Evolutionary formulations for design of heterogeneous Earth observing constellations

The constellation and orbit design problem for Distributed Spacecraft Missions is challenging due to the extremely large space of alternatives, the presence of multiple conflicting objectives, and the expensive evaluation functions required to assess these objectives. Prior work in the literature has typically constrained this problem to simple constellation patterns that generally provide good coverage performance, such as the homogeneous Walker Delta pattern. However, this ignores potentially good architectures formed by heterogeneous constellations mixing satellites at different altitudes and inclinations. Thus, this paper studies methods to explore this region of the constellation design space which is understudied and has potential to be a cost-efficient way of satisfying regional coverage requirements and solving orbit design problems with disjoint areas of interest. We adopt a framework based on multi-objective evolutionary algorithms, which have been used extensively in the literature to solve similar problems due to their ability to deal with mixed-integer problems with highly non-linear and nonconvex objective functions such as the numerical simulations required to compute coverage performance for multi-satellite systems. Moreover, we focus on formulations (i.e., chromosomes and operators) rather than algorithmic details, since it is well known that formulations are at least as critical in driving performance as the details of the optimization algorithm used itself. Specifically, we explore two new formulations for heterogeneous constellation design: (1) A fixed-length chromosome containing $m$ 5-tuples, which encode a hybrid architecture with a fixed number of $m$ Walker constellations at different altitudes and inclinations, where each constellation is defined by a 5-tuple consisting of altitude, inclination, number of satellites, number of planes and relative spacing parameter. In this formulation, the number of satellites in each constellation can be set to 0 to effectively reduce the number of constellations. (2) A variable-length chromosome, which encodes a heterogeneous constellation defined by the total number of satellites n and $p$ 2-tuples, where each plane is defined by a 2-tuple consisting of its altitude and inclination and includes $n$ / $p$ satellites. All planes are assumed to be equally spaced in RAAN and satellites within a plane are equally spaced in mean anomaly. These two formulations are compared to a third formulation proposed earlier in the literature, which consists of a variable-length chromosome containing $n$ 4-tuples to encode a constellation of $n$ satellites, where each satellite is defined by a 4-tuple with its altitude, inclination, RAAN, and mean anomaly. The performance of these formulations is compared by solving a constellation design problem where the objectives are to optimize coverage performance and lifecycle cost. Search performance is assessed by looking at the evolution of hypervolume with the number of function evaluations. The formulations discussed in this paper were developed in the context of a NASA-funded project to develop a tool called Tradespace Analysis Tool for Constellations using Machine Learning (TAT-C ML). This tool is intended to be used during early stages of the design of Earth observation missions, and it is planned to be released open source within the next year.