2016 l8.2 propeller analysis and dynamic modeling...robot dynamics -rotary wing uas: propeller...

||Autonomous Systems Lab

151-0851-00 V

Marco Hutter, Roland Siegwart 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

||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 2

Contents | Rotary Wing UAS

1. Introduction - Design and Propeller Aerodynamics

2. Propeller Analysis and Dynamic Modeling

3. Control of a Quadrotor

4. Rotor Craft Case Study


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 jointsexJ

S

, T Tex ex actb g F M J S


The Quadrotor Example

F1

1

F4

4

F3

3 F2

2

, TTe 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 energy27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 8

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

VndA

(2)p ndA VV ndA F : surface pressurep

21 (3)2

dEV V ndA Pdt

: energy

: powerEP


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

10 1

2 30 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

223

0 3

220 3 3

1 3

= = (6)

Thrust

R

F V dA V u dA

A V A V uA V u u

p ndA

331 3

0 3

330 3 3

1 3 3

1 12 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


212 (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

33

22 = (12)2 2

(13)

Thrust

R R

Thrust

mgFPA A

F mg

With eq. (10)


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

dT dL

dDdQ/r

V

VT = ωRrVP

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 element27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 19

BEMT | blade element2

2 LdL C cdrV 2

2 LdD C cdrV

)sincos( dDdLNdT b rdDdLNdQ b )cossin(

TVV T

P

VV '

dLNdT b

0( )L LC C

2LC

5.7LC

'2

0( ) (14)2

Pbe b L R T

T

VdT N C cdrVV

Linearize polar for Reynolds number at 2/3 R

α0: zero lift angle of attack. Used for asymmetric profiles

dT dL

dDdQ/r

V

VT = ωRrVP

uind θR αϕ


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 RcN

RV

RV

Rrx

, , ,

'

20( ) ( ) 0

8 8L L

V RC C x

)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 | Approachesu

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 (-<<)Pitch (-/2<</2)Yaw (-<<)

CEB: Rotation matrix from B to E

cos0sin010

sin0cos),(12 yC

2B


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) CT2B (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

sin01rJ

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

MF

ωIωvω

ωv

I B

B

BB

BB

B

Bx mmE 0

033

vCx BEBE ωJ Br

Aero

AeroG

BB

BBB

MMFFF

mgE

TEBGB 0

0CF

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

II

I

000000

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 theadvance 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,i

2 b: thrust constant Qi=dωp,i

2 d: drag constant


Modeling a Quadrotor | Simplifications


Hover forces Thrust forces in the shaft

direction Ti=bωp,i

2

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

iBi

BT004

1T

hB

hBi

iB H

vvH

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

iiB

B

PopB

GB

PB

opCTB

iHR

II

vvpM

vvR

ωMM

4

1

131

24

)1(00

0

)()(

i

ii

B

B

B

TB

QTTlTTl

QM

2016 l8.2 propeller analysis and dynamic modeling...robot dynamics -rotary wing uas: propeller...

Documents