Industrial Applications of Genetic Algorithms:Using a Genetic Algorithm to Determine the Optimum Two-Impulse Transfer Between Coplanar, Elliptical Orbits

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MOTIVATION FOR USING GENETIC ALGORITHMS

Since this problem is dependent on three unknown variables (eccentricity, semi-latus rectum, and angular orientation of the transfer orbit), the search space for the minimum ΔV is three-dimensional. The overriding concern in the selection of an algorithm for optimization is the nature of the fitness function. If the fitness function value is unimodal and easily differentiable, then a gradient method such as Newton’s method has significant advantage over others. But the fitness functions encountered in practice are rarely of the above nature. Instead they are discontinuous and jagged. While it is possible to design a gradient-based algorithm to suit a specific problem, there is little incentive for using such methods where simplicity and generality of implementation are needed.

The fitness function for this problem has a rather peculiar nature that prevents the use of gradient-based methods and their variants. Note that every point in three-space with 0 < e < 1 and positive p corresponds to an elliptical transfer orbit. Not all transfer orbits necessarily intersect the initial and final orbits, implying that, at these points, the fitness function cannot be defined. Although it is desirable that the fitness function be defined everywhere over the variation of the three parameters, in reality, it is defined only over some limited range of the three parameters. The absences of definition of the fitness function over certain ranges may be visualized as “holes” in a graphical representation of the fitness function. That is, if the fitness function were plotted as a surface in four dimensions, that surface would have holes at the points where the function is not defined. Since gradient-based methods use only local information (i.e., they seek an optimum in a neighborhood of the initial guess), it is difficult to use such methods to find the optimal transfer ellipse.

Genetic algorithms are immune to problems created by discontinuity and “holes in domain” such as this, because they use global information. Discontinuities and holes in domain generate “useless” ellipses that are overlooked by the genetic algorithm, while desirable cases — ellipses that intersect both initial and final orbits — are found. Unlike other methods, the search direction is determined from function evaluation at several randomly generated points. This “population approach” is unique to genetic algorithms and helps avoid the pitfalls of local methods.

FITNESS FUNCTION

The parameters that are required by the fitness function are the orbital parameters stated previously: e, p, and ω. Given these defining properties of an ellipse in a plane, the two ΔV impulses that would be required to make a transfer from the initial to final ellipse, using the specified transfer trajectory, are calculated. The total ΔV required defines the fitness function value and is the value to be minimized by the genetic algorithm. However, because a genetic algorithm maximizes by nature, the minimum total ΔV is calculated by specifying the fitness function in the form of the following equation:

This approach will allow the GA to find the minimum ΔV when the fitness function value is maximized.

Once the GA picks values for the three parameters (e, p, and ω), they are passed to the fitness function to be evaluated. Since this ellipse is to be the trajectory that the satellite is to follow while traveling from the initial orbit to the final, it must first be determined that this transfer ellipse intersects both the initial and final orbits. This is done by systematically calculating the location of the points of intersection between the initial and transfer ellipse, and the transfer and final ellipse. If an intersection point does not exist for one of these cases, an orbital transfer using this trajectory is impossible and the fitness function is given the lowest possible value of zero.

In order to determine the intersection points of two orbits, the number of possible intersection points must first be determined. It can be shown that two ellipses sharing a common focus may only have a maximum of two points of intersection. For example, one may say that given the orbit configuration displayed in Figure 7.1, a maximum of four intersection points are possible. However, the orbits illustrated are not possible because the ellipse in the vertical position has a semi-latus rectum that is less than the periapsis. When the trajectory equation [3],

is solved at the periapsis location, the radius at periapsis is given by:

By solving (7.3) for p, in terms of eccentricity and radius at periapsis,

it can be seen that the semi-latus rectum must always be greater than the radius at periapsis, because eccentricity must always be between 0 and 1 for an ellipse. Therefore, only a maximum of two points of intersection is possible between two ellipses sharing a common focus.

Figure 7.1 Incorrect representation of orbits.

Finding the exact location of the points of intersection is complicated by the fact that the equation for an ellipse is second order in both the x and y directions. To solve this problem, an incremental search is used in conjunction with a bisection root-finder to determine an accurate solution for the intersection points.

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