99162-01 The Engineering Analysis and Design of the Aircraft Dynamics Model For the FAA Target Generation Facility Mark Peters Michael A. Konyak Prepared for: Scott Doucett ANG-E161 Simulation Branch, Laboratory Services Division Federal Aviation Administration William J. Hughes Technical Center Atlantic City, NJ 08405 Under: Air Traffic Engineering Co., LLC 1027 Route 9 South Palermo, NJ 08223 FAA Prime Contract No. DTFAWA-10A-00020 October 2012 Acknowledgments This project was supported under contract No. DTFA03-94-C-0042 (CADSS) with the Federal Aviation Administration William J. Hughes Technical Center, Atlantic City, NJ, under subcontract to System Resources Corporation (SRC). The author acknowledges the insightful project direction provided by Mr. Dan Warburton of the FAA and Mr. George Chachis of SRC, during the course of this work. The author also wishes to acknowledge the various technical contributions and reviews from the members of the Seagull team. Specifically, the author acknowledges Dr. John Sorensen and Tysen Mueller for their contributions to the error modeling sections of the Navigator development, and George Hunter and JC Gillmore for their leadership and guidance throughout the project. Mark Peters September, 1999 i THIS PAGE INTENTIONALLY LEFT BLANK ii Table of Contents 1. INTRODUCTION....................................................................................................1 1.1 BACKGROUND .........................................................................................................1 1.2 SCOPE ......................................................................................................................2 1.3 ORGANIZATION .......................................................................................................2 2. THE AIRCRAFT EQUATIONS OF MOTION ...................................................5 2.1 THE DEFINITION OF THE BODY FRAME AND THE INERTIAL FRAME .........................5 2.2 DEFINITION OF FLIGHT MECHANICS NOMENCLATURE .............................................8 2.3 THE WIND AND STABILITY REFERENCE FRAMES .....................................................9 2.4 THE DERIVATION OF THE SIX DEGREE OF FREEDOM EQUATIONS OF MOTION .......12 2.5 THE MODAL PROPERTIES OF THE SIX DEGREE OF FREEDOM MODEL .....................16 2.5.1 The Short Period Mode ..................................................................................18 2.5.2 The Phugoid Mode .........................................................................................18 2.5.3 The Dutch Roll Mode .....................................................................................18 2.5.4 The Roll Mode ................................................................................................19 2.5.5 The Spiral Mode .............................................................................................19 2.6 SIMPLIFYING THE EQUATIONS OF MOTION TO FOUR DEGREES OF FREEDOM .........20 2.7 THE ADDITION OF WINDS ......................................................................................27 2.8 TRAJECTORY PROPAGATION OVER AN EARTH MODEL ..........................................31 2.8.1 Elliptic Earth Reference Frames ...................................................................31 2.8.2 The Earth Model and its Kinematic Parameters ...........................................35 2.8.3 Kinematics Relating the Coordinate Frames .................................................39 2.8.4 Transforming DIS to Lat/Lon ........................................................................43 2.9 THE DERIVED STATE VARIABLES ..........................................................................45 2.10 THE AIRFRAME MODEL ......................................................................................47 2.11 THE ENGINE MODEL ..........................................................................................48 2.12 THE STANDARD ATMOSPHERE MODEL ..............................................................50 2.13 INTEGRATION TECHNIQUES ................................................................................54 2.13.1 The Second Order Runge-Kutta Method ....................................................54 iii 2.13.2 The First Order Euler Method ....................................................................56 2.14 THE INTEGRATION OF THE ROLL EQUATION .......................................................56 2.14.1 The open loop roll rate and roll angle equations .......................................57 2.14.2 The closed loop system ...............................................................................57 2.14.3 An Analytical Solution in the Discrete Time Interval .................................59 2.15 DESIGN OF THE LATERAL CONTROL SYSTEM .....................................................63 3. THE EXAMINATION OF THE LONGITUDINAL DYNAMICS ............. - 57 - 3.1 THE LINEAR MODEL OF THE LONGITUDINAL DYNAMICS .................................. - 57 - 3.2 THE ANALYSIS OF LONGITUDINAL AIRCRAFT MODAL PROPERTIES ................. - 66 - 3.2.1 The Characteristic Polynomial of an LTD System ................................... - 66 - 3.2.2 Modal Properties of the Longitudinal Dynamics...................................... - 67 - 3.3 TRANSFER FUNCTION ANALYSIS OF THE LONGITUDINAL DYNAMICS ............... - 72 - h 3.3.1 The CL Transfer Function ........................................................................ - 73 - M 3.3.2 The CL Transfer function ........................................................................ - 76 - h 3.3.3 The T Transfer function ........................................................................... - 77 - 4. THE FEEDBACK CONTROL SYSTEM FOR LONGITUDINAL CONTROL79 4.1 THE GENERAL CONTROL LAW ...............................................................................79 4.2 MANIPULATING AN LTD STATE -SPACE WITH INTEGRAL CONTROL ......................81 4.3 AN ANALYSIS OF THE EFFECTS OF FEEDBACK CONTROL ON THE MODAL PROPERTIES ....................................................................................................................83 4.4 FEEDBACK CONTROLLER DESIGN ..........................................................................88 4.4.1 Lift Coefficient Control of Altitude Rate ........................................................89 4.4.2 Lift Coefficient Control of Speed ...................................................................96 4.4.3 Controlling Speed and Altitude Rate Simultaneously ..................................106 5. THE SUPPORTING FUNCTIONAL LOGIC OF THE LONGITUDINAL CONTROL SYSTEM ............................................................................................... - 115 - 5.1 CONTROL STRATEGIES ................................................................................... - 116 - 5.1.1 Altitude Rate Controller - Feeding Back Altitude Rate Only ................. - 116 - 5.1.2 Speed Controller - Feeding Back Speed Only ........................................ - 117 - 5.1.3 Dual Controller - Feeding Back Speed and Altitude Rate ...................... - 117 - iv 5.1.4 Altitude Capture ...................................................................................... - 118 - 5.1.5 Speed Capture ......................................................................................... - 120 - 5.2 DIVIDING THE FLIGHT ENVELOPE INTO REGIONS ........................................... - 121 - 5.2.1 The Speed-Altitude Plane ........................................................................ - 122 - 5.2.2 Taking off ................................................................................................ - 134 - 5.2.3 Landing Region Management ................................................................. - 138 - 5.3 THROTTLING IN REGIONS 1 THROUGH 6 ......................................................... - 140 - 6. THE SELECTION OF GAINS ..........................................................................139 6.1 THE AIRCRAFT ’S FLIGHT ENVELOPE ...................................................................139 6.2 DETERMINING ACCEPTABLE MODAL PROPERTIES ...............................................142 6.3 CHOOSING A SINGLE REFERENCE CONDITION FOR GAIN CALCULATION .............143 6.4 EVALUATING SYSTEM PERFORMANCE WITH SCHEDULED GAINS ........................144 7. THE LATERAL DIRECTIONAL CONTROL LAWS ...................................149 7.1 THE BANK ANGLE CAPTURE ALGORITHM ...........................................................149 7.2 THE HEADING CAPTURE ALGORITHM ..................................................................150 7.3 USING THE BANK ANGLE CAPTURE AND HEADING CAPTURE ALGORITHMS TO EXECUTE A TURN ..........................................................................................................151 8. LATERAL GUIDANCE AND NAVIGATION ................................................161 8.1 ROUTE LEGS AND WAYPOINTS ............................................................................161 8.1.1 Path and Terminator Concept .....................................................................161 8.1.2 Leg Terminators – Geometric Nodes of a Route .........................................162 8.1.3 Direct-To-Fix Leg ........................................................................................162 8.1.4 Track-To-Fix Guidance ...............................................................................162 8.1.5 Course-To-Fix Guidance .............................................................................162