Robot Dynamics Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling
Robot Dynamics Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling
151-0851-00 V
Marco Hutter, Roland Siegwart, and Thomas Stastny Autonomous Systems Lab
Autonomous Systems Lab Robot Dynamics - Fixed Wing UAS: Basics of Aerodynamics | 20.12.2016 | 1 Contents | Fixed Wing UAS
1. Introduction/(brief) Historical Overivew 2. Basics of Aerodynamic 3. Aircraft Dynamic Modeling 4. Aircraft Performance (wrap-up) 5. Aircraft Stability 6. Simulation 7. Modeling for Control 8. Fixed-wing Control
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 2 Historical Overview
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 3 http://en.wikipedia.org/wiki/Montgolfier
Historical Overview
. First Flight: Montgolfier Brothers 1783 . Ballon filled with hot air . First unmanned demonstrations . Later with animals . Finally manned
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 4 Historical Overview
. Otto Lilienthal . First person to make repeated successful short flights . Used a fixed wing glider . Died after a crash in 1896, saying “Sacrifices must be made” http://en.wikipedia.org/wiki/Otto_Lilienthal
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 5 Historical overview
. Wright brothers . Started as glider engineers and pilots . First engine powered flight in 1903 . First to actively manipulate the plane by control surfaces
http://en.wikipedia.org/wiki/Wright_Brothers
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 6 Historical Overview | Small fixed-wing UAVs
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 7 Basics of Aerodynamics
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 8 Basics of Aerodynamics | Basic Principles
Analysis on differential volumes: . With viscosity: Navier-Stokes Equation . Without viscosity: Euler Equation . Incompressible along streamline: Bernoulli Equation v2 p gh const 2
www.speedace.info/pito t_tube.htm as on 29th July 2009
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 10 Basics of Aerodynamics | Basic Principles
V=FASTER P=LOWER LIFT!
V=SLOWER P=HIGHER
https://commons.wikimedia.org/wiki/File:Streamlines_around_a_NACA_0012.svg
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 11 Basics of Aerodynamics | Basic Principles
. But…Bernoulli isn’t the whole story! . Watch out for common misconceptions of lift . E.g. the “distance traveled” argument for speed difference . What is really going on? –streamline curvature induced pressure gradients.
p fluid particle out centripetal force: p > p v out in pin
Nice lecture by Dr. Holger Babinsky, University of Cambridge streamline https://www.youtube.com/watch?v=XWdNEGr53Gw
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 12 Basics of Aerodynamics | Basic Principles
. pupper < patm patm . plower > patm
. Therefore: pupper < plower
LIFT! pupper
plower
patm
https://commons.wikimedia.org/wiki/File:Streamlines_around_a_NACA_0012.svg
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 13 Basics of Aerodynamics | Wing Geometry
Wing Geometry x b: Wingspan y c: Chord
c0: Root Chord c 0 ct: Tip Chord c A
t A: Reference Area c AR: Aspect Ratio b b2 AR A
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 14 Basics of Aerodynamics | Wing Geometry
Wing Geometry x b: Wingspan y
Mean geometric chord c: Chord
푐 c0: Root Chord c 0 ct: Tip Chord c A
t A: Reference Area c AR: Aspect Ratio b b2 AR A
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 15 Wing Geometry
. Various types of wing . Biplanes & vertical composition of wing . Lift not proportional to the number of wings. Biplane: factor ~ 1.5 . Drag also increased . Advantage of higher stiffness and less Inertia around x-axis (Aerobatics)
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 16 Basics of Aerodynamics | Airfoils
2-Dimensional Flow Analysis . Flow field (pressure distribution, laminar/turbulent) highly dependant on angle of attack, Reynolds number and Mach number
Transition point Turbulent Separated Laminar boundary layer boundary boundary layer layer Suction
Overpressure
Stagnation point www.thuro.at/aerodynamik2.htm http://www.thuro.at/anims/abloesung.gif
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 17 Basics of Aerodynamics | Airfoils Leading edge dL c
Angle of attack a dM dD Trailing edge Chord
v 25 % Chord Thickness Pressure distribution can be reduced to two forces and one moment per unit length:
2 : Density of fluid (air) [kg/m3] Lift force dL C cdy V l 2 c : Chord length [m] V : Flight speed (w.r.t. air) [m/s] 2 Drag force dD C cdy V C : Airfoil lift coefficient [-] d 2 l Cd : Airfoil drag coefficient [-] C : Airfoil moment coefficient [-] Moment dM C c2 dy V 2 m m 2 Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 18 Basics of Aerodynamics | Airfoils
Coefficients Cl, Cd and Cm depend on angle of attack a . As long as flow is attached: Separation point dC . C – linear: l 2 l da
. Cm – almost constant
. At stall: flow separation
. Cl – stops to increase Flow field highly depending on Re (and Ma), . Cd – increases dramatically in particular: . Location of laminar/turbulent transition point . Separation point . Stall angle
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 19 Basics of Aerodynamics | Airfoils
Reynolds' number influence Polars of Airfoil we3.55-9.3 W. Engel and A. Noth 2005 Re V c Cl Re Inertial Forces Viscous Forces
at low speed, Re and Cd
McMasters, J. H. and M. L. Henderson (1980). "Low Speed Single C a Element Airfoil Synthesis." Technical Soaring 6(2): 1-21 d
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 20 Symmetric Airfoils Airfoils
. The choice of an airfoil depends on: Reflexed Airfoils . Flying speed . Wing loading
. Construction method Flat-Bottom Airfoils . Kind of flight (acrobatic, glide,…) . Placement on the airplane . Standard airfoils (some examples) Semi-Symmetrical Airfoils . Goettingen . Eppler
. Wortmann Under-Cambered Airfoils . NACA Example: NACA 2412
Thickness (% of chord) Position of maximum camber deflection (tenths of chord) Maximum camber deflection (% of chord)
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 21 Airfoil Lift, Drag and Moment Methods to determine airfoil lift, drag and moment coefficients: . Theoretically using 2D-CFD software . Javafoil Javafoil http://www.mh-aerotools.de/ . Xfoil http://raphael.mit.edu/xfoil/ . … . Experimentally in a wind tunnel . Extruded airfoil mounted on a measurement system . Laminar flow produced by fans
www.uwal.org/publicdata/photos
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 22 Induced Drag aerospaceweb.org
From 2D to 3D: the wing is not infinite… . Vortices are created at wing extremities . Tip vortices induce NASA Dryden Flight Research Center downward flow (w) dDi and thus reduce the effective angle of attack dL V (free stream) w
. Approx. induced drag: 2 CL . e: Oswald Factor < 1 for C Di non-elliptic lift distribution e AR
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 23 Blended Winglets: Less Induced Drag
Modern Glider Designs
www.aviationpartners.com Spiroids
http://airpigz.com/blog/2010/8/27/poll-spiroids-funky-circular- winglets-love-em-or-hate-em.html Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 24 How to Reduce Induced Drag: Winglets
. Ideally, winglets… (free V stream) . … reduce induced drag at low speeds . … reduce spanwise flow . … increase the Reynolds Bound number near wing tip vortex Upward v . … do not increase the Wing winglet parasite drag too much (relevant for high speed Top view
performance)
vortex Tip
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 25 Parasite Drag
. Wing: integrate Cd along the wing . Fuselage: highly Re number
and geometry depending… Drag total . Friction drag . Form drag . Interference drag Speed . e.g. at the transition between fuselage and wing . Can also be negative
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 26 Control surfaces
. For small airplanes, the standard control surfaces are: . Ailerons (rolling) . Elevator (pitching) . Rudder (yawing) . For larger airplanes, they can be more complex… Ailerons: 2. Low-Speed Aileron 3. High-Speed Aileron Lift increasing flaps and slats: 4. Flap track fairing 5. Krüger flaps 6. Slats 7. Three slotted inner flaps 8. Three slotted outer flaps Spoilers: 9. Spoilers 10. Spoilers-Air brakes
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 27 Propulsion Group | Types
. Propeller driven by piston engine or electrical motor are most common for small fixed-wing UAVs
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 28 Propulsion Group | Placement
In the front… In the back…
On the wings…
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 29 Power Required and Available for Level Flight 2푚𝑔 . Given: drag coeff. as a function of lift coeff.: 퐶 푉 = , 퐶 퐶 퐿 퐴휌푉2 퐷 퐿
1 3 . Required power: Prequired DV V AC D 2
. Specific Excess Power: 푆퐸푃 = 푃푎푣푎𝑖푙푎푏푙푒 − 푃푟푒푞푢𝑖푟푒푑 푚푔 ≈ 푉푐푙𝑖푚푏,푎푐ℎ𝑖푒푣푎푏푙푒 E Max. Range*(v ): s VT V max rmax P /
P P D CD CL available D mg mg min max
V L CL CD Power Pexcess Best glide ratio
Max. Endurance*(vemax): P / min
Prequired CD 2mg CD P VD V mg mg CL ACL CL Vstall Vemax Vrmax Vmax Vne C 3 True Airspeed L max Minimum sink in C 2 * Assuming constant propulsive efficiency η D gliding mode
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 30 Aircraft Dynamic Modeling
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 31 Why Model the Dynamics of an Airplane?
. System analysis: model allows evaluating future flight characteristics . Stability . Controllability . Power required fuel needs . Controllability in the case of actuator failure . Autopilot design and simulation: model allows comparing different control techniques and autopilot parameter tuning . Gain of time and money . Higher performance of the autopilot . No risk of damage compared to real tests . Pilot training (in Simulator) . Allows simulating and training especially emergency situations
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 32 Aircraft Dynamic Modeling | Intro
. Dynamics of an airplane ... Are very different from an acrobatic aircraft to a line jet airplane ... but the principles remain the same for all . Wings, stabilizers . control surfaces (ailerons, rudder, elevator, flaps,spoilers) . propulsion group (motor-gearbox-propeller, turbine, rocket,…)
. In this lecture, we will model a typical fixed-wing UAV . Steps for the creation of the model 1. Define the coordinate frames 2. List all the physical effects acting on the airplane 3. Set assumptions, make simplifications 4. Express the physical effects into equations 5. Derive the equations of motion (here: Newton)
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 33 Aircraft Dynamic Modeling | Reference axes
Inertial (local) reference frame • North, East, Down (or NED) • Flat Earth assumption • Where to define the origin? • E.g. UTM (Universal Transverse Mercator) coordinates
Body-fixed frame • x out the nose • y out the right wing • z (with right hand rule) down • Origin typically located at center of gravity
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 34 Aircraft Dynamic Modeling | Reference axes
Body-axis
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 35 Aircraft Dynamic Modeling | Reference axes
Body-axis Body velocity:
• Air-mass relative speed (airspeed):
• V is always positive
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 36 Aircraft Dynamic Modeling | Reference axes
Body-axis Body velocity:
• Air-mass relative speed (airspeed):
• V is always positive
Body rates:
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 37 Aircraft Dynamic Modeling | Reference axes
Body-axis Body velocity:
• Air-mass relative speed (airspeed):
Wind-axis • V is always positive Airflow angles: Body rates: • Angle of attack • Sideslip angle • Wind frame is opposite to “free-stream” velocity, or “wind” (i.e. the air-mass)
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 38 Aircraft Dynamic Modeling | Reference axes
Body-axis Body velocity: *Possible point-of-confusion: wind-axis does NOT consider wind in the inertial• Air-mass relative sense. Both body and wind frames onlyspeed (airspeed): consider air-mass relative motion
Wind-axis • V is always positive Airflow angles: Body rates: • Angle of attack • Sideslip angle • Wind frame is opposite to “free-stream” velocity, or “wind” (i.e. the air-mass)
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 39 Aircraft Dynamic Modeling | Coordinate transformation
. Euler angles, roll, pitch, and yaw, are used to transform between inertial and body axes.
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 40 Earth fixed frame Aircraft Dynamic Modeling(regarded | Coordinate as inertial): Body fixed frame: e ,e ,e e ,e ,e transformation xE yE zE xB yB zB Rotation Matrix ( to ) is parametrized with 3 successive rotations using the ZYX Tait-Brian Angles (specific kind of Euler Angles):
Yaw: Pitch: Roll: 1 around - 2 around - 3 around - Frame 1 Frame 2 Frame B
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 41 Aircraft Dynamic Modeling | A note on angular rates
. Angular Rates: Time variation of Tait-Bryan angles ≠ Body angular rates p,q,r
p 1 0 sin q J J 0 cos sin cos r r r 0 sin cos cos . Singularity: for (Jr becomes singular) . « Gimbal Lock » 2
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 42 Aircraft Dynamic Modeling | Polar coordinates
Longitudinal Inertial Frame (*side view) • : Flight path angle, defined from horizon to • : Pitch angle, from horizon to (푁, 퐸) x-body axis • : Roll angle, rotation about x- (푁, 퐸) body axis (note: pointing out of slide) • (퐷)
(푁, 퐸)
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 43 Aircraft Dynamic Modeling | Polar coordinates (푁) Lateral-directional Inertial Frame (*top view) • : Heading angle, defined from North to • : Yaw angle, from North to x- body axis • Note: this STILL does not consider wind.
(퐸)
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 44 Aircraft Dynamic Modeling | Polar coordinates (푁) Lateral-directional Inertial Frame (*top view) • : Heading angle, defined from North to • : Yaw angle, from North to x- body axis Adding a constant, horizontal wind: • : Course angle, defined from (퐸) North to • : Ground-based inertial velocity (or “ground speed”) • : Wind velocity • Note: the constant wind assumption effects only position dynamics.
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 45 Aircraft Dynamic Modeling | Polar coordinates
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 46 Aircraft Dynamic Modeling | Forces & Moments
. Forces and moments acting on the airplane . Weight at the center of gravity . Thrust of propeller: complex task will not be presented here . Sum aerodynamic forces and moments from each part of the airplane: L Note: Thrust . Wing D force can be . Tail offset from FT x-body axis. . Fuselage Typically Lm denoted by Y angle ε. M m mg
Nm Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 47 Aircraft Dynamic Modeling | Forces & Moments Lift and Drag always . Forces and moments acting on the airplane perpendicular . Lift and parallel, . Drag L respectively, to free-stream. . Thrust D . Gravity FT For geometry definitions Lm (i.e. S, b, 푐 ) see “Wing Y Geometry” slides (p14-15) M . Moments m mg . Rolling moment: N . Pitching moment: m Free-stream velocity . Yawing moment:
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 48 Definitions, Assumptions and Simplifications
. Definitions . Remember: origin of body-fixed coordinate frame set into center of gravity . Assumptions and simplifications . Rigid and symmetric structure:constant, (almost) diagonal inertia matrix . Constant mass . Motor without dynamics and without gyroscopic effects (can be adapted) . Aerodynamics (list not complete):
. We don’t enter stall (operation in the linear cl domain) . Neglect fuselage lift/sideslip force (may be easily included, if modeled correctly) . Inputs/Outputs/States Elevator Velocities (Body Fr.): u,v,w u,v,w; Propulsion, Nonlinear Aileron Forces Turn rates (Body Fr.): p,q,r p,q,r Mechanics, Rudder Moments Aircraft Position (Earth Fr.): x,y,z x,y,z; Aerodynamics Throttle Dynamics Tait-Bryan angles: ,, ,,
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 49 On the Rigidity…
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 50 On the Rigidity…
NASA Helios Crash: www.nasa.gov/centers/dryden/history/pastprojects/Helios
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 51 Development of the Model
. Forces and moments represented in body frame, attacking at the CoG:
D cosa Lsina F cos 0 T Ftot Y 0 mCBI 0 Dsina Lcosa FT sin g F cos D cosa Lsina mg sin T Y mg sin cos FT sin Dsina Lcosa mg cos cos
L L m mT M M M tot m mT N N m mT
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 52 Development of the Model
. Application of Newton‘s Second Law u p u u qw rv d F m v m v q v mv ru pw tot B dt w r w w pv qu
Euler I xx 0 I xz p d d Derivatives M I 0 I 0 q tot B yy dt dt Typically I xz 0 I zz r small I 0 I p p I 0 I p I p I r qrI I qpI xx xz xx xz xx xz zz yy xz 0 I 0 q q 0 I 0 q I q pr I I r2 p2 I yy yy yy xx zz xz I xz 0 I zz r r I xz 0 I zz r I xz p I zzr pqI yy I xx qrI xz
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 53 Development of the Model
. Summarized equations of motion: . Translation 1 u rv qw F cos D cosa Lsina g sin m T 1 v pw ru Y g sin cos m 1 w qu pv F sin Dsina Lcosa g cos cos m T
x u y C v IB z w
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 54 Development of the Model
. Rotation (simplified with Ixz≈0): 1 p L L qrI I m mT zz yy I xx 1 q M M prI I m mT xx zz I yy 1 r N N pqI I m mT yy xx I zz
p p q tan sin r tan cos 1 J q qcos r sin r sin cos r q r cos cos
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 55 References
. B.W. McCormick. Aerodynamics, Aeronautics, and Flight Mechanics. Wiley, 1979. ISBN: 9780471030324. . B. Etkin. Dynamics of Atmospheric Flight . Wiley, 1972. ISBN: 9780471246206. . G.J.J. Ducard. Fault-Tolerant Flight Control and Guidance Systems: Practical Methods for Small Unmanned Aerial Vehicles . Advances in Industrial Control. Springer, 2009. ISBN: 9781848825611. . R.W. Beard and T.W. McLain. Small Unmanned Aircraft: Theory and Practice. Princeton University Press, 2012. ISBN: 9780691149219. . R.F. Stengel. Flight Dynamics. Princeton University Press, 2004. ISBN: 0-691-11407-2
Autonomous Systems Lab Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling | 20.12.2016 | 56 Exercise tomorrow morning!
Autonomous Systems Lab Robot Dynamics - Fixed Wing UAS: Basics of Aerodynamics | 20.12.2016 | 57