>
<P>Because
</P>
<P ALIGN="CENTER"><IMG SRC="images/07-11d.jpg"></P>
<P>the difference of the two flight path angles may be specified in either order, <IMG SRC="images/07-18i.jpg"> or <IMG SRC="images/07-19i.jpg">, and the result will be the same.</P>
<P><A NAME="Fig3"></A><A HREF="javascript:displayWindow('images/07-03.jpg',300,327)"><IMG SRC="images/07-03t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-03.jpg',300,327)"><FONT COLOR="#000077"><B>Figure 7.3</B></FONT></A> Velocities and flight path angles for two orbits at an intersection point.</P>
<P><A NAME="Fig4"></A><A HREF="javascript:displayWindow('images/07-04.jpg',220,298)"><IMG SRC="images/07-04t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/07-04.jpg',220,298)"><FONT COLOR="#000077"><B>Figure 7.4</B></FONT></A> Flight path angles with the same sign.</P>
<P>The above process explains how to determine the Δ<I>V</I> required to maneuver a spacecraft from the initial to the transfer orbit. The process of determining the required Δ<I>V</I> must be performed for each intersection point of the initial and transfer orbit, and for each intersection point of the transfer and final orbit. Then, the minimum Δ<I>V</I> required to go from the initial to the transfer orbit is added to the minimum Δ<I>V</I> required to go from the transfer to the final orbit, resulting in the minimum total Δ<I>V</I> required to transfer from the initial to the final orbit. The fitness function is then found using a modification of equation (7.1), shown below in equation (7.13). For the fitness function, a constant value of 10 TU/DU is used because Δ<I>V</I> is usually of the order of 0 to 5DU/TU. (The canonical units of distance units (<I>DU</I>) and time units (<I>TU</I>) are used throughout this chapter). Making the constant value well above the expected range is important to avoid any negative fitness function values that may arise. Finally, the fitness function is multiplied by a factor (1,000) to give the GA a more substantial number to evaluate. The fitness function is</P>
<P ALIGN="CENTER"><IMG SRC="images/07-12d.jpg"></P>
<P>The GA calculates this fitness function for many populations in order to systematically search for the parameters of <I>e</I>, <I>p</I>, and ω that lead to the optimum fitness function value, thus minimizing total Δ<I>V</I>.</P><P><BR></P>
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