STABILITY HILLER STABILIZER BAR

7/28/2019 STABILITY HILLER STABILIZER BAR

1/11

DYNAMIC MODELING AND STABILITY

ANALYSIS OF MODEL-SCALE HELICOPTERS

WITH BELL-HILLER STABILIZING BAR

R. Cunha and C. Silvestre

Instituto Superior Tecnico,

Institute for Systems and Robotics,

Lisbon, Portugal.

This paper presents an accurate self-contained helicopter dynamic model, derivedfrom first-principles, and that is specially tailored for model-scale helicopters. Thesimulation model includes the rigid body, main rotor flapping, and Bell-Hiller stabi-lizing bar dynamics. Particular emphasis is placed on the analysis of the stabilizingbar and on the evaluation of its impact on the overall helicopter dynamics. Themodel is parameterized for the Vario X-Treme acrobatic helicopter, and solutions

for a set of trimming trajectories are identified and discussed. Different simplifica-tions, needed to derive models for control system design, are presented, compared,and their influence on the resultant dynamics evaluated. The effect of changing thephysical parameters of the stabilizing bar is also assessed. An LQ state feedbackcontroller is synthesized to stabilize the vehicle in forward flight. Simulation resultsobtained with the full nonlinear dynamic model and the forward flight control systemare presented and discussed.

Introduction

Over the last few years, model-scale unmanned he-licopters have become a major topic of research. The

wide and valuable range of applications due to thehelicopter high maneuvering capabilities, togetherwith the recent availability of increasingly accurate,reliable and miniaturized sensors justify the exten-sive research and development effort undertaken bynumerous groups. Dynamic modeling of helicoptersplays an essential role in this effort. Without accu-rate and reliable models, the design of guidance andcontrol systems for autonomous flight is seriously un-dermined.

The main contribution of this paper is the presen-tation of a simple yet accurate self-contained heli-copter dynamic model, whose derivation was based

on theoretical principles, and that has the particularPh.D. candidate, Department of Electrical Engineering

and Computer Science, and Institute for Systems and Robotics(ISR), Instituto Superior Tecnico, Av. Rovisco Pais, 1, 1046-001, Lisboa, Portugal. [email protected]

Assistant Professor, Department of Electrical Engineeringand Computer Science, and Institute for Systems and Robotics(ISR), Instituto Superior Tecnico, Av. Rovisco Pais, 1, 1046-001, Lisboa, Portugal. [email protected]

feature of including a detailed description of the Bell-Hiller stabilizing bar. Its equations of motion, whichexplicitly include the inertial, gyroscopic and aero-dynamic effects, are incorporated in the helicoptermodel, taking into account the geometry of the Bell-Hiller mixing device. An analysis and explanationof the Bell-Hiller system stabilizing effect is devel-oped, based both on the analytical expressions andon linear models obtained by linearization of the fullnonlinear model about specific trimming conditions.The impact of changing the physical parameters ofthe stabilizing bar is also assessed.

As a result of the work presented hereafter, a he-licopter dynamic model simulator, named SimMod-Heli,5 implemented in Matlab, using Simulink and CMEX-file S-functions, will be made freely available

for the scientific community. This simulator is com-pletely parameterizable and describes the dynamicsof helicopters with any number of blades, with orwithout Hiller or Bell-Hiller stabilizing bar. The sim-ulation model includes the rigid body, main rotorflapping, and stabilizing bar dynamics.

Several reference books on the theory of heli-copter flight dynamics can be found in the litera-ture. The reader is referred to Padfield11, Johnson7,

1

American Institute of Aeronautics and Astronautics

AIAA Guidance Navigation and Control Conference, Texas, USA, 2003


2/11

Bramwell et al.2, and Prouty12, for a thorough andextensive coverage of full-scale helicopters aerody-namic and dynamic modeling. However, none ofthem covers the modeling of the Bell-Hiller system.Only Bramwell et al.2 give a very brief and simplifiedexplanation of the Bell stabilizing device that resem-bles the Bell-Hiller stabilizing bar, but does not use

aerodynamic paddles nor does it include a mixing de-vice. In addition, the analysis presented covers onlythe longitudinal plane dynamics.

Although there are a few publications that ex-plicitly address the topic of model-scale unmannedhelicopter modeling, including the Bell-Hiller system(see, for example, 6,10 and references therein), theseare mostly based on a black box system identificationapproach. Results from linear frequency identifica-tion and time-domain analysis of flight-data are usedto develop the model. In general, the obtained mod-els are restricted to particular flight regimes, suchas, hover or forward flight, and use very crude ap-proximations to describe the fundamental effects onhelicopter behavior. Only Kim et al.9 present amodeling approach based on first-principles and ba-sic aerodynamics, however it is not clear how is thestabilizing device included in the overall helicopterdynamics. To the best of the authors knowledge, nostability analysis of the Bell-Hiller system effect onhelicopter dynamics has been presented so far.

The paper is organized as follows. The first sec-tion introduces a general model for the dynamicsof helicopters, presents the main rotor flapping dy-namics, and describes in detail the modeling of the

Bell-Hiller stabilizing bar dynamics. The second sec-tion presents the trimming results obtained for theVario X-Treme model-scale helicopter followed by astability analysis of the stabilizing bar impact on thehelicopter dynamics. The third section focuses onthe design and implementation of a forward flightcontrol system for the Vario X-Treme helicopter, andpresents the simulation results obtained with the fullnonlinear dynamic model. The last section sum-marizes the contents of the paper and points outdirections for future work.

Helicopter dynamic model

This section presents the dynamic model of a sin-gle main rotor and tail rotor helicopter equipped witha Bell-Hiller stabilizing bar, as the one depicted inFig. 1. A comprehensive study of the helicopter dy-namic model can be found in 3. For in depth coverageof helicopter flight dynamics, the reader is referredto Johnson7 and Padfield11.

Fig. 1 Vario X-Treme helicopter

The dynamics of the helicopter can be describedusing a 6 DoF rigid body model driven by forcesand moments that explicitly include the effects ofthe main rotor, Bell-Hiller stabilizing bar, tail rotor,fuselage, horizontal tailplane, and vertical fin.

To derive the equations of motion, the following

notation is required:{U} - universal coordinate frame;

{CM} - body-fixed coordinate frame, with ori-gin at the vehicles centre of mass;

p =

x y zT

- position of the vehicles centerof mass, expressed in {U};

=

T

- Z-Y-X Euler angles thatparametrize locally the orientation of the vehiclerelative to {U};

v = u v w T

- body-fixed linear velocity vec-

tor;

=

p q rT

- body-fixed angular velocity vec-tor.

Fig. 2 captures the general structure of the heli-copter model that can be written as

v = v + [f(v,,u) + fg ()] /m = I1 ( I) + I1n (v,,u)p = R ()v

= Q ()

, (1)

where m is the vehicle mass, I is the tensor of inertia

about the {CM} frame, u is the command vector, fand n are the vectors of external forces and momentsrespectively along the same frame, fg is the gravita-tional force also expressed in {CM}, R is the rotationmatrix from {CM} to {U}, and Q is the transforma-tion from angular rates to Euler angle derivatives.

The command vector u =

0 1c 1s 0tT

com-prises the main rotor collective input 0, main rotor

2




3/11

Fig. 2 Helicopter model - block diagram

and flybar cyclic inputs 1c and 1s, and tail rotor col-lective input 0t. The total force and moment vectorsaccount for the contributions of all helicopter com-ponents, and can be decomposed as

f= fmr + ftr + ffus + ftp + ffnn=nmr +ntr +nfus +ntp +nfn

, (2)

where subscript mr stands for main rotor, tr for tailrotor, f us for fuselage, tp for horizontal taiplane, andfn for vertical fin.

As the primary source of lift, propulsion and con-trol, the main rotor dominates helicopter dynamicbehaviour. The Bell-Hiller stabilizing bar improvesthe stability characteristics of the helicopter. Thetail rotor, located at the tail boom, provides the mo-ment needed to counteract the torque generated bythe aerodynamic drag forces at the rotor hub. Theremaining components have less significant contri-butions and simpler models as well. In short, thefuselage produces drag forces and moments and thehorizontal tailplane and vertical fin act as wings inforward flight, increasing flight efficiency.

The following sections present mathematical mod-

els for the main rotor and Bell-Hiller stabilizing bar.Detailed models for the remaining components canbe found in Padfield11 and Prouty12.

Main rotor

In rotary-wing aircraft, the main rotor is not onlythe dominant system, but also the most complexmechanism. It is the primary source of lift, whichcounteracts the body weight and sustains the heli-copter on air. Additionally, the main rotor generatesother forces and moments that enable the control ofthe aircraft position, orientation and velocity. This

section presents a simplified rotor dynamic model,whose main building blocks are depicted in Fig. 3.

The main rotor actuation system consists in chang-ing the blade pitch angle , along each blade revolu-tion. Through this variation, control over the bladeaerodynamic loads, which ultimately determines themain rotor force and moment contributions (fmr andnmr), is attained. Without the Bell-Hiller system

Fig. 3 Main rotor block diagram

and neglecting the servo actuators dynamics, theblade pitch angle is given by

() = 0 + 1c cos() + 1s sin(). (3)

where = t is the blade azimuth angle and is the rotor speed. In systems equipped with theBell-Hiller stabilizing bar, only the collective input0 is directly applied to the main rotor. The cyclicinputs are mixed with the motion of the bar to de-

termine the actual cyclic components (1c and 1s)applied to blade pitch links. The equations govern-ing the motion of these components (represented bythe Bell-Hiller block in Fig. 3) are described in thenext section.

The aerodynamic forces, generated at the surfaceof the rotating blades, produce the rotor thrust thatis responsible for sustaining and propelling the heli-copter. In reaction to this thrust force, by conserva-tion of momentum, the air is accelerated downwards.The generated flowfield can be approximated by themean component perpendicular to the rotor disk,usually called induced downwash, see Fig. 3. The

present model uses momentum theory combined withblade element theory7 to compute the induced down-wash, in the normal working state (climb, hover andlow rates of descent) and windmill brake state (highrate of descent). In the vortex ring and turbulentwake states, where momentum theoty is not applica-ble, empirical approximations are used.4

Rotor blade loads are not uniquely determined bythe applied inputs. They also depend on helicoptervelocities, induced downwash velocity, and on themotion of the blades themselves, which besides rota-tion about the hub and blade pitch angle variation,also includes flap and lag bending and pitch torsional

motions.7 The model adopted to describe rotorblades is standard and assumes that these are rigidand linked to the hub through flap hinge springs,with stiffness k.

11 The dynamic behaviour is thusconfined to the flapping motion (Fig. 3), describedby vector =

0 1c 1s

T, where 0 denotes the

collective mode (also denominated coning), and 1cand 1s the longitudinal and lateral cyclic modes,respectively. This vector, which corresponds to the

3




4/11

constant and first-order harmonics of the Fourier Se-ries expansion of (), comprises the fundamentalcomponents of the flapping motion.

The flapping equation of motion for a single bladeis obtained from the moment equilibrium expressedon a frame attached to the rotating and flappingblade, mounted on the moving helicopter. To derive

the aerodynamic contributions to this equilibrium,expressions for the air velocity at each blade sec-tion were established. Using standard notation inhelicopter theory, the helicopter velocities are nor-malized and expressed in the main rotor wind frame,with and z denoting the forward and vertical ve-locities, respectively, and p and q the roll and pitchrates, respectively.11 The induced downwash is alsonormalized and decomposed into constant 0 and si-nusoidal components 1c and 1s.

The resulting equation of motion, expressed in thewind-aligned frame, can be expanded in Fourier Se-ries and the constant and first-order harmonic termsextracted to obtain the following second-order sys-tem of differential equations

+ A() + 2A() = 2B()

01c

1s

+

2B()

pq

+ 2B()

z 01c

1s

.

(4)It should be noted that, for control system design

purposes, the flapping motion as described by (4)

preserves a high degree of accuracy, while renderinga much more tractable system. For instance, thecoefficient matrices in (4) depend solely on the heli-copter forward velocity.4

In steady-state, equation (4) reduces to

= [A()]1

B() 01c

1s

+ B()

pq

+ B()

z 01c

1s

,

(5)with coefficient matrices given by

[A()]1 = 1S2 + 1

4

4

S2+1

4

4

8S+1

0 0

43 S

8S+1

8S

8

1 +

2

2

43 1

2

28S+1

8

1

2

2

8S

.

(6)

B() =

8

1 + 2 0 43

0 1 + 2 083 0 1 +

32

2

, (7)

B() =

8

23

0

28

8 2

, (8)

and

B() =

8

43 0 23 0 1 0

2 0 1

. (9)

The Lock number and Stiffness number S arestructural adimensional constants that give the ra-tio of aerodynamic to inertial forces, and the ratioof stiffness to aerodynamic moments, respectively.They embed all the physical characteristics of therotor relevant for an accurate description of its flap-ping behaviour.

The major effects in flapping motion become moreperceptible if the conditions on helicopter motion arefurther constrained to include no forward velocity atthe hub ( = 0). Then, expressions (6) and (7) showthat 0 commands the coning mode 0 and that thecyclic modes 1c and 1s are determined by an S-dependent combination of the cyclic inputs.

In the case of articulated rotors (k = 0 and thusS = 0), with no forward velocity at the hub ( = 0),A becomes the identity matrix and expression (5)reduces to

0 =

8

0 +

43 z

43 0

1c = 1s p + 16 q+ 1s

1s = 1c +16 q+ q 1c

. (10)

The cyclic modes are now fully decoupled and cor-respond to the solution of a second order dynamicsystem excited at the resonant frequency (maximummagnitude amplification, 90o input-output phaseshift).

Using either the dynamic or the steady-state so-lution for the flapping, the main rotor forces andmoments at the hub (last block in Fig. 3) can bewritten as

fmr = n2

Y1sY1c2Z0

+ n2

Z1c Z0 0Z1s 0 Z00 0 0

01c1s

,(11)

and

nmr = n

00

N0

+ n

2

N1c N0 kN1s k N0

0 0 0

01c

1s

.(12)

4




5/11

The Y(.), Z(.), and N(.) terms, in (11) and (12),represent the force and moment components gener-ated by the blades. These quantities are functions ofthe helicopter state variables and main rotor inputs(see 3, for further details).

The main rotor thrust and torque, Z0 and N0 re-spectively, have dominant out-of-plane components

(along the hub z axis), and smaller in-plane compo-nents, which are due to the main rotor tilt. TermsZ1c0 and Z1s0 represent the in-plane contribu-tions of the blade lift forces due to the rotor coning,while Y1c and Y1s account for the in-plane contribu-tions of the drag forces acting on the blades. In (12),the spring moments, due to the cyclic flap angles, areexplicitly given by k1s for the roll moment andk1c for the pitch moment.

Bell-Hiller stabilizing bar

The Bell-Hiller stabilizing bar, a mechanical blade

pitch control system that improves helicopter stabil-ity, is currently a standard component of model-scalehelicopters. From a control point of view, the stabi-lizing bar can be interpreted as a dynamic feedbackof the roll and pitch rates. The system consists of aso-called flybar, a teetering rotor placed at a 90o ro-tation interval from the main rotor blades and tippedon both ends by aerodynamic paddles, see Fig. 4.

Paddle

Flybar

Paddle

Fig. 4 Bell-Hiller stabilizing bar

With this mechanism, the blade pitch cyclic com-

mands do not go directly from the swashplate to theblade pitch links. Instead, the cyclic commands areapplied to the flybar whose flapping motion deter-mines the blade pitch angles. The system derivesfrom a combination of the Bell stabilizing bar, fit-ted with a mechanical damper and weights at eachtip, and the Hiller stabilizing bar, which instead ofweights uses small airfoils with incidence commandedby the cyclic inputs.2 In the Hiller system, the blade

pitch angle is determined by the flybar flapping only.The Bell-Hiller system introduces the mixing devicethat allows some of the swashplate input to be di-rectly applied to the blades.

The flybar and main rotor flapping motions aregoverned by the same effects, namely the gyroscopicmoments due the helicopter roll and pitch rates. But

unlike the main rotor, the flybar is not responsiblefor providing lift or maneuvering ability. Thus, it canbe designed to have a slower response and providethe desired stabilization effect. The flybar responsecan be optimized by varying the ratio of aerody-namic to inertial loads on the paddles. Changingthe shape, weight or distance between the paddlesare all straightforward ways of tailoring the system.

The notation used to describe the Bell-Hiller sys-tem is presented in Fig. 5.

Fig. 5 Bell-Hiller system with angular displace-ments

Due to the Bell-Hiller system, the flybar flappingand blade pitching angles are physically constrainedto satisfy

0 ()1 () =l

l

l1

l1 + l2 c1

(l4 h ()) /l1 () +

lfl

l2l1 + l2 c2

0

f

+ 2 , (13)

where 1 is the differential pitch input, given by

1() = 1c cos() + 1s sin(). (14)

5




6/11


7/11


8/11

flybar. Fig. 9 shows evolution of eigenvalues forthe X-Treme helicopter linearized dynamic model asa function of forward speed, and identifies the as-sociated modes that characterize helicopter motionunder the small perturbation assumption.

80 60 40 20 040

30

20

10

0

10

20

30

40

Re(s)

Im(s)

hover8 m/s

10 5 06

4

2

0

2

4

6

roll subsidence

pitch subsidence

Dutch roll

phugoid

heave and yawsubsidences

Fig. 9 Eigenvalues for X-Treme helicopter modellinearized at different forward velocities

To assess, in terms of stability, the impact of theBell-Hiller system on the overall helicopter dynamicmodel, the following analysis focuses on the unstablephugoid mode and on the comparison between highand reduced order models. As shown in Fig. 10,the introduction of the flybar has a stabilizing ef-fect on the phugoid mode, whose damping and fre-quency factors change from (0.50; 1.30 rad/s) to

(0.61; 0.89 rad/s). These eigenvalues are identi-cal, considering either the first or the second-orderflybar flapping dynamics (or main rotor blade pitch-ing dynamics). In fact, the first-order system, whichassumes that the inertial terms in (16) are negligi-ble, constitutes an adequate approximation, since theonly mode that is being neglected is fast (322 rad/s)and does not couple with the fuselage body motions.The same is not true for the slower flybar flappingmode, which couples with the roll and pitch subsi-dence modes (see Fig. 10).

A standard approach followed in helicopter sta-

bility analysis assumes that the coupling betweenlateral and longitudinal modes can be neglected,and that the helicopter motion can be describedby two independent lower-order systems. In heli-copters fitted with the Bell-Hiller flybar, this par-titioning proves to be inadequate. An unstable 8th-order model may correspond to two stable reduced-order models. This is due to the fact that lat-eral/longitudinal couplings increase the instability of

80 60 40 20 0400

300

200

100

0

100

200

300

400

Re(s)

Im(s)

flybarlessflybar (1st order)flybar (2nd order)

2 1 0 1

1

0

1

Fig. 10 Eigenvalues for systems with and withoutBell-Hiller stabilizing bar (hover)

the long-period oscillatory modes, see Fig.11.

3 2 1 0 11

0.5

0

0.5

1

1.5

Re(s)

Im(s)

6DoF (no flybar)6DoF (flybar)Long.+Lat. (no flybar)Long.+Lat. (flybar)

Fig. 11 Comparison between 6DoF, longitudinaland lateral models (hover)

Another relevant issue, addressed in this analysis,consists in quantifying the impact of the physicalparameters of the Bell-Hiller system on the over-all helicopter stability. The obtained results are inagreement with expressions for the blade pitchingdynamics given in (16)-(21), confirming that an in-crease in paddle weight mf improves stability, whileincreasing the flybar radius R2 or the lever arm l1have the opposite effect, see Fig. 12.

From the conducted analysis, it can be concludedthat, for an adequate description of the Bell-Hillerstabilizing effect, the model should include both thefirst-order blade pitching dynamics and the couplingsbetween longitudinal and lateral modes.

8




9/11

0.4 0.45 0.5 0.55 0.6 0.650.6

0.65

0.7

0.75

0.8

Re(s)

Im(s)

l1: 0.018> 0.030 (m)

R2: 0.365> 0.620 (m)

mf: 0.029> 0.045 (Kg)

(l1

= 0.030 m)

(R2

= 0.0620 m)

(mf

= 0.045 Kg)

Fig. 12 Variation of Bell-Hiller stabilizing barparameters (hover)

Control system design and simulation

This section focuses on the design, implementa-tion and simulation of a forward flight control sys-tem for the Vario X-Treme helicopter. The lin-ear state feedback controller was required to meetthe following design specifications: i) Zero SteadyState Error, achieve zero steady state error in re-sponse to constant input commands in the vectore = [zc z, c , uc u, vc v]

, four ex-tra integrators were added, one to each channel ine; ii) Actuator Bandwidth Requirements, the controlloop bandwidth for all actuators should not exceed30 rad/s to ensure that the main and tail rotor com-mand servos are not driven beyond their normalactuation bandwidth.

The forward flight controller was obtained by re-sorting to the solution of the standard continuoustime Linear Quadratic Regulator problem,1 wherethe state and control weighting matrices Q and R,respectively, were selected as to achieve a reasonabletracking performance for the channels in e withoutviolating the actuator bandwidth requirements.

KTz

z1z1Tz

iEE E

ic

T

E

'

EE

E

+

+

x

eu

Fig. 13 Controller implementation with an anti-windup mechanism.

The controller was discretized using a samplingfrequency of 50 Hz and the actuators were saturatedat 8o to avoid blade stall. The implementation of the

resulting discrete time controller, was done by usingthe D-methodology,8 which guarantees the followingfundamental linearization property: the linearizationof the nonlinear feedback control system about eachequilibrium trajectory preserves the internal as wellas the input-output properties of the correspondinglinear closed loop designs. This methodology moves

all integrators to the plant input, and adds derivatorswhere they are needed to preserve the transfer func-tions, making straightforward the implementation ofanti-windup schemes, see Fig. 13. Furthermore, theinput trimming values are naturally provided by theintegrator block, which is a major issue in this ap-plication where the constant terms present in modelhave to be compensated. In the figure, e representsthe state variables that are required to achieve goodtracking performance in steady state, vector x thehelicopter state variables including the main rotorblade pitching, and u = [0, 1s, 1c, 0t]

the heli-copter actuation vector.

The results of the simulation, presented in Figs. 14through 20, were obtained with the full nonlinearclosed loop system that includes the nonlinear dy-namic model of the Vario X-Treme helicopter andthe D implementation of the controller. The maneu-ver was performed about a horizontal forward flighttrimming condition (Vc = 2 m/s, c = 0, c = 0) andcan be divided in five stages: i) keep the helicopterin level flight during one second; ii) track a positiveramp in altitude, Fig. 15; iii) keep the new altitudefor three seconds; iv) track an increase in forwardvelocity, Fig. 14; v) maintain the helicopter at the

new equilibrium (Vc = 2.6 m/s, c = 0,

c = 0).

0 2 4 6 8 10 121.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

Time (s)

Forwardvelocity(m)

vc

v

Fig. 14 Forward velocity vc and v

9




10/11

0 2 4 6 8 10 12

2

1

0

Time (s)

Altitud

e(m)

zc

z

Fig. 15 Altitude zc and z

0 2 4 6 8 10 122.5

3

3.5

4

4.5

5

Time (s)

Collectivepitchcommands(deg) 0

0t

Fig. 16 Collective actuation commands

0 2 4 6 8 10 120.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Cyclicpitchcommands(deg)

1c

1s

Fig. 17 Cyclic actuation commands

Between the first and third seconds of the maneu-ver, the actuation variable 0, that corresponds tothe main rotor collective, increases to impart thedesired ascending rate to the vehicle, Fig. 16. Theremaining actuation variables, the longitudinal andlateral cyclics,

1c,

1s respectively, and the collective

tail rotor, 0t, react as to compensate for the modelcoupling. In the third stage of the maneuver, the ac-tuation acquires the initial trimming values, requiredto keep the vehicle in the commanded altitude.

0 2 4 6 8 10 121.2

1

0.8

0.6

0.4

0.2

0

0.2

Time (s)

Linearv

elocities(m/s)

v

w

Fig. 18 Linear velocities

0 2 4 6 8 10 120.06

0.04

0.02

0

0.02

0.04

0.06

Time (s)

Angularvelocities(rad/s)

p

q

r

Fig. 19 Angular velocities

In the fourth stage, the controller responds to thecommanded increase in forward velocity by loweringand raising 1s, which has the effect of tilting the he-licopter forward (decrease in pitch angle , Fig. 20)and redirecting the thrust vector to increase forwardspeed. As the vehicle enters the final stage of the

10




11/11

maneuver, the reverse action takes place and the ac-tuation acquires the trimming values associated withthe new equilibrium condition.

0 2 4 6 8 10 123

2

1

0

1

2

3

Time (s)

Eulerangles(deg)

Fig. 20 Euler angles

Conclusions

The paper presented an accurate self-contained he-licopter dynamic model, derived from first-principles,and that was specially tailored for model-scale heli-copters. Particular emphasis was placed on the anal-ysis of the stabilizing bar and on the evaluation ofits influence on the overall helicopter dynamics. Thetrimming trajectories were identified and discussed.Different approximations, needed to derive simplifiedmodels for control system design, were presented,compared, and their impact on the resultant dynam-

ics evaluated. The model was parameterized for thecase of the Vario X-Treme acrobatic helicopter, andthe effect of changing the physical parameters of thestabilizing bar on the helicopter dynamics was as-sessed. An LQ state feedback controller was designedand implemented, and its performance evaluated insimulation along a typical maneuver. Future workwill focus on adjusting and validating the model sothat it can be used to exploit the particular dy-namic characteristics of model-scale helicopter in itswhole flight envelope. Extra effort will be placed onstudying, developing, and testing advanced controlstrategies to achieve good performance characteris-

tics in highly demanding maneuvers.

Acknowledgements

This work was supported by the Portuguese FCTPOSI programme under framework QCA III and bythe POSI/SRI/41938/2001 ALTICOPTER project.

The work of R. Cunha was supported by a PhDStudent Scholarship, SFRH/BD/5034/2001, fromthe Portuguese FCT POCTI programme.

References

1B. D. O. Anderson, J. B. Moore, Optimal Con-trol, Linear Quadratic Methods, Prentice Hall, NewJersey, 1990.

2A. R. S. Bramwell, G. Done, D. Balm-ford, Bramwells Helicopter Dynamics, 2nd Edition,

Butterworth-Heinemann, Oxford, Great Britain,2001.

3R. Cunha, Modeling and control of an au-tonomous robotic helicopter, MSc thesis, Depart-ment of Electrical and Computer Engineering, Insti-tuto Superior Tecnico, Portugal, 2002, in english.

4R. Cunha, C. Silvestre, Modeling and simu-lation of model-scale helicopters, Internal Report,Institute for Systems and Robotics, Portugal, 2003.

5R. Cunha, C. Silvestre, SimModHeli: A Dy-namic Simulator for Model-Scale Helicopters, 11th

Mediterranean Conference on Control and Automa-tion MED03, Rhodes, Greece, June 2003.

6V. Gavrilets, B. Mettler, E. Feron, Nonlin-ear Model for a Small-Size Acrobatic Helicopter,Proc. of the AIAA Guidance, Navigation, and Con-trol Conference, Montreal, Quebec, Canada, August2001.

7W. Johnson, Helicopter Theory, Dover Publica-tions, New York, USA, 1994.

8I. Kaminer, A. Pascoal, P. Khargonekar, E. Cole-man, A Velocity Algorithm for the Implementa-

tion of Gain-Scheduled Controllers, Automatica,31(8):11851191, 1995.

9S. K. Kim, D. M. Tilbury. MathematicalModeling and Experimental Identification of an Un-manned Helicopter with Flybar Dynamics, Submit-ted to the Journal of Robotic Systems, 2001.

10B. Mettler, M. Tischler, T. Kanade, SystemIdentification Modeling of a Model-Scale Helicopter.Technical report CMU-RI-TR-00-03, Robotics In-stitute, Carnegie Mellon University, Pittsburg, PA,USA, January 2000.

11G. D. Padfield, Helicopter Flight Dynamics:

The Theory and Application of Flying Qualities andSimulation Modeling, AIAA Education Series, Wash-ington, USA, 1996.

12R. W. Prouty, Helicopter Performance, Sta-bility, and Control, Krieger Publishing Company,Florida, USA, 1995.

11



Documents

STABILITY HILLER STABILIZER BAR