NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society

Combining fuzzy and PID control for an unmannedhelicopter

Edgar N. SanchezCINVESTAV, Unidad Guadalajara

Apartado Postal 31-438. Plaza la Luna.Guadalajara, Jalisco, Mexico. C. P. 45091

Email: sanchez@gdl.cinvestav.mx

Hector M. BecerraCINVESTAV, Unidad Guadalajara

Apartado Postal 31-438. Plaza la Luna.Guadalajara, Jalisco, Mexico. C. P. 45091

Email: hbecerra@gdl.cinvestav.mx

Carlos M. VelezUniversidad EAFIT.

Carrera 49 No. 7 Sur 50.Medellin, Colombia, South America.

E-mail: cmvelez@eafit.edu.co

Abstract-This paper reports the synthesis of a controller forthe X-Cell mini-helicopter. It is developed on basis of the mostrealistic mathematical model actually available ([1]). A combinedcontrol structure is proposed: Mamdani controllers keep setpoints for an altitude/attitude controller. These controllers aredesigned in the simplest rule base. Altitude/attitude controller isconstituted for conventional SISO PII) controllers for z-positionand roll, pitch and yaw angles. This control scheme mimics theaction of an expert pilot. The proposed scheme is tested viasimulations; it presents a good performance for hover flight, andcontrol position in slow speed.

I. INTRODUCTION

The research reported in this paper is a contribution tothe overall objective of the Colibri project [6]: Design, im-plementation and experimentation of a control system for amini-helicopter robot. The Colibri project is one of the mostimportant research projects under development in Colombia,South America.

Helicopter control requires first, to guarantee stability atdifferent points to hold the helicopter in a desired trim state;and second, to change the helicopter velocity, position andorientation in order to track a desired trajectory. In mini-helicopters this is usually done through a remote control by ahuman pilot.The particular problem addressed in this paper is the syn-

thesis of a control structure to reproduce the pilot action inits simplest form. So, an altitude and attitude (roll, pitch,and yaw) controller which achieves stable behavior for anunmanned helicopter must be designed. The flight controlstructure proposed in this paper is tested in simulation using anonlinear multiple-input-multiple-output (MIMO) model of areal unmanned helicopter platform, the X-Cell. The simulationmodel is implemented on basis of the mathematical modelproposed in [1], which is the most realistic model actuallyavailable for this mini-helicopter.The paper is organized as follows. Section II briefly in-

troduces the mathematical model used. Section III presentsthe control structure. Section IV discusses simulation results,displaying how simplicity in the control designed is achieved.Finally, Section V provides relevant conclusions.

Fig. 1. X-Cell mini-helicopter.

II. MATHEMATICAL MODEL

The platform for research is an X-Cell mini-helicopter,which is shown in Fig 1. This helicopter is part of the Colibriproject. X-Cell is a representative vehicle for today research.It is highly maneuverable as a result from its stiff rotor head,allowing the transmission of large control moments from therotor to the fuselage, a large thrust-to-weight ratio, and afast rotor speed. It is equipped with a powerful engine andan electronic governor for maintaining a near constant rotorspeed. Like most miniature helicopters, it is also equippedwith a stabilizer bar, which acts like a lagged attitude ratefeedback, and is designed to help the remote human pilot tocontrol attitude dynamics. As an activity of the Colibri project,a X-Cell helicopter is being instrumented to performanceautonomous flight in real time.The X-Cell mathematical model is based on [1]. This model

is described for 17 state variables. Several parameters mustbe estimated experimentally and others are measured directly.The model covers from hover to about 20 rn/sec forward flight(low advance ratio It < 0.15). This pennits to consider thatthe thrust is perpendicular to the rotor disk

A. Rigid-Body Equation ofMotionThe helicopter is a vehicle that is free to simultaneously ro-

tate and translate in all six degrees of freedom (6 DOF), whichare: rolling, pitching, yawing, surging, swaying and heaving.The rigid-body dynamics of such vehicles are described by theNewton-Euler equations of motion. There are two reference

0-7803-9187-X/05/$20.00 02005 IEEE. 235

frames which in general are used to determine the motion ofa helicopter: body-fixed and inertial. The equations of motionwith respect to the body-fixed reference frame are:

mv+m(w x v) = F (1)I'+ (w x Iw) = M (2)

where v = [ u v w ]T is the vector of body velocities,w = [ p q r ]T is the vector of angular rates, F =[ X y Z ]T is the vector of external forces acting on thevehicle center of gravity (c.g.), M = [ L M N ]T is thevector of external moments, m is the helicopter mass and Iis the inertial tensor. The external forces and moments areproduced by the main and the tail rotors, the gravitationalforces, and the aerodynamic forces produced by the fuselageand the tail surfaces used as stabilizers.From equation (1), the three differential equations de-

scribing the helicopter translational motion (3) about its threereference body-fixed axes with origin at the vehicle c.g, are:

a = vr-wq-g sinO+ (Xmr +Xfus)/m (3)v = wp-ur + gsinqcosO+

+(Ymr + Yfus + Ytr + Yvf)/mw = uq-vp +gcos4 cosOS+ (Zmr + Zfus + Zht)/m

where q is the roll angle, 0 is the pitch angle and gis the gravity acceleration. Similarly, from equation (2), thethree ordinary differential equations describing the helicopterrotational motion (4), are derived. These equations do notdepend on the reference frame.

T- 2

- Y zzzz xz qr+ (4)Ixxlz-I x,z

+Ix -I + IzzIX

=-I(IPIq)+L(+ Izz LL+ ixz NX(I. - 1 2 (Iz -IX+z-Xqz

=(Izz - I*z Pr _ixzZ(p2 -r2) + I AIYY IYY IYY

r IXV y)x_x-Ijz2Zxpq-(1- lyI IZZ-I XZ qr+'xx zz xz Xxqzz -x

+IXjIzzZ--1-2- IXX I - Iz2Nxzz xz xx zz -x

where L = Lmr + L,f + Ltr, M = Mmr + Mht,N = -Qe + Nvf + Ntr. The subscript references to thecomponent generating the respective force or moment: mainrotor (mr), tail rotor (tr), fuselage (fus), and the stabilizers;vertical fin (vf) and horizontal tail (ht). Qe is the torqueproduced by the engine to counteract the aerodynamic torqueon the main rotor blades. In the above equations, we assumedthat the fuselage center of pressure coincides with the c.g.,therefore the moments created by the fuselage aerodynamic

forces are neglected. See [1] for details about forces andmoments expressions.The six previous first order differential equations constitute

the rigid-body equations. We can know the inertial position(x, y, z) and the Euler angles (X, 0, O) from them, using thesuitable transfornation. Euler angles give information aboutattitude. Solving the following three differential equationsallows to know inertial position from body velocities:

xY

[.]R[v]

y~ Lw

[cOcOR = c0so

[ -sO

sbsOc4' -COSO CqSOC4 + S0S'V 1S0080 + COC) CqSOSO - SOc-

SqCO CqCcO

(5)

(6)

R is the transformation from body-fixed to inertial refer-ence frame. cO and sO are abbreviations for cosO and sinO,respectively. R is a rotation matrix; it has some importantproperties:

* It can be described by the product of individual rotationmatrices.

* It is a orthogonal matrix.* Its determinant is the unity.Finally, to know the Euler angles from angular rates, the

following transformation is used:

[ i]

t0s/

sq5/cO-s q

c4/cO J LrJ(7)

B. Extension of the Rigid-Body ModelTo improve the rigid-body model accuracy higher-order

effects are taken into account, which are coupled to therigid-body dynamics. These extensions are rotor dynamics,engine-drive train and actuators dynamics. The first ones arecritical because on the X-Cell the rotor forces and momentslargely dominate the dynamic response. The coupled rotorand stabilizer bar equations are lumped into one first-orderequation ofmotion. This procedure is done for both lateral (b1)and longitudinal (a,) tip-path-plane flapping. The equationsare expressed as follows:

= _ _ b1 _ 1 abl v-v +vlB1t (8)Te TTe altv QR Te

qal I Aaa u- ub &al w-Www

al= -q- -'(~-Te-9 QRft z QR++ Jlon

where BiSlat and A310n are effective steady-state lateral andlongitudinal gains from the cyclic inputs to the main rotorflap angles; 6lat and Jlon are the lateral and longitudinal cycliccontrol inputs; uw, vw and wq, are the wind components along,respectively, X, Y and Z helicopter body axes; Te is theeffective rotor time constant for a rotor with the stabilizer bar.

236

The partial derivatives are approximated by some functions,which are determined experimentally. See [1] for details.

The other inputs 5coi and aped do not explicitly appear in anyequation shown in this paper, but are used for the computationof main and tail rotor thrust (Tmr and Ttr, respectively). Thisinvolves an iterative computation of thrust coefficients whichis included in the simulation model.The rotor speed dynamic is also modeled, but the important

fact is the existence of a electronic governor to regulate rotorspeed by changing the throttle. This governor can not bemodified. Note that the rotor speed is not constant. It isincluded in the dynamics proposed in [1].

Actuators are accurately modeled as third order transferfunctions for the collective and cyclic (lateral and longitudinal)inputs. A second order transfer function is used for the pedalinput. No inflow dynamics are necessary because the settlingtime for the inflow is significantly faster than the rigid bodydynamics.

Equations (3), (4), (5), (7) and (8) constitute the mathemat-ical model for the X-Cell. The rotational kinematic equations(7) are calculated using quaternions. Thus, 15 state variablesare included in equations (3), (4), (5), (7), (8). By adding onestate for rotor speed and one state for the rotor speed trackingerror, the total model is a 17 state variables one. This mathe-matical model is implemented in Matlab/Simulink, which is atrademark of The MathWorks Inc. For the simulation, the fuelconsumption is constant.

III. CONTROL STRUCTURE

There already exist different types of control modes for anunmanned helicopter [5], [3], [2], [4]. Each control mode isnamed according to the states which are considered as outputs.For example, to keep the helicopter at a specific location,position and heading (x, y, z, 7b) control mode is used. For asensor failure such as the absence ofposition information fromthe global positioning system (GPS), altitude and attitude (z,, 0, O) control mode is more desirable to stabilize the vehiclesince the on-board inertial navigation system (INS) would stillbe able to provide altitude and attitude information. The setof outputs x, y, z, 4, constitutes the position and side slipangle mode. In particular, if the controller tries to keep slideslip angle, 4, at zero, the system is said to be operated incoordinatedflight mode.The X-Cell helicopter is equipped with a remote-control

operated by a human pilot to maintain desired attitude, i. e.,as any kind of VTOL it is maneuvered by controlling theirattitude angles. The objective of the present paper is substitutethe action of a pilot for an on-board automatic control system.An altitude/attitude control mode is used. Hence, we can movethe helicopter to a specific location tilting it adequately suchthat the rotor produces sufficiently large horizontal forces.The larger the pitch or roll angle the bigger the longitudinal,respectively lateral, propulsive force.

Fig. 2 shows the control structure proposed. It is constitutedby two control loops in cascade. The inner loop controls the

Fig. 2. Control scheme.

faster dynamics: altitude and attitude, and the outer loop con-trols the slower dynamics: lateral and longitudinal translation.

The synthesis procedure is also sequential. First it is neces-sary to ensure attitude control and then, translational control.

A. Altitude / Attitude Control

An altitude/attitude control mode allows to control thehelicopter closely to a human pilot. The helicopter attitudedynamics are conditionally stable; a minimum amount ofattitude feedback is required for the system to be stable;however, too much feedback will destabilize the system [4].

Taking into account the main control objective, it is pro-posed to develop the altitude/attitude control using SISO PIDcontrollers, where altitude (z) is the vertical position in inertialreference frame. There are four independent PID controllers,one for each angle (X, 0, O) and one for altitude. It is requiredto consider for the maximum ranges accepted by actuators, inorder to avoid saturation.The tuning procedure is difficult due the coupling between

the state variables. The four PID controllers have to be tuningat the same time. Beginning to set proportional feedback in allthe loops is sufficient to stabilize the attitude angles and to takez-position close to the desired value. However, every variablehas an offset, which is eliminated including integral part in thecontrollers. To avoid abrupt changes, it is necessary to includea derivative part in the controllers. Finally, all the controllershave proportional, integral and derivative component, whosegains are fixed according of a trial and error procedure.The transfer function considered for each PID controller is

as:

C(s) = K + Ki +KdsU(s) " s (9)

The gains for each controller are shown in Table I:

TABLE IGAIN VALUES FOR EACH PID CONTROLLER

Kt KdzPID 3 0.2 0.6

0 PID 3 0.5 0.20 PID 13 1 0.2X PID I10 1 .5 -110j

237

B. Lateral / Longitudinal ControlWe use fuzzy logic controllers of Mamdani type to control

x, y position. This type of controller has a heuristic nature,which reflect the experience of a human pilot.An important issue in the design of the Mamdani controllers

are the trim conditions. A helicopter is said to be at trim ifall the forces, aerodynamic and gravitational, and aerodynamicmoments acting on the helicopter about its center of gravity arein balance. Hence, by solving the nonlinear equations F = 0and M = 0, one can solve for the system trim condition. Thetrim condition does not depend on positions, velocities andyaw. Six values are obtained as trim conditions: T3 6TpedTafT, bT, XT and oT. In our case, it is important to know thelast two ones, because these values allow us to obtain lateraland longitudinal steady positions.To compute trim condition, it is necessary to approximate

the expressions for total force and total moment (given in[1]) without considering contribution of the stabilizers; this ispossible because the contribution of stabilizers on total forceand total moment is small. Besides, an approximation of therotor thrust is made for solving the trim condition. As result ofthis computation, the attitude trim in radians is <iT = 0.074665and ST = 0. The real attitude trim used is <iT = 0.07988 and0T = 0 in radians.

Finally, two Mamdani controllers are designed to adequatelyfix roll and pitch angle reference values. Both controllers aredesigned as the simplest possible forn.

C. Mamdani rules for lateral positionGiven a reference y-position, yd, this controller infers a

desired roll angle, <d, using y-position error, ey, and velocity,vY. Both inputs to the lateral Mamdani controller are in inertialreference frame. Fig. 3 displays the membership functionsused for this Mamdani controller. There are nine rules tocompute the desired value for roll. An example of these rulesis as follows:

If ey is Negative and vy is Negative Then qd is Positive

D. Mamdani rules for longitudinal positionGiven a reference x-position, xd, this controller infers a

desired pitch angle, 0d, using a x-position error, ex, and a ve-locity, vx. Both inputs to the longitudinal Mamdani controllerare in inertial reference frame. Fig. 3 displays the membershipfunctions used for this Mamdani controller. There are ninerules to compute the desired value for pitch. An example ofthese rules is as follows:

If ex is Negative and vx is Negative Then 0d is Negative

IV. SIMULATION RESULTSIn this Section we report the performance of the PID

controllers by tracking a square signal for each attitude angleindependently. Also, it is shown how the overall control systemis able to control the helicopter in hovering at a desired x, y,z position. Finally, sinusoidal tracking is done by z-position,while x and y positions reach constant references.

03

coIa)

0

C

a)

C]

l! x1 x x30.8 - ---- -- r- --- r -- - -T - - - - - - - - - T - - - - - - -

0.6 -- -- -- -- -- ---08 ~~~~~~~~~~~~~~~~~~~------ ---

0.4 ------------ ----------- - ------- -L' ,----0.2 -------r-----------r--------- - ----T----------- -----------0

,__ j,,__j__ ____j ____-60 -40 -20 0 20 40 63

Position

vxl Xx3

)

3 -8 ------ -------r ---------------------------~ " ~~~-§---- -- --

1.4 -- ---- ----- ----'----- '~~~~~~ i <~~'~, \

]3.2 ------ ---.- -- -- -1-i- -- -- -.-- - - - - - -- - TX---- ----- n - -- -- - -r -- -- --r------

I I I I_ _-10 -8 -6 -4 -2 0 2 4 6 8 13

Velocity

Fig. 3. Membership function used for each fuzzy controller.

coarIa

0.u5

-0.060 10 20 30 40 50 60 70 80 90 10

Yaw Anglel t^ --

200~~~~~I0 10 20 30 40 50 60 70

Time (sec)

10

80 90 100

Fig. 4. Tracking for roll angle.

A. PID Controllers Performance

Fig. 4, Fig. 5 and Fig. 6 displays roll, pitch and yaw attitudeangles. In each figure, the respective angle tracks a squaresignal while the other two angles has zero as reference. Eachfigure shows how attitude angles are strongly coupled.

Fig. 7 presents attitude angles, each one with zero reference.The system is subject to disturbance. A white noise acts aswind disturbance on each body axe. It is incepted at 50 sec.with a maximum value unity. This maximum value representsa wind velocity of 1 m/sec. It has little effect over the attitudeangles.

Fig. 8 shows inertial position for the last attitude controlcondition, i.e. stabilization of attitude angles. It is shown asthe z-position is taken to an 20 m reference. Note as the xand y-positions increase fast.

238

t

0

Roll Angle

rI-I'- - N-1-UV.. 11

0 10 20 30 40 50 60 70 90 101Pitch Angle

0.2

0

-0.20 10 20 30 40 50 60 70 80 90 101

Yaw Angle

0

"I-100

30

x-Position

.1.--

o io 20 30 40 50 60 70 80 90 10(y-Position

-500-- - -~---------I------- ------ ----- - --I -----

0 10 20 30 40 50 60 70 80 90 101z-Position

-0 10 20 30 40 50 80 70 80 90

Time (sec)

Fig. 5. Tracking for pitch angle.

10 20 30 40 50 60 70 80 90 1Time (sec)

Fig. 8. x and y-posftions without control and tracking in z-posftion.

x-Position

10

10

J0

0.02-

-6 0

cc

Roll Angle

9010 20 30 A 50 60 70 6Pitch Angle

0 20 40 80 80 100 120 140 160 180 20y-PositZ]0o

0 10 20 30 40 50 60 70 80 90 100

Time (sec)

Fig. 6. Tracldng of yaw angle.

Roll Angle0.021

-0.060 10 20 30 40 50 60 70 80 90 10o

Pitch Angle

30 - zu.2CD2 10

n0 20 40 60 80 100 120

Time (sec)

10

0

140 160 180 200

Fig. 9. Stabilization of inertial position.

B. Fuzzy Control Perfornance

Fig. 9 shows inertial position for the same previous condi-

tion, but now including fuzzy control. The z-position reachs a

20 m reference and the x and y-positions return to zero after

a transitory.

Fig. 10 displays attitude angles for the previous 20 m z

reference and stabilization of x and y-positions; roll angle,

,and pitch angle, 0 , are practically maintained on their

trim values 0.07988 and 0, respectively. Because the constant

condition of position and attitude angles the helicopter is

maintained in hover flight.

C. Position Tracking

Fig. 11 shows inertial position for constant tracking in x

and y-positions and sinusoidal tracking in z-position, with a

good performance for the vertical (z) and longitudinal (x)

movements. The angular frequency of the signal tracked byz is 0.1 rad/sec, and the x reference is -8 m. Because the

coupling between collective input and lateral cyclic input, the

y-position oscillates around the 5 m reference.

0 10 20 30 40 50 60 70 80 90 100

White Noise

102I0I I~ ~ ~~~i

Fig. 7. Stabilization of attitude angles.

239

w0.1

cr-

-6co

waC13

0.-

II

0-------

~~I

15 -----

10 20 30 40 50 60 70 80 90

Yaw Angle

(10r-Co.0mm

2-- ---.-- --.-

0 ---'-

-2-I

-10 -- -t-

0 20 40 60 80 100 120 140 160 180 201z-Position

(ACc

0 10 20 30 40 50 60 70 80 90 10Yaw Angle

7 1 4~~----------------

b

13 -r------

E20)(1)M

U2:F

(1)

O.C

Attitude Angles0.2 I- , -Ir02

: F;t~ollWo ----

;

Pitch:-0.2 t - -- --- ----- L ......

-0.4 ----------;I------- ------ -.--------;---- ----------'-------'------r------

Y w-0.6 , , , , ,

-0. ---- -- - - -- -- - - -- ---- -------- ---r -- -- - - -- -- - -- --- - -- -- -- -r--- - --- r-- - -----

-1.2

0 20 40 60 80 100 120 140 160 180 200Time (sec)

Fig. 10. Attitude angles to stabilization of positions.

x-Position and reference10

o s0- ----*1 I- -|-- -L- -L--10- _- -

-200 20 40 60 80 100 120 140 160 180 201

y-Position and reference10 _ ,a

0 ------, r --

X -

0 20 40 60 80 100 120 140 160 180 201z-Position and reference

40

20

0 fXon An ~r-nn mrcan I4n -tr 4 Arn cr-n 4Qo -i nU 40 60 80 100 120Time (sec)

Attitude Angles

-0.

-0.26- - -L-0.-0.4 ----t - - - - - - '-- - - - - - - -- ------ -- - - - - - -I ----- ---r - - - - --

-14

-1.4Di-I'--5---'---r------n---.___ _____

0 20 40 60 80 100 120 140 160 180 20Time (sec)

O

Fig. 12. Attitude angles to make tracking in all the 3 positions.

co

a)

14U 1Iu 1 BU 2uu

Control Inputs

Colloctive

Lat-era

7-l-OgitudiNal-20 20 40 60 60 100 120 140 160 180 20(

Time (sec)O

Fig. 11. Tracking for all the 3 positions.

Fig. 12 portrays attitude angles for the previous trackingcondition. It is shown as roll and pitch angles oscillate tomaintain the constant position. Yaw angle oscillates becausethe coupling between collective and pedals inputs.

Fig. 13 presents control inputs for the previous condition;the cyclic inputs move sinusoidally to maintain the desiredposition and also react when the disturbance begins to affect.

V. CONCLUSIONS

The control structure proposed in this work presents a goodperformance for hovering and forward flight at slow velocities.The control structure uses conventional PID controllers in an

altitude/attitude control mode and Mamdani fizzy controllersfor translational movement. It is tested via simulation.The control structure shows its effectivity over the most re-

alistic mathematical model actually used for a mini-helicopter.Moreover, this control offers the advantage of rapid imple-mentation. As a part of Colibri project, this control structureis going to be implemented in real time.

Fig. 13. Control inputs to make tracking in all the 3 positions.

We are working to simplify the mathematical model andthen to design a controller based on this simplified model.This new controller might be designed as a Takagi-Sugenoregulator.

REFERENCES[1] V. Gavrilets, B. Mettler and E. Feron, "Dynamic model for a miniature

aerobatic helicopter".[2] B. Kadmiry and D. Driankov, "A fizzy gain-scheduler for the attitude

control of an unmanned helicopter", in IEEE Transactions on FuzzySystems, Vol. 12, No. 4, August 2004.

[3] T. Koo and S. Sastry, "Output tracking control design of a helicoptermodel based on approximate linearization", in Proc. 37th IEEE Conf on

Decision and Control (CDC'98), pp. 3635-3640, Tampa, FL., USA, Dec.1998.

[4] B. Mettler, Identification Modeling and Characteristics of MiniatureRotorcraft, Kluwer Academic Publishers, Boston, MA., USA, 2003.

[5] M. Sugeno, "Development of an intelligent unmanned helicopter", inFuzzy Modeling and Control, Selected Works ofM Sugeno, T. H. Nguyenand N. R. Prasad (Eds.), CRC Press, pp. 13-43, Boca Raton, FL., USA,1999.

[6] www.control-systems.net/colibri.htin

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