David B. Spencer

Optimal Spacecraft Rendezvous Using Genetic Algorithms

The optimal rendezvous of two spacecraft are examined using a genetic algorithm. The minimum fuel solution of the optimal rendezvous contains many local optima, as well as discontinuities in the solution. These local optima and discontinuities make locating a global optimal solution difficult. Genetic algorithms are effective in solving these kinds of problems. The goal is to find the thrust time history that includes the magnitude and direction of the velocity change and the burn position, such that the boundary conditions are satisfied to an acceptable level and in a reasonable time. In addition, the number of thrust arcs and the maximum magnitude of the velocity change are regulated. This method was used on three test cases: 1) the Hohmann transfer, 2) the bielliptic transfer, and 3) rendezvous with two impulses. The results of the Hohmann and the bielliptic transfers match the analytical solutions within the resolution of the variables of the genetic algorithm. Although the result from the rendezvous with two impulses is not exact, the configuration of the trajectory is similar to the analytical solution.

Optimal, Low-Thrust Earth-Orbit Transfers Using Higher-Order Collocation Methods

A trajectory optimization technique based upon higher-order collocation is used to solve optimal, low-thrust, Earth-orbit transfer problems. The optimal control problem solved is defined, and the solution method for solving this problem is described. For several example cases analyzed, a spacecraft is transferred from low Earth orbit to a variety of final mission orbits. A range of thrust accelerations from approximately 1 to 10 - 3 g was used. A comparison is made between the optimal transfers found in this work and the transfers found by using analytical blended control methods. Finally, conclusions drawn from this work are discussed.

Satellite constellation design tradeoffs using multiple-objective evolutionary computation

Multiple-objective evolutionary computation provides the satellite constellation designer with an essential optimization tool due to the discontinuous, temporal, and/or nonlinear characteristics of the metrics that architectures are evaluated against. In this work, the nondominated sorting genetic algorithm 2 (NSGA-2) is used to generate sets of constellation designs (Pareto fronts) that show the tradeoff for two pairs of conflicting metrics. The first pair replicates a previously published sparse-coverage tradeoff to establish a baseline for tool development, whereas the second characterizes the conflict between temporal (revisit time) and spatial (image quality) resolution. A thorough parameter analysis is performed on the NSGA-2 for the constellation design problem so that the utility of the approach may be assessed and general guidelines for use established. The approximated Pareto fronts generated for each tradeoff are discussed, and the trends exhibited by the nondominated designs are revealed. Nomenclature a = semimajor axis, km e = eccentricity F = focal length, m i = inclination, deg K = units conversion constant M = mean anomaly, deg P = pixel size, μm e = elevation, deg ρ = range, km � = right ascension of the ascending node, deg ω = argument of perigee, deg

Many-objective reconfiguration of operational satellite constellations with the Large-Cluster Epsilon Non-dominated Sorting Genetic Algorithm-II

A general framework for the reconfiguration of satellite constellations is developed for the operational scenario when a loss of capacity has occurred and the future configuration must be constructed from the remaining assets. A multi-objective evolutionary algorithm, ε-NSGA-2, adapted for use on large heterogeneous clusters, facilitated the exploration of a six-dimensional fitness landscape for several loss scenarios involving the Global Positioning System Constellation. An a posteriori decision support process was used to characterize and evaluate non-traditional but innovative constellation designs identified. The framework has enhanced design discovery and innovation for extremely complex space domain problems that have traditionally been considered computationally intractable.

Visual Steering Commands and Test Problems to Support Research in Trade Space Exploration

Designers can simulate thousands, if not millions, of design alternatives more cheaply and quickly than ever before with today’s computing power; however, the resulting data can overwhelm designers without proper tools to support multi-dimensional data visualization. In this paper, we discuss the use of a multi-dimensional data visualization tool and visual steering commands which allow designers to navigate multi-attribute trade spaces. The novelty in our work is providing designers with a set of visual steering commands to simultaneously explore the trade space and exploit new information and insights as they are gained. Specifically, designers can explore the entire design space (either sampled randomly or manually) or along the entire Pareto front using the Basic Sampler, Point Sampler, and/or Pareto Sampler. Alternatively, they can exploit information they have gained during the exploration process by searching near a specific point of interest or within a region of high preference using the Attractor, Preference Sampler, and/or Guided Pareto Sampler. Examples of each are included in this paper. Meanwhile, a suite of test problems is being formalized to support our trade space exploration – algorithmic development as well as empirical studies involving human decision-makers. This work supports our long-term goal of quantifying the benefits of putting humans back “in-the-loop” during design optimization.

Transit-Orbit Search for Planar Restricted Three-Body Problems with Perturbations

A new class of trajectory search methods for the planar circular restricted three-body problem (CR3BP) with perturbations is presented. In the phase space of the CR3BP, there exist bundles of trajectories involving the transition from one region to another inside invariant manifolds of libration point orbits. Under the influence of perturbing forces, although these trajectory bundles can change their distributions and the locations, they still remain as orbit bundles for a considerable time in the phase space of the perturbed CR3BP. This paper presents a simple procedure of locating these orbit bundles directly in the CR3BP with perturbations. It is shown that by formulating the circular restricted problem of six bodies with the sun and the elliptical restricted problem of four bodies as perturbed CR3BP systems, the bundles of these solution orbits can be systematically and directly found on an arbitrarily chosen Poincare section.

Pareto-hypervolumes for the Reconfiguration of Satellite Constellations

A satellite constellation is designed to perform its mission with a nominal number of spacecraft. When a reduction in capacity is experienced, for whatever reason, the remaining constellation may be able to restore performance through reconfiguration. In this work we present a general framework that exploits recent efforts in parallel multiobjective evolutionary computation, to reconfigure satellite constellations that have suffered the loss of one or more of their vehicles. The framework is illustrated through several loss scenarios for the Global Positioning System constellation. Pareto-hypervolumes are constructed which are the set of solutions that approximate the optimum tradeoff between minimizing cost and risk while maximizing performance. The decision making processes using the high-dimensional data sets is illustrated. The results demonstrate a pragmatic approach to optimization wherein the insights gained from a multi-objective view of the design space tradeoffs allow for informed decision making. Nomenclature

Designing Continuous-Thrust Low-Earth-Orbit to Geosynchronous-Earth-Orbit Transfers

This paper examines a method that can he used to transfer a spacecraft from a circular low Earth orbit (LEO), inclined to the equator, to a circular geosynchronous Earth orbit (GEO) with no inclination. The principle is to minimize the propulsive-mass cost for a continuously thrusting vehicle with the capability for multiple on-off thrusting cycles. The analysis was conducted for a large range of initial accelerations, and it was found that the method is best used to bridge the gap between very low-thrust transfers and high-thrust, impulsive transfers. The simulation of a LEO-GEO transfer showed that the results varied from 1 % over the optimal cost for a high-thrust transfer, to 2.5% over the optimal cost for an intermediate-thrust transfer, to 0.3% over the cost of a low-thrust, spiral transfer. This makes this technique a good first estimate algorithm for the entire range of high- to low-thrust transfers.

Coupling of Estimation and Sensor Tasking Applied to Satellite Tracking

The tracking of resident space objects has been a topic of recent concern due to the numerous objects that must be monitored with respect to relatively few sensors that can observe them. This discrepancy creates situations in which sensors have multiple objects within their view and must decide which to observe and which to ignore at a particular time, a process known as sensor tasking or sensor network management. Previous studies have suggested calculating information- (or covariance-) based metrics to aid in this process, which rely on uncertainty estimates of an object’s location obtained using a nonlinear estimator. In these studies, the coupling between estimation and sensor tasking is investigated using two covariance-based tasking strategies in conjunction with two nonlinear estimators applied within a simple planar satellite-tracking simulation. Results demonstrate that the use of more accurate estimators leads to better overall estimates, not only due to the advantages within the estimation meth...

Identifying Optimal Interplanetary Trajectories through a Genetic Approach

†‡ The goal of this paper was to design a trajectory from Earth to Jupiter using a single flyby of either Venus or Earth to obtain a trajectory which is lower in terms of ∆v than the Hohmann transfer. The Hohmann transfer ∆v from Earth to Jupiter is 14.43 km/s, and its minimum time of arrival (TOA) is 1081 Earth days after the epoch time of 01 January, 2005 00:00 UT. Using two Evolutionary Algorithms (EAs): Differential Evolution (DE) and Particle Swarm Optimization (PSO), two trajectories were identified which were lower than the Hohmann ∆v with one requiring an Earth swing-by, and the second a Venusian swingby. The trajectory requiring the Venusian fly-by required a total ∆v of 11.82 km/s, and its TOA was 1250 Earth days. This trajectory requires 18% less ∆v than the Hohmann transfer, but its drawback is that the vehicle arrives 169 Earth days later. The second transfer required an Earth flyby, and its total ∆v is 13.43 km/s, with a TOA of 1700 Earth days. This transfer also requires less fuel than the Hohmann transfer, but its TOA is 450 Earth days later, which is certainly substantial. Both DE and PSO were used to determine the optimal trajectory, and PSO outperformed DE because it was able to arrive at the optimal trajectory in much fewer function evaluations.