Bion L. Pierson

OPTIMAL EARTH-MOON TRAJECTORIES USING NUCLEAR ELECTRIC PROPULSION

Minimum-fuel, two-dimensional and three-dimensional, Earth ‐moon trajectories are obtained for a nuclear electric propulsion spacecraft. The initial state is a circular low Earth parking orbit and the terminal state is a circular low lunar parking orbit. For the three-dimensional problem, the lunar orbit is inclined 90 deg to the Earth‐moon plane. The trajectory has a e xed thrust-coast-thrust engine sequence and is governed by the classical restricted three-body problem dynamic model. An analytical expression for quasicircular low-thrust transfers is usedtoapproximateandreplacethehundredsofinitialande nalorbitsabouttheEarthandmoon.Acostate-control transformation is also utilized to enhance convergence to the optimal solution. Numerical results are presented for the optimal Earth ‐moon trajectories.

Optimal Planar Evasive Aircraft Maneuvers Against Proportional Navigation Missiles

A comparative study of evasive strategies of an aircraft against a missile with fixed, gravity-limited, proportional navigation is conducted for both subsonic and supersonic confrontations. A complete point-mass aircraft model and a variable-mass missile model that includes missile dynamics are used. No linearization is required in the analysis, and all motion is constrained to a horizontal plane. Sequential quadratic programming is used to solve the optimal control problem. Numerical results are presented for an early model of the F-4 fighter aircraft. In particular, the effects of varying the aircraft/missile initial velocity ratio, the missile initial heading angle, and the missile guidance time constant are determined.

An Optimal Control Approach to Minimum-Weight Vibrating Beam Design

ABSTRACT A gradient projection algorithm is applied to a class of vibrating cantilever beam optimization problems which are formulated as optimal control problems. The cross-section area is distributed along the beam for minimum total weight subject to fixed natural frequency constraints and a minimum allowed area limit. Three topics receive major emphasis: the effects of shear deformation and rotary inertia, higher-mode frequency constraints, and multiple frequency constraints. Computational results are presented for the cases of fixed fundamental frequency, fixed second-mode frequency, and fixed fundamental and second-mode frequencies under a variety of conditions.

An optimal control approach to the design of vibrating elastic-viscoelastic sandwich beams

Abstract This work features the application of an optimal control algorithm to a new class of continuous one-dimensional structural design problems. A sandwich beam of rectangular cross-section is considered. It has a variable-thickness core and two equal variable-thickness cover layers and is subjected to harmonic forced vibrations. The objective is to distribute both the core and layer mass so as to minimize a measure of dynamic compliance for forced steady-state vibration and fixed material volumes. Either or both materials may be viscoelastic. Any constitutive relation may be used provided it is linear and time-invariant. The design problem is formulated as an optimal control problem. The resulting problem contains ten state variables, two control functions, four control parameters, and six terminal state constraints. Simple transformations are used to treat the minimum-gage constraints. A conjugate gradient/gradient projection optimal control algorithm is then used to obtain numerical solutions. Several optimal beam designs are presented and compared for a variety of problem parameter values.

Optimal Aeroelastic Design of an Unsymmetrically Supported Panel

Abstract The problem of weight minimization for an infinite-span panel subject to a supersonic flutter constraint and constant in-plane loading is formulated as an optimal control problem. One end of the panel is simply supported while the other end is clamped. Values of the critical aerodynamic parameter are determined for a uniform reference panel. Numerical solutions are then obtained using a gradient projection optimal control algorithm for both homogeneous and sandwich panels. Upper and lower bounds on the panel thickness are enforced using control transformations.

A SINGULAR-ARC APPROXIMATION TO A DYNAMIC SAILPLANE FLIGHT PATH OPTIMIZATION PROBLEM

A dynamic sailplane performance problem is investigated using optimal control theory. The problem is to minimize the total flight time between successive thermals subject to zero altitude loss. From the original nonlinear optimal control problem, a singular linear/quadratic problem is derived and solved. A relationship between the original optimal control problem and a certain parameter optimization problem is explored, and it is shown that the solution to this parameter optimization provides a lower bound for the minimum flight time of the original optimal control problem. The parameter optimization solution is adopted as the reference trajectory for the linear/quadratic problem. Finally, the linear/quadratic problem is shown to provide a good approximation to the original optimal control problem at a small fraction of the computing cost.

CLUSTER ANALYSIS AFTER A PARTIAL GENETIC ALGORITHM SEARCH

After a few generations of a genetic algorithm minimization, the resulting population of design vectors will typically cluster around a small number of prospective global or relative minima. If these clusters can be identified, sequential search methods can often be used to more efficiently locate the isolated minima as part of a broad-based design process. Two such “cluster analysis” algorithms are proposed here: a frequency distribution technique and a community technique. The first sorts the genetic algorithm population into intervals for each design variable, while the second identifies those individuals with many close neigbors which are also distant from other such population centers. Numerical results from two small test problems are presented to demonstrate the use of these techniques.

A MODEL COMPARISON FOR OPTIMAL MARS ASCENT TRAJECTORIES

Maximum payload Mars launch trajectories are computed for a nominal two-stage Mars ascent vehicle. This vehicle transfers a payload from the surface of Mars to a circular parking orbit about Mars in preparation for transferring the payload to an Earth return vehicle. Both gravity turn and pitch-rate control models for controlling the ascent vehicles trajectory are investigated. For each case, a rather complicated nonlinear programming problem is obtained and then solved using a sequential quadratic programming algorithm.

Optimal aircraft landing-approach trajectories: A comparison of two dynamic models

Abstract Sequential quadratic programming is used to solve a minimum-noise aircraft landing-approach problem for two dynamic models. The “exact” model features the usual point-mass equations of motion for flight in a vertical plane. The other model is based on two simplifying assumptions: (1) small angle of attack and flight path angle and (2) no flight path angle dynamics (lift equals weight). Range is used to replace time as the independent variable. The resulting models are of order three and two, respectively; each model involves two control functions. The primary objective is to compare the solutions for each model with regard to accuracy and computational effort. Numerical results are presented for a variety of boundary conditions and path constraints.