# Modeling Examples of Variable-Size Design Space Problems

The mathematical modeling of an engineering optimization problem is crucial for effective optimization. Modelling here means to mathematically express the objective and the constraint functions. These mathematical expressions depend on the physics of the problem and on how the problem is formulated. The problem formulation usually refers to the selection of the design variables and the objective function.

A design optimization problem may have multiple mathematical models that can be used. The model dictates the optimization methods that can solve the problem. The efficiency and/or the effectiveness of a solution method to a problem, in general, could be affected by the mathematical model being used. This chapter presents some case studies for engineering optimization problems, highlighting the thought process of formulating the problem and developing the mathematical models. Some of these case studies will be further addressed when discussing the system architecture optimization methods in subsequent chapters.

## Satellite Orbit Design Optimization

In Earth orbiting space missions, the orbit selection dictates the mission parameters such as the ground resolution, the area coverage, and the frequency of coverage parameters. Earths regions of interest are identified, and one way for a

Figure 2.1: Spacecraft at high altitudes have larger coverage on Earth as compared to those at lower altitudes. The spacecraft speed at higher altitudes, however, is less than those at lower altitudes.

spacecraft to cover these regions is to continuously maneuver between them. This method is expensive: it requires a propulsion system onboard the spacecraft, working throughout the mission lifetime. It also requires a longer time to cover all the regions of interest, due to the very weak thrust forces compared to that of the Earth’s gravitational field. A natural orbit, in which the regions of interest are visited depending only on the gravitational forces without the use of a propulsion system, would be more economic and would need a short time period for visiting all sites. Lower altitude orbits enable a spacecraft to provide higher resolution measurements, but lack wide coverage and orbital perturbations are non-negligible due to atmospheric drag.

The problem can be formulated as an optimization problem. A penalty function is constructed to quantify the error of a given orbit from the ideal case of visiting all the sites within a given time frame. This penalty function is minimized to find the orbit for a given set of sites and a given time frame.

*Problem Formulation*

Assume we have *n* sites to be visited. Each site is defined by its geodetic longitude and latitude. A* and respectively, where

_{k},

*к =*1 , ...,/t. The difference between geodetic and geocentric latitudes is usually very small, and is neglected in this analysis. The position vector for the

*k'*site in the Earth Centered Inertial

^{h}(ECI) frame is:

The only variable in the position vector of the Ath site is the time, r, at which this site will be visited.

A space orbit can be completely specified using five orbital parameters; these are: the semi-major axis *a,* the eccentricity *e,* the inclination i, the argument of perigee ft), and the right ascension of the ascending node *Cl.* The position of the spacecraft on this orbit is determined using one additional angle, called the true anomaly *(p.* The spacecraft position vector can be computed at any location on the orbit using the previous six orbit parameters as briefed here. The satellite position r" can be expressed in the perifocal coordinate system as:

where *p = h*^{2}* jp* is the orbit parameter, *h* is the specific angular momentum of the spacecraft, and *p* is a constant gravitational parameter. This vector is transformed to the ECI coordinate system through the transformation matrix, *R ^{,!}° (C_{x} =* cos* and S.v = sin*):

by

The time at which a ground site is visited (in Eq. (6.3)) is coupled with the spacecraft true anomaly (in Eq. (6.12)) through the Kepler equation.

where

Consider a satellite with an observing instrument (radar, camera, etc.) with an aperture of t9pov- Two possible objectives are considered here: one is to maximize the resolution, and one is to maximize the observation time. For the best resolution case, a candidate optimality criterion is to minimize a weighted sum of squares of the distances between each site and the satellite at the nearest ground track point. The objective function

will drive a solution orbit to pass as near as possible to each site and also have the best achievable resolution since the resolution is proportional to ||(r£ — r^{7}) ||. For the observation time, the objective function is then expressed as:

where *Tj _{k}* is the angle between

**r**and

**г***(see

**Fig.**2.2).

*H(x)*is the Heaviside unit step function [

*H{x) =*0 if

*x*< 0,

*H(x)*= 1 if

*x*> 0], and

*S*is the nadir angle, measured at the satellite from the nadir to the site.

_{k}The goal is to minimize the objective function *L* which is a function of a state vector (design variables to be optimized) whose elements are the orbital parameters a, *e*, (, *со, Q.* and all the visiting times *t _{k}.* A visiting time

*t*is the time of closest approach to the site

_{k}*k.*The minimizing state vector dictates the solution orbit. Reference [9] presents the detailed problem formulation for this problem as well as solution results when solving this problem. Reference [7] also presents the problem formulation for a more complex problem where the mathematical model takes into account the size of the field of view of the spacecraft sensor and the perturbation on the spacecraft motion due to the Earth oblateness.

**Figure **2.2: The angle *rj _{k}* as defined for the Л-th site.