Education

Abstract

This issue contains two articles in the Education section. The first one is “An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation,” by Matthew Kelly. The author presents a tutorial on discretization ideas for optimal control problems with ordinary differential equations. The article starts with a simple example, which is used to introduce some basic notions and a general formulation of an optimal control problem, where the dynamics is described by an ordinary differential equation. That formulation includes general state and control constraints. The main focus is put on methods, which construct a polynomial approximation of the state and control functions. The state and control spaces are discretized, and the infinite-dimensional control problem becomes a finite-dimensional nonlinear optimization problem. Piecewise linear and quadratic approximations of the state and the control are explained and illustrated by figures. Attention is brought to the error accumulated in the numerical calculation of integrals according to several methods. The author offers some practical suggestions for the initialization of the numerical optimization algorithm, as well as for the interpretation of the solution provided by the optimization method. The optimization models obtained as a result of the discretization procedure typically lack convexity properties; this entails that the optimization procedure would provide a local solution, which would not be globally optimal, in general. Some ideas about how to look for a better solution or confirm the quality of the one at hand are provided. Further discussion includes suggestions for smoothing of nondifferentiable functions that may occur in the problem and a comparison of open loop vs. closed loop solutions. The last section of the paper briefly surveys methods based on discretization and numerical treatment of the conditions of optimality. The author has made his software---a general-purpose MATLAB library for solving trajectory optimization problems, complete with documentation---publicly available at https://GitHub.com/MatthewPeterKelly/OptimTraj. The electronic supplement is described in Appendix A. In addition to the optimization software, it includes all example problems, which are presented in the paper. The second article is “How to Mitigate Sloshing,” authored by H. Ockendon and J. R. Ockendon. The authors start with an example from everyday life familiar to most of us: carrying a mug of coffee can often lead to spillage. How should we reduce this danger? The authors devise a model of this dynamical system under certain simplifying assumptions. A point (representing the hand carrying the coffee) is oscillating horizontally with a given frequency and amplitude (representing the walk) about a fixed mean position. A spring connects the hand to the coffee mug, which can slide freely on a smooth table. The effect of that spring on the motion of the liquid is the focus of attention. It is assumed additionally that the mug is rectangular and that no motion perpendicular to the direction of the action of the spring is present. The goal becomes to model and analyze the two-dimensional motion of liquid in a rectangular container of given width and depth. To this end, the authors introduce and analyze a mathematical model based on partial differential equations. Although the article is compact, the authors make a number of suggestions for future investigation and extensions.