Games Against the Wind: Guided Parafoil Accuracy David W. Carter * and Scott A. Rasmussen † Draper Laboratory, Cambridge, MA 02139, USA Viewing parafoil guidance as a differential game against the wind, we show how to compute the landing accuracy that can be guaranteed by a perfect guidance system in the presence of worst case variable wind. What perfect guidance can guarantee depends on distance and relative heading to the target, remaining time to impact, system airspeed and maximum turn rate, and wind speed. We use the theory to bound the landing accuracy that can be guaranteed for the “Snowflake” parafoil system. Nomenclature x, y Horizontal coordinates of parafoil position with respect to the target ψ, ˙ ψ Parafoil heading and heading rate v Parafoil horizontal airspeed w x ,w y Horizontal components of wind velocity Δt A small increment of time C(x, y, t) The accuracy (smallest miss distance) guidance can guarantee when play starts from position (x, y) at time t I. Parafoil Equations of Motion The mathematics is easiest if we work in a right-handed coordinate frame whose x axis is in the direction of the horizontal component of parafoil velocity with respect to the surrounding air mass and whose z axis points down. We’ll call this the body-referenced local vertical local horizontal (LVLH) frame. Let x and y denote the horizontal coordinates of parafoil position with respect to the target expressed in this frame, let ψ denote the azimuth of the x axis relative to north— the parafoil heading — let v denote parafoil horizontal airspeed, and let w x and w y denote the horizontal components of wind velocity. Here, to first order, is what happens when time t increases by the small increment Δt: • ψ increases by ˙ ψ Δt, where ˙ ψ is heading rate • x increases by y ˙ ψ Δt because of the change in heading • y decreases by x ˙ ψ Δt because of the change in heading • x increases by v Δt because of parafoil velocity with respect to the air • x and y increase by w x Δt and w y Δt respectively because of the wind velocity As Δt approaches zero we obtain the equations of motion in the body-referenced LVLH frame: ˙ x = + ˙ ψy + w x + v (1) ˙ y = - ˙ ψx + w y (2) In this note, we assume that parafoil airspeed and sink rate are known constants, at least approximately, and that vertical wind and terrain may be neglected. So time to impact can be calculated from altitude above the target and equations 1 and 2 tell the whole story. * Member of the Technical Staff, Information and Decision Systems Division, Senior Member AIAA. † Member of the Technical Staff, Embedded Navigation and Sensor Systems Division, Member AIAA. 1 of 11 American Institute of Aeronautics and Astronautics Downloaded by David Carter on April 3, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2159 23rd AIAA Aerodynamic Decelerator Systems Technology Conference 30 Mar - 2 Apr 2015, Daytona Beach, FL AIAA 2015-2159 Copyright © 2014 by The Charles Stark Draper Laboratory, Inc.. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Aerodynamic Decelerator Systems Technology Conferences