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Aeronautical and Astronautical Engineering

Astrodynamics
^ Minimum-Fuel Recovery of Satellites that Fail to Achieve Geostationary Transfer Orbit
B. A. Conway*
University of Illinois

In 1995, a partial failure of a launch vehicle upper stage left the Koreasat 1 spacecraft in an orbit 6351 km lower than the planned geostationary transfer orbit (GTO). A Lockheed-Martin company team performed a rescue that put the satellite on station but consumed about half of the satellite's fuel supply for attitude control and station keeping. The sequence of maneuvers used was not determined optimally. There have been several subsequent failures of spacecraft to reach GTO. In this research, scientists are developing an algorithm to find optimal, intermediate-thrust trajectories for such spacecraft, so that they may be recovered using minimum fuel and thus, maximizing their lifetime.

^ Optimal Control for Atmospheric Reentry
B. A. Conway*
University of Illinois; NASA Langley Research Center

The problem of optimal control for atmospheric reentry is examined with the objective of minimizing the number and size of the controllers. For example, the trajectory of a future Space Shuttle type of vehicle might be determined in such a way as to simultaneously minimize the fuel required for reentry and the size of flight control surfaces such as ailerons, elevators, and rudder needed for the vehicle to maneuver in the atmosphere. The DCNLP (direct collocation with nonlinear programming) method will be used to convert the continuous problem into a discrete problem.

^ Optimal Low-Thrust Trajectories for Asteroid Interception and Rendezvous
B. A. Conway*
The Boeing Company

The recent NEAR spacecraft mission demonstrated dramatically the feasibility of asteroid rendezvous and landing. Asteroids are interesting bodies in their own right, but also pose concern regarding future impact of an asteroid with the Earth. Researchers must know much more about Earth-approaching asteroids in order to develop hazard mitigation strategies. This team has already developed a tool for optimal low-thrust asteroid interception. In this research, optimal low-thrust trajectories will be developed for two scenarios: the spacecraft is to be placed in orbit about the asteroid, or the spacecraft is to land on the surface of the asteroid.

^ Optimal Very-Low-Thrust Orbit Raising
B. A. Conway*
The Boeing Company

Electric propulsion may soon be used for spacecraft maneuvering in low Earth orbit because of its very high efficiency. Approximate analytical solutions exist for very low thrust accelerations (< 100 micro-g's) for the circular orbit to circular orbit case. But there are no analytical solutions for elliptical orbit to elliptical orbit cases; numerical methods must be used. In this research, the method of direct collocation with nonlinear programming is used, with an improved, sparse NLP problem solver, to find optimal trajectories for such cases and to determine the lower bound on thrust acceleration for which such problems are tractable.

^ Application of Parallel Recombinative Simulated Annealing to Propellant Minimizing Low-Thrust Trajectories
V. Coverstone,* J. Hartmann, W. Mason
University of Illinois

Parallel recombinative simulated annealing is a hybrid computer algorithm combining both simulated annealing and genetic algorithm concepts and attempts to incorporate and augment the traditional strengths of each while simultaneously eliminating their respective weaknesses. This optimization technique is applied to interplanetary missions. Trade studies involving minimum transfer time and maximum delivered mass are being performed. Preliminary results show that parallel recombinative simulated annealing combines the speed and parallel nature of genetic algorithms with the probabilistic hill-climbing ability, selectivity, and guaranteed convergence of simulated annealing.

^ Earth Escape Using a Solar Sail
V. Coverstone,* J. E. Prussing,* J. W. Hartmann
NASA/Caltech Jet Propulsion Laboratory

The feasibility of escaping Earth orbit using a solar sail is investigated. Starting in geosynchronous transfer orbit, the very small solar sail force per unit mass (less than 1 mm/s/s) can be used to gradually accelerate the spacecraft over several months until escape energy is attained. Optimal orientation of the sail at each point in the orbit is required to achieve the desired performance. Several constraints are important for an actual implementation, such as shadowing and sail attitude/angular rate constraints.

^ Optimal Constellation Design through Genetic Algorithms
V. Coverstone,* W. Mason
University of Illinois

Modern satellite systems have started using large constellations; however, little investigation has been performed to optimize their configuration. Historically, continuous global coverage has been the only consideration, but launch cost, manufacturing difficulty, and payload and communications performance are all issues directly affected by the choice of orbital configuration. A general tool for constellation design optimization is being developed to include software that uses a genetic algorithm with flexible cost considerations and indicators of merit.

^ Optimal Control of a Multibody Spacecraft Using Averaging Theory
V. Coverstone,* W. T. Cerven
University of Illinois; National Science Foundation

The problem of reorienting multibody spacecraft through joint motion is addressed. The multibody space system presents a nonholonomic underactuated mechanical system. It is shown that, for small periodic controls, the nonlinearity of the system results in a secular variation of the states. This secular variation provides a mechanism to reorient the multibody spacecraft to any arbitrary attitude. Optimal control theory is then applied to determine an analytic control algorithm that minimizes the required control effect. This control law is developed for a general four-body spacecraft system actuated by three revolute joints.

^ Optimal Interplanetary Spacecraft Trajectories via a Pareto Genetic Algorithm
V. Coverstone,* J. Hartmann
University of Illinois; NASA Jet Propulsion Laboratory

A Pareto genetic algorithm is applied to the optimization of low-thrust interplanetary spacecraft trajectories. A multiobjective, nondominated sorting algorithm is developed following existing methodologies. Verification of expected performance is accomplished through application of the algorithm to a suite of multiobjective test functions. Reshearchers have designed a hybridized scheme that integrates the Pareto genetic algorithm with a calculus-of-variations-based trajectory optimization algorithm. "Families" of Pareto optimal trajectories are generated for the cases of Earth-Mars flyby, Earth-Mars rendezvous, and Earth-Mercury rendezvous trajectories.

^ Optimal Orbit Transfer Analysis for Advanced Space Systems
V. Coverstone,* J. Hargens, J. Hartmann, W. Mason
University of Illinois; Spectrum Astro Inc.; NASA Goddard Space Flight Center

Techniques for effectively optimizing orbit transfers of advanced space systems employing low-thrust propulsion are investigated. The techniques of collocation and parallel shooting are used to solve boundary-value problems that result from applying the calculus-of-variations to optimal control problems. Genetic algorithms allow for the optimal selection of discrete parameter values such as thruster on/off cycles. The orbit transfer dynamics are modeled using variational equations based upon modified equinoctial elements. This model includes third-body, solar radiation pressure and shadowing effects. The techniques will be used to optimize low-thrust planetary orbit insertion and escape trajectories, low-thrust transfers from a planetary orbit to natural satellite orbit, and low-thrust transfers to liberation point orbits.

^ Optimal Spacecraft Trajectories via Higher Order Differential Inclusions
V. Coverstone,* J. Hargens
University of Illinois; Spectrum Astro Inc.; U.S. Air Force Phillips Laboratory

Higher order differential inclusions is a new modeling technique that is applied to the modeling and optimization of spacecraft trajectories. The spacecraft equations of motion are mathematically manipulated into differential constraints that remove explicit appearance of the control variables (thrust direction and magnitude) from the problem statement. These constraints are transformed into a nonlinear programming problem by using higher order approximations for the derivatives of the states. The new method is applied to three-dimensional propellant-minimizing low-Earth orbit to geosynchronous Earth orbit spacecraft transfer. Comparisons are made with results that were obtained using established modeling and optimization techniques.

^ Trajectory Modeling and Optimization for Future Space Systems
V. Coverstone*
University of Illinois; Spectrum Astro Inc.; U.S. Air Force Phillips Laboratory

In an effort to minimize the cost of deploying and operating space systems and to increase competitiveness, mission planners are examining a variety of new technologies for boosting space systems into orbit and for performing propulsive maneuvers to achieve and maintain mission orbits. In addition, spacecraft builders are producing smaller bus designs to allow for the use of smaller launch vehicles and for multiple launches aboard a single launch vehicle. Research into developing mathematical and computational modeling and optimization techniques to effectively perform the needed mission analysis and operational planning is being performed.

^ Trajectory Optimization of a Solar Sail Spacecraft in a Three-Body System
V. Coverstone,* J. Hartmann
University of Illinois

Propulsion of a spacecraft using a solar sail is an old concept that has recently gained popularity due to an industry drive for a cost-effective propulsion system. Most of the research to date has studied the effect of a solar sail on the position and stability of the equilibrium points in three-body systems, such as the Earth-Sun spacecraft or Earth-Moon spacecraft. This research computes a variety of trajectories for a solar sail. Total flight time, solar array size, and payload capacity are varied to establish a trade space for mission-planning purposes.

^ Optimal Continuous-Thrust Escape Trajectories
J. E. Prussing,* M. C. Tanzillo
University of Illinois

Two types of continuous-thrust escape trajectories are investigated: constant-thrust acceleration and power-limited. For constant-thrust acceleration, maximization of the instantaneous rate of energy increase is obtained by thrusting along the velocity vector. However, a larger energy increase in a specified time (equivalent to minimum time for a specified energy increase) can be achieved using the optimal thrust direction obtained from primer vector theory. For power-limited trajectories, the optimal thrust direction is along the velocity vector. The existence of conjugate points on these optimal trajectories is also being investigated.

^ Second-Order Necessary and Sufficient Conditions for Optimal Control Problems
J. E. Prussing*
University of Illinois

A procedure is derived and applied to test second-order necessary and sufficient conditions for a weak local minimum of the Bolza optimal control problem. For a system with n state variables, an improved method is used to transform a test for the unboundedness of an n x n matrix (indicating a conjugate point) into a test for a scalar being zero. The procedure is being applied to various spacecraft trajectories that satisfy first-order necessary conditions.

^ Second-Order Necessary and Sufficient Conditions for Optimal Impulsive Spacecraft
J. E. Prussing,* B. S. J. Jea
University of Illinois

Primer vector theory for optimal (minimum-fuel) spacecraft trajectories involves only first-order necessary conditions. This theory is being extended to include second-order necessary conditions and sufficient conditions. A practical test is sought for determining locally minimizing solutions. These second-order conditions may also lead to more efficient iteration algorithms.


Summary of Engineering Research