1109.4457v1

Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3)

Taeyoung Lee, Melvin Leok, and N. Harris McClamroch

Abstract This paper provides nonlinear tracking controlsystems for a quadrotor unmanned aerial vehicle (UAV) thatare robust to bounded uncertainties. A mathematical model of aquadrotor UAV is defined on the special Euclidean group, andnonlinear output-tracking controllers are developed to follow(1) an attitude command, and (2) a position command for thevehicle center of mass. The controlled system has the desirableproperties that the tracking errors are uniformly ultimatelybounded, and the size of the ultimate bound can be arbitrarilyreduced by control system parameters. Numerical examplesillustrating complex maneuvers are provided.

I. INTRODUCTION

A quadrotor unmanned aerial vehicle (UAV) consists oftwo pairs of counter-rotating rotors and propellers. It hasbeen envisaged for various applications such as surveillanceor mobile sensor networks as well as for educational pur-poses, and several control systems have been studied.

Linear control systems have been widely used to enhancethe stability properties of an equilibrium of a quadrotorUAV [1], [2], [3]. In [4], the quadrotor dynamics is modeledas a collection of simplified hybrid dynamic modes, andreachability sets are analyzed to guarantees the safety andperformance for larger area of operating conditions.

Several nonlinear controllers have been developed as well.Backstepping and sliding mode techniques are applied in [5],[6], and a nonlinear H controller is studied in [7]. Anadaptive neural network based control system is developedin [8]. Since all of these controllers are based on Eulerangles, they exhibit singularities when representing complexrotational maneuvers of a quadrotor UAV, thereby signif-icantly restricting their ability to achieve complex flightmaneuvers.

An attitude control system based on quaternions is appliedto a quadrotor UAV [9]. Quaternions do not have singulari-ties, but they have ambiguities in representing an attitude,as the three-sphere S3 double-covers SO(3). As a result,in a quaternion-based attitude control system, convergenceto a single attitude implies convergence to either of thetwo disconnected, antipodal points on S3 [10]. Therefore,depending on the particular choice of control inputs, aquaternion-based control system may become discontinuous

Taeyoung Lee, Mechanical and Aerospace Engineering, The GeorgeWashington University, Washington DC 20052 [email protected]

Melvin Leok, Mathematics, University of California at San Diego, LaJolla, CA 92093 [email protected]

N. Harris McClamroch, Aerospace Engineering, University of Michigan,Ann Arbor, MI 48109 [email protected] research has been supported in part by NSF under grants CMMI-1029551.This research has been supported in part by NSF under grants DMS-

0726263, DMS-1001521, DMS-1010687, and CMMI-1029445.

when applied to actual attitude dynamics [11], and it mayalso exhibit unwinding behavior, where the controller rotatesa rigid body through unnecessarily large angles [12], [13].

Attitude control systems also have been developed directlyon the special orthogonal group, SO(3) to avoid the sin-gularities associated with Euler-angles and the ambiguityof quaternions [14], [15], [16], [17]. By following thisgeometric approach, the dynamics of a quadrotor UAV isglobally expressed on the special Euclidean group, SE(3),and nonlinear control systems are developed to track outputsof several flight modes, namely an attitude controlled flightmode, a position controlled flight mode, and a velocitycontrolled flight mode [18]. Several aggressive maneuversof a quadrotor UAV are also demonstrated based on ahybrid control architecture. This is particularly desirablesince complicated reachability set analysis is not required toguarantee a safe switching between different flight modes,as the region of attraction for each flight mode covers theconfiguration space almost globally.

In this paper, we extend the results of [18] to con-struct nonlinear robust tracking control systems on SE(3)for a quadrotor UAV. We assume that there exist unstruc-tured, bounded uncertainties, with pre-determined bounds,on the translational dynamics and the rotation dynamicsof a quadrotor UAV. Output tracking control systems aredeveloped to follow an attitude command or a positioncommand for the vehicle center of mass. We show thatthe tracking errors are uniformly ultimately bounded, andthe size of the ultimate bound can be arbitrarily reduced.The robustness of the proposed tracking control systems arecritical in generating complex maneuvers, as the impact ofthe several aerodynamic effects resulting from the variationin air speed is significant even at moderate velocities [2].

The paper is organized as follows. We develop a globallydefined model for the translational and rotational dynamics ofa quadrotor UAV in Section II. A hybrid control architectureis introduced and a robust attitude tracking control systemis developed in Section III. Section IV present results for arobust position tracking, followed by numerical examples inSection V.

II. QUADROTOR DYNAMICS MODEL

Consider a quadrotor UAV model illustrated in Figure1. This is a system of four identical rotors and propellerslocated at the vertices of a square, which generate a thrustand torque normal to the plane of this square. We choosean inertial reference frame {~e1, ~e2, ~e3} and a body-fixedframe {~b1,~b2,~b3}. The origin of the body-fixed frame islocated at the center of mass of this vehicle. The first and

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~e1

~e2~e3

~b1

~b2

~b3

f1

f2

f3

f4

x R

Fig. 1. Quadrotor model

the second axes of the body-fixed frame, ~b1,~b2, lie in theplane defined by the centers of the four rotors, as illustratedin Figure 1. The third body-fixed axis ~b3 is normal to thisplane. Each of the inertial reference frame and the body-fixed reference frame consist of a triad of orthogonal vectorsdefined according to the right hand rule. Define

m R the total massJ R33 the inertia matrix with respect to the

body-fixed frameR SO(3) the rotation matrix from the body-fixed

frame to the inertial frame R3 the angular velocity in the body-fixed

framex R3 the position vector of the center of mass

in the inertial framev R3 the velocity vector of the center of mass

in the inertial framed R the distance from the center of mass to

the center of each rotor in the ~b1,~b2plane

fi R the thrust generated by the i-th pro-peller along the ~b3 axis

i R the torque generated by the i-th pro-peller about the ~b3 axis

f R the total thrust magnitude, i.e., f =4i=1 fi

M R3 the total moment vector in the body-fixed frame

The configuration of this quadrotor UAV is defined by thelocation of the center of mass and the attitude with respect tothe inertial frame. Therefore, the configuration manifold isthe special Euclidean group SE(3), which is the semidirectproduct of R3 and the special orthogonal group SO(3) ={R R33 |RTR = I, detR = 1}.

The following conventions are assumed for the rotors andpropellers, and the thrust and moment that they exert on thequadrotor UAV. We assume that the thrust of each propelleris directly controlled, and the direction of the thrust of eachpropeller is normal to the quadrotor plane. The first andthird propellers are assumed to generate a thrust along thedirection of ~b3 when rotating clockwise; the second andfourth propellers are assumed to generate a thrust along

the same direction of ~b3 when rotating counterclockwise.Thus, the thrust magnitude is f =

4i=1 fi, and it is

positive when the total thrust vector acts along ~b3, andit is negative when the total thrust vector acts along ~b3. Bythe definition of the rotation matrix R SO(3), the totalthrust vector is given by fRe3 R3 in the inertial frame.We also assume that the torque generated by each propelleris directly proportional to its thrust. Since it is assumed thatthe first and the third propellers rotate clockwise and thesecond and the fourth propellers rotate counterclockwise togenerate a positive thrust along the direction of ~b3, thetorque generated by the i-th propeller about~b3 can be writtenas i = (1)icffi for a fixed constant cf . All of theseassumptions are common [3], [9].

Under these assumptions, the moment vector in the body-fixed frame is given by

fM1M2M3

=

1 1 1 10 d 0 dd 0 d 0cf cf cf cf

f1f2f3f4

. (1)The determinant of the above 4 4 matrix is 8cfd2, so itis invertible when d 6= 0 and cf 6= 0. Therefore, for giventhrust magnitude f and given moment vector M , the thrust ofeach propeller f1, f2, f3, f4 can be obtained from (1). Usingthis equation, the thrust magnitude f R and the momentvector M R3 are viewed as control inputs in this paper.

The equations of motion of the quadrotor UAV can bewritten as

x = v, (2)mv = mge3 fRe3 + x, (3)

R = R, (4)

J + J = M + R, (5)where the hat map : R3 so(3) is defined by the conditionthat xy = x y for all x, y R3 (see Appendix A). Theinverse of the hat map is denoted by the vee map, :so(3) R3. Unstructured uncertainties in the translationaldynamics and the rotational dynamics of a quadrotor UAVare denoted by x and R R3, respectively. We assumethat uncertainties are bounded:

x x, R R (6)for known, positive constants x, R R.

Throughout this paper, m() and M () denote the min-imum eignevalue and the maximum eigenvalue of a matrix,respectively.

III. ATTITUDE CONTROLLED FLIGHT MODEA. Flight Modes

Since the quadrotor UAV has four inputs, it is possible toachieve asymptotic output tracking for at most four quadrotorUAV outputs. The quadrotor UAV has three translationaland three rotational degrees of freedom; it is not possibleto achieve asymptotic output tracking of both attitude andposition of the quadrotor UAV. This motivates us to introduce

several flight modes, namely (1) an attitude controlled flightmode, and (2) a position controlled flight mode.

A complex flight maneuver can be defined by specifyinga concatenation of flight modes together with conditionsfor switching between them; for each flight mode one alsospecifies the desired or commanded outputs as functions oftime. Unlike a hybrid flight control system that requiresreachability analyses [4], the proposed control system isrobust to switching conditions since each flight mode hasalmost global stability properties, and it is straightforward todesign a complex maneuver of a quadrotor UAV.

In this section, an attitude controlled flight mode is consid-ered, where the outputs are the attitude of the quadrotor UAVand the controller for this flight mode achieves asymptoticattitude tracking.

B. Attitude Tracking Errors

Suppose that an arbitrary smooth attitude commandRd(t) SO(3) is given. The corresponding angular velocitycommand is obtained by the attitude kinematics equation,d = R

Td Rd.

We first define errors associated with the attitude dynamicsof the quadrotor UAV. The attitude error function studiedin [14], [19], [20], and several properties are summarized asfollows.Proposition 1: For a given tracking command (Rd,d),

and the current attitude and angular velocity (R,), wedefine an attitude error function : SO(3)SO(3) R, anattitude error vector eR R3, and an angular velocity errorvector e R3 as follows:

(R,Rd) =1

2tr[I RTdR

], (7)

eR =1

2(RTdRRTRd), (8)

e = RTRdd, (9)

Then, the following statements hold:

(i) is locally positive-definite about R = Rd.(ii) the left-trivialized derivative of is given by

TIR (DR(R,Rd)) = eR. (10)

(iii) the critical points of , where eR = 0, are {Rd} {Rd exp(pis), s S2}.

(iv) a lower bound of is given as follows:

1

2eR(R,Rd)2 (R,Rd), (11)

(v) Let be a positive constant that is strictly less than 2.If (R,Rd) < < 2, then an upper bound of isgiven by

(R,Rd) 12 eR(R,Rd)

2. (12)

Proof: See [20].

C. Attitude Tracking Controller

We now introduce a nonlinear controller for the attitudecontrolled flight mode, described by an expression for themoment vector:

M = kReR ke + J J(RTRdd RTRdd) + R, (13)

R = 2ReA

ReA+ R , (14)eA = e + c2J

1eR, (15)

where kR, k, c2, R are positive constants.In this attitude controlled mode, it is possible to ignore

the translational motion of the quadrotor UAV; consequentlythe reduced model for the attitude dynamics are given byequations (4), (5), using the controller expression (13)-(15).We now state the result that the tracking errors (eR, e) areuniformly ultimately bounded.Proposition 2: (Robustness of Attitude Controlled Flight

Mode) Suppose that the initial attitude error satisfies

(R(0), Rd(0)) < 2 < 2 (16)

for a constant 2. Consider the control moment M defined in(13)-(15). For positive constants kR, k, the constants c2, Rare chosen such that

c2 < min

{k,

4kkRm(J)2

k2M (J) + 4kRm(J)2,kRm(J)

},

(17)

R 0. All of these results can be applied toa nonlinear robust control problem for the attitude dynamicsof any rigid body.

Asymptotic tracking of the quadrotor attitude does notrequire specification of the thrust magnitude. As an auxil-iary problem, the thrust magnitude can be chosen in many

different ways to achieve an additional translational motionobjective. For example, it can be used to asymptotically tracka quadrotor altitude command [18].

Since the translational motion of the quadrotor UAV canonly be partially controlled; this flight mode is most suitablefor short time periods where an attitude maneuver is to becompleted.

IV. POSITION CONTROLLED FLIGHT MODE

We now introduce a nonlinear controller for the positioncontrolled flight mode. This flight mode requires analysis ofthe coupled translational and rotational equations of motion;hence, we make use of the notation and analysis in the priorsection to describe the properties of the closed loop systemin this flight mode.

A. Position Tracking Errors

An arbitrary smooth position tracking command xd(t) R3 is chosen. The position tracking errors for the positionand the velocity are given by:

ex = x xd, (20)ev = v xd. (21)

Following the prior definition of the attitude error and theangular velocity error, we define

eR =1

2(RTc RRTRc), e = RTRcc, (22)

and the computed attitude Rc(t) SO(3) and computedangular velocity c R3 are given by

Rc = [b1c ; b3c b1c ; b3c ], c = RTc Rc, (23)where b3c S2 is defined by

b3c = kxex kvev mge3 +mxd + xkxex kvev mge3 +mxd + x , (24)

and b1c S2 is selected to be orthogonal to b3c, therebyguaranteeing that Rc SO(3). The constants kx, kv arepositive, and the control input term x is defined later in(29). We assume that

kxex kvev mge3 +mxd + x 6= 0, (25)and the commanded acceleration is uniformly bounded suchthat

mge3 +mxd < B (26)for a given positive constant B.

B. Position Tracking Controller

The nonlinear controller for the position controlled flightmode, described by control expressions for the thrust mag-nitude and the moment vector, are:

f = (kxex + kvev +mge3 mxd x) Re3, (27)M = kReR ke + J

J(RTRcc RTRcc + R), (28)

Forcecontroller

Momentcontroller

--

-

- -QuadrotorDynamics

-f

M

b3cxd

(b1d )

x, v,R,

6 -q qController

Fig. 2. Controller structure for position controlled flight mode

x = +2x eBeB

+1x eB+1 + +1x, (29)

eB = ev +c1mex, (30)

R = 2ReA

ReA+ R , (31)eA = e + c2J

1eR, (32)

where kx, kv, kR, k, c1, c2, x, R, are positive constants,and > 2.

The nonlinear controller given by equations (27), (28)can be given a backstepping interpretation. The computedattitude Rc given in equation (23) is selected so that thethrust axis b3 of the quadrotor UAV tracks the computeddirection given by b3c in (24), which is a direction of thethrust vector that achieves position tracking. The momentexpression (28) causes the attitude of the quadrotor UAV toasymptotically track Rc and the thrust magnitude expression(27) achieves asymptotic position tracking.

The closed loop system for this position controlled flightmode is illustrated in Figure 2. The corresponding closedloop control system is described by equations (2)-(5), us-ing the controller expressions (27)-(32). We now state theresult that the tracking errors (ex, ev, eR, e) are uniformlyultimately bounded.Proposition 3: (Robustness of Position Controlled Flight

Mode) Suppose that the initial conditions satisfy

(R(0), Rc(0)) < 1 < 1, (33)ex(0) < exmax , (34)

for positive constants 1, exmax . Consider the control inputsf,M defined in (27)-(32). For positive constants kx, kv , wechoose positive constants c1, c2, kR, k, x, R such that

c1 < min

{kv(1 ), 4mkxkv(1 )

2

k2v(1 + )2 + 4mkx(1 ) ,

kxm

},

(35)

c2 < min

{k,

4kkRm(J)2

k2M (J) + 4kRm(J)2,kRm(J)

},

(36)

m(W2) >W122

4m(W1), (37)

x + R x+1x eB+1

+1x eB+1 + +1x+1x

(xeB) (

xxeB

)x x.

Now we find a bound of X given by (70). Since f =A(eT3 RTc Re3), we haveX A (eT3 RTc Re3)Re3 Rce3

(kxex+ kvev+B + x) (eT3 RTc Re3)Re3 Rce3.The last term (eT3 RTc Re3)Re3 Rce3 represents the sineof the angle between b3 = Re3 and bc3 = Rce3, since

(b3c b3)b3 b3c = b3 (b3 b3c).The magnitude of the attitude error vector, eR representsthe sine of the eigen-axis rotation angle between Rc and R(see [18]). Therefore, we have (eT3 RTc Re3)Re3Rce3 eR. It follows that(eT3 RTdRe3)Re3 Rde3 eR =

(2)

{

1(2 1) , }< 1.

(75)

Therefore, X is bounded by

X (kxex+ kvev+B + x)eR (kxex+ kvev+B + x). (76)

Substituting (74), (76) into (73),

V1 (kv(1 ) c1)ev2 c1kxm

(1 )ex2

+c1kvm

(1 + )exev+ eR

{(B + x)(

c1mex+ ev) + kxexev

}+ x. (77)

In the above expression for V1, there is a third-order errorterm, namely kxeRexev. Using (75), it is possibleto choose its upper bound as kxexev similar to otherterms, but the corresponding stability analysis becomes com-plicated, and the initial attitude error should be reducedfurther. Instead, we restrict our analysis to the domainD1 defined in (66), and its upper bound is chosen askxexmaxeRev.

c) Lyapunov Candidate for the Complete System:: LetV = V1 + V2 be the Lyapunov candidate of the completesystem.

V = 12kxex2 + 1

2mev2 + c1ex ev

+1

2e Je + kR(R,Rd) + c2eR e. (78)

Using (67), the bound of the Lyapunov candidate V can bewritten as

zT1 M11z1 + zT2 M21z2 V zT1 M12z1 + zT2 M 22z2, (79)

where z1 = [ex, ev]T , z2 = [eR, e]T R2, andthe matrices M11,M12,M21,M22 are given by

M11 =1

2

[kx c1c1 m

], M12 =

1

2

[kx c1c1 m

],

M21 =1

2

[kR c2c2 m(J)

], M 22 =

1

2

[2kR

21 c2c2 M (J)

].

Using (63) and (77), the time-derivative of V is given byV zT1 W1z1 + zT1 W12z2 zT2 W2z2 + x + R, (80)

where W1,W12,W2 R22 are defined as follows:

W1 =

[c1kxm (1 ) c1kv2m (1 + ) c1kv2m (1 + ) kv(1 ) c1

], (81)

W12 =

[c1m (B + x) 0

B + x + kxexmax 0

], (82)

W2 =

[c2kRM (J)

c2k2m(J) c2k2m(J) k c2

]. (83)

d) Boundedness: Under the given conditions (35), (36),all of the matrices M11, M12, M21, M 22, W1, and W2are positive-definite. Therefore, the Lyapunov function V ispositive-definite and decrescent to obtain

min{m(M11),m(M21)}z2 V max{M (M12), M (M 22)}z2, (84)

where z = [z1, z2]T R2, and the time-derivative ofV is bounded byV m(W1)z12 + W122z1z2 m(W2)z22

+ x + R

= zTWz + x + R m(W )z2 + x + R. (85)

where the matrix W R22 is given by

W =

[m(W1) 12W122 12W122 m(W2)

]. (86)

Similar to the proof of Proposition 2, we can show thatthe tracking errors are uniformly ultimately bounded if theconstants x, R are sufficiently small, as given in (38), andthe corresponding ultimate bound is given by (39).

D. Proof of Proposition 4

The given assumptions satisfy the assumption of Propo-sition 2, from which the tracking error z2 = [eR, e]Tis guaranteed to exponentially decrease until it satisfies thebound given by (19). But, (42) guarantees that the attitudeerror enters the region defined by (33) in a finite time t.

Therefore, if we show that the tracking error z1 =[ex, ev]T is bounded in t [0, t] as well, then the com-plete tracking error (z1, z2) is uniformly ultimately bounded.

The boundedness of z1 is shown as follows. The errordynamics or ev can be written as

mev = mge3 fRe3 mxd + x.Let V3 be a positive-definite function of ex and ev:

V3 = 12ex2 + 1

2mev2.

Then, we have ex

2V3, ev

2mV3. The time-

derivative of V3 is given byV3 = ex ev + ev (mge3 fRe3 mxd + x) exev+ ev(B + x) + evRe3|f |.

From (27), we obtain

V3 exev+ ev(B + x)+ ev(kxex+ kvev+B + x)

= kvev2 + (2(B + x) + (kx + 1)ex)ev d1V3 + d2

V3,

where d1 = kv 2m + 2(kx + 1)1m

, d2 = 2(B + x)

2m .

Suppose that V3 1 for a time interval [ta, tb] [0, t]. Inthis time interval, we have

V3 V3. Therefore,V3 (d1 + d2)V3 V3(t) V3(ta)e(d1+d2)(tta).

Therefore, for any time interval in which V3 1, V3is bounded. This implies that V3, and therefore z1 =[ex, ev]T , are bounded for 0 t t.

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I INTRODUCTIONII QUADROTOR DYNAMICS MODELIII ATTITUDE CONTROLLED FLIGHT MODEIII-A Flight ModesIII-B Attitude Tracking ErrorsIII-C Attitude Tracking Controller

IV POSITION CONTROLLED FLIGHT MODEIV-A Position Tracking ErrorsIV-B Position Tracking ControllerIV-C Direction of the First Body-Fixed Axis

V NUMERICAL EXAMPLESAppendixA Properties of the Hat MapB Proof of Proposition ??C Proof of Proposition ??D Proof of Proposition ??

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