Quadrotor Aircraft control without Velocity Measurements

Ruifeng Zhang, Xinhua Wang and Kai-Yuan Cai

Abstract— In this paper a quadrotor aircraft is developedwith the signals of attitude angle and angular rate being filteredby using frequency analysis. A PD sliding mode controlleris designed to stabilize the attitude and position while atracking-differentiator is employed to obtain the estimate of therequired velocity. Experiments are conducted to demonstratethe developed quadrotor can be appropriately controlled interms of attitude as well as position.

I. INTRODUCTION

In recent years, quadrotor aircraft has become as a popularresearch topic [1-9]. The major advantage of this type ofaircraft includes heavy loading capacity, and simple butreliable mechanical structure. Moreover, the correspondinggyroscope moments generated by motors can be counteractedeach other. However, the attitude movement of this aircraft isinstable, so it is important to study the control of its attitudemovement. Wen and Kreutz-Delgado presented a kind of PDcontroller for rigid body based on quaternions which can alsowork in model-independent case. This controller is provedto be able to stabilize attitude movement of a rigid bodywith “guaranteed transient performance, and robustness”[1]. Because a quadrotor can be treated as a rigid bodyfor the purpose of controller design, many studies on thecontrol of quadrotor aircraft follow the line of quaternions asdemonstrated in [1]. For example, a control method based onlinearization was presented in [2] to make a quadrotor aerialrobot ‘almost fly’, while [3] proposed a quaternion-basedfeedback control scheme which compensated the Coriolisand gyroscopic torques.

Position control is also necessary to make quadrotoraircraft fly properly. Both the position and velocity mea-surements or their estimates of the aircraft are needed inposition control. To this end, Pollini exploited a GPS re-ceiver which was augmented with Initial Measurement Unit(IMU) by an Extended Kalman Filter Navigation System[4]. In [5], Hoffmann and Waslander compared the positionmeasurement accuracy of different sampling rates of GPSfused with IMU. Experimental results showed that highersampling rate would produce higher accuracy of positionmeasurement. Ultrasonic sensors fusing with IMU [6] andmonocular vision with Moire Patterns [7] were also proposedto obtain estimates of the position and velocity. Differentfrom the apparatus employed in [4-7], a motion capturesystem (Vicon Positioning System) was adopted in the works

This work is supported by National Nature Science Foundation of China(60774008) and Aviation Science Foundation of China (2008ZG51092).

The authors are with the National Key Laboratory of Science andTechnology on Integrated Control, the Department of Automatic Control,Beijing University of Aeronautics and Astronautics, Beijing, 100191, Chinazhangruifengd635@gmail.com.

presented in [8, 9]. Although the Vicon Positioning Systemis capable of obtaining highly accurate position informationand achieving a high sampling rate, the way of calculatingthe velocity using the position information was not clearlyprovided in [8, 9].

In this paper we propose a tracking-differentiator [10,11] to estimate the velocity of quadrotor aircraft by usingthe position measurements collected by a Vicon PositioningSystem available in the National Key Laboratory of Scienceand Technology on Integrated Control, BUAA. In order toverify the validity of the proposed tracking-differentiator, wedevelop a small quadrotor prototype in the Laboratory, inwhich a PD sliding mode controller based on quaternionsis designed to stabilize the attitude of the quadrotor, and acontrol method based on linearization is adopted to controlthe corresponding position. The information of attitude andangle rate provided for the attitude controller is obtainedwith an IMU and filtered by an integer-coefficient digitalfilter. A tracking-differentiator [10, 11] is used to calculatethe velocity of the aircrafts in this paper.

The rest of this paper is organized as follows. In section2, the mathematical model of the aircraft is given and thecontrol problem is stated. In section 3, an integer-coefficientdigital filter is proposed to restrain the effect of measurementdisturbances. In sections 4 and 5, the attitude controllerand position controller are given respectively. In section 6,the indoor flight experiment results are shown. And, theconclusions are given in section 7.

II. PROBLEM STATEMENT

The quadrotor aircraft under consideration consists of arigid cross frame equipped with four rotors as shown inFig. 1. The up (down) motion is achieved by increasing(decreasing) the total thrust while maintaining an equalindividual thrust. The forward/backward, left/right and theyaw motions are achieved through a differential controlstrategy of the thrust generated by each rotor. In order toavoid the yaw drift due to the reactive torques, the quadrotoraircraft is configured such that the set of rotors (right-left)rotates clockwise and the set of rotors (front-end) rotatescounterclockwise. There is no change in the direction ofrotation of the rotors. If a yaw motion is desired, one has toreduce the thrust of one set of rotors and increase the thrust ofthe other set while maintaining the same total thrust to avoidan up-down motion. Hence, the yaw motion is then realizedin the direction of the induced reactive torque. On the otherhand, forward (backward) motion is achieved by pitching inthe desired direction by increasing the end (front) rotor thrustand decreasing the front (end) rotor thrust to maintain the

Joint 48th IEEE Conference on Decision and Control and28th Chinese Control ConferenceShanghai, P.R. China, December 16-18, 2009

ThBIn6.10

978-1-4244-3872-3/09/$25.00 ©2009 IEEE 5213

total thrust. Finally, a sideways motion is achieved by rollingin the desired direction by increasing the left (right) rotorthrust and decreasing the right (left) rotor thrust to maintainthe total thrust.

right

left

end

e1Be2B

e3B

e1I

e3I

e2I

F3 F4

front roll

pitch

yaw F1F2

Sb

Sg

Fig. 1. Sketch map of quadrotor aircraft

In Fig.1, Sg denotes the inertial frame; Sb denotes the bodyframe; ξ =

[x y z

]T ∈ Sg and v =[

x y z]T

de-note the mass center position and the velocity of the aircraftin the inertial frame, respectively; The Fi lift generated bythe rotor in the free air (expressed in Sb). The linear motionequations are given as follows:

ξ = v

v = gez − THReZ

m

where m denotes the aircraft mass; g denotes the accelerationdue to the gravity; the vector ez =

[0 0 1

]Tdenotes the

unit vector in the frame Sg; the orthogonal matrix R ∈ <3×3

depends on the pitch angle ϑ, the yaw angle ψ and the rollangle φ, and is further expressed as

R =

cϑcψ cψsϑsφ− sψcφ cψsϑcφ + sψsφsψcϑ sψsϑsφ + cψcφ sψsϑcφ− cψsφ−sϑ cϑsφ cϑcφ

(1)

with c· = cos(·), s· = sin(·). By quaternion, the kinematicsdynamics of the aircraft are described as

q =12

(q × Ω + q0Ω)

q0 = −12ΩT q (2)

R = R · sk(Ω)

where the vector (q, q0) ∈ <4 represents the so-called unitquaternion representation of the orientation (see [3] and thereferences therein) with the q0 and q subject to the constraintcondition qT q + q2

0 = 1; the Ω denotes the angular velocityof the airframe expressed in the frame Sb; the sk(Ω) denotesthe skew-symmetric matrix given by

sk(Ω) =

0 −Ωz Ωy

Ωz 0 −Ωx

−Ωy Ωx 0

The attitude dynamics equations are

JΩ = −Ω× JΩ + Γ (3)

or, equivalently,

Ω = −J−1Ω× JΩ + J−1Γ

where J ∈ <3×3 denotes the inertial matrix and the Γ ∈<3×3 denotes the total control moment vector.

Use F1, F3, F2, F4 to denote the lift forces generated bythe front, end, left and right rotors, respectively. Then, thetotal force FH satisfies

FH = F1 + F3 + F2 + F4

In the quadrotor aircraft, if the four rotors rotate at thesame speed, there will be no yaw motion as the reactivetorques are cancelled out (the reader is referred to [3] fordetailed discussion about this point). Therefore, we can get

(Γ

FH

)= M

F1

F3

F2

F4

M =

0 0 d + ∆dl −d−∆dr

d + ∆df −d−∆de 0 0k1/α1 k2/α2 k3/α3 k4/α4

1 1 1 1

where d denotes the average distance between lift forcesand mass center; df , de, dl, dr are the distances betweenthe operating points of the four lift forces and the centerof gravity, respectively; ∆dl = |dl − d|, ∆dr = |dr − d|,∆df = |df − d|, ∆de = |de − d|; the term ki/αi( i =1, ..., 4) denotes the ratio between the lift force generatedby the rotor and the corresponding reactive torque. Define

MF , M[

F1 F3 F2 F4

]T

then, we obtain

MF =[

ΓFH

]

Now, we are ready to state the main objective of thispaper. That is, design appropriate filters and attitude/positioncontrollers for the aircraft based on the above mathematicalmodels.

III. FILTER DESIGN

In this section, an integer-coefficient digital filter is pro-posed to restrain the measurement disturbances.

The frequency property of integer coefficient digital filteris described by

H(ejω) =+∞∑

k=−∞h(k)e−jωk (4)

where h(k) represents the impulse response of the filter’stransfer function.

In this paper, we choose

H(z) =1− z−N

1− z−1(5)

ThBIn6.10

5214

then the (4) can be written as

H(ejω) = e−j(N−1)ω/2 sin(ωN/2)sin(ω/2)

, N = 1, 2, 3... (6)

Noting (6), we can get an appropriate filter by selectingN properly. For thirty percent of the original PWM wavesignal, the corresponding power spectrum is shown in Fig.2.

From Fig.2, we see that the high frequency componentshave obvious saliences. To avoid big time delays, the degreeof the filter (i.e. N ) should not be too high. And, it is betterto make the coefficient be integer. In our design, accordingto the frequency-magnitude curve of the filet plotted in Fig.3,we select N = 5.

The power spectrum of filtered signal with N = 5 isshown in Fig. 4.

From Fig.2 and Fig.4, it is seen that the filtered signal hasa better spectrum property than the original one.

Fig. 2. The frequency spectrum of a signal without filter

N=5

Fig. 3. The magnitude and frequency curve of the filter

IV. ATTITUDE CONTROLLER DESIGN

Rewrite the equation (3) as follows:

JΩ = −Ω× JΩ + Γt + d

Fig. 4. The power spectrum of the signal after filtering

where Γ = Γt + d with the disturbance d and the controltorque Γt denoted by

Γt =[

MT1 MT2 MT3

]T

Choose the attitude control law as

Γt = Γm + Γa (7)

whereΓm = −Ω− q (8)

Γa = −Kssign(Ω + q) (9)

For the attitude control law (7), the sliding mode controlterm Γa expressed in (9) is used to restrain the disturbanced and the other term Γm expressed in (8) represents a PDcontroller. In fact, without considering the disturbance (i.e.considering d ≡ 0), the PD control term can render theaircraft stable. However, to deal with big disturbances, thesliding mode control term is required.

To show the stability of the resulting closed-loop attitudesystem with the controller (7), we consider the slidingvariable s = Ω + q. Let the Lyapunov function be

V = sT s

From (2), (3) and (7) we obtain

V = sT[q × Ω + q0Ω− 2J−1(Γt + d)

]

= sT [sk(q) + q0I] Ω− 2J−1Ω× JΩ

+ 2J−1(Γt + d),where I ∈ R3×3 is a identical matrix. It can be derived that

Γt = J [−s−Kssign(s)]

where

sign(s) =[

sign(s1) sign(s2) sign(s3)]

Let

∆ =[

K′1 K ′

2 K′3

]T

= [sk(q) + q0I3] Ω− 2J−1Ω× JΩ + 2J−1d

ThBIn6.10

5215

and

Ks = diag(K1,K2,K3),

(K1 >∣∣∣K ′

1

∣∣∣ ,K2 >∣∣∣K ′

2

∣∣∣ ,K3 >∣∣∣K ′

3

∣∣∣)Therefore, we get

V = sT (∆ + J−1Γt)

= sT [∆− s−Kssign(s)]

= −sT s−3∑

i=1

(Ki |si| −K

′isi

)

≤ −‖s‖2 −3∑

i=1

(Ki |si| −K

′i |si|

)(10)

= −‖s‖2 −3∑

i=1

(Ki −K′i) |si|

< −K ‖s‖ < 0

where K = mini=1,2,3

(Ki −∣∣∣K ′

i

∣∣∣).From (10), we know that there exists a time ts such that

s = q + Ω = 0 for t ≥ ts, i.e. q = −Ω for t ≥ ts. It thenfollows from (2) that, for t ≥ ts,

q = −12q0q, q0 =

12qT q

Select the Lyapunov function as W = qT q+(1−q0)2, whosetime derivative along the solution of above equation satisfies

W = −qT q0q − (1− q0)qT q= −qT q< 0

Therefore, q → 0,Ω → 0 as t →∞.In the following section, we will present a differentiator-

based position controller for the quadrotor aircraft.

V. POSITION AND VELOCITY FEEDBACK CONTROL

The key of feedback control is to obtain the position andvelocity signals of the aircraft. We use Vicon system to catchthe position of the aircraft. The velocity of the aircraft isobtained by the tracking-differentiators proposed in [10] and[11]. And, the second-order differentiator is

x1 = x2

ε2x2 = −a10(x1 − v(t))− a11 |x1 − v(t)|α sign(x1 − v(t))− a20εx2 − a21 |εx2|α sign |x2|

y = x2

where 0 < ε < 1 is the perturbation parameter; 0 < α < 1,a10, a11, a20 and a21 are positive design constants, v is theposition signal obtained from Vicon system. Therefore, thereexists a positive number γ satisfying ργ > 2 (where ρ = α),such that

x1 − v = O(εργ), x2 − ˙v = O(εργ−1)

where ˙v denotes the derivative of the signal v, and O(εργ−1)denotes the approximation of order εργ−1 between x2 andv.

In the following, we will design the position and velocityfeedback controller.

Since the roll and pitch angle are relatively small, thelinearization method can be adopted. And, we choose theposition control law as

Fg =

Fxg

Fyg

Fzg

=

−K1xx−K2xx−K1yy −K2y y

−mg

(11)

where K1x, K2x, K1y and K2y are positive parameters tobe specified.

We can obtain Fb expressed in the body frame. Notingthat

Fb =[

0 0 −mg]T

,mg = FH

we obtain

fg = RgbFb = −

cψsϑcφ + sψsφsψsϑcφ− cψsφ

cϑcφ

FH (12)

where fg denotes the gravity expressed in the body frame,the transfer matrix Rgb (from the body frame to the inertialframe) is equal to the R matrix defined in (1).

To get the desired values of ϑ, φ noting (11) and (12), welet

Fxg

Fyg

Fzg

= −

cψsϑcφ + sψsφsψsϑcφ− cψsφ

cϑcφ

FH (13)

Since ϑ, φ are relatively small, by linearizing (13) we get

Fxg

Fyg

Fzg

= −

ϑ cos ψ + φ sinψϑ sinψ − φ cos ψ

1

FH (14)

From (14) we obtain

ϑd = Fxgcψ+FygsψFzg

φd = Fxgsψ−FygcψFzg

(15)

With the Fxg , Fyg , Fzg given in (11) and the measured ψ,we can use the equation (15) to compute the desired valuesof ϑ, φ denoted by ϑd, φd respectively.

By regulating the PD controller parameters, the acceptableresponse can be carried out even without the model ofquadrotor aricraft. As a result, the closed-loop system isstable. Physically, the aircraft hovers in the given position.

VI. EXPERIMENT

A. Hardware Design of Quadrotor Aircraft.

We develop a quadrotor aircraft prototype shown in Fig.5.In this aircraft, an Electric Speed Controller (ESC, RCE-BL35X) based on Pulse width modulation (PWM) is adoptedto regular the speed of Brushless Direct Current (BLDC)motors. Its peaking current is 45A, and constant current is35A, moreover, the input voltage is 5.5-16.8v. This typeof ESC is relatively cheap and reliable. Its shortest regular

ThBIn6.10

5216

period we can get is 2.5ms by experiments. IMU (XSENSMTi) is selected as the mail attitude sensor. Therefore, the3-axis angles and angular rates can be obtained. A DSP(TMS320LF2812) is taken as the driven board, which hasmultiple PWM output channels. The control period is 8ms.

Fig. 5. Prototype of quadrotor aircraft

B. The parameters of the aircraft

The parameters are: m = 2.33kg, g = 9.81m/s2. Themoment of inertia is

J =

Jxx 0 00 Jyy 00 0 Jzz

where Jxx = 0.16Nm, Jyy = 0.16Nm,Jzz = 0.32Nm.

Voltage range of BLDC motors is 6-18v, weight is 78g,max power is 340w, and maximum thrust is 1400g. The fourrotors are all airscrew woody. The differentiator parametersis chosen as ε = 0.002, a10 = 0.1, a11 = 0.015, b10 =0.3, b11 = 0.3.

Flying test with motion capture system (Vicon system) onguaranteeing plane is shown in Fig.6, while flying test withVicon system is shown in Fig.7.

In Fig.8, x is the X-axis position signal obtained by Viconsystem, whose time derivative, denoting the X-axis velocity,is obtained from tracking-differentiator. In Fig.9, y is the Y-axis position signal obtained by Vicon system, whose timederivative, denoting the Y-axis velocity, is obtained fromtracking-differentiator as well. Fig.10 shows roll and pitchangles. Fig.11 and Fig.12 show the front, end, left and rightforces generated by the four rotors, respectively. Though theposition signals have frequent erroneous readings from Viconsystem, the velocities obtained from tracking-differentiatorstill have satisfying quality, and the noises are restrainedsufficiently by the tracking-differentiator. At the same time,excellent stability performance is obtained.

Fig. 6. Flying test with Vicon system on guaranteeing plane

Fig. 7. Flying test with Vicon system

Fig. 8. The position and velocity in x-axis

Fig. 9. The position and velocity in y-axis

ThBIn6.10

5217

Fig. 10. Roll and pitch angles

Fig. 11. Front and end forces

Fig. 12 Left and right forces

The experiment video of the aircraft is shown in thefollowing web address:http://you.video.sina.com.cn/b/19086673-1314556583.html,or http://v.youku.com/v show/id XNzgzODE1MDA=.html.

VII. CONCLUSIONS

In this paper a quadrotor aircraft is developed, and a PDsliding mode controller is designed to stabilize the attitudeand position based on tracking-differentiator. Through theexperiments, the tracking-differentiator can overcome the fre-

quent erroneous readings, and excellent position and attitudestability of aircraft are guaranteed.

REFERENCES

[1] J. T. Y. Wen and K. Kreutz-Delgado, “The attitude control problem,”IEEE Transactions on Automatic Control, Vol. 36, No. 10, pp.1148–1162, 1991.

[2] P. Pounds, R. Mahony, P. Hynes, and J. Roberts, “Design of afour-rotor aerial robot,” Australasian Conference on Robotics andAutomation, pp.145–150, November 2002.

[3] A. Tayebi and S. McGilvray, “Attitude stabilization of a four-rotoraerial robot,” 43rd IEEE Conference on Decision and Control, De-cember 14–17, 2004, Atlantis, Paradise Island, Bahamas.

[4] L. Pollini. “Simulation and robust backstepping control of a quadrotoraircraft,” AIAA Modeling and Simulation Technologies Conference andExhibit, August 18–21, 2008, Honolulu, Hawaii

[5] G. M. Hoffmann, S. L. Waslander, and C. J. Tomlin, “DistributedCooperative Search using Information-Theoretic Costs for ParticleFilters, with Quadrotor Applications”, AIAA Guidance, Navigation,and Control Conference and Exhibit, 21–24 August 2006.

[6] F. Kendoul, D. Lara, I. Fantoni-Coichot, and R. Lozano, “Real-Time Nonlinear Embedded Control for an Autonomous QuadrotorHelicopter”, Journal of Guidance, Control, and Dynamics, Vol. 30,No. 4, pp.1049–1051, July–August 2007.

[7] M. Valenti, B. Bethke, G. Fiore, J. P. How, E. Feron, “Indoor Multi-Vehicle Flight Testbed for Fault Detection, Isolation, and Recovery”,AIAA Guidance, Navigation, and Control Conference and Exhibi,. 21–24 August 2006, Keystone, Colorado.

[8] G. P. Tournier, M. Valenti, J. P. How, E. Feron, ”Estimation andControl of a Quadrotor Vehicle Using Monocular Vision and MoirePatterns”, AIAA Guidance, Navigation, and Control Conference andExhibit, 21–24 August 2006, Keystone, Colorado.

[9] J. J. Troy, C. A. Erignac, and P. Murray, “Closed-Loop MotionCapture Feedback Control of Small-Scale Aerial Vehicles”, AIAAInfotech@Aerospace 2007 Conference and Exhibit, 7–10 May 2007,Rohnert Park, California.

[10] X. Wang, Z. Chen, and G. Yang, “Finite-time-convergent differentiatorbased on singular perturbation technique”, IEEE Transactions onAutomatic Control, Vol. 52, No. 9, pp.1731–1737, 2007.

[11] X. Wang, Z. Chen, and Z. Yuan, “Nonlinear tracking-differentiatorwith high speed in whole course”. Control Theory &Applications (inChinese), Vol. 20, No. 6, pp.875–878, 2004.

ThBIn6.10

5218