Mathematical Modeling and Experimental Identification of a Model

Helicopter

S. K. Kim and D. M. Tilbury

Department of Mechanical Engineering and Applied Mechanics

2250 G. G. Brown

University of Michigan

Ann Arbor, MI 48109-2125

sungk@engin.umich.edu

tilbury@umich.edu

Submitted to the 36th CDCApril 9, 1997

Abstract

In this paper, we develop a new mathematical model for the nonlinear dynamical equations of a model

helicopter, and describe some preliminary experiments which validate this model. We also describe our

experimental setup and overall project goals.

1 Introduction

It has been more than 20 years since the first commercial model helicopter was conceived, and since then,the design has significantly improved. Model helicopters are now well within the reach of many hobbyistsand are also often used for commercial purposes, such as crop dusting or sport-event broadcasting. However,even with improved stability augmentation devices, model helicopters are inherently unstable. A skilled,experienced pilot is required to control them during flight.

As a small, dynamically fast, unstable system, a model helicopter makes an excellent testbed for nonlinearcontrol experiments. As a highly maneuverable machine, a model helicopter makes an excellent testbed forpath planning algorithms for autonomous robots. The integration of nonlinear control and path planningis our main interest in this project. We plan to examine energy-based methods for nonlinear control, andto examine the potential of these energy-based schemes for controlling autorotation maneuvers in a modelhelicopter.

As a preliminary step toward these goals, this paper describes the new mathematical model that we havederived for a model helicopter control system, as well as some preliminary system identification experimentswe have conducted which validate this model. The outline of this paper is as follows. First, we briefly reviewsome previous work on model helicopter control and energy-based methods for control. We then describethe new mathematical model that we have derived, working from first principles and basic aerodynamics.In Section 3, we describe the system identification algorithm that we have used and present our preliminaryresults. We end with conclusions and a description of our future work.

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Figure 1: The coordinates defined. The 3 orientation variables, pitch, roll, and yaw (φ, θ, ψ) are relative tothe body frame (fixed to the helicopter). The 3 position variables (x, y, z) are relative to an inertial frame(fixed to the earth).

Previous Work

In the past several years, there have been number of researchers interested in model helicopter control, andthey have had various degree of success. A group of researchers at Caltech [2, 12] performed the systemidentification on a “hovering” model helicopter and were able to control the pitch, roll, and yaw attitude,and rotor angular velocity using LQG method. Their helicopter was affixed to a stand, and had only therotational degrees of freedom.

At Purdue [10], a student derived the dynamic equation of a model helicopter’s vertical motion (with thehelicopter affixed to a stand) and was able to control its motion using linearized state feedback.

In the unmanned aerial vehicle competition, various universities have been successful in completing agiven task of acquiring and moving an object. Success in this competition has often depended more on theimage recognition and sensing than on the helicopter control algorithms. For this competition, USC useda concept called “behavior based control” [3, 7, 8], which implements a number of complex tasks with acollection of simple, interacting behaviors in parallel. The three tasks they used are stabilization, movement,and picking up an object.

Dr. Sugeno at Tokyo Institute of Technology [11] had a considerable amount of success in flying a voice-controlled crop-dusting model helicopter using fuzzy control.

2 Mathematical model of the model helicopter

In this section, we will derive the dynamic equations of motion for the model helicopter including its actuatordynamics.

2.1 Differences between model and full scale helicopters

Before considering the helicopter dynamics, it is important to consider the differences between a full-scalehelicopter and a model helicopter. First of all, a model helicopter has a much faster time-domain response

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due to its small size. Therefore, without employing an extra stability augmentation device, it would beextremely difficult for a human pilot to control it. A large control gyro with an airfoil, often referred to asa flybar, is almost always used nowadays to improve the stability characteristic around the pitch and rollaxes and to minimize the control stick force required. Also, the tail rotor control for the model helicopter isassisted by an electronic gyro to further stabilize the yaw axis. Most full scale helicopters do not have sucha control gyro on the rotor system.

Secondly, most model helicopters do not have a flapping hinge on the rotor to maximize the controlpower. Therefore, the blade flapping motion in a model helicopters can be neglected. Full scale helicoptersoften use either a free flapping rotor hinge or a spring-mounted hinge.

2.2 Rigid Body Equations in SE(3)

As shown in Figure 1, we represent the position of the helicopter relative to a fixed ground reference as(x, y, z). We use the variables (φ, θ, ψ) to represent the pitch, roll, and yaw angles respectively.

The helicopter can be viewed as a rigid body moving in space, with forces and moments acting on it. Assuch, it will obey the standard rigid body dynamical equation:

M(q)q + C(q, q)q + g(q) = B(q)τ (1)

where the state vector q = (x, y, z, φ, θ, ψ). These equations can be written in coordinates for the helicopteras follows.

x = − 1m

cosψ(T sin θ +DFx) (2)

y =1m

cosψ(T sinφ+DFy ) (3)

z =1m

cosφ(T cos θ +DFz )− g (4)

φ =1Ixx

Mφ (5)

θ =1Iyy

(Mθ − T`r) (6)

ψ =1Izz

(Mψ + Tm − IrΩ) (7)

Note that the inertia matrix M(q) is diagonal and constant. The D terms represent drag forces; we willtreat these as disturbances in our model. The mass of the helicopter is given by m, and the fuselage inertiasare Ixx, Iyy, Izz. The rotor rotational inertia is Ir. The rotor angular velocity is Ω, and the offset betweenthe rotor axis and the helicopter’s center of gravity is `r. The gravitational acceleration constant is g. It isalso assumed that the helicopter’s center of gravity is in-line with the rotor axis laterally.

The four independent inputs are T , the net thrust generated by the rotor, and Mφ,Mθ,Mψ, the netmoments acting on the helicopter. The torque applied by the motor, Tm, is related to the thrust T andcannot be controlled independently. The mechanisms for creating these inputs will be described morethoroughly in the following section.

2.3 Actuator dynamics

A model helicopter moves forward when a pitching moment is first applied and the fuselage is tilted forward.The thrust vector T then gives a forward component for a forward thrust. A full scale helicopter only requiresa forward tilt of the rotor disk to move forward while the fuselage stays level [6], but this type of maneuveris not possible with a model helicopter.

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Figure 2: The top view of the helicopter. The lift distribution on the rotor disk when a forward cyclic (pitchforward) input is applied. The precession effect will pitch the helicopter forward.

There are four inputs available to the pilot of a model helicopter. These are physically controlled by twojoysticks on the radio transmitter, each with two degrees of freedom. The left joystick commands throttle(up/down) and yaw (left/right), and the right joystick commands pitch (up/down) and roll (left/right). Thefour values representing the positions of the sticks are encoded in a pulse-width modulated signal, and sentvia radio link to the helicopter. We will use these four positions (xpitch, xroll, xyaw, xthrottle) as the inputsto our actuator dynamic equations.

The throttle command (xthrottle) controls the power to the main motor as well as the collective pitch (θ0)of the rotor blades. As the blade pitch increases, more lift is created, and the rotational motion of the mainrotor blade is converted into vertical thrust. The yaw command controls the pitch of the tail rotor blade.The tail rotor on a helicopter is used to counteract the yaw moment created by the main rotor blade; thus,altering the amount of pitch on the tail rotor can create more or less total yaw moment for the helicopter.The pitch and roll commands influence the cyclic control, which varies the cyclic pitch (θcyc) of the rotorblades around each cycle of rotation, creating different amounts of lift in different regions (as shown in Figure2). These differing amounts of thrust create a moment around the rotor hub, and can thus create pitch androll moments on the helicopter.

The actuator dynamics include as states the flapping angle and velocity (β, β) of the flybar and theangular velocity (Ω) of the main rotor blade. Before developing the dynamic equations, we introduce somebasic aerodynamic terms that will be required. The advance ratio, µ, and the descent ratio, ν, represent theairspeed components parallel to and perpendicular to the rotor disk respectively [9]. They are close to zerowhen the helicopter is hovering. Both quantities are non-dimensionalized by RΩ.1

µ =1RΩ

[(x− hr θ) cos θ + (y + hrφ) cosφ+ (z − `r θ) sin θ

](8)

ν =1RΩ

[(x− hr θ) sin θ + (y − hrφ) sinφ− (z − `r θ) cos θ

](9)

The constants hr and `r are the offset of the rotor from the helicopter’s center of gravity.The rotor solidity, σ, is the area ratio between the rotor blade area and the rotor disk. It indicates how

“solid” the rotor disk is [6], and is taken to be equal to twice the area of a rotor blade (cR) divided by the1Strictly speaking, R is the rotor span, excluding the rotor hub length. However, since the model helicopter’s rotor blade

remains nearly rigid due to its short length (0.4 to 0.8 meter), high rotor speed (1300 to 1900 rpm), and hingeless hub design,

R is assumed to be the distance between the rotor axis and the rotor tip.

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area defined by the rotor disk (πR2).

σ =2cπR

The inflow ratio, λ, is the net value of the descent ratio ν and the induced air velocity ratio vi.

λ = −ν +viRΩ

The lift curve slope, a, is the slope of the function of the lift vs. angle of attack of the main rotor blade.This is one of the parameters that we will identify using the system identification algorithm described inSection 3. The constant tip loss factor B takes into account the fact that a finite length rotor blade wouldlose some of the lift generated due to the wing tip vortex effect. We will use the value B = 0.97, as given by[9].

The dynamics of the rotor and the flybar are the most significant nonlinearities involved in the creationof the forces and moments on the helicopter. We denote the rotor angular velocity by Ω [9].

Ω = Kthrottlexthrottle −1IrρπR5Ω2

[18σCdo +

12aσλ

(B3

3θo −

B2

2λ+

B2

4µ(xroll + xpitch

L1))]

(10)

The throttle stick position xthrottle also influences the collective pitch angle θ0 through a proportionalconstant. Under normal operating conditions, these two effects balance each other, and Ω remains constant[9]. Recall that θ0 is the collective pitch angle, and xroll and xpitch are the roll and pitch inputs respectively.The profile drag coefficient Cd0 is a constant.

The flapping angle of the flybar is denoted as β; see Figure 3. As mentioned before, the flybar plays amajor role in augmenting the stability of the helicopter. This system is often called as a Bell-Hiller mixer,because it takes advantage of two different cyclic control systems. Cyclic control is the mechanism by whichthe rotor blade’s pitch is changed in a rotation so that an unequal distribution of the lift applies a momentaround the rotor hub. This moment then provides pitch and roll attitude control as in Figure 2. The Bell-mixer allows the blade pitch to be changed directly from the cyclic servo actuator. It is fast in response, butlacks stability. Meanwhile, the Hiller-mixer allows the pitch of the flybar to be changed instead of the pitchof the blade. The flybar then flaps, and this flapping motion causes the pitch of the main blade to change.

We denote by If the inertia of the flybar. The lengths Li are defined in Figure 3. There is a directrelationship between the cyclic input xcyc (which will be the stick command xpitch or xroll) and the cyclicangle of the rotor blades θcyc and the flapping angle of the flybar β given by the following geometric equation.

θcycL1 = (L3

L2 + L3)xcyc + (

L2L4

L2 + L3)β (11)

The dynamic equations governing the flapping of the flybar are then given as follows.

If β + Ω2Ifβ =18ρΩ2ac(

xcycL5− β + ω

Ω)(R2

4 −R14) (12)

In the equation, ω will be equal to either φ or θ, depending on whether the helicopter is rolling or pitching.We can now finish defining the actuator dynamics. Once again, T is the thrust generated by the rotor

blade [9], and Mφ and Mθ represent the moments created by the rotor blade around the roll and the pitchaxes respectively.

T =12ρπR4Ω2aσ

[B3

3θo −

B2

2λ

(B3

3θo −

B2

2λ+

B2

2µ(xroll + xpitch

L1))]

(13)

Mφ =12ρΩ2acR4θcyc|xcyc=xroll, ω=φ (14)

Mθ =12ρΩ2acR4θcyc|xcyc=xpitch, ω=θ (15)

Mψ =12ρπR4

TΩ2TaTσT

B3

3θoTLT (16)

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The subscripts T indicate that the values pertain to the tail rotor. LT is the distance between the tail rotoraxis and the main rotor axis.

The yaw command directly influences the collective pitch of the tail rotor blades, and the throttle inputis directly coupled both to the motor torque and the collective pitch of the main rotor blades. We modelthese relationships as linear because their dynamics are fast compared to the main rotor dynamics.

θoT = KθoT · xyaw (17)

θo = Kθo · xthrottle (18)

Tm = KTm · xthrottle (19)

ΩT = KΩ · Ω (20)

2.4 Reduced model for identification

As a first step for identification, we will limit our consideration to the roll dynamics. We will use as an inputxroll, and as an output the roll angle φ.

The system identification (described in the following section) will allow us to preliminarily validate ourmodel, and also to find values for the inertias Ixx, If , and the lift curve slope a.

For our identification, we will use equations (5), (11), (12), and (14). We assume that Ω is constant.Restricting ourselves to roll motion only, we arrive at the following set of equations, which will result in afourth-order transfer function to be identified.

β =1

8IfρΩ2ac(R2

4 −R14)(

xrollL5− β + φ

Ω)− Ω2β (21)

φ =1

2IxxρΩ2acR4 1

L1(L2 + L3)(L3xroll + L2L4β) (22)

3 System Identification

3.1 Algorithm

At this point in time, we are attempting to identify some of the physical parameters while the helicopter isin roll motion only. The indirect recursive least square method will be applied for parameter identification.The output data will be collected while a pilot gives a roll control input only (the other inputs are held tozero). Therefore, we only need to consider the dynamics related to the roll motion, as noted in equations (21)and (22). The input is the roll command, xroll, and the output is the roll angle, φ. The parameters to beidentified are the inertias Ixx and If , along with the lift curve slope a (we are able to directly measurethe rotor angular velocity Ω. The continuous time transfer function relating the roll angle to the roll stickcommand is given by

H(s) =φ(s)

xroll(s)=

d1s2 + d2s+ d3

s(s3 + c1s2 + c2s+ c3)(23)

where the coefficients ci, di are functions of both the unknown parameters and measurable physical con-stants. Because the input from the transmitter is held constant over each sample time, the output from thecontinuous-time transfer function will be equal to the output from the zero-order hold equivalent discretetime transfer function Hd(z):

Hd(z) =b1z

4 + b2z3 + b3z

2 + b4z + b5z4 + a1z3 + a2z2 + a3z + a4

(24)

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Figure 3: The basic structure of the model helicopter’s cyclic control system. The flapping angle of theflybar, β, is defined as the angle of the flybar measured with respect to the horizontal. Ball joints areshown as “”, and fixed joints are shown as “•”. Note that the cyclic pitch input to the main rotor blade iscontrolled by the combination of the Bell input from the swashplate and the Hiller input from the flybar.

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Figure 4: A sketch of the experimental setup for the system identification of the helicopter’s physical param-eters. The helicopter is controlled through the radio transmitter by a human pilot. The computer is used torecord the input data from the transmitter and the output data from the sensor.

with appropriate coefficients ai, bi.We will use a recursive least squares scheme to estimate the discrete transfer function Hd(z) [1]. This

will give us estimates of the coefficients of the discrete transfer function, ai and bi. The measured output isy(k) = φ(k), the roll angle measured from the sensor, and the input is u(k) = xroll(k), the stick position ofthe radio transmitter. We include the most recent measured values of input and output in the Φ vector:

ΦT (k) = [y(k − 1) y(k − 2) · · · y(k − 4) u(k) u(k − 1) · · ·u(k − 4)] (25)

The actual values of the coefficients of the discrete-time transfer function are included in the vector Θ; wewill represent their current estimates by Θ(k).

Θ = [−a1 − a2 − a3 − a4 b1 b2 b3 b4 b5] (26)

Note that the discrete transfer function specifies that the current output is a linear function of theprevious output and input values

y(k) = −a1y(k − 1)− a2y(k − 2)− a3y(k − 3)− a4y(k − 4)

+b1u(k) + b2u(k − 1) + b3u(k − 2) + b4u(k − 3) + b5u(k − 4) (27)

= ΦT (k)Θ (28)

Thus, our on-line estimate of y is given by

y(k) = ΦT (k)Θ(k − 1) (29)

The parameters Θ are updated by the rule that the next value is equal to the previous value plus a gainmatrix K(k) times the difference between actual and predicted output.

Θ(k) = Θ(k − 1) +K(k) [y(k)− y(k)] (30)

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The covariance matrix P (k) is initially set to be a large diagonal matrix. The gain matrix K(k) is definedto be the covariance matrix times the current values of input and output [1].

K(k) = P (k)Φ(k) (31)

The update of the covariance matrix P (k) is defined as [1]

P (k) =[I −K(k)ΦT (k)

]P (k − 1) (32)

which, when coupled with the definition of the gain matrix (31), gives the update rule for the gain matrixK(k).

K(k) = P (k − 1)Φ(k)[I + ΦT (k)P (k − 1)Φ(k)

]−1(33)

These update rules will guarantee that the estimate converges (with suitable persistency of excitation con-ditions on the input values [1]).

Once the estimates Θ(k) have converged, the coefficients of the continuous time transfer function (23)can be determined using the equivalence of the discrete and continuous transfer function. The knowledge ofthese coefficients will allow us to estimate the physical parameters of the helicopter.

3.2 Experimental setup

We use a sensor made by Polhemus [5] to measure the position of the helicopter. As shown in Figure 2.4,the sensor consists of a board connected to the PC’s ISA slot, a transmitter and a receiver. The transmitteris fixed to the ceiling, and it sends out a magnetic field via three orthogonal inductors. The receiver, fixedto the helicopter, senses the strength and the orientation of the magnetic field and sends this informationback to the PC. The sensor is therefore capable of getting the position data for six degrees of freedom(x, y, z, φ, θ, ψ). For the preliminary experiments described here, we have fixed the helicopter to a the standto be able to concentrate on the roll motion. The input and output data are taken at 50 Hz, for a total ofapproximately 2 minutes duration.

3.3 Results

The comparison of the measured roll angle with the simulated output of the estimated discrete-time transferfunction is shown in Figure 3.3. The two graphs match reasonably well.

It was determined that the discrete transfer function (24) is given by

Hd(z) =1.2553s4 + 1.1580s3 + 1.0541s2 + 0.9502s+ 0.8514

z4 − 0.6187z3 + 0.5405z2 − 0.2446z − 0.4079(34)

which gives us the equivalent continuous time transfer function

H(s) =φ(s)

xroll(s)=

1.25z4 + 181.1z3 + 11676z2 + 780900z + 19800000s4 + 44.83s3 + 5648s2 + 209600s+ 1012000

(35)

The above expression does not match with (23); there is not a pole at the origin and there are four finitezeros. We recognized that the stand has a the slight pendulum effect, which serves to move the pole awayfrom the origin. Two of the identified zeros were much further to the left than the poles; we took these tobe numerical approximations of the two zeros at infinity of the theoretical transfer function. In addition,the disturbances in our system are significant. As shown in Figure 3.3, when the roll input is held constant,the helicopter still has a significant roll motion. This effect will make it very difficult to have an accurateestimation.

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True Simulated

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time step

Rol

l Ang

le (

rad)

True vs. Simulated output

Figure 5: The comparison between the simulated output based on the identified discrete time transferfunction and the actual output from the experiment, given the identical input command history.

Our final estimates for the unknown parameters are given by

Ixx = 0.024 kg ·m2

If = 0.00017 kg ·m2

a = 5.1 (36)

These values are reasonable; the helicopter’s inertia Ixx should be much larger that the inertia of the flybarIf , and a is typically between 4 and 7.

4 Conclusions and Future Work

In this paper, we derived a new mathematical model for the dynamics of a model helicopter. We alsopresented some preliminary experiments to identify some of the physical parameters of the model helicoptersystem. The values that we identified were of the same order of magnitude as we expected them to be. Thesepreliminary experiments also give a useful validation of our model.

We plan to continue the system identification to identify the other physical parameters of the helicopter,in pitch and yaw motions as well as couplings between the rotational and translational degrees of freedom.

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Figure 6: The roll angle measured when the roll control input remained neutral. The output shows aslowly varying fluctuation mainly due to aerodynamic disturbances. The amplitude of the output due to thedisturbance is almost as large as the output amplitude from the identification experiment.

Once we have a fairly accurate nonlinear model, we will begin our feedback control experiments. A short-term goal of this project is to autonomously hover the helicopter in the lab. A longer-term goal is to usethe complete nonlinear model and an energy-based control algorithm to control the model helicopter in anautorotation maneuver.

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Appendix: Nomenclature

a = main rotor lift slopeaT = tail rotor lift slopeB = tip loss factorc = main rotor blade chord lengthCdo = profile drag coefficient of blade sectionDFx , DFy , DFz = fuselage profile drag forcesg = gravitational accelerationhr = distance between rotor disk and CG, parallel to the rotor axisIcyc = rotor blade pitch inertiaIf = flybar moment of inertia in flappingIr = rotor moment of inertia around rotor axisIxx, Iyy, Izz = fuselage rotational inertia along x, y, and z directionKθo = proportional constant relating xthrottle to θoKTm = proportional constant relating xthrottle to TmKthrottle = proportional constant relating xthrottle to Ω`r = distance between rotor axis and CG, parallel to the rotor diskLT = distance between tail rotor axis and CGL1 . . . L5 = lengths for various linkages in the rotor hub assemblym = helicopter total massMφ, Mθ, Mψ = net moment applied around the roll, pitch, and yaw axis, respectivelyR = length of the main rotor bladeRT = length of the tail rotor bladeR1 = the distance between the rotor axis and the flybar tipR2 = the distance between the rotor axis and the flybar rootT = net thrust generated by the rotorTm = torque applied by the motorx, y, z = helicopter position coordinates relative to the groundxcyc = cyclic input displacementxpitch, xroll, xyaw = pitch, roll, and yaw command input displacementxthrottle = throttle command input displacementvi = average induced air velocity through rotor diskβ = flybar flapping angleθo = collective pitch angle of main rotor bladesθoT = collective pitch angle of tail rotor bladesθcyc = cyclic pitch angle of a main rotor bladeλ = inflow ratioµ = advance ratioν = descent ratioρ = air densityσ = main rotor solidityσT = tail rotor solidityφ, θ, ψ = helicopter angular position relative to the body coordinates fixed on the helicopterΩ = main rotor angular velocityΩT = tail rotor angular velocity

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References

[1] K. Astrom and B. Wittenmark. Adaptive Control. Addison-Wesley Publishing Company, 1995.

[2] P. Bendotti and J. Morris. Identification and stabilization of a model helicopter in hover. Journal ofGuidance, Control, and Dynamics, 1995.

[3] A. Fagg, M. Lewis, J. Montgomery, and G. Bekey. The USC autonomous flying vehicle: An experimentin real-time behavior-based control. IEEE Intl Conference on Intelligent Robots and Systems, 1993.

[4] R. Hostetler. RAY’s complete helicopter manual. R/C modeler corporation, 1991.

[5] Polhemus Incorporated. 3SPACE User’s Manual. 1993.

[6] W. Johnson. Helicopter Theory. Dover Publications, inc., 1980.

[7] M. Lewis. AFV-II:robotic aerial platform for autonomous robot research. 1994.

[8] J. Montgomery, A. Fagg, and G. Bekey. AFV-I:a behavior-based entry in the 1994 international aerialrobotics competition. 1994.

[9] Y. Okuno, K. Kawachi, A. Azuma, and S. Saito. Analytical prediction of height-velocity digram of ahelicopter using optimal control theory. Journal of Guidance, Control, and Dynamics, 14(2):453–459,1991.

[10] T. Pallett and S. Ahmad. Real-time Helicoper Flight Control. Master’s Thesis, School of ElectricalEngineering, Purdue University, 1991.

[11] M. Sugeno. Fuzzy hierarchical control of an unmanned helicopter. 1993.

[12] X. Zhu and M. Nieuwstadt. The Caltech helicopter control experiment. CDS Technical Report 96–009,1996.

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