Survey of Numerical Methods for Trajectory Optimization

This paper attempts to define the Multidisciplinary Design Optimization (MDO) as a new field of research endeavor and as an aid in the design of engineering systems. It examines the MDO conceptual components in relation to each other and defines their functions.

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Multidisciplinary Design Optimization: An Emerging New Engineering Discipline

A new method is described for the determination of optimal spacecraft trajectories in an inverse-square field using finite, fixed thrust. The method employs a recently developed optimization technique which uses a piecewise polynomial representation for the state and controls, and collocation, thus converting the optimal control problem into a nonlinear programming problem, which is solved numerically. This technique has been modified to provide efficient handling of those portions of the trajectory which can be determined analytically, i.e., the coast arcs. Among the problems that have been solved using this method are optimal rendezvous and transfer (including multirevolution cases) and optimal multiburn orbit insertion from hyperbolic approach.

Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming

The most effective numerical techniques for the solution of trajectory optimization and optimal control problems combine a nonlinear iteration procedure with some type of parametric approximation to the trajectory dynamics. Early methods attempted to parameterize the dynamics using a small number of variables because the iterative search procedures could not successfully solve larger problems. With the development of more robust nonlinear programming algorithms, it is now feasible and desirable to consider formulations of the trajectory optimization problem incorporating a large number of variables and constraints. The purpose of this paper is to address the manner in which a trajectory is parameterized and the design of the nonlinear programming algorithm to effectively deal with this formulation.

Application of sparse nonlinear programming to trajectory optimization

A review of optimization techniques in spacecraft flight trajectory design

SUMMARY A direct transcription method is applied to the problem of multiple-pass aeroassisted orbital transfer from geostationary orbit to low Earth orbit with a large inclination change. The objective is to provide minimum-impulse requirements and corresponding optimal trajectories for a gliding vehicle with a high lift-to-drag ratio subject to constraints on heating rate, angle of attack, and transfer time. The multiplepass aeroassisted orbital transfer problem is set up as a multi-phase optimal control problem. All relevant parameters, including de-orbiting, intermediate, and circularizing impulses, are optimized. Copyright # 2002 John Wiley & Sons, Ltd.

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Numerical optimization study of multiple‐pass aeroassisted orbital transfer

Sensitivity analysis for trajectory optimization problems

Numerical optimization study of multiple-pass aeroassisted orbital transfer

The adoption of robust numerical optimization techniques in trajectory simulation programs has resulted in powerful design and analysis tools. These trajectory simulation/optimization programs are widely used, and a representative list includes the GTS system, the POST program, and newer collocation methods such as OTIS and FONPAC. All of these programs rely on optimization algorithms which require objective function and constraint gradient data during the iteration process. However, most trajectory optimization problems lack simple analytical expressions for these derivatives. In the general case a function evaluation involves integrating aerodynamic, propulsive, and gravity forces over multiple trajectory phases with complex control models. With the newer collocation methods, the integration is replaced by defect constraints and cubic approximations for the state. While analytic gradient expressions can sometimes be derived for trajectory optimization problems, the derivation is cumbersome, time consuming, and prone to mistakes. Fortunately, an alternate method exists for the gradient evaluation, namely finite difference approximations. In this paper some finite difference gradient techniques developed for use with the GTS system are presented. These techniques include methods for computing first and second partial derivatives of single and multiple sets of functions. A key feature of these methods is an error control mechanism which automatically adjusts the perturbation size to obtain accurate derivative values.

Numerical derivative techniques for trajectory optimization

This paper describes the design of a “sparse” nonlinear programming algorithm for a “large scale” flight optimization and sizing code. This program is designed to solve trajectory optimal control problems with path constraints. The direct transcription technique is applied to discretize an optimal control problem into a sparse, large scale parameter optimization problem for subsequent numerical solution by a nonlinear programming algorithm. This paper focuses on the design of a generalized reduced gradient algorithm to exploit the sparsity structure of the collocation equations.

Wayne P. Hallman

Papers

Numerical optimization study of multiple‐pass aeroassisted orbital transfer

Sensitivity analysis for trajectory optimization problems

Numerical optimization study of multiple-pass aeroassisted orbital transfer

Numerical derivative techniques for trajectory optimization

A Generalized Reduced Gradient Algorithm for Large-scale Trajectory Optimization Problems