2017 rotorcraft - dynamic modeling of rotorcraft & …€œif you are in trouble anywhere, an...

||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 DynamicsRotorcrafts: Dynamic Modeling of Rotorcraft & Control

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

Contents | Rotorcrafts (Rotary Wing UAV)

1. Introduction Rotorcrafts

2. Dynamics Modeling & Control

3. Thrust Force of Propeller / Rotor

4. Rotor Craft Case Study


IntroductionRotorcrafts



Rotorcraft: Aircraft which produces lift from a rotary wing turning in a plane close to horizontal

17.01.2018Robot Dynamics: Rotary Wing UAS 4

Rotorcraft: Definition

“A helicopter is a collection of vibrations held together by differential equations” John Watkinson

Advantage Disadvantage

Ability to hover High maintenance costs

Power efficiency during hover Poor efficiency in forward flight

“If you are in trouble anywhere, an airplane can fly over and drop flowers, but a helicopter can land and save your life” Igor Sikorsky


Helicopter Autogyro Gyrodyne

Power driven main rotor Un-driven main rotor, tilted away

Power driven main propeller

The air flows from TOP to BOTTOM

The air flows from BOTTOM to TOP

The air flows from TOP to BOTTOM

Tilts its main rotor to fly forward

Forward propeller for propulsion

Main propeller cannot tilt

No tail rotor required Additional propeller for propulsion

Not capable of hovering

17.01.2018Robot Dynamics: Rotary Wing UAS 5

Rotorcraft | Overview on Types of Rotorcraft

||Autonomous Systems Lab 17.01.2018Robot Dynamics: Rotary Wing UAS 6

Rotorcraft | Rotor Configuration 1

Single rotor Multi rotor

Most efficient Reduced efficiency due to multiple rotors and downwash interference

Mass constraint Able to lift more payload

Need to balance counter-torque Even numbered rotors can balance counter-torque


Rotorcraft | at UAS-MAV Size 1

Quadrotor / Multicopter Std. helicopter

Four or more propellers in cross configuration Very agile

Direct drive (no gearbox) Most efficient design

Very good torque compensation Complex to control

High maneuverability


Rotorcraft | at UAS-MAV Size 2

Ducted fan Coaxial

Fix propeller Complex mechanics

Torques produced by control surfaces Passively stable

Heavy Compact

Compact Suitable for miniaturization


Dynamics Modeling & ControlRotorcrafts



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


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


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. .


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



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


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


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 = ( ,0,0)T

Angular velocity due to change in pitch angle Bωpitch = CT

2B (x,ϕ) (0, ,0)T

Angular velocity due to change in yaw angle Bωyaw = [C12(y,θ) C2B (x, ϕ)] T (0,0, )T


Modeling of Rotorcraft | Rotational Velocity


Relation between Tait-Bryan angles and angular velocities

Linearized relation at hover

17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 21

Modeling of Rotorcraft | Rotational Velocity

r r BE ω

rrE ; coscossin0cossincos0

sin01

ωBrE 100010001

0,0

Singularity for = 90o


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

MF

ωIωvω

ωv

I B

B

BB

BB

B

Bx mmE 0

033

vCx BEBE ωBrE

Aero

AeroG

BB

BBB

MMFFF

mgE

TEBGB 0

0CF

E3x3: Identity matrix


Torque!!


Hover forces Thrust forces in the shaft

direction

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

iAeroB

TF 0

04

1

hB

hBi

iB H

vvH

4

1

2,i i p iT b


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

2 2, , i i p i i i p iT b Q d

4 2

1 34

1

1

00

01

B Aero B T B

iBi

iB

l T TM M Q l T T

Q


Rotational dynamics near hover

Full control over all rotational speeds Not dependent on position states


Control of a Quadrotor | Analysis of the Dynamics 1

2 2,4 ,2

2 2,1 ,3

2 2 2 2,1 ,2 ,3 ,4

( ) ( )

( ) ( )

( )

;

xx yy zz p p

yy zz xx p p

zz p p p p

r r B r

B

I p qr I I lb

I q rp I I lb

I r d

pE q

r

,

: thrust constant: drag constant

: distance of propeller from the CoG: rotational speed of propellor p i

bdl

i

MF

ωIωvω

ωv

I B

B

BB

BB

B

Bx mmE 0

033

0 00 00 0

zz yyxx

B B yy xx zz

zz yy xx

qr I Ip I pI q I q rp I I

r I r pq I I

0


Translational dynamics in hover

Only direct control of the body z velocity Use attitude to control full position


Control of a Quadrotor | Analysis of the Dynamics 2

2 2 2 2,1 ,2 ,3 ,4

( ) sin ( ) sin cos ( ) cos cos ( )p p p p

EB EB B

B

mu m rv qw mgmv m pw ru mgmw m qu pv mg b

ux C v C v

w

,

: thrust constant: rotational speed of propellor p i

bi

MF

ωIωvω

ωv

I B

B

BB

BB

B

Bx mmE 0

033

B B

p u m qw rvm v q m v m ru pw

r w m pv qu

0 sin0 sin cos

cos cos

TB G EB

E

F C mgmg


Goal: Move the Quadrotor in space Six degree of freedom Four motor speeds as system input Under-actuated system Not all states are independently controllable Fly maneuvers

Control using full state feedback Assume all states are known Needs an additional state estimator

Assumption Neglect motor dynamics Low speeds

→ only dominate aerodynamic forces considered→ linearization around small attitude angles


Control of a Quadrotor | General Overview


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


Define virtual control inputs to simplify the equations Get decoupled and linear inputs Moments along each axis

Total thrust


Control of a Quadrotor | Virtual Control Input

2

3

4

( )

( )xx yy zz

yy zz xx

zz

I p qr I I U

I q rp I I U

I r U

)( 24,

23,

22,

21,1 ppppbU

2 22 ,4 ,2

2 23 ,1 ,3

2 2 2 24 ,1 ,2 ,3 ,4

( )

( )

( )

p p

p p

p p p p

U lb

U lb

U d

1

( ) sin ( ) sin cos ( ) cos cos

mu m rv qw mgmv m pw ru mgmw m qu pv mg U


Rotational dynamics independent of translational dynamics Use a cascaded control structure Inner loop: Control of the attitude Input: U2, U3, U4

Outer loop: Control of the position Input: U1, ϕ, θ


Control of a Quadrotor | Hierarchical Control

rqp

wvuZYX

432 ,, UUU

1U


Linearize attitude dynamic at hover Equilibrium point:


Control of a Quadrotor | Linearization of Attitude Dynamics

This model has no coupling We use 3 individual controller

2

3

4

( )

( )xx yy zz

yy zz xx

zz

r

B

I p qr I I U

I q pr I I U

I r U

pE q

r

21 U

I xx

31 U

I yy

41 UIzz

2 3 4 10 ; 0 ; p q r U U U U mg


PID control strategy Generic non model based control Consist of three terms P - Proportional I - Integral D - Derivative

Different tuning methods Ziegler-Nichols method Pole placement

Widely used


Control of a Quadrotor | PID Control

0

( )( ) ( ) ( )t

p i dd e tu t k e t k e t dt k

dt

PlantCr(t) y(t)e(t) u(t)


Consider roll subsystem

P controller can‘t stabilize the system Harmonic oscillation

Use a PD controller


Control of a Quadrotor | PID Attitude Control

21 U

I xx

)(1 despxx

kI

1 ( ) ( )p des d desxx

k kI


3 separate control loops 1 additional thrust input

Aspect on the implementation Get angle derivatives from

transformation of the body angular rates

Avoid integral element in controller


Control of a Quadrotor | Attitude Control Design

1

2

3

4

( )

( )

( )

des

des pRoll dRoll

des pPitch dPitch

des pYaw dYaw

U T

U k k

U k k

U k k

real

real

real

2Ud

d

d

3U

4U


kp is chosen to meet desired convergence time kd is chosen to get desired damping In theory any convergence time can be reached Actuator dynamics allow

for a limited control signal bandwidth Closed loop system dynamics

must respect the bandwidth of actuator dynamics

Actuator signals shall not saturate the motors


Control of a Quadrotor | Parameter Tuning


Transformation of virtual control inputs to motor speeds Rewrite virtual control inputs in matrix form Invert matrix to get motor speeds

Limit motor speeds to feasible (positive) values17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 36

Control of a Quadrotor | Control Allocation

4

3

2

1

24,

23,

22,

21,

14101

211

41

1411

2101

41

14101

211

41

1411

2101

41

UUUU

dlbb

dlbb

dlbb

dlbb

p

p

p

p

24,

23,

22,

21,

4

3

2

1

0000

p

p

p

p

ddddlblb

lblbbbbb

UUUU


A possible control flow for altitude control

Control attitude and the altitude Attitude controller to control the angles Parallel control loop. Altitude controller to control the horizontal thrust

and nonlinear transformation to calculate the corresponding thrust input


Control of a Quadrotor | Altitude Control

ϕdes,θdes,ΨdesU2,U3,U4

zdes Tz U1

ωi,des


Rewrite translational dynamicsin the inertial frame

Only look at the altitude dynamics Virtual Control

PD controller for the height

Transformation to the body frame


Control of a Quadrotor | Altitude Control 2

1

0010

0

UmgB

EB

E

CV

VX

11cos cos z g Um

11 cos cos Z Zz g T T Um

( )z p des dT k z z k z mg

coscos1zTU


Possible control structure for full position control

Control position and yaw angle Hierarchical controller Outer loop: Position control Thrust vector control. Desired thrust vector in inertial frame Transform to thrust and roll and pitch angles Inner loop: Attitude control


Control of a Quadrotor | Position Control

U1

Ψdes

ϕ, θ ωi,desxdesydeszdes

TxTyTz

U2U3U4


Thrust vector control Assume that the control input is the thrust in the inertial frame Inertial position system dynamics

Use 3 separate PD controller for each direction Transform to desired thrust, roll and pitch


Control of a Quadrotor | Position Control 2

gTTT

mzyx

z

y

x

00

1

2 2 21

sin cos1 ( , ) sin

cos cos

xT

x y z yE

z

TT T T T C z T

TT


Thrust Force of Propeller / RotorRotorcrafts



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 (Tip Path Plan)


Modeling of Rotorcraft | Rotor/Propeller Aerodynamics


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


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 46

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 (angle of attack) Thin airfoil theory (plate) Experimental results Linearization of polar

Thrust at blade element27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 57

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


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)

2017 rotorcraft - dynamic modeling of rotorcraft & …€œif you are in trouble anywhere, an...

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