Reichert [5] used a GA to minimize the total ΔV in three coplanar orbit transfer cases. The first was the general transfer between circular orbits studied by Hohmann in 1925, which can typically be solved by hand. The second and third were the two cases studied by Lawden. The classical solution to these cases was used to verify the accuracy of the genetic algorithm.

In 1995, Pinon and Fowler [6] used a GA to generate two-dimensional lunar trajectories. The goal was to analyze orbital transfers from the earth to the moon. Their results were comparable to the trajectories used by the Apollo missions. They concluded that the two-dimensional trajectories produced by the GA could be used as a starting point for three-dimensional transfers.

The above-mentioned studies were simplified to have either coplanar or circular initial and final orbits or both. The purpose of the current effort is to examine the use of a GA in non-coplanar orbit transfers. A GA is effective in finding near optimum solutions, but does not guarantee to find the optimum solution. Because of the GA’s capability, test cases will include both circular and elliptical problems. This situation is encountered more often in orbital maneuvers. However, this problem, like the coplanar transfer, is not easily solved. In this case, not only must an optimum transfer orbit be found, but also the amount of plane change at each ΔV.

PROBLEM STATEMENT

In this chapter, orbital perturbations will be ignored, impulsive maneuvers will be assumed, and test cases will be limited to bodies orbiting the earth. In a real world analysis, these issues would need to be addressed. However, neglecting them will be sufficient for this study. This investigation will limit the number of velocity changes to two, the first when leaving the initial orbit and the second at the intersection of the transfer orbit and the final orbit. It will also be assumed that the semi-major axes of the initial, final and transfer orbits are aligned with one another.

The analysis will begin by user specification of the initial and final orbits, which will be limited to elliptical or circular. Parameters used to define an orbit will be the eccentricity (e), the argument of perigee (ω), the semi-major axis (a), the inclination with respect to the equatorial plane (i), and the longitude of the ascending node (Ω). The GA will be used to search for defining parameters for the transfer orbit and an initial plane change (α₁) that minimizes the velocity change needed.

This study assumes the first velocity change will occur at the perigee point of the initial orbit and of the transfer orbit. Therefore, the perigee radius of the initial and transfer orbits will correspond. Since the perigee radius of the transfer orbit is defined by the initial orbit, the semi-major axis can be calculated (Equation 8.1) and need not be one of the search parameters for the GA.

where

The perigee velocity on both the initial and transfer orbits can then be calculated using Equation 8.3.

where

The velocity change required to go from the initial orbit to the transfer orbit (ΔV₁) can be found by applying the law of cosines (Figure 8.5, Equation 8.5).

Figure 8.5  Physical representation of the law of cosines at the first velocity change.

where

= perigee velocity on the transfer orbit

= perigee velocity on the initial orbit

In order for the transfer orbit to be sufficient, it must intersect the final orbit (Figure 8.6). The next step will be to determine the intersection location. The initial, random population will undoubtedly contain transfer orbits that do not intersect the final orbit and are not possible solutions. In these situations, the objective function will be penalized according to how badly the transfer orbit misses the final orbit (details will be discussed later).

Figure 8.6  (a) transfer orbit does not intersect final orbit, (b) transfer orbit intersects final orbit at two locations, (c) transfer orbit intersects final orbit at one location.

The process of finding the intersection point or points of two non-coplanar elliptical orbits is not a trivial task. The point will be defined by the following two values: a true anomaly angle on the transfer orbit, v_t, and a true anomaly angle on the final orbit, v_f (Figure 8.7).

Figure 8.7  Representation of intersection points.

The first step is to represent the radius vector to the intersection point on both the transfer orbit and the final orbit in a “perifocal coordinate system” (Figure 8.8). The perifocal coordinate system is defined as having a fundamental plane in the plane of the orbit. Positive x points toward perigee of the orbit and positive y is rotated 90° in the direction of the orbital motion. This is one of the most convenient coordinate systems for describing an orbit because the z component is always zero (Equation 8.6).

Figure 8.8  Perifocal coordinate system.

where