Modeling and Attitude Control of Tri-Tilt Ducted Fan Vehicle

Flight Dynamics and Control Lab.Department of Mechanical and Aerospace Engineering

Seoul National University, Republic of Korea

Modeling and Attitude Control ofTri-Tilt Ducted Fan Vehicle

SciTech 2016, Manchester Grand Hyatt, San Diego, California

Yongjun Seo*, and Youdan Kim

4 January 2016

Yongjun Seo, et al. “Modeling and Attitude Control of Tri-Tilt Ducted Fan Vehicle,”AIAA SciTech 2016, Manchester Grand Hyatt, San Diego, California, January 2016. / 332

Contents

1. Introduction

2. Vehicle Configuration and Dynamics

3. Control Allocation

4. Attitude Controller

6. Conclusion

5. Simulation

Yongjun Seo, et al. “Modeling and Attitude Control of Tri-Tilt Ducted Fan Vehicle,”AIAA SciTech 2016, Manchester Grand Hyatt, San Diego, California, January 2016.

Introduction

1. Introduction


Introduction (1/2)

Tri Tilt Rotor Vehicle

Properties

Each engine rotates along the y-axis of fuselage frame.

Tilting engines to make yawing moment enables rapid response compared to the conventional quad rotor configuration.

It does not need to tilt the fuselage to advance forward or backward.

The small amount of aerial drag occurs during the cruising flight.

The additional aerodynamic apparatus such as a wing is available to the vehicle.

tilting axis

thrust and moment

center of gravity

Ducted Fan 1

Ducted Fan 2

Ducted Fan 3

rotation


Introduction (2/2)

Several VTOL Concept vehicles

Proprotor VTOL Tri‐tilt rotor vehicles

Tri‐Fan V‐STOL Quad‐Tilt‐Wing UAV


Vehicle Dynamics

2. Vehicle Dynamics


Configuration

Installation of duct Location of the tilting axis Arm vector

• The relative position vector of the point which is the intersection of the plane containing and perpendicular to and the tilting axis.

Tilting axis• The rotation axis of a duct

Axis deviation• The deviation of from

/

Tb bx y z A Gr r ré ù= =ê úë ûr p

D

r

[ ]0 1 0 Td =n1=n

n

0⋅ =n a [ ] /0 Td dx z A Da a= =a p

D A

n

1A

2A

3A

G1μ

2μ

3μ

1n

1rω

1r

2n2

rω

2r

3n

3rω

3r

bi

bj

bk


Coordinate transform matrices

Angular Velocities

Dynamics (1/5)

/ / / /

/ / / /

d d bd i d b d b b ir r dr i r d r d d i

= +

= +

ω ω C ω

ω ω C ω

/ / / /

/ / / /

d d bd i d b d b b i

r r dr i r d r d d i

= +

= +

ω ω C ω

ω ω C ω

2 2 2 20 1 2 3 1 2 0 3 1 3 0 2

2 2 2 2/ 1 2 0 3 0 1 2 3 2 3 0 1

2 2 2 21 3 0 2 2 3 0 1 0 1 2 3

2

/ 1 1

1 1

2 22 22 2

cos 0 sin 1 0 0 cos si0 1 0 0 cos sin

sin 0 cos 0 sin cos

b i

d b

q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q

C

C2

2 2

n 0sin cos 0

0 0 1

/

1 0 00 cos sin0 sin cos

r d

C ( )0

t

rtd = ò

A

r

rω

bibjbk

n

G

zθx

θzθ μ


Approximation of moment of inertia

Angular momentum

Forces and moment

Fuselage Duct Rotor

Dynamics (2/5)

00 0

0

xx zxbG yy

zx zz

J JJ

J J

é ù-ê úê ú= ê úê ú-ë û

J

00 0

0

Dxx DzxdD Dyy

Dzx Dzz

J JJ

J J

é ù-ê úê ú= ê úê ú-ë û

J

0 00 00 0

RarR Rr

Rr

JJ

J

é ùê úê ú= ê úê úë û

J

Rotor Duct Duct‐rotor assembly Fuselage

/

r r rR R r i=h J ω

/d d dD D d i=h J ω :d d d

DR D R= +h h h /

b b bG G b i=h J ω

Duct‐rotor assembly Fuselage

k k k kd d d dk k D km= + +åF N G T

k k k k kd d d d dk k k k k= + ´ +åM τ a N Q

b b b bb b bm=- + +åF N G F

( ){ }b b b b bb k k b

k

=- + ´ - +å åM τ r N M


Dynamics (3/5)

Euler’s laws of motion

Elimination of the actions and reactions by joint

Forces

Moments

( ){ }/ jk k

k k

d d bb d u k b

k

+ =å C j D τ τ

/k

k

d b bb d k k b

k k

= =å åC N N N

k k

k

d dk u⋅ =τ j

( ){ }/ jk k

k k

d d bb d u k b

k

+ =å C j D τ τ j

1 0 00 0 00 0 1


DA

r

G

xτtτ

zτ

xτ−

tτ−

zτ−

N

−N

A′


/k k k k kd d d d di

D D i k k D km m= = + +åv F N G Tk k k k k k

k

d d d d d diDR k k k k k= = + ´ +åh M τ a N Q

/

i b b b b bb G i b b bm m= =- + +åv F N G F

( ){ }i b b b b b bG b k k b

k

= =- + ´ - +å åh M τ r N M


Dynamics (4/5)

Several simpler model Simplification on the multibody model

• Force

• Moment

Tilting axis alignment

Single rigid body dynamics with additional angular momenta

The action-reaction parameter is not introduced in this case, and thus the load on the servo motor and the precession of the ducts with respect to the fuselage are not considered. But the gyroscopic effect caused by the rotation of the rotors are still under consideration.

Zero tilting axis deviation ( kdk =a 0 )

( )( ) ( )/ / /k

k

di b i bb D G i b i b b d k

k

m nm+ - = +åv C G F C T

{ }k k k

k k

d d diDR k u- ⋅ =h Q j ( ){ } ( ){ }/ j /

k k k k

k k k k

d d d di b i b bGA b d u DR k k b d k b

k k

é ù=- + - + ´ +ê úë ûå åh C j D h Q r C T M

( 1 0 = and 2 0 = )

( kdk =a 0 )

( ), ,kk s k k cf =


( )( ) ( )/ / /k

k

di b i bb D G i b i b b d k

k

m nm+ - = +åv C G F C T V G D R= + +h h h h / /+ k

k k

db b bV GA b i b d DR

k

= åh J ω C h

( ){ }/ /k k

k k

d di b b bV k b d k b d k b= ´ + +åh r C T C Q M


Dynamics (5/5)

Actuator dynamics Tilting actuator

Ducted fan

• Power model

∫ ∫1K

2Krμ μ

μ�

( )k kf f r dt T + = -

2

2

d T f

d Q f

T C

Q C

=

=

32

d PT dP C T=

32PT

t k kk kd

CP P T

= =å å

tilting axis

thrust and moment

center of gravity

Ducted Fan 1

Ducted Fan 2

Ducted Fan 3

rotation


System Description (1/2)

State space description System of minimal coordinates

The functions and are computed via MATLAB symbolic math toolbox.

( ) ( ) ( )1 , ,-=- =x E x n x u f x u

( )E x ( ),n x u

The minimal coordinate description of the dynamics of given multibody system is directly obtained bycanceling the reaction forces and moments. The result is

( ) ( ),+ =E x x n x u 0

or ( ) ( ),=-E x x n x u

where nÎx , mÎu .

The rows of ( )E x are linearly independent and thus it is invertible.


System Description (2/2)

Assignment of state and input variables States

• Duct-rotor assembly

• Quaternion

• Position in the NED coordinate system

Inputs• Rotor speed

• Tilt angles

1 1 2 2 3 31 1 2 2 3 3

T

r r r r r r é ù= ê úë ûx

[ ]0 1 2 3Tq q q q=q

[ ]/Ti

G O x y z=p

/ / /

Tb T b T T T i TG i b i G O

é ù= ê úë ûx v ω x q p

1 2 3

T

u c c c é ù= ê úë ûω

1 2 3

T

u u u u é ù= ê úë ûμ

[ ]Tu u=u ω μ


Control Allocation

3. Control Allocation


Control Allocation (1/4)

Control Allocation Over-actuated system The total number of force and moment components that can be generated by the configuration of actuator is 5

while the actuators have total degree of freedom of 6.

Reaction moment cancelation Fuselage moment equation

Moment to cancel

Fuselage moment equation after cancelation

( ){ } ( ){ }/k k

k k

d di b i b b bG b d DR k k k A

k k

=- - + ´ - +å åh C h Q r N M

( ){ } ( ){ }/ /k k k

k k k

d d di b i b bGA b d DR k k b d k A

k k

=- - + ´ +å åh C h Q r C T M

/:b b b

GA GA b i=h J ω 2b b

GA G D kk

m= - åJ J R

( ){ }/: k k

k k

d db ic b d DR k

k

=- -åM C h Q kdk =Q 0

( )/ / /i b b b b b b b b

GA GA b i b i GA b i= + ´ =h J ω ω J ω M



Input vectors

The coordinate transformation of the actual input vector is required to avoid nonlinear configuration mapping.

Configuration mapping

Virtual input vector Actual input vector

[ ]Td d d d dX Z L M N=τ 1 2 3

T

1 2 3c c cT T T é ù= ê úë ûu

[ ]T1 1 2 2 3 3c x z x z x zT T T T T T=u ( )arctan 2 ,kc kz kxT T =- [ ]Tk kx kzT T=T

[ ]

[ ] ( ){ }/

/

k

k

k

k

T dbd d d b d k

k

T db b bd d d c k b d k

k

X Y Z

L M N

= =

= - = ´

å

å

F C T

M M r C T

( )( ) ( )

( )

2 1 1 3 1 1 1 1 3 3 3

2 1 1 3 3

1 3

1 2 3 1 2 3

3 1

1 1 1 3 3 3

1

cos cos sin sin sin sincos cos

2 2

cos sin sin 1 0 cos sin sin0 cos 0 1 0 co

x x z x z z x z z x z

z z x z x

z z y

x x x z z z z x

x x y

z x z z x z

x

T T T T TT T T

T T rT T T r T T T r

T T r

é ù+ + + +ê úê ú+ +ê úê ú-= ê úê ú- + - - +ê úê ú-ê úë û

=

τ

1

13

2

2

3

3

s0 0 0 0

2 20 0 0 0

x

zx

xy y c

zz x z x z x

xy y

z

TTT

r rT

r r r r r rT

r rT

é ùé ù ê úê ú ê úê ú ê úê ú ê úê ú ê ú- =ê ú ê úê ú ê ú- - -ê ú ê úê ú ê ú-ê ú ê úë û ê úë û

Cu



Control allocation methods• Minimizing control efforts

• Mixed optimization

• Partially constrained mixed optimization (PCMO)

arg minc

Top c c

U VÎ Ç=

uu u Qu { }c cU = - =u τ Cu 0 ( ){ }c cV = £u h u 0

( ) ( )( )arg minc

T Top c c c c

VÎ= - - +

uu τ Cu W τ Cu u Qu

2

21 1 2

arg minc

op c cU V

kP

u

u τ C u u { }2 2 2c cU u τ C u 0= - =

1

2

ττ

τ

1

2

CC

C

1T

T TX Zτ 2T

T T TL M Nτ

1 1 1 3 3 3

11 3

cos sin sin 1 0 cos sin sin0 cos 0 1 0 cos

z x z z x z

x x

é ùê ú= ê úë û

C

2

0 0 0 02 2

0 0 0 0

y y

z x z x z x

y y

r rr r r r r rr r

é ù-ê úê ú= - - -ê úê ú-ê úë û

C



Several solutions

Rolling Pitching Yawing

Hovering phase

Forward flight phase


Attitude Controller

4. Attitude Controller


Attitude Controller (1/2)

Attitude Controller

Quaternion feedback regulator

Principal axis transformation

Controller design

QuaternionConversion

AttitudeController

VehicleDynamicscΦ

cq

q

τ u

x, x�

ControlAllocation

MomentCancelation

0

/ / / / /eb

e b r i r b i r i b ibe

qq q q q q q

é ùê ú= = = * = *ê úë ûq

( )/ / /b b b b b b

b i b i b i e= ´ - -u ω J ω Dω Kq

p T b=J Q J Q 1 T- =Q Q / /

p T bb i b i=ω Q ω

/

/ /

/ / /

b b bb i

p b pb i b i

p T b p p pb i b i b i

=

=

= =

h J ω

Qh J Qω

h Q J Qω J ω

/ / / / / /pe p r i r p i r i b i p bq q q q q q q= = * = * *

( ){ }/ / / /b p p p p p

b p b i b i b i e= ´ - -u C ω J ω Dω Kq

, p pk d= =K J D J 22 0

2 n nkd + + = + + =

( ) 1pk -

= +K J I pd=D J

( ){ }21

tr p pPI -æ ö÷ç= + - ÷ç ÷÷çè ø

J I J


Attitude Controller (2/2)

Vehicle SpecificationLength 1,207mm Width 1,800mm Height 445mm

bm 7.437kg

Dm 0.87kg

bGAJ 2

0.5989 0 0.09220 0.5295 0 kg m

0.0922 0 1.0658

é ù-ê úê ú ⋅ê úê ú-ë û

dDRJ

4 8

8 2

7.965 10 7.647 10 07.647 10 0.002 0 kg m

0 0 0.002

- -

-

é ù´ - ´ê úê ú- ´ ⋅ê úê úê úë û

RaJ 4 21.245 10 kg m-´ ⋅

1br [ ]T0.240 0.345 0.060 m

1x , 2x , 2z , 3x 0

1z 2-

3z 2 0.1s

dT 0.02s

maxT 60N

max 6.16rad/s

TC 6 2 24.1614 10 N s / rad-´ ⋅

QC 2 20N m s / rad⋅ ⋅

PTC 3/22.8963W / N


Simulation

5. Simulation


Simulation I

Simulation I

Simulation dL ( N m⋅ ) dM ( N m⋅ ) dN ( N m⋅ ) dX ( N ) dZ ( N )

I 0 0.5 0 0 100- II 0.5 0 0 0 100-

III 0 0.5 0 ( )20sin t ( )100 20cos t- +

dLN m⋅dMN m⋅dNN m⋅dXNdZN00.500 100-0.5000 100-00.50 ( )20sin t ( )100 20cos t- +


Simulation II

Simulation II


I 0 0.5 0 0 100- II 0.5 0 0 0 100-

III 0 0.5 0 ( )20sin t ( )100 20cos t- +


Simulation III

Simulation III


I 0 0.5 0 0 100- II 0.5 0 0 0 100-

III 0 0.5 0 ( )20sin t ( )100 20cos t- +


Simulation IV

Simulation IV

0.5814 0 00 0.5295 00 0 1.0833


D ,

0.5673 0 00 0.5405 00 0 1.0880


K 0.82 = , 3.2rad/sn =

1 0 0 0 00 1 0 0 00 0 400 0 00 0 0 400 00 0 0 0 400

é ùê úê úê úê ú= ê úê úê úê úê úë û

W

0NdX =

200NdZ =-

Case 1Partially constrained mixed optimization(PCMO)

Case 2PCMO withoutmoment cancelation

Case 3Mixed optimization


Simulation IV

Simulation IV


Simulation IV

Simulation IV

Mixed optimization Partially constrained mixed optimization


Simulation V

Simulation V ( )100 100cos NdX t= + ( )100 100sin NdZ t=- +

Case 1Partially constrained mixed optimization(PCMO)

Case 2PCMO withoutmoment cancelation

Case 3Mixed optimization


Simulation V

Simulation V

PCMO without moment cancelation PCMO


6. Conclusion

Conclusion


Conclusion

Conclusion The precise model for the tri-tilting ducted fan vehicle considering the gyroscopic coupling by multibody

dynamics with Euler’s laws of motion was derived. The formulation enables the simplification of the original complex model to a single body system with forces

and moments as control input through control allocation. The partially constrained mixed optimization was also proposed to improve attitude stability utilizing simple

quaternion feedback regulator.

Further research Development of the landing gear suspension model and tyre friction model Aerodynamics model Consideration on the nonlinear controllers


Any Question will beappreciated.

Documents

Modeling and Attitude Control of Tri-Tilt Ducted Fan Vehicle