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AIRCRAFT NAVIGATION and SURVEILLANCE ANALYSIS for a SPHERICAL EARTH Michael Geyer

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AIRCRAFT NAVIGATION and SURVEILLANCE ANALYSIS for a SPHERICAL EARTH Michael Geyer AIRCRAFT NAVIGATION AND SURVEILLANCE ANALYSIS FOR A SPHERICAL EARTH Michael Geyer S π/2 - α - θ d h α N U Re Re θ O Boeing Michael Geyer Navigation Vertical Plane Model Boeing Peter Mercator Federico Rostagno Spherical Surface Model Surveillance Project Memorandum — October 2014 DOT-VNTSC-FAA-15-01 Prepared for: Federal Aviation Administration Wake Turbulence Research Office DOT Volpe Center FOREWORD This memorandum addresses a basic function of aircraft (as well as marine, missile and satellite) surveillance and navigation systems analyses — quantifying the geometric relationship of two or more locations relative to each other and to a spherical earth. Here, geometry simply means distances (ranges) and angles. Applications that fit well with the methods presented herein include (a) planning a vehicle’s route; (b) determining the coverage region of a radar or radio navigation installation; or (c) calculating a vehicle’s position from slant-ranges, spherical-ranges, slant- or spherical-range differences, azimuth/elevation angles and/or altitude. The approach advocated is that, to simplify and clarify the analysis process, the three-dimen- sional problems inherent in navigation and surveillance analyses should, to the extent possible, be re-cast as the most appropriate set/sequence of sub-problems/formulations: . Vertical-Plane Formulation (two-dimensional (2D) problem illustrated in top right panel on cover) — Considers the vertical plane containing two problem-specific locations and the center of the earth, and utilizes plane trigonometry as the primary analysis method; provides a closed-form solution. Spherical-Surface Formulation (2D problem illustrated in bottom left panel on cover) — Considers two or three problem-specific locations on the surface of a spherical earth; utilizes spherical trigonometry as the primary analysis method; provides a closed-form solution. Three-Dimensional Vector Formulation — Utilizes 3D Cartesian vector frame- work; best-suited to situations involving four or more problem-specific points and slant-range or slant-range difference measurements; provides a closed-form solution. Linearized Least-Squares Iterative Formulation — When warranted by the distan- ces involved, the accuracy required, and/or the need to incorporate empirical data, the least-squares iterative method is employed based on an ellipsoidal earth model. These techniques are applied to a series of increasing complex situations, starting with those having two problem-specific points, then extending to those involving three or more problem- specific points (e.g., two or more sensor stations and an aircraft). Closed-form (non-iterative) solutions are presented for determining an aircraft’s position based on virtually every possible combination of ranges, pseudoranges, azimuth or elevation angles and altitude measurements. The Gauss-Newton Linearized Least-Squares (LLS) iterative methodology is employed to address the most complex situations. These include any combination of the following circum- stances: more measurements than unknown variables, measurement equations are too complex to be analytically inverted (including those for an ellipsoidal-shaped earth), or empirical data is utilized in the solution. Also, the capability of the LLS methodology to provide an estimate of the accuracy of any solution to the measurement equations is presented. i DOT Volpe Center TABLE OF CONTENTS FOREWORD.................................................................................................................... I 1. INTRODUCTION...................................................................................................... 1 1.1 Overview of Methodologies and Their Applications ..................................................... 1 1.1.1 Overview of Methodologies ........................................................................................................1 1.1.2 Overview of Application of Methodologies................................................................................2 1.2 Vertical Plane Formulation.............................................................................................. 3 1.3 Spherical Surface Formulation........................................................................................ 3 1.4 Applicability and Limitations of Analysis ...................................................................... 5 1.5 Outline of this Document.................................................................................................. 6 2. MATHEMATICS AND PHYSICS BASICS............................................................... 8 2.1 Exact and Approximate Solutions to Common Equations ........................................... 8 2.1.1 The Law of Sines for Plane Triangles .........................................................................................8 2.1.2 The Law of Cosines for Plane Triangles .....................................................................................8 2.1.3 Solution of a Quadratic Equation ................................................................................................9 2.1.4 Computing Inverse Trigonometric Functions..............................................................................9 2.1.5 Expansions of arcsin, arccos and arctan for Small Angles........................................................11 2.1.6 Secant Method for Root Finding ...............................................................................................12 2.1.7 Surface Area on a Sphere ..........................................................................................................13 2.2 Shape of the Earth .......................................................................................................... 13 2.2.1 WGS-84 Ellipsoid Parameters...................................................................................................13 2.2.2 Radii of Curvature in the Meridian and the Prime Vertical.......................................................14 2.2.3 Methods for Addressing an Ellipsoidal Earth............................................................................16 2.3 Accounting for User Altitude......................................................................................... 17 2.3.1 Accounting for Known User Altitude .......................................................................................17 2.3.2 Conditions for Unblocked Line-of-Sight...................................................................................17 3. TWO-POINT / VERTICAL-PLANE PROBLEM FORMULATION .......................... 20 3.1 Mathematical Problem and Solution Taxonomy ......................................................... 20 3.1.1 Mathematical Formulation ........................................................................................................20 3.1.2 Taxonomy of Solution Approaches...........................................................................................20 3.1.3 Detailed Geometry.....................................................................................................................21 3.2 Computing Geocentric Angle......................................................................................... 22 3.2.1 Altitude and Elevation Angle Known – Basic Method .............................................................22 3.2.2 Altitude and Elevation Angle Known – Alternative Method ....................................................23 3.2.3 Altitude and Slant Range Known ..............................................................................................23 3.2.4 Elevation Angle and Slant Range Known .................................................................................24 3.3 Computing Elevation Angle........................................................................................... 24 3.3.1 Altitude and Geocentric Angle Known .....................................................................................24 3.3.2 Altitude and Slant Range Known ..............................................................................................25 3.3.3 Geocentric Angle and Slant Range Known...............................................................................25 3.4 Computing Slant Range ................................................................................................. 25 3.4.1 Altitude and Geocentric Angle Known .....................................................................................25 3.4.2 Altitude and Elevation Angle Known .......................................................................................26 ii DOT Volpe Center 3.4.3 Geocentric Angle and Elevation Angle Known ........................................................................26 3.5 Computing Altitude ........................................................................................................ 27 3.5.1 Slant Range and Geocentric Angle Known...............................................................................27 3.5.2 Slant Range and Elevation Angle Known .................................................................................27 3.5.3 Elevation Angle and Geocentric Angle Known ........................................................................27 3.6 Example Applications..................................................................................................... 28 3.6.1 Example 1: En Route Radar Coverage .....................................................................................28
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