The Engineering Analysis and Design of the Aircraft Dynamics - FAA

321 Pages·2012·4.28 MB·English

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

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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
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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 -
3.3.1 The h(cid:1) Transfer Function ........................................................................ - 73 -
C
L
3.3.2 The M Transfer function ........................................................................ - 76 -
C
L
3.3.3 The h(cid:1) Transfer function ........................................................................... - 77 -
T
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 -
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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
8.1.6 Radius-To-Fix Guidance ..............................................................................163
8.2 LATERAL GUIDANCE ...........................................................................................164
8.2.1 Tracking a Geographic Path .......................................................................164
8.2.2 Following a Course Law ..............................................................................170
8.2.3 Transition .....................................................................................................174
8.3 LATERAL GUIDANCE MANAGER ..........................................................................179
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8.3.1 Route Following ...........................................................................................181
8.3.2 Route Capture ..............................................................................................181
8.3.3 Hold..............................................................................................................189
8.3.4 Heading Capture ..........................................................................................189
8.4 BASIC ALGORITHMS REQUIRED FOR COMPLETE FUNCTIONALITY .......................189
8.4.1 Calculating the Aircraft Turn Radius ..........................................................190
8.4.2 Determining Distance to go Along a Segment .............................................190
8.4.3 Rhumb Line Bearing ....................................................................................190
8.4.4 Rhumb Line Distance ...................................................................................194
8.4.5 Creating Vectors Representing Segments ....................................................194
9. NAVIGATION ERROR MODELING ..............................................................187
9.1 VOR/DME NAVIGATION ....................................................................................187
9.1.1 Determining Aircraft Position from a VOR/DME Station ...........................187
9.1.2 Determining the Proper VOR/DME Station to use for Navigation .............193
9.2 GPS NAVIGATION ...............................................................................................196
9.3 ILS LOCALIZER ERROR MODEL ...........................................................................197
10. THE LONGITUDINAL GUIDANCE
SYSTEM 201
10.1 AIRCRAFT DEVICE DEPLOYMENT .....................................................................201
10.1.1 Flaps on Approach ...................................................................................201
10.1.2 Flaps on Takeoff .......................................................................................202
10.1.3 Speed Brakes .............................................................................................202
10.1.4 Spoiler .......................................................................................................203
10.1.5 Landing Gear ............................................................................................203
10.1.6 Ground Braking ........................................................................................203
10.2 PREPARING FOR APPROACH AND LANDING ......................................................204
10.2.1 Energy Gradient .......................................................................................204
10.2.2 Distance Remaining to Runway ................................................................208
10.3 CROSSING RESTRICTIONS IN DESCENT .............................................................214
11. FLIGHT TECHNICAL ERROR .......................................................................203
11.1 OPERATIONAL DETAILS ...................................................................................204
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11.1.1 Piloted Flight Technical Error .................................................................204
11.1.2 FMS Flight Technical Error .....................................................................204
11.1.3 ILS Flight Technical Error .......................................................................204
12. MODEL VERIFICATION AND VALIDATION .............................................206
12.1 CONSTANT AIRSPEED CLIMBS AND DESCENTS .................................................208
12.2 MACH/CAS DESCENTS AND CAS/MACH CLIMBS ............................................209
12.3 SPEED CHANGES ..............................................................................................211
12.4 SPEED CHANGES DURING CLIMBS AND DESCENTS ...........................................214
12.5 AUTOMATIC ROUTE CAPTURE ..........................................................................216
12.6 VECTORED ROUTE CAPTURE ............................................................................220
12.7 INITIAL FIX ROUTE CAPTURE ...........................................................................222
12.8 SEGMENT TRANSITION .....................................................................................223
12.9 FLIGHT TECHNICAL ERROR ..............................................................................224
12.10 NAVIGATION ERRORS ...................................................................................228
12.10.1 GPS Navigation Error ..............................................................................228
12.10.2 VOR/DME Error.......................................................................................229
12.11 TERMINAL FLIGHT PHASES ...........................................................................231
12.11.1 Take-Off ....................................................................................................231
12.11.2 Landing .....................................................................................................232
12.12 CONCLUSIONS ...............................................................................................234
APPENDIX A .................................................................................................................237
ANALYSIS OF THE TRANSFER FUNCTIONS OF THE LONGITUDINAL
DYNAMICS....................................................................................................................237
APPENDIX B ............................................................................................................ - 245 -
2ND ORDER GAUSS-MARKOV MODEL FOR GPS CLOCK ERROR DUE TO
SELECTIVE AVAILABILITY ............................................................................... - 245 -
11.2 DISCRETIZING THE CONTINUOUS 2ND ORDER GAUSS MARKOV PROCESS ... - 248 -
11.3 CALIBRATION OF PARAMETERS ................................................................... - 250 -
GLOSSARY....................................................................................................................254
ACRONYMS ..................................................................................................................256
REFERENCES .......................................................................................................... - 257 -
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List of Figures

Figure 2.1. The body fixed reference frame aligned with an aircraft ................................5
Figure 2.2. The relationship between body and inertial reference frames.......................6
Figure 2.3. The 3-2-1 Euler sequence of rotations used to quantify the aircraft's
orientation ............................................................................................................................6
Figure 2.4. The forces, moments, velocity components and angular rates of an aircraft ..8
Figure 2.5. Illustration of the stability and wind coordinate systems ............................10
Figure 2.6. Illustration of the Short Period mode causing oscillations about the
aircraft’s center of gravity .................................................................................................18
Figure 2.7. Illustration of the Phugoid mode .................................................................18
Figure 2.8. Illustration of the Dutch Roll mode ..............................................................19
Figure 2.9. Illustration of the Roll mode .........................................................................19
Figure 2.10. Illustration of an unstable Spiral mode .......................................................20
Figure 2.11. The DIS frame and the surface frame ..........................................................33
Figure 2.12. The longitude and latitude rotations ..........................................................33
Figure 2.13. The Spheroidal Earth model .......................................................................36
Figure 2.14. Triangle Representing Slope of Meridional Ellipse and Incremental
Latitude Change .................................................................................................................36
Figure 2.15. Alternate Triangle of Relationship Between Geodetic and Parametric
Latitude on the Meridional Ellipse ....................................................................................37
Figure 2.16 Position Vector on the Spheroid ....................................................................39
Figure 2.17. Maximum thrust vs altitude for a DC-9/MD80..........................................48
Figure 2.18. Fuel Consumption at maximum thrust (both engines) for a DC-9/MD80 ..50
Figure 2.19. Temperature vs altitude for the standard day atmosphere .........................51
Figure 2.20. The speed of sound variation with altitude for the standard day
atmosphere .........................................................................................................................52
Figure 2.21. Pressure variation with altitude for the standard atmosphere ..................53
Figure 2.22. Density variation with altitude for the standard atmosphere ......................54
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