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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 DynamicsRotary Wing UAS: Propeller Analysis and Dynamic Modeling
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 2
Contents | Rotary Wing UAS
1. Introduction - Design and Propeller Aerodynamics
2. Propeller Analysis and Dynamic Modeling
3. Control of a Quadrotor
4. Rotor Craft Case Study
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 3
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 4
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 5
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
Propeller / Rotor AnalysisRotary Wing UAS: Propeller Analysis and Dynamic Modeling
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 6
||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 7
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 8
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 9
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 10
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 11
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 12
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 13
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 14
BEMT (Blade Elemental and Momentum Theory)
ωR
RR
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 15
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 16
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 17
BEMT | Example of Lift and Drag Coefficient
cL
cD
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 18
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 Thin airfoil theory (plate) Experimental results Linearization of polar
Thrust at blade element27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 19
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 20
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
Dynamic Modeling of Rotorcraft
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 21
Rotary Wing UAS: Propeller Analysis and Dynamic Modeling
||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 22
Modeling | Introduction
Modeling and simulation are important, but they must be validated in reality
||Autonomous Systems Lab
Whitebox modeling Use a priori information to
model dynamics Blackbox modeling Use input and output
measurements to deduce dynamics
Greybox modeling Use as much of a priori
information as available Identify unknown part with
measurements (e.g. unknown parameter 1, 2)
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 23
Modeling | Approachesu
u
u
y
y
y
g(u,x)
f(x)
g(u,x)
f(x)1
2
||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 24
Modeling of Rotorcraft | Overview
ωR,cmd ωR F, M v, ω. .
||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 25
Modeling of Rotorcraft | Coordinate System
||Autonomous Systems Lab 27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 26
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 = (ϕ,0,0)T
Angular velocity due to change in pitch angle Bωpitch = CT
2B (x,ϕ) (0,θ,0)T
Angular velocity due to change in yaw angle Bωyaw = CT
12(y,θ) CT2B (x, ϕ) (0,0,Ψ)T
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 27
Modeling of Rotorcraft | Rotational Velocity
.
.
.
||Autonomous Systems Lab
Relation between Tait-Bryan angles and angular velocities
Linearized relation at hover
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 28
Modeling of Rotorcraft | Rotational Velocity 2
ωJ Br
coscossin0cossincos0
sin01rJ
ωJ Br 100010001
0,0
Singularity for = 90o
||Autonomous Systems Lab
Consider fuselage as one rigid body Recall the rigid body dynamics Rigid body can be completely describes by the velocity at the CoG and
the rotational velocity Change of momentum p = mvCoG
ṗ = ∑F Change of spin N = Iω Ṅ = ∑M
Euler differentiation rule for vector (c) representations B(dc/dt) = dBc/dt + ω x Bc
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 29
Modeling of Rotorcraft | Body Dynamics 1
||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 30
Modeling of Rotorcraft | Body Dynamics 2
MF
ωIωvω
ωv
I B
B
BB
BB
B
Bx mmE 0
033
vCx BEBE ωJ Br
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
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
27.10.2015Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 31
Modeling of Rotorcraft | Rotor/Propeller Aerodynamics
||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 32
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 33
Modeling of Rotorcraft | Motor Dynamics
i(t)
U(t)
ωm(t) Q(t)
||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 34
Modeling a Quadrotor | Control OverviewF1
1
F4
4
F3
3 F2
2
||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 35
Modeling a Quadrotor | Properties
zz
yy
xx
II
I
000000
I
||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 36
Modeling a Quadrotor | Assumptions
||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 37
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 38
Modeling a Quadrotor | Propeller Aerodynamics 2
ωP
Vrel
||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 39
Modeling a Quadrotor | Simplifications
||Autonomous Systems Lab
Hover forces Thrust forces in the shaft
direction Ti=bωp,i
2
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 40
Modeling a Quadrotor | Aerodynamic Forces
iBi
BT004
1T
hB
hBi
iB H
vvH
4
1
||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 41
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
4
1
131
24
)1(00
0
)()(
i
ii
B
B
B
TB
QTTlTTl
QM