Identification of system parameters in a nonlinear helicopter model

24th EUROPEAN ROTORCRAFT FORUM Marseilles, France - 15th-17th September 1998

FM05

Identification Of System Parameters In A Nonlinear

Helicopter Model

1

Cv

Cvo

CL

CLo

CLu

CAo,(}

c,<.o

c,,,o

Cx

Cxo

c,.

!vi.

Rohlfs

Deutsches Zentrum fiir Luft- und Raumfahrt e.V. (DLR)

Institut fiir Flugmechanik

38108 Braunschweig, Germany

Since regulations for the qualification of helicopter flight simulators were pub-lished in the Advisory Circular 120-63 (AC 120-63) of the Federal Aviation Administration (FAA), the need of increasing the simulation fidelity of the heli-copter 1nathematical models becomes more and more relevant to meet the new and more restricted requirements.

In the past the DLR Institute of Flight Mechanics used mainly two approaches to develop helicopter mathematical models. The first one determines coefficients in a linear derivative model using system identification methods. In this case the model is derived from the information which is contained in the used flight test data. The second method is a physical approach where the model is generically derived. Here detailed vehicle characteristics are used to derive the model and flight test data is only needed for the evaluation.

In this paper an advanced integrated approach is introduced. A physically based modular structured nonlinear helicopter model is developed. To improve the overall simulation fidelity, in particular the off-axis responses, parametric ex-tensions are added to the model. The introduced parameters are determined using system identification methods.

The paper shortly describes the helicopter modeling activities at the DLR Insti-tute of Flight Mechanics. The used system identification procedure is explained and the requirements of the AC 120-63 are shortly described. Some comments about the modeled helicopter BO 105 and the used flight test data are made. The helicopter model is discussed in detail and identification and verification results are presented in both, the time and frequency domain.

NOTATION

Cyo

Lateral force derivatives

Cz

Vertical force coefficient

Drag coefficient

Cz()

Vertical force derivatives

Drag zero derivative g Acceleration due to gravity

Lift coefficient

Io

l\1oments of inertia

Lift zero derivative fAtR l\~Iain rotor moment of inertia

Lift o: derivative

ko

Drag derivatives

Zero downwash derivative kcng,() Engine derivatives

Cos downwash derivative Laero Aerodynamic roll moment

Sin downwash derivative Linertia Inertial roll moment

Longitudinal force coefficient m Mass

Longitudinal force derivatives Mach Mach number

J.Vfinertir1 Naero Ninertitl p Qeng Qrot q 1" 1l v 111

v;.

X aero X inertia Yaero Yinertill Zaero Zinertia a

!3o

Ao

A,

A,

At Ao,PP Ac,PP As,PP Ar_,PP

n

6,(1 <!>

8

lji ljiR

()

0

Inertial pitch moment Aerodynamic yaw moment Inertial yaw moment Roll rate

Engine torque Rotor torque Pitch rate Yaw rate

Longitudinal velocity component Lateral velocity component Vertical velocity component Rotor tip speed

Aerodynamic longitudinal force Inertial longitudinal force Aerodynamic lateral force Inertial lateral force Aerodynamic vertical force Inertial vertical force Angle of attack Blade flapping angle

Extended do\vmvash velocity, zero camp. Extended down\vash velocity, cos camp. Extended dowmvash velocity, sin camp. Extended downwash velocity, tail rotor Original dowmvash velocity, zero camp. Original downwash velocity, cos camp. Original downwash velocity, sin camp. Original dowmvash velocity, tail rotor Rotor angular rate

44.4-

n

Roll angle Pitch angle Yaw angle Rotor azimuth

d()j dt

d

2

_()jdt

2

2

INTRODUCTION

Fig. 1 shows the different helicopter modeling ap-proaches at the DLR Institute of Flight Mechanics.

2.1

System identification approach

The first approach, represented by the left column in Fig. 1, is the classical system identification (SID) pro-cedure. In this approach coefficients in linear deriva-tive models are determined using system identification methods. In this case extensive flight test data is needed for the model determination and validation. The ob-tained global models are mainly used for the devel-opment of control systems [1, 2] and for stability and control analysis. Since the structure of these models is linear, they cover only a small range of the flight enve-lope, namely the range around the trim condition and they are valid only for small control excitations.

High-Fidelity Simulation State Space Models

Integrated Approach to Rotoruaft Modeling and Simulation SID Models _I SIM & SID Models SIM Models Cla11ical SID approach Advanced integrated Cla11ical SIM approach

approach

Derivative model~ Generic models ba~d

Gene"' models on modular cll!ments Linear aerodynamics augmented with

~

pJrametric sub models

~

Nonlinear Extensive ilight data aerodynamics lot Point-modeiiO Nonlinear

and val'ldat'1on aerodynamics Fl'lght data only for model validation Stability & Control Flight data fot

sub-analysis and control modeiiD and global Simulation, 1ystem de1ign model validation performance,

and vchide d~ign

r

System Identification

l

System Simulation &

I

Identification Syrtem Simulation

I

Figure 1: Rotorcraft modeling and validation, the three columns approach

2.2

System simulation approach

The second approach, the classical system simulation (SIM) approach, is represented by the right column in Fig. L Here the model is generically derived [3]. The model is modular structured and it is based on detailed vehicle design data. This model is mainly used for real time flight simulation tasks and it is implemented in the institute's helicopter ground based flight simulator. Depending on the model complexity, it can also be used during the design phase for detailed prediction and for overall performance analysis. The main advantages of this model are that it covers the whole flight envelope and that flight test data is only needed to validate the modeL The modeling status and fidelity is described in [4].

2.3

Advanced integrated approach

In the DLR Institute of Flight lvlechanics a great amount of experience exists in the area of system iden-tification for both, rotary and fixed wing aircraft. One of the latest most challenging tasks was the identifi-cation of the model database of the Dornier D0-328 aircraft for a level D flight simulator [5]. Despite the very complex mathematical model and a high number of parameters it turned out, that the system identifica-tion methodology is well suited to determine a database \Vhich meets the stringent requirements for a level D fixed wing aircraft simulator [6]. In parallel to fixed wing aircraft application, system identification activi-ties also resulted in the determination of accurate linear helicopter models from flight test data [7, 8, 9].

In contrast to fixed wing aircraft models it is not pos-sible to separate a nonlinear helicopter mathematical model in a longitudinal and lateral motion because the helicopter is a highly coupled system. This is mainly due to the rotating blades which have additional de-grees of freedom. Therefore it is not possible to identify separated parts of the helicopter modeL But first suc-cessful! steps [10, 11] showed, that it should be possi-ble to identify system parameters in a nonlinear highly

T\1anocuvre Actual T\lensurements Optitni,_cd

Input

Input Rcspomc _{Data Collection} Flight Vehicle & Compatibility

---

--

Data Analysis -~-Methods Rc.,pon-'>t lldcntifocatinn I ldenlificalion '~ Alt<>mhm Criteria Models

J

ParJmctcr A<lju>tmcnl<

..

---.

_--1

_Mmhcmaticnl Model Rc.,pomc : APriori !~

: Values : Model

'---..!

I

Model Ycrifoc:uion

I

APPLICATIONS: (!)Data Ba_o;c Gcncr.uion _{(2) Flying Qualities Evaluation} (3) F1i£hl C<Jntml _{(4) Simulation Vati,l:uinn} Optimi,..:~tinn

Figure 2: The principle of parameter identification

coupled helicopter simulation model.

In regard of this development it was considered to combine the two helicopter mathematical model devel-opment approaches to an integrated system simulation and identification approach. This is represented by the middle column of Fig. 1. The used model is, similar to the model of the 'system simulation approach' gener-ically derived. In addition1 parametric extensions are

introduced to provide improved descriptions of inaccu-rately known model components. Flight test data are needed for the determination of the introduced param-eters with system identification methods and for the validation of the model.

3 PARAMETER

IDENTIFI-CATION PROCEDURE

Fig. 2 shows the principle of the parameter identifi-cation [12]. Specially designed input signals are used to excite the aircraft. Both, control inputs and air-craft response are measured and recorded. The data is checked for compatibility, and errors are corrected as far as possible. The identification techniques can be im-plemented for working either in the time or frequency domain. Consequently the measured data has the form of time histories or it is transferred into the frequency domain. The aircraft model is formulated as a set of differential equations or respectively as transfer func-tions. Unknown parameters in the model are adjusted using the differences between measured and computed data to obtain a better agreement. The identification process is usually an iterative process. The adjustment of the parameters is repeated until an accuracy require-ment is accomplished or a certain number of iterations is reached.

4 FLIGHT TEST DATABASE

The flight test program was conducted with the BO 105 research helicopter from DLR (Fig. 3). The flight test

Figure 3: BO 105 research helicopter

database consists of tests at different speeds from hover to llO kts and different input signals like sweep and 32ll signals for all four control inputs. Before using the data for identification and verification1 its consistency

was checked by performing a flight path reconstruction.

5

NONLINEAR HELICOPTER

MODELING

The mathematical model described in this paper is de-veloped for the BO 105 research helicopter from DLR (Fig. 3). The BO 105 is designed as multipurpose light helicopter. Typical use of the highly manoeuverable twin engine vehicle are transport, police and military missions. An important design feature is the hingeless main rotor system. In the following the mathematical modeling of this helicopter is described.

The basic problem of a flight dynamics simulation program is to describe the motion of the vehicle's center of gravity in space. This description is governed by the following system of differential equations.

m (1£

+

qw-rv) X aero+ X inertia

+

mgsin

e

m ( iJ

+

ru - pw) Yaero

+

YineTI.ia

m.g cos

e

sin <I>

m (w

+

pv- qu) Zaero

+

Zinertia

mg cos

e

cos <I>

lxxP lx,r

+

qr (!,. - !yy) - lx.pq =

Laero

+

Linert.ia

lyyrj

+

pr Uxx- !,.)

+

lx.

(p

2

- r2) A1aero

+

ldinertia

I,.r lx.P

+

pq (Iyy - Ixx)

+

Ix.qr =

J\Taero

+

Ninert.ia (1)

In a generic approach t.he right hand sides of these equations are filled up with physically based descrip-tion of the vehicle with all components contribut-ing to the balance of the three forces and moments. In contrast to a classical 6 degrees of freedom ap-proach for a helicopter simulation program1 \vhere only

----~=-.:"--llelicoplcr model

Input: Cuntrnl Yariahk~ und 'late vari:lhl<..-; {f~l~lhack)

Oulpot: Furn':'l und mnmcnl•

llody motion

Input: Sum nrrnrc~s and mumcn~' nrall rnudcllcd cumpuncnL> Output: ln!t):raliun nr,talc vurh1hll"S ur CG muliun {6D0f)

'"""'""'"""

ln••.,'-''"'''"''""'~"" ;-:~~~·.::-~~;;:,,

,,,,

....

~"~''

Figure 4: Principle structure of a helicopter simulation program

aerodynamic forces (XaerOl YaeTOl Zaero) and moments

(Laero₁ IYiaero, Naero) are taken into account, the inter-nal inertia forces (Xinertia1 Yinertia, Zinertia) and mo-ments ( Linertia, lvh,tertia, Ninertia) and their internal reactions have to be described in detail.

The most common approach is a modular descrip-tion where the model of the vehicle is divided up into its components or modules and their individual contri-butions are added to the right hand side balance of the above system (Eq. 1). For a helicopter these compo-nents are

• main rotor including flapping dynamics, • tail rotor,

• fuselage,

• empennage-horizontal stabilizer, • fin-vertical stabilizer,

• engine and

• dynamic downwash models for the main and tail rotor.

The principle structure is shown in Fig. 4 as a block diagram.

5.1

Helicopter model modules

In the following the modules are described in more de-tail.

5.1.1 Main rotor

The standard approach for the main rotor description is the blade element formulation. The rotor is discreti-sized into the individual blades which are then devided into several segments. The local aerodynamic and in-ertia forces are then summed up for all segments and all blades to determine the total rotor forces and mo-ments. In addition, the flapping motion of each rotor blade is modeled in a so-called rigid blade formulation where the blade elasticity is neglected.

5.1.2 Tail rotor

The modeling of the tail rotor is the same as for the main rotor. Again the blade element formulation is used to model the aerodynamic and inertia forces of the tail rotor. Due to the high rotational speed of the tail ro-tor, its flapping motion can be neglected or assumed quasisteady.

5.1.3 Fuselage, empennage, fin

In this model an integrated derivative formulation is made for the helicopter fuselage, empennage and fin.

5 .1.4 Engine

The helicopter yaw response, \vhen considered as cou-pling response, is highly influenced by the dynamic en-gine and drive train torque. Therefore the basic enen-gine dynamics and the behavior of the rpm governor have to be modeled.

5.1.5 Dynamic downwash

The calculation of the distribution of the induced ve-locities of the main rotor is based on momentum the-ory either global or local. The basic description of us-ing trapezoidal downwash distribution in wind axis was given by Glauert as quasisteady description. The exten-sion to a dynamic formulation was developed by Pitt

& Peters

[13]

using the dynamics of thrust and aedynamic pitch and roll moments produced by the ro-tor. Their perturbation approach was recently extended to describe the influence of the helicopter motion onto the dynamics of the downwash shape. One of the ap-proaches can be referenced as parametric wake distor"' tion approach. For the downwash of the tail rotor a similar dynamic formulation is used.

5.2

State variables

The mathematical model of the BO 105 described in this paper is of 25th order and has 16 degrees of free-dom. The state variables are given in the following list.

• 9 first order differential equations for the rigid body motion (U,V,W,

1

\<j,f,~,0,

ti!)

• 4 second order differential equations for the blade flapping motion (ilblue 1 ~green' i3yellow 1 l3red)

• 4 first order differential equations for the dynamic downwash (_.\o,

Ac,

.\s,

..\t.)

• 1 second order differential equation for the rotor azimuth (;i) R = fl)

• 1 second order differential equation for the engine torque

(Q,n

9 )

5.3

Model parts with parameters to be

estimated

In the following, the parts of the model where param-eters are estimated, are described in more detail.

5.3.1 Blade element aerodynamics

In contrast to most helicopter simulation models the aerodynamic parameters of the rotor and tail rotor blades are not taken from wind tunnel data tables. In this work a derivative approach is made.

CL = CLo

+ CLaO:,

Co= CDO

+ krCL

+ k,Cf.

(2)

(3)

Eq. 2 provides the derivative formulation of the lift efficient. Eq. 3 describes the modeling of the drag co-efficient .. All five derivatives, CLo, CLeo Cvo, k1 and k2 can be determined in the identification process. The

same aerodynamic formulation is used for the tail rotor. That leads to a total of only ten aerodynamic param-eters for both, the main and the tail rotor.

5.3.2 Extended dynamic downwash model Most of the parameters are used to extend the dynamic downwash model from Pitt and Peters mainly to im-prove the off a..'is coupling. The downwash model from Pitt and Peters is described in [13]. Nowadays it is used in most of the efficient helicopter simulation models. A proposal for extending the model is given in [14]. In this paper a more complex approach is made. As an exam-ple the extended equation for the cosine dmvnwash is written.

+

+

+

( 4)

Eq. 4 explain the approach.

5.,,PP

is the original Pitt and Peters dynamic down wash approach for the cosine part. This approach is then parametrically extended and the derivatives are determined in the estimation process.

5.3.3 Combined fuselage and empennage for-lnulation

For the fuselage and empennage a combined parametric formulation is made.

Cx = Cxo

+

Cx,Machlviach

Cy _{= Cyo}

₊

_{CY,M acl)\1 ach}

Cz = Czo

+

Cz,!1Jachll1ach

CL = CLo

+

CL,Machl\1 ach

eM

= CMo

+

CM,MachMach

eN

= CNo

+

CN,Machll1 ach (5)

Eq. 5 shows the very simple approach which is made for the forces and moments. The six coefficients are mod-eled only with the zero and Mach-number dependent derivatives. For the investigated flight test data it was not necessary to introduce other derivatives which for example take the angle of attack or sideslip angle into account.

5.3.4 Engine formulation

The first order differential equation of the rotor speed of rotation is given by:

. 1

n

=

r

+

-1-(Qwg- Q,ot)

MR

(6) Assuming the engine dynamics due to fuel flow as first order system and the engine governor (fuel flow due to changes in rotor speed of rotation) as PI governor the second order differential equation of the engine torque follows as:

6

IDENTIFICATION

SULTS

(7)

RE-Twelve runs of flight test data were concatenated. Each run has a duration of 12 seconds. Three velocities: hover, 40 lets and 80 lets were evaluated together. For each velocity all four control inputs, longitudinal, lat-eral, collective and pedal \Vere used. The input signal for all runs is the DLR 3211-signal. The identification was conducted in the time domain with the NLlviLKL program of the Institute of Flight Mechanics. In the following the identification results are presented. First overall results for the velocities, engine states, rates and Euler angles are shmvn. Then, results which are required to fullfil! the criteria given in the AC 120-63 are presented in more detail.

Fig. 5 shows the identification results of the longi-tudinal, lateral and vertical velocities for all four con-trol inputs, longitudinal, lateral, collective and pedal at all three investigated velocities, hover, 40 kt.s and 80 kts. The solid line represents the measured data and the dashed line represents the identified results. It can be seen that the overall performance of the identified model is very good. Some discrepancies can be seen. This is possibly due to unknown gust influences, which could not be modeled.

Fig. 6 shows the identification results of the rotor speed of rotation and the engine torque again for all investigated data. In addition the yaw rate is shown.

It can be seen, that the engine dynamics are very well modeled although the structure of the model is linear and only three engine parameters are needed (Eq. 7).

Fig. 7 shows the identification results of the roll, pitch and yaw rates for all velocities and all control inputs. Again a very good overall performance of the model can be seen. All cross couplings for all velocities

longitudinal velocity [m/s] hover 40 kts 80 kts

sot-8~x~_o~_,~oo~~8L-~8~x_,~o~~8~o~~8L-~~~~,_~,~~

i I I

___

,

- - . 01~-'-· lOt--,--.--,--t---,--~-,--+--,--,--->7~ lateral 0 velocity [m/s] vertical velocity [m/s] -10t--L-~L---~-t--~--~~L---~-L--+---~-r4 2+--,-~-,-~-~r--,---,--,-_,--.---,--,-_, 0 """'·~~,...

~n_J'

~

!

i

I :1

,

'

.

I

.

-10+----L-~~--~--~-r---~~----~-L--+---~~ 0 20 40 60 80 100 120 140 time [s] ~-measured _identified

Figure 5: Identification results: velocities

hover 40 kts 80 kts

ox

8x

8

80

8

8x

8

80

op

45 .5 +--""---.-''""--,--.;'-'--,--"L-+-~-,-"L_,-=--,--""'---'I--'-"-,--:L--,--=-=-.-,--.::.t:..-j rotor rpm [rad/s] engine torque [%] 30t--L-~L---~--t--r---~~~--~-L---~-L-~ 34+--,---,-~-~r--.--,---,-~---,---,-~ yaw rate 0 [o /s] 20 40 60 80 100 time [s] -measured _identified

Figure 6: Identification results: engine states

FM05- 6

34 roll rate 0 [o /s] -34 23 pitch rate 0 [o /s] -23 34 ox

h

"'

1r

hover ov 80 or ox

l

1.1 _-

_.AA...

I~

{\f

-'Vv

_~

-~ 40 kts 80 kts ov 80 op ox oy oo op V'

~

~

I"""'"

r/v.-

.Jtv---"~

_-v·

lv----

~

J

_II'"'

~- ~

_l!~fvv-.

yaw rate 0 [o /s]

~

1/V

rv

\0

~tF\

. I "

_'Y

1/'-"

rl

~

_I'

••

,.

lA v -...

\f\ov

-34 0 34 roll angle [0] 0 -23 17 pitch angle 0 [OJ -17 86 yaw angle [o /s] 0

-29

20 40 60 80 100 120 140 time [s] -measured _{---- identified}

Figure 7: Identification results: rates

hover 40 kts 80 kts ox ov 80

Op

ox oy 80 or ox oy 80 op

f./\

rv

... ,

....

r

r

_;

_~

_lfr

_i\

""

.,~

iV>"=

,r-

_~

---"\..

h

rr

v

~

~

~

~

r\

~

~

_,.}\

IV

J

~

_~

v!

_If'--

_.V~tv....

0 20

40

60 80 100 120 140 time [s] -measured _{---- identified}

and all control inputs arc modeled correctly and show the right trend as required in the AC 120-63. Runs for which error boundaries are defined are underlayed white. They will be presented in more detail later.

Fig. 8 shows the identification results of the roll, pitch and yaw angles. Since the rates are well mod-eled, the fit of attitude angles must be good, too. Some minor discrepancies can be seen which result from the integration of the rates. Small errors in the rates are summed up by the integration and lead to the errors in the attitude angles. Runs for which error boundaries arc defined in the AC 120-63 are underlayed white. They will be presented in more detail later.

6.1

Performance with respect to the

AC 120-63

In the following the white underlayed runs of Fig. 7 and Fig. 8 for which criteria are given in the AC 120-63 are presented in more detail.

For the presentation of the results, tolerances given in the Advisory Circular AC 120-63 on Helicopter Sim-ulator Qualification [15] of the Federal Aviation Ad-ministration (FAA) will be applied. They help to give a quantitative impression about the model accuracy and needs for further improvements. The tolerances are added to the measured flight test data, shown as thin solid line, and the obtained range is plotted as shaded area. The calculated model response is shown as thick solid line which has to stay within the error boundaries to fullfil! the AC 120-63 criteria.

The AC 120-63 criteria require step control inputs. Here 3211 signals were used. However, since the 3211 signal consists of five steps, it should be justified to use it instead of the step input signaL

Fig. 9 to Fig. 11 show the direct responses for the hover flight condition. Fig. 9 shows the longitudinal 3211 input signal, the helicopter pitch rate response and the corresponding pitch angle. It can be seen that the pitch rate response is well within the tolerance. The pitch angle fullfills not always the AC 120-63 cri-teria. Fig. 10 shows the lateral 3211 input signal, the helicopter roll rate response and the corresponding roll angle. Again it can be seen that the rate response is well within the tolerance, whereas the roll angle shows some minor discrepancies. Fig. 11 shows the pedal 3211 input signal, the helicopter yaw rate response and the corresponding yaw angle. It can be seen that the results are similar to the others discussed before.

Fig. 12 to Fig. 14 show the direct responses for the 40 kts flight condition. Again the input signals and corresponding rate responses and attitude angles are shown. The results are similar to the results achieved in hover. The rates and attitude angles are most of the time within the tolerances.

Fig. 15 to Fig. 17 show the direct responses for the 80 kts flight condition. As for the other velocities, the rate and attitude angle results are mostly within the tolerances. 90 85 : 80

i

~;~

' > - '

-~~

~ 65

i

~ ~~I

-3211 input 50

:~ Lr~·~,-~ ---~··---··

---,---0 3 t1me[s] 10 11

, :: I

AC 120-63 boundary -measured -ident<fied

~ ~-~ i ~ . o L ' - -... g -0.1 i 0. ·0.2 ~ -0.3

i

·04

1---"

8 10 11 12 time[s] 0.251 021 ~ 0.15 i

AC 120·63 boundary -measured -identilied

; O.ll ~ o.os 1

-...._.="'

~ 0 a. ·0.05 ·0.1 -0.15 2 -··---···--,---, 6 7 10 11 12 time [s]

Figure 9: Identification results: direct response clue to longitudinal input at hover

-3211 input g_ 40 i ~ 30

i

20•

,I

o

L.-- ,

·-~--·---·---

_

··-~---10 11 12 time[s]

AC 120·63boundary -measured -identified

•

"

·0.2 2 _·0.4 ·0 6

.,_,I .. _____

-·---·---·-3 ' 6 7 time [sj 10 11 12

0.3 . ·• AC 120-63 boundary -measured -identified

0.2 ~ 0.1 ~ 0

'

2 ·0.1 ·0.2 6 tim&[s] 8 10 11 12

Figure 10: Identification results: direct response due to lateral input at hover

-3211 input 1.6 • 1.4 ~ 1.2 1

t

0.6. 2 3 AC 120-63 boundary 3 AC 120-63 boundary

~

, : 1' 04

~ 0.~

"'-' - - -.... ·0.2 ~ 7 9 10 11 12 timejs] -measured -identified 10 11 12 time Is] -measured -identified ·0.4 ~---~--- -~---·-~---~---~---0 2 3 5 9 10 11 12 time ]s]

Figure 11: Identification results: direct response due to pedal input at hover

''I

70 -3211 input 65 1

£:

60 _; :; 55 • ~

-~!~I

40 J 35

I

30 _:·---~----0 2

'·'

0.3 - 0.2

i

0.1 • 0 ~ -01 j ~ . I a. -0.2 " AC 120·63 boundary 031 -0.4,--~ 0 1 2 3 4 5 7 10 11 12 time Is] -measured -identified 7 8 10 11 12 time js] 0.1 -1 I _ 0.05-1

AC 120·53boundary -measured -ldenti~ed ~ 0-1 - I ~-0.05

i

~ -0.1 J K i ·0.15 l -0.2 _J_--0 3 7 8 10 11 12 time js]

Figure 12: Identification results: direct response due to longitudinal input at 40 kts

,, I

60

i

'50

l

'i

40

c:

3o 1 !!l 20 ~ , j I -3211 input 0 -i ·-···---·r·-2 3 4 5 6 7 time (s] 10 11 12

AC 120·63 boundary -measured -identified

0.3

i

0.1 • 0 '§ ·0.1

e

-o.2 -0.3 ·0.4 ~ ·0.5 ~----· 03 0.2 ~ 0.1

•

~

e

·0.1

I

2 AC 120-63 boundary

::::

L-.---~--.-2 10 11 12 time ]s] -measured -identilied 8 9

"

time ]s]

Figure 13: Identification results: direct response due to lateral input at 40 kts 65 l if

:~

j

~50 -1 -~ 45 _I

~

40 ,I 35

!

30 ~--·r 06 04

i

0.2.

•

0

"

I ~ -0.2 _i I ·0.4 l ·0.6 j L2

"

08 "- 0.6

t

04 ~ 0.2 J ·0.2 ·0.4 -3211 input 10 11 12 time ]s]

AC 120·63 boundary -me~sured -identified

7 10 11 12

time (s)

AC 120·63 boundary -measured -identified

9 10 11 12

time ]s]

Figure 14: Identification results: direct response due to pedal input at 40 kts

70 65

z

_{~50 ~---J}

:~

~ 45 £

"

35 30 . -3211 input 3 10 11 12 lime{s] 0.3 ' 02

AC 120-63 boundary -measured -identified

~ 0.1 g_ 0 '-',

---.J

~ ·0.1 ~ -g_ ·0.2 . -0.3

i

-0.4 l 0 ~-·--- . --. -~--~----,---0.25 0.2 . ' [ 0.15 1 -=- 0.\ ' AC 120-63 boumlary 10 11 12 time[s) -measuTed -idenlilied E! I ~ o.os I

~

0 · · --~-0.05 :-·0.1 : -0.15 -~--- ---,-~---·· 0 3 10 11 ~me [s]

Figure 15: Identification results: direct response due to longitudinal input at 80 kts 60 j 0' 55 ' £':.. 50 ' g_ 45 : -~ 40 ' E 35 -j -321tinput 30 i 25 ' 20 1---·· 4 7 8 10 11 12 lime [s)

AC 120-63 boundary -measured -identified

2 -0.4

-0.6 :

·0.8 .;_---~~---,~-.-,-...---~~---···---r

1 2 3 5

time[s]

0.3 j :AC t20-63boundary -measvred -identified

''I

i

0 1 0 _i

t

.Q 1

l

~

,_, I

-{}.3 0.4 -0 3 10 11 12 lime [s]

Figure 16: Identification results: direct response due to lateral input at 80 kts 70 65

-"'

i. 55 ~50 ai 45 ~ 40

I

~ ~ 35 30 0.6 OA 02 0 ·0.2 ·0.4 -0.6 -3211 input 8 time (s]

'

AC 120-63 boundary -measured

I

!

'

:

·O.B 1--~--- ··-,---o. 15 ; 0.1 0.05 :

~-~d

I

~ ·0.15 ~ -~0 2~ I 6 8 lime (s] -measuwi;l

"

11 -1dentiried ---·-· 10 11 ·031 ·0.35 ,----~-.·-·----r----,----, - - , - -·-- --~---r--·-0 3 5 6 7 8 9 10 11 time\S} 12 12 12

Figure 17: Identification results: direct response clue to pedal input at 80 kts

It can be seen that the achieved results not ahvays fullfil! the AC 120-63 criteria. This depends mainly on gust influences which could not be taken into account. In addition, data with high amplitudes in the control inputs and consequently high amplitudes in the vehi-cle response is investigated. 3211 control inputs instead of step control inputs with relative long durations are used. When flight test data with less wind, lower con-trol input amplitudes and shorter duration is investi-gated, it should be easier to fullfil! all AC 120-63 crite-ria.

7

VERIFICATION RESULTS

To validate the obtained model and to show its predic-tion capability, a verificapredic-tion simulapredic-tion was done \Vith data not used in the identification process and with a different input signaL In this case only offsets in the control inputs and initial conditions of the state vari-ables were estimated. Examplary results in the time and frequency domain of the direct responses due to sweep inputs are shown.

7.1

Time domain results

Fig. 18 shows the roll response and the corresponding roll angle due to a sweep input for the hover case as an example. It can be seen that there is a satisfying

::1

60 'l 55 &_ 50

=

45

•

40 35 -sweep input 30 '---~---~~---0 0.3 02

i

~

e

·0.1 -0.2

i

-0.3 0 0.2 1 0.1

i

" 0 ~ -0.1 -l ~ -0.2 -1

e

-o.3 ! -0.4 : 2 4 6 10 12 14 16 18 20 time [s)

AC 120-63 boundary -measured -simulated

4 6 10 12 14 16 18 20

time [s]

AC 120-63 boundary -measured -simulated

-0.5 ~---... ----~-~

-~---10 12 14 16 18 20

time[s]

Figure 18: Verification result: direct response due to lateral input at hover

65 ; 60

!

~55 ~

~ .s_ 50 .§ ' 45 ; -sweep input 40 _: ---~---·-··· 0 0.2 ' 0.15 1 ';;;' 0.1 1 ~

o.osl

• 0 ~ ·0.05 '1 C.. -0. I ~ ·0.15 ! 4 10 time [s} 12 AC 120-63boundary -measumd ·0.2 J _____ , -2 6 10 12 14 time [s) 0.06 1 AC 120-63 boundary -measured 0.04 _) ';:;;' 0.02 ~ ~ 0 -; -o.oz

I

~ •0.04 -, "5 ·0.06 l '5,_ -0.08 I -0.1 16 18 20 -simulated 16 18 20 -simulated ·0.12 ; -· - - -· --6 10 12 14 16 18 w time js)

Figure 19: Verification result: direct response due to longitudinal input at 40 kts

"I

60 ~55 j -~

t

50 45 -sweep input 40 ---~---.-- ---~, --,--~ 0 10 12 14 16 18 20

i

_{• 0}

"

~ -0.1 ' >--0.2 ~ ·0.3l ·0.4 ~ 0 0.15 1 0.1 ~

!

0.05

i

• 0

!

.o.o5 . 1 g_ ·0.1

l

-0.15 I ·0.2 • 0 time js)

AC 120·63 boundary -measured -simulated

10 12 14 16 18 20

lime js)

AC 120-63 bour'idary -measured -simulated

4 8 10 12 14 16 18 20

time[s}

Figure 20: Verification result: direct response due to pedal input at 80 kts

match of the roll rate. The match of the roll angle is acceptable although a drift is seen. As example for the 40 kts case Fig. 19 shows the pitch response and pitch angle due to a longitudinal sweep input. In this case the match of the pitch rate and of the pitch angle is very good. For the 80 kts case the results of the yaw rate and yaw angle due to a pedal sweep input are shown in Fig. 20. Again a very good fit between measured data and simulated data is seen for the rate response and for the attitude angle.

7.2

Frequency domain results

In the following the frequency domain results of the above shown sweep responses are presented.

Fig. 21 shows the transfer functions of the measured and simulated roll rate due to the lateral sweep input.

It can be seen that there are some deficiencies in the match for higher frequencies. But considering that the data was not used for identification the result is still acceptable.

As a further tool for evaluating the model fidelity, a technique was independently proposed by Hamel [16] and Tischler [9]. In their proposal the relationship be-tween a measured response variable and the corre-sponding model response is used. When, for example, the measured pitch rate is considered as output and the model pitch rate as input, a 'frequency response' can be calculated. Then, for an ideal model, the

mag--25 I ' -301 "'"gnitudc ~~~ --~ -[till[ -IS: -50

_,

I' -M'

'"

45 i ---'""

0---' ~'' pi'~'c 'XI: 101

.m!

-1~11! _ nmcga [rJd-"1 -2:'.5 [1 .. 0 . --~,-;-n-~g;;-~Jt:~i-·-- ---- "'"-'-'""'t! ---,;rnulatcd _ _ I 10.0

1"

10.0 20.0

Figure 21: Verification result: transfer functions of mea-sured and simulated roll rate due to lateral input at hover

;::~,"''""''~[~ ~~

-IO l.O omc~J [<"'"II.<[ W.O 211.U

~~~F==I

========d

!.0 omega [r"J<II~] W.O 20.0 - - - lr.uL,fcr functioo1

Figure 22: Verification result: transfer function of mea-sured and simulated roll rate at hover, unnoticeable dynamic effects

i I

~

~lUI

Figure 23: Verification result: transfer function of mea-sured and simulated pitch rate due to longitudinal in-put at 40 kts

nitude is one and the phase angle is zero. Deviations are caused by differences between model and flight test. Using the frequency domain format, boundaries \Vere used as defined in [17], symbolizing acceptable errors a pilot would not notice in a simulation.

Fig. 22 shows the transfer function of measured and simulated roll rate at hover for a lateral sweep input. It

is the same data shown in Fig. 18 which was not used

~,"-

":r========::;;;;;=:g

·W \,;;, : , , -00-,,-;:,,-,,,,-,,,,----,,,-,,-:-, _ _ _ j211.0

~~t~ ~~

l.O 10.0 200 - - - trJll.'fcr futleti'~'

Figure 24: Verification result: transfer function of mea-sured and simulated pitch rate at 40 kts, unnoticeable dynamic effects -25;·---·30] . . . ----~-·351 ... ;:;~titudc ~~

r ...

-51)1 -55: -611~--- ... ----··--45 !.(!_ ______ _

~~-,.

-45 j pi'"'c 'XI! l'l

.mJ

-1~01 ·225 ! __ , !.() - - - simubtcd I

- j

2()_()

Figure 25: Verification result: transfer function of mea-sured and simulated yaw rate due to pedal input at 80 kts w r - - - •

;::~t"'"":t~ ~q

-W Ll.l,-1 ---,m-,,-.,c-[r;c-,,c-~l---~~~~I.H---~21W

r,-~f

_d

I.U omega [rmll>] 111.0 20.0

Figure 26: Verification result: transfer function of mea-sured and simulated yaw rate at 80 kts, unnoticeable dynamic effects

for identification. In addition the shaded area of the so-called unnoticeable dynamic effects is shown. It can be seen that the magnitude of the transfer function stays within the boundary. The phase is out of the boundary at higher frequencies. This effect can also be seen in Fig. 21. The match of the transfer functions in this frequency region is not so good, either.

Fig. 23 and Fig. 24 show the results for the 40 kts case corresponding to the results shown in Fig. 20, Fig. 25 and Fig. 26 show the results for the 80 kts case corre-sponding to the results shown in Fig. 21. It can be seen

that the results for higher velocities are better than for the hover case. The match of the transfer functions is good and the results stay always within the area of the unnoticeable dynamic effects.

8

CONCLUSIONS

Based on the experience in the field of helicopter model development and system identification at the DLR In-stitute of Flight Mechanics, a nonlinear modular struc-tured helicopter model was developed. The goal was to combine the good simulation fidelity of pure identified parametric models with the advantages of generically derived nonlinear model formulations in particular the wide speed range in which the nonlinear model is valid. To improve the overall performance, parametric exten-sions were introduced to the model. The simulation fi-delity of the model was successfully optimized using system identification methods.

The model was extracted from information in flight test data of three different velocities and four different control inputs at each velocity. It was successfully vali-dated with data that was not used in the identification process.

The model shows a very good overall performance at all considered flight tests for the velocities, rates, atti-tude angles and engine parameters although the crite-ria given in the Advisory Circular AC 120-63 could not always be fullfilled. The verification simulations show good results, too. Investigations performed in the fre-quency domain showed that a pilot would not feel a difference between model and real helicopter for the considered verification cases.

It could be shown t.hat the method of parameter identification is well suited to determine parameters in very complex nonlinear and highly coupled dynamic systems.

A great part of flight simulator qualification is the comparison of flight test data with simulated data in the time domain where the simualtion has to stay within certain boundaries. Because of the nature of the parameter identification method, to achieve the best mathematical fit between measured and simulated data, it is best suited to do this work.

References

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Following Control System, American Helicopter Society 45'" Annual Forum, May 1989 and Ver-t.ica, Vol. 13, No 3, 1989

[2] von Griinhagen, W., Bouwer, G., Pausder, H.-J., Henschel, F., Kaletka, J.: A High Bandwidth Con-trol System for a Helicopter In Flight Simulator, International Journal of Control, Vol. 59, No. 1, 1994

[3] von Griinhagen, W.: Nonlinear Helicopter Model

Validation Applied to Realtime Simulations, The American Helicopter Society Aeromechanics Spe-cialist Conference 1995, Fairfield County, CT, Oc-tober 1995

[4] Hamers, M., von Griinhagen, W.: Nonlinear Heli-copter Model Validation Applied to Realtime

Sim-ulations, American Helicopter Society 53rd Annual Forum, Virginia Beach, Virginia, April 1997 [5] Jategaonkar, R., Monnich, W.: Identification Of

D0328 Aerodynamic Database For A Level D Flight Simulator, AIAA Modeling and Simulation Technologies Conference and Exhibit, New Or-leans, LA, 1997

[6]

N.N.: Airplane Flight Simulator Evaluation

Hand-book, The Ro:yal Aeronautical Society, London, February 1995

[7] Fu, K.-H., Kaletka, J.: Frequency-Domain Identi-fication of EO 105 Derivative Models with Rotor Degrees of Freedom, 16'" European Rotorcraft Fo-rum, Glasgow, UK, 1990 and Journal of the Amer-ican Helicopter Society, Volume 38, No 1, January 1993

[8] Fletcher, J. W.: Identification of UH-60 Stability

Derivative Models in Hover from Flight Test Data,

American Helicopter Society 49th Annual Forum, May 1993

[9] Tischler, M. B.: System Identification Methods for Aircraft Flight Control Development and Valida-tion, NASA TM110369, Army TR-95-A-007, Oc-tober 1995

[10]

[11]

Rohlfs, M.: Identification of Nonlinear Derivative Models from EO 105 Flight Test Data, 22"d Euro-pean Rotorcraft Forum, Brighton, November 1996 Rohlfs, M., von Griinhagen, W., Kaletka, J.: Non-linear Rotorcraft Modeling and Identification, Sys-tem Concepts and Integration Panel, 'SysSys-tem Identification for Integrated Aircraft Development and Flight Testing', Paper 23, Madrid, May 1998 [12] Hamel, P. G., Jategaonkar, R .. V.: The Evolution

of Flight Vehicle System Identification, AGARD Stuctures and Materials Panel Specialists' Meet-ing on ADVANCED AEROELASTIC TESTING AND DATA ANALYSIS, Rotterdam, The Nether-lands, May 1995

[13] Pitt D. M., Peters D. A.: Theoretical prediction of

(14] Curtiss Jr., H. C., Keller, J. D., Kothrnann,

B. D.: On Aerodynamic Modelling for Rotorcraft

Flight Dynamics, 22nd European Rotorcraft Fo-rum, Brighton, November 1996

(15] N.N.: Advisory Circular· AC 120-63, U.S

Depart-ment of Transportation, Federal Aviation

Admin-istration, October 1994

(16] Hamel, P. G., Kaletka,

J.:

Advances in mtorcraft

system identification, AGARD CP 552, 1995

(17] Hodgkinson, J., Wood, J. R.: Equivalent System Criteria for the Flying Qualities MIL Standard, in Fuller, S. G., Potts, D. W. Design Criteria for the Future of Flight Controls. Proceedings of the Flight Dynamics Laboratory Flying Qualities and Flight Controls Symposium, 2. - 5. March 1982. Technical Report AFWAL- TR-82-3064

(18] Hamel, P. G. (editor): Parameter identification, AGARD LS-104, 1991

(19] Hamel, P. G. (editor): Rotorcraft system

identifi-cation, AGARD AR-280, 1991

[20] Hamel, P. G. (editor): Rotorcraft system

identifi-cation, AGARD LS-178, 1991

(21] Hamel, P. G., Kaletka, J.: Advances in rotorcmft

system identification, Progress in Aerospace

Sci-ences, Vol.33, 1997

(22] von Griinhagen, W., Bouwer, G., Pausder, H.-.J., Henschel, F., Kaletka, .J.: A high bandwith con-trol system for the helicopter in-flight simulator ATTHeS-modelling, performance and applications in Advances in Aircraft Flight Control, Taylor &

Francis, 1996, pp 73-101