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||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
||Autonomous Systems Lab
IntroductionRotorcrafts
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 3
||Autonomous Systems Lab
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
||Autonomous Systems Lab
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
||Autonomous Systems Lab 17.01.2018Robot Dynamics: Rotary Wing UAS 7
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
||Autonomous Systems Lab 17.01.2018Robot Dynamics: Rotary Wing UAS 8
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
||Autonomous Systems Lab
Dynamics Modeling & ControlRotorcrafts
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 9
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 10
Modeling | Introduction
Modeling and simulation are important, but they must be validated in reality
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 11
Modeling a Quadrotor | Assumptions
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 12
Modeling of Rotorcraft | Overview
ωR,cmd ωR F, M v, ω. .
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 13
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
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 14
The Quadrotor Example
F1
1
F4
4
F3
3 F2
2
, TTe cte ax xb Fg M SJ
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 15
The Quadrotor Example
F1
1
F4
4
F3
3 F2
2
xR
yR
zR
xI
yI
zI
, T Tex ex actb g F M J S
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 16
Modeling a Quadrotor | Properties
zz
yy
xx
II
I
000000
I
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 17
Modeling a Quadrotor | Simplifications
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 18
Modeling of Rotorcraft | Coordinate System
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 19
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
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 20
Modeling of Rotorcraft | Rotational Velocity
||Autonomous Systems Lab
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
||Autonomous Systems Lab
Change of momentum and spin in the body frame
Position in the inertial frame and the attitude
Forces and moments Aerodynamics and gravity
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 22
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
, T Tex ex actb g F M J S
Torque!!
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 23
Modeling a Quadrotor | Aerodynamic Forces
iB
iAeroB
TF 0
04
1
hB
hBi
iB H
vvH
4
1
2,i i p iT b
||Autonomous Systems Lab
Hover moments Thrust induced moment Drag torques
Additional moments Inertial counter torques Propeller gyro effect Rolling moments Hub moments
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 24
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
||Autonomous Systems Lab
Rotational dynamics near hover
Full control over all rotational speeds Not dependent on position states
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 25
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
||Autonomous Systems Lab
Translational dynamics in hover
Only direct control of the body z velocity Use attitude to control full position
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 26
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
||Autonomous Systems Lab
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
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 27
Control of a Quadrotor | General Overview
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 28
Modeling a Quadrotor | Control OverviewF1
1
F4
4
F3
3 F2
2
||Autonomous Systems Lab
Define virtual control inputs to simplify the equations Get decoupled and linear inputs Moments along each axis
Total thrust
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 29
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
||Autonomous Systems Lab
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
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 30
Control of a Quadrotor | Hierarchical Control
rqp
wvuZYX
432 ,, UUU
1U
||Autonomous Systems Lab
Linearize attitude dynamic at hover Equilibrium point:
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 31
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
||Autonomous Systems Lab
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
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 32
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)
||Autonomous Systems Lab
Consider roll subsystem
P controller can‘t stabilize the system Harmonic oscillation
Use a PD controller
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 33
Control of a Quadrotor | PID Attitude Control
21 U
I xx
)(1 despxx
kI
1 ( ) ( )p des d desxx
k kI
||Autonomous Systems Lab
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
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 34
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
||Autonomous Systems Lab
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
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 35
Control of a Quadrotor | Parameter Tuning
||Autonomous Systems Lab
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
||Autonomous Systems Lab
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
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 37
Control of a Quadrotor | Altitude Control
ϕdes,θdes,ΨdesU2,U3,U4
zdes Tz U1
ωi,des
||Autonomous Systems Lab
Rewrite translational dynamicsin the inertial frame
Only look at the altitude dynamics Virtual Control
PD controller for the height
Transformation to the body frame
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 38
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
||Autonomous Systems Lab
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
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 39
Control of a Quadrotor | Position Control
U1
Ψdes
ϕ, θ ωi,desxdesydeszdes
TxTyTz
U2U3U4
||Autonomous Systems Lab
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
17.01.2018Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 40
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
||Autonomous Systems Lab
Thrust Force of Propeller / RotorRotorcrafts
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 41
||Autonomous Systems Lab
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)
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 42
Modeling of Rotorcraft | Rotor/Propeller Aerodynamics
||Autonomous Systems Lab
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) …
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 43
Modeling a Quadrotor | Propeller Aerodynamics 1
Ap
Similar to blade element
||Autonomous Systems Lab
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
…
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 44
Modeling a Quadrotor | Propeller Aerodynamics 2
ωP
Vrel
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 45
The Propeller Thrust Force | Momentum Theory
||Autonomous Systems Lab
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
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 47
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
||Autonomous Systems Lab
Conservation of momentum Unconstrained flow:
→ net pressure force is zero
Conservation of energy
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 48
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)
With eq. (4) and (5)
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 49
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
With eq. (6) and (7)
212 (10)Thrust RF A u
0 3 ; = (11)2
RAA A
||Autonomous Systems Lab
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 …
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 50
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)
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 51
Propeller/Rotor Efficiency
1hover torequiredpower Actual
hover torequiredpower IdealFM
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 52
BEMT (Blade Elemental and Momentum Theory)
ωR
RR
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 53
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
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 54
BEMT | Example of Lift and Drag Coefficient
xcord
cLcL cD
cD α
α α
cM cL/cD
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 55
BEMT | Example of Lift and Drag Coefficient
cL
cD
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 56
BEMT | Example of Lift and Drag Coefficient
cL
α
cL/cD
α
||Autonomous Systems Lab
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
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 58
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(
With eq. (14) and (15)
and/or indu
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 59
Modeling of Rotorcraft | Motor Dynamics
i(t)
U(t)
ωm(t) Q(t)
||Autonomous Systems Lab
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 60
Modeling of Rotorcraft | Motor Dynamics
i(t)
U(t)
ωm(t) Q(t)