Trajectory Optimization - History

History

Trajectory optimization began in earnest in the 1950s as digital computers became available for the computation of trajectories. The first efforts were based on optimal control approaches which grew out of the calculus of variations developed at the University of Chicago in the first half of the 20th century most notably by Gilbert Ames Bliss. Pontryagin in Russia and Bryson in America were prominent researchers in the development of optimal control. Early application of trajectory optimization had to do with the optimization 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 the key to maximizing climb to altitude performance. Optimal control based trajectories were responsible for some of the world records. In these situations, the pilot followed a Mach versus altitude schedule based on optimal control solutions.

In the early phase of trajectory optimization; many of the solutions were plagued by the issue of singular subarcs. For such problems, the term in the Hamiltonian linearly multiplying the control variable goes to zero for a finite time and it becomes impossible to directly solve for the optimal control. The Hamiltonian is of the form: and the control is restricted to being between an upper and a lower bound: . To minimize, we need to make as big or as small as possible, depending on the sign of, specifically:

If is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a bang-bang control that switches from to at times when switches from negative to positive.

The case when remains at zero for a finite length of time is called the singular control case and the optimal trajectory follows the singular subarc.

In this case, 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 breakthrough occurred with a condition sometimes referred to as the Kelley condition. While not a sufficient condition, this provided an additional necessary condition that allowed downselection to a trajectory that is usually the optimal singular control.

An example of a problem with singular control is the optimization of the thrust of a missile flying at a constant altitude and which is launched at low speed. Here the problem is one of a bang-bang control at maximum possible thrust until the singular arc is reached. Then the solution to the singular control provides a lower variable thrust until burnout. At that point bang-bang control provides that the control or thrust go to its minimum value of zero. This solution is the foundation of the boost-sustain rocket motor profile widely used today to maximize missile performance.

Many of the early triumphs of trajectory optimization have moved into the background knowledge of the modern flight mechanicist and the origins of these discoveries are not widely known. The development of a mathematical way to divide time into infinitely small packets, so that a rocket's trajectory could be computed in an uninterrupted continuum, is credited to the Japanese mathematician Kiyoshi Ito.