Trajectory Control of Miniature Helicopters Using a Unified Nonlinear Optimal Control Technique

Ming Xin1

Department of Aerospace Engineering,

Mississippi State University,

Starkville, MS 39759,

e-mail: [email protected]

Yunjun XuDepartment of Mechanical, Materials, and

Aerospace Engineering,

University of Central Florida,

Orlando, FL 32816

Ricky HopkinsNordam Transparency Division,

Tulsa, OK 74117

Trajectory Control of MiniatureHelicopters Using a UnifiedNonlinear Optimal ControlTechniqueIt is always a challenge to design a real-time optimal full flight envelope controller for aminiature helicopter due to the nonlinear, underactuated, uncertain, and highly couplednature of its dynamics. This paper integrates the control of translational, rotational, andflapping motions of a simulated miniature aerobatic helicopter in one unified optimalcontrol framework. In particular, a recently developed real-time nonlinear optimal con-trol method, called the h� D technique, is employed to solve the resultant challengingproblem considering the full nonlinear dynamics without gain scheduling techniques andtimescale separations. The uniqueness of the h� D method is its ability to obtain anapproximate analytical solution to the Hamilton–Jacobi–Bellman equation, which leadsto a closed-form suboptimal control law. As a result, it can provide a great advantage inreal-time implementation without a high computational load. Two complex trajectorytracking scenarios are used to evaluate the control capabilities of the proposed methodin full flight envelope. Realistic uncertainties in modeling parameters and the wind gustcondition are included in the simulation for the purpose of demonstrating the robustnessof the proposed control law. [DOI: 10.1115/1.4004060]

1 Introduction

Research in the topic of unmanned aerial vehicle (UAV) controlis highly motivated and has a very renewed interest as the commer-cial demand for such vehicles is growing rapidly. UAVs to date,both military and commercial, have been allowed to operate onlywithin restricted airspace. Recently, the Department of Defense hasconstructed a plan for UAVs to operate within the National Air-space System (NAS) by 2010 [1]. The Federal Aviation Adminis-tration has requirements for UAVs to maintain flight qualities thatare at least as stringent as those placed upon manned aircraft so thatUAVs can be operated safely within the NAS. One of the majorrequirements facing the incorporation of UAVs into the NAS is theability to track desired trajectories precisely, no matter the circum-stances. Thus, UAVs must be able to quickly maneuver even underadverse environmental conditions or with degraded systems.

As a subset of UAVs, unmanned rotorcrafts have implicationsfor both commercial and military uses. In the International AerialRobotics Competition, which began in 1991, the most commonvehicle to successfully complete simulated tasks of toxic wasteidentification and mapping, survivor or hostage detection in volatileenvironments, and other lifelike reconnaissance missions is minia-ture helicopters [2]. Miniature helicopters can offer unique advan-tages over traditional fixed-wing craft such as vertical takeoff, verti-cal landing, hover, and movement independent of heading. Theaforementioned capabilities lend miniature helicopters to be bettersuited than other aircraft for specific missions or environments. Forexample, hovering may be an advantage in urban settings.

To date, the majority of the control techniques applied in rotor-craft attitude and velocity tracking has some forms of linearizationand gain scheduling [3–10]. Position tracking adds more complexityto the problem due to the extra nonlinearities that arise in translatingthe velocity vector from a body fixed coordinate system to the iner-tial system [6]. Controllers such as Proportional-Integral-Derivative

(PID) [11], H1 [10,12], and H2 [4,6] have been designed for linearmotions, but the model under control is only valid in a small regionaround its trim condition. These controllers are less reliable whenthe flight path deviates from the predetermined trim conditions.Specific flight conditions such as forward flight or hover have beencontrolled adequately with linear controllers [13]. However,challenges will arise for linear controllers when significant nonli-nearities exist due to rapid changing aerodynamics over certainoperations such as ascending, descending, and turning maneuvers.

To control nonlinear motions of a miniature helicopter, dynamicinversion [14], neural networks [15], and sliding mode control(SMC) [16] have been investigated. For example, Zeng and Zhu[14] applied the combined PID and dynamic inversion technique toa simplified/reduced model (4 degrees of freedom) with adaptivecompensators to account for the modeling uncertainties and distur-bances. However, only a limited flight envelope that includes hoverand forward flight has been shown. Oh et al. [17] investigated posi-tion tracking using a zero finding algorithm with a reduced dynam-ics model and assumed that aerodynamic effects do not play asignificant role. In addition, only a simple ascending motion issimulated.

When full dynamics are considered, which is an underactuatedsystem in the case of miniature helicopters, pseudo-inverses arerequired in the input–output feedback linearization and SMCapproaches. Numerical errors coming from this inversion will causeinternal state instability. The common practice in aircraft control tohandle the underactuated system is to use the inner–outer loop ormultitimescale strategy [15,18–23]. The flight dynamics is sepa-rated into a slow mode for translational motion and a fast mode forattitude motion. The control law is designed accordingly in eachmode. For example in Ref. [23], a chattering free nonlinear control-ler combined with a zero-finding algorithm has been investigatedfor three timescale full envelope flight. Different flight modes havebeen tested; however, the computational cost is high. In addition,this conventional design paradigm normally considers the robust-ness but is not optimized because the separate designs cannot takethe coupling from other elements into full consideration and canresult in excessive iterations during the integration process. Johnsonand Kannan [15] proposed a novel pseudocontrol hedging approach

1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the

JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT AND CONTROL. Manuscript received De-cember 28, 2009; final manuscript received March 5, 2011; published online Septem-ber 6, 2011. Assoc. Editor: Rama K. Yedavalli.

Journal of Dynamic Systems, Measurement, and Control NOVEMBER 2011, Vol. 133 / 061001-1Copyright VC 2011 by ASME

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 09/28/2013 Terms of Use: http://asme.org/terms

to enable neural network adaptation in the outer loop without inter-acting with the attitude dynamics. This method will alleviate tosome extent the coupling effect in the two-loop design process.

The primary contribution of this work is to integrate the controlof helicopter trajectory, attitude motion, and flapping dynamics inone unified real-time optimal control framework. The conven-tional separated control design for different timescale modes isavoided so that the overall system performance can be optimizedwith one single design process. The resultant problem involveshighly nonlinear kinematics and dynamics, and thus traditionallinear control designs are unsuitable for a precise control. Bogda-nov and Wan [24] used a similar optimal control formulation withthe state-dependent Riccati equation (SDRE) technique and acomplex nonlinear compensator. This method (validated throughexperiments) shows its capability of tracking position over anentire flight envelope with the assumption of a quasistatic flappingmotion. The results are promising. However, the SDRE approachdemands more computational resources because it needs to solvethe algebraic Riccati equation numerically at every integrationstep, which involves a complex and iterative process. The compu-tational load will be particularly high when dealing with the fulldegree-of-freedom dynamics and higher-order systems. In thispaper, a recently emerging nonlinear suboptimal control method,called the h� D technique [25,26], is employed to design thisintegrated nonlinear control law. Based on an approximate solu-tion to the Hamilton–Jacobi–Bellman (HJB) equation, the h� Dtechnique provides a closed-form feedback controller. Comparedwith the SDRE method, the h� D approach offers a great compu-tational advantage for onboard implementation without solvingthe Riccati equation repetitively at every instant.

Rest of the paper is organized as follows. In Sec. 2, the dynamicsmodel of the miniature helicopter is described. The integrated con-trol law is designed using the proposed h� D nonlinear optimalcontrol technique in Sec. 3. Simulation results and analysis are pre-sented in Sec. 4. Some concluding remarks are given in Sec. 5.Detailed model parameters are described in Appendix andNomenclature.

2 Miniature Helicopter Dynamics

The prevailing way of treating aircraft dynamics, especiallywith respect to miniature helicopters, breaks the dynamics intodifferent timescales with the assumption that the flapping dynam-ics are quasistatic [18–22]. In this paper, the flapping dynamics is

explicitly considered and the optimal control formulation does notdifferentiate these timescales. However, for the convenience ofpresentation, the helicopter equation of motion will still bedescribed in terms of translational (slow) motion (six states), rota-tional (middle) motion (six states), and the flapping (fast) motion(two states). The model has a total of 14 states and is modifiedand reorganized according to Gavrilets et al. [27]. Note that theexplanation of the symbols used and their values for this particularminiature helicopter can be found in the Nomenclature section. Asketch of such a miniature helicopter is shown in Fig. 1, in whichsome of the state and control variables are illustrated.

2.1 Translational Motion. The states of the translationalmotion include the inertial position vector ½xp; yp; zp�T and the in-ertial velocity vector ½u; v;w�T expressed in the helicopter’s bodyframe. The kinematic relation between the body frame velocity (u,v, and w) and the velocity in the inertial coordinate ( _xp, _yp, and _zp)is given by

_xp

_yp

_zp

24

35 ¼ R Hð Þ

uvw

24

35 (1)

where H ¼ /; h; w½ �T are the Euler angles and

R Hð Þ ¼cos h cos w sin / sin h cos w� cos / sin w cos / sin h cos wþ sin / sin w

cosh sinw sin / sin h sin wþ cos / cos w cos / sin h sin w� sin / cos w

� sin h sin / cos h cos / cos h

264

375 (2)

is the direct cosine matrix from the body coordinate to the inertialcoordinate using a 3-2-1 Euler rotation sequence. The transla-tional dynamics is governed by

_u_v_w

24

35 ¼ vr � wqþ Xfus=m� g sin h

wp� ur þ Yfus þ Ytr þ Yvf

� �=mþ g sin / cos h

uq� vpþ Zfus=mþ g cos / cos h

24

35

þ�a1Tmr=mYtr

dpeddped þ b1Tmr=m

�Tmr=m

24

35 (3)

where g is the gravitational constant;m is the mass of the miniaturehelicopter; a1 and b1 are longitudinal and lateral rotor flappingangles, respectively; and p; q; r are roll, pitch, and yaw rates. Theexternal forces contain the aerodynamic forces acting on the fuse-lage, Xfus, Yfus, and Zfus, the force on the vertical fin Yvf , the forcefrom the tail rotor Ytr , the main rotor thrust Tmr , and the gravitationalforce. dped is the pedal input. The detail description of these forcesin terms of this helicopter’s physical and aerodynamic parameters isgiven in the Appendix A and Nomenclature following Ref. [27].

2.2 Rotational Motion. The states in the rotational motionxrot include the angular velocities ½p; q; r�T and Euler angles½/; h;w�T . The attitude dynamics model is represented by

Fig. 1 Illustration of the miniature helicopter with the axis sys-tem and forces

061001-2 / Vol. 133, NOVEMBER 2011 Transactions of the ASME


_p_q_r

2435 ¼ qr Iyy � Izz

� �=Ixx þ Lvf þ Ltr

� �=Ixx

pr Izz � Ixxð Þ=Iyy þMht=Iyy

pq Ixx � Iyy

� �=Izz þ Nvf þ Ntr � Qe

� �=Izz

24

35

þhtrmYtr

dpeddped=Ixx þ Kb þ Tmrhmr

� �b1=Ixx

Kb þ Tmrhmr

� �a1=Iyy

�mYtrdped

dpedltr=Izz

264

375 (4)

and the kinematic relation between the angular velocities andEuler angular rates are

_/_h_w

24

35 ¼ pþ q sin / tan hþ r cos / tan h

q cos /� r sin /q sin / sec hþ r cos / sec h

24

35 (5)

In this motion, Ltr and Ntr are the moments produced by the tailrotor about the roll and yaw axes, respectively; Lvf and Nvf are themoments generated by the vertical fin about the roll and yaw axes,respectively; Mht is the moment generated by the horizontal tail;Qe is the engine torque. The detail description of these momentsin terms of this helicopter’s physical and aerodynamic parametersis given in the Appendix B and Nomenclature following Ref. [27].

2.3 Flapping Dynamics. The states of the fast mode are theflapping angles, xf ¼ a1; b1½ �T , and the control inputs are the col-lective pitch control and the lateral and longitudinal cyclic controlinputs, uf ¼ dcol; dlat; dlon½ �T . a1 and b1 are the longitudinal and lat-eral rotor flapping angles, respectively. The longitudinal flappingangle represents a fore–aft tilt of the rotor tip path plane, while thelateral flapping angle represents a left–right tilt of the rotor tippath plane. The collective pitch control adjusts the pitch of themain rotor blades, effectively increasing or decreasing the altitudeof the miniature helicopter. The cyclic control adjusts the pitch ofthe main rotor blade disk. The angle of attack and lift generatedby each blade are thus dependent upon the position of the blade asit rotates around the main rotor hub.

The flapping dynamics are described by

_a1

_b1

" #¼�q�a1

seþ 16l2Kl

8 lj jþarsign lð Þ

� �wa

seXmrRmr�2Klk0

uaseXmrRmr

�p�b1se�2Klk0

vaseXmrRmr

26664

37775

þ

8Klua

3seXmrRmrdcolþ

Anomdlon

se

Xmr

Xnom

� 2

dlon

8Klva

3seXmrRmrdcolþ

Bnomdlat

se

Xmr

Xnom

� 2

dlat

266664

377775 (6)

where se is a time constant for this mode, Kl is the scaling of theflap response to the speed variation, and a is the main rotor bladelift curve slope. signðlÞ is defined as

sign lð Þ ¼ 1 if l � 0

�1 if l < 0

(7)

Anomdlon

and Bnomdlat

are the longitudinal and lateral gains between thecyclic inputs and flapping angles, respectively.

2.4 Uncertainties and Wind Gust Model. There exist vari-ous uncertainties in the miniature helicopter model and environmen-tal conditions. In order to test the robustness of the later designednonlinear controller, different from other approaches [24,27], theparameter uncertainties and perturbations will be included in themodel as shown in Table 1, which contains 45 types of uncertain-ties. The wind gust may have a significant effect on the miniaturehelicopter and must be considered. The wind gust model is scaleddown from the one given in AC120-28D [28]. Specifically, in the xdirection, the wind gust model has a mean value of 1:258þ0:949log10 zð Þ m=s and a standard deviation of 0:199 m=s, wherez is the altitude of the helicopter. In the y direction, the wind gustmodel has a zero mean and a standard deviation of 0:199 m=s. Theuncertainties considered in the helicopter model can be categorizedas measurement noise, estimation error, parameter uncertainties,and control input uncertainties, which are shown in Table 1. Forexample, there is 5% noise assumed in the estimated flapping angles(a1; b1); 5% uncertainty is considered in the main rotor thrust (Tmr);and 10% noise is considered in the induced main rotor velocity.

3 Unified Nonlinear Optimal Control Design via h� DMethod

3.1 Unified Optimal Control Formulation. The approachto controlling the miniature helicopter modeled in Sec. 2 is to for-mulate the full dynamics of motion into one unified state-spaceand optimal control framework. The control design will be simpli-fied via one single optimization process and the overall systemperformance can be optimized through one cost function. Further-more, since trajectory control is the focus of this work, the accu-racy of the position tracking is important. Thus, three integralstates of the position components xp, yp, and zp, denoted as xi, yi,and zi, are augmented into the state-space to improve the trackingperformance. Therefore, the state variables are chosen to be

x ¼xp yp zp u v w / h w p q

r a1 b1 xi yi zi

� �T

2 <17�1

(8)where xi, yi, and zi satisfy

_xi ¼ xp; _yi ¼ yp; _zi ¼ zp (9)

The state-space equation of motion including the full dynamics,Eqs. (1), (3)–(6), and (9), can be written in a general form as

_x ¼ f ðxÞ þ BðxÞu (10)

where the control vector is

u ¼ Tmr dped dcol dlon dlat½ �T 2 <5�1 (11)

The trajectory tracking problem is formulated as an optimal con-trol problem by minimizing a quadratic cost function

J ¼ 1

2

ð10

xTQxþ uTRu� �

dt (12)

The advantage of this unified optimal control formulation is thatsystem concerns, such as tracking errors, control limits, angle andangular rate constraints, and velocity constraints, can be easilyaddressed in one cost function by imposing proper weights on thecorresponding variables. In addition, the design and implementationis easier to conduct without much iterative process in the multiti-mescale approach, which typically needs multiple processors inimplementation. However, Eq. (10) includes 17 highly nonlinear

Table 1 Uncertainties considered

ParametersUncertainty

(%) ParametersUncertainty

(%) ParametersUncertainty

(%)

a1; b1 5 dpedal 5 amr 5Anom

dlon5 Bnom

dlat5 cmr 5

g 0.5 hmr 5 htr 5Ixx 5 Iyy 5 Izz 5Kb 5 Kl 5 ltr 5m 0.5 p; q; r 5 Rmr 0.5Tmr 5 u; v;w 5 uw; vw;ww 5Vimr 10 Vitr 10 xp; yp; zp 5Ytr

dped10 dcol; dlon; dlat 5 /; h;w 5

gw 1 k0 1 q 0.5se 5 Xnom 5

Journal of Dynamic Systems, Measurement, and Control NOVEMBER 2011, Vol. 133 / 061001-3


and coupled equations and is an underactuated system. Thus, itimposes a challenging control problem. In this work, we willemploy a new nonlinear optimal control technique, h� D method,to solve this problem.

3.2 h� D Nonlinear Suboptimal Control Technique. Theh� D technique [25,26] is a systematic nonlinear optimal controlmethod to address a class of nonlinear time-invariant systemsdescribed by

_x ¼ f ðxÞ þ BðxÞu (13)

The objective is to find a stabilizing control u to minimize thequadratic cost functional

J ¼ 1

2

ð10

xTQxþ uTRu� �

dt (14)

where x 2 Xc � Rn; f 2 Rn;B 2 Rn�m;u 2 Rm;Q 2 Rn�n;R 2Rm�m; Assume that Xc is a compact subset in Rn; Q is a positivesemidefinite matrix and R is a positive definite constant matrix.It is assumed that f is of class C1in x on Xc and f(0) 5 0.

The optimal solution to this infinite-horizon nonlinear regulatorproblem can be obtained by solving the HJB partial differentialequation [29]

@V

@xf ðxÞ � 1

2

@V

@x

� T

BðxÞR�1BTðxÞ @V

@xþ 1

2xTQx ¼ 0 (15)

where V(x) is the optimal value function, i.e.,

VðxðtÞÞ ¼ minu1

2

ð1t

xTQxþ uTRu� �

dt (16)

Assume that V(x) is continuously differentiable and V(x) > 0 withV(0)¼ 0. The optimal control is given by

u ¼ �R�1BTðxÞ @V

@x(17)

The HJB equation is extremely difficult to solve in general. Theh� D method gives an approximate closed-form solution byintroducing perturbations to the cost function, i.e.,

J ¼ 1

2

ð10

xT QþX1i¼1

Dihi

" #xþ uTRu

( )dt (18)

whereP1

i¼1 Dihi is a perturbation series in terms of an instrumen-

tal variable h. The construction of this series will be discussedafterward. Rewrite the state Eq. (13) in a linearlike structure[25,26]:

_x ¼ f ðxÞ þ BðxÞu ¼ FðxÞxþ BðxÞu

¼ A0 þ hAðxÞ

h

� � �xþ g0 þ h

gðxÞh

� � �u (19)

where A0 and g0 are constant matrices such that ðA0; g0Þ is a con-trollable pair and FðxÞ;BðxÞ½ � is pointwise controllable.

Remark 3.1. The state-dependent factorization in Eq. (19) isexact and does not involve any linearization process. The form offactorization is not unique [30], which provides extra flexibility inthe control design.

The new optimal control problem Eqs. (18) and (19) can besolved through the perturbed HJB equation

@V

@x

� T

A0 þ hAðxÞ

h

� �x� 1

2

@V

@x

� T

g0 þ hgðxÞh

� � �

R�1 g0 þ hgðxÞh

� � �T @V

@xþ 1

2xT Qþ

X1i¼1

Dihi

!x ¼ 0

(20)

Assuming a power series expansion of @V=@x ¼P1

i¼0 Tiðx; hÞhix,the optimal control becomes

u ¼ �R�1BTðxÞX1i¼0

Tiðx; hÞhix (21)

where Tiðx; hÞ(i ¼ 0;…; n;…) is a symmetric matrix andis solved recursively by the following algorithm [Eq. (22)],which is obtained by substituting @V=@x ¼

P1i¼0 Tiðx; hÞhix in

the perturbed HJB Eq. (20) and equating the coefficients of hi

to zero

T0A0 þ A0TT0 � T0g0R�1g0

TT0 þ Q ¼ 0 (22a)

T1ðA0 � g0R�1g0TT0Þ þ ðA0

T � T0g0R�1g0TÞT1 ¼

� T0AðxÞh� ATðxÞT0

hþ T0g0R�1 gTðxÞ

hT0

þ T0

gðxÞh

R�1g0TT0 � D1

(22b)

T2ðA0 � g0R�1g0TT0Þ þ ðA0

T � T0g0R�1g0TÞT2 ¼

� T1AðxÞh� ATðxÞT1


hT1 þ T0

gðxÞh

R�1g0TT1 þ T0

gðxÞh

R�1 gTðxÞh

T0 þ T1g0R�1g0TT1

þ T1g0R�1 gTðxÞh

T0 þ T1

gðxÞh

R�1g0TT0 � D2

(22c)

TnðA0� g0R�1g0TT0Þþ ðA0

T �T0g0R�1g0TÞTn ¼

�Tn�1AðxÞh

�ATðxÞTn�1

hþXn�2

j¼0

TjgðxÞh

R�1 gTðxÞh

Tn�2�j

þXn�1

j¼0

Tj g0R�1 gTðxÞhþ gðxÞ

hR�1g0

T

� �Tn�1�j

þXn�1

j¼1

Tjg0R�1g0TTn�j�Dn

(22d)

Note that Eq. (22a) is an algebraic Riccati equation and the rest ofequations are Lyapunov equations that are linear in terms ofTi(i ¼ 1;…; n).

The steps of applying the h� D algorithm to solve Ti recur-sively are summarized as follows:

(1) Solve the algebraic Riccati Eq. (22a) to obtain T0 onceA0,g0, Q, and R are determined. Note that the resulting T0 isa positive definite constant matrix under the controllabilityand observability conditions.

(2) Solve the Lyapunov Eq. (22b) to obtain T1ðx; hÞ. Note thatthis is a linear algebraic equation in terms of T1ðx; hÞ and aunique property of this equation is that the coefficient mat-rices A0 � g0R�1g0

TT0 and A0T � T0g0R�1g0

T are constantmatrices. Assume that Ac0

¼ A0 � g0R�1g0TT0. Through

linear algebra, Eq. (22b) can be brought into the form ofA0vecðT1ðx; hÞÞ ¼ vec M1ðx; h; tÞ½ �, where M1ðx; h; tÞincludes all the nonlinear terms on the right-hand side ofthe Eq. (22b); vecðM1ðx; h; tÞÞ denotes stacking the ele-ments of the matrix M1ðx; h; tÞ by rows in a vector form;A0 ¼ In � AT

c0þ Ac0

� In is a constant matrix and the sym-bol � denotes the Kronecker product. Thus, the resultingsolution of T1ðx; hÞ can be written in a closed-form expres-sion vecðT1ðx; hÞÞ ¼ A0

�1vec M1ðx; h; tÞ½ �.

(3) Solve Eqs. (22c) and (22d) for T2;…; Tn following the sim-ilar procedure in (2).



Since all the coefficients of Ti; i ¼ 1;…; n on the left-hand sideof the Eqs. (22b)–(22d) are the same constant matrices, i.e.,A0 � g0R�1g0

TT0 and A0T � T0g0R�1g0

T , closed-form solutionfor each Tiðx; hÞ can be easily obtained with only one matrixinverse operation, i.e., A0

�1.

The perturbation matrix Di; i ¼ 1;…; n is constructed asfollows:

D1 ¼ k1e�l1t

��T0AðxÞ

h� ATðxÞT0


hT0

þ T0

gðxÞh

R�1g0TT0

� (23a)

D2¼k2e�l2t½�T1AðxÞh�ATðxÞT1

hþT0g0R�1 gTðxÞ

hT1

þT0

gðxÞh

R�1g0TT1:þT0

gðxÞh

R�1 gTðxÞh

T0þT1g0R�1 gTðxÞh

T0

þT1

gðxÞh

R�1g0TT0þT1g0R�1g0

TT1�

(23b)

Dn ¼ kne�lnt

(� Tn�1AðxÞ

h� ATðxÞTn�1

h

þXn�1

j¼0

Tj½g0R�1 gTðxÞhþ gðxÞ

hR�1g0

T �Tn�1�j

þXn�2

j¼0

TjgðxÞh

R�1 gTðxÞh

Tn�2�j þXn�1

j¼1

Tjg0R�1gT0 Tn�j

)

(23c)

where ki and li > 0; i ¼ 1;…; n are adjustable design parameters.The Di are chosen such that

�Ti�1AðxÞh

�ATðxÞTi�1

hþXi�2

j¼0

TjgðxÞh

R�1 gTðxÞh

Ti�2�j

þXi�1

j¼0

Tjðg0R�1 gTðxÞhþgðxÞ

hR�1g0

TÞTi�1�j

þXi�1

j¼1

Tjg0R�1g0TTi�j�Di¼ eiðtÞ�

Ti�1AðxÞh

�ATðxÞTi�1

h

þXi�1

j¼0

Tj½g0R�1 gTðxÞhþgðxÞ

hR�1g0

T �Ti�1�j:

þXi�2

j¼0

TjgðxÞh

R�1 gTðxÞh

Ti�2�jþXi�1

j¼1

Tjg0R�1g0TTi�j

)(24)

where

eiðtÞ ¼ 1� kie�lit (25)

ei is chosen to be a small number used to overcome the initiallarge control gain problem because the state-dependent terms A(x)and g(x) on the right-hand side of the Eqs. (22b)–(22d) may causea large magnitude of Tiðx; hÞ if the initial states are large. Let ussuppose there is no ei or Di in Eq. (22). For example, in Eq. (22b),if there exists a cubic term in A(x) and the initial x is large, thislarge value will be reflected in the solution of T1. Since T1 andA(x) will be used in solving for T2 in the ensuing Eq. (22c), thislarge value will be propagated and amplified. As a result, it causesthe large control gain or even instability. Therefore, the small

number ei is used to suppress this large value from propagating inEqs. (22b)–(22d).

ei is also required in the proof of convergence and stability ofthe above algorithm [25]. The exponential term e�lit in Di lets theperturbation terms in the cost function [Eq. (18)] diminish as timeevolves.

Remark 3.2. ki and li are design parameters used to modulatesystem transient responses, which can be systematically deter-mined by the method in Refs. [25, 26] and will be discussed indetail at the end of Sec. 3.

Remark 3.3. h is merely an intermediate variable. The introduc-tion of h is for the convenience of power series expansion, and itis cancelled when Tiðx; hÞ multiply hi in the final control calcula-tions, i.e., Eq. (21). Note that in every Ti [Eq. (22)] and Di expres-sion [Eq. (23)], 1

�hi factor appears linearly on the right-hand side

of the equations. Consequently, 1�hi will appear linearly in the so-

lution of Ti, i.e., Ti ¼ Ti

�hi where Ti is the solution without 1

�hi.

When Ti is multiplied by hi in the control Eq. (21), hi gets can-celled. The cancellation is also true for the perturbation seriesP1

i¼1 Dihi.

In the h� D algorithm, retaining the first three termsT0; T1, and T2 in the control Eq. (21) has been sufficient toachieve satisfactory performance in the problems that havebeen solved [25,26,31]. Theoretical results concerning theconvergence of the series

P1i¼0 Tiðx; hÞhi, closed-loop stabil-

ity, and optimality of truncating the series can be referred toRef. [25].

3.3 Trajectory Control Design Using the h� DMethod. The problem we are addressing is to control the minia-ture helicopter to track desired trajectories. The unified state varia-bles are defined in Eq. (8) and the control variables are defined inEq. (11). The optimal control problem is described by Eqs. (10)and (12).

In order to apply the h� D method, f(0)¼ 0 needs to be satis-fied in order to factorize f(x) into the linearlike structure FðxÞx. Ifthere are state-independent terms that make f ð0Þ 6¼ 0, they pre-vent a direct C1 factorization of f(x) and they are called bias terms.For example, the terms such as Xfus=m, Yfus þ Ytr þ Yvf

� �=m, and

Zfus=mþ g cos / cos h are bias terms because they may not go tozero when the states become zero. One of the ways to handle thisproblem is to augment the system with a stable state “s” satisfying[26,30,31]

_s ¼ �kss (26)

in which ks is a positive number. With the augmented state s, thebias term denoted by b(t) can then be factorized as

bðtÞ ¼ bðtÞs

� �s (27)

The introduction of s does not change the actual helicopter dy-namics. Each time through the controller, the initial value s(0) isused in the state-dependent coefficient matrices and in calculatingthe control. Usually we set s(0)¼ 1.

Including the augmented state s, the total state-space contains18 states and is denoted by

xa ¼xp yp zp u v w / h w p q r

a1 b1 xi yi zi s

� �T

2 <18�1

(28)

The state-dependent coefficients FðxaÞ and BðxaÞ of the h� D for-mulation, Eq. (19), can be written as



FðxaÞ ¼

03�3 R Hð Þ3�3 03�1 03�1 03�1 03�1 03�1 03�1 03�1 03�1 03�3 03�1

03�3 ~x3�3

0

a57

0

24

35 a48

0

0

24

35 03�1 03�1 03�1 03�1 03�1 03�1 03�3

a4 18

a5 18

a6 18

24

35

01�3 01�3 0 0 0 1 a7 11 a7 12 0 0 01�3 0

01�3 01�3 0 0 0 0 a8 11 a8 12 0 0 01�3 0

01�3 01�3 0 0 0 0 a9 11 a9 12 0 0 01�3 0

01�3 01�3 0 0 0 0 a10 11 a10 12 0 a10 14 01�3 a10 18

01�3 01�3 0 0 0 a11 10 0 a11 12 a11 13 0 01�3 a11 18

01�3 01�3 0 0 0 a12 10 a12 11 0 0 0 01�3 a12 18

01�3 01�3 0 0 0 0 �1 0 a13 13 0 01�3 a13 18

01�3 01�3 0 0 0 �1 0 0 0 a14 14 01�3 a14 18

I3�3 01�3 0 0 0 0 0 0 0 0 01�3 03�1

01�3 01�3 0 0 0 0 0 0 0 0 01�3 �ks

266666666666666666666664

377777777777777777777775

(29)

where

~x3�3 ¼0 r �q�r 0 pq �p 0

24

35;

a48 ¼ �gsin h

h; a4 18 ¼

Xfus

m � s ; a57 ¼ g cos hsin /

/;

a5 18 ¼Yfus þ Ytr þ Yvf

m � s ; a6 18 ¼Zfus

m � sþg cos / cos h

s

a7 11 ¼ sin / tan h; a7 12 ¼ cos / tan h; a8 11 ¼ cos /;

a8 12 ¼ � sin /; a9 11 ¼ sin / sec h;

a9 12 ¼ cos / sec h;

a10 11 ¼rIyy

Ixx; a10 12 ¼ �

qIzz

Ixx; a10 14 ¼

Kb

Ixx; a10 18 ¼

Lvf þ Ltr

Ixx � s;

a11 10 ¼ �rIxx

Iyy; a11 12 ¼

pIzz

Iyy; a11 13 ¼

Kb

Iyy; a11 18 ¼

Mht

Iyy � s;

a12 10 ¼qIxx

Izz; a12 11 ¼ �

pIyy

Izz; a12 18 ¼

Nvf þ Ntr � Qe

Izz � s;

a13 13 ¼�1

se; a13 18 ¼

16l2Kl

8 lj j þ arsign lð Þ

� �wa

seXmrRmr � s

� 2Klk0

ua

seXmrRmr � s;

a14 14 ¼ �1

se; a14 18 ¼ �2Klk0

va

seXmrRmr � s;

and

BðxaÞ ¼

03�1 03�1 03�1 03�1 03�1

b41 0 0 0 0

b51 b52 0 0 0

b61 0 0 0 0

03�1 03�1 03�1 03�1 03�1

b10 1 b10 2 0 0 0

b11 1 0 0 0 0

0 b12 2 0 0 0

0 0 b13 3 b13 4 0

0 0 b14 3 0 b14 5

04�1 04�1 04�1 04�1 04�1

2666666666666666664

3777777777777777775

(30)

where

b41 ¼ �a1

m; b51 ¼

a1

m; b52 ¼ Ytr

dped; b61 ¼ �

1

m;

b10 1 ¼hmrb1

Ixx; b10 2 ¼

htrmYtrdped

Ixx; b11 1 ¼

hmra1

Iyy;

b12 2 ¼ �mYtr

dpedltr

Izz; b13 3 ¼

8Klua

3seXmrRmr; b13 4 ¼

Anomdlon

se

Xmr

Xnom

� 2

;

b14 3 ¼8Klva

3seXmrRmr; b14 5 ¼

Bnomdlat

se

Xmr

Xnom

� 2

The A0, AðxaÞ, g0, and gðxaÞ are chosen to be A0 ¼ Fðxaðt0ÞÞ,AðxaÞ ¼ FðxaÞ � Fðxaðt0ÞÞ, g0 ¼ Bðxaðt0ÞÞ, and gðxaÞ ¼ BðxaÞ�Bðxaðt0ÞÞ, i.e.,

_xa ¼ Fðxaðt0ÞÞ þ hFðxaÞ � Fðxaðt0ÞÞ

h

� � �xa

þ Bðxaðt0ÞÞ þ hBðxaÞ � Bðxaðt0ÞÞ

h

� � �u (31)

Remark 3.4. The state-dependent factorization approach(FðxaÞ;BðxaÞ) captures the nonlinearities and dynamic changes inthe helicopter model as the states vary. Thus, the optimal controlgains based on this live information can be computed adaptivelyeven under uncertain operating conditions.

The cost function for this problem is chosen to be a quadraticfunction

J ¼ 1

2

ð10

xaTQxa þ uTRu

� �dt (32)

where Q and R are the weighting matrices. After tuning, they arechosen to be

Q ¼ diagð½104; 2� 104; 107; 100; 100; 1; 100; 100; 103; 200; 1;

200; 105; 105; 2� 103; 2� 103; 1� 106; 0�Þ(33)

R ¼ diag 1; 200; 107; 105; 105� ��

(34)

where diag �ð Þ represents a diagonal matrix.In order to perform the command tracking, the h� D controller

is implemented as an integral servomechanism as shown in[26,31]

u ¼ �R�1BTðxaÞ T0 þ T1 xa; hð Þ þ T2 xa; hð Þ½ �Vtrack (35)

where

Vtrack ¼½xp� xc;yp� yc; zp� zc;u;v;w;/;h;w;p;q; r;a1;b1;xi

�ð

xcdt;yi�ð

ycdt; zi�ð

zcdt;1�Tð36Þ

xc yc zc½ �T is the desired position vector that the helicopterneeds to track. T0, T1 xa; hð Þ; and T2 xa; hð Þ are solved from



the h� D algorithm Eqs. (22) and (23). Using these threeterms is found to be sufficient to produce good trackingperformance.

The parameters in the perturbation matrix (23) are chosen to bek1¼ 1, k2¼ 1, l1¼ 1, and l2¼ 1. Recall that the ki and li parame-ters are used to tune the system transient performance as men-tioned in Remark 3.2. ki are usually chosen to be close to 1 such

that eiðtÞ ¼ 1� kie�lit is a small number to mitigate the large ini-

tial control problem discussed in Sec. 3.2. li control how fast theperturbation terms Di diminish and the transient response. Theselection of these parameters can be done systematically [26,31]by applying least-squares curve fitting such that the errorsbetween rmax½

Pni¼0 Tiðxa; hÞ� and rmax PðxaÞ½ � are minimized,

where rmax denotes maximum singular value and PðxaÞ is the so-lution to the following SDRE

PðxaÞFðxaÞþFTðxaÞPðxaÞ�PðxaÞBðxaÞR�1BTðxaÞPðxaÞþQ¼ 0

(37)This strategy to determine the parameters is based on the observa-tion that the h� D method is similar to the SDRE method in thesense that both bring the nonlinear equation into a linearlike struc-ture and solve a quadratic optimal control problem. The formergives an approximate closed-form solution while the latter solvesthe algebraic Riccati equation numerically online. Therefore, itcan be expected that the h� D solution is close to the SDRE solu-tion, i.e., Pðh� DÞ PðSDREÞ where Pðh� DÞ ¼

Pni¼0

Tiðxa; hÞhi is obtained by solving Eq. (22) and PðSDREÞ ¼ PðxaÞis obtained by solving Eq. (37). An SDRE controller can be firstused to generate a state trajectory of xa and then the maximumsingular value of PðxaÞ, i.e., rmax PðxaÞ½ �, is computed at each statepoint xa. Similarly, the ki and li parameters determinePðh� DÞ ¼

Pni¼0 Tiðxa; hÞhi and its associated rmax

Pni¼0

�Tiðxa; hÞ� along the trajectory xa. Thus, these parameters can beselected offline to minimize the difference between rmax

Pni¼0

�Tiðxa; hÞ� and rmax PðxaÞ½ � in the least-squares sense.

Note that we have designed an optimal controller for this highlynonlinear helicopter system with 18 state variables. The h� Dtechnique is particularly useful for solving this type of high-ordernonlinear control problems since the optimal controller, Eq. (35),can be obtained in a closed-form by virtue of the h� D algorithm,Eqs. (22) and (23). This closed-form law facilitates the real-timeimplementation because it does not demand intensive and iterativecomputations.

4 Simulation Results and Analysis

In this section, the simulation results of the position trackingusing the unified optimal control formulation are demonstrated. Inorder to show the tracking performance and the robustness of theproposed controller, 45 uncertainties and the wind gust, which aredefined in Sec. 2.4, are included in the simulation. The nominalvalues of the aerodynamic coefficients, geometry parameters, andphysical constants are used in the control law design whereas theuncertainties are added in testing scenarios. All the feedbackstates are assumed to be measurable. However, measurementuncertainties are considered into the control design by addingwhite Gaussian noises with the standard deviation of 5% of themagnitude of the feedback states as given in Table 1. In addition,when the computed control law is applied to the helicopterdynamics, noises with the level of 5% of the control magnitudeare added to represent the actuator uncertainties. Position trackingover the entire flight envelope includes trajectories in any direc-tion with the speed up to the maximum allowed by the physicalcapabilities of the helicopter (20 m=s [27]).

Two simulation scenarios are used below. The trajectories arecombinations of ramps, sinusoidal waves, and step inputs in threedirections. These complex paths demonstrate a full flight envelopebecause they require (1) vertical takeoff; (2) vertical landing; (3)hover; (4) forward, rearward, and sideward flight at a constant alti-

tude; (5) forward, rearward, and sideward flight at varying alti-tudes; and (6) tracking in all three dimensions simultaneously.

The first desired trajectory starts with a vertical takeoff to analtitude of 3.5 m, followed by a 5-s hover, then a figure “8” trajec-tory in the x–y plane, followed by a 10-s hover, and then a verticallanding.

Figures 2–7 present the results of the first simulation scenario.Figure 2 gives a three dimensional view of the trajectory and Fig.3 shows the time histories of the x, y, and z components of the in-ertial position vector, respectively. In the figure 8 maneuver, thedesired trajectory has the helicopter travel in the negative y direc-tion first and it eventually returns to the start of the figure 8. Ascan be seen, the tracking in the x and z directions is very preciseeven though there are two abrupt command changes. The smalldeviations on the transient y direction occur during transitionsbetween ascending or descending flight and the figure 8 maneu-ver. The helicopter either is already hovering and then catching upwith the figure 8 trajectory or is trying to halt and resume hoveringfollowing the maneuver in these regions. The trajectory showsslight overshoots at these points in the y direction because of theabrupt change of the velocity commands.

In Fig. 4, the three Euler angles are shown. A few quick jumpscan be observed in the roll and yaw angles due to the start andstop of the figure 8 maneuver. But, in between, they show smoothtransient responses. The angular velocity responses are shown inFig. 5. The similar abrupt jumps can be seen during the trajectorymaneuvers. Oscillations on the curve stem from the uncertaintyand random noises in the dynamics model because angular veloc-ities are fast changing variables with larger bandwidth and thuseasier to be affected by the noise. Translational velocities areshown in Fig. 6 and are for the most part smooth, with a few largejumps occurring in the “w” response, corresponding to the abruptchanges in altitude commands at these moments. Figure 7 showsthe flapping angle responses. The high frequency oscillation isbecause the flapping mode is the fastest dynamic mode with alarge bandwidth and the uncertainties and noises manifest them-selves more significantly. But the magnitude is in a reasonablerange. The control deflections and the main rotor thrust are shownin Figs. 8 and 9, respectively. These curves show the similar trendwith large jumps occurring at the transition periods of the trajec-tory. Also, the oscillations are due to the uncertainties and noisesadded in the simulation.

In the second simulation scenario, the helicopter also startswith a vertical takeoff to 3.5 m and then traverses a large circle inthe x–y plane while descending and then ascending, as seen from

Fig. 2 Scenario I: three-dimensional view of the trajectory



Fig. 3 Scenario I: time histories of the position tracking

Fig. 4 Scenario I: Euler angle responses

Fig. 5 Scenario I: angular velocity response



Fig. 7 Scenario I: flapping angle responses

Fig. 6 Scenario I: translational velocity responses

Fig. 8 Scenario I: control deflections



the three dimensional view in Fig. 10. The time histories of thethree position components are shown in Fig. 11. The miniaturehelicopter follows the trajectory in a counterclockwise motion.The result is similar to the first scenario with an excellent trackingin the x and z directions and very small deviations in the y direc-tion due to the same reason as that in the first scenario. Figures12–15 give the responses of the Euler angles, angular velocities,translational velocities, and flapping angles, respectively. Theresults are very similar to the corresponding ones in the first sce-nario. The large jumps occur because of the abrupt changes in thecommanded signal. The high frequency oscillations are due to thenoises and uncertainties in the simulations. Figures 16 and 17show the control deflections and the main rotor thrust responses,which are also similar to the results in the first scenario.

The above two simulation results demonstrate the capabilitiesof the unified optimal control design in full envelope trajectorytracking since all possible helicopter motions are considered. It isnecessary to recapitulate the two significant advantages of theapproach proposed in this paper. First of all, the single optimalcontrol formulation integrating the translational motion and atti-tude motion as well as the fast flapping motion simplifies thedesign process, which is much easier to conduct than the tradi-

tional timescale separation design paradigm since it avoids exces-sive iterations and tuning process between different dynamicmodes. Second, the closed-form optimal control law provided bythe h� D method (algorithm Eqs. (22) and (23)) facilitates thereal-time implementation because it does not demand intensivecomputational load. This feature becomes more important for theminiature helicopter application since the computational resourcesonboard the vehicle may be very limited.

5 Conclusion

In this paper, trajectory control of a miniature helicopter overcomplex paths was addressed using a unified optimal control for-mulation and the h� D nonlinear optimal control technique. Theunified formulation greatly simplifies the design process and opti-mizes the overall system performance. The significant benefit ofusing the h� D method is to provide an approximate closed-formfeedback controller, which does not require intensive computa-tions. The advantages of the control design in this underactuated

Fig. 9 Scenario I: main rotor thrust

Fig. 10 Scenario II: 3-dimensional view of the trajectory

Fig. 11 Scenario II: time histories of the position tracking



Fig. 12 Scenario II: Euler angle responses

Fig. 13 Scenario II: angular velocity responses

Fig. 14 Scenario II: translational velocity responses



miniature helicopter have been demonstrated by complex trajec-tory tracking including vertical takeoff, vertical landing, hover,forward, rearward, and sideward flight. The robustness of the con-troller was also validated in the simulation by taking into account45 functional and parametric uncertainties as well as the windgust. Furthermore, as compared with the conventional timescaleseparation based control systems, the proposed control formula-tion is much easier to design and implement.

Acknowledgment

This work was supported by the U.S. National Science Founda-tion CAREER Award No. ECCS-0846877.

Appendix A: Helicopter Model Parameters—

Translational Motion

The external forces contain the aerodynamic forces acting on thefuselage, Xfus, Yfus, and Zfus, the force on the vertical fin

Yvf ¼ � 1=2ð ÞqSvf ðCvfLa

Vtr1þ vvf

Þvvf , the force from the tail rotor

Ytr ¼ mYtrv vtr , the main rotor thrust Tmr, and the gravitational

force. Vtr1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

a þ w2tr

p, vvf ¼ va � etr

vf Vitr � ltrr, and

Fig. 15 Scenario II: flapping angle responses

Fig. 16 Scenario II: control deflections

Fig. 17 Scenario II: main rotor thrust



wtr ¼ wa þ ltrq� KkVimr , where ua ¼ u� uw, va ¼ v� vw, andwa ¼ w� ww, are relative velocities with respect to the wind andthe subscript w denotes the wind. Vimr is the induced main rotorvelocity and Vitr is the induced tail rotor velocity. Svf is the effec-tive vertical fin area; etr

vf is the vertical fin area portion exposed to

the induced tail rotor velocity; ltr is the tail rotor hub locationbehind the center of gravity. The side force on the vertical fin is

constrained by Yvf

0:5qSvf ððVtr1Þ

2 þ v2vf Þ to compensate for

potential stall of the vertical fin [23,27].Kk is the wake intensity factor calculated according to

Kk ¼

0 if Vimr wa or uaVimr � wa

gi

1:5 if uaVimr � wa

� gf

else 1:5

uaVimr � wa

�gi

gf � gi;

8>>><>>>:

(A1)

with

gi ¼ltr � Rmr � Rtr

htr(A2)

and

gf ¼ltr � Rmr þ Rtr

htr(A3)

where Rtr and Rmr are the radius of the tail rotor and the mainrotor, respectively. The main rotor engine speed Xmr is calculatedby solving the collection of equations below iteratively using thezero-finding algorithm [23] once the main rotor thrust Tmr is com-puted from the control law

2gwk0qpR4mrX

2mr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2 þ k0 � lzð Þ2

q� Tmr ¼ 0 (A4)

l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

a þ v2a

pXmrRmr

(A5)

lz ¼wa

XmrRmr(A6)

Here, gw is the coefficient to account for the nonideal and nonuni-form wake; k0 is the inflow ratio; l is the advance ratio; lz is thenormal airflow component.The forces acting on the fuselage are given by

Xfus ¼ �0:5qCDxSfusx uaV1 (A7)

Yfus ¼ �0:5qCDySfusy vaV1 (A8)

and

Zfus ¼ �0:5qCDzSfusz wa þ Vimrð ÞV1 (A9)

where the axial velocity magnitude is V1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

a þ v2a þ wa þ Vimrð Þ2

qand Sfus

x , Sfusy , and Sfus

z are the front fuselage drag area, side fuse-lage drag area, and vertical fuselage drag area, respectively.

Appendix B: Helicopter Model Parameters—Rotational

Motion

In this motion, the moments produced by the tail rotor are calculatedby Ltr ¼ htrmYtr

v vtr and Ntr ¼ �ltrmYtrv vtr , where vtr ¼ va � ltrr

þhtrp. The moments of the vertical fin are Lvf ¼ Yvf htr andNvf ¼ �Yvf ltr . The moment generated from the horizontal tail is

Mht ¼ Zhtlht, where Zht ¼ 0:5qShtðChtLa

uaj jwht þ whtj jwhtÞ is the lift

from the horizontal tail with Sht the horizontal tail area, ChtLa

the hori-

zontal tail lift curve slope, and wht ¼ wa þ lhtq� KkVimr . The liftfrom the horizontal stabilizer is constrained by Zhtj j 0:5qSht

u2a þ w2

ht

� �to accommodate for stall. Kk is defined in Eq. (A1) with

different values depending upon the relation between Vimr and wa.The engine torque is modeled as

Qe ¼ �Irot _r þ Qmr þ ntrQtr (B1)

where the main rotor torque is calculated by Qmr ¼ CQmrq Xmrð

RmrÞ2pRmr3 and CQmr

¼ CT k0 � lzð Þ þ CD0r=8 1þ 7

3l2

� �. In this

equation, r ¼ 2cmr=ðpRmrÞ and CT ¼ 2gwk0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2 þ k0 � lzð Þ2

q.

Xtr ¼ ntrXmr is the tail rotor speed and ntr is the gear ratio betweenthe main rotor and the tail rotor.

Nomenclature

a1; b1¼ longitudinal and lateral rotor flapping anglesa; amr ¼ main rotor blade lift curve slope, 5.5 rad�1

atr ¼ tail rotor blade lift curve slope, 5.0 rad�1

Anomdlon¼ nominal longitudinal cyclic to flap gain, 4.2 rad/

radBnom

dlat¼ nominal lateral cyclic to flap gain, 4.2 rad/rad

CDx;CDy

;CDz¼ drag coefficients in three directions, 0.1, 0.5, 0.2

CmrD0¼ main rotor blade zero lift drag coefficient, 0:024

CtrD0¼ tail rotor blade zero lift drag coefficient, 0:024

ChtLa¼ horizontal tail lift curve slope, 3.0 rad�1

CvfLa¼ vertical fin lift curve slope, 2.0 rad�1

cmr ¼ main rotor chord, 0.058 mctr ¼ Tail rotor chord, 0.029 m

CQ ¼ CQmr¼ main rotor torque coefficient

CT ¼ thrust coefficientCmr

Tmax¼ maximum main rotor thrust coefficient, 0:0055

g¼ gravity acceleration, 9.81 m/s2

hmr ¼ main rotor height above c.g., 0.235 mhtr ¼ tail rotor height above c.g., 0.08 mIrot¼ rotating inertia of the main rotor blade, 2:5Ibmr

Ixx¼ rolling moment of inertia, 0.18 kg m2

Iyy¼ pitching moment of inertia, 0.34 kg m2

Izz¼ yawing moment of inertia, 0.28 kg m2

Ibmr¼ main rotor blade flapping inertia, 0.038 kg m2

Kb¼ hub torsional stiffness, 54 N m/radKk¼ main rotor wake intensity factorKl¼ scaling of flap response to speed variation, 0:2

L;M;N¼ rolling, pitching, and yawing momentslht¼ location of stabilizer bar behind c.g., 0.71 mltr ¼ tail rotor location behind c.g., 0.91 mm¼ helicopter mass, 8.2 kg

ntr ¼ tail rotor to main rotor gear ratio, 4:66p; q; r¼ roll, pitch, and yaw rate

Qe¼ engine torqueRmr ¼ main rotor radius, 0.775 mRtr ¼ tail rotor radius, 0.13 m

R Hð Þ¼ direct cosine matrix from body to inertialcoordinates

Sht¼ horizontal fin area, 0.01 m2

Svf ¼ effective vertical fin area, 0.012 m2

Sfusx ¼ front fuselage drag area, 0.1 m2

Sfusy ¼ side fuselage drag area, 0.22 m2

Sfusz ¼ vertical fuselage drag area, 0.15 m2

Tmr ¼ main rotor thrustTmax

mr ¼ maximum main rotor thrustu; v;w¼ body-axis velocities

V1¼ axial velocity magnitudeVtr1¼ axial velocity at the tail rotor

Vimr ¼ induced main rotor velocity, 4.2 m/sVitr¼ induced tail rotor velocity, 0.42 m/s

xp; yp; zp¼ inertial positionX;Y; Z¼ body-axis forces

Ytrdpedal¼ tail rotor side force coefficient due to pitch, 5.4

m/(s2 rad)



Ytrv ¼ tail rotor side force coefficient due to velocity,

0.06 s�1

dcol¼ main rotor collective inputdlat¼ lateral cyclic inputdlon¼ longitudinal cyclic inputdped¼ pedal inputetr

vf ¼ portion of vertical fin exposed to tail rotorinduced velocity, 0.2

gw¼ coefficient of nonideal wake contraction, 0:9k0¼ inflow ratio of the main rotor, 0:033l¼ advance ratiolz¼ normal airflow component

Xnom¼ nominal main rotor speed, 167 rad/sX¼ rotor speedq¼ atmospheric density, 1.225 kg/m3

H ¼ h;/;w½ � ¼ Euler anglesr¼ solidity ratio, 0.0361se¼ damping time constant for the flapping motion,

0.1 s

Subscripta¼ actuald¼ desirede¼ engine

fus¼ fuselageht¼ horizontal tail

mr¼ main rotortr¼ tail rotorvf ¼ vertical finw¼ wind

References[1] Department of Defense, Airspace Integration Plan for Unmanned Aviation

(Office of the Secretary of Defense, Washington, DC, November 2004), pp. 1–5, 21–28.

[2] “Association for Unmanned Vehicle Systems International’s Aerial RoboticCompetition,” http://avdil.gtri.gatech.edu/AUVS/IARCLaunchPoint.html,Document accessed: March 21, 2008.

[3] Kaminer, I., Pascoal, A., Hallberg, E., and Silvestre, E., 1998, “TrajectoryTracking for Autonomous Vehicles: An Integrated Approach to Guidance andControl,” J. Guid. Control Dyn., 21(1), pp. 29–38.

[4] Shin, J., Nonami, K., Fujiwara, D., and Hazawa, K., 2005, “Model-BasedOptimal Attitude and Positioning Control of Small-Scale Unmanned Heli-copter,” Robotica, Cambridge University Press, United Kingdom, Vol. 23, pp.51–63.

[5] Perhinschi, M., and Prasad, J., 1998, “A Simulation Model of an Autono-mous Helicopter,” Proceedings of the RPV/UAV Systems Thirteenth BristolInternational Conference and Exhibition, Bristol, United Kingdom, March,pp. 1–13.

[6] Cheviron, T., Chriette, A., and Plestan, F., 2006, “Robust Control of an Autono-mous Reduced Scale Helicopter in Presence of Wind Gusts,” AIAA Guidance,Navigation, and Control Conference and Exhibit, Keystone, CO, Aug. 21–24, pp.1–22.

[7] Civita, M., Papageorgiou, G., Messner, W., and Kanade, T., 2006, “Design andFlight Ttesting of an Controller for a Robotic Helicopter,” J. Guid. ControlDyn., 29(2), pp. 485–494.

[8] Bendotti, P., and Morris, J., 1995, “Robust Hover Control for a Model Heli-copter,” Proceedings of the 1995 American Control Conference, Seattle, WA,June, pp. 682–687.

[9] Kato, A., Kobayashi, T., Katayama, H., and Ichikawa, A., 2003, “Attitude Con-trol of a Helicopter Model by Nonlinear Sampled-Data Control,” SICE 2003Annual Conference, Fukui University, Japan, Aug. 4–6, pp. 930–935.

[10] Weilenmann, M., Christen, U., and Geering, H., 1994, “Robust Helicopter Posi-tion Control at Hover,” Proceedings of the 1994 American Control Conference,Baltimore, MD, June, pp. 2491–2495.

[11] Perhinschi, M., and Prasad, J., 1998, “A Simulation Model of an Autono-mous Helicopter,” Proceedings of the RPV/UAV Systems Thirteenth BristolInternational Conference and Exhibition, Bristol, United Kingdom, March,pp. 1–13.

[12] Takahashi, M., 1994, “H-Infinity Helicopter Flight Control Law Design Withand Without Rotor State Feedback,” J. Guid. Control Dyn., 17(6), pp. 1245–1251.

[13] Shim, D., Kim, H., and Sastry, S., 2000 “Control System Design for Rotocraft-Based Unmanned Aerial Vehicles Using Time-Domain System Identification,”Proceedings of the 2000 IEEE International Conference on Control Applica-tions, Anchorage, AK, Sept. 25–27, pp. 808–813.

[14] Zeng, S., and Zhu, J., 2006, “Adaptive Compensated Dynamic Inversion Con-trol for a Helicopter With Approximate Mathematical Model,” InternationalConference on Computational Intelligence for Modeling Control and Automa-tion, and International Conference on Intelligent Agents, Web Technologiesand Internet Commerce, IEEE, Sydney, Australia, Nov. 28–Dec. 1, pp. 1–6.

[15] Johnson, E., and Kannan, S., 2005, “Adaptive Trajectory Control for Autono-mous Helicopters,” J. Guid. Control Dyn., 28(3), pp. 524–538.

[16] Spurgeon, S., Edwards, C., and Foster, N., 1996, “Robust Model ReferenceControl Using a Sliding Mode Controller/Observer Scheme With Applicationto a Helicopter Problem,” 1996 IEEE Workshop on Variable Structure Systems,Tokyo, Japan, Dec. 5–6, pp. 36–41.

[17] Oh, A., Ryu, J., and Agrawal, S., 2006, “Dynamics and Control of a HelicopterCarrying a Payload Using a Cable-Suspended Robot,” ASME J. Mech. Des.,128, pp. 1113–1121.

[18] Verhulst, F., 2005, Methods and Applications of Singular Perturbations Bound-ary Layers and Multiple Timescale Dynamics Springer ScienceþBusinessMedia, Inc., New York, NY, USA.

[19] Avanzini, G., and Matteis, G., 2001, “Two-Timescale Inverse Simulation of aHelicopter Model,” J. Guid. Control Dyn., 24(2), pp. 330–339.

[20] Reiner, J., Balas, G., and Garrard, W., 1996, “Flight Control Design Using Ro-bust Dynamic Inversion and Time-Scale Separation,” Automatica, 32(11), pp.1493–1504.

[21] Marconi, L., and Naldi, R., 2006, “Robust Nonlinear Control for a MiniatureHelicopter for Aerobatic Maneuvers,” Proceedings 32nd Rotorcraft Forum,Maasctricht, The Netherlands, pp. 1–16.

[22] Avanzini, G., and Matteis, G., 1999, “Two-Timescale-Integration Method forInverse Simulation,” J. Guid. Control Dyn., 22(3), pp. 330–339.

[23] Hopkins, R., and Xu, Y., 2008, “Position Tracking Control for a SimulatedMiniature Hhelicopter,” AIAA Guidance, Navigation, and Control Conferenceand Exhibit, Honolulu, Hawaii, Aug. 18–21.

[24] Bogdanov, A., and Wan, E., 2007, “State-Dependent Riccati Equation Con-trol for Small Autonomous Helicopters,” J. Guid. Control Dyn., 30(1), pp.47–60.

[25] Xin, M., and Balakrishnan, S. N., 2005, “A New Method for SuboptimalControl of a Class of Nonlinear Ssystems,” Opt. Control Appl. Methods,26(3), pp. 55–83.

[26] Xin, M., Balakrishnan, S. N., Stansbery, D. T., and Ohlmeyer, E. J., 2004,“Nonlinear Missile Autopilot Design With Technique,” J. Guid. Control Dyn.,27(3), pp. 406–417.

[27] Gavrilets, V., Mettler, B., and Feron, E., 2003, “Dynamic Model for a MiniatureAerobatic Helicopter,” MIT-LIDS Report, Report No. LIDS-P–2580.

[28] Federal Aviation Administration, 1999, “Criteria for Approval of Category IIIWeather Minima for Takeoff, Landing, and Rollout: Appendix 4—Wind Modelfor Approach and Landing Simulation,” AC120–28D.

[29] Bryson, A. E., and Ho, Y. C., Applied Optimal Control (Hemisphere Publish-ing, Corp., 1975).

[30] Cloutier, J. R., and Stansbery, D. T., 2002, “The Capabilities and Art of State-Dependent Riccati Equation-Based Design,” Proceedings of the 2002 AmericanControl Conference, pp. 86–91.

[31] Xin, M., Balakrishnan, S. N., and Ohlmeyer, E. J., 2006, “Integrated Guidanceand Control of Missiles With Method,” IEEE Trans. Control Syst. Technol.,14(6), pp. 981–992.



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Trajectory Control of Miniature Helicopters Using a Unified Nonlinear Optimal Control Technique

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