birotors sumary
TRANSCRIPT
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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
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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
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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
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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
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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
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
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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
sin sin sin cos cos cos
sin sin sin cos cos cos
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
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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
sin sin sin cos cos cos
sin sin sin cos cos cos
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
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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
Rotors MomentGyroscopic moments due to tilting rate:
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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
Rotors MomentGyroscopic moments due to tilting rate:
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
sin sin sin cos cos cos
sin sin sin cos cos cos
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
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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.
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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.
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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.
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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
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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.
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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.
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Input from Controller
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
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Input from Controller
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
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Input from Controller
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
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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