Folienmaster ETH Zürich · 2015. 12. 3. · Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 27.10.2015 19 BEMT | blade element 2 2 L dL C cdrV U 2 2 L dD

||Autonomous Systems Lab

151-0851-00 V

Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas Stastny

27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 1

Robot DynamicsRotary Wing UAS: Propeller Analysis and Dynamic Modeling


1. Introduction

2. Mechanical Design

3. Propeller Aerodynamics

4. Propeller Modeling

5. Dynamics Modeling

6. Control

7. Case Study: Airobots

Part 2: Propeller Analysis and Dynamic

Modeling

1. Propeller/Rotor Analysis

Momentum Theory

Figure of Merit

BEMT

2. Dynamic Modeling

Modeling Overview

Fuselage Dynamics

Propeller/Rotor Dynamics

Motor Dynamics

3. Modeling Examples

Quadrotor

Coaxial Helicopter


Contents | Rotary Wing UAS

||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 3

Multi-body Dynamics | General Formulation

Generalized coordinates

Mass matrix

, Centrifugal and Coriolis forces

Gravity forces

Actuation torque

External forces (end-effector,

act

ex

b

g

F

M

ground contact, propeller thrust, ...)

Jacobian of external forces

Selection matrix of actuated joints

exJ

S

, T Tex ex actb g F M J S


The Quadrotor Example

F1

1

F4

4

F3

3 F2

2

, TT e cte ax xb Fg M SJ


The Quadrotor Example

F1

1

F4

4

F3

3 F2

2

xR

yR

zR

xI

yI

zI



Propeller / Rotor AnalysisRotary Wing UAS: Propeller Analysis and Dynamic Modeling



Analysis of an ideal propeller/rotor

Power put into fluid to change its momentum downwards

Actio-reactio: Thrust force at the propeller/rotor

Assumptions

Infinitely thin propeller/rotor disc area AR Thrust and velocity distribution is uniform over disc area

One dimensional flow analysis

Quasi-static airflow

Flow properties do not change over time

No viscous effects

No profile drag

Air is incompressible


The Propeller Thrust Force | Momentum Theory


Conservation of fluid mass

Mass flow inside and outside control volume must be equal (quasi-

static flow)

Conservation of fluid momentum

The net force on the fluid is the change of momentum of the fluid

Conservation of energy

Work done on the fluid results in a gain of kinetic energy


Momentum Theory | Recall of Fluid Dynamics

0 (1)V ndA

: density of fluid

: flow speed at surface element

: suface normal unity vector

: patch of control surface area

V

n

dA

(2)p ndA VV ndA F : surface pressurep

21 (3)2

dEV V ndA P

dt

: energy

: power

E

P


One-dimensional analysis

Area of interest 0,1,2 and 3

Area 0 and 3 are on the far field with

atmospheric pressure

Conservation of mass

No change of speed across rotor/propeller disc,

but change in pressure


Momentum Theory | Derivation 1

1

0 1

2 3

0 2 0 3

0

V dA V u dA

V dA V u dA V dA V u dA

0 1 2 3 3 (4)R RA V A V u A V u A V u

1 2 (5)u u

0 0, , V A p

1 1, , RV u A p

2 2, , RV u A p

3 3 0, , V u A p


Conservation of momentum

Unconstrained flow:

→ net pressure force is zero

Conservation of energy


Momentum Theory | Derivation 2

22

3

0 3

22

0 3 3

1 3

=

= (6)

Thrust

R

F V dA V u dA

A V A V u

A V u u

p ndA

33

1 3

0 3

33

0 3 3

1 3 3

1 1

2 2

1 1

2 2

1 2 (7)

2

Thrust Thrust

R

P F V u V dA V u dA

A V A V u

A V u V u u

0 0, , V A p

1 1, , RV u A p

2 2, , RV u A p

3 3 0, , V u A p

With eq. (4) and (5)



Comparison between momentum and energy

Far wake slipstream velocity is twice the induced velocity

Thrust force formula

Hover case (V = V0 = 0)

Thrust force

Slipstream tube


Momentum Theory | Derivation 30 0, , V A p

1 1, , RV u A p

2 2, , RV u A p

3 3 0, , V u A p

3 12 (8)u u

1 12 (9)Thrust RF A V u u


2

12 (10)Thrust RF A u

0 3 ; = (11)2

RAA A


Ideal power used to produce thrust

Hover case

Power depends on disc loading FThrust / AR

Decreasing disc loading reduces power

Mechanical constraint: Tip Mach Number

More profile/structural drag

Longer tail

…


Momentum Theory | Power consumption in hover0 0, , V A p

1 1, , RV u A p

2 2, , RV u A p

3 3 0, , V u A p

1 (11)ThrustP F V u With eq. (10)

(13)

)12( 2

)(

2

23

23

mgF

A

mg

A

FP

Thrust

RR

Thrust


Defining the efficiency of a rotor

Figure of Merit FM

Ideal power: Can be calculated using the momentum theory

Actual power: Includes profile drag, blade-tip vortex, …

FM can be used to compare different propellers with the

same disc loading


Propeller/Rotor Efficiency

1hover torequiredpower Actual

hover torequiredpower IdealFM


Combined Blade Elemental and Momentum Theory

Used for axial flight analysis

Divide rotor into different blade elements (dr)

Use 2D blade element analysis

Calculate forces for each element and sum them up


BEMT (Blade Elemental and Momentum Theory)

ωR

RR


BEMT | introduction

Forces at each blade element

With the relative air flow V we can

determine angle of attack and Reynolds number

The corresponding lift and drag coefficient

are found on polar curves for blade shape (see next slide)

Problem: What is the induced velocity uind ?

Calculation of angle of attack (AoA) depends on axial velocity VP and

induced velocity uind (influence of profile)

Axial velocity VP’=VP+uind

Can be calculated with the help of the Momentum Theory

dTdL

dD

dQ/r

V

VT = ωRr

VP

uind θR αϕ


BEMT | Example of Lift and Drag Coefficient

xcord

cLcL cD

cD α

α α

cM cL/cD



cL

cD



cL

α

cL/cD

α


Lift and drag at blade element

Thrust and drag torque element

Nb: Number of blades

VT: ωRr

c : cord length

Approximation (small angles)

Describing the lift coefficient

Assume a linear relationship

with AoA

Thin airfoil theory (plate)

Experimental results

Linearization of polar

Thrust at blade element


BEMT | blade element2

2LdL C cdrV

2

2LdD C cdrV

)sincos( dDdLNdT b

rdDdLNdQ b )cossin(

TVV

T

P

V

V '

dLNdT b

0( )L LC C

2LC

5.7LC

'2

0( ) (14)2

Pbe b L R T

T

VdT N C cdrV

V

Linearize polar for Reynolds number at 2/3 R

α0: zero lift angle of attack.

Used for asymmetric profiles

dTdL

dD

dQ/r

V

VT = ωRr

VP

uind θR α

ϕ


Vp’ is still unknown

Use momentum theory at each

blade annulus to estimate the

induced velocity

Equate thrust from blade

element theory and

momentum theory

Solve equation for Vp’

With solved Vp’ calculate the

thrust and drag torque with the

blade element theory

Integrate dT and dQ over the

whole blade


BEMT | Momentum Theory

' '2 ( ) 4 ( ) (15)mt P ind ind P P PdT V u u dA V V V rdr

bemt dTdT

R

b

RR

PV

RR

P

R R

cN

R

V

R

V

R

rx

, , ,

'

2

0( ) ( ) 08 8

L LV R

C Cx

)sincos( dDdLNdTT b rdDdLNdQQ b )cossin(


and/or indu


Dynamic Modeling of Rotorcraft


Rotary Wing UAS: Propeller Analysis and Dynamic Modeling


Two main reasons for dynamic modeling

System analysis: the model allows evaluating the characteristics

of the future aircraft in flight or its behavior in various conditions

Stability

Controllability

Power consumption

…

Control laws design and simulation: the model allows comparing

various control techniques and tune their parameters

Gain of time and money

No risk of damage compared to real tests


Modeling | Introduction

Modeling and simulation are important, but they must be validated in reality


Whitebox modeling

Use a priori information to

model dynamics

Blackbox modeling

Use input and output

measurements to deduce

dynamics

Greybox modeling

Use as much of a priori

information as available

Identify unknown part with

measurements (e.g.

unknown parameter 1, 2)


Modeling | Approaches

u

u

u

y

y

y

g(u,x)

f(x)

g(u,x)

f(x)1

2


The model of the rotorcraft consist of three main parts

Input to the system are commanded rotor speeds or the input

voltage into the motor

Motor dynamics block: Current rotor speeds

Due to fast dynamics w.r.t. body dynamics. Not necessary for control

design if bandwidth is taken into account

Propeller aerodynamics: Aerodynamic forces and moments

Body dynamics: Speed and rotational speed


Modeling of Rotorcraft | Overview

ωR,cmd ωR F, M v, ω. .


Coordinate systems

E Earth fixed frame

B Body fixed frame

The information required is the position and orientation of B (Robot)

relative to E for all time


Modeling of Rotorcraft | Coordinate System


Modeling of Rotorcraft | Representing the Attitude

cossin0sincos0001

),(2 xC B

1000cossin0sincos

),(1

zCE

1Yaw1 2 Pitch2 3 Roll3

Roll (-


Split angular velocity into the three

basic rotations

Bω = Bωroll + Bωpitch + Bωyaw

Angular velocity due to change in roll

angle

Bωroll = (ϕ,0,0)T

Angular velocity due to change in pitch

angle

Bωpitch = CT

2B (x,ϕ) (0,θ,0)T

Angular velocity due to change in yaw

angle

Bωyaw = CT

12(y,θ) CT

2B (x, ϕ) (0,0,Ψ)T


Modeling of Rotorcraft | Rotational Velocity

.

.

.


Relation between Tait-Bryan angles and angular velocities

Linearized relation at hover


Modeling of Rotorcraft | Rotational Velocity 2

ωJ Br

coscossin0cossincos0

sin01

rJ

ωJ Br 100010001

0,0

Singularity for = 90o


Consider fuselage as one rigid body

Recall the rigid body dynamics

Rigid body can be completely describes by the velocity at the CoG and

the rotational velocity

Change of momentum

p = mvCoG

ṗ = ∑F

Change of spin

N = Iω

Ṅ = ∑M

Euler differentiation rule for vector (c) representations

B(dc/dt) = dBc/dt + ω x Bc


Modeling of Rotorcraft | Body Dynamics 1


Change of momentum and spin in the body frame

Position in the inertial frame and the Attitude

Forces and moments

Aerodynamics and gravity


Modeling of Rotorcraft | Body Dynamics 2

M

F

ωIω

vω

ω

v

I B

B

BB

BB

B

Bx mmE

0

033

vCx BEBE ωJ Br

Aero

AeroG

BB

BBB

MM

FFF

mgE

T

EBGB 00

CF

E3x3: Identity matrix


Torque!!


Highly depends on the structure

Propeller: Generation of forces and torques

Thrust force T

Hub force H (orthogonal to T)

Drag torque Q

Rolling torque R

Rotor: Generation of forces and torques. Additional tilt dynamics of

rotor disc

Modeling of forces and moments similar to propeller

Modeling the orientation of the TPP


Modeling of Rotorcraft | Rotor/Propeller Aerodynamics


Use DC motor model

Consists of a mechanical and

an electrical system

Mechanical system

Change of rotational speed

depends on load Q and

generated motor torque Tm Imdωm/dt =Tm-Q

Generated torque due to

electromagnetic field in coil

Tm=kti

kt: Torque constant


Modeling of Rotorcraft | Motor Dynamics

i(t)

U(t)

ωm(t)Q(t)


Electrical System

Voltage balance

Ldi/dt = U-Ri-Uind

Induced voltage due to back

electromagnetic field from the

rotation of the static magnet

Uind=ktωm

Motor is a second order system

Torque constant kt given by the

manufacturer

Electrical dynamics are usually

much faster than the mechanical

Approximate as first order system


Modeling of Rotorcraft | Motor Dynamics

i(t)

U(t)

ωm(t)Q(t)


Four propellers in cross

configuration

Two pairs (1,3) and (2,4), turning

in opposite directions

Vertical control by simultaneous

change in rotor speed

Directional control (Yaw) by rotor

speed imbalance between the

two rotor pairs

Longitudinal control by converse

change of rotor speeds 1 and 3

Lateral control by converse

change of rotor speeds 2 and 4


Modeling a Quadrotor | Control OverviewF1

1

F4

4

F3

3 F2

2


Arm length l

Planar distance between CoG and

rotor plane

Rotor height h

Height between CoG and rotor plane

Mass m

Total lift of mass

Inertia I

Assume symmetry in xz- and yz-plane


Modeling a Quadrotor | Properties

zz

yy

xx

I

I

I

00

00

00

I


Main modeling assumptions

The CoG and the body frame origin are assumed to coincide

Interaction with ground or other surfaces is neglected

The structure is supposed rigid and symmetric

Propeller is supposed to be rigid

Fuselage drag is neglected


Modeling a Quadrotor | Assumptions


Propeller in hover

Thrust force T

Aerodynamic force perpendicular to

propeller plane

│T │=ρ∕2APCT(ωPRP)2

Drag torque Q

Torque around rotor plane

│Q │=ρ∕2APCQ(ωPRP)2RP

CT and CQ are depending on

Blade pitch angle (propeller

geometry)

Reynolds number (propeller speed,

velocity, rotational speed)

…


Modeling a Quadrotor | Propeller Aerodynamics 1

Ap

Similar to blade element


Propeller in forward flight

Additional forces due to force unbalance

at position 1. and 2.

Hub force H

Opposite to horizontal flight direction VH

│H │=ρ∕2APCH(ωPRP)2

Rolling moment R

Around flight direction

│R │=ρ∕2APCR(ωPRP)2RP

CH and CR are depending on the

advance ratio μ

μ=V∕ωPRP …


Modeling a Quadrotor | Propeller Aerodynamics 2

ωP

Vrel


Quadrotor shall only be considered in near hover

condition

Main forces and moments come from propellers

Depend on flight regime (hover or translational flight)

In hover: Hub forces and rolling moment small compared to thrust

and drag moment

Since hover is considered, thus thrust and drag are proportional to

propellers‘ squared rotational speed:

Ti=bωp,i2 b: thrust constant

Qi=dωp,i2 d: drag constant


Modeling a Quadrotor | Simplifications


Hover forces

Thrust forces in the shaft

direction

Ti=bωp,i2

Additional Forces: Can be neglected

Hub forces along the

horizontal speed

Ti = ρ/2ACH(Ωi Rprop)2

vh = [u v 0]T


Modeling a Quadrotor | Aerodynamic Forces

iB

iB

T004

1

T

hB

hBi

iB H

v

vH

4

1


Hover moments

Thrust induced moment

Drag torques

Additional moments

Inertial counter torques

Propeller gyro effect

Rolling moments

Hub moments


Modeling a Quadrotor | Aerodynamic Moments

4

1

14

1

Pr

Pr

)1(

00

00

i hB

hBPBiHB

hB

hBi

i

iB

B

PopB

GB

PB

opCTB

iHR

II

v

vpM

v

vR

ωMM

4

1

1

31

24

)1(

00

0

)(

)(

i

i

i

B

B

B

TB

Q

TTl

TTl

QM


Modeling the CoaX

helicopter

Two brushless DC motors

drive a rotor each

Hingeless upper and lower

rotors

Flybar on upper rotor to

reduce dynamics and

stabilize system

Swashplate controls lower

rotor blade pitch angles


Modeling a Coaxial Helicopter | Overview


Two rotors in coaxial configuration

Rotors are turning in opposite directions

Directional control () by rotor

speed imbalance

Vertical control (z) by

simultaneous change in

rotor speed

Longitudinal and lateral

control (x, y) by tilting the

lower rotor tip path plane


Modeling a Coaxial Helicopter | Control Overview

Going up

Rotate left Rotate right

Move right


Main modeling assumptions

The CoG and the body frame origin are assumed to coincide

Interaction with ground or other surfaces is neglected

The structure is supposed rigid and symmetric

Diagonal inertia matrix

Fuselage drag is neglected

Rotor dynamics are faster than main body dynamics

Only consider steady state solution of the rotor blade dynamics


Modeling a Coaxial Helicopter | Assumptions


Aerodynamic forces acting on the upper and lower rotor-

head

Thrust forces


Coaxial Helicopter | Rotor Aerodynamic Forces 1

1 1

1 1

1 1

2

,

cos sin

cos sin

cos cos

B i i B i

i i

s c

i i

B i c s

i i

c sB

i i r i

T

T b

T n

n

G

E

Tu

Tl

Qu

Ql

Mflap,u

Mflap,l

Normal to the tip-path plan

(upper and lower)

, (lower and upper rotor)i l u


Upper rotor produces an

airflow on lower rotor

Lower rotor has an additional

downward velocity

component on the rotor

Reduces the efficiency of the

lower rotor

Can be adjusted by different

rotor speeds (or blade pitch

angles of lower rotor)


Coaxial Helicopter | Rotor Aerodynamic Forces 2


Rotor drag torques

Thrust induced moment

Flap moment

κ: Spring constant


Coaxial Helicopter | Rotor Aerodynamic Moments

2

1)1(

00

i

i

iB

B

Q

Q

lui

iBiBTB

,

TrM

lui

i

c

i

s

B

flapB

,

1

1

0

MG

E

Tu

Tl

Qu

Ql

Mflap,u

Mflap,l

Due to axis-offset of thrust force

rl


Model of the blade flapping dynamics

Consider one blade only and introduce the hinge forces

Conservation of spin at point E

Moment due to lift

Varies with blade pitch angle θR

Varies with velocities , ω, v

Moment due to gravity

Moment due to gyroscopic effects


Modeling a Coaxial Helicopter | Blade Flapping 1

fl

, , x yf f


Rewrite the flapping equations

General solution of nonlinear 2nd order system with harmonic input

Damped oscillator

Simplification

Consider only first harmonic

Blade flapping is fully described by the three parameters β0, β1c and β1s



( , ) ( , , , , , ) (16)fl R R L R R fl flGyroscopic Lift

ω v

0

1

( ) ( ) ( ( ) cos( ) ( )sin( )) (17)nc nsn

t t t n t n

0 1 1( ) ( ) ( ) cos( ) ( )sin( ) (18)c st t t t

Gravity force ignored (steady state)


Describe the parameter variation in time

Put the general solution in flapping model dynamics

Matrices A depend on geometric, structural and inertial parameters

and on the advance ratio

Estimated using appropriate experiments



0 0

1 1

1 1

dimensionless inflow velocity

body rotation

(19)

c c

s s

p

q

p

q

A A A A A A

0

1

1

blade pitch angelc

s


1c → cos ; 1s → sin


Blade flapping depends on three main terms

Blade flapping due to change of blade pitch angle

System input for the lower rotor

Defined by the stabilizer bar on the upper rotor

Blade flapping due to rotational velocities of the fuselage

Blade flapping due to different relative inflow distribution

Flapping dynamics are faster than fuselage dynamics

Approximate blade flapping with its steady state solution



A A A A A A

0 and 0


The orientation of the stabilizer bar needs to be modeled

Can be modeled the same way as for blade flapping

Passive system.

Gyroscopic terms dominant

Analog to eq. (19) but

without aerodynamic forces

Connected dampers tries to

align the stabilizer bar

perpendicular to shaft

Stabilizer bar directly coupled with the blade pitch angle


Coaxial Helicopter | Stabilizer Bar Dynamics 1

)sin()()cos()()( tttsb

s

sb

c

sb

pq sbssp

sb

s

sb

c

sp

sb

c 11111

1

rotation


Coupling between stabilizer bar and upper blade pitch

angle

According to phase lag of

blade flapping motion

Fixed angle. Angle is derived from

phase lag at nominal rotational speed

If driven at different speed, coupling effects


Coaxial Helicopter | Stabilizer Bar Dynamics 2

)cos()sin(

)sin()cos(

111

111

sb

sb

ssbsb

sb

csb

u

s

sb

sb

ssbsb

sb

csb

u

c

kk

kk


Books:

[1] Christoph Hürzeler: Modeling and Design of Unmanned

Rotorcraft Systems for Contact Based Inspection, PhD Dissertation

Nr. 21083. Autonomous Systems Lab, ETH Zürich, 2013.

[2] Samir Bouabdallah: Design and control of quadrotors with

application to autonomous flying, PhD Dissertation Nr. 3727,

Autonomous Systems Lab, EPFL Lausanne, 2007.

Papers:

[3] Péter Fankhauser et al., Modeling and decoupling control of the

coax micro helicopter, International Conference on Intelligent

Robots and Systems. IEEE, 2011.


References

Documents

Folienmaster ETH Zürich · 2015. 12. 3. · Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 27.10.2015 19 BEMT | blade element 2 2 L dL C cdrV U 2 2 L dD