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Multi-Timescale Nonlinear

Robust Control for a Miniature

Helicopter

YUNJUN XU

University of Central Florida

A new nonlinear control approach, which is applied to

a miniature aerobatic helicopter through a multi-timescale

structure, is proposed. Because of the highly nonlinear, unstable,

and underactuated nature of a miniature helicopter, it is a

challenge to design an autonomous flight control system that

is capable of operating in the full flight envelope. To deal with

unstable internal dynamics, the translational, rotational, and

flapping dynamics of the helicopter (eleven degrees of freedom)

are organized into a three-timescale, nonlinear model. The

concepts of dynamic inversion and sliding manifold are combined

together such that 1) the controller proposed is robust with

respect to functional and parametric uncertainties, and 2)

the settling time in faster modes is guaranteed to be less than

the fixed step size of slower modes. A time-varying feedback

gain, derived according to global stability and sliding manifold

variations, is proved to be uniquely solvable based on the

Perron-Frobenius Theorem. Partial uncertainties are explicitly

taken into account in the nonlinear robust control design, and

Monte Carlo simulations are used for validations under other

sensor noises, model uncertainties, and a Federal Aviation

Administration suggested gust condition.

Manuscript received August 14, 2007; revised January 17 and May

28, 2008; released for publication December 10, 2008.

IEEE Log No. T-AES/46/2/936804.

Refereeing of this contribution was handled by M. Tahk.

This work was supported by Rockwell Collins, Inc. under the

Advanced Technology Center Alliances Fund (105052000).

Authors address: Dept. of Mechanical, Material, and Aerospace

Engineering, University of Central Florida, 4000 Central Florida

Blvd., Orlando, FL 32816, E-mail: ([email protected]).

0018-9251/10/$26.00 c 2010 IEEE

I. INTRODUCTION

The evolution of the unmanned aerial vehicle(UAV), whether rotary wing or fixed wing, isdeveloping at a tremendous pace, as represented bythe Predator or Global Hawk in military usage andby the Centurion in civil usage. The UAV offers awide range of services to support further researchand commercial implementation. However, in order

to respond to a growing markets demand and operatein the U.S. National Airspace System (NAS), UAVsrequire a Certificate of Authorization from the FederalAviation Administration (FAA). The certificationindirectly requires a mechanism whereby the UAVwill be able to reliably avoid other air traffic inthe operational area by using agile maneuvers [1].Specifically, the Air Traffic Management AdvisoryCommittee (ATMAC) has recommended a concept ofthe Airborne Conflict Management (ACM) that canbe followed by autonomous UAVs [2]. Two differentsafety zones that surround the UAV are proposed [1].The outer zone is the conflict detection zone (CDZ),

whereas the inner zone is the collision avoidancezone (CAZ). As per one of the examples, the CDZas defined by [3] is a cylinder of 9.26 km radius and609.6 m thickness. The CAZ has a 0.28 km radiusand a 182.88 m thickness.

In terms of this requirement, a rotary-wing orfixed-wing UAV needs the ability to quickly andautomatically adjust the flight controls to avoidhazardous situations, even under certain systemdegradations and unknown environment. Due to itsnonlinear, unstable, and underactuated nature, it isalways a challenge to design an agile controller fora miniature helicopter that meets these requirements.

Much work has been done in controllingrotary-wing UAVs. Typically, the practice inrotary-wing UAV flight control is to linearize theaircraft dynamics about different trim conditions(such as hover and cruise) and use gain-scheduledlinear control techniques to control the helicopterduring different flight conditions. Mismatchesand uncertainties among these operating pointsare captured through different robust controlmethodologies such as H1 [4, 5], control [6],neural nets [7], or fuzzy logic [8]. However,there are several clear drawbacks. More than onecontroller (numerous control gains in terms ofproportional-integral-derivatives or PIDs) needs tobe tuned, and performance degrades if the UAVmoves away from predefined trim conditions.Aerodynamic forces acting on a rotary-wing UAVand the corresponding state change dramatically whenthe UAV operates between different flight conditions(e.g. hover, cruise, ascending, and descending flight).At a precise operating point (e.g. cruise, hover, orforward flight) and within a certain region around thatpoint, linear models can accurately capture vehicle

656 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 2 APRIL 2010

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dynamics. However, the nonlinearities arising fromchanging aerodynamics over the range of usefuloperating conditions are significant.

To design a controller for operations in the fullflight envelope, dynamic inversion, input-outputfeedback linearization, or sliding mode control(SMC) is typically used [6, 911]. Because of theunderactuated nature, a pseudoinverse is typicallyexperienced in miniature helicopter control problems.

Numerical errors cannot be avoided, and the inducedinternal dynamics instability may occur even withthe help of singular value decomposition techniques.Two of the typical methodologies used to address thisproblem are timescale separation (TSS) [12, 13] andthe state-dependent Riccati equation control (SDRE)[14].

In the SDRE approach only the part of the systemthat is not able to be presented in the state dependentcoefficient (SDC) form needs to be inverted. ARiccati equation needs to be solved for each step.The mismatch between the original dynamics andits linearized version through extended linearization

is compensated by nonlinear correction terms.However, there are several drawbacks in this techniquethat need improvement. The most important oneis that linear quadratic (LQ)-type methods used instate-dependent, pseudolinear models can be designedto have certain functional gain and phase marginsafter extensive tuning. However, the uncertainties inthe pseudolinear form and the nonlinear compensatorare not considered in the control system design.

This paper is mainly concerned with thedevelopment of a general multiple-input-multiple-output (MIMO) nonlinear robust controller. Thiscontroller is combined with a three-timescale structure

(three resolvable problems) and applied in a miniaturehelicopter control problem. Different from [12] and[13], where a two-timescale model is used and wherethe flapping motion is assumed to be quasi-static,three timescales are used where the flapping motionis regarded as the fastest mode. In each timescalea uniform, nonlinear, robust controller is designedwith guaranteed stability, and the controller is robustwith respect to explicitly considered parametricuncertainties and unmodeled dynamics. Unlikea typical SMC [1519], there is no discontinuityfunction or nonideal switch, which typically resultin the chattering phenomenon [20, 21]. Removal ofthe discontinuity function guarantees a fixed settlingtime, which is crucial in multi-timescale controlmethodologies. In the mean time the controller isguaranteed to be insensitive to bounded parametricand functional uncertainties, which is better than atypical dynamic inversion approach.

The rest of the paper is organized as follows. First,the Lyapunov, stability-based, nonlinear, robust controlis derived, and the existence and uniqueness of thefeedback control gain are proved. Second, led by the

analytical stability study, some characteristics of theproposed nonlinear control method are discussed.After that, the three-timescale miniature helicopterdynamics model is briefly listed. Then controllersare designed and validated through Monte Carlosimulations for each timescale. Finally, the overallsimulation results are demonstrated, and a conclusionis drawn.

II. NONLINEAR ROBUST TRACKING CONTROL

A. General Case

Let us consider a nonlinear system with a statefunction of

x(ni)i = fi(x1, : : : ,xn, t) +mX

j=1

bij(x1, : : : ,xn)uj,

i = 1, : : : ,n, j= 1, : : : , m (1)

and an output function of

yi = hi(x1, : : : ,xn), i = 1, : : : ,p (2)

where xi = [xi, : : : ,x(ni1)i ]

T 2

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functions is bounded by F = [F1, : : : ,Fp]T 2

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Therefore the gain is calculated by

F + D

dryd

dtrLr

fh(x) +1 e +

r2Xk=0

k e(k+1)

+ s

= (ID)k s (14)

when si(t) = 0, vi = 0, and _vi = 0. Therefore the

reaching and sliding on the sliding manifold si = 0

are guaranteed. Here vi is lower bounded by zero,and _vi is negative semi-definite. Because _vi 0 and

vi(t) vi(0), the output and output tracking errors

are bounded. Thus _si and si are also bounded. For

the continuous functions _si and si, vi = sisi + _si _si are

also bounded, which means _vi(x1, : : : ,xn, t) is uniformly

continuous over time. According to Barbalats Lemma

[22], _v(x, t)! 0 as t !1, and thus the controlled

system is asymptotically stable.

B. Existence of a Solution to k

Is there always a unique solution for k thatsatisfies (14)?

LEMMA 1 There exists a unique positive solution for

the control gain k that satisfies (14) for any positive

numbers and .

PROOF For 8si > 0, the Lyapunov stability requires

Fi +

pXj=1

Dij

y

rjj,d L

rj

fhj + 1,jej +

rj2Xk=0

k,je(k+1)j

+ isi

= kisi pX

j=1

Dijjkjsjj (15)

while for 8si < 0, the Lyapunov stability requires

Fi

pXj=1

Dij

yrjj,d Lrjf hj + 1,jej +

rj2Xk=0

k,je(k+1)

j

+ isi

= kisi +

pXj=1

Dijjkjsjj (16)

or

Fi +

pXj=1

Dij


rj2Xk=0

k,je(k+1)j

isi

=kisi

pXj=1

Dijjkjsjj: (17)

Case 1 si > 0 or si < 0, 8i 2 [1, : : : ,p].

Under this case both (15) and (17) can be writtenas

Fi +

pXj=1

Dij


rj2Xk=0

k,je(k+1)j

+ jisij

= jkisij

pXj=1

Dijjkjsjj: (18)

Let us define &i = jkisij> 0 and = [1, : : : , 1] 2 0and the maximum eigenvalue of matrix D is less than1, there exists a unique solution of & and & > 0. Theunique and positive k can be found by

ki =

&i=si si > 0

&i=si si < 0: (21)

In case of si ! 0, the magnitude of kisi (&i = jkisij)instead of ki is calculated using (19) because theproposed controller (8) only uses kisi. The sign of kisiis determined by si since ki > 0. It is different from theconventional SMC or BASMC methodologies, where

kisign(si) or kisat(si) is used and where ki has to becalculated explicitly.In case of si 6= 0, kisi switches its sign as si

switches signs. It is quite similar to SMC, which hasan equivalent kisign(si).

Case 2 sl > 0, sq < 0, l,q 2 [1, : : : ,p], and q 6= l.Let us define &l = klsl for sl > 0 and &q =kqsq for

sq < 0. Equations (15) and (17) can be simplified as

Fl +

pXj=1

Dlj

y

rjj,d L

rj

fhj + 1,jej +

rj2Xk=0

k,je(k+1)j

+ lsl

= klsl

pXj=1

Dljjkjsjj (22)

and

Fq +

pXj=1

Dqj


rj2Xk=0

k,je(k+1)j

qsq

=kqsq

pXj=1

Dqjjkjsjj: (23)

XU: MULTI-TIMESCALE NONLINEAR ROBUST CONTROL FOR A MINIATURE HELICOPTER 659

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The left side of both (22) and (23) are positivevalues. Using the above definition for &, the first termof the right side is also positive. Equations (22) and(23) can be written in vector form as = (ID)&.Based on the same derivation as shown in Case 1,there exists a unique and positive solution for &, fromwhich a unique k can be explicitly calculated.

C. Special Case-System with ni = 1 and ri = 2, i =1, : : : , n

Let us see a system with a state function

_xi = fi(x1, : : : ,xn, t) +mX

j=1

bij(x1, : : : ,xn)uj,

i = 1, : : : , n, j= 1, : : : ,m: (24)

The output is

yi = hi(x1, : : : ,xn), i = 1, : : : ,p: (25)

The predicted mode is shown as

_xi =fi(x1, : : : ,xn, t) +

m

Xj=1

bij(x1, : : : ,xn)uj,

i = 1, : : : ,n, j= 1, : : : ,m (26)

and the predicted output is yi = hi(x1, : : : ,xn).According to Theorem 1, the robust controller can besimplified as

u = [LBL

fh(x)]+[yd L

2

fh(x) +1 e +0 _e + k s]:

(27)The feedback gain k can be found from

Fi +

p

Xj=1

Dijjyj,d L2

fhj + 1,jej + 0,j_ejj+ isi

= kisi

pXj=1

Dijjkjsjj if si > 0

Fi +

pXj=1

Dijjyj,d L2

fhj + 1,jej + 0,j_ejj isi

=kisi

pXj=1

Dijjkjsjj if si < 0:

(28)

In the case of si = 0, jkjsjj will be solved together toavoid the infinite gain problem.

D. Special Case-System with ni = 1 and ri = 1,i = 1, : : : , n

If the relative degree of the first-order system isone

_xi = fi(x1, : : : ,xn, t) +mX

j=1

bij(x1, : : : ,xn)uj,

i = 1, : : : ,n, j= 1, : : : ,m (29)

the output is

yi = hi(x1, : : : ,xn), i = 1, : : : ,p (30)

and the predicted mode is shown as

_xi =fi(x1, : : : ,xn, t) +

mXj=1

bij(x1, : : : ,xn)uj,

i = 1, : : : , n, j= 1, : : : , m: (31)

According to Theorem 1, the first-order robustcontroller can be simplified as

u = [LB

h(x)]+[ _yd L1

fh(x) + 1 e + 0 _e + k s]:

(32)

The feedback gain k can be found from

Fi +

pXj=1

Dijj _yj,d L1

fhj + 1,jejj+ isi

= kisi

p

Xj=1

Dijjkjsjj if si > 0

Fi +

pXj=1

Dijj _yj,d L1

fhj + 1,jejj isi

=kisi

pXj=1

Dijjkjsjj if si < 0:

(33)

In the case of si = 0, jkjsjj will be solved together toavoid the infinite gain problem.

III. CHARACTERISTICS OF THE CONTROLLER

Several advantages of the proposed controller arediscussed here. The rationale of using this method indesigning a multi-timescale controller for a miniaturehelicopter is also elaborated.

A. Chattering Free

Like a typical dynamic inversion approach, theproposed nonlinear robust control law eliminatesthe discontinuous function. Thus there is no highspeed switching involved as in SMC or BASMC.

Furthermore, the settling time of the faster modesmight be larger than the fixed step size of the slowermodes if the discontinuity function exists.

B. Saturation Protection

Because the initial errors tend to be large, it isoften true that saturation happens in the initial stageof a tracking problem and that the correspondinginitial gain is large. This kind of saturation problem


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Fig. 1. Saturation protection.

cannot be solved by the traditional SMC or BASMC.Let us use the second-order system as an example.The control feedback gain k (in SMC) is calculatedby

F + Djyd L2

fh(x) +1 e +0 _ej+ = (ID)k

(34)

and is not affected by the sliding manifold. Wecan simplify the control gain calculation as k =

(ID)

1

p + (ID)

1

where p is defined asF + Djyd L2

fh(x) + 1 e + 0 _ej. As can be seen

the control feedback gain k does not reflect the slowtransient effects due to the saturation.

However, the solution of k in the proposedcontroller is found through (14) as

(ID)1p:=s + (ID)1 = k (35)

where

p:=s = [p1=s1, : : : ,pm=sm]T: (36)

An intuitive explanation of the saturation protectionmechanism is shown in Fig. 1. Outside of theboundary layer si = 1, the control ki is smaller thanthose of SMC and BASMC. Whereas inside of theboundary, the controller ki is larger than those of SMCand BASMC, as illustrated. This mechanism couldhelp to solve the large gain issue in the initial stage.In the mean time when the sliding surface is reached,a relatively larger gain ki does not necessarily meanthat the control signal is larger when compared withthe traditional SMC or BASMC. The term kisat(si=i)(in BASMC) or kisign(si=i) (in SMC) amplified theoriginal signal kisi.

C. Implementation

The proposed controller works for cases wherethe number of control variables is larger than orequal to the number of outputs. Although stabilityderivatives are required for this controller, theirrelative degrees are less than two for most applications(i.e., the acceleration information may be needed). Ascompared with the higher order sliding mode method[23], which involves integrations of the higher order

sliding manifold, the proposed controller is easier toimplement.

In the following sections, this controller iscombined with a three-timescale structure andapplied in designing a fully nonlinear controllerfor a miniature helicopter in three resolvableproblems.

IV. MINIATURE HELICOPTER DYNAMICS IN THREE

TIMESCALES

In order to take into account uncertaintiesexplicitly in the control design, the multiple-timescaleconcept is used.

Different from [12] and [13], three timescalesare used in this paper. Separations are made amongthe fast mode (i.e., flapping dynamics), the middlemode (i.e., attitude dynamics), and the slow mode(i.e., translational dynamics). The idea is to simulatethe flapping dynamics at the fastest rate (1 kHz), andthe control command, in the fast mode, is generatedat 100 Hz, where the outputs are the longitudinaland lateral flapping angles a1 and b1, respectively.The control inputs uf for the fast mode are themain rotor collective col, longitudinal cyclic lon,and lateral cyclic lat. The pedal ped, a1, and b1are used as the control vector um for the middlemode (100 Hz) to track the desired Euler angles[d,d, d]

T. In the slow mode two of the Eulerangles [,]T and the main engine thrust Tmr willbe regarded as the control us for controlling thebody velocity. In the slow mode (10 Hz), the roll,pitch, and yaw rates ped, a1, and b1 are all in theaverage sense (within the fixed step size of the slow

mode).Detailed information of the miniature helicopter

model can be found in [24]. To simplify the robustcontrol design, the dynamics model has beenreorganized in a much clearer way. The description,symbols, acronyms, and parameters used in the modelare listed in Appendices A and B.

The assumptions used in the control designs are1) the induced velocities of the main rotor and tailrotor Vimr and Vitr are assumed to be constants; 2)the inflow rate of the main rotor and tail rotor areassumed to be constants. The mismatches comingfrom these assumptions are captured in Monte Carlo

simulations conducted in each timescale.

A. Fast Mode

The flapping dynamics response is much fasterthan the rigid body dynamics (translational androtational motion). Typically, the flapping dynamicsare assumed to be quasi-static and are neglected in thecontrol system design [14]. Instead it is consideredas the fast mode in this paper. The dynamics model


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is

_a1

_b1

=

264q

a1e

+

162K

8jj+ asign()

wa

eR2K0

uaeR

pb1e

2K0e

vaR

375

+

2

6648K

3e

uaR

Anormlon

e

norm

20

8K

3e

va

R 0

Bnormlat

e

norm

23

775

24collon

lat

35 (37)

where the state is xf = [a1, b1]T, the advance ratio is

=p

u2a + v2a =(R), and the control input vector is

uf = [col, lon,lat]T. Also the relative velocities of the

helicopter, with respect to the gust, are ua = uuw,va = v vw, and wa = www. Since the dimensionof the control variables is larger than that of thestate variables (output variables in this case), thepseudoinverse used in the fast mode does not induceany numerical error.

B. Middle Mode

The middle mode (attitude dynamics) is governedby26666666664

_

_

_

_p_q

_r

37777777775

=

26666666664

p + q sin tan+ r cos tan

q cos r sin

q sinsec+ r cos sec

qr(Iyy Izz )=Ixx + (Lvf+Ltr)=Ixxpr(Izz Ixx)=Iyy +Mht=Iyy

pq(IxxIyy )=Izz + (Nvf +Ntr Qe)=Izz

37777777775

+

26666666664

0

0

0

htrmYtr

r=Izz ped + [K + Tmr()hmr]=Ixxb1

[K + Tmr()hmr]=Iyy a1

mYtrv ltr=Ixxped

37777777775

(38)

where the state vector is xm = [,,,p, q, r]T and

where the control input is [ped,a1,b1]T. The moments

generated by the vertical fin can be calculated byLvf = Yvfhtr and Nvf =Yvfltr. y component of the

side force is Yvf =0:5SvfCvf

L(Vtr1 + jvvfj)vvf, where

Vtr1 =q

u2a + w2tr, vvf = va "

trvfVitr ltrr, and wtr = wa +

ltrqK Vimr. The moment from the horizontal tail isMht = Zhtlht, where Zht = 0:5ShtC

htL

(juajwht + jwhtjwht)and where wht = wa + lhtqK Vimr. The intensity factor

K can be found from [24]. The torque contributionsfrom the tail rotor are Ltr = Ytrhtr and Ntr =Ytrltr.The force component from the tail rotor is calculatedby Ytr = mY

trped

ped + mYtr

v trz trRtr, where tr = ntr,

vtr = va ltrr + htrp, and trz = vtr=(trRtr).

The equation for calculating the engine torque issimplified to be

Qe =_rIrot + Q + ntrQtr (39)

where the main rotor torque is Q = CQ(R)2R3.The torque coefficient, tail rotor torque, and tail rotorspeed are calculated by

CQ = CT(0 z) + CmrD0

=8(1+7=32) (40)

Qtr = CtrQ(trRtr)

2R3tr (41)

andtr = ntr (42)

numerically, where z = wa=(R).

C. Slow Mode

In the slow mode (translational motion), the stateand the control vectors are defined as xs = [u, v,w]

T

and as us = [,,Tmr]T. The governing equation is2

64_u

_v

_w

375=

264

vrwq + Xfus=m

w pur + [Yfus + Yvf + Ytr]=m + Ytr

ped

uq v p +Zfus=m

375

+

264

g sin+ (a1=m)Tmr

g sin cos+ (b1=m)Tmr

g cos cos+ (1=m)Tmr

375 (43)

where symbols with overbars denote variables that areevaluated using the average values from the middlemode. The forces generated by the fuselage are

Xfus =0:5Sfus

x CfusD,xuaV1 (44)

Yfus =0:5Sfus

y CfusD,yvaV1 (45)

andZfus =0:5S

fusz C

fusD,z(wa + Vimr)V1 (46)

where V1 =p

u2a + v2a + (wa + Vimr)

2.

V. FAST MODE CONTROL AND MONTE CARLOVALIDATION

A. Nonlinear Robust Control for the Fast Mode

The fast mode controller is designed to trackreference lateral and longitudinal flapping angles,a1,d and b1,d respectively, commanded by the middlemode. The settling time needs to be less than the fixedstep size of the middle mode (0.01 s), and the steadystate error should be within 5% (uncertainty bounds


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considered by the middle mode). The fast mode iscontrolled through the main rotor collective col, mainrotor longitudinal lon, and lateral cyclic lat.

Let us denote the fast mode to be _xf = ff(x) +Bfuf and yf = xf, where subscript f designatesthe fast mode. According to (37) the state functionff = [ff,1,ff,2]

T and the input function Bf are

ff =2664q

a1e

+

162K

8jj+ asign()

waeR

2K0ua

eR

pb1e

2K0e

vaR

3775(47)

and

Bf =

2664

8K3e

uaR

Anormlone

norm

20

8K3e

vaR

0Bnormlat

e

norm

23775 :

(48)

The uncertainties in the fast mode state functionare modeled as

j ff,i ff,ij Ff,i, i = 1, 2 (49)

and those of the input matrix are

Bf = (I + f)Bf, jf,ijj Df,ij < 1,

i = 1,2, j= 1, 2 (50)

where represents the predicted modes.Uncertainties in the state function of the fast mode Ff,imay come from the following sources: 1) gust model,sensor noise associated with the helicopter bodyvelocity [u,v,w]T and helicopter attitude rate [p, q, r]T;2) errors in stability derivatives (e.g. Anormlon and B

normlat

)

and other parameters (such as K, e, and 0, etc.);and 3) variation of the main and tail rotors inflowrates and induced velocities. Uncertainties associatedwith the input matrix bound Df may come from themain rotor actuation system noise, misalignments, orpartial failures, in addition to uncertainties similar tothose of the state function.

Since the relative degree for the fast mode is one,the nonlinear robust controller can be simplified as

uf = B+f[ _yd ff + kf sf] (51)

without the integral gain, where sf = ef = yf,d yf.To satisfy the asymptotical stability requirement underbounded uncertainties, the control parameter kf needsto satisfy

Ff,i +

2Xj=1

Df,ijj _yf,j,d ff,jj+ f,isf,i

= kf,isf,i 2X

j=1

Df,ijkf,isf,i (52)

where f,i > 0, i = 1,2.

Fig. 2. Settling time of fast mode (Monte Carlo).

Note that the uncertainty bounds derived basedon (49) and (50) may become too conservative, suchthat the performances (such as steady state error,settling time, and rising time) might be degraded, asthe number of parametric uncertainties increase. Acommon practice is to derive the bounds accordingto (49) and (50), followed by extensive Monte Carlosimulations to achieve less conservative boundswith satisfying performances. Therefore controllersdesigned for all three timescales are validated throughMonte Carlo runs.

B. Monte Carlo Simulation

The following random noise and uncertainties(all assumed to be Gaussian) are considered inthe Monte Carlo simulation for the fast mode.The gust model is scaled (10%) from the onedesigned for conventional large aircraft [25]. Thelongitudinal gust velocity has a mean value of1:258+0:949log10(z) m/s and a standard deviationof 0.199 m/s, where z is the altitude. The lateraldirection gust is modeled with a zero mean anda 0.199 m/s standard deviation. The onboard

airspeed indicator and attitude sensors are assumedto have 2% noise. The estimated states a1 andb1 are assumed to have 1% noise. There are 5%uncertainties in the helicopter parameters K, e,mr, cmr, amr, A

normlon

, and Bnormlat , whereas Rmr and0 have 0.5% and 1% noises, respectively. Thecontrol uf has a noise with a zero mean and a 1

standard deviation. If the settling time or steadystate error is larger than the requirement, a failureis reported. With 12,500 Monte Carlo simulations,the success rate was 99.97%. The mean andvariance of the settling time (a1) are 0.0040 s and0.0039 s, whereas those of b1 are 0.0054 s and0.0054 s (Fig. 2). Note that there are multiple dataat each point in the figure. The mean and varianceof the steady state error (a1) are 0.6602% and0.2411%, whereas those of b1 are 1.2286% and1.6420% (Fig. 3). In the Monte Carlo simulation,the estimation precision of the flapping dynamics(a1 and b1), the speed indicator, the attitude sensors,the radius of the main rotor, and the inflow rate ofthe main rotor are found to be the driving factors forperformance.


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VI. MIDDLE MODE CONTROL AND MONTE CARLOVALIDATION

A. Nonlinear Robust Control for the Middle Mode

Referring to (38) the state vector in the middlemode is [,, ,p,q,r]T, and the control variablesare [ped, a1, b1]

T. The steady state error in trackingthe commanded Euler angles (by the slow mode)needs to be less than 10%, while the settling time is

within the slow mode fixed step size (0.1 s). Let usrewrite the fast mode to be xm = fm(x) + Bmum, wherethe subscript m means the middle mode. The statefunction fm = [fm,i]i=1,:::,6 and the input function Bmare

fm =

26666666664

p + q sin tan+ r cos tan

q cos r sin

q sinsec+ r cos sec

qr(Iyy Izz )=Ixx + (Lvf+Ltr)=Ixx

pr(Izz Ixx)=Iyy +Mht=Iyy

pq(IxxIyy )=Izz + (Nvf +Ntr Qe)=Izz

37777777775

(53)

and

Bm =

2666666666666664

0 0 0

0 0 0

0 0 0

htrmYtr

r

Izz0

[K + Tmr(mr)hmr]

Ixx

0[K + Tmr(mr)hmr]

Iyy0

mYtrv ltrIxx

0 0

3777777777777775

:

(54)

The output for the middle mode is ym = hm =[,,]T, and the relative degree is two. The numberof control inputs and system outputs are equal (squaresystem), and the pseudoinverse in the controller canbe avoided.

To take into account modeling uncertainties andachieve asymptotic stability, the following controller isused

um = [LBmLfm hm(x)]1[yd L

2

fmhm(x) + 2em + _em + km sm]

(55)

where em = yd,m ym. The denotes a predictedmodel. The corresponding Lie derivates are shownbelow (56)(61)

Lfm

hm =

264

p + q sin tan+ r costan

q cos r sin

q sin sec + r cos sec

375 (56)

@Lfm

hm,1@xm

= [q costan r sin tan,(q sin + r cos)

sec2 ,0,1,sin tan,cos tan]T (57)

@Lfm

hm,2@xm

= [q sin r cos,0,0,0,cos,sin]T

(58)

@Lfm

hm,3@xm

= [q cos sec r sin sec, (q sin + r cos)

tan sec,0,0,sin sec,cos sec]T

(59)

LBmLfm hm(xm) = [@Lfm hm=@xm]Bm (60)

andL2

fmhm(x) = [@L fm hm(xm)=@xm]fm: (61)

In order to reduce the steady state error andsettling time, an integral gain is used in the slidingsurface

s = [si]i=1,:::,3 =

"Zeidt +

1Xk=0

kdkeidtk

#i=1,:::,3

:

(62)

The uncertainties in the state equation and input

matrix are modeled as

jL2fh(x) +L2

fh(x)j =

@L1fh(x)@x (f f)

= F 2

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Fig. 3. Steady state error of fast mode (Monte Carlo).

Fig. 4. Settling time of middle mode (Monte Carlo).

The success rate is 99.48% with 13,000 MonteCarlo simulations. The driving factors associatedwith the middle mode performance are the mass mand the attitude sensor precision. The mean valuesof the settling time are 0.0569 s () and 0.0570 s(). The variances of the settling time are 0.0002 sand 0.0003 s (Fig. 4). The mean values of the steadystate error are 2.5854% () and 2.1767% (), and thevariances are 8.9045% () and 10.0823% () (Fig. 5).

VII. SLOW MODE CONTROL AND MONTE CARLOVALIDATION

A. Nonlinear Robust Control for the Slow Mode

At this level the Euler angles are used to drivethe translational velocity to desired values. The statevector in the slow mode is xs = [u,v,w]

T, and thecontrol variables are [,, Tmr]

T. Let us define

us =

2

64&1

&2

&3

3

75=

2

64g sin+ (a1=m)Tmr

g sin cos+ (b1=m)Tmr

g cos cos

+ (1=m)Tmr

3

75(66)

fs =

264

vrwq + Xfus=m

w p ur + [Yfus + Yvf + Ytr]=m + Ytr

ped

uq v p + (Zfus)=m

375

(67)

and Bs to be a 33 identity matrix. Now the systemdynamics can be written as _xs = fs + Bsus where

us = [&1, &2,&3]T. xm = [ p, q, r]

T and um = [a1, b1, ped]T

Fig. 5. Steady state error of middle mode (Monte Carlo).

represent the average state and control from themiddle mode.

The control designed for the slow mode involvestwo steps: 1) a nonlinear, robust, control law usto track desired body velocity [ud, vd, wd]

T and 2)using the control commands us to solve for desiredEuler angles and thrust levels through a zero findingalgorithm.

Since the output in the slow mode is ys = xs, the

full state feedback controller is designed for step 1.The uncertainty model for the slow mode is

jfs fsj Fs (68)

and

Bs = (I + s)Bs, (s)ij (Ds)ij < 1: (69)

These uncertainties may come from 1) sensor noise,2) average value of the middle mode state and controlvariables, 3) steady state error of the Euler angle, and4) zero-finding algorithm error.

Since the relative degree is one in this case, thenonlinear robust control can be simplified as

us = B1s [ _xs,d fs + ks ss] (70)

where ss = es = xs,d xs and the control parameterneeds to satisfy

ks ss = (Is Ds)1(Fs + Dsj _xs,d fsj+s ss)

(71)

where s > 0.The true control variables [,,Tmr]

T can be foundby solving (66). First the controls found in (70) and[,, Tmr]

T have the following relations

264&1 + (Tmr=m)a1

&2 (Tmr=m)b1

&3 + Tmr=m

375= 264g sin

g sin cos

g cos cos

375 : (72)The magnitude of both sides is equal as

(&1 + a1Tmr=m)2 + (&2 b1Tmr=m)

2 + (&3 + Tmr=m)2 = g2

(73)and the thrust can be found as

Tmr = (2q

22 413)=(21) (74)


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where

1 = [(a1=m)2 + (b1=m)

2 + (1=m)2]T2mr (75)

2 = (2&1a1=m 2&2b1=m + 2&3=m)Tmr (76)

and

3 = &21 + &

22 + &

23 g

2: (77)

To guarantee a valid solution for the main rotor thrust,

it is required to have 3 = &2

1 + &2

2 + &2

3 g2

0 and thevelocity command is shaped to avoid rapid changes.The Euler angles ( and ) can be calculated from

(72) as

sin =[&1 + (Tmr=m)a1]=g (78)

and

tan = [&2 (Tmr=m)b]=[&3 + Tmr=m]: (79)

The main engine speed and the true inflow rate0 can be found through a zero finding algorithmusing the relation described in [24].

B. Monte Carlo Simulation

The Euler angle controlled by the middle modehas 10% steady state error. In addition to theuncertainties and noises already mentioned in thefast and middle modes, the gravity coefficient g andcorresponding density are assumed to have 0.5%noise. w has a zero-mean noise with a 1% standarddeviation. In the 10,500 Monte Carlo runs, the successrate is 97.10%. The mean values for the settlingtime in velocity tracking (u, v, and w) are 1.9994 s,1.9937 s, and 0.4059 s, respectively, whereas thevariances are 0.0111 s, 0.0068 s, and 0.0196 s (Figs. 6

and 7). The mean values of the steady state errors inthree directions are 0.0141%, 0.0074%, and 0.0602%.The corresponding variances are 3.8931%, 1.7115%,and 17.0253% (Figs. 8 and 9).

VIII. OVERALL CONTROL SYNTHESIS

The overall structure of the three-timescale controlframework is discussed here.

Fig. 10 shows a block diagram of the fast modeand shows the exchange of information among thefast, middle, and slow modes. The fast mode is

controlled through col, lon, and lat. The informationcoming from the middle mode are [p, q,r]T and yf,d,

and the parameters from the slow mode is [, u, v,w]T.The information transmitted to the middle mode fromthe fast mode is [a1,b1]

T. The fixed step size of thefast mode is 0.001 s, and the settling time is less than0.01 s.

In Fig. 11 the block diagram of the middle modeis shown. The middle mode is controlled though thepedal ped and the lateral and longitudinal flapping

Fig. 6. Settling time of slow mode (Monte Carlo, u v).

Fig. 7. Settling time of slow mode (Monte Carlo, uw).

Fig. 8. Steady state error of slow mode (Monte Carlo, u v).

Fig. 9. Steady state error of slow mode (Monte Carlo, uw).

angles a1 and b1. The information from the slowmode is [, u,v, w]T and desired Euler angles. Theinformation transmitted to the fast mode from themiddle mode is the desired flapping dynamics yf,dand attitude rate [p, q, r]T. The slow mode requires


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Fig. 10. Fast mode control architecture.

Fig. 11. Middle mode control architecture.

Fig. 12. Slow mode control architecture.

the average attitude rate [ p, q, r]T, average flapping

dynamics, and pedal input [a1, b1, ped]T from the

middle mode. Also the actual Euler angle is sent tothe slow mode and control input. The fixed step sizefor this mode is 0.01 s, and the settling time needs tobe less than 0.1 s.

The information flow of the slow mode is shownin Fig. 12. The slow mode is controlled by the Euler

angles [,]T

and the main rotor thrust Tmr. Theinformation from the middle mode is [ p, q, r]T and

[ped, a1, b1]. The information transmitted to the middle

mode from the slow mode is [d,d, d]T. The fixed

step size of the fast mode is 0.1 s.

IX. SIMULATION RESULTS

The proposed three-timescale nonlinear controlfor the miniature helicopter is tested here. The

Fig. 13. Velocity tracking in u.

Fig. 14. Velocity tracking in v.

Fig. 15. Velocity tracking in w.

miniature helicopter is assumed to fly at an altitudeof 300 m. Initially, the helicopter is in a hovermode with zero velocity. As shown in Figs. 1315,the helicopter is able to track the desired velocity(ud = 1 m/s, vd = 1 m/s, and wd = 1 m/s) for thefirst five seconds, and it is able to go back tohover mode for the later five seconds without anychatter.

The Euler angles controlled by the middle modeduring this maneuver are shown in Figs. 1618.Notice that to track the instantaneous jumps of thevelocity command (as shown in Figs. 1416 bydashed lines), relatively high jumps are shown in thebank and pitch angles (Fig. 16 and 17). Practically,this can be mitigated by smoothing the speedcommands. (A rate limiter can be added to shape thecommands.)

The flapping dynamics performance is shown inFigs. 19 and 20. The main rotor collective and the


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Fig. 16. Euler angle (bank angle).

Fig. 17. Euler angle (pitch angle).

Fig. 18. Euler angle (heading angle).

Fig. 19. Flapping dynamics a1.

tail rotor pedal input are shown in Fig. 21, wherethe pedal input (middle mode variable) has moreoscillations in the slow mode scale. The longitudinalcyclic and lateral cyclic of the main rotor are shownin Fig. 22. The main rotor thrust and engine speedperformance are show in Fig. 23.

Fig. 20. Flapping dynamics b1.

Fig. 21. Main rotor collective and pedal input.

Fig. 22. Main rotor longitudinal and lateral cyclic.

Fig. 23. Main engine speed and main rotor thrust.

X. CONCLUSION

A nonlinear, robust control algorithm is proposedhere, which could enable a miniature helicopterto operate in the full flight envelop. In additionthe controller is valid for general nth order MIMO


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systems, with coupled uncertainties in state and inputfunctions and parameters. The miniature helicopteris controlled through a three-timescale (flapping,rotational, and translational modes), nonlinearrobust controller. The advantage of the proposedcontroller is to have a finite time convergence andguaranteed settling time, which is preferred for themulti-timescale control structure. The unique solutionof the control feedback gain is guaranteed. Monte

Carlo simulations are conducted for the validation ofthe proposed controller for each of the three modes.

APPENDIX A. NOMENCLATURE

a1, b1 Longitudinal and lateral flapping angles

amr = a 5.5 rad1, main rotor blade lift curve

slope

Anormlon 4.2 rad=rad, longitudinal cyclic to flapgain at nominal rpm

Bnormlat 4.2 rad=rad, lateral cyclic to flap gain atnominal rpm

C

mr

D0 0.024, main rotor blade zero lift dragcoefficientCfusD,x 0.1, drag coefficient (the fuselage in x)

CfusD,y 0.5, drag coefficient (the fuselage in y)

CfusD,z 0.2, drag coefficient (the fuselage in z)

CtrQ Tail rotor torque coefficient

cmr = c 0.058 m, main rotor chord

ChtL 3.0 rad1, horizontal tail lift curve slope

CvfL 2.0 rad1, vertical fin lift curve slope

CQ Main rotor torque coefficient

CT Thrust coefficient

g Gravitational coefficient

hmr 0.235 m, main rotor hub height above thec.g.

htr 0.08 m, tail rotor height above the c.g.

Irot 2.5I mr , rotation inertia of the main rotorblade

Ixx 0.18 kg m2, rolling moment of inertia

Iyy 0.34 kg m2, pitching moment of inertia

Izz 0.28 kg m2, yawing moment of inertia

I mr 0.038 kg m2, main rotor blade flapping

inertiaK Intensity factor

K 54 N m=rad, hub torsional stiffness

K 0.2, scaling of flap response to speedvariation

L,M,N Roll, pitch, and yaw momentslht 0.71 m, horizontal tail aerodynamic center

location behind the c.g.

ltr 0.91 m, tail rotor hub location behind thec.g.

m 8.2 kg, helicopter massntr 4.66, gear ratio of tail rotor to main rotor

p, q, r Roll, pitch, and yaw rate

Q = Qe Engine torque

Rmr = R 0.775 m, main rotor radius

Rtr 0.13 m, tail rotor radius

Sht 0.01 m, horizontal tail area

Svf 0.012 m2, effective vertical fin area

Sfusx 0.1 m2, frontal fuselage drag area

Sfusy 0.22 m2, side fuselage drag area

Sfusz 0.15 m2, vertical fuselage drag area

Tmr Thrust generated by the main rotoru, v, w Translational velocity in the body frameuw,vw,ww Gust velocity in the body frame

Vimr,Vitr Main rotor and tail rotor induced velocity

X, Y,Z Force in the helicopter body frameYtrv 0.06 1/s, linearized tail rotor sideforce due

to side velocity

Ytrr 5.4 m=s2, linearized tail rotor sideforce

due to the tail rotor pitch,, Euler anglese Damping time constant for the flapping

motion

Advance ratioz Normal airflow component

Atmospheric density 2cmr=(Rmr), solidity ratio

"trvf 0.2, fraction of vertical fin area exposed totail rotor induced velocity

Main engine speednorm 167 rad/s, nominal main rotor speed

tr Tail rotor speed

0 Inflow rate of the main rotor

tr0 Inflow rate of the tail rotor

col Main rotor collective input

lat Lateral cyclic inputlon Longitudinal cyclic input

ped Pedal (tail rotor collective) input

w 0.9, coefficient of nonideal wakecontraction.

APPENDIX B. ACRONYM

c.g. center of gravitySubscript:e Enginefus Fuselage

ht Horizontal tailmr Main rotortr Tail rotorvf Vertical fin.

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Yunjun Xu received the B.S. and M.S. degrees from Nanjing University ofAeronautics and Astronautics, Nanjing, China, in 1996 and 1999, respectively.He also received the M.S. and Ph.D. degrees from the University of Florida,Gainesville, FL, in 2002 and 2003, respectively.

Currently he is an assistant professor in the Department of Mechanical,Materials, and Aerospace Engineering at the University of Central Florida. Hisresearch interests include linear and nonlinear control theory and application,dynamics modeling and simulation, robotics, navigation, guidance and controlsystems, and virtual environments.


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