143,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Get a complete understanding of aircraft control and simulation
Aircraft Control and Simulation: Dynamics, Controls Design, and Autonomous Systems, Third Edition is a comprehensive guide to aircraft control and simulation. This updated text covers flight control systems, flight dynamics, aircraft modeling, and flight simulation from both classical design and modern perspectives, as well as two new chapters on the modeling, simulation, and adaptive control of unmanned aerial vehicles. With detailed examples, including relevant MATLAB calculations and FORTRAN codes, this approachable yet detailed reference also provides access to supplementary materials, including chapter problems and an instructor's solution manual.
Aircraft control, as a subject area, combines an understanding of aerodynamics with knowledge of the physical systems of an aircraft. The ability to analyze the performance of an aircraft both in the real world and in computer-simulated flight is essential to maintaining proper control and function of the aircraft. Keeping up with the skills necessary to perform this analysis is critical for you to thrive in the aircraft control field.
- Explore a steadily progressing list of topics, including equations of motion and aerodynamics, classical controls, and more advanced control methods
- Consider detailed control design examples using computer numerical tools and simulation examples
- Understand control design methods as they are applied to aircraft nonlinear math models
- Access updated content about unmanned aircraft (UAVs)
Aircraft Control and Simulation: Dynamics, Controls Design, and Autonomous Systems, Third Edition is an essential reference for engineers and designers involved in the development of aircraft and aerospace systems and computer-based flight simulations, as well as upper-level undergraduate and graduate students studying mechanical and aerospace engineering.
Das E-Book können Sie in Legimi-Apps oder einer beliebigen App lesen, die das folgende Format unterstützen:
Seitenzahl: 1130
Veröffentlichungsjahr: 2015
Ähnliche
Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
CHAPTER 1: THE KINEMATICS AND DYNAMICS OF AIRCRAFT MOTION
1.1 INTRODUCTION
1.2 VECTOR OPERATIONS
1.3 MATRIX OPERATIONS ON VECTOR COORDINATES
1.4 ROTATIONAL KINEMATICS
1.5 TRANSLATIONAL KINEMATICS
1.6 GEODESY, COORDINATE SYSTEMS, GRAVITY
1.7 RIGID-BODY DYNAMICS
1.8 ADVANCED TOPICS
REFERENCES
PROBLEMS
CHAPTER 2: MODELING THE AIRCRAFT
2.1 INTRODUCTION
2.2 BASIC AERODYNAMICS
2.3 AIRCRAFT FORCES AND MOMENTS
2.4 STATIC ANALYSIS
2.5 THE NONLINEAR AIRCRAFT MODEL
2.6 LINEAR MODELS AND THE STABILITY DERIVATIVES
2.7 SUMMARY
REFERENCES
PROBLEMS
CHAPTER 3: MODELING, DESIGN, AND SIMULATION TOOLS
3.1 INTRODUCTION
3.2 STATE-SPACE MODELS
3.3 TRANSFER FUNCTION MODELS
3.4 NUMERICAL SOLUTION OF THE STATE EQUATIONS
3.5 AIRCRAFT MODELS FOR SIMULATION
3.6 STEADY-STATE FLIGHT
3.7 NUMERICAL LINEARIZATION
3.8 AIRCRAFT DYNAMIC BEHAVIOR
3.9 FEEDBACK CONTROL
3.10 SUMMARY
REFERENCES
PROBLEMS
CHAPTER 4: AIRCRAFT DYNAMICS AND CLASSICAL CONTROL DESIGN
4.1 INTRODUCTION
4.2 AIRCRAFT RIGID-BODY MODES
4.3 THE HANDLING QUALITIES REQUIREMENTS
4.4 STABILITY AUGMENTATION
4.5 CONTROL AUGMENTATION SYSTEMS
4.6 AUTOPILOTS
4.7 NONLINEAR SIMULATION
4.8 SUMMARY
REFERENCES
PROBLEMS
CHAPTER 5: MODERN DESIGN TECHNIQUES
5.1 INTRODUCTION
5.2 ASSIGNMENT OF CLOSED-LOOP DYNAMICS
5.3 LINEAR QUADRATIC REGULATOR WITH OUTPUT FEEDBACK
5.4 TRACKING A COMMAND
5.5 MODIFYING THE PERFORMANCE INDEX
5.6 MODEL-FOLLOWING DESIGN
5.7 LINEAR QUADRATIC DESIGN WITH FULL STATE FEEDBACK
5.8 DYNAMIC INVERSION DESIGN
5.9 SUMMARY
REFERENCES
PROBLEMS
CHAPTER 6: ROBUSTNESS AND MULTIVARIABLE FREQUENCY-DOMAIN TECHNIQUES
6.1 INTRODUCTION
6.2 MULTIVARIABLE FREQUENCY-DOMAIN ANALYSIS
6.3 ROBUST OUTPUT FEEDBACK DESIGN
6.4 OBSERVERS AND THE KALMAN FILTER
6.5 LINEAR QUADRATIC GAUSSIAN/LOOP TRANSFER RECOVERY
6.6 SUMMARY
REFERENCES
PROBLEMS
CHAPTER 7: DIGITAL CONTROL
7.1 INTRODUCTION
7.2 SIMULATION OF DIGITAL CONTROLLERS
7.3 DISCRETIZATION OF CONTINUOUS CONTROLLERS
7.4 MODIFIED CONTINUOUS DESIGN
7.5 IMPLEMENTATION CONSIDERATIONS
7.6 SUMMARY
REFERENCES
PROBLEMS
CHAPTER 8: MODELING AND SIMULATION OF MINIATURE AERIAL VEHICLES
8.1 INTRODUCTION
8.2 PROPELLER/ROTOR FORCES AND MOMENTS
8.3 MODELING ROTOR FLAPPING
8.4 MOTOR MODELING
8.5 SMALL AEROBATIC AIRPLANE MODEL
8.6 QUADROTOR MODEL
8.7 SMALL HELICOPTER MODEL
8.8 SUMMARY
REFERENCES
PROBLEMS
CHAPTER 9: ADAPTIVE CONTROL WITH APPLICATION TO MINIATURE AERIAL VEHICLES
9.1 INTRODUCTION
9.2 MODEL REFERENCE ADAPTIVE CONTROL BASED ON DYNAMIC INVERSION
9.3 NEURAL NETWORK ADAPTIVE CONTROL
9.4 LIMITED AUTHORITY ADAPTIVE CONTROL
9.5 NEURAL NETWORK ADAPTIVE CONTROL EXAMPLE
9.6 SUMMARY
REFERENCES
PROBLEMS
Apppendix A: F-16 MODEL
A.1 MASS PROPERTIES
A.2 WING DIMENSIONS
A.3 REFERENCE CG LOCATION
A.4 CONTROL SURFACE ACTUATOR MODELS
A.5 ENGINE ANGULAR MOMENTUM
A.6 STANDARD ATMOSPHERE MODEL
A.7 ENGINE MODEL
A.8 AERODYNAMIC DATA
Appendix B: SOFTWARE
B.1 AIRCRAFT STEADY-STATE TRIM CODE
B.2 NUMERICAL LINEARIZATION SUBROUTINE
B.3 RUNGE-KUTTA INTEGRATION
B.4 OUTPUT FEEDBACK DESIGN
Index
End User License Agreement
Pages
iv
v
xi
xii
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
Guide
Cover
Table of Contents
Begin Reading
List of Illustrations
CHAPTER 1: THE KINEMATICS AND DYNAMICS OF AIRCRAFT MOTION
Figure 1.2-1 Rotation of a vector.
Figure 1.3-1 A plane rotation of coordinates.
Figure 1.4-1 A vector derivative in a rotating frame.
Figure 1.5-1 Velocity and acceleration in moving frames.
Figure 1.6-1 The geoid and definitions of height.
Figure 1.6-2 The oblate spheroidal model of the Earth.
Figure 1.6-3 The geometry of a point above the spheroid.
CHAPTER 2: MODELING THE AIRCRAFT
Figure 2.2-1 Definitions associated with an airfoil.
Figure 2.2-2 Airfoil moment about different axes.
Figure 2.2-3 Typical plots of lift, drag, and moment coefficients.
Figure 2.2-4 Types of aircraft wing planform.
Figure 2.3-1 Definitions of axes and aerodynamic angles.
Figure 2.3-2a Transonic drag rise for a fighter aircraft.
Figure 2.3-2b Drag coefficient of a fighter aircraft.
Figure 2.3-3a Lift coefficient of a low-speed transport aircraft.
Figure 2.3-3b Lift coefficient of a fighter aircraft.
Figure 2.3-3c Normal force coefficient of the F-4E aircraft.
Figure 2.3-4 Sideforce coefficient of the F-4B, C aircraft.
Figure 2.3-5 Rolling moment coefficient of the F-4B aircraft.
Figure 2.3-6 Roll damping derivative of the F-4C aircraft.
Figure 2.3-7a Pitching moment coefficient of a low-speed transport aircraft.
Figure 2.3-7b Pitching moment coefficient of a jet trainer aircraft.
Figure 2.3-8a Yawing moment coefficient of the F-4B, C aircraft.
Figure 2.3-8b Yawing moment coefficient of a jet trainer aircraft.
Figure 2.3-9 Sideforce coefficient of the F-16 model.
Figure 2.4-1 Diagram for calculating pitching moment.
Figure 2.6-1 Pitching moment derivatives of a jet trainer aircraft.
CHAPTER 3: MODELING, DESIGN, AND SIMULATION TOOLS
Figure 3.1-1 State-space operations.
Figure 3.2-1 A mechanical system.
Figure 3.2-2 State variables for an electrical circuit.
Figure 3.2-3 A SISO simulation diagram.
Figure 3.2-4 A general second-order SISO simulation diagram.
Figure 3.2-5 State equation simulation diagram.
Figure 3.3-1 -Plane vectors representing pole-zero factors.
Figure 3.3-2 Simulation diagram for a simple lead.
Figure 3.3-3 Bode plots for a simple lead.
Figure 3.3-4 Geometrical properties of a quadratic lag.
Figure 3.3-5 Bode gain plot for a quadratic lag.
Figure 3.3-6 Bode plots for a lead compensator.
Figure 3.3-7 Step response of a quadratic lag.
Figure 3.4-1 Time-history simulation.
Figure 3.4-2 A Van der Pol limit cycle.
Figure 3.5-1 Transport aircraft model.
Figure 3.5-2 Model of the F-16 aircraft.
Figure 3.6-1 Steady-state trim flowchart.
Figure 3.6-2 Terminal display for trim.
Figure 3.6-3 F-16 model, trimmed power curve.
Figure 3.6-4 Transport aircraft, elevator-doublet response.
Figure 3.6-5 Transport aircraft, throttle-doublet response.
Figure 3.6-6 Simulation results for F-16 model.
Figure 3.6-7 The ground track of a coordinated turn.
Figure 3.7-1 Flowchart for linearization.
Figure 3.7-2 Jacobian matrices for a pull-up.
Figure 3.7-3 Jacobian matrix for 4.5 turn.
Figure 3.8-1a Bode gain plot, elevator to pitch rate.
Figure 3.8-1b Bode phase plot of the pitch-rate transfer function.
Figure 3.9-1 Feedback control: single-loop configuration.
Figure 3.9-2 Feedback control with an inner loop.
Figure 3.9-3 Cascaded state-space systems.
Figure 3.9-4 Transforming to unity feedback.
Figure 3.9-5 Disturbance input, transfer function description.
Figure 3.9-6 Disturbance input, state-space description.
Figure 3.9-7 Integrator windup protection.
Figure 3.9-8 (
a
) A Nyquist D-contour; (
b
) a Nyquist plot.
Figure 3.9-9 Stability margins on the Nyquist plot.
Figure 3.9-10 PI compensation with no closed-loop zero.
Figure 3.9-11 Lead compensation on the root-locus plot.
Figure 3.9-12 PI compensation on the root-locus plot.
Figure 3.9-13 Step response with PI compensation.
Figure 3.9-14 Lead compensator polar plot.
Figure 3.9-15a Lag-compensated Bode plots.
Figure 3.9-15b Lag-compensated Nyquist plot.
Figure 3.9-16 Lead-compensated Bode plots.
Figure 3.9-17 Lead-compensated step response.
Figure 3.9-18 Step response with feedback lead compensation.
CHAPTER 4: AIRCRAFT DYNAMICS AND CLASSICAL CONTROL DESIGN
Figure 4.1-1 Aircraft altitude-Mach envelope.
Figure 4.1-2 An electromechanical control system.
Figure 4.3-1 The -star envelope.
Figure 4.3-2 Closed-loop frequency response for Neal-Smith criterion.
Figure 4.3-3 Neal-Smith evaluation chart.
Figure 4.4-1 Pitch-axis stability augmentation.
Figure 4.4-2a Inner-loop root-locus plot for pitch SAS.
Figure 4.4-2b Expanded inner-loop root-locus plot for pitch SAS.
Figure 4.4-3 Outer-loop root-locus plot for pitch SAS.
Figure 4.4-4 Lateral-directional augmentation.
Figure 4.4-5 F-16 model roll time constant versus alpha in degrees.
Figure 4.4-6 Root-locus plot for the roll damping loop.
Figure 4.4-7 Root-locus plot for the yaw-rate loop.
Figure 4.4-8 Alternate yaw-rate root locus.
Figure 4.4-9 Roll-rate response to an aileron doublet.
Figure 4.5-1 Pitch-rate control augmentation.
Figure 4.5-2 Root-locus plot for the pitch-rate CAS.
Figure 4.5-3 Step response of the pitch-rate CAS.
Figure 4.5-4 Normal acceleration control augmentation.
Figure 4.5-5 Root-locus plot for the normal acceleration CAS.
Figure 4.5-6 Normal acceleration CAS; step and C-star responses.
Figure 4.5-7 A lateral-directional CAS.
Figure 4.5-8 Root-locus plot for lateral acceleration feedback.
Figure 4.6-1 A pitch-attitude autopilot.
Figure 4.6-2 Root-locus plot for pitch-rate feedback.
Figure 4.6-3 Step response of the pitch-attitude controller.
Figure 4.6-4 Bode plots for the pitch-attitude controller.
Figure 4.6-5 Step response of the pitch-attitude controller.
Figure 4.6-6 An altitude-hold autopilot.
Figure 4.6-7 Bode plots for the altitude-hold controller.
Figure 4.6-8 Step-response of the altitude-hold controller.
Figure 4.6-9 Glide-slope geometry for autoland.
Figure 4.6-10 Control system for automatic landing.
Figure 4.6-11a Bode gain plot for the automatic-landing
d
-loop.
Figure 4.6-12 A roll angle control system.
Figure 4.6-13 Root-locus plot for roll-angle-hold controller.
Figure 4.6-14 A heading-hold control system.
Figure 4.6-15 A VOR-hold autopilot.
Figure 4.7-1 Aircraft trajectory in the vertical plane.
Figure 4.7-2 Aircraft pitch-rate response along the trajectory.
Figure 4.7-3a Aircraft trajectory in the vertical plane.
Figure 4.7-3b Angle of attack versus time.
Figure 4.7-3d Aircraft body-axes angular rates along trajectory.
Figure 4.7-3e Control surface deflections along trajectory.
Figure 4.7-3f Sideslip angle variation along trajectory.
Figure 4.7-3g True airspeed variation along trajectory.
Figure 4.7-4a Automatic landing; elevation profile.
Figure 4.7-4b Automatic landing; deviation from glide path.
Figure 4.7-4c Automatic landing; controlled variables.
Figure 4.7-4d Automatic landing; throttle variation.
Figure 4.7-5 Landing-flare geometry
Figure 4.7-6a Automatic landing; glide slope and flare trajectory.
Figure 4.7-7a Ground track during roll angle steering.
Figure 4.7-7b Bank angle during roll angle steering.
Figure 4.7-7e Aileron deflection during roll angle steering.
Figure 4.7-8a Simulation of limiting; alpha variation.
Figure 4.7-8b Simulation of limiting; control surface deflections.
Figure 4.7-8c Simulation of limiting; sideslip variation.
Figure 4.7-8d Simulation of limiting; aircraft angular rates.
Figure 4.7-8e Simulation of limiting; roll angle variation.
Figure 4.7-8f Simulation of limiting; pitch-attitude variation.
Figure 4.7-8g Simulation of limiting; airspeed variation.
CHAPTER 5: MODERN DESIGN TECHNIQUES
Figure 5.2-1 Plant with regulator.
Figure 5.2-2 Closed-loop response to angle-of-attack initial condition.
Figure 5.3-1 Closed-loop lateral response: (
a
) dutch roll states and ; (
b
) roll-mode states and .
Figure 5.3-2 Effect of PI weighting parameters: (
a
) sideslip as a function of ; (
b
) sideslip as a function of .
Figure 5.4-1 Plant with compensator of desired structure.
Figure 5.4-2 Plant/feedback structure.
Figure 5.4-3 G-command system.
Figure 5.4-4 Normal acceleration step response.
Figure 5.5-1 Closed-loop lateral response: (
a
) dutch roll states and
r
; (
b
) spiral and roll subsidence states and
p
.
Figure 5.5-2 Normal acceleration step response.
Figure 5.5-3 Pitch-rate control system.
Figure 5.5-4 Pitch-rate step response using time-dependent weighting design.
Figure 5.5-5 Pitch-rate step response using derivative weighting design.
Figure 5.5-6 Wing-leveler lateral control system.
Figure 5.5-7 Closed-loop response to a command of , . Bank angle (rad) and washed-out yaw rate (rad/s).
Figure 5.5-8 Glide-slope coupler.
Figure 5.5-9 Glide-slope coupler responses: (
a
) altitude
h
(ft); (
b
) off-glide path distance
d
(ft). (
c
) Angle of attack
α
and pitch angle (deg); (
d
) velocity deviation . (
e
) Control efforts (rad) and (per unit).
Figure 5.6-1 Explicit model-following command generator tracker for .
Figure 5.6-2 Flare-path geometry.
Figure 5.6-3 Automatic flare control system.
Figure 5.6-4 Controlled flare, altitude in feet.
Figure 5.6-5 Aircraft response during controlled flare: (
a
) flight-path angle (deg); (
b
) elevator command (deg).
Figure 5.8-1 Dynamic inversion controller.
Figure 5.8-2 Dynamic inversion controller and simulation code.
Figure 5.8-3 Modified controlled variable.
Figure 5.8-4 Pitch rate .
Figure 5.8-5 Nonlinear dynamic inversion controller.
Figure 5.8-6 Nonlinear model of aircraft longitudinal dynamics.
Figure 5.8-7 Nonlinear dynamic inversion controller (Part I). Nonlinear dynamic inversion controller (Part II).
CHAPTER 6: ROBUSTNESS AND MULTIVARIABLE FREQUENCY-DOMAIN TECHNIQUES
Figure 6.2-1 Standard feedback configuration.
Figure 6.2-2 Typical Bode plots for the uncertain signals in the system: (
a
) disturbance magnitude; (
b
) measurement noise magnitude.
Figure 6.2-3 MIMO Bode magnitude plot of SVs versus frequency.
Figure 6.2-4 SISO Bode magnitude plots for F-16 lateral dynamics.
Figure 6.2-5 Singular values for F-16 lateral dynamics.
Figure 6.2-6 Magnitude specifications on , , and .
Figure 6.2-9 Frequency-domain performance specifications.
Figure 6.2-7 Plant augmented with integrators.
Figure 6.2-8 MIMO Bode magnitude plot for augmented plant.
Figure 6.2-10 MIMO Bode magnitude plots of SVs: (
a
) actual plant; (
b
) reduced-order approximation.
Figure 6.2-11 High-frequency stability robustness bound: (
a
) ; (
b
) .
Figure 6.3-1 Frequency-domain magnitude plots and robustness bounds.
Figure 6.3-2 Optimal pitch-rate step response.
Figure 6.4-1 State observer.
Figure 6.4-2 Actual and estimated states.
Figure 6.4-3 Gaussian PDF.
Figure 6.4-4 Vertical wind gust spectral density.
Figure 6.4-5 Regulator design using observer and full state feedback.
Figure 6.5-1 Typical polar plot for optimal LQ return difference (referred to the plant input).
Figure 6.5-2 Definition of multivariable phase margin.
Figure 6.5-3 Guaranteed phase margin of the LQR.
Figure 6.5-4 (
a
) Loop gain with full state feedback; (
b
) regulator using observer and estimate feedback; (
c
) regulator loop gain.
Figure 6.5-5 Aircraft turn coordinator control system.
Figure 6.5-6 Singular values of the basic aircraft dynamics.
Figure 6.5-7 Singular values of aircraft augmented by integrators.
Figure 6.5-8 Singular values of aircraft augmented by integrators and inverse dc gain matrix
P
.
Figure 6.5-9 Multiplicative uncertainty bound 1/m(ω) for the aircraft dynamical model.
Figure 6.5-10 Singular values of Kalman filter open-loop gain
C
Φ(
s
)
L
: (
a
) for
r
f
= 1, including robustness bounds; (
b
) for various values of
r
f
.
Figure 6.5-11 Step responses of target feedback loop : (
a
) ; (
b
) ; (
c
) .
Figure 6.5-12 Singular-value plots for the LQG regulator: (
a
) LQG with ; (
b
) LQG with ; (
c
) LQG with , including robustness bounds.
Figure 6.5-13 Closed-loop step responses of the LQG regulator: (
a
) LQG with ; (
b
) LQG with ; (
c
) LQG with .
CHAPTER 7: DIGITAL CONTROL
Figure 7.2-1 Digital controller.
Figure 7.2-2 Data reconstruction using a ZOH: (
a
) discrete control sequence ; (
b
) reconstructed continuous signal .
Figure 7.2-3 Digital control simulation scheme.
Figure 7.2-4 Digital control simulation driver program.
Figure 7.4-1 Sampling in the frequency domain: (
a
) spectrum of ; (
b
) spectrum of sampled signal .
Figure 7.3-1 Continuous pitch-rate controller.
Figure 7.3-2 Digital pitch-rate controller.
Figure 7.3-3 Digital simulation software: (
a
) FORTRAN subroutine to simulate digital pitch-rate controller; (
b
) subroutine
to simulate continuous plant dynamics.
Figure 7.3-4 Effect of sampling period: (
a
) step response ; (
b
) control input .
Figure 7.4-2 Example of aliasing in the time domain.
Figure 7.4-3 ZOH impulse response.
Figure 7.4-4 ZOH Bode plots: (
a
) magnitude; (
b
) phase.
Figure 7.4-5 Modified continuous plant with anti-aliasing filter and compensation to model hold device and computation delays.
Figure 7.4-6 Step response using modified continuous-time pitch-rate controller.
Figure 7.4-7 Response of digital controller using modified and unmodified continuous-time design.
Figure 7.4-8 Control input required for modified digital pitch-rate controller.
Figure 7.5-1 Actuator saturation function.
Figure 7.5-2 Flight control system including actuator saturation.
Figure 7.5-3 Fortran code implementing proportional-integral controller with anti-windup compensation.
Figure 7.5-4 Implementations of second-order digital filters: (
a
) direct form 1, D1; (
b
) direct form 3, D3; (
c
) cross-coupled form 1, X1.
CHAPTER 8: MODELING AND SIMULATION OF MINIATURE AERIAL VEHICLES
Figure 8.1-1 Typical fixed-pitch propeller used for small airplanes and multirotors.
Figure 8.1-2 A Multirotor with four fixed-pitch propellers (quadrotor) is able to independently control thrust, roll moment, pitch moment, and yaw moment.
Figure 8.1-3 Variable-pitch propeller mechanism schematic typical for a small helicopter tail rotor.
Figure 8.1-4 Swashplate mechanism schematic typical for a helicopter main rotor.
Figure 8.1-5 A Helicopter is able to independently control thrust, roll moment, pitch moment, and yaw moment via the collective and cyclic pitch of the main rotor and the pitch of the tail rotor.
Figure 8.1-6 Helicopter main rotor with a stabilizer bar.
Figure 8.2-1 Modeling the rotational degree of freedom of a propeller.
Figure 8.2-2 Idealized flow through a propeller.
Figure 8.2-3 Idealized flow through a rotor in forward flight.
Figure 8.2-4 Using the blade element method to estimate propeller/rotor thrust.
Figure 8.2-5 Finding rotor thrust and induced velocity numerically.
Figure 8.2-6 H-force acting on a propeller/rotor.
Figure 8.3-1 Blade first-order harmonic flapping motion (exaggerated).
Figure 8.3-2 Effective hinge offset of a rotor blade.
Figure 8.3-3 Stabilizer bar geometry.
Figure 8.4-1 Notional engine torque/power curves.
Figure 8.5-1 33% Zivko Scale Edge 540T made by Aeroworks.
Figure 8.6-1 Modified AscTec Pelican Quadrotor as configured at the Georgia Institute of Technology.
Figure 8.7-1 Yamaha RMAX as configured at the George Institute of Technology.
CHAPTER 9: ADAPTIVE CONTROL WITH APPLICATION TO MINIATURE AERIAL VEHICLES
Figure 9.2-1 Use of adaptation to correct model error in a dynamic inversion controller.
Figure 9.3-1 Parametric (left) and nonparametric (right) NN structures.
Figure 9.3-2 State history (top) and input history (bottom) for Example 9.3-1.
Figure 9.3-3 Neural network parameters for Example 9.3-1.
Figure 9.4-1 Use of PCH to address input saturation.
Figure 9.4-2 State history (top) and input history (bottom) for Example 9.4-1.
Figure 9.4-3 Neural network parameter for Example 9.4-1.
Figure 9.4-4 Use of PCH to address systems in cascade (for clarity, full state/input feedback is not shown to all elements).
Figure 9.5-1 Integrated adaptive guidance, navigation, and control systems.
Figure 9.5-2 Adaptive GNC position tracking for rapidly flown small box pattern from simulation of quadrotor.
Figure 9.5-3 Adaptive GNC attitude angles for rapidly flown small box pattern from simulation of quadrotor.
Figure 9.5-4 Adaptive GNC motor states for rapidly flown small box pattern from simulation of quadrotor.
Figure 9.5-5 Visualization of Yamaha RMAX in simulation environment.
Figure 9.5-6 Adaptive GNC 180° heading command step response (left) from simulation of RMAX.
Figure 9.5-7 Adaptive GNC 30 ft longitudinal position command step response (forward) from simulation of RMAX.
Figure 9.5-8 Adaptive GNC 30 ft lateral position command step response (left) from simulation of RMAX.
Figure 9.5-9 Adaptive GNC 20 ft altitude command step response (down) from simulation of RMAX.
Figure 9.5-10 Adaptive GNC position and altitude tracking step response on all axes simultaneously from simulation of RMAX.
Figure 9.5-11 Adaptive GNC velocity during step response on all axes simultaneously from simulation of RMAX.
Figure 9.5-12 Adaptive GNC altitude during step response on all axes simultaneously from simulation of RMAX.
Figure 9.5-13 Adaptive GNC inputs during step response on all axes simultaneously from simulation of RMAX.
Figure 9.5-14 Adaptive GNC position tracking during complex flight pattern from simulation of RMAX.
Figure 9.5-15 Adaptive GNC altitude and speed tracking during complex flight pattern from simulation of RMAX.
Figure 9.5-16 Adaptive GNC altitude during complex flight pattern from simulation of RMAX.
Figure 9.5-17 Adaptive GNC inputs during complex flight pattern from simulation of RMAX.
Figure 9.5-18 Adaptive GNC 180° heading command step response (left) from flight test of RMAX.
Figure 9.5-19 Adaptive GNC 30 ft longitudinal position command step response (forward) from flight test of RMAX.
Figure 9.5-20 Adaptive GNC 30 ft lateral position command step response (left) from flight test of RMAX.
Figure 9.5-21 Adaptive GNC 20 ft altitude command step response (down) from flight test of RMAX.
Figure 9.5-22 Adaptive GNC tracking during racetrack pattern at 50 ft/s from flight test of RMAX.
Figure 9.5-23 Adaptive GNC speed during racetrack pattern from flight test of RMAX.
Figure 9.5-24 Adaptive GNC altitude during racetrack pattern from flight test of RMAX.
Figure 9.5-25 Adaptive GNC inputs during racetrack pattern from flight test of RMAX.
List of Tables
CHAPTER 2: MODELING THE AIRCRAFT
Table 2.2-1 Important Wing Planform Parameters
Table 2.3-1 Force, Moment, and Velocity Definitions
Table 2.3-2 Aircraft Drag Components
Table 2.5-1 The Flat-Earth, Body-Axes 6-DoF Equations
Table 2.6-1 The Force Dimensional Derivatives
Table 2.6-2 The Moment Dimensional Derivatives
Table 2.6-3 Longitudinal Dimensional versus Dimensionless Derivatives
Table 2.6-4 Lateral-Directional Dimensional versus Dimensionless Derivative
Table 2.6-5 Importance of Longitudinal Stability Derivatives
Table 2.6-6 Importance of Lateral-Directional Derivatives
CHAPTER 3: MODELING, DESIGN, AND SIMULATION TOOLS
Table 3.3-1 Network Transfer Functions and State Equations
Table 3.5-1 Aircraft Control Surface Sign Conventions
Table 3.5-2 F-16 Model Test Case
Table 3.6-1 Trim Data for the Transport Aircraft Model
Table 3.6-2 Trim Data for the F-16 Model
Table 3.6-3 Trimmed Flight Conditions for the F-16
Table 3.8-1 F-16 Model, Elevator-to-Pitch-Rate Transfer Function
CHAPTER 4: AIRCRAFT DYNAMICS AND CLASSICAL CONTROL DESIGN
Table 4.2-1 Accuracy of Short-Period and Phugoid Formulas
Table 4.2-2 Effect of Flight-Path Angle on F-16 Modes
Table 4.2-3 Effect of Speed and Altitude on F-16 Modes
Table 4.3-1 Pilot Opinion Rating and Flying Qualities Level
Table 4.3-2 Definitions—Flying Qualities Specifications
Table 4.3-3a Short-Period Damping Ratio Limits
Table 4.3-3b Limits on
Table 4.3-4 Maximum Roll-Mode Time Constant (s)
Table 4.3-5 Spiral-Mode Minimum Doubling Time
Table 4.3-6 Dutch-Roll-Mode Specifications
Table 4.5-1 Transfer Function Zeros versus Accelerometer Position
Table 4.5-2 Trim Conditions for Determining ARI Gain
CHAPTER 5: MODERN DESIGN TECHNIQUES
Table 5.2-1 Desired and Achievable Eigenvectors
Table 5.3-1 LQR with Output Feedback
Table 5.3-2 Optimal Output Feedback Solution Algorithm
Table 5.4-1 LQ Tracker with Output Feedback
Table 5.5-1 LQ Tracker with Time-Weighted PI
Table 5.7-1 LQR with State Feedback
CHAPTER 6: ROBUSTNESS AND MULTIVARIABLE FREQUENCY-DOMAIN TECHNIQUES
Table 6.4-1 The Kalman Filter
CHAPTER 7: DIGITAL CONTROL
Table 7.4-1 Padé Approximants to
e
−
s
Δ
for Approximation of Computation Delay
Table 7.4-2 Approximants to (
1
−
e
−
sT
)
sT
for Approximation of Hold Delay
Table 7.5-1 Elements of Second-Order Modules
Table 7.5-2 Difference Equation Implementation of Second-Order Modules
CHAPTER 8: MODELING AND SIMULATION OF MINIATURE AERIAL VEHICLES
Table 8.1-1 Common Propeller and Rotor Classes
Table 8.5-1 Aeroworks Edge Example Model Parameters: Mass Properties
Table 8.5-2 Aeroworks Edge Example Model Parameters: Wing
Table 8.5-3 Aeroworks Edge Example Model Parameters: Horizontal Tail
Table 8.5-4 Aeroworks Edge Example Model Parameters: Vertical Tail
Table 8.5-5 Aeroworks Edge Example Model Parameters: Fuselage
Table 8.5-6 Aeroworks Edge Example Model Parameters: Engine and Propeller
Table 8.6-1 AscTec Pelican Example Model Parameters: Mass Properties
Table 8.6-2 AscTec Pelican Example Model Parameters: Rotor (each)
Table 8.6-3 AscTec Pelican Example Model Parameters: Motor and Speed Control (each)
Table 8.7-1 Yamaha RMAX Example Model Parameters: Mass Properties
Table 8.7-2 Yamaha RMAX Example Model Parameters: Main Rotor
Table 8.7-3 Yamaha RMAX Example Model Parameters: Tail Rotor
Table 8.7-5 Yamaha RMAX Example Model Parameters: Fuselage and Tail
CHAPTER 9: ADAPTIVE CONTROL WITH APPLICATION TO MINIATURE AERIAL VEHICLES
Table 9.5-1 AscTec Pelican Controller Parameter Choices for the Results Presented in this Section
Table 9.5-2 Yamaha RMAX Controller Parameter Choices for the Results Presented in this Section
AIRCRAFT CONTROL AND SIMULATIONThird Edition
Dynamics, Controls Design, and Autonomous Systems
BRIAN L. STEVENSFRANK L. LEWISERIC N. JOHNSON
Cover image: Space Shuttle Landing© 3Dsculptor/Shutterstock
Cover design: Wiley
Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section~107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with the respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for damages arising herefrom.
For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com.
Library of Congress Cataloging-in-Publication Data is available.
978-1-118-87098-3 (hardback)
978-1-118-87099-0 (epdf)
978-1-118-87097-6 (epub)
To Deane, Bill, and Richard
B.L.S.
To my sons, Chris and Roma
F.L.
To Amy, Elliot, and Theresa
E.N.J.
Preface
This book is primarily aimed at students in aerospace engineering, at the senior and graduate level. We hope that it will also prove useful to practicing engineers, both as a reference book and as an update to their engineering education. In keeping with the rising importance of autonomous aircraft systems in the world today, the third edition includes two new chapters that cover principles of unmanned aerial vehicle design and control.
As the subtitle suggests, the book can be viewed as having three Parts. Part I comprises Chapters 1–4 and presents aircraft Kinematics and Dynamics, Modeling, and Simulation, with numerous design examples using classical control methods. Part II, consisting of Chapter 5–7, covers Modern design techniques including Linear Quadratic design, which is based on optimality principles. Also included are LQG/Loop-Transfer Recovery and digital control implementation. Part III contains two newly added Chapters 8 and 9 that that detail the modeling, simulation, and control of small unmanned aerial vehicles.
In addressing simulation of aerospace vehicles we have reviewed the relevant parts of classical mechanics and attempted to provide a clear, consistent notation. This has been coupled with a thorough treatment of six-degrees-of-freedom (6-DOF) motion, including a detailed discussion of attitude representation using both Euler angles and quaternions. Simulation of motion over and around the Earth requires some understanding of geodesy and the Earth's gravitation, and these topics have also been discussed in some detail within the framework of the WGS-84 datum. Familiarity with these topics is indispensable to many of the engineers working in the aerospace industry. Given this background the student can independently construct 6-DOF simulations and learn from them.
High-speed motion within the Earth's atmosphere entails aerodynamic forces and moments. We have reviewed aerodynamic modeling, and provided many graphical examples of such forces and moments for real aircraft. The small-perturbation theory of aerodynamic forces and moments is also described in detail. This study of 6-DOF motion and aerodynamic effects culminates in two realistic nonlinear aircraft models, which are then used for design and simulation examples in the rest of the book.
We have provided computer code in both MATLAB and Fortran to perform simulation and design with these models. Involvement with the models and designs will demonstrate many ideas in simulation, control theory, computer-aided design techniques, and numerical algorithms. The design examples are easily reproducible, and offer a great deal of scope to a class of students.
Before starting feedback control design we have reviewed linear systems theory, including the Laplace transform, transfer functions, and the state-space formulation. Transform theory views dynamic systems through their poles and zeros and leads to many convenient graphical and back-of-the-envelope design techniques, while state-space techniques are ideally suited to computer-aided design. We have attempted to pass “seamlessly” between the two formulations.
Classical control design is illustrated through many examples performed on the aircraft models using transform domain techniques supported by an underlying state-space model. Modern design in the later chapters simply uses the state-space models.
Finally, we note that the choice of topics herein is influenced by our experience in the broader area of guidance, navigation, and control (GNC). Very few engineers entering the aerospace industry will find themselves designing flight control systems, and those few will take part in the design of only two or three such systems in their careers. Instead, they will find themselves involved in a broad spectrum of projects, where a good grasp of classical mechanics, dynamics, coordinate transformations, geodesy, and navigation will be invaluable. The importance of modeling and simulation cannot be overstated. Large sums of money are spent on mathematical modeling and digital simulation before any hardware is built.
The first and third authors wish to acknowledge the help of colleagues in Aerospace Engineering at Georgia Tech. Prof. C. V. Smith provided invaluable help with Chapter 1 during many hours of interesting discussion. The computer support of B. H. Hudson at the Georgia Tech Research Institute is also gratefully acknowledged. Both authors wish to thank the staff of John Wiley & Sons for their painstaking preparation of the manuscript.
Brian L. StevensGeorgia Institute of TechnologyFrank L. LewisUniversity of Texas at ArlingtonEric N. JohnsonGeorgia Institute of Technology
CHAPTER 1THE KINEMATICS AND DYNAMICS OF AIRCRAFT MOTION
1.1 INTRODUCTION
In this chapter the end point will be the equations of motion of a rigid vehicle moving over the oblate, rotating Earth. The flat-Earth equations, describing motion over a small area of a nonrotating Earth, with constant gravity, are sufficient for many aircraft simulation needs and will be derived first. To reach this end point we will use the vector analysis of classical mechanics to set up the equations of motion, matrix algebra to describe operations with coordinate systems, and concepts from geodesy (a branch of mathematics dealing with the shape of the Earth), gravitation (the mass attraction effect of the Earth), and navigation, to introduce the effects of Earth's shape and mass attraction.
The moments and forces acting on the vehicle, other than the mass attraction of the Earth, will be abstract until Chapter 2 is reached. At this stage the equations can be used to describe the motion of any type of aerospace vehicle, including an Earth satellite, provided that suitable force and moment models are available. The term rigid means that structural flexibility is not allowed for, and all points in the vehicle are assumed to maintain the same relative position at all times. This assumption is good enough for flight simulation in most cases as well as for flight control system design provided that we are not trying to design a system to control structural modes or to alleviate aerodynamic loads on the aircraft structure.
The vector analysis needed for the treatment of the equations of motion often causes difficulties for the student, particularly the concept of the angular velocity vector. Therefore, a review of the relevant topics is provided. In some cases we have gone beyond the traditional approach to flight mechanics. The introduction of topics from geodesy, gravitation, and distance and position calculations allows us to accurately simulate the trajectories of aircraft that can fly autonomously at very high altitudes and over long distances, including “point-to-point suborbital flight” (e.g., White Knight 2 and SpaceShipTwo). Some topics have been reserved for an “optional” advanced section (e.g., quaternions), Section 1.8.
The equations of motion will be organized as a set of simultaneous first-order differential equations, explicitly solved for the derivatives. For n independent variables, Xi (such as components of position, velocity, etc.), and m control inputs, Ui (such as throttle, control surface deflection, etc.), the general form will be
where the functions fi are the nonlinear functions that can arise from modeling real systems. If the variables Xi constitute the smallest set of variables that, together with given inputs Ui, completely describe the behavior of the system, then the Xi are a set of state variables for the system, and Equations (1.1-1) are a state-space description of the system. The functions fi are required to be single-valued continuous functions. Equations (1.1-1) are often written symbolically as
where the state vector X is an (n × 1) column array of the n state variables, the control vector U is an (m × 1) column array of the control variables, and f is an array of nonlinear functions. When U is held constant, the nonlinear state equations (1.1-1), or a subset of them, usually have one or more equilibrium points in the multidimensional state and control space, where the derivatives vanish. The equations are usually approximately linear for small perturbations from equilibrium and can be written in matrix form as the linear state equation:
Here, the lowercase notation for the state and control vectors indicates that they are perturbations from equilibrium, although the derivative vector contains the actual values (i.e., perturbations from zero). The “A-matrix” is square and the “B-matrix” has dimensions determined by the number of states and controls.
The state-space formulation will be described in more detail in Chapters 2 and 3. At this point we will simply note that a major advantage of this formulation is that the nonlinear state equations can be solved numerically. The simplest numerical solution method is Euler integration, described by
in which is the kth value of the state vector computed at discrete times , starting from an initial condition X0. The integration time step, , must be made small enough that, for every interval, U can be approximated by a constant value, and provides a good approximation to the increment in the state vector. This numerical integration allows the state vector to be stepped forward, in time increments of , to obtain a time-history simulation.
1.2 VECTOR OPERATIONS
Definitions and Notation
Kinematics can be defined as the study of the motion of objects without regard to the mechanisms that cause the motion. The motion of physical objects can be described by means of vectors in three dimensions, and in performing kinematic analysis with vectors we will make use of the following definitions:
Frame of Reference
: A rigid body or set of rigidly related points that can be used to establish distances and directions (denoted by
, etc.). In general, a subscript used to indicate a frame will be lowercase, while a subscript used to indicate a point will be uppercase.
Inertial Frame
: A frame of reference in which Newton's laws apply. Our best inertial approximation is probably a “helio-astronomic” frame in which the center of mass (cm) of the sun is a fixed point, and fixed directions are established by the normal to the plane of the ecliptic and the projection on that plane of certain stars that appear to be fixed in position.
Vector
: A vector is an abstract geometrical object that has both magnitude and direction. It exists independently of any coordinate system. The vectors used here are Euclidean vectors that exist only in three-dimensional space and come in two main types:
Bound Vector
: A vector from a fixed point in a frame (e.g., a position vector).
Free Vector
: Can be translated parallel to itself (e.g., velocity, torque).
Coordinate System
: A measurement system for locating points in a frame of reference. We may have multiple coordinate systems (with no relative motion) within one frame of reference, and we sometimes loosely refer to them also as “frames.”
In choosing a notation the following facts must be taken into account. For position vectors, the notation should specify the two points whose relative position the vector describes. Velocity and acceleration vectors are relative to a frame of reference, and the notation should specify the frame of reference as well as the moving point. The derivative of a vector depends on the observer's frame of reference, and this frame must be specified in the notation. A derivative may be taken in a different frame from that in which a vector is defined, so the notation may require two frame designators with one vector. We will use the following notation:
Vectors will be in boldface type fonts.
Right subscripts
will be used to designate two points for a position vector, and a point and a frame for a velocity or acceleration vector. A “
” in a subscript will mean “with respect to.”
A
left superscript
will specify the frame in which a derivative is taken, and the dot notation will indicate a derivative.
A
right superscript
on a vector will specify a coordinate system. It will therefore denote an array of the components of that vector in the specified system.
Vector length will be denoted by single bars, for example,
.
Examples of the notation are:
The individual components of a vector will have subscripts that indicate the coordinate system or be denoted by the vector symbol with subscripts x, y, and z to indicate the coordinates. All component arrays will be column arrays unless otherwise indicated by the transpose symbol, a right superscript T. For example, arrays of components in a coordinate system b could be shown as
Vector Properties
Vectors are independent of any Cartesian coordinate system. Addition and subtraction of vectors can be defined independently of coordinate systems by means of geometrical constructions (the “parallelogram law”). Thus, we can draw vectors on charts to determine the track of a vehicle through the air or on or under the sea. Some vector operations yield pseudovectors that are not independent of a “handedness” convention. For example, the result of the vector cross-product operation is a vector whose direction depends on whether a right-handed or left-handed convention is being used. We will always use the right-hand rule in connection with vector direction.
It is usually most convenient to manipulate vectors algebraically by decomposing them into a sum of appropriately scaled unit-length vectors usually written as (i.e., ). These unit vectors are normally chosen to form a right-handed orthogonal set, that is, the right-hand rule applied to i and j gives the direction of k (i.e., ). The use of orthogonal unit vectors leads naturally to using Cartesian coordinate systems for their scaling factors and thence to manipulating the coordinates with matrix algebra (next section).
The direction of a vector p relative to a coordinate system is commonly described in two different ways: first by rotations in two orthogonal planes, for example, an azimuth rotation to point in the right direction and then an elevation rotation above the azimuth plane (used with El-over-Az mechanical gimbals), and second by three direction angles α, β, γ to the coordinate axes (used with some radar antennas). The direction cosines of p——give the projections of p on the coordinate axes, and two applications of the theorem of Pythagoras yield
The dot product of two vectors, say u and v, is a scalar defined by
where is the included angle between the vectors (it may be necessary to translate the vectors so that they intersect). The dot product is commutative and distributive; thus,
The principal uses of the dot product are to find the projection of a vector, to establish orthogonality, and to find length. For example, if (1.2-2) is divided by |v|, we have the projection of u on v,
If , , and the vectors are said to be orthogonal. If a vector is dotted with itself, then , and we obtain the square of its length. Orthogonal unit vectors satisfy the dot product relationships
Using these relationships, the dot product of two vectors can be evaluated in terms of components in any convenient orthogonal coordinate system (say a, with components ),
The cross-product of u and v, denoted by , is a vector w that is normal to the plane of u and v and is in a direction such that (in that order) form a right-handed system (again, it may be necessary to translate the vectors so that they intersect). The length of w is defined to be , where θ is the included angle between u and v. It has the following properties:
As an aid for remembering the form of the triple products, note the cyclic permutation of the vectors involved. Alternatively, the vector triple product can be remembered phonetically using “ABC = BAC-CAB.”
The cross-products of the unit vectors describing a right-handed orthogonal coordinate system satisfy the equations
and, using cyclic permutation,
Also remember that , and so on.
An example of the use of the cross-product is finding the vector moment of a force F acting at a point whose position vector is r.
Rotation of a Vector
It is intuitively obvious that a vector can be made to point in an arbitrary direction by means of a single rotation around an appropriate axis. Here we follow Goldstein (1980) to derive a formula for vector rotation.
Consider Figure 1.2-1, in which a vector u has been rotated to form a new vector v by defining a rotation axis along a unit vector n and performing a left-handed rotation through around n. The two vectors that must be added to u to obtain v are shown in the figure and provide a good student exercise in using the vector cross-product (Problem 1.2-4). By doing this addition, we get
or
Equations (1.2-5) are sometimes called the rotation formula; they show that, after choosing n and μ, we can operate on u with dot and cross-product operations to get the desired rotation; no coordinate system is involved, and the rotation angle can be arbitrarily large.
Figure 1.2-1 Rotation of a vector.
1.3 MATRIX OPERATIONS ON VECTOR COORDINATES
As noted earlier, the coordinate system components of a vector will be written as a (3 × 1) column array. Here, we shall show how those components are manipulated in correspondence with operations performed with vectors.
The Scalar Product
If and are column arrays of the same dimension, their scalar product is , and, for example, in three dimensions,
This result is identical to Equation (1.2-3) obtained from the vector dot product. The scalar product allows us to find the 2-norm of a column matrix:
In Euclidean space this is the length of the vector.
The Cross-Product Matrix
From the unit-vector cross-products, given earlier, we can derive a formula for the components of the cross-product of two vectors by writing them in terms of a sum of unit vectors. A convenient mnemonic for remembering the formula is to write it so that it resembles the expansion of a determinant, as follows:
where subscripts , indicate components in a coordinate system whose axes are aligned respectively with the unit vectors . We often wish to directly translate a vector equation into a matrix equation of vector components. From the above mnemonic it is easy to see that
A skew-symmetric matrix of the above form will be denoted by the tilde overbar and referred to as the tilde matrix or cross-product matrix. An example of the use of the cross-product matrix involves the centripetal acceleration at a point described by a position vector r rotating with an angular velocity vector ω (see also Equation 1):
In the case of a vector triple product, the vector operation in parentheses must be performed first, but the corresponding matrix operations may be performed collectively in any order:
Here, the third term requires only postmultiplication by a column array and hence fewer operations to evaluate than the second term.
Coordinate Rotation, the DCM
When the rotation formula (1.2-5b) is resolved in a coordinate system , the result is
where is a square matrix, I is the identity matrix, and is a cross-product matrix. This formula was developed as an “active” vector operation in that a vector was being rotated to a new position by means of a left-handed rotation about the specified unit vector. In component form, the new array can be interpreted as the components of a new vector in the same coordinate system, or as the components of the original vector in a new coordinate system, obtained by a right-handed coordinate rotation around the specified axis. This can be visualized in Figure 1.3-1, which shows the new components of a vector v after a right-handed coordinate system rotation, θ, around the -axis. Instead, if the vector is given a left-handed rotation of the same amount, then will become the components of the vector in the original system. Taking the coordinate system rotation viewpoint and combining the matrices in (1.3-4) into a single coefficient matrix, this linear transformation can be written as
Here Cb/a is a matrix that transforms the coordinates of the vector u from system a to system b and is called a direction cosine matrix (DCM), or simply a rotation matrix.
Figure 1.3-1 A plane rotation of coordinates.
In Figure 1.3-1 a new coordinate system is formed by a right-handed rotation around the z-axis of the original orthogonal coordinate system; the DCM can easily be found by applying Equation (1.3-4) using
The DCM and the components ofuin system b are then found to be
The direction cosine matrix is so called because its elements are direction cosines between corresponding axes of the new and old coordinate systems. Let i, j, k, with appropriate subscripts, be unit vectors defining the axes of our orthogonal coordinate systems a and b. The -component of an arbitrary vector r can be written as
