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Halphen's Theorem states that, "A necessary and sufficient condition for every dynamical trajectory in a positional field of force in E3 to be planar is that the field of force is either parallel or central." This result has been known for some time, however only the sufficiency part of the theorem is widely documented. A new analytic proof of the necessity part of Halphen's Theorem was developed. The details of this proof motivated the new concepts of a flat point in a field of force and a flat point on a dynamical trajectory in a positional field of force. / Ph. D.

The terminal stage of a minimum-time mission of a high- performance aircraft is studied using both a reduced-order "energy" model formulation and a point-mass model formulation of the aircraft. The mission is confined to vertical plane maneuvers, and is defined as consisting of three stages; a climb to the dash point,a steady-state dash at the high velocity point, and finally, a terminal transient from the dash point to the final state. This terminal maneuver evolves outside of the flight envelope, rapidly decreasing altitude while increasing the velocity to values greater than the dash velocity. The velocity then decreases from this maximum value as required in order to meet the final state specification. Some of the trajectories that are generated during this terminal transient maneuver experience dynamic pressures that will exceed the dynamic pressure limit unless a constraint is placed on the state variables. Because of the need for enforcing this state constraint, a direct adjoining method for handling state constraints in the optimal control problem is studied. A numerical example is given to demonstrate the application of this method of handling state constraints for the case of the dynamic pressure limit. Finally, trajectories are generated that lead from the dash point to a final state having lower altitude and energy values than those of the dash point, and observations are made concerning the characteristics of these maneuvers. / Master of Science

Consider the problem of developing an algorithm that computes optimal preprogrammed evasive maneuvers for a Maneuvering Reentry Vehicle (MaRV) attacking a target defended with Anti-Ballistic Missiles (ABMs). The problem is large in terms of the number of optimization parameters, and perhaps in terms of the number of nonlinear constraints. Since both MaRV and ABM trajectories are expensive to compute, rapid convergence of the optimization algorithm is of prime concern. This paper examines a discontinuity in the cost function that degrades both the speed and the reliability of optimizer convergence. A solution is offered, proposing that the optimization algorithm be operated in a new parameter space, in which the discontinuity occurs at infinity. Effectively, the mapping prevents the optimization algorithm from crossing the discontinuity thereby improving optimizer convergence. Results comparing convergence with and without the parameter mapping demonstrate the effectiveness of the procedure. / Master of Science

Approved for public release, distribution is unlimited / This thesis explores using a direct pseudospectral method for the solution of optimal control problems with mixed dynamics. An easy to use MATLAB optimization package known as DIDO is used to obtain the solutions. The modeling of both low thrust interplanetary trajectories as well as aerocapture trajectories is detailed and the solutions for low thrust minimum time and minimum fuel trajectories are explored with particular emphasis on verification of the optimality of the obtained solution. Optimal aerocpature trajectories are solved for rotating atmospheres over a range of arrival Vinfinities. Solutions are obtained using various performance indexes including minimum fuel, minimum heat load, and minimum total aerocapture mass. Finally, the problem formulation and solutions for the mixed dynamic problem of low thrust trajectories with a terminal aerocapture maneuver is addressed yielding new trajectories maximizing the total scientific mass at arrival. / Lieutenant, United States Navy

Thesis. 1977. M.S.--Massachusetts Institute of Technology. Dept. of Aeronautics and Astronautics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERO. / Bibliography : leaves 51-53. / M.S.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERONAUTICS. / Bibliography: leaf 72. / by Yuji Hanawa. / M.S.

The reconstruction of entry, descent and landing (EDL) trajectories is significantly affected by the knowledge of the atmospheric conditions during flight. Away from Earth, this knowledge is generally characterized by a high degree of uncertainty, which drives the accuracy of many important atmosphere-relative states. One method of obtaining the in-flight atmospheric properties during EDL is to utilize the known vehicle aerodynamics in deriving the trajectory parameters. This is the approach taken by this research in developing a methodology for accurate estimation of ambient atmospheric conditions and atmosphere-relative states. The method, referred to as the aerodynamic database (ADB) reconstruction, performs reconstruction by leveraging data from flight measurements and pre-flight models. In addition to the estimation algorithm, an uncertainty assessment for the ADB reconstruction method is developed. This uncertainty assessment is a unique application of a fundamental analysis technique that applies linear covariance mapping to transform input variances into output uncertainties. The ADB reconstruction is applied to a previous mission in order to demonstrate its capability and accuracy. Flight data from the Mars Science Laboratory (MSL) EDL, having successfully completed on August 5th 2012, is used for this purpose. Comparisons of the estimated states are made against alternate reconstruction approaches to understand the advantages and limitations of the ADB reconstruction. This thesis presents a method of reconstruction for EDL systems that can be used as a valuable tool for planetary entry analysis.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

The problem of developing a generalized impulse as a function of a set of parameters is investigated. The proposed generalized impulse alters an existing orbit by producing, over some period of time, a change in velocity, ΔV, as well as a change in position, Δr. The generalized impulse is described by parameters associated with an instantaneous change in velocity as well as parameters associated with an atmospheric skip trajectory. Closed form solutions are obtained through several changes of independent variable, the use of modified Chapman variables and the consequent analytical integration of the uncoupled equations. The closed form solutions contain between two and six parameters depending on the complexity of the desired skip trajectory. Fuel optimal transfer orbits are obtained using the generalized impulse along with Keplerian arcs and instantaneous changes in velocity. Families of coplanar and noncoplanar transfers for circular orbit to circular orbit are numerically generated. The generated transfer trajectories involve the rendezvous of two vehicles. The orbits are not globally optimal but rather optimal for the specified number and type of velocity impulses specified. The optimal solution to the nonlinear problem is determined via sequential quadratic programming which satisfies the Kuhn-Tucker optimality conditions for constrained minimization. It is found that for transfer between coplanar and noncoplanar orbits, solutions using the generalized impulse compare favorably with solutions obtained by optimal control theory. Numerical solution to complex problems involving transfer from general orbit to general orbit were not obtained. / Master of Science