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modeling coax helicopter RC
uar
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egy
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o air vehicles (MAV) for tasksarch and rescue or inspectionmanythe foheir poxamplCoaXSchafrhe Unines,
mainly on full scale helicopters, with various control techniques
adaptive neural networks (Leitner, Calise, & Prasad, 1997) andmany other methodologies. A strong interest exists in the
ationoverownthe
ica-
system identication techniques to model-scale helicopters exist
ARTICLE IN PRESS
Contents lists available at ScienceDirect
sev
Control Enginee
Control Engineering Practice 18 (2010) 700711able to cover all the existing effects. For example linearizationE-mail address: [email protected] (D. Schafroth).such as linear quadratic Gaussian control (Zhao & Murthy, 2009), and the results are, compared to full-scale helicopters, limited(Mettler, Kanade, & Tischler, 2000). Additionally, the models aremainly identied on the linearized state space system aroundhover. In general this gives good results but linear models are not
0967-0661/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.conengprac.2010.02.004
Corresponding author. Tel.: +41446328949; fax: +41446321181.choice. It is more likely that higher order, model based controllersare a better choice and more effective. There exist many works,
tion on the ground and on test benches, identication on realight data is needed. Only a few examples of the application ofalready very difcult, the micro helicopter additionally suffersfrom faster dynamics, inaccurate actuators and low output qualityof lightweight sensors. Nevertheless, a very tight feedback controlis necessary for the foreseen missions and it is questionable iftraditional control approaches like PID controllers are the right
DC motors. This model is based on the rigid body equincluding all the attacking forces and moments around hstate. An important part of the control design and often not shin literature is the verication of the used model andparameter identication. In addition to the parameter identEuropean Framework project muFly is to develop a fullyautonomous micro helicopter in the size and mass of a smallbird. A big challenge and key point in the design of anautonomous helicopter in such a size is the feedback control.While the control of a full size and normal size RC-helicopter is
multiple output (MIMO) H1- controllers for an accurate control ofthe helicopter.
In order to have an appropriate model for the control andsimulations, a nonlinear dynamic model for muFly is developed,including all the specialties like the stabilizer bar or the brushlessThe interest in autonomous micrsuch as surveillance and security, seand exploration is still growing, andEspecially micro helicopters are indue to their ability to hover and tExisting micro helicopters are for edeveloped by Epson (2004), the(Bermes, Leutenegger, Bouabdallah,and the MICOR developed by t(Bohorquez, Rankinsy, Baederz, & Pvehicles are developed.cus of the researcherswer to carry payload.e the muFR helicopter2 developed at ETHZoth, & Siegwart, 2008)iversity of Maryland2003). The goal of the
H1-loop shaping method due to its robustness and structureddesign method. In Walker (2003) a comprehensive overview onthe works on H1- controllers in the eld of helicopters during thelast 20 years is given, showing the complexity but also thestrength of the design method. A work on the control of a coaxialmicro helicopter is (Wang, Song, Nonami, Hirata, & Miyazawa,2006) where PID control is combined with H1- control techni-ques. The goal of this work is to use different multiple input1. Introduction fuzzy logic adaptive control (Wade & Walker, 1996), backsteppingand sliding mode control (Bouabdallah & Siegwart, 2005),Modeling, system identication and robcoaxial micro helicopter
D. Schafroth , C. Bermes, S. Bouabdallah, R. Siegw
Autonomous Systems Lab, Tannenstr. 3, ETH Zurich, 8092 Zurich, Switzerland
a r t i c l e i n f o
Article history:
Received 2 April 2009
Accepted 2 February 2010Available online 16 February 2010
Keywords:
Helicopter
Control
H-innity
Identication
Model
Mixed sensitivity problem
a b s t r a c t
In this paper the design pr
process starts with the d
elements of the helicopte
data from test benches an
Adaptation Evolution Strat
H1- controllers for attitud
journal homepage: www.elst control of a
t
ss of robust H1- control for a coaxial micro helicopter is presented. Thelopment of a nonlinear dynamic model reecting all the important
he corresponding system parameters are identied using measurement
al ight, as well as a nonlinear identication tool, the Covariance Matrix
(CMA-ES). The identied and veried model is then used for the design of
d heave control, which are successfully tested in ight on the real system.
& 2010 Elsevier Ltd. All rights reserved.
ier.com/locate/conengprac
ring Practice
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cancels cross couplings between the angular rates. Moreover it isdesirable to have an identied nonlinear model for accuratesimulations and controller evaluation, even though the controlleritself is linear. Another advantage of the nonlinear identication isthat operation points for hover do not have to be estimated. Inthis work a nonlinear parameter identication based on theCovariance Matrix Adaptation Evolution Strategy (CMA-ES)(Hansen & Ostermeier, 1996) is presented. This randomizedsearch algorithm allows for an accurate identication of systemswith high dimensions and multiple parameters.
The paper is organized as follows. In Section 2 the muFly microhelicopter and its hardware setup is shown, followed by thenonlinear model in Section 3. The identication process with theCMA-ES is presented in Section 4. Section 5 presents the controldesign for muFly.
same package as the sensor data. The same microprocessor is usedfor the control.
D. Schafroth et al. / Control Engineering Practice 18 (2010) 700711 7012. The muFly helicopter
muFly is a 17 cm in span, 15 cm in height coaxial helicopter(Fig. 1) with a mass of 95 g. The two rotors are driven by twolightweight brushless DC (BLDC) motors and are counter rotatingto compensate the resulting torque due to aerodynamical drag.This allows to control the yaw by differential speed variation ofthe two rotors, whereas the altitude can be controlled varying therotor speeds simultaneously. The motor speed is reduced by agear in order to achieve a higher torque on the rotor side.
A benet of the coaxial setup is that one rotor can be used tohelp stabilizing the helicopter using a stabilizer bar. Such devices,mounted on the upper rotor, are often found on RC-Models. Thehelicopter is steered by a conventional swash plate on the lowerrotor actuated by two servos and powered by a lithium polymerbattery. All the signals to the servos and motor controllers arepulse position modulated (PPM) signals.
Sensors mounted on the platform are an inertial measurementunit (IMU) and an ultrasonic distance sensor for the measurementof the distance to the ground. With this setup, the attitude anglesand the height over ground can be measured. Further, anomnidirectional camera in combination with lasers is in devel-opment, that will be used together with a down-looking camerafor position measurement. The sensor data is processed by a dsPICmicroprocessor and sent to the ground station by a serialconnection using a bluetooth module. In order to minimize timedelays for the identication, the actuator inputs are sent in theFig. 1. The second prototype of the muFly helicopter.3. Nonlinear model
The nonlinear model developed for the muFly project is aphysical model based on the rigid body motion being presented indetail in Schafroth, Bermes, Bouabdallah, and Siegwart (2009). Inthis work, an overview is given, pointing out the specialties of themuFly helicopter. The goal of the model is to be as simple aspossible, since it has to be used for the controller design. On theother hand, the physics and dynamics of the different componentsmounted on the muFly helicopter have to be reected accurately.
As common in aeronautics, an inertial J frame and a body xedframe B are introduced, resulting in the transformation equationfor the position, velocities, angles and angular rates. UsingNewtonian mechanics the differential equations for the rigidbody motion in the body-xed frame located at the helicopterscenter of gravity (CoG) are
_u
_v
_w
264
375 1
mf
p
q
r
264
375
u
v
w
264
375
and
_p
_q
_r
264
375 I1 m
p
q
r
264
375 I
p
q
r
264
375
0B@
1CA;
with the body velocities u, v, w, the system mass m, the angularvelocities p, q, r the body inertia tensor I and the total externalforce and moment vectors f andm. So far, the equations of motionare independent of the ying platform and can be found inliterature, e.g. (Mettler, 2003). Now the platform dependent totalexternal force f and moment m vectors have to be dened.
The forces and moments acting on muFly can be summarizedas
f tuptdwgwhub;
m qupqdwrCup tuprCdw tdwqgyro;dwqgyro;up;
with the terms: upper and lower rotor thrust vector tup and tdw,gravity vector g, aerodynamic fuselage drag whub and rotor dragtorques qup and qdw, gyroscopic torques of the rotors qgyro,dw,qgyro,up and the moments due to the cross product of the forcesnot aligned with the CoG. Aerodynamic forces and moments onthe fuselage due to translation in the air are neglected since thehelicopter will mainly operate around hover condition. The nextstep is to dene the single forces and moments.
The rotor thrust vector is dened as ti and rotor torque vectorqi as ti Ti nTi and qi Qi nQi, with iAfdw;upg for the lower andupper rotor. In hover, the thrust and torque magnitude Ti and Qi ofa rotor with radius R can be dened as (Bramwell, 2001)
Ti cTiprR4O2i cTikTO2i ;
Qi cQiprR5O2i cQikQO2i ;with air density r, the thrust and torque coefcient cT and cQ, andthe rotor speed Oi.
The thrust vectors can be described using two tilt angles ai andbi around the x- and y-axis Fig. 2 as
nTi cosaisinbi
sinaicosaicosbi
264
375:
The rotor torque vectors are assumed to act only in the rotoraxis (z-axis), but it has to be considered that the lower rotor turnsclockwise, while the upper rotor turns counterclockwise as seen
from above.
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[]
D. Schafroth et al. / Control Engineering Practice 18 (2010) 700711702The last torques result from the acceleration of the rotors. For
Fig. 2. Visualization of the tilted thrust vector with tilt angles a and b.
Fig. 3. The principle of the stabilizer bar. Due to the high inertia the stabilizer barlags behind the roll/pitch movement, applies a cyclic pitch input to the rotor and
creates a redress moment.the case of the lower rotor this might be negligible, but the highinertia of the stabilizer bar, mounted on the upper rotor, causes anotable torque while accelerating the rotor. The gyroscopic torquevectors are assumed to act only in the rotor axis direction with themagnitude
Qgyro;i Jdrive;i _Oi:Now the only missing part in the model is the dynamics of the
stabilizer bar, swash plate, servos and electro motors.An important part of the system model is the stabilizer bar. In
simple words this stabilization mechanism gives cyclic pitchinputs, similar to the swash plate, to the upper rotor to stabilizethe helicopter in ight. The stabilizer bar has a high inertia andlags behind a roll or pitch movement of the fuselage as shown inFig. 3. Through a rigid connection to the rotor, this time delayresults in a cyclic pitching of the rotor blades and therefore to atilting of the tip path plane (TPP) (Leishman, 2006). If thestabilizer bar is adjusted correctly, the thrust vector points inthe opposite direction of the roll or pitch movement causing aredress moment.
The stabilizer bar following the roll/pitch movement can bemodeled as a rst order element (Mettler, 2003) as
_Zbar 1
Tf ;upfZbar;
_zbar 1
Tf ;upyzbar;
with the angles between the rotor axis and the normal of thestabilizer bar plane Zbar and zbar, and the time constants Tf,up. Thetilt angles of the thrust vector in the body-xed frame are thedifferences between the two angles Zbar and zbar and the roll andpitch angles scaled by a factor lup, since the interest is in the tiltangle of the TPP and not that of the stabilizer bar. Thus theequations for the tilting angles for the thrust vector are
aup lupfZbar;
bup lupyzbar:The inuence of the modeled stabilizer bar is shown in Fig. 4,
where the reaction of the helicopter to an initial displacement inthe roll angle is plotted. After a short time period, the helicopter isback in the hover position.
The idea of the swash plate model is the same as for thestabilizer bar. The reaction from the servo input (PPM signal) tothe change in the TPP is also modeled by a rst order system.Hereby all the dynamics of the servos and rotor are covered by thetime constant Tf,dw. The tilting angles of the lower rotor aremodeled as
_adw 1
Tf ;dwldwuserv2 ySPmaxadw;
_bdw 1
Tf ;dwldwuserv1 ySPmaxbdw;
with the time constant Tf,dw, scaling factor ldw, maximal swashplate tilting angle ySPmax and servo inputs userv,i.
Since the motor-controllers for the BLDC motors do not have aspeed control and no speed measurement output, the electromotors have to be modeled as well. The equations for the rotorspeeds Oi are based on the well known electro motor equation(Mueller & Ponick, 2006) and extended by the gear resulting inthe nal equation
Jdrive _Oi kMUbatumot;ikMkEigearOi
igearROdROi
cQikQO2i
i2gear Zgear;
with the moment of inertia Jmot, the electrical and mechanical
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 25
0
5
10
Time [s]
Ang
le
Fig. 4. The reaction of the helicopter to an initial roll displacement of 203 with themodeled stabilizer bar.15
20 Roll Pitch Yaw motor constants kE and kM, the electrical resistance RO, thefriction coefcient dR the gear ratio igear, the efciency of the gearZgear, the battery voltage Ubat and the motor input umot,i.
All the equations result in a nonlinear model with 18 states
x x; y; z;u; v;w;f; y;c; p; q; r;adw;bdw;aup;bup;Odw;OupT
and four inputs (the two motors and the two servos)
u umot;dw;umot;up;userv1;userv2T
scaled to uservAf1;1g and umotAf0;1g. The nonlinear model isshown in Fig. 5 as a block diagram.
4. Parameter identication
In order to use the model for the controller design andsimulations, the missing parameters have to be identied and
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Gra
ic m
0 1 2 3 4 5 6 7 80
0.5
1
1.5
2
2.5
3
Torque Q [Nmm]
Cur
rent
I [A
]
MeasurementLeastSquare
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5012345678
RPM []
Torq
ue [N
mm
]
MeasurementLeastsquare
x 104
Fig. 6. Least square result plots for the electro motor parameter identication.
D. Schafroth et al. / Control Engineering Practice 18 (2010) 700711 703adjusted to the real helicopter. The identication process is a non-trivial task especially since most of the parameters are coupled. Inorder to minimize the complexity of the identication on realight data it is necessary to measure or estimate as manyparameters as possible beforehand.
All the mechanical properties such as mass, maximal swashplate angle, gear ratio, rotor diameter or body inertias can easilybe measured or determined from the CAD design. The aero-dynamical coefcients cT and cQ, determining the thrust andtorque values of the rotors, are identied by measuring the torqueand thrust of the lower and upper rotor on a coaxial rotor test-bench designed for blade optimization (Schafroth, Bouabdallah,Bermes, & Siegwart, 2008). Running the test bench in coaxialconguration allows to include the thrust loss on the lower rotordue to the down wash of the upper. The parameters for the lowcost off-the-shelf electro motors are not available, thus experi-mental data has to be used for the identication. Those motormeasurements have been done by our partner CEDRAT (2009) byapplying a constant voltage, varying the external torques on themotors and measuring the rotational speed and current. Themotor constants are identied using the stationary solution _o 0of the motor equation and the least-square method (LS) (Schwarz& Kaeckler, 2004). Result plots for the identication are shown inFig. 6.
The remaining unknown parameters are identied dynami-cally using real ight data. Since the equations are not staticanymore, a dynamic identication process has to be used.
In this work the Covariance Matrix Adaptation EvolutionStrategy (CMA-ES) (Hansen & Ostermeier, 1996) approach is used
Motor UP
Motor DW
Rotor UP
Rotor DW
Stabbar
SwashPlate
umot, up
umot, dw
up
dw
TupQup
TdwQdw
userv1
userv2
up, up
dw, dw
Fig. 5. muFly dynamto identify the parameters of the nonlinear model. Similar toquasi-Newton methods (Nocedal &Wright, 2006), the CMA-ES is asecond order approach estimating a positive denite matrixwithin an iterative procedure. It differs from the quasi-Newtonmethods in the way that instead of the inverse Hessian matrixH1 the covariance matrix C is estimated. This matrix is, onconvex-quadratic functions, closely related to the inverse HessianH1. The key idea of the CMA-ES is to adapt the covariance matrixin a way that the probability to reproduce successful mutationsteps is increased. Randomized search algorithms like the CMA-ESare regarded to be robust in a rugged search landscape, which cancomprise discontinuities or local optima. The CMA-ES in parti-cular is designed to tackle, additionally, ill-conditioned, non-separable, nonlinear and non-convex problems in dimensions ofthree to one hundred and thus it constitutes an adequate tool toidentify the parameters using multiple sets of measurement data.The exact formulation of the optimization algorithm is welldescribed in Hansen (2009). For the practical application of theCMA-ES it is important to set adequate initial conditions and, evenRigidBodyMotion
Trans-formations
vity FuselageDrag
G Whub
p, q, r
u, w, v
,
N, E, D
odel block diagram.more important, feasible boundaries for the parameters. Other-wise the algorithm ts the simulation output to the measurementdata while neglecting the physics.
For recording the ight data, the helicopter is steered by apilot. In order to cover as much frequency bandwidth as possible,a chirp signal is generated and superimposed on the pilot input.However, the helicopter is not adequately controllable in openloop by a pilot. Therefore, an additional PID controller is used tocontrol attitude and help the pilot. Effectively, the pilot controlsthe set point values of the PID controller. The controller is notproblematic for the parameter identication, since the actuatorsignals are recorded and sent together with the sensor data.Hence the identication of the system is independent of thecontroller. As a visualization the identication process is shown inFig. 7. It shows the reference value input r(t) of the pilot as theinput of the PID controller. The output of the controller u(t), themotor and servo inputs, is sent to the helicopter and the nonlinearmodel, where a new output y^t is predicted. This output is
ARTICLE IN PRESS
compared to the respective sensor measurement y(t) and theerror e is build and minimized by adjusting the parameters Yusing the CMA-ES.
Since only sensors for the attitude angles and the distance tothe ground are mounted on the helicopter, the model is reducedby the horizontal linear motion states, resulting in a state-spacesystem of 14 states. The identication is split into two subsystemsheave-yaw and pitch-roll. For the heave-yaw subsystem, theparameters are identied on two datasets. In the rst dataset, thealtitude is kept constant and the yaw angle reference is excited,and in the second dataset vice versa. This allows a correctadjustment of the parameters since most of them act on the yawand heave dynamics. Similar to the heave-yaw the roll and pitchare coupled due to gyroscopic effects and thus identiedsimultaneously. The results for the identied subsystems areshown in Figs. 811, respectively.
The results show a good match between the nonlinear modeland the experimental data. It mainly deviates in the amplitude,which does not impose a problem in closed-loop operation, sinceit is covered by the gain of the controller. A table with all theidentied parameters is shown in Table 1 in the appendix.
5. Controller design
The control structure for full position control of muFly isshown in Fig. 12. It consists of three multi-input multi-output(MIMO) controllers: the rst controls the heave z and the yaw
PID muFly
NonlinearModel
Parameter Adaptation(CMA-ES)
Minimizing
r e u y
()
Fig. 7. The identication process block diagram.
55 56 57 58 59 60 61 62 63 64 650.550.6
0.650.7
0.750.8
0.850.9
0.951
Time [s]
Mot
or P
PM
[]
Motor DownMotor Up
55 56 57 58 591.251.2
1.151.1
1.051
0.950.9
0.850.8
T
Alti
tude
z [m
]
ExperimentModel
594321
01234
T
Yaw
rate
r [ra
d/s]
ExperimentModel
put
D. Schafroth et al. / Control Engineering Practice 18 (2010) 70071170455 56 57 58
Fig. 8. CMA-ES identication result plots for an actuation in heave. UP: Motor in
measurement and model.60 61 62 63 64 65ime [s]
60 61 62 63 64 65ime [s]
signals. MID: Ultra sonic altitude measurement and model. DW: IMU yaw rate
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246 246.5 247 247.5 248 248.5 249 249.5 2500.6
0.650.7
0.750.8
0.850.9
0.951
Time [s]
Mot
or P
PM
[]
Motor DownMotor Up
246 246.5 247 247.5 248 248.5 249 249.5 2501.65
1.61.55
1.51.45
1.41.35
1.3
Time [s]
Alti
tude
z [m
]
ExperimentModel
246 246.5 247 247.5 248 248.5 249 249.5 2508642
02468
10
Time [s]
Yaw
rate
r [ra
d/s]
ExperimentModel
Fig. 9. CMA-ES identication result plots for an actuation in yaw. UP: Motor inputsignals. MID: Ultra sonic altitude measurement and model. DW: IMU yaw rate
measurement and model.
25.5 26 26.5 27 27.5 28 28.5 29 29.5 300.8
0.6
0.4
0.2
0
0.2
0.4
0.6
Time [s]
Ser
vo P
PM
[]
Servo 1Servo 2
25.5 26 26.5 27 27.5 28 28.5 29 29.5 300.2
0.15
0.1
0.05
0
0.05
0.1
0.15
Time [s]
Rol
l ang
le
[rad]
ExperimentModel
Fig. 10. CMA-ES identication result plots for an actuation in roll. UP: Servo inputsignals. DW: IMU roll angle measurement and model.
157.5 158 158.5 159 159.5 160 160.5 161 161.5 1620.40.30.20.1
00.10.20.30.4
Time [s]
Ser
vo P
PM
[]
Servo 1Servo 2
157.5 158 158.5 159 159.5 160 160.5 161 161.5 1620.2
0.15
0.1
0.05
0
0.05
0.1
0.15
Time [s]
Pitc
h an
gle
[ra
d]
ExperimentModel
Fig. 11. CMA-ES identication result plots for an actuation in pitch. UP: Servoinput signals. DW: IMU pitch angle measurement and model.
muFlyhelicopter
umot, dwumot, up
userv1userv2
Heave-YawController
Roll-PitchController
PositionController
xy
z
r
pq
Fig. 12. Control structure for muFly.
GsK
Wi
Wi
zi
ziwi
Fig. 13. Illustration of the weighting for the H1- control design.
Kf
Kb Gs
W
WM
Wd
Wr
Wr
Wd
Controller K
wd
wr
wd
wr r
e
u y
zu
zy
Fig. 14. The dual 2-dof GS/T-weighting scheme.
D. Schafroth et al. / Control Engineering Practice 18 (2010) 700711 705
ARTICLE IN PRESS
angle c via the two rotor speeds, the second controls the roll fand pitch y angles using the swash plate inputs (servos), and thelast sets the reference inputs f and y for the roll-pitch controllerfor a tight position reference tracking. As mentioned in Section 3the sensor hardware for the position measurement is not readyyet, so the focus in this paper is on the heave-yaw and roll-pitchcontroller.
The designated missions for muFly do not require fastmaneuvers of the helicopter. Slow maneuvers and tight controlaround the hover point are more important. Hence it seems a
reasonable choice to use linear control techniques for thecontroller design. In this work a mixed sensitivity H1 optimiza-tion is used for the two controllers. The theory of H1 controllerscan be found in various sources (e.g. Skogestad & Postlethwaite,2005), thus only an outline and the specic parts of the controllersused in this work are given.
For the H1- control design, the plant Gs to be controlled isaugmented by the dynamical weights Wi described in thefrequency domain. Using those weights allows to shape the (ingeneral closed loop) behavior of the control system by minimizing
105 104 103 102 101 100 101 102 103 104 105300
200
100
0
100
200
300
Frequency [rad/s]
Valu
e [d
B]
Fig. 15. Singular value plot of the heave-yaw subsystem.
105 104 103 102 101 100 101 102 103 104 105400
300
200
100
0
100
200
300
Frequency [rad/s]
Valu
e [d
B]
-100
-50
0
50
D. Schafroth et al. / Control Engineering Practice 18 (2010) 70071170610-5 10-4 10-3 10-2 10-1-400
-350
-300
-250
-200
-150Fig. 16. Singular value plots of the heave-yaw plant loop gain Lu (UP), the se100 101 102 103 104 105nsitivity Su and the complementary sensitivity Tutransfer function (DW).
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used to shape the sensitivity Su(s). The two weightsWM(s) andWr(s)are used as a reference model for Tyr(s) and as an upper boundary forthe model-matching error, respectively.
Solving the H1 problem results in a g 1:38. The resultingtransfer functions for the loop-gain Lu, sensibility Su andcomplementary sensibility Tu are shown in Fig. 16. The cross-over frequency is at oc 3:26 rad=s while as the maximumsensitivity and the maximum complementary sensitivity peak areat Su,max = 3.7398dB and Tu,max = 3.93dB.
The result of the closed-loop simulation to a step input inheave (0.5m, negative means upwards) on the nonlinear plantis plotted in Fig. 17. In this condition the helicopter can easilyfollow the reference step in heave without overshoots (the sameresult was found for a step response in yaw).
Measurements of a corresponding ight experiment on thereal system are plotted in Fig. 18. As expected, the real systemshows a slightly worse performance due to small modeldeviations and noisy data. Especially the control in yawdirection is not as tight as predicted by simulation. A reasonmight be the strong drift of the yaw measurement from the IMU.The heave control shows a good correlation with the prediction ofthe simulation, having only a small overshoot, but considering thescale the deviation is very small. Overall the performance for theheave-yaw control is satisfying.
Furthermore, the robustness of the H1 controller is tested insimulation by varying threes system parameters from their
0 1 2 3 4 5 6 7 8 9 10
0.60.50.40.30.20.1
00.1
Alti
tude
z [m
]
ReferenceNonlinear Model
D. Schafroth et al. / Control Engineering Practice 18 (2010) 700711 707the H1- norm of the transfer function Tzw resulting in thecontroller K. An illustration of the weighting principle is displayedin Fig. 13.
Both controllers are designed using the dual (y4u) GS/T-weighting scheme (Geering, 2004), where the product of the plantGs with the sensitivity S and the complementary sensitivity T areshaped by the weights Wi. However, the controllers differ in theirstructure. While for the roll-pitch control a pure feedbackcontroller is used, a two degree of freedom (2-dof) approach(Horowitz, 1963) is taken for the heave-yaw control. This meansthe controller K consists of two sub-controllers, a controller Kf forfeedforward reference input and a controller Kb for feedbackcontrol as shown in Fig. 14. The reason for the choice of afeedforward part is the desired tight reference following foraltitude and heading. The feedforward part minimizes overshootsin the altitude, which is a critical aspect in the foreseen missions.On the downside it needs more processing power, which islimited on the micro-controller. The main task of the pitch-rollcontroller is to stabilize the system, and only small referencechanges will be given by the position controller, so a purefeedback controller should be adequate.
For the application of the H1- technique the nonlinear modelfrom Section 3 has to be linearized around the operation point, inthis case hover. Similar to Section 4 the model is split in twosubsystem and linearized resulting in two state-space system of 6states for heave-yaw and 8 states for roll-pitch.
5.1. Heave-yaw control
The dual 2-dof GS/T-weighting scheme of the heave-yawcontroller is shown in Fig. 14 and has the corresponding transferfunction Tzw
Tzw
W ~uTuWd GsSuWdW ~uSuKfWr TyrWrW ~uSuKfWr TyrWMWrW ~uSuKbWd TeWd
266664
377775
T
:
If the H1- problem has a solution JTzwJ1rg with gr1, thespecications for the sensitivities are fullled. However, techni-cally it is not necessary to achieve the value g 1, a small valuefor g is sufcient for an asymptotically stable and robust system(Geering, 2004).
In Fig. 15the singular values of the linearized heave-yaw plantGs is shown. The cross over frequency is at 2.82 rad/s and the plantshows an integrating character. This means that theoretically anintegrating part of the controller is not needed, but neverthelessthe controller is designed to have poles close to the imaginaryaxis, thus having integrating character. This compensates for thebattery voltage drop during operation, requiring a higher motorinput signal with time.
The controllers Kf and Kb are then designed using followingweights augmented to their right dimensions
W ~u s 0:001; Wds s10
0:1s20 ;
Wds 0:7s1
s0:7=100 ; Wr s s7
0:01s14 ;
Wrs s215s14
104s214s0:0014; WMs
14
s14 :
The meaning of the weights is as follows: the weight W ~u s isused to minimize the inuence of the second row (can be neglected).The weightWd
1(s) is a boundary for the complementary sensitivity
Te(s), W1r s for the complementary sensitivity Tyr(s) and Wd s isTime [s]
0 1 2 3 4 5 6 7 8 9 100.060.050.040.030.020.01
00.010.020.03
Time [s]
Yaw
ang
le
[rad
]
ReferenceNonlinear Model
0 1 2 3 4 5 6 7 8 9 100.20.40.60.8
11.21.41.6
Time [s]
Con
trol O
utpu
t []
Motor DownMotor UpFig. 17. Simulation result plots for a closed-loop step response in heave.
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0.3
0.1 ReferenceNonlinear Model
D. Schafroth et al. / Control Engineering Practice 18 (2010) 70071170824 25 26 27 28 29 300.8
0.7
0.6
0.5
0.4
Time [s]
Alti
tude
z [m
]
3 Reference0.2
0.1 ReferenceMeasurementnominal value, the upper thrust and torque coefcients cT up, cQ upand the upper drive train inertia Jdrive,up. This corresponds to adeviation of the identied system parameters from the realvalues. In Fig. 19, the results for a step input in heave and yaw fora deviation range of 720% for the three parameter is plotted. Itshows that the controller can handle strong deviations in thesystem parameters. It mainly suffers in the beginning due tothe implied difference in the operation points. As soon as thedifference is compensated, the performance is similar tothe simulation with correct parameters. Additionally, a smallsteady state error is seen, due to the chosen weights, which placethe poles only close to the imaginary axis and not on theimaginary axis itself (no pure integrator).
5.2. Roll-pitch control
The procedure for the roll-pitch control is similar to the heave-yaw control, it differs only in the weighting scheme due to the
24 25 26 27 28 29 30-3
-2
-1
0
1
2
Time [s]
Yaw
ang
le
[rad
]
Measurement
24 25 26 27 28 29 300.20.40.60.8
11.21.41.61.8
Time [s]
Con
trol O
utpu
t []
Motor DownMotor Up
Fig. 18. Measurement result plots for a step response in heave.missing feedforward controller Kf. Thus, it is abandoned to show ithere and present only the key elements.
The singular value plot of the linearized roll-pitch plantis shown in Fig. 20. The bandwidth of the system is very low,which is caused by the stabilizer bar. Additionally the resonancefrequency peak at 14.5 rad/s leads to problems in the closed-loopdesign, as it can be seen in Fig. 20, where the inuence of the
0 2 4 6 8 10 12 14 16 18 200.15
Time [s]
0 2 4 6 8 10 12 14 16 18 200.8
0.6
0.4
0.2
0
0.2
0.4
0.6
Time [s]Y
aw a
ngle
[r
ad]
ReferenceNonlinear Model
Fig. 19. Robustness simulation for a step input in heave (2 s) and yaw (6 s).0.050
0.050.1
0.150.2
0.25
Alti
tude
z [m
]peaks on the loop-gain Lu, sensibility Su and complementarysensibility Tu is clearly visible. It is desirable to have those peaksafter the cut off frequency, therefore the controller is designed tohave a low bandwidth of 2 rad/s. Since the main task of the roll-pitch control is to support the insufcient stabilization of thestabilizer bar and follow moderate reference angles from theposition controller, this should be adequate. Another solution toreduce the effect of the resonance peak might also be to modifythe weights Wi (Fig. 21).
In Fig. 22 the simulation results for a reference trackingin pitch is shown as a comparison to the measured data inFig. 23, where the reference values are set by the pilots remotecontrol. In this measurement the real system is able to follow thereference slightly better than the model prediction. Here a reasonmight be that the nonlinear model underestimates the dynamicsof the system in pitch. Nevertheless, the inuence of the stabilizerbar on the dynamics is strong and it might be considered toremove it from the helicopter in future for better controlauthority.
6. Conclusions
In this paper the design procedure for a robust control of thecoaxial micro helicopter muFly using H1- controller techniques ispresented. As a rst step, a nonlinear model is developed. Anappropriate model is essential for the controller design andsimulations. The model is based on the rigid body dynamics,where all the possible acting forces and moments are discussed.Further, the dynamics of the mechanical devices such as the
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D. Schafroth et al. / Control Engineering Practice 18 (2010) 700711 70950
100swash plate, electro motor and stabilizer bar are derived, resultingin a complete structured model. However, a good model withincorrect parameters is worthless, hence those parameters haveto be identied. While most parameters are measured using CAD
105 104 103 102 101150
100
50
0
Freque
Valu
e [d
B]
Fig. 20. Singular value plot of the roll-p
105 104 103 102 101200
150
100
50
0
50
100
150
Freque
Frequen
Valu
e [d
B]
Valu
e [d
B]
10-5 10-4 10-3 10-2 10-1-180
-160
-140
-120
-100
-80
-60
-40
-20
0
20
u
Fig. 21. Singular value plot of the roll-pitch subsystem loop gain Lu (UP), sendata or test benches, some parameters have to be identied fromreal ight data applying identication methods. For this task arandomized search algorithm, the Covariance Matrix AdaptationEvolution Strategy (CMA-ES), is used. This allows to identify
100 101 102 103 104 105
ncy [rad/s]
itch subsystem with stabilizer bar.
100 101 102 103 104 105
ncy [rad/s]
cy [rad/s]100 101 102 103 104 105
sitivity Su and the complementary sensitivity Tu transfer functions (DW).
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In conclusion it can be said that the controllers aresuccessfully tested on the real system showing the appropriate-ness of the overall design procedure. It showed that this designprocedure ts well for this application, but it can also be used forother more general applications. On the other hand there is stillsome margin to improve the implemented controllers in near
D. Schafroth et al. / Control Engineering Practice 18 (2010) 7007117100.080.060.040.02
00.02
ngle
[r
ad]multiple parameter on multi-dimensional data. The parametersare successively identied showing the appropriateness of thenonlinear model.
The identied model is then used for the design of thecontrollers for the heave-yaw and roll-pitch subsystems. Thecontrollers are designed applying a mixed sensitivity H1-optimization using the GS/T-weighting scheme.
future.As another future work the performance might be improved
using nonlinear control techniques such as feedback linearizationor gain scheduling. In addition there is the idea to remove thestabilizer bar from the helicopter and stabilize it actively. Beside abetter control authority this would increase the autonomy timedue to reduced drag force. The last task is to design the positioncontrol as soon as the sensor for position measurement isavailable.
Acknowledgments
The authors would like to thank Mr. S. Leutenegger, Mr. T.Baumgartner, Mr. F. Haenni, Mr. M. Buehler and Mr. D. Fenner.muFly is a STREP Project under the Sixth FrameworkProgramme of the European Commission, Contract no. FP6-2005-IST-5-call 2.5.2 Micro/Nano Based Sub-Systems FP6-IST-034120. The authors gratefully acknowledge the contributionof our muFly Project partners BeCAP at Berlin University of
Appendix A
0 1 2 3 4 5 6 7 8 9 100.160.140.120.1
Time [s]
Pitc
h a
ReferenceNonlinear Model
0 1 2 3 4 5 6 7 8 9 100.8
0.6
0.4
0.2
0
0.2
0.4
0.6
Time [s]
Con
trol O
utpu
t []
Servo 1
Fig. 22. Simulation result plot for a reference following in pitch.
91 92 93 94 95 960.2
0.150.1
0.050
0.050.1
0.150.2
0.25
Time [s]
Pitc
h an
gle
[ra
d]
ReferenceMeasurement
91 92 93 94 95 960.30.20.1
00.10.20.30.40.5
Time [s]
Ser
vo In
put
[]
Servo 1
Fig. 23. Measurement result plots for a reference following in pitch. The referenceis applied by the pilot with the RC stick.
Izz Inertia around z-axis 2.66e5 kgm2
zdw Distance CoG lower rotor hub 0.051 mzup Distance CoG upper rotor hub 0.091 m
YSP;max Maximal swash plate angle 15 degR Rotor radius 0.0875 m
cT dw Thrust coefcient lower rotor 0.0117
cT up Thrust coefcient upper rotor 0.0138
cQ dw Torque coefcient lower rotor 0.0018
cQ up Torque coefcient upper rotor 0.0025
Jdrive,dw Drive train inertia (down) 7.04e6 kgm2
Jdrive,up Drive train inertia (up) 2.11e5 kgm2
kE Electrical motor constant 0.0045 V1 s1
kM Mechanical motor constant 0.0035 NmA1
dR Motor friction 5.2107e7 Nms
RO Resistance 1.3811 Oigear Gear ratio 1.5
Zgear Gear efciency 0.84 Whub Drag force on the fuselage 0.009 N
Tf,dw Following time upper rotor 0.001 s
Tf,up Following time upper rotor 0.24 s
ldw Linkage factor upper rotor 0.77
lup Linkage factor lower rotor 0.48 A table with all the identied parameters is shown inTable 1.
Table 1Identied parameters.
Parameter Description Value Unit
m Mass 0.095 kg
Ixx Inertia around x-axis 1.12e4 kgm2
Iyy Inertia around y-axis 1.43e4 kgm2Technology, CEDRAT Technologies, CSEM, Department ofComputer Science at University of Freiburg and XSENS MotionTechnologies.
ARTICLE IN PRESS
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D. Schafroth et al. / Control Engineering Practice 18 (2010) 700711 711
Modeling, system identification and robust control of a coaxial micro helicopterIntroductionThe muFly helicopterNonlinear modelParameter identificationController designHeave-yaw controlRoll-pitch control
ConclusionsAcknowledgmentsReferences