Trajectory optimization

Trajectory optimization is the process of designing a trajectory that minimizes (or maximizes) some measure of performance while satisfying a set of constraints. Generally speaking, trajectory optimization is a technique for computing an open-loop solution to an optimal control problem. It is often used for systems where computing the full closed-loop solution is either impossible or impractical.

Although the idea of trajectory optimization has been around for hundreds of years (calculus of variations, ), it only became practical for real-world problems with the advent of the computer. Many of the original applications of trajectory optimization were in the aerospace industry, computing rocket and missile launch trajectories. More recently, trajectory optimization has also been used in a wide variety of industrial process and robotics applications.

Trajectory optimization first showed up in 1697, with the introduction of the Brachystochrone problem: find the shape of a wire such that a bead sliding along it will move between two points in the minimum time. The interesting thing about this problem is that it is optimizing over a curve (the shape of the wire), rather than a single number. The most famous of the solutions was computed using calculus of variations.

In the 1950s, the digital computer started to make trajectory optimization practical for solving real-world problems. The first optimal control approaches grew out of the calculus of variations, based on the research of Gilbert Ames Bliss and Bryson in America, and Pontryagin in Russia. Pontryagin's maximum principle is of particular note. These early researchers created the foundation of what we now call indirect methods for trajectory optimization.

Much of the early work in trajectory optimization was focused on computing rocket thrust profiles, both in a vacuum and in the atmosphere. This early research discovered many basic principles that are still used today. 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.

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