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Marco Hutter, Roland Siegwart, and Thomas Stastny

Autonomous Systems Lab

20.12.2016Robot Dynamics - Fixed Wing UAS: Basics of Aerodynamics 1

Robot DynamicsFixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling


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

Contents | Fixed Wing UAS

20.12.2016Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling 2

||Autonomous Systems Lab 20.12.2016 3

Historical Overview

Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling


First Flight:

Montgolfier Brothers

1783

Ballon filled with hot air

First unmanned

demonstrations

Later with animals

Finally manned

Historical Overview

http://en.wikipedia.org/wiki/Montgolfier

20.12.2016 4Robot Dynamics – Fixed Wing UAS: Basics of Aerodynamics & Dynamic Modeling


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”

Historical Overview

http://en.wikipedia.org/wiki/Otto_Lilienthal



Wright brothers

Started as glider engineers and pilots

First engine powered flight in 1903

First to actively manipulate the plane by control surfaces

Historical overview

http://en.wikipedia.org/wiki/Wright_Brothers



Historical Overview | Small fixed-wing UAVs



Basics of Aerodynamics



Analysis on differential volumes:

With viscosity: Navier-Stokes Equation

Without viscosity: Euler Equation

Incompressible along streamline: Bernoulli Equation

Basics of Aerodynamics | Basic Principles

www.speedace.info/pito

t_tube.htm as on 29th

July 2009

constp

ghv

2

2




https://commons.wikimedia.org/wiki/File:Streamlines_around_a_NACA_0012.svg

V=FASTER

V=SLOWER

P=HIGHER

P=LOWERLIFT!



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.

20.12.2016 12


streamline

fluid particle centripetal force:

pout > pinv

pout

pin

Nice lecture by Dr. Holger Babinsky, University of Cambridge

https://www.youtube.com/watch?v=XWdNEGr53Gw




https://commons.wikimedia.org/wiki/File:Streamlines_around_a_NACA_0012.svg

LIFT!

patm

pupper

plower

patm

pupper < patm

plower > patm

Therefore: pupper < plower



Wing Geometry

Basics of Aerodynamics | Wing Geometry

b: Wingspan

c: Chord

c0: Root Chord

ct: Tip Chord

A: Reference Area

AR: Aspect Ratio

x

y

c

ct

c 0

b

A

A

bAR

2



Wing Geometry

Basics of Aerodynamics | Wing Geometry

b: Wingspan

c: Chord

c0: Root Chord

ct: Tip Chord

A: Reference Area

AR: Aspect Ratio

x

y

c

ct

c 0

b

A

A

bAR

2


𝑐

Mean geometric chord


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)

Wing Geometry



Suction

Overpressure

2-Dimensional Flow Analysis

Flow field (pressure distribution, laminar/turbulent) highly dependant

on angle of attack, Reynolds number and Mach number

Basics of Aerodynamics | Airfoils

http://www.thuro.at/anims/abloesung.gifwww.thuro.at/aerodynamik2.htm http://www.thuro.at/anims/abloesung.gif

Laminar

boundary layer

Transition point Turbulent

boundary layer

Separated

boundary

layer

Stagnation point



Pressure distribution can be reduced to two forces and one moment

per unit length:


v

Angle of attack

a

Leading edge

Trailing edge

Chord

c

25 % Chord Thickness

dL

dDdM

22

2VdycCdM m

2

2VdycCdD d

2

2VdycCdL l

Lift force

Drag force

Moment

: Density of fluid (air) [kg/m3]

c : Chord length [m]

V : Flight speed (w.r.t. air) [m/s]

Cl : Airfoil lift coefficient [-]

Cd : Airfoil drag coefficient [-]

Cm : Airfoil moment coefficient [-]



Coefficients Cl, Cd and Cm depend on

angle of attack a As long as flow is attached:

Cl – linear:

Cm – almost constant

At stall: flow separation

Cl – stops to increase

Cd – increases dramatically Flow field highly depending on Re (and Ma),

in particular:

Location of laminar/turbulent transition point

Separation point

Stall angle


Separation point

a

2d

dCl



Reynolds' number influence

at low speed, Re and Cd


Cl

Cda

Polars of Airfoil we3.55-9.3

Re

cV Re

Forces Viscous

Forces Inertial

McMasters, J. H. and M. L. Henderson (1980). "Low Speed Single

Element Airfoil Synthesis." Technical Soaring 6(2): 1-21

W. Engel and A. Noth 2005



The choice of an airfoil depends on:

Flying speed

Wing loading

Construction method

Kind of flight (acrobatic, glide,…)

Placement on the airplane

Standard airfoils (some examples)

Goettingen

Eppler

Wortmann

NACA

Example: NACA 2412

Airfoils

Maximum camber deflection (% of chord)

Symmetric Airfoils

Semi-Symmetrical Airfoils

Under-Cambered Airfoils

Reflexed Airfoils

Flat-Bottom Airfoils

Thickness (% of chord)

Position of maximum camber deflection (tenths of chord)



Methods to determine airfoil lift, drag and

moment coefficients:

Theoretically using 2D-CFD software

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

Airfoil Lift, Drag and

Moment

www.uwal.org/publicdata/photos

Javafoil



From 2D to 3D: the wing is not

infinite…

Vortices are created at wing

extremities

Tip vortices induce

downward flow (w)

and thus reduce the

effective angle of attack

Approx. induced drag:

e: Oswald Factor < 1 for

non-elliptic lift distribution

Induced Drag

2

i

LD

CC

e AR

aerospaceweb.org

dL

V (free stream)

w

dDi

20.12.2016 23

NASA Dryden Flight Research Center



Winglets: Less Induced Drag

www.aviationpartners.com

http://airpigz.com/blog/2010/8/27/poll-spiroids-funky-circular-

winglets-love-em-or-hate-em.html

Blended

Spiroids

Modern Glider

Designs



Ideally, winglets…

… reduce induced drag at

low speeds

… reduce spanwise flow

… increase the Reynolds

number near wing tip

… do not increase the

parasite drag too much

(relevant for high speed

performance)

How to Reduce Induced Drag: Winglets

V (fre

e s

tream

)

v

Wing

Upward

winglet

Bound

vortex

Tip

vort

exTop view



Wing: integrate Cd along

the wing

Fuselage: highly Re number

and geometry depending…

Friction drag

Form drag

Interference drag

e.g. at the transition between

fuselage and wing

Can also be negative

Parasite Drag

Dra

g

Speed

total



For small airplanes,

the standard control surfaces are:

Ailerons (rolling)

Elevator (pitching)

Rudder (yawing)

For larger airplanes, they can be more complex…

Control surfaces

20.12.2016 27

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



Propeller driven by piston engine or electrical motor are

most common for small fixed-wing UAVs

Propulsion Group | Types



Propulsion Group | Placement

20.12.2016 29

In the front… In the back…

On the wings…



Given: drag coeff. as a function of lift coeff.: 𝐶𝐿 𝑉 =2𝑚𝑔

𝐴𝜌𝑉2, 𝐶𝐷 𝐶𝐿

Required power:

Specific Excess Power: 𝑆𝐸𝑃 = 𝑃𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 − 𝑃𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑚𝑔 ≈ 𝑉𝑐𝑙𝑖𝑚𝑏,𝑎𝑐ℎ𝑖𝑒𝑣𝑎𝑏𝑙𝑒

20.12.2016 30

Power Required and Available for Level Flight

Drequired ACVVDP 3

2

1

Pavailable

Prequired

Pexcess

Vstall Vmax Vne

Pow

er

True Airspeed

Vemax Vrmax

min

L

D

C

Cmg

L

DmgD

V

Pmax

D

L

C

C

Best glide ratio

max/

P

EVTVs

min/ P

max2

3

D

L

C

C

L

D

LL

D

C

Cmg

CA

mg

C

CmgVVDP

2

Minimum sink in

gliding mode

Max. Range*(vrmax):

Max. Endurance*(vemax):

* Assuming constant propulsive efficiency η



Aircraft Dynamic Modeling



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

Why Model the Dynamics of an Airplane?



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)

Aircraft Dynamic Modeling | Intro



Aircraft Dynamic Modeling | Reference axes

20.12.2016 34

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




20.12.2016 35

Body-axis




20.12.2016 36

Body velocity:

• Air-mass relative

speed (airspeed):

• V is always positive

Body-axis




20.12.2016 37

Body velocity:


speed (airspeed):


Body rates:

Body-axis



Body-axis


20.12.2016 38

Body velocity:


speed (airspeed):


Body rates:Airflow angles:

• Angle of attack

• Sideslip angle

• Wind frame is opposite to “free-stream”

velocity, or “wind” (i.e. the air-mass)

Wind-axis



Body-axis


20.12.2016 39

Body velocity:


speed (airspeed):


Body rates:Airflow angles:

• Angle of attack

• Sideslip angle

• Wind frame is opposite to “free-stream”

velocity, or “wind” (i.e. the air-mass)

Wind-axis

*Possible point-of-confusion: wind-axis

does NOT consider wind in the inertial

sense. Both body and wind frames only

consider air-mass relative motion



Aircraft Dynamic Modeling | Coordinate

transformation

20.12.2016 40

Euler angles, roll, pitch, and yaw, are used to transform

between inertial and body axes.



Earth fixed frame

(regarded as inertial): Body fixed frame:exB,eyB,ezBexE,eyE,ezE

Rotation Matrix ( to ) is parametrized with 3 successive rotations using

the ZYX Tait-Brian Angles (specific kind of Euler Angles):

Roll:

around -

Frame B

3Pitch:

around -

Frame 2

2Yaw:

around -

Frame 1

1

20.12.2016 41

Aircraft Dynamic Modeling | Coordinate

transformation



Angular Rates:

Time variation of Tait-Bryan angles

Body angular rates

Singularity: for (Jr becomes singular)

« Gimbal Lock »

Aircraft Dynamic Modeling | A note on angular

rates

≠ rqp ,,

coscossin0

cossincos0

sin01

rJ

r

r

q

p

J

2



Aircraft Dynamic Modeling | Polar coordinates

(𝑁, 𝐸)

(𝑁, 𝐸)

(𝑁, 𝐸)

(𝐷)

Longitudinal Inertial Frame

(*side view)

• : Flight path angle, defined

from horizon to

• : Pitch angle, from horizon to

x-body axis

• : Roll angle, rotation about x-

body axis (note: pointing out of

slide)

•



Aircraft Dynamic Modeling | Polar coordinates(𝑁)

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.

(𝐸)



Lateral-directional Inertial Frame

(*top view)

• : Heading angle, defined

from North to

• : Yaw angle, from North to x-

body axis

20.12.2016 45

Aircraft Dynamic Modeling | Polar coordinates(𝑁)

(𝐸)

Adding a constant, horizontal

wind:

• : Course angle, defined from

North to

• : Ground-based inertial

velocity (or “ground speed”)

• : Wind velocity

• Note: the constant wind

assumption effects only

position dynamics.



Aircraft Dynamic Modeling | Polar coordinates



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:

Wing

Tail

Fuselage

Aircraft Dynamic Modeling | Forces & Moments

20.12.2016 47

L

D

YLm

Nm

Mm

FT


mg

Note: Thrust

force can be

offset from

x-body axis.

Typically

denoted by

angle ε.


Forces and moments acting on the airplane

Lift

Drag

Thrust

Gravity

Moments

Rolling moment:

Pitching moment:

Yawing moment:

Aircraft Dynamic Modeling | Forces & Moments

20.12.2016 48

L

D

YLm

Nm

Mm

FT


mg

For geometry definitions

(i.e. S, b, 𝑐) see “Wing

Geometry” slides (p14-15)

Free-stream velocity

Lift and Drag

always

perpendicular

and parallel,

respectively, to

free-stream.


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

Definitions, Assumptions and Simplifications

Velocities (Body Fr.): u,v,w

Turn rates (Body Fr.): p,q,r

Position (Earth Fr.): x,y,z

Tait-Bryan angles: ,,

Nonlinear

Aircraft

Dynamics

Forces

Moments

u,v,w;

p,q,r

x,y,z;

,,

Propulsion,

Mechanics,

Aerodynamics

Elevator

Aileron

Rudder

Throttle



On the Rigidity…



On the Rigidity…

NASA Helios Crash: www.nasa.gov/centers/dryden/history/pastprojects/Helios



Forces and momentsrepresented in body frame, attacking at the CoG:

Development of the Model

aa

aa

aa

aa

coscoscossinsin

cossin

sinsincoscos

0

0

sin

0

cos

cossin

sincos

mgLDF

mgY

mgLDF

g

m

F

F

LD

Y

LD

T

T

BI

T

T

tot CF

T

T

T

m

m

m

m

m

m

tot

N

M

L

N

M

L

M



Application of Newton‘s Second Law


Euler

Derivatives

xzxxyyzzxz

xzzzxxyy

xzyyzzxzxx

zzxz

yy

xzxx

zzxz

yy

xzxx

qrIIIpqrIpI

IprIIprqI

qpIIIqrrIpI

r

q

p

II

I

II

r

q

p

r

q

p

II

I

II

22

0

00

0

0

00

0

vF Btot mdt

d

qupvw

pwruv

rvqwu

m

w

v

u

r

q

p

w

v

u

m

r

q

p

II

I

II

dt

d

dt

d

zzxz

yy

xzxx

Btot

0

00

0

IMTypically

small



Summarized equations of motion:

Translation


aa

aa

coscoscossinsin1

cossin1

sinsincoscos1

gLDFm

pvquw

gYm

rupwv

gLDFm

qwrvu

T

T

w

v

u

z

y

x

IBC



Rotation (simplified with Ixz≈0):


xxyymm

zz

zzxxmm

yy

yyzzmm

xx

IIpqNNI

r

IIprMMI

q

IIqrLLI

p

T

T

T

1

1

1

cos

cos

cos

sinsincos

costansintan1

rq

rq

rqp

r

q

p

rJ



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

20.12.2016 56

References



Exercise tomorrow morning!

20.12.2016Robot Dynamics - Fixed Wing UAS: Basics of Aerodynamics 57

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