Birotors sumary

7/31/2019 Birotors sumary

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Modeling of Birotor

Reference:

Farid Kendoul, Isabelle Fantoni, Rogelio LozanoModeling and Control of A Small Autonomous Aircraft

Having Two Tilting Rotors

Sumarized by:

tata


2/33

Birotor with OLT


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Birotor with LT

(vectors only)


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Description

The main components of a birotor-craft are the two rotorsattached on both side of the airframe (lateral-wise), equi-distantto airframes symmetric axis. Each rotor-airframe attachment reston a 2 DoF angular joint so that each rotors can be inclined (tilted)with respect to airframe body (see slide 2 & 3). Each joint is drivenby an actuator.

Both rotors act as thrust-actuating device. Moments producedwhile the rotors are operating are made to diminished each otherby setting the rotor pair rotation opposing each other and byapplying tilt angle ratio constraint as will be shown in thispresentation.

Controlling input is achieved by applying rotor output differentialsbetween rotor pair, tilting each rotors attitude, or a combinationof both.


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Bi-Rotor System

Functional Diagram


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Equation of Motion

G G

d d

d d

d d

d d

p hF M

t t

Vm V

t t

J J

WW W W

Force and Moment as momentum rate

Vectors are stated in Body Reference Frame


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Equation of Motion

T

P grav aeB

T

P grav aeB

F F F F V u v w

M M M M p q r

W

External force, external moment, vector of translational

velocity, and vector of angular velocity

Tensor of moment of inertia, on body reference frame with origin atcenter of gravity.

G G G

G G G G

G G G

xx xy xz

yx yy yz

zx zy zz

J J J

J J J

J J J

J


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Equation of Motion

(Rigid Body Kinematics)

I BI/B I/BV V

DCM KM

f

q W

y

I/ B

I/ B

cos cos cos sin sin sin cos sin sin cos sin cos

cos sin cos cos sin sin sin sin cos cos sin sinsin sin cos cos cos

1 0 0

0 cos sin

sin sin cos cos cos

DCM

KM

q y f y f q y f y f q y

q y f y f q y f y f q yq f q f q

f f

q f q f q


9/33

Equation of Motion

I/ B

I/ B

G

direct cosine matrix of local inertial reference frame with respect to body reference frame

kinematic matrix of local inertial reference frame with respect to body reference frame

tensor of

DCM

KM

J moment of inertia at center of gravity (G)

, vectors of external force and moment

, vectors of translational and angular momentum

vectors of translational and angular displacement ra,

F M

p h

V W

G B

R R B

te

vertical distance of vehicle's mass center (G) from body frame's origin (O )

horizontal distance of rotor frame origin (O ) from body frame's origin (O )

vehicle's mass

, , components of

i i

h

l

m

p q r W, expressed in body frame

components of , expressed in body frame, ,

longitudinal and lateral inclination angle of rotor reference frame to body reference frame,

Vu v w

a b

variable description


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Equation of Motion

3 3 3 3

3 3 CG CG

3 3 3 3

1CG

3 3 CG

d

d

dd

or

d 1d

d

d

V

m VmF t

Mt

VF m Vt

mM

t

I 0

0 JJ

I 0

J0 J

W

WW W

W

W W W

6 DoF equation of motion


11/33

Rotors Reference Frame Orientation with

respect to Body Reference Frame

R BR /B1 1i i DCM

B-R R B

RB-R

Note: is angular speed of rotor 's reference frame 1 with respect to body reference frame 1 ,

and its -component is NOT the rotation speed by which it produces thrust and

expe

i i

ii

i

z r

W

riences respective torque.

RB

R

B-R B-RB/R R

R

,

i

i ii i

ii

p

q

r

KM

b

W a W

y

Each rotor reference frame has 2 degree of freedom with respect to vehicle's body reference frame.

Its attitude can be derived by applying 2 consecutive rotations: one on its -axis followed by one ony -axis.

The corresponding rotation angles and rotation rates are ( , ), and ( , ).

x

a b a b


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Rotors Direct Cosine Matrix and Kinematics,

with respect to Body Reference Frame

R1 R1 R1 R1 R1

B/R1 R1 R1

R1 R1 R1 R1 R1

R 2 R 2 R 2 R 2 R 2

B/R 2 R 2 R 2

R 2 R 2 R 2 R 2 R 2

cos sin sin cos sin

0 cos sin

sin sin cos cos cos

cos sin sin cos sin

0 cos sin

sin sin cos cos cos

DCM

DCM

a - b a - b a

b b

a b a b a

a b a - b a

b b

a - b a b a

R1 R1

R1 R1 R1

R1 R1 R1R1 B

R 2 R 2

R 2 R 2 R 2

R 2 R 2 R 2R 2 B

1 0 0

0 cos 00 sin 0

1 0 0

0 cos 0

0 sin 0

p

qr

p

q

r

b

- b ab y

- b

- b a

- b y


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Rotors ForceRotor force is generated by thrust. Other sources of

force are periodic and have zero sums at anyinstantaneous time, assuming that each rotor is

perfectly symmetric.

Distribution of force from rotor thrust along each axisin body reference frame is regulated by tilt angles a

andb.

2

P R

1

TR B/ R R

0 0 , 1, 2

T i

i

T i i i

F F

F T i

DCM


14/33

Rotors Force

R R R1 R 2

P R R1 R 2

R R R1 R 2

imbalance

imba

cos sin

sin

lance

equilibrium

cos cos

T T

F T T

T T

b a

b

b a


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Rotors Moment

There are several sources moment is generated by a rotor:

Moment from the product of moment arm and thrust ( )

Moment due to torque ( ).

Moment due to gyroscopic effect ( , , , ).

Moment d

T

Q

p q

M

M

M M M M

a b

ue to tilting mechanism's reaction ( ).M

a

R2

P R R R R R

1

iT i Q i i i i

i

M M M M M M

W W a

Total moment generated by rotor system:


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Rotors Moment

R R G R , 1, 2T i i T iM R R F i

From the product of moment arm and thrust:

T

R B/R R0 0 , 1, 2Q i i iM Q i DCM

Due to torque:

R RB/R R , 1, 2i ii iM i DCM JW WGyroscopic moments:

R R R RB/R R , 1, 2i i i ii iM i DCM JW W

T T T

R1 R2 G G0 0 0 0 0 0R l R l R h

Tilting mechanisms reaction:

R T R B0 0 , 1, 2i iM J i a a


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Rotors MomentFrom the product of moment arm and thrust:

R1 R1 G R1 R 2 R 2 G R2

T TR1 G R2 GB/R1 R1 B/R 2 R2

R1 R1 R1 R 2

R1 R1

G R1 R1 R1 G

0 0 0 0

0 cos sin 0 co

sin

cos cos

T T T T

M R R F M R R F

R R T R R T

T T

l T l

h T h

DCM DCM

b a

b

b a

R 2 R 2

R 2 R 2

R2 R2 R 2

R1 R1 R1 G R1 R1 R 2 R 2 R 2 G R 2 R2

G R1 R1 R1 G R2 R 2 R 2

R1 R1 R1 R2B

s sin

sin

cos cos

cos cos sin cos cos sin

cos sin cos sin

cos sin co

T

T

l T h T l T h T

h T h T

l T l T

b a

b

b a

b a b b a b

b a b a

b a R2 R2 Bs sin

b a

R 2 R1 R R 2 R1 R R 2 R1 R

R 2 R1 R R 2 R1 R R 2 R1 R

sin sin sin cos cos cos


a a a a a a a a a

b b b b b b b b b

G R R R

R1 R 2

R R G R R1 R2

R

R R R1 2 B

2

R

1 Rcos sin i

cos cos sin equilibrium

cos sin equilibr

mbalan e

i m

c

u

T T T

T

M M M

l h T T

h T TM

l T T

b a b

b

b a

a


18/33

Rotors MomentDue to rotor torque:

T TR1 R 2B/R1 R1 B/R 2 R 2

R1 R1 R1 R 2 R 2 R 2

R1 R1 R 2 R2

R1 R1 R1 R 2 R 2 R 2B B

0 0 0 0

cos sin cos sin

sin sin

cos cos cos cos

Q QM Q M Q

Q Q

Q Q

Q Q

DCM DCM

- b a b a

b b

b a b a

R 2 R1 R R 2 R1 R R 2 R1 R

R 2 R1 R R 2 R1 R R 2 R1 R



a a a a a a a a a

b b b b b b b b b

R R1 R 2

R R R1 R 2

R

R R R

R R1 R 2

1 R 2 B

cos sin equilibrium

cos cos equi

sin

libri

im

um

balance

Q Q Q

Q

M M M

Q Q

M

Q Q

Q Q

b a

b

b

a


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Rotors Moment

TR1 R1 R1B/R1 R1 R1 R1

R1 R1 R1 R1 R1

R1 R1 R1 R1 R1 R1 R1

R1 R1 R1 R1 R1

, 0 0

cos cos sin

cos sin cos cossin cos sin

M h h J

q r J

M r p Jp q J

DCMW

W

W

b a b

b a b ab b a

Gyroscopic moments due to body angular rate:

T

R 2 R 2 R 2B/R 2 R 2 R 2 R 2

R 2 R 2 R 2 R 2 R 2

R 2 R 2 R 2 R 2 R 2 R 2 R 2

R 2 R 2 R 2 R 2 R 2

, 0 0cos cos sin

cos sin cos cos

sin cos sin

M h h Jq r J

M r p J

p q J

DCMW

W

Wb a b

b a b a

b b a


20/33

Rotors MomentGyroscopic moments due to body angular rate:

R 2 R1 R

R 2 R1 R R 2 R1 R R 2 R1 R

R 2 R1 R R 2 R1 R R 2 R1 R



J J J

a a a a a a a a a

b b b b b b b b b

R R1 R 2

R R R R1 R 2

R R R R R R1 R 2

R R R R1 R 2

R R R1 R 2

R R R1 R 2

imbalance

imb

equ

al

i br um

a

li i

cos cos

cos sin cos cos

cos sin

sin

s in

M M M

J q

r p J

J q

J r

J p

W W W

b a

b a b

b

b

a

b a nce


21/33

R1 R1 R1 R1B/R1

R1

B/R1 R1 R1

R1 R1 R1 R1 R1R1

R1 R1 R1 R1 R1 R1 R1 R1

R1 R1 R1 R1

R1 R1 R1

0

cos 0

sin

cos cos sin sincos

sin cos

M h

J

JJ

DCM

DCM

W W

b

a b

a b

a a b b b a

b b

a a b R1 R1 R1 R1 R1B

sin cos J

b b a

Rotors MomentGyroscopic moments due to tilting rate:


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R2 R2 R2 R2B/R2

R2

B/R2 R 2 R2

R2 R2 R2 R2 R2R2

R2 R 2 R2 R 2 R 2 R 2 R2 R2

R2 R 2 R2 R2

R2 R 2 R

0

cos 0

sin

cos cos sin sin

cos

sin cos

M h

J

J

J

DCM

DCM

W W

b

a b

a b

a a b b b a

b b

a a b 2 R 2 R2 R 2 R2 R2B

sin cos J

b b a



23/33

R1 R 2

R R R R R R R R1 R2

, R

R R R R R R

R R R R

R R

1 2

1 R2B

R

equilibrium

equili

imbalanc

bri

e

um

cos cos sin sin

sin cos sin cos

cos

J

JM

J

W W

a b a b a b

a b a b a b

b b


R 2 R1 R

R 2 R1 R R 2 R1 R R2 R1 R

R 2 R1 R R 2 R1 R R 2 R1 R

R2 R1 R

R2 R1 R



J J J

a a a a a a a a a

b b b b b b b b b

a a a

b b b


24/33

Rotors MomentReactionary moment:

R T R B0 2 0M J a a

Reactionary moment occurs when actuators in tilting

mechanism system is repositioning rotor tilt. Lateral tilt for

both rotors are in oppossing direction to each other, hence

resulting reactionary moments that are eliminating each

other for symmetric action. Longitudinal tilt for both rotors

are in the same direction, resulting reactionary moment that

are the sum of both.


25/33

Rotors MomentIt can be seen that for moment equilibrium, moment

imbalance from rotor pairs thrust-coupled momentand moment imbalance from rotor pairs torque must

counter-act each other:

R R1 R 2 G R R R1 R 2

R R1 R 2G / /

R R1 R 2

sin cos sin

tan,

sin

T Q T Q

Q Q h T T

T Th C C

Q Q

b b a

b

a

This is the constraint that must be satisfied for rotor

pairs moment equilibrium on y-axis.


26/33

Rotors MomentMoment imbalance due to gyroscopic effect.

R1 R 2

R R R1 R 2

, , R R R R R1 R2

R R R1 R 2 B

sin

cos

sin

J r

M J

J p

W W W

b

b b

b

This imbalance occurs when the birotor experienceslateral or radial maneuvering, or when the OLT

mechanism is repositioning rotor pair lateral tilt.


27/33

Rotors Thrust and Torque

R R

R

R R

2

R

Rotor thrust ( ) and torque ( ) can be identified

to be functions of its rotation speed ( ). And it can

be determined that and vary proportionally

with .

i i

i

i i

i

T Q

T Q

2

2

2

R R/

2

R R/

, 1, 2

i iT

i iQ

T C

Q C i

Hence, 2

2

/

R R

/

, 1, 2Q

i i

T

CQ T i

C


28/33

Rotors Tilting Mechanism

T

TR T R T T

T T

T

1 12 2

0

x

x y y

M

M M M MJ J

b a

Rotor tilt position is done by exerting moment of

corresponding direction to each rotor simultaneously.

MxT and MyT are nominal moments of correspondingnominal tilt angles that must be exerted to both rotor

by mechanisms actuators.


29/33

Input from Controller

Input of birotor system is done either by

differentiating rotor-pairs speed to drive thrust and

torque differentials, collectively changing rotor-pairs

speed to drive collective thrust and torque,maneuvering rotor-pairs tilt angle to distribute rotor-

pairs force and moment to each body frame axis. As

shown in rotors moment analysis previously, rotorsmoment has cross-couple relation among the three

axis due to rotor tilting.


30/33


Selection for system input is done by evaluating rotors forceand moment at equilibrium condition while birotor is in levelhover flight with small tilt angles.

From force and moment analysis, one may identify four

variables by which force and moment can be exerted to birotorsystem: collective thrust (T

coll), differential torque (Q

diff),

opposed lateral tilt angle (bR), and longitudinal tilt angle (a

R).

2 2

coll R1 R 2 diff R1 R2 R R1 R 2 R R1 R2

/ coll diff

/

2 2 2 2

R1 R 2 R1 R2/ /

1T Q

T Q

T Q

T T T Q Q Q

C Q TC

C C

b b b a a a


31/33


C P

R R1 R 2 R coll

C R R1 R 2 R / diff

R1 R 2 coll

eqR eqcollcoll

R

/ eqdiff eqR /

R

diff

1 10 0

2 21 10 0

2 2

1 0 0 0

T Q

T Q T Q

F F

T T T

F T T C Q

T T T

T T

C Q C

Q

a a

b b

a

bb

a

Translational channel


32/33


C R R

R diff G R R1 R 2 R R1 R 2 G R / diff

collG R R1 R2 R R1 R 2 G R coll R

/

R R1 R2 R1 R 2 R / diff

diff

T Q

T Q

T Q

T Q

M M M

Ql h T T Q Q l h C Q

Th T T Q Q h T

Cl T T Q Q l C Q

Q

ab a b

a b a b

a a

G / eqdiff eqdiff / G / eqR eqR

coll

eqR eqcoll R

G eqR G eqcoll

R/ /

diff

/ eqdiff eqR /

1 1 10 2

2 2 2

1 1 10

2 2 2

1 10 0 2

2 2

T Q T Q T Q

T Q T Q

T Q T Q

h C Q Q l C h C

T

Th h T

C C

Ql C Q l C

b a

b ba

a

a

Angular channel


33/33

Input from ControllerAll channel

eqR eqcoll

/ eqdiff eqR /

C

G / eqdiff eqdiff / G / eqR eqRC

eqR eqcoll

G eqR G eqcoll

/ /

/ eqdiff

1 10 0

2 2

1 10 0

2 2

1 0 0 0

1 1 10 2

2 2 2

1 1 10

2 2 2

1 10 02 2

T Q T Q

T Q T Q T Q

T Q T Q

T Q

T

C Q C

F

h C Q Q l C h C M

Th h T

C C

l C Q l

a

b

b a

ba

a

coll

R

R

diff

eqR /

TTR

TR

2

10

2

1

0 2

T Q

x

y

T

Q

C

MJ

M

J

b

a

b

a

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Birotors sumary