IN DEGREE PROJECT , FIRST CYCLE, 15 CREDITS STOCKHOLM SWEDEN 2020,
Turbulence Modeling in Urban Air Mobility
Applications
TOVE ÅGREN
KTH ROYAL INSTITUTE OF TECHNOLOGY
Acknowledgements
This work was made possible thanks to the great support and guidance of Shannah Withrow Maser, Carlos Malpica and William Warmbrodt, as well as the Swedish National Space Agency for enabling this opportunity. An expression of thanks is also due to the entire Aeromechanics Branch at NASA Ames research center for a warm welcome and inspiring work place. Lastly, I would like to thank Raffaello Mariani and Martin Viklund at KTH Royal Institute of Technology for assistance and guidance in turbulent times.
Abstract
A proof-of-concept turbulence modeling approach for rotorcraft in low-altitude, low-speed condi- tions based on parametric Control Equivalent Turbulence Input (CETI) models was developed. In its original form, CETI-model outputs are disturbance control inputs to replicate the vehicle motion response as if it was operating in turbulent conditions. Consequently, a CETI-model is directly tied to a specific vehicle, and model extraction requires flight data in turbulent conditions. The purpose of the new model was to find Atmospheric Disturbance Equivalent (ADE) inputs that would be applicable to all rotorcraft. The equivalent turbulence components were obtained through linear combinations of the control signals generated through white-noise driven transfer functions that constitute the original CETI-model.
A preliminary evaluation of the proposed modeling approach was conducted by simulating turbu- lence vehicle response in low-altitude, low-speed conditions. The considered vehicle was a NASA reference concept aircraft designed for Urban Air Mobility applications, namely a 1-passenger elec- tric quadrotor using collective controls. The results were compared to those generated through conventional theoretical turbulence models, i.e the Von Karman model.
Preliminary results showed promising implications that an empirical atmospheric turbulence model with the proposed approach is feasible. Computed Atmospheric Disturbance Equivalent turbulence components generated vehicle response consistent with the original CETI-model in the vehicle lin- ear axes. Vehicle response in angular rates showed less consistency. The NASA Quadrotor vehicle response showed over-all validating resemblance between the new CETI-ADE model and the exist- ing von Karman model, but with observable differences that legitimated the development of a new model. To meet the final objectives, the new model needs to be refined and validated further by pilot evaluation.
Modellering av turbulens f¨or till¨ampningar inom Urban Air Mobility
Sammanfattning
En konceptuell metod f¨or att modellera turbulens f¨or rotordrivna fordon i urbana flygf¨orh˚allanden utvecklades. Metoden baserades p˚a existerande parametriska Control Equivalent Turbulence In- put (CETI) modeller. I sin ursprungliga form utg¨ors CETI-modellens utsignal av styrsignaler med syfte att replikera fordonets r¨orelserespons som om det opererade i turbulenta f¨orh˚allanden.
F¨oljaktligen s˚a ¨ar en CETI-modell direkt bunden till ett specifikt fordon och att extrahera en mod- ell kr¨aver tillg˚ang till relevant flygdata i turbulens. Syftet med den nya modellen var att finna ADE (Atmospheric Disturbance Equivalent)-utsignaler som vore till¨ampbara till en bredare klass av rotordrivna fordon. De ekvivalenta komponenterna f¨or ett turbulent hastighetsf¨alt erh¨olls genom linj¨arkombinationer av utdata fr˚an en CETI-modell. Utdatan genererades genom att filtrera vitt gaussiskt brus genom ¨overf¨oringsfunktionerna som utg¨or den ursprungliga CETI-modellen.
En prelimin¨ar utv¨ardering av den f¨oreslagna modelleringsmetoden utf¨ordes genom att simulera r¨orelsesvaret hos ett multi-rotorfordon under l˚agh¨ojd och l˚aghastighetsf¨orh˚allanden. Det betrak- tade fordonet var ett NASA-referensfordon designat f¨or till¨ampningar inom Urban Air Mobility, specifikt en elektrisk quadrotor f¨or en passagerare. Som referensram j¨amf¨ordes resultaten med de som genererades genom konventionella teoretiska turbulensmodeller, h¨ar Von Karman-modellen.
De prelimin¨ara resultaten visar lovande indikationer att en empirisk turbulensmodell med den f¨oreslagna metoden ¨ar g˚angbar. F¨or att uppfylla de slutliga m˚alen b¨or modellen f¨orfinas och valideras ytterligare genom pilotutv¨ardering.
Table of Contents
1 Introduction 1
1.1 Theoretical background . . . . 1
1.1.1 Statistical theory for turbulent flows . . . . 1
1.1.2 Theoretical Turbulence Models for Aircraft Simulation . . . . 2
1.1.3 Dryden model . . . . 2
1.1.4 von Karman model . . . . 3
1.1.5 Scale length and intensity . . . . 3
1.1.6 Comparison of theoretical models . . . . 3
1.1.7 Issues with Theoretical Models Related to Rotorcraft . . . . 4
1.1.8 Control Equivalent Turbulence Input (CETI) model . . . . 4
1.2 Objectives . . . . 5
2 Development of a Reformulated Empirical Model 6 2.1 Atmospheric Disturbance Equivalence (ADE) from UH-60 Black Hawk helicopter CETI model . . . . 6
2.1.1 UH-60 Helicopter vehicle response to CETI and CETI-ADE model . . . . 7
2.2 ADE from small quadrotor CETI model . . . . 14
2.2.1 Iris quadrotor longitudinal response to CETI and CETI-ADE model . . . . . 15
2.2.2 Construction of Hybrid model . . . . 16
3 Simulations in UAM applications 17 3.1 Atmospheric turbulence components . . . . 17
3.1.1 Von Karman model . . . . 17
3.1.2 UH-60 CETI-ADE . . . . 18
3.1.3 Hybrid model . . . . 20
3.2 Aircraft for simulations: NASA Quadrotor . . . . 22
3.3 NASA Quadrotor vehicle response . . . . 23
4 Conclusions and Future Work 28 4.1 Conclusions . . . . 28
4.2 Future work . . . . 28
List of Figures
1 CETI block diagram . . . . 5
2 Vehicle response u-axis with ˙θ = qt, ˙Φ = pt . . . . 8
3 Vehicle response v-axis with ˙θ = qt, ˙Φ = pt . . . . 8
4 Vehicle response w-axis with ˙θ = qt, ˙Φ = pt . . . . 9
5 RMS of UH-60 vehicle response with ˙θ = qt, ˙Φ = pt. . . . 9
6 Vehicle response roll rate with ˙θ = qt, ˙Φ = pt . . . . 10
7 Vehicle response pitch rate with ˙θ = qt, ˙Φ = pt . . . . 11
8 Vehicle response yaw rate with ˙θ = qt, ˙Φ = pt . . . . 11
9 RMS of UH-60 vehicle response with ˙θ = qt, ˙Φ = pt. . . . 12
10 Vehicle response without kinematic relationship . . . . 13
11 Vehicle response angular rates without kinematic relationships . . . . 13
12 Frequency response Wnoiseq . . . . 15
13 Longitudinal dynamic system block diagram with CETI-ADE model . . . . 15
14 Iris longitudinal response to turbulence input . . . . 16
15 Atmospheric disturbance components, Von Karman model . . . . 17
16 PSD, Von Karman model . . . . 18
17 Atmospheric disturbance components, UH60 CETI-ADE model . . . . 19
18 PSD’s for UH60 CETI-ADE model . . . . 19
19 Atmospheric disturbance components, Hybrid model . . . . 21
20 PSD:s for Hybrid model . . . . 21
21 Single-passenger quadrotor with electric propulsion . . . . 22
22 Simplified block diagram of quadrotor simulation with atmospheric disturbance . . 23
23 Quadrotor vehicle response, u-axis . . . . 23
24 Quadrotor vehicle response, v-axis . . . . 24
25 Quadrotor vehicle response, w-axis . . . . 24
26 Quadrotor vehicle response, roll rate . . . . 25
27 Quadrotor vehicle response, pitch rate . . . . 26
28 Quadrotor vehicle response, yaw rate . . . . 26
List of Tables 1 Parameter values used in UH-60 vehicle response simulations . . . . 7
2 Parameter values to generate Von Karman turbulence . . . . 17
3 Parameter values to generate UH-60 CETI-ADE turbulence . . . . 18
Nomenclature
δlat,lon,ped,col Control inputs
Ω Spatial frequency [rad/m]
ω Circular frequency [rad/s]
φ, θ, ψ Roll, pitch, yaw euler body-axis attitude angles
Φu,v,w(Ω) Power Spectral Densities (PSD) of turbulence velocity components
σu,v,w Turbulence intensities along body axis (standard deviation in velocity) [ft/s]
A Bare-airframe stability derivative matrix B Bare-airframe control derivative matrix Bt Atmospheric turbulence disturbance matrix
BCET I Control Equivalent Turbulence Input disturbance matrix C Bare-airframe state output matrix
D Bare-airframe control output matrix h Altitude [ft]
Lu,v,w Turbulence characteristic length scale in body axis [ft]
p, q, r Roll, pitch and yaw rates [rad/s]
pt, qt, rt Roll, pitch and yaw turbulence disturbance rates [rad/s]
R Rotor radius
u, v, w Longitudinal, lateral and vertical body-axis velocities [ft/s]
U0 Mean wind speed [ft/s]
ut, vt, wt Longitudinal, lateral and vertical body-axis turbulence wind velocities [ft/s]
W20 Wind speed at 20 feet (6 m) [ft/s]
Wnoise Gaussian white noise signal
ADE Atmospheric Disturbance Equivalence CETI Control Equivalent Turbulence Input PSD Power Spectral Density
V True air speed [ft/s]
1 Introduction
The emerging field of Urban Air Mobility (UAM) is inducing the development of new electric vertical take-off and landing (eVTOL) concepts and innovations. Consequently, these concepts and designs need to be investigated and evaluated to focus and guide successful aircraft development. An essential part of evaluating aircraft performance and handling qualities is the modeling and simulation of atmospheric turbulence and its effect on the vehicle. In turbulence modeling, one must consider and specify under which conditions the vehicle will be simulated, i.e altitude, terrain, speed, range and mission tasks, as this highly directs the appropriate approach. To this day, there exists no generic approach to the modeling of turbulence and its effect on rotorcraft. However, it is a topic widely and extensively investigated. Introducing the dynamics associated with rotorcraft to the already complex concepts of turbulence modeling leads to an array of issues that must be addressed.
Conventional theoretical turbulence models for fixed wing aircraft are not sufficient to rotorcraft in low-speed low-altitudes conditions that are typical for UAM applications. Recent work has taken on an empirical approach with Control Equivalent Turbulence Input (CETI) models. The conceptual idea of a CETI-model is to use flight data acquired in turbulent conditions to extract a realistic model for a specified vehicle. However, when flight data is not available, as is the case at the moment for concept vehicle designs proposed for UAM applications, there is strong motivation to develop a turbulence model that is not tied to a specific vehicle, but applicable to a class of similar rotorcraft.
1.1 Theoretical background
1.1.1 Statistical theory for turbulent flows
Using tools from statistical theory is a way to fathom the complexity of turbulent flow. Averag- ing velocities and considering statistical variances allows simplifications necessary in computational applications. With a statistical description, turbulent flows can be characterised by typical eddy lengths scales, L, and velocity variance, σ.
Theoretical turbulence modeling requires some insight in correlations between random variables, in particular correlation functions. This study is mainly restricted to single point correlations between turbulent velocity field components. Consider a total, instantaneous turbulent velocity ˜ui= Ui+ ui, where Ui denotes mean flow and ui the fluctuation, that will be referred to as turbulence. Now define the velocity auto-covariance, ui(t + ∆t)uito be the covariance between a velocity component and itself separated in time with an increment ∆t. Now, the correlation function, f (x, t), can be defined by the auto-covariance normalized by the variance u2,
f(x, t) =ui(t + ∆t)ui
u2
In the mathematical representation of homogeneous atmospheric turbulence, the fundamental re- lation is the Fourier transform pair between the correlation function and Power Spectral Density (PSD). Fundamentally, a PSD is a description of how the power of a signal is distributed over its frequency content. It characterises a stochastic process in the frequency domain, hence in this con- text the PSD serves as a statistical characterisation of the turbulent flow field [1]. The two following concepts will be of importance for understanding theoretical turbulence modeling:
Gaussian white noise
The use of gaussian white noise in turbulence modeling is motivated by the assumption that the ve- locity components in a turbulent flow field are normally distributed. A desired property of Gaussian white noise is that the power spectral density is equal to 1.
Kolmogorov -5/3 law
Within an inertial subrange, the energy of the eddies in a turbulent flow increases with eddy scale size to the 23 power. In the Fourier space, this becomes −53 . This correlation is derived using dimensional analysis and is a way to verify the validity of a turbulence model.
1.1.2 Theoretical Turbulence Models for Aircraft Simulation
This review is restricted to the conventional approach to statistically analysing atmospheric tur- bulence, that is a continuous Power Spectral Density (PSD) approach. An important concept and conventional assumption is that of Taylor’s Frozen Atmosphere Hypothesis, in short that the tur- bulence is assumed to be frozen in time and space. Also, the approximation that free atmospheric turbulence is isotropic and homogeneous is fundamental in classical turbulence analysis [1]. The two frequently used theoretical models to simulate atmospheric turbulence in fixed-wing aircraft applications are the von Karman and the Dryden models. The mathematical representation of these are presented in the Military Specification MIL-F-8785C and Military Handbook MIL-HDBK-1797 [2]. The differences between the two models originate from the different forms of power spectra. The difference in spectra characteristics governs the difference frequency validity range and implementa- tion methods [3].
From the Power Spectral Densities, wind velocity components are generated. The following method for modeling von Karman or Dryden turbulence relies on driving Gaussian white noise through linear forming filters. Essentially, the shaping filters here denoted by Hi(s) serve to generate an output signal with approximately the same power spectra characteristics as the desired theoretical PSD [1].
Φi(ω) ≈ |Hi(s)|2s=jωΦWnoise(ω) (1)
Hi(s)
Wnoise xi
In essence, these filters define ordinary differential equations. Hence, there are different approaches to generate turbulence velocities from the derived filters. Solving methods include zero-order hold z-transforms, sum-of-sinusoids and recursive methods [3] [4]. Different applications and constraints govern the choice of method.
The mathematical representations of the different PSD’s are presented in Section 1.1.3 and 1.1.4.
Consider an aircraft flying at a velocity V through a frozen turbulence velocity field with a spatial frequency Ω. Since the turbulence field is approximated frozen, the spatial and circular frequencies can be related as ω = Ω.
1.1.3 Dryden model
Equation 2-4 presents the Dryden form of the power spectral densities outlined in MIL-F8785C [2].
As described in Section 1.1.2, turbulence wind velocities with desired spectral characteristics are generated by driving forming filters Hi(Ω) with Gaussian white noise signals. These forming filter are derived from the power spectral densities Φi(Ω) in Eq. 2-4 .
Φu(Ω) = σu22Lu
π
1
[1 + (LuΩ)2 (2)
Φv(Ω) = σv2Lv
π
1 + 3(LvΩ)2
[1 + (LvΩ)2]2 (3)
Φw(Ω) = σ2wLw
π
1 + (3)(1.339LwΩ)2
[1 + (LwΩ)2]2 (4)
1.1.4 von Karman model
Equation 5-7 gives the von Karman form of the power spectral densities outlined in MIL-F8785C.
The main difference from the Dryden model in Section 1.1.3 is that the power spectral densities are irrational in the von Karman model.
Φu(Ω) = σu22Lu
π
1
[1 + (1.339LuΩ)2]5/6 (5)
Φv(Ω) = σ2vLv π
1 + (8/3)(1.339LvΩ)2
[1 + (1.339LvΩ)2]11/6 (6)
Φw(Ω) = σw2Lw π
1 + (8/3)(1.339LwΩ)2
[1 + (1.339LwΩ)2]11/6 (7)
1.1.5 Scale length and intensity
The spectral densities are parameterized with turbulence scale lengths, Li, and standard deviations that correspond to turbulence intensities, σi, that govern the form of the spectrum. As proximity to the ground invalidates the assumptions about complete isotropy and homogenity, the MIL-F-8785C outlines a separation into a Low-altitude model and Medium/High altitude model. Note that the altitude dependence is defined differently in MIL-F-8785C and MIL-HDBK-1797. The MIL-F-8785C definitions are presented in this study.
Low-altitude model
In the Low-altitude model, the turbulence parameters in the Power Spectral Densities presented in Section1.1.3 and Section 1.1.4 depend on altitude, h, and mean wind speed W20. Here, W20denotes the wind speed at 20 feet (6 meters). This dependence is contained in Equations 8-9.
Lw= h, Lu= Lv= h
(0.177 + 0.000823h)1.2 (8)
σw= 0.1W20, σu= σv= σw
(0.177 + 0.000823h)0.4 (9)
Typically for light turbulence, the wind speed at 20 feet is 15 knots; for moderate turbulence, the wind speed is 30 knots; and for severe turbulence, the wind speed is 45 knots [2].
Medium/High altitude model
In the Medium/High altitude model, homogenity and isotropy is assumed and the turbulence pa- rameters are independent of altitude. In the MIL-F-8785C, the parameters are set as presented in Equation 10-11.
Lw= Lu= Lv= 2500f t (10)
σw= σu= σv (11)
1.1.6 Comparison of theoretical models
The Von Karman model has shown to have better correlation with experimental data, particularly at higher spatial frequencies [4]. The valid frequency range is greater in the von Karman model. The Dryden model has a valid normalized frequency range of less than 10 rad/s while the von Karman model, with forming filters recommended in MIL-F-8785C, has a valid normalized frequency range of less than 50 rad/s. Note that this range can be improved depending on implementation method [4].
The most prominent reason to use the Dryden model is the computational efficiency gained over the von Karman model, hence explaining the frequent use of the Dryden form in many applications. For
this spectrum, the desired filters for generating turbulence velocities can be analytically derived [1].
The von Karman method, on the contrary, due to its irrational power form must be implemented approximately. There are different methods to do this that will affect the validity range of the model.
Von Karman is the preferred choice in the military standards. In addition, the von Karman method obeys the Kolmogorov −5/3 law, Dryden form does not [4].
1.1.7 Issues with Theoretical Models Related to Rotorcraft
The two theoretical models have traditionally been used in fixed-wing aircraft simulations. However, mainly due to blade rotational velocity, fixed wing aircraft and rotorcraft differ regarding how tur- bulence is experienced. In addition to the translational motion of the wing element of a fixed-wing aircraft, a rotorcraft blade element experiences rotational motion. Gaonkar [5] discusses the differ- ences between blade and body-fixed atmospheric turbulence models. At certain flight conditions, the rotor blade velocity effects become more evident, more specifically at lower altitudes and low speed conditions. A more involved approach that addresses this issue is to use a blade-element sampling model such as in the SORBET model. In pilot evaluation tests, the SORBET model has received favorable pilot reviews because of its over-all attenuation of high-frequency content in the vehicle response [6]. However, a blade-element sampling model implies greater implementation complexity compared to a hub-sampling method.
Because of the frozen field assumption, experienced turbulence varies only due to a non-zero air speed V when implementing a Dryden or von Karman model. When simulating a rotorcraft in hover, the turbulence output will then be zero with a hub-element model if airspeed is zero. According to pilot reviews in the SORBET-study by MacFarland, the theoretical models were undesirably predictive at low-speed conditions [6]. In a traditional fixed-wing model, wing span is required to compute turbulence angular rates. This parameter must be reconfigured if applied to rotorcraft.
1.1.8 Control Equivalent Turbulence Input (CETI) model
Alongside the theoretical approach to simulate rotorcraft vehicle response to atmospheric turbu- lence, recent work has taken on an empirical approach. The conceptual idea of a CETI model is to use actual flight data acquired in turbulent conditions to extract a realistic model. Control in- puts are reproduced to generate equivalent motion as would be seen during flight in atmospheric turbulence. As described in Section 1.1.2, the Dryden and von Karman models are based on the frozen turbulence field assumption, that fails to be applicable during hover/low-speed flight. This empirical approach is a way to bypass this issue and fathom the differences in how rotorcraft ex- periences atmospheric turbulence compared to fixed-wing aircraft. Although the approach differs, extracted CETI models for rotorcraft have been shown to model turbulence in the same low-order form as other existing models such as Dryden [7]. The measured aircraft rate time histories originate from control inputs coming both from pilot commands and turbulence disturbances. The extraction methodology utilizes the identified model of the aircraft to identify the control input signals that would be required to produce the same aircraft rates in calm atmospheric conditions.
In simulations, the extracted transfer functions are driven by random white-noise processes to gener- ate equivalent control input characteristics. A conceptual block diagram of a CETI model is shown in Fig 1.
Gc(s) δc H(s) CET I
−1
Y δt
Wnoise
Figure 1: CETI block diagram - δi are control inputs.
- H(s) is the aircraft bare-airframe system.
- Gc(s) denotes control system.
- Wnoiseis generated Gaussian white noise.
- Y is measured output, i.e aircraft state.
Thereby, the ultimate goal when extracting a CETI model is to determine the corresponding transfer function Gδt, where δt denotes the turbulence control input. For a first order method, the generic form takes the shape:
Gδt(s) = K
s+ a (12)
Gain, K, and break frequency, a, should be functions of turbulence Root Mean Square (RMS), σ, mean velocity, U0, and turbulence scale length, L. In order to extract controller equivalent turbu- lence inputs, the accuracy of the vehicle dynamics model is crucial. Therein, the identification of the bare-airframe system is usually a big part of extracting a CETI model for a specific aircraft [8]. Thereby, a CETI-model is inherently bound to the specific vehicle for which it was extracted.
As bare-airframe and control systems differ significantly between rotorcraft, a CETI-model in its original form is not applicable to any other vehicle than for which it was designated.
1.2 Objectives
The main objective of this study is to investigate the feasibility of an empirically based hover/low- speed atmospheric turbulence model that:
- Generates velocity and rate disturbance components that produce aircraft response rates for rotorcraft vehicles consistent with those recorded in hovering flight of rotorcraft vehicles in atmospheric turbulence.
- Is scalable to different level of turbulence, proposedly with mean wind speed and turbulence intensity parameters.
- Is compatible with linear aircraft math models and linear-control-system analysis tools.
The study also aims to compare the developed model to the existing theoretical atmospheric turbu- lence models in UAM simulations applications. Again, the existing theoretical model will only serve as a frame of reference but not as an accurate model, as it is stated not to be representative in the considered conditions, but rather a best case approximation.
2 Development of a Reformulated Empirical Model
The general approach in this study to develop an empirically based atmospheric turbulence model, is to utilize a CETI-model as a starting point. Again, the goal is to establish Atmospheric Disturbance Equivalent (ADE) inputs, so that the new model is no longer bound to a specific vehicle as is the original CETI-model, but applicable to a class of rotorcraft as a whole. Ultimately, this allows for a more practical solution when one desires to analyse different control systems or conceptual UAM vehicles with similar aerodynamic properties. The proposed modeling approach will be denoted by CETI-ADE in the manuscript.
2.1 Atmospheric Disturbance Equivalence (ADE) from UH-60 Black Hawk helicopter CETI model
An extracted CETI model for the UH-60 Black Hawk helicopter is presented by Lusardi [9]. What follows is the approach developed to establish an equivalent atmospheric disturbance input model applied to this CETI model. The final disturbance model transfer functions (outputs in inches of mixer) presented in Lusardi [9] are for lateral/longitudinal:
Gδlat/lon= 0.08192σ−0.6265w
rσv2U0 πL
1
s+ 2U0/L (13)
For directional:
Gδped = 0.08464σv−0.6493 rσ2vU0
πL 1
s+ U0/L (14)
And for collective:
Gδcol = 0.01286σw−0.7069
r3σ2wU0
πL
(s + 33.91UL0)
(s + 1.46UL0)(s + 9.42UL0) (15) In this study, σw= σv and the characteristic length scale, L, is set to the rotor radius, R.
The UH-60 Black Hawk vehicle math model considered is a fourteen Degrees of Freedom (DoF) model presented in Lusardi [9]. The vehicle model is an unstable Multiple Input Multiple Output (MIMO) system. In order to simulate vehicle turbulence response, the system was stabilized through state feedback by controllable decomposition and pole placement, with right-half plane poles mirrored in the imaginary axis. Note that it is not the same control system as used in Lusardi [9], but merely a simple state feedback with the main purpose of stabilizing the system. The original system as in Lusardi [9] is presented in Equation 16, alongside the proposed equal system.
(˙x = Ax + Bu + BCET IδCET I
y= Cx + Du ⇐⇒
(˙x = Ax + Bu + Btxt
y= Cx + Du (16)
In this case BCET I = B, whereas Btdetermines the resultant forces on the aircraft due to experi- enced atmospheric turbulence. The disturbance inputs are defined as
δCET I =
δlat
δlon
δped
δcol
xt=
ut
vt
wt
pt
qt
rt
With the feedback u = −Kx, the closed-loop system is now governed by the equations (˙x = (A − BK)x + BCET IδCET I
y= (C − BK)x ⇐⇒
(˙x = (A − BK)x + Btxt
y= (C − BK)x (17)
By the assumption that the two systems are equal and that a unique solution exists, the equivalent gust disturbance is obtained by solving the equation
BCET IδCET I= Btxt (18)
In general, the least-square fit solution to Eq. 18 is obtained through the pseudo inverse:
xt= Bt+BCET IδCET I (19)
From Eq. 19, it becomes evident that the construction of the disturbance matrix, Bt, highly impacts the wind velocity outputs and, in particular, the vehicle response.
The time histories are obtained through simulations using lsim in MATLAB. Disturbance control inputs are generated by driving four independently seeded white noise signals with sampling rate 100 Hz through the extracted CETI-model transfer functions in Eq. 13-15. The corresponding atmo- spheric turbulence components are then obtained by linearly combining the control signal samples according to Eq. 20
ut
vt wt pt qt rt
= Bt+BCET I
δlat δlon δped δcol
(20)
As a starting point, the Btmatrix is constructed by selecting the six columns corresponding to the velocity and angular rate states. However, the A matrix contains kinematic relationships between the Euler angle states and the angular rate states that are not necessarily true for the atmospheric disturbance states. More specifically, the equalities ˙θ = q, ˙φ= p are contained in the original state space formulation. As this is not an aerodynamic relationship, it is of interest to compare the model outputs when this relationship is eliminated from the Bt matrix.
2.1.1 UH-60 Helicopter vehicle response to CETI and CETI-ADE model
The vehicle response when (Bt)i,j = Ai,j for i = 1, 2, .., 37 and j = 1, 2, .., 6 are shown in Fig. 2-6.
Turbulence intensity parameters are shown in Table 1.
Table 1: Parameter values used in UH-60 vehicle response simulations U0 15 ft/s
σw= σv 3 ft/s
Results in linear axes are shown in Fig. 2-4.
Figure 2: Vehicle response u-axis with ˙θ = qt, ˙Φ = pt
Figure 3: Vehicle response v-axis with ˙θ = qt, ˙Φ = pt
Figure 4: Vehicle response w-axis with ˙θ = qt, ˙Φ = pt
The Root Mean Square (RMS) of the velocity component time histories are shown in Fig.5.
Figure 5: RMS of UH-60 vehicle response with ˙θ = qt, ˙Φ = pt
The simulation results presented in Fig.2-5 indicate that the vehicle response in linear axes shows great but not complete consistency between the compared models. Time domain analysis, i.e. a
qualitative assessment of the time histories implies that the level of agreement is best in longitu- dinal and vertical axes. Comparing RMS-values, the smallest absolute difference between the two models are observed in the longitudinal axis with 0.024 ft/s, followed by lateral axis with 0.26 ft/s.
However, if normalized with the RMS value from the original CETI-model, the relative difference is most significant in the lateral axis with 39%. Moreover, the resulting time histories in the lateral axis shown in Fig. 3 also exhibits the most significant difference between the compared systems.
Although there are similarities in characteristics, e.g. resemblance in gust amplitudes, the samples do not overlap in the same way seen in Fig.2 and Fig.4. This discrepancy is unexpected, considering the consistency in the other linear axes. A proposed cause is the somewhat crude way in which the Bt-matrix was constructed.
Validation checks are also performed in the frequency domain. The frequency range of interest is primarily 0.1 − 10 rad/s [10], which corresponds to 0.02 − 2.4 Hz. The resulting power spectral den- sities from the simulations show good level of agreement between the models, but exhibit differences especially for frequencies over 10 Hz throughout the three axes. For higher frequencies, the original CETI-model seems to attenuate high frequencies to a greater extent. In particular, the CETI-ADE model exhibits a peak for frequencies around 10 Hz. In this stage of the study, it is difficult to deduce the cause of this peak more in detail. However, it is once again likely to originate from the construction of the Bt-matrix and the characteristics of its pseudo-inverse. Spectral content shows best level of agreement in the frequency range 0.1-1 Hz throughout the three linear axes.
Angular rate response is shown in Fig.6-8.
Figure 6: Vehicle response roll rate with ˙θ = qt, ˙Φ = pt
Figure 7: Vehicle response pitch rate with ˙θ = qt, ˙Φ = pt
Figure 8: Vehicle response yaw rate with ˙θ = qt, ˙Φ = pt
The Root Mean Square (RMS) of the angular rate time histories are shown in Fig.9.
Figure 9: RMS of UH-60 vehicle response with ˙θ = qt, ˙Φ = pt
The results in Fig.6-9 indicate larger inconsistencies between the two models compared to the re- sults in the linear axes. In particular, roll and pitch rates are perturbed to a larger extent with the CETI-ADE model. Quantitatively, the relative differences in RMS between the original CETI- and CETI-ADE model are 140% and 290% in roll and pitch respectively.
In the frequency domain, the same discrepancies can be observed as the CETI-ADE model PSD’s show greater amplitude in the lower frequency range, compared to the original CETI-model. The same 10 Hz peak is distinguishable also for the angular rate response. It becomes evident that differences in the frequency range of flight control, which is the frequency range of interest, causes significant differences in the time domain.
Regarding the coherence estimates between the resulting vehicle responses to the compared models, the results are similar in all but the vertical axis, as coherence above 0.6 is mainly observed in the frequency range 6 − 12 Hz. In the vertical axis, the two compared signals show great coherence over the considered frequency range. These results mainly indicate the differences in how aircraft rates depend on control inputs, i.e the identified bare-airframe system, but does not relate as much to validation of the approach and is therefor not discussed further.
The vehicle response when the kinematic relation is eliminated from the Btmatrix is shown in Fig.
10 - 11.
Figure 10: Vehicle response without kinematic relationship
Figure 11: Vehicle response angular rates without kinematic relationships
The results in Fig.10-11 indicate that the vehicle response shows very little consistency in both linear axes and angular rates with the modified Bt-matrix. An interesting observation in terms of spectral content however, is that the PSD for this setup exhibits similar PSD shape between the compared
systems but with what seems to be a scaled amplitude. In conclusion, it becomes evident that modifying the Bt matrix highly impacts the simulated vehicle response to the CETI-ADE model.
Eliminating the kinematic relationship outputs less desirable results in terms of approach validation, as the simulated vehicle response with the CETI-ADE is not at all consistent with the results from the original CETI-model.
2.2 ADE from small quadrotor CETI model
In the study by Juhasz, an extracted CETI model for a small quadrotor, namely the 3DR Iris+
(herein Iris) quadrotor, is presented [7]. What follows herein is a proposed approach to compare a CETI model to a CETI-ADE model, and a first validation. The state space longitudinal dynamics in hover for Iris identified in Juhasz [7] are displayed in Eq. 21.
˙u
˙q
˙θ δ0lon˙
=
Xu 0 −g 0
Mu Mq Mδlon
0 1 0 0
0 0 0 −lag
u q θ δ0lon
+
0 0 0 lag
δlon(t − τ ) (21)
Further on, the control inputs are considered to be caused only by equivalent turbulence input.
Following the same procedure as in Section 2.1, the proposed equivalent system to Eq. 21 is presented in Eq. 22.
˙u
˙q
˙θ
=
Xu 0 −g
Mu Mq 0
0 1 0
u q θ
+
−Xu 0
−Mu −Mq
0 0
ut
qt
(22)
Note that the lagged control input is no longer a state in the space-state model, but rather it is governed by Eq. 23 and is interpreted as a different system of equations.
δ0lon˙ = −lag · δ0lon+ lag · δlon(t − τ ) (23) A Laplace transform of Eq. 23 gives
sδlon0 = lag(δlon· e−τ s− δ0lon) =⇒ δ0lon
δlon = lag · e−τ s
lag+ s (24)
Now, by assuming equality, the gust disturbances are then given by
ut qt
=
−Xu 0
−Mu −Mq
0 0
+
0 Mδlon
0
δlon0 (25)
Given the extraced CETI model, we have that δlon Wnoise
= Gδt(s) = K
s+ a (26)
To validate that this approach is equivalent, the frequency response Wnoiseq should be identical, given that Mq 6= 0. This is confirmed in Fig. 12.
Figure 12: Frequency response Wq
noise
2.2.1 Iris quadrotor longitudinal response to CETI and CETI-ADE model
The block diagram in Fig. 13 presents the simulation set-up when treating the generated longitudinal wind velocity as a disturbance input.
Gc(s) δc [Hδ, Hut] Gut(s)
−1
δlon q
xt
Wnoise
Figure 13: Longitudinal dynamic system block diagram with CETI-ADE model
Lagged disturbance control inputs are generated by driving a white noise signal through the transfer function, Gδt(s) = lag·elag+s−τ s, and wind velocity time histories are obtained via Eq.25. As it turns out, the value of the control derivative, Mq, has crucial impact on the model. In particular, if Mq = 0 =⇒ qt = 0, always. If Mq 6= 0 =⇒ ut = 0, always. In this study, Mq is set to zero as in the original identified system in Juhasz [7]. The unstable identified bare-airframe system is stabilized analogously as described in Section 2.1. Note once again that it is not the same control system as in Juhasz [7] but a preliminary measure to enable time domain analysis. Time simula- tions of the longitudinal vehicle response to CETI model turbulence and the equivalent atmospheric disturbances are shown in Figure 14.
Figure 14: Iris longitudinal response to turbulence input
The simulated vehicle response of the Iris quadrotor shown in Fig.14 is in almost complete coherence for the longitudinal control inputs and corresponding atmospheric disturbance. These results are promising in terms of approach validation. In all, the longitudinal Iris model is less complex and elaborate than the fourteen DoF UH-60 math model, which is one probable explanation to why the somewhat simplified proof-of-concept modeling approach is more compatible with the less complex model. In particular, the Iris sim is ultimately a single axis model, so the method of finding atmo- spheric equivalence was reduced to a single multiplicative inversion which is a one-to-one mapping.
Consequently, it would be surprising if the vehicle response differed significantly between the original CETI and the CETI-ADE model.
2.2.2 Construction of Hybrid model
As only the results in longitudinal axis are presented in Juhasz [7], this study further relies on the assumption of symmetry in longitudinal and lateral axis. Consequently, the same transfer function is considered to generate lateral turbulence wind velocities vt, but driven with an independently seeded white noise signal.
For a more comprehensive model, a vertical wind disturbance input is included. Vertical wind veloc- ities are obtained by once again utilizing the UH-60 CETI-model in Section 2.1. This is motivated by the fact that resulting forces in the z-axis are mainly dominated by the collective control inputs and vertical velocity, meaning the equivalent equation becomes significantly less intricate, since
˙
w= Zww+ Zδcolδcol⇔ ˙w= Zww+ Zwwt =⇒ wt= Zδcol
Zw
δcol
Hence, the transfer function in Eq. 15 is used to generate vertical wind velocities. Including this additional component, the model based on the two different CETI-models will further on be referred to as the Hybrid model.
3 Simulations in UAM applications
3.1 Atmospheric turbulence components
In this section, generated turbulence components from the constructed UH-60 and Hybrid CETI- ADE models are presented. As no pilot evaluation of the reformulated empirical models has been conducted, the results from the theoretical Von Karman model will serve as a frame of reference.
Note that these results are not necessarily physically correct or representative in the considered conditions. The relevant parameters in the different models are set to generate light turbulence in low-altitude conditions in hover.
3.1.1 Von Karman model
Turbulence wind velocities are generated by driving independently seeded gaussian white noise signals with sampling rate 100 Hz through the transfer functions in Eq. 5-7, implementing the Low- altitude model. Relevant parameters to simulate the desired flight conditions are shown in Table 2.
The time histories are obtained using lsim in MATLAB.
Table 2: Parameter values to generate Von Karman turbulence
h 20 ft
W20 15 ft/s
V 15 ft/s
b 4R = 26 ft
The generated time histories of turbulence wind velocities and angular rates from the Von Karman model are shown in Fig. 15; the corresponding Power Spectral Densities are shown in Fig. 16.
Figure 15: Atmospheric disturbance components, Von Karman model
Figure 16: PSD, Von Karman model
The results shown in Fig.15-16 do indicate some, but not complete, isotropy; in this particular sample the vertical gusts exhibits somewhat greater magnitudes. Regarding regularity, there are distinguishable periodic tendencies that would be of interest to evaluate further in piloted simula- tions. The time sample is not considered sufficient for more definite conclusions. Regarding spectral content, the shape of the PSD’s in Fig.16 is a shared characteristic among the turbulence components with a roll-off of approximately 20 dB/decade. The u,v,w-components can be grouped together in terms of spectral content, separated in amplitude from the p,q,r-components.
3.1.2 UH-60 CETI-ADE
Turbulence wind velocities are generated from the transfer functions and method described in Section 2.1 with the kinematic relationship left in the Btmatrix. Used parameter values shown in Table 3.
U0 15 ft/s σw= σv 3 ft/s
Table 3: Parameter values to generate UH-60 CETI-ADE turbulence
The generated time histories of turbulence wind velocities and angular rates from the UH-60 CETI- ADE model are shown in Fig.17; the corresponding Power Spectral Densities are shown in Fig.18.
Figure 17: Atmospheric disturbance components, UH60 CETI-ADE model
Figure 18: PSD’s for UH60 CETI-ADE model
Some first observations can be noted from the simulation results in Fig.17-18. In Section 2.1.1, it was concluded that modifying the Btmatrix highly impacts the simulated vehicle response. An interest- ing observation from the results in this section however, is that the output turbulence components are identical for the two versions of the Bt-matrix discussed in Section 2.1.1. The pseudo-inverse, Bt+, is apparently independent of the particular modification that was made.
As for the simulated wind velocity time histories in Fig.17, the resulting turbulence velocity com- ponents show great variation in amplitude in a way that contradicts the conventional assumptions that the turbulence would be isotropic and homogeneous. The most surprising result is the signif- icant difference in lateral and longitudinal wind velocities, that seems difficult to explain from a physical standpoint, at least in comparison to the Von Karman model outputs. It is also evident in the frequency domain, as the PSD in the lateral axis shows a much greater amplitude than other components. Although the same first-order filter shape with a 20 dB/decade roll-off is seen in Fig.18 as with he Von Karman model in Fig.16, the discrepancy in amplitude between the linear axes and angular rates seen with t is not as clearly distinguishable in 18.
It shall be acknowledged that the disturbances not necessarily represent physical gusts in the sense that they model the actual atmospheric turbulence, but merely the disturbance signals needed for the aircraft to respond as it would operate in such conditions. That being said, the notable asymme- try in longitudinal and lateral inputs does imply that there is a need to re-iterate the construction of the Bt matrix to evaluate reliability. The aerodynamic accuracy of the Bt matrix will probably be the most prominent factor in obtaining physically credible atmospheric turbulence and thereby a model that is less biased by the original vehicle. Also, the existence and uniqueness of a solution to the ADE Eq.18 is crucial.
In addition, the model description given in Lusardi [9] contains a mixing unit that has not been considered in this simulation. It would be of interest to investigate the impact of the mixer on the results.
3.1.3 Hybrid model
Longitudinal and lateral wind velocities are scaled with a factor 0.2 to achieve light turbulence level in longitudinal and lateral axis. Vertical outputs are generated with the same parameter values as in Section 3.1.2. The generated time histories of turbulence wind velocities and angular rates from the Hybrid model are shown in Fig.19; the corresponding Power Spectral Densities are shown in Fig.20.
Figure 19: Atmospheric disturbance components, Hybrid model
Figure 20: PSD:s for Hybrid model
As only the lateral and longitudinal wind velocities origin from the IRIS CETI-model, these are the ones discussed herein. In the construction of this model, lateral and longitudinal symmetry was assumed to be valid for the CETI transfer functions, meaning that the two control input signals were generated through the same transfer functions but with independently seeded white noise sig- nals. This was motivated by the natural symmetry in the aerodynamic response of a quadrotor.
