IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006 1297

that the target of interest disappears for a while, but tracking is main-tained in this situation. The reason of success is explained as follows.

It is well known that the target detection process used, SAD templatematching, is weak when it comes to tracking with occlusion. When oc-clusion happens, the reference template can not find similar measure-ments in the sensed image frame before the template is updated. How-ever, this case is also considered in VPDAF by assuming event �0(k)in (4). The corresponding association probability �0(k) dominates theestimate. The updated state x0(kjk) in (2) increases the effect on thefinal weighted average x(kjk) at the same time. The result is that theestimation only depends on the object motion model. Once the target ofinterest appears around this estimate again, the SAD template matchingdetects candidates, and the functions of VPDAF are recovered.

V. CONCLUSION

The VPDAF proposed in this paper is an extension of the traditionalPDAF [1], which is designed mainly for target tracking from a clusterof measured data. It constructs the data-associated relationships of themeasurements based on their positions. In the field of visual tracking,the measured data obtained through target detection in the image do-main include not only position but also visual similarity information.Thus, in this paper, the significant visual characteristics are consid-ered, while augmented association probabilities are derived. On theother hand, nontarget objects or some patterns in the background aremajor disturbances for visual target tracking. Fortunately, these per-sistent noises can be eliminated by classifying their motions. Throughpersistent noise removal, VPDAF generates modified data-associationprobabilities and outputs the final suboptimal target estimate.

VPDAF with template matching applied to target detection has beeninvestigated through several experiments. The visual tracking systemis constructed on either a fixed or 2-DOF camera platform. Reliabletarget-tracking performance has been achieved, even in a highly clut-tered environment or one subjected to occasional occlusions. AlthoughVPDAF takes a bit more time to evaluate the measurement variationsand to filter out the stationary noise than the traditional PDAF, it stillcan be processed in about 30 ms for every image frame. Hence, real-time visual tracking is indeed achieved in our work.

REFERENCES

[1] Y. Bar-Shalom and T. E. Fortmann, Tracking and Data Association.New York: Academic, 1988.

[2] H. M. Shertukde and Y. Bar-Shalom, “Tracking of crossing targets withimaging sensors,” IEEE Trans. Aerosp. Electron. Syst., vol. 27, no. 4,pp. 582–592, Jul. 1991.

[3] A. Kumar, Y. Bar-Shalom, and E. Oron, “Precision tracking based onsegmentation with optimal layering for imaging sensors,” IEEE Trans.Pattern Anal. Mach. Intell., vol. 17, no. 2, pp. 182–188, Feb. 1995.

[4] T. Kirubarajan, Y. Bar-Shalom, and K. R. Pattipati, “Multiassignmentfor tracking a large number of overlapping objects and application tofibroblast cells,” IEEE Trans. Aerosp. Electron. Syst., vol. 37, no. 1, pp.2–21, Jan. 2001.

[5] Y. S. Yao and R. Chellappa, “Dynamic feature point tracking in animage sequence,” in Proc. 12th IAPR Int. Conf. Comput. Vis. ImageProcess., 1994, vol. 1, pp. 654–657.

[6] C. Rasmussen and G. D. Hager, “Probabilistic data association methodsfor tracking complex visual objects,” IEEE Trans. Pattern Anal. Mach.Intell., vol. 23, no. 7, pp. 560–576, Jul. 2001.

[7] J. C. Nascimento and J. S. Marques, “Robust shape tracking in the pres-ence of cluttered background,” IEEE Trans. Multimedia, vol. 6, no. 6,pp. 852–861, Dec. 2004.

[8] D. Liu and L. C. Fu, “Target tracking in an environment of nearly sta-tionary and biased clutter,” in Proc. IEEE/RSJ Int. Conf. Intell. RobotsSyst., 2001, vol. 3, pp. 1358–1363.

[9] S. B. Colegrove and S. J. Davey, “PDAF with multiple clutter regionsand target models,” IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 1,pp. 110–124, Jan. 2003.

[10] M. Boshra and B. Bhanu, “Predicting performance of object recog-nition,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 22, no. 9, pp.956–969, Sep. 2000.

[11] K. B. Sarachik, “The effect of Gaussian error in object recognition,”IEEE Trans. Pattern Anal. Mach. Intell., vol. 19, no. 4, pp. 289–301,Apr. 1997.

[12] J. Chen, H. Leung, T. Lo, J. Litva, and M. Blanchette, “A modifiedprobabilistic data association in a real clutter environment,” IEEETrans. Aerosp. Electron. Syst., vol. 32, no. 1, pp. 300–313, Jan. 1996.

[13] T. W. Ridler and S. Calvard, “Picture thresholding using an iterativeselection method,” IEEE Trans. Syst., Man. Cybern., vol. SMC-8, pp.630–632, 1978.

[14] Y. S. Chen, Y. P. Hung, and C. S. Fuh, “A fast block matching algo-rithm based on the winner-update strategy,” in Proc. 4th Asian Conf.Comput. Vis., 2000, vol. 2, pp. 977–982.

[15] Y. Bar-Shalom and X. R. Li, Multitarget-Multisensor Tracking: Prin-ciples and Techniques. Storrs, CT: Yaakov Bar-Shalom, 1995.

[16] G. Gennari and G. D. Hager, “Probabilistic data association methodsin visual tracking of groups,” in Proc. Int. Conf. Comput. Vis. PatternRecog., 2004, vol. 2, pp. II-876–II-881.

[17] C. Haworth, A. M. Peacock, and D. Renshaw, “Performance of refer-ence block updating techniques when tracking with the block matchingalgorithm,” in Proc. IEEE Int. Conf. Image Process., 2001, vol. 1, pp.365–368.

[18] A. Cavallaro, O. Steiger, and T. Ebrahimi, “Tracking video objects incluttered background,” IEEE Trans. Circuits Syst. Video Technol., vol.15, no. 4, pp. 575–584, Apr. 2005.

Modeling and Control of a Small AutonomousAircraft Having Two Tilting Rotors

Farid Kendoul, Isabelle Fantoni, and Rogelio Lozano

Abstract—This paper presents recent work concerning a small tiltrotoraircraft with a reduced number of rotors. The design consists of two pro-pellers which can tilt laterally and longitudinally. A model of the full birotordynamics is provided, and a controller based on the backstepping proce-dure is synthesized for the purposes of stabilization and trajectory tracking.The proposed control strategy has been tested in simulation.

Index Terms—Backstepping methodology, helicopter dynamics mod-eling, tiltrotor rotorcraft, trajectory tracking.

I. INTRODUCTION

Since the beginning of the 20th century, many research efforts havebeen done to create effective flying machines with improved perfor-mance and capabilities. The tiltrotor aircraft configuration [1] has thepotential to revolutionize air transportation by providing an economicalcombination of vertical takeoff and landing (VTOL) capability with ef-ficient high-speed cruise flight. Indeed, the Bell Eagle Eye unmanned

Manuscript received October 2, 2005; revised February 14, 2006. This paperwas recommended for publication by Associate Editor G. Sukhatme and EditorF. Park upon evaluation of the reviewers’ comments. This work was supportedin part by the ONERA (French Aeronautics and Space Research Centre), in partby the DGA (French Arms Procurement Agency of the Ministry of Defence),and in part by the French Picardie Region Council. This paper was presentedin part at the 44th IEEE Conference on Decision and Control/European ControlConference, Seville, Spain, December 2005. Color versions of Figs. 1–4 areavailable online at http://ieeexplore.org.

The authors are with Laboratoire Heudiasyc, UMR CNRS 6599, Universitéde Technologie de Compiègne, BP 20529–60205 Compiègne cedex, France(e-mail: fkendoul@hds.utc.fr; ifantoni@hds.utc.fr; rlozano@hds.utc.fr).

Digital Object Identifier 10.1109/TRO.2006.882956

1552-3098/$20.00 © 2006 IEEE

1298 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006

Fig. 1. Tiltrotor-based rotorcraft.

aerial vehicle (UAV) [2], which is based on tiltrotor technology, has alarge success in the civil and military domains. The tiltrotor technologyis generally reserved for full-scale VTOL planes. But this solution hasusually not been considered because of the transition problems and theaerodynamic effects [1], [3]. Furthermore, degradation of stability isusually observed at high-speed forward flight (airplane mode) [3], andthe involved equations of motion are highly coupled and nonlinear.

An interesting alternative consists of exploiting and adapting thetiltrotor mechanism for small-UAV control. Effectively, Gress provedin [4] that the total number of reactive devices, namely, propellers,could in fact be reduced to two by using their gyroscopic nature. Heconsidered tilting rotors in order to obtain the three required moments(roll, pitch, yaw). However, this mechanism suffers from a weakpitching torque and some parasitic moments. In this paper, we proposean alternative configuration where the center of mass of the UAVis located below the tilting axes, resulting in a significant pitchingmoment.

We have constructed two prototypes of a birotor rotorcraft, as shownin Fig. 1, inspired by Gress’s mechanism. The experimental resultsshowed that this aerodynamical configuration is very promising. Theadvantage of this design is that the reaction torque of the motors iscancelled, since the noncyclic propellers (propellers with a fixed bladeangle) rotate in different directions. This configuration is also adaptedfor the miniaturization of the UAV, and it results in a simple mechanicalrealization.

In addition, our main contribution is to provide a complete modelof the birotor rotorcraft, and to develop a control algorithm for sta-bilization and trajectory tracking based on backstepping methodology[5]–[8]. This control strategy turned out to be methodologically simplefor the proposed system.

In this paper, we first motivate the use of tilting rotors, and describesuch a mechanism in Section II. Next, in Sections III and IV, we presentthe model of the birotor rotorcraft whose dynamical model is obtainedvia a Newton approach. Finally, we address a control strategy for stabi-lization and trajectory tracking in Section V. Furthermore, robustnesswith respect to parameters uncertainty and unmodeled dynamics hasbeen observed in simulation, as shown in Section VI. Final remarksare given in Section VII.

II. TILTROTOR-BASED MECHANISM

In this section, we describe and show briefly the capabilities of thetiltrotor-based mechanism for providing the three required moments(roll, pitch, and yaw). We will focus on certain basic aspects of theproposed configuration, namely, that our aircraft model is distinguishedfrom the one proposed by Gress by its center of mass, which is movedbelow the tilt axes. This arrangement has the advantage of providing asignificant pitching moment. But this modification results in a complexdynamical model of the vehicle. This issue is solved by synthesizingan appropriate controller which takes into account these model nonlin-earities.

Fig. 2. OLT and the associated moments.

For more details on system design and technical analysis, refer to[4], [5], and [9].

Differential Propeller Speeds: Since it is possible to independentlymodify the speed of each propeller, then the rolling moment can begiven by

�� = l(P1 � P2) (1)

where l is the distance from the motors to the aerodynamic center O(see Fig. 2). P1 and P2 are the lifts generated by the two propellers.

Longitudinal Tilting (LT): The yaw moment is obtained by differ-ential LT of the rotors

� = l(P1 sin�1 � P2 sin�2) (2)

where �1 and �2 are the LT angles.Since the center of gravity G is located below the tilt axes, the sum

of longitudinal components of P1 and P2 generates a pitching momentwith respect to G. It can be expressed as

�� = h(P1 sin�1 + P2 sin�2) (3)

where h is the distance from G to the tilt axes.Opposed Lateral Tilting (OLT): Referring to Fig. 2, forcing pro-

pellers to precess laterally in opposite directions will create gyroscopicmoments (M1� ;M2�) [9], which are directed as shown in Fig. 2. Theirtotal magnitude, calculated in Section III, is given by

M� = M1� +M2� = Ir _�(!1 + !2) cos� (4)

with Ir is the fan inertia moment about its spin axis.Also, a nonzero tilt angle � will generate a component of the fan

torques (Q1; Q2) along the lateral axisE2, creating a pitching momenton the aircraft in the same direction as M� . Then, �� = Ir _�(!1 +!2) cos� + (Q1 + Q2) sin�.

The key idea consists of using both LT and OLT for pitch/forwardmotion control. More precisely, the collective LT generates a pitchingmoment (�� ) by directing the combined thrust vector, and the OLTcreates complementary pitching moments (�� ) and minimizes oreliminates the resulting adverse reaction Ma�r .

IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006 1299

III. BIROTOR MODEL

The complete dynamics of a helicopter is quite complex, and some-what unmanageable for the purpose of control [10]. Therefore, we con-sider a helicopter model as a rigid body incorporating a force/torques-generation process [6].

A. Rigid-Body Dynamics

The equations of motion for a rigid body subject to body force F0 23 and torque � 2 3 applied at the center of mass and specified with

respect to the body coordinate frame A = (E1; E2; E3) (see Fig. 2)are given by the Newton–Euler equations inA, which can be written as

m _�A + �m�A = F0

J _ + � J = �(5)

where �A 23 is the body velocity vector, 2

3 is the body angularvelocity vector, m 2 specifies the mass, and J 2 3�3 is an inertiamatrix.F0 combines the force of gravity and the lift vectorF generatedby the propellers.

The first equation in (5) can be also expressed in the inertial frameI = (Ex; Ey; Ez). By defining � = (x; y; z) and �I as the positionand the velocity of the helicopter relative to I , we can write

_� = �I ; m _�I = RF �mgEz

J _ + � J = �(6)

where R 2 SO(3) is the rotation matrix of the body axes relative toI , satisfying R�1 = RT and det(R) = 1. It can be obtained by usingthe Euler angles � = ( ; �; �) which are yaw, pitch, and roll (cf. [5]and [11] for its expression).

B. Force/Torques-Generation Process

In the following, we express the force/torques pair (F; � ) exertedon the helicopter. To develop our analysis, we use two additional co-ordinate frames, A1 = (A1x; A1y; A1z) and A2 = (A2x; A2y; A2z),which are associated with the rotor n�1 and rotor n�2, respectively(see Fig. 2). Therefore, the orientation of the rotors n�1 and n�2 withrespect toA can be defined by the rotational matrices T1 and T2 whoseexpressions are given, respectively, by

c� s� s� s� c�

0 c� �s�

�s� c� s� c� c�

;

c� �s� s� s� c�

0 c� s�

�s� �c� s� c� c�

: (7)

Force Vector: The forceF generated by the rotorcraft is the resultantforce of the thrusts generated by the two propellers

PA1

= (0; 0; P1)T; P

A2

= (0; 0; P2)T: (8)

Pi is the lift that the propeller i produces by pushing air in a directionperpendicular to its plane of rotation [10]. This thrust can be modeledas Pi = Cl!

2

i , where Cl > 0 is the lift coefficient depending on bladegeometry and fluid density. Thus, the vector F expressed inA is givenby

F = T1PA1

+ T2PA2

=

(P1s� + P2s� )c�(P2 � P1)s�

(P1c� + P2c� )c�

: (9)

Gyroscopic Moments: As explained in Section II, tilting the pro-pellers around the axesE2 andAix creates gyroscopic moments which

are perpendicular to these axes and to the spin axes (Aiz). Indeed,these moments are defined by the cross-product of the kinetic moments(Ir!iAiz) of the propellers and the tilt velocity vectors. They are firstexpressed in the rotor frames as [9] MA

� = �Ir!1 _�1A1x; MA� =

Ir!2 _�2A2x for the LT, and MA

1� = Ir!1 _�A1y , MA

2� = Ir!2 _�A2y

for the lateral tilting.Multiplying the above equations by T1 and T2, we obtain the expres-

sions of the gyroscopic moments in frame A

M� =

2

i=1

TiMA� = Ir

�!1 _�1c� + !2 _�2c�0

!1 _�1s� � !2 _�2s�

(10)

M� =

2

i=1

TiMA

i� = Ir _�

(!1s� � !2s� )s�(!1 + !2)c�

(!1c� � !2c� )s�

: (11)

Fan Torques: As the blades rotate, they are subject to drag forceswhich produce torques around the aerodynamic center O. These mo-ments act in opposite directions relative to !

QA1

= (0; 0;�Q1)T; Q

A2

= (0; 0; Q2)T: (12)

The positive quantities Qi can be written as a function of propellerspeeds Qi = Ct!

2

i ; Ct > 0.Similarly, these torques can be written in A as

Q =

2

i=1

TiQA

i =

�(Q1s� �Q2s� )c�(Q1 +Q2)s�

�(Q1c� �Q2c� )c�

: (13)

Thrust-Vectoring Moment: Denote O1 (O2) as the applica-tion point of the thrust P1 (P2). From Fig. 2, we can defineO1G = (0;�l;�h)T and O2G = (0; l;�h)T as positional vectorsexpressed in A.

Then, the moment exerted by F on the airframe is

MF = (T1P1)�O1G+ (T2P2)�O2G: (14)

After some development, we obtain

MF =

l(P1c� � P2c� )c� + h(P1 � P2)s�h(P1s� + P2s� )c��l(P1s� � P2s� )c�

: (15)

Adverse Reactionary Moment: As described in Section II, this mo-ment appears when forcing the rotors to tilt longitudinally. It dependsespecially on the propeller inertia It and on tilt accelerations. This mo-ment acts as a pitching moment, and can be expressed in A as follows[9]:

Ma�r = �It( ��1 + ��2) � E2: (16)

The complete expression of the torque vector � acting with respectto the center of mass of the helicopter and expressed in A is

� =M� +M� +Q+MF +Ma�r: (17)

Finally, the explicit expression of � could be obtained by replacing theright-hand terms in (17) by the formulae (10)–(16).

1300 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006

IV. MODEL ANALYSIS FOR CONTROL DESIGN

The nonlinear model (6) and (17) is very complex, and it seems dif-ficult to exploit it directly for control-design purposes. Indeed, it hashigh nonlinearities and strong coupling of control inputs. It is, thus, es-sential to consider some approximations.

In order to minimize the loss in vertical thrust, the lateral tilt angle� is imposed to be small (< 15�). Therefore, we can consider thatcos� � 1 and sin� � �.

The lateral component (P1 � P2)� of the force vector F in (9) is aparasitic force which occurs only when applying simultaneously lateraltilting and differential rotor speeds. The nominal inputs considered arethen defined as

ux = P1s� + P2s� ; uz = P1c� + P2c� : (18)

The torque vector � given by (17) includes the principal control in-puts (��; ��; � )T and parasitic terms, which are expected to be of amuch smaller magnitude.

Analyzing the expression of � , we can define the nominal controlsassociated to the rolling and yawing torques as

�� = l(P1c� � P2c� ); � = �l(P1s� � P2s� ): (19)

The pitching moment can be expressed in the following form:

�� = �� + hux +Ma�r: (20)

From the previous considerations, control vectors can be representedas follows:

F = (ux; 0; uz)T +�F; � = (��; ��; � )

T +�� (21)

where �F and �� contain the parasitic terms which are not includedin the nominal inputs. These small body forces/moments will allowarbitrary (though bounded) control action, so they are considered asperturbations and set to zero in the control design.

In comparison with other rotorcraft models (four-rotor [11], conven-tional helicopter [6]), the birotor model given by (6) and (21) presentssome difficulties for control design, due to the coupling of the trans-lational force F and the torque � through the term ux. Furthermore,the pitch movement is controlled by two independent inputs (hux and�� ). Therefore, the control strategy must take into account this “over-actuation” in order to avoid the cases where the two control inputs,described above, could compensate themselves mutually. The synthe-sized controller also has to be robust against unmodeled dynamics andinput uncertainties presented in (21).

V. BACKSTEPPING-BASED TRACKING-CONTROL DESIGN

In this section, a control law is derived for the purpose of stabiliza-tion and trajectory tracking. This algorithm exploits the procedure ofbackstepping that we found appropriate for the birotor model. In orderto develop our backstepping procedure, let us rewrite system (6) in thefollowing form [7]:

_� = �I

_�I =1

mRF � gEz

_R = R sk()

_ = J�1� � J

�1� J (22)

where sk(v) is the associated skew-symmetric matrix of a vector v suchthat sk(v) = v � .

Note that system (22) is not in pure strict-feedback structure, andcontrol vectors F and � have different relative degree with respect tothe position �. Another difficulty is the coupling ofF and � via the termux (cf. Section IV). All these facts motivate us to consider a dynamicextension of control vector F by a double integrator

�F = ~F : (23)

Thus, both the vector F and its first time derivative _F become internalvariables of a dynamic controller.

Now, it is possible to define a transformation that results in alge-braically less complicated equations

X :=1

mRF � gEz

Y := _X =R

m(sk()F + _F )

� := J~� + � J: (24)

Without loss of generality and recalling (24), we reformat the system(22) to

_� = �I

_�I = X

_X = Y

_Y =R

m( ~F � sk(F )~� + 2sk() _F + sk()sk()F) := �: (25)

Now, both new inputs ~F and ~� have a relative degree 4 with respect tothe position �. Thus, they can be assigned at the last stage performingalso a nonaggressive control for the translation compared with the con-trol of the rotation.

Let (�; ) = (x(t); y(t); z(t); (t)) 2 4 be the desired trajecto-ries. Then, the control strategy aims at finding a feedback control ( ~F ; ~�)such that the tracking error " := (�(t) � �(t); (t) � (t)) 2 4 isasymptotically stable.

The control strategy can be divided into two steps. In the first step,a standard backstepping methodology [8] is applied for system1 (25).The only difference is the introduction of the yaw error "3 := �

in the third stage of the backstepping procedure [5].The backstepping methodology guarantees exponential convergence

of the original tracking error " to zero, and results in the followingcontrol [5], [7], [8]:

� = �5(� � �)� 10( _� �_�)� 9(X �

��) (26)

� 4(Y � �(3)) + �

(4)

� := � = �2( � )� 2( _ �_ ) +

� : (27)

It is important to note that the auxiliary inputs � and � are functionsof known signals. It remains to express the control inputs ~F and ~�according to the known variables � and � .

From the last equality of (25), we obtain

~F � sk(F )~� = mRT�� 2sk() _F � sk()sk()F := ~�: (28)

The vector ~� 2 3 is an auxiliary variable which is function of knownvariables, and can be computed from (26) and (28).

Concerning the yaw angle control, let us recall the kinematic rela-tionship between the generalized velocities _� = ( _ ; _�; _�) and the an-gular velocity vector

_� =W�1 (29)

where the expression of W�1 can be found in the literature [6], [7].

1System (25) is in the form of a chain of four integrators.

IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006 1301

Differentiating (29) and replacing the matrixW�1 by its expression,one obtains a relation between the desired input � and the other avail-able control inputs

cos�

cos �~� +

sin�

cos �~�� = � �ET1 _W�1 := ~�: (30)

The new variable ~� 2 is a function of known parameters. Thus,the control component ~� can be used for yaw control as long as thefollowing condition is valid: (�; �) 2](��)=(2); (�)=(2)[.

Rewriting (28) and (30), we obtain

�ux + uz~�� = ~�1

ux~� � uz~�� = ~�2

�uz � ux~�� = ~�3

cos�

cos �~� +

sin�

cos �~�� = ~�: (31)

Now, it remains to solve system (31) in order to find inputs ( ~F ; ~�), i.e.,(�ux; �uz ; ~��; ~��; ~� ).

As explained in Section IV, pitch angle is overactuated, so it is nec-essary to consider the equation governing this movement, i.e., �ux +uz~�� = ~�1.

This equation contains two unknown variables, �ux and ~�� , so, math-ematically, it admits an infinite number of solutions. However, some ofthem may induce inputs to diverge or oscillate. Let us choose

~�� =1

uz(k1 _ux + k2ux) (32)

where k1 and k2 are well-chosen positive constants, and uz is assumedto be different from zero.

This choice is motivated by the fact that (32) represents a stable sub-system with respect to the variable ux. Moreover, solving the differ-ential equation (32), we can show that the two inputs ux and ~�� arecomplementary, thereby preventing them from diverging once the con-trol objective is achieved.

From (31) and (32), we obtain

�ux = ~�1 � (k1 _ux + k2ux): (33)

Remark 1: From Section II and (20), pitch load can be divided into�� and hux by a suitable choice of k1 and k2. These gains shouldalso be chosen such that subsystem (33) is stable when ~�1 convergesto zero.

From the third equation of system (31), we deduce

�uz = ~�3 + ux~�� = ~�3 +uxuz

(k1 _ux + k2ux): (34)

The expression of ~� is given from (31) by

~� =cos �

cos�(~� �

sin�

cos �

1

uz(k1 _ux + k2ux)): (35)

Finally, ~�� is deduced from the second equation of (31)

~�� =1

uz(ux~� � ~�2); with ~� is given by (35): (36)

TABLE IMODEL AND CONTROLLER PARAMETERS

One notes that all the terms at the left-hand side of (32)–(36) arewell-defined, provided that

uz 6= 0

� 6=�

2+ k�

� 6=�

2+ k� with k 2 : (37)

Using (23) and (24), the original inputs F and � can be recovered,which, of course, are themselves functions of all the primitive variables(P1; P2; �; �1; �2) [see (18)–(20)].

VI. SIMULATION RESULTS

In this section, two simulation results are presented in orderto observe the performances of the proposed control law. Weconsidered the case of stabilization and a trajectory-tracking problem.The parameters used for the aircraft model and the controllerare given in Table I.

In the first simulation, the stabilization of the rotorcraft’s po-sition and attitude is considered. We took the initial conditions(x(0); y(0); z(0); (0); �(0); �(0)) = (1; 1; 0; 0; 0:3; 0:1) and_�(0) = _�(0) = (0; 0; 0)T . We also adopted the followingchoice of initial conditions for the force inputs: ux(0) = 0 anduz(0) = mg = 9:81. The desired values of altitude and yaw are fixedat (z; ) = (3; 0:5). The results in Fig. 3, obtained by consideringthe complete model of the birotor, illustrate the performance of thecontrol law developed in this paper. Moreover, the last plot in Fig. 3shows that the moments �� and �� both contribute to generate thepitch moment, and the adverse reaction moment Ma�r is compensatedby �� when the former is not desired as at the beginning of tilting.

We also ran simulations for trajectory tracking. The desired trajec-tory was chosen as a helix, i.e., � = (2 cos ; 2 sin ; 0:5t)T and =0:6� t. The rotorcraft was initially at the origin, i.e., �(0) = �(0) = 0.The results presented in Fig. 4(a) show that the position converges to thedesired trajectory after a short transient time with a small error �. Ro-bustness against unmodeled inputs and external forces [see Fig. 4(b)]is also observed, even if a mathematical analysis of the controller ro-bustness is not provided in this paper.

VII. CONCLUSION

The analysis presented here has shown that hover control of atwo-propeller VTOL aircraft is possible using two tilting rotors.The pitch stability is increased by combining the OLT with theLT of the two rotors. The resulting oblique tilting appears to beeffective and practical. Indeed, the experimental results obtainedon prototypes constructed by our team [see Fig. 1(b)] are verypromising.

In this paper, we have also developed a 6-degree-of-freedommodel of the birotor aircraft and synthesized a nonlinear controller

1302 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006

Fig. 3. Birotor stabilization in presence of small body forces/moments.

Fig. 4. Trajectory tracking when considering the complete model.

which leads to a satisfactory control. To the authors’ knowledge,no other control law has been derived for a small tiltrotor-basedrotorcraft model.

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