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).

Figure 8.9  Geocentric equatorial coordinate system.

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].

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.

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).

Figure 8.10  Physical representation of the law of cosines at the second velocity change.

where

= velocity on the transfer orbit at intersection point

= velocity on the final orbit at intersection point

where

= the inclination of the initial orbit measured from the equatorial plane

= the inclination of the final orbit measured from the equatorial plane

Hence, the total ΔV is:

It should be noted that for those transfer orbits which intersect the final ellipse twice, two values for ΔV_TOT will exist. However, since the semi-major axes were assumed to be aligned, the magnitude of each ΔV_TOT is equal.

GENETIC ALGORITHM PARTICULARS

Fitness Function

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).

where

a weighting factor of 1500 (determined by numerical experimentation)

the radius of apogee for the final orbit

the distance the proposed transfer orbit misses the final orbit (Figure 8.11)

Figure 8.11  Diagram of how penalty function is assessed when the transfer orbit does not intersect the final orbit.

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.

Coding Scheme

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 (α₁). 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).

where

the accuracy of the parameter

for a binary representation

length of substring

maximum value the parameter can take

minimum value the parameter can take

The following table shows the accuracy chosen for each parameter and the substring length needed based on solving Equation 8.14.

The substring length for α₁ 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.

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: