TRAJECTORY SOLUTION Treatises
Optimization
In a mathematical
context, the trajectory optimization problem is a nonlinear, multi-point
boundary-value problem requiring a numerical solution. This solution is best
obtained by the intensive, iterative calculations of a high-speed digital
computer.
The good news is that there are several viable methods that can
be programmed on a computer to obtain solutions. Usually, a truly optimum
solution is not cost-effective, and the engineer or scientist must be satisfied
with a quasi-optimum solution. However, a thoughtful construction of the
trajectory-optimization problem will usually result in a quasi-optimum solution
that is not significantly different from the truly optimum solution.
A
method I developed in the 1970's, and used successsfully for many years after
that, is based on the iterative solution to a cleverly-constrained linear
approximation of the non-linear optimization problem. On each iterate, the
Simplex algorithm (of linear programming) is used to solve the linear
approximation. The user defines the cost function and constraint equations and
inequalities, and he also characterizes the control functions in terms of
parameters (constants) whose optimum values are to be determined.
In practice, this adaptation of the Simplex algorithm has proven to be a robust,
effective method for obtaining quasi-optimum solutions for a wide variety of
missions involving rockets. Over the years, the method has been enhanced as it
has been applied to more and more mission types, involving different objectives
and constraints. What has been learned has, in large part, been incorporated
into the first edition of a program named ZOOM. This
first edition is termed the "conceptual design and analysis" edition because
the propulsion and aerodynamic models are relatively simple. However, it can be
a very useful tool to an aerospace engineer in trajectory shaping and rocket
sizing tasks.
---- To
Be Continued ---