Trajectory optimization

Trajectory optimization is the process of designing a trajectory that minimizes or maximizes some measure of performance.

The selection of flight profiles that yield the greatest performance plays a substantial role in the preliminary design of flight vehicles, since the use of ad-hoc profile or control policies to evaluate competing configurations may inappropriately penalize the performance of one configuration over another. Thus, to guarantee the selection of the best vehicle design, it is important to optimize the profile and control policy for each configuration early in the design process.

Consider this example. For tactical missiles, the flight profiles are determined by the thrust and load factor (lift) histories. These histories can be controlled by a number of means including such techniques as using an angle of attack command history or an altitude/downrange schedule that missile must follow. Each combination of missile design factors, desired missile performance, and system constraints results in a new set of optimal control parameters.

History

Trajectory optimization began in earnest in the 1950s as computers such as the IBM NORC became available for computation of traejctories. The first efforts were based on optimal control optimal control approaches which grew out of inventions of variational calculus at the University of Chicago in the first half of the 20th century most notably by Gilbert Ames Bliss. Pontyragin's methods Pontryagin's minimum principle were developed in the East at about the same time but remained unknown in the West until the 60's. . Early application of trajectory optimzation had to do with the optimzation of rocket thrust profiles in a vacuum and in an atmosphere. From the early work, much of the givens about rocket propulsion optimization were discovered. Another successful application was the climb to altitude trajectories for the early jet aircraft. Because of the high drag associated with the transonic drag region and the low thrust of early jet aircraft, trajectory optimization was key to maximizing climb to altitude performance. Optimal control based trajectories were responsible for some of the world records. In these situations, the pilot was given a Mach versus altitude schedule based optimal control solutions to follow. In the early phase of trajectory optimization; many of the solutions were plagued by the issue of singular arcs. For such problems, the control "disappears" in the solution and it becomes impossible to directly solve for the optimal control. Instead one is left with a family of feasible solutions. At that point, the investigators had to numerically evaluate each member of the family to determine the optimal solution. A breaktrhough occurred with a condition sometimes referred to as the Kelley condition in the East. While not a sufficient condition, this provided an additional necessary condition that allowed downselection to a trajectory that is usually the optimal.

Solution Techniques

The techniques available to solve optimization problems fall into two broad categories: the optimal control methodology that allows solution by either analytical or numerical procedures and an approximation to the optimal-control problem through the use of nonlinear programming that allows solution by numerical procedures. The optimal control problem is an infinite dimensional proglem while the nonlinear programming approach approximates the problem by a finite dimensional problem. Trajectory optimization shares the same optimization algorithms as other optimization problems. The optimal control methodology can produce the best answers but is difficult for it to achieve a solution. Once the problem has been set up, the optimal control method can find close solutions very quickly but large variations in the problem definitio may reulst in failed searches. Alternately, the nonlinear programming methods such as BFGS BFGS and SQP SQP may be used to solve the finite dimensional problem. The nonlinear programming approach is generally more robust in terms of finding a solution than numerical optimal control but many of the gradient or Newton-Raphson methods require "smoothness" in the function algorithms to be successful. Smoothness is continuity in the first derivative. The smoothness requirement imposes a burdent on flight trajectory analysts in that most highly detailed trajectory simulations do not exhibit smmothness. This restriction was a problem in the early days of trajectory optimization when computer computation speed was an issue. Often, special approximate trajectory models had to be used to work with non-linear programming models. As computation time has become cheap compared to manpower; direct sample methods have evolved as the optimmization algorihms of choice. These algorithms may require orders of magnitude increases in the number of functional samples but exhibit robustness to non-smoothness in the trajectory code. Examples include: genetic algorithms, stochastic sampling methods, and hill climbing algorithms. An excellent overview of the state of the art in numerical methods is given in Betts.

Persons

Persons in trajectory optimization .

• Arthur E. Bryson..
• .

• Eugene Cliff
• Hans Seywald
• Henry J. Kelley
• Bertrand, R.
• Oberel, H. J.
• Miele, Angelo
• Ernest Ohlmeyer
• Craig Phillips
• Betts, J. T.
• Uday Shankar
• Navabi, M. R.