Trajectory Optimization - Solution Techniques

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 former technique is "indirect" in that it finds a solution where the total differential of the performance measure is zero. The latter technique is "direct" in that it finds a solution where the performance measure is smaller (or greater) than that of any other solution in the neighborhood.

The optimal control problem is an infinite dimensional problem 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 numerical optimal control methodology can produce the best answers but converging to a solution is difficult. Convergence is rapid when the initial guess is good, otherwise the search may fail. The ascent trajectories for the US space program (Gemini and Apollo) were designed using numerical optimal control. The very tight tolerances associated with space launchers allowed optimal control to be a useful tool. For systems with less controlled environments such as missiles, numerical optimal control would not prove as useful.

The analytic solution of the optimal control often involves extensive approximations but can still produce useful algorithms. An example is given in Ohlmeyer & Phillips. In this example, linear assumptions are made and yet the algorithm can produce near optimal trajectories. Another example of an analytic solution is the "Iterative Guidance Mode (IGM)", the guidance algorithm used by the two exo-atmospheric stages of the Saturn V rocket. The IGM algorithm is an analytical calculus-of-variations solution of the two-point boundary value problem posed by the ascent of the rocket to prescribed orbit-injection conditions. The analytical solution requires that gravitational acceleration be approximated as a constant vector, and an iteration of the solution is required to improve the accuracy of this approximation.

Many numerical procedures exist to solve parameter optimization problems. The simplest procedures use the gradient descent technique, sometimes also known as the method of steepest descent. Second-order methods are also available to improve the rate of convergence, for example, the Newton–Raphson iteration, which requires the evaluation of the Hessian matrix. Quasi-Newton or variable-metric methods avoid the evaluation of the Hessian matrix by using iterative evaluation of first-order information to approximate the Hessian matrix. The nonlinear programming methods such as BFGS and SQP may be used to solve the finite dimensional problem. An effective and robust nonlinear programming method employing the Simplex algorithm was developed in the 1970s. It was first used to determine quasi-optimum reentry trajectories for the Space Shuttle and has subsequently been used to solve a wide variety of rocket trajectory optimization problems. 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 burden on flight trajectory analysts in that most highly detailed trajectory simulations do not exhibit smoothness. 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 optimization algorithms 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 overview of the state of the art in numerical methods is given in Betts.