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In a trajectory optimal control or prescribed path control problem, a trajectory is determined which satisfies simultaneously both the equations of motion for the vehicle and additional mission constraints. In general, the equations of motion form a nonlinear system of ordinary differential equations (ODEs). In a prescribed path control problem, additional path constraints are imposed to dictate the shape of the trajectory. In an optimal control problem, a performance index (or cost functional) is minimized or maximized subject to the satisfaction of the differential equations, path constraints, and associated boundary conditions. In both cases, the control profile along the trajectory, as well as the state profile, must be determined.
The first-order necessary conditions for a solution to the optimal control problem generate the Euler-Lagrange system of equations. The Euler-Lagrange system is most generally a boundary value problem for a system of differential-algebraic equations (DAEs). Typically, these DAEs have the semi-explicit form of a set of differential equations coupled with some algebraic equations. The differential equations include the equations of motion and are dependent on both the state and control variables in general. The algebraic equations arise from explicit path constraints specified in the problem, or more subtly from the necessary conditions for an optimal solution. The algebraic constraints may or may not involve the control variables. Hence, there is the potential that the underlying DAE system has an index greater than one.
Over the past 35 years, many methods have been developed to solve these trajectory problems. One such technique, known as the direct transcription method, has been implemented in the trajectory optimization and sizing code FONSIZE. In the direct transcription method, a discretization based on a collocation formula is applied to the differential equations and mission constraints to obtain a parameter optimization problem.
In theory, any nonlinear programming (NLP) algorithm can be used to solve the parameter optimization problem resulting from the direct transcription of an optimal control problem. However, it is critical to the efficiency and ultimate success of this approach to employ an NLP algorithm designed for {\em sparse, large-scale} parameter optimization problems. The sparsity has two origins: the collocation method and the inherent sparse character of the trajectoryproblems (i.e., each variable is involved in relatively few constraints). It is important to exploit the sparsity properties in order to reduce storage requirements and to increase efficiency of the solution of linear systems required by the NLP algorithm.
We based our sparse NLP code on a generalized reduced gradient (GRG) algorithm. It is specifically designed to exploit the structure of the collocation equations.
FONSIZE is used to design new space vehicles, specifically the size of fuel tanks ("sizing") in conjunction with optimally flying the trajectory. For example, in a launch vehicle, one might want to minimize the ascent fuel required, satisfy some trajectory constraints along with the equations of motion, and design the fuel tank lengths.
The equations of motion of a space vehicle are systems of ordinary differential equations. One may wish to solve an initial value problem, say where the initial position and velocity of the spacecraft is given and you want to determine the trajectory for some period of time. Boundary value problems also arise, for example when you want to design an orbit transfer maneuver between two different orbits. In that case you have beginning and end point constraints on the maneuver. Parameter optimization problems arise as well. Engineers frequently want to know the sensitivity of their solution with respect to some system parameters.
Curve fitting of tabular data requires knowledge of least squares processes as well as splines and other types of fitting techniques. In particular, I once developed some smooth curve fits to tabular atmospheric data using a weighted least squares method and cubic splines. These smooth fits have been used regularly in trajectory optimization simulations where smoothness of the function evaluations is critical to the successful convergence of the optimization algorithms.