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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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<P><FONT SIZE="+1"><B><I>Case I</I></B></FONT></P>
<TABLE WIDTH="100%" BORDER>
<CAPTION ALIGN=LEFT><B>Table 8.5</B> Initial and final orbits parameters for two non-coplanar elliptical orbits.
<TR>
<TH WIDTH="30%">
<TH WIDTH="35%"><U>Initial Orbit</U>
<TH WIDTH="35%"><U>Final Orbit</U>
<TR>
<TD>Eccentricity
<TD ALIGN="CENTER">0.1
<TD ALIGN="CENTER">0.2
<TR>
<TD>Inclination
<TD ALIGN="CENTER">0°
<TD ALIGN="CENTER">10°
<TR>
<TD>Semi-major axis
<TD ALIGN="CENTER">6500 km
<TD ALIGN="CENTER">6800 km
</TABLE>
<P><A NAME="Fig16"></A><A HREF="javascript:displayWindow('images/08-16.jpg',640,347)"><IMG SRC="images/08-16t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-16.jpg',640,347)"><FONT COLOR="#000077"><B>Figure 8.16</B></FONT></A> Total <IMG SRC="images/08-28i.jpg"> Produced by the GA vs. number of generations.</P>
<P>When the GA is run on this case, the results shown in Table 8.6 were obtained. These results are also shown in Figure 8.17.
</P>
<TABLE WIDTH="100%" BORDER><CAPTION ALIGN=LEFT><B>Table 8.6</B> GA results for transfer orbit parameters for non-planar elliptical initial and final orbits. (GA population size=150, number of generations=15, probability of crossover=0.9, probability of mutation=0.001)
<TR>
<TH WIDTH="50%">
<TH WIDTH="50%"><U>GA Results</U>
<TR>
<TD>Eccentricity
<TD ALIGN="CENTER">0.1652
<TR>
<TD>α<SUB><SMALL>1</SMALL></SUB>
<TD ALIGN="CENTER">0.6222°
<TR>
<TD>Semi-major axis
<TD ALIGN="CENTER">7007.6 km
<TR>
<TD><IMG SRC="images/08-29i.jpg"><TD ALIGN="CENTER">2.6019 km/sec
</TABLE>
<P><A NAME="Fig17"></A><A HREF="javascript:displayWindow('images/08-17.jpg',640,223)"><IMG SRC="images/08-17t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-17.jpg',640,223)"><FONT COLOR="#000077"><B>Figure 8.17</B></FONT></A> Initial and final orbits along with GA-produced transfer orbit.</P>
<P><FONT SIZE="+1"><B><I>Case II</I></B></FONT></P>
<TABLE WIDTH="100%" BORDER>
<CAPTION ALIGN=LEFT><B>Table 8.7</B> Initial and final orbits parameters for two non-coplanar elliptical orbits.
<TR>
<TH WIDTH="30%">
<TH WIDTH="35%"><U>Initial Orbit</U>
<TH WIDTH="35%"><U>Final Orbit</U>
<TR>
<TD>Eccentricity
<TD ALIGN="CENTER">0.01
<TD ALIGN="CENTER">0.5
<TR>
<TD>Inclination
<TD ALIGN="CENTER">5°
<TD ALIGN="CENTER">12°
<TR>
<TD>Semi-major axis
<TD ALIGN="CENTER">6500 km
<TD ALIGN="CENTER">7500 km
</TABLE>
<P><A NAME="Fig18"></A><A HREF="javascript:displayWindow('images/08-18.jpg',500,266)"><IMG SRC="images/08-18t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-18.jpg',500,266)"><FONT COLOR="#000077"><B>Figure 8.18</B></FONT></A> Total ΔV produced by the GA vs. number of generations.</P>
<P>When the GA is run on this case, the following results (Table 8.8) (Figure 8.19), were obtained:
</P>
<TABLE WIDTH="100%" BORDER><CAPTION ALIGN=LEFT><B>Table 8.8</B> GA results for transfer orbit parameters for non-planar elliptical initial and final orbits, (GA population size=150, number of generations=15, probability of crossover=0.9, probability of mutation=0.001.)
<TR>
<TH WIDTH="50%">
<TH WIDTH="50%"><U>GA Results</U>
<TR>
<TD>Eccentricity
<TD ALIGN="CENTER">0.2717
<TR>
<TD>α<SUB><SMALL>1</SMALL></SUB>
<TD ALIGN="CENTER">4.2387°
<TR>
<TD>Semi-major axis
<TD ALIGN="CENTER">8835.6 km
<TR>
<TD><IMG SRC="images/08-30i.jpg"><TD ALIGN="CENTER">5.736 km/sec
</TABLE>
<P><A NAME="Fig19"></A><A HREF="javascript:displayWindow('images/08-19.jpg',640,226)"><IMG SRC="images/08-19t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-19.jpg',640,226)"><FONT COLOR="#000077"><B>Figure 8.19</B></FONT></A> Initial and final orbits along with GA-produced transfer orbit.</P>
<P><FONT SIZE="+1"><B><I>Case III</I></B></FONT></P>
<TABLE WIDTH="100%" BORDER>
<CAPTION ALIGN=LEFT><B>Table 8.9</B> Initial and final orbits parameters for two non-coplanar elliptical orbits.
<TR>
<TH WIDTH="30%">
<TH WIDTH="35%"><U>Initial Orbit</U>
<TH WIDTH="35%"><U>Final Orbit</U>
<TR>
<TD>Eccentricity
<TD ALIGN="CENTER">0.2
<TD ALIGN="CENTER">0.2
<TR>
<TD>Inclination
<TD ALIGN="CENTER">5°
<TD ALIGN="CENTER">5°
<TR>
<TD>Semi-major axis
<TD ALIGN="CENTER">8000 km
<TD ALIGN="CENTER">8500 km
</TABLE>
<P>When the GA is run on this case, it produced the following results (Table 8.10) (Figure 8.20):
</P>
<TABLE WIDTH="100%" BORDER><CAPTION ALIGN=LEFT><B>Table 8.10</B> GA results for transfer orbit parameters for non-planar elliptical initial and final orbits. (GA population size=150, number of generations=10, probability of crossover=0.9, probability of mutation=0.01.)
<TR>
<TH WIDTH="30%">
<TH WIDTH="35%"><U>Analytical Results</U>
<TH WIDTH="35%"><U>GA Results</U>
<TR>
<TD>Eccentricity
<TD ALIGN="CENTER">0.2289
<TD ALIGN="CENTER">0.241
<TR>
<TD>α<SUB><SMALL>1</SMALL></SUB>
<TD ALIGN="CENTER">0°
<TD ALIGN="CENTER">0°
<TR>
<TD>Semi-major axis
<TD ALIGN="CENTER">8300 km
<TD ALIGN="CENTER">8432 km
<TR>
<TD><IMG SRC="images/08-31i.jpg"><TD ALIGN="CENTER">0.5963 km/sec
<TD ALIGN="CENTER">0.6643 km/sec
</TABLE>
<P><A NAME="Fig20"></A><A HREF="javascript:displayWindow('images/08-20.jpg',500,309)"><IMG SRC="images/08-20t.jpg"></A>
<BR><A HREF="javascript:displayWindow('images/08-20.jpg',500,309)"><FONT COLOR="#000077"><B>Figure 8.20</B></FONT></A> (a) Initial and final orbits along with analytical transfer orbit (b) Initial and final orbits along with GA produced transfer orbit.</P>
<P><FONT SIZE="+1"><B>CONCLUSIONS</B></FONT></P>
<P>A genetic algorithm was used to search for transfer orbits which minimize the velocity change needed to transverse from one orbit to another. The effectiveness of using this approach was tested through comparison to two problems with known solutions. The GA produced near optimum results for the coplanar and non-coplanar Hohmann transfer problems. The GA was then used to search for transfer orbits between non-circular orbits which are inclined. While no analytical solutions exist for such problems, the resulting GA solution appeared quite reasonable. Therefore, it can be concluded that a genetic algorithm can be used successfully to find near optimum transfer orbits for ΔV<SUB><SMALL>TOT</SMALL></SUB> requirements.</P>
<P><FONT SIZE="+1"><B>REFERENCES</B></FONT></P>
<DL>
<DD><B>1</B> Chobotov, V. A. (1991). <I>Orbital mechanics</I>, Washington, DC: American Institute of Aeronautics and Astronautics, Inc.
<DD><B>2</B> Bender, D.F. (1962). Optimum coplanar two-impulse transfers between elliptic orbits. <I>Aerospace Engineering</I>, <B>21</B>(Oct.), 44-52.
<DD><B>3</B> Lawden, D. S. (1962). Impulsive transfer between elliptical orbits. <I>Optimization Techniques</I> (Chapter 11), edited by G. Leitmann, New York: Academic Press.
<DD><B>4</B> Baker, J. M. (1966). Orbit transfer and rendezvous maneuvers between inclined circular orbits. <I>Journal of Spacecraft and Rockets</I>, 3, 1216-1220.
<DD><B>5</B> Reichert, A. (1993). Optimum two-impulse transfer between coplanar, nonaligned elliptical orbits. Paper presented at the 1993 Southeast Regional AIAA Conference, Tuscaloosa, AL.
<DD><B>6</B> Pinon, E. III, & Fowler, W. T. (1995). Use of a genetic algorithm to generate earth to moon trajectories. AAS Paper Number 95-141.
<DD><B>7</B> Bate, R. R., Mueller, D. D., & White, J. E. (1971). <I>Fundamentals of astrodynamics</I>. New York: Dover Publications Inc.
<DD><B>8</B> Gerald, C. F., & Wheatley, P. O. (1989). <I>Applied numerical analysis, 4<SUP>th</SUP> Edition</I> (pp. 141-145). Reading, MA: Addison-Wesley Publishing Company.
<DD><B>9</B> Goldberg, D. E. (1989). <I>Genetic algorithms in search, optimization, and machine learning</I>. Reading, MA: Addison-Wesley Publishing Company.
</DL>
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