Geometric Tracking Control of a QuadrotorUAV on SE(3) for Extreme

Maneuverability ?

Taeyoung Lee ∗ Melvin Leok ∗∗ N. Harris McClamroch ∗∗∗

∗Mechanical and Aerospace Engineering, Florida Institute ofTechnology, Melbourne, FL 39201 (e-mail: taeyoung@fit.edu)

∗∗Mathematics, University of California at San Diego, La Jolla, CA92093 (e-mail: mleok@math.ucsd.edu)

∗∗∗Aerospace Engineering, University of Michigan, Ann Arbor, MI48109 (e-mail: nhm@umich.edu)

Abstract: This paper provides new results for the tracking control of a quadrotor unmannedaerial vehicle (UAV). The dynamics of the quadrotor UAV are expressed globally on theconfiguration manifold of the special Euclidean group, and we construct geometric controllers totrack outputs that correspond to each of three flight modes, namely (1) almost global asymptotictracking of the attitude of the quadrotor UAV, (2) almost global asymptotic tracking of theposition of the center of mass of the quadrotor UAV, and (3) almost global asymptotic trackingof the velocity of the center of mass of the quadrotor UAV. Since the control approach iscoordinate-free, it completely avoids singularities and complexities that arise when using localcoordinates. Based on a hybrid control architecture, we show that the proposed control systemcan generate complex acrobatic maneuvers of a quadrotor UAV.

1. INTRODUCTION

A quadrotor unmanned aerial vehicle (UAV) consists oftwo pairs of counter-rotating rotors and propellers, locatedat the vertices of a square frame. Due to its simplemechanical structure, it has been envisaged for variousapplications (see Valenti et al. [2006], Pounds et al. [2010],Hoffmann et al. [2007]).

Despite the substantial interest in quadrotor UAVs, littleattention has been paid to constructing nonlinear controlsystems. Linear control systems are widely used to enhancethe stability properties of an equilibrium (see, for example,Valenti et al. [2006], Hoffmann et al. [2007], Castillo et al.[2005], Bouabdalla et al. [2005], Nice [2004]). A nonlinearcontroller is developed for the linearized dynamics of aquadrotor UAV by Guenard et al. [2005]. Backsteppingand sliding mode techniques are applied by Bouabdallaand Siegward [2005]. Since all of these controllers are basedon Euler angles, they exhibit singularities when represent-ing complex rotational maneuvers of a quadrotor UAV,thereby significantly restricting their ability to achievecomplex flight maneuvers.

Geometric control, as utilized in this paper, is concernedwith the development of control systems for dynamic sys-tems evolving on nonlinear manifolds that cannot be glob-ally identified with Euclidean spaces (see Bloch [2003],Bullo and Lewis [2005]). By characterizing geometric prop-erties of nonlinear manifolds intrinsically, geometric con-trol techniques provide unique insights to control theory

? This work was supported in part by NSF under grants CMMI-1029551, DMS- 0726263, DMS-1001521, DMS-1010687, and CMMI-1029445.

that cannot be obtained from dynamic models representedusing local coordinates. This approach has been applied tofully actuated rigid body dynamics on Lie groups by Bulloand Lewis [2005], Cabecinhas et al. [2008], Chaturvediet al. [2009].

In this paper, we make use of geometric methods to defineand analyze controllers that can achieve complex aerobaticmaneuvers for a quadrotor UAV. The dynamics of thequadrotor UAV are expressed globally on the configurationmanifold of the special Euclidean group SE(3). Based on ahybrid control architecture, we construct controllers thatcan achieve output tracking for outputs that correspondto each of several flight modes. In particular, we introducethree flight modes, each defined by a nonlinear controllerthat achieves: (1) almost global asymptotic tracking of theattitude of the quadrotor UAV, (2) almost global asymp-totic tracking of the position of the center of mass, and (3)almost global asymptotic tracking of the velocity of thecenter of mass. Since the control approach is coordinate-free, it completely avoids singularities and complexitiesthat arise when using local coordinates. This paper is anextension of the prior work of Lee et al. [2010b], introduc-ing a hybrid control structure, a velocity tracking mode,convergence properties of the first body fixed axis, anda new example. Due to page limit, all of the proofs arerelegated to Lee et al. [2010a].

2. QUADROTOR DYNAMICS MODEL

Consider a quadrotor UAV model illustrated in Fig. 1.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 choose

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 18th IFAC World Congress. Received October 18, 2010.

e1

e2e3

b1

b2

b3

f1

f2

f3

f4

x R

Fig. 1. Quadrotor model

an 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 andthe second axes of the body-fixed frame, b1, b2, lie in theplane defined by the centers of the four rotors. The thirdbody-fixed axis b3 is normal to this plane. Define

m ∈ R the total massJ ∈ R3×3 the inertia matrix with respect to the

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

frame to the inertial frameΩ ∈ 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, b2

planefi ∈ R the thrust generated by the i-th pro-

peller along the −b3 axisτi ∈ R the torque generated by the i-th pro-

peller about the b3 axis

f ∈ R the total thrust, 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 bythe location of the center of mass and the attitude withrespect to the inertial frame. Therefore, the configurationmanifold is the special Euclidean group SE(3), which is thesemidirect product of R3 and the special orthogonal groupSO(3) = R ∈ R3×3 |RTR = I, detR = 1.The following conventions are assumed for the rotors andpropellers and the thrust and moment that they exert onthe quadrotor UAV. We assume that the thrust of eachpropeller is directly controlled, i.e., we do not consider thedynamics of rotors and propellers, and the direction ofthe thrust of each propeller is normal to the quadrotorplane. The first and third propellers are assumed togenerate a thrust along the direction −b3 when rotatingclockwise; the second and fourth propellers are assumed togenerate a thrust along the direction −b3 when rotatingcounterclockwise. Thus the total thrust magnitude is f =∑4i=1 fi along the direction of −b3. According to the

definition of the rotation matrix R ∈ SO(3), the totalthrust vector is given by −fRe3 ∈ R3 in the inertialframe. We also assume that the torque generated by eachpropeller is directly proportional to its thrust. Since itis assumed that the first and the third propellers rotate

clockwise and the second and the fourth propellers rotatecounterclockwise to generate a positive thrust along thedirection −b3, the torque generated by the i-th propelleris given by τi = (−1)icτffi for a fixed constant cτf .

Under these assumptions, the total thrust, and the totalmoment vector in the body-fixed frame can be written as f

M1

M2

M3

=

1 1 1 10 −d 0 dd 0 −d 0−cτf cτf −cτf cτf

f1

f2

f3

f4

. (1)

The determinant of the above 4× 4 matrix is 8cτfd2, so it

is invertible when d 6= 0 and cτf 6= 0. Therefore, for giventotal thrust and moment vector f,M , the thrust of eachpropeller f1, f2, f3, f4 can be obtained from (1). Using thisequation, the total thrust f ∈ R and the moment vectorM ∈ 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, (3)

R = RΩ, (4)

JΩ + Ω× JΩ = M, (5)

where hat map · : R3 → so(3) is defined by the conditionthat xy = x× y for all x, y ∈ R3 (see Appendix A).

3. GEOMETRIC TRACKING CONTROL OF AQUADROTOR UAV

Since the quadrotor UAV has four inputs, it is possibleto achieve asymptotic output tracking for at most fourquadrotor UAV outputs. This motivates us to introduceseveral flight modes. Each flight mode is associated with aspecified set of outputs for which exact tracking of thoseoutputs define that flight mode.

• Attitude controlled flight mode: the outputs are theattitude of the quadrotor UAV and the controllerfor this flight mode achieves asymptotic attitudetracking.

• Position controlled flight mode: the outputs are theposition vector of the center of mass of the quadrotorUAV and the controller for this flight mode achievesasymptotic position tracking.

• Velocity controlled flight mode: the outputs are thevelocity vector of the center of mass of the quadrotorUAV and the controller for this flight mode achievesasymptotic velocity tracking.

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 time func-tions. For example, one might define a complex aerobaticflight maneuver for the quadrotor UAV that consists ofa hovering flight segment with specified constant positionvector, a reorientation segment with specified vehicle atti-tude, and a surveillance flight segment with specified time-varying position vector. These types of complex aerobaticmaneuvers, involving large angle transitions between flightmodes, have not been much studied in the literature. Sucha hybrid flight control architecture, for longitudinal flight

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 18th IFAC World Congress. Received October 18, 2010.

maneuvers only, has been proposed by Oishi and Tomlin[1999], Ghosh and Tomlin [2000], Oishi and Tomlin [2002].

4. ATTITUDE CONTROLLED FLIGHT MODE

We now introduce a nonlinear controller for the atti-tude controlled flight mode. We show that this controllerachieves almost globally asymptotic attitude tracking, thatis the output attitude of the quadrotor UAV asymptoti-cally tracks the commanded attitude.

An arbitrary smooth attitude tracking command t →Rd(t) ∈ SO(3) is given as a function of time. The cor-responding angular velocity command is obtained by theattitude kinematics equation as Ωd = RTd Rd. We firstdefine errors associated with the attitude dynamics ofthe quadrotor UAV. The attitude and angular velocitytracking error should be carefully chosen as they evolve onthe tangent bundle of SO(3). First, define the real errorfunction on SO(3)× SO(3):

Ψ(R,Rd) =1

2tr[I −RTdR

]. (6)

This function is locally positive-definite about R = Rdwithin the region where the rotation angle between R andRd is less than 180 (see Bullo and Lewis [2005]). For agiven Rd, this set can be represented by the sublevel setL2 = R ∈ SO(3) |Ψ(R,Rd) < 2, which almost coversSO(3).

The variation of a rotation matrix can be expressed asδR = Rη for η ∈ R3, so that the derivative of the errorfunction is given by

DRΨ(R,Rd) ·Rη = −1

2tr[RTdRη

]=

1

2(RTdR−RTRd)∨ · η, (7)

where the vee map ∨ : so(3) → R3 is the inverse of thehat map. We used a property of the hat map given byequation (A.3) in Appendix A. From this, the attitudetracking error eR is chosen to be

eR =1

2(RTdR−RTRd)∨. (8)

The tangent vectors R ∈ TRSO(3) and Rd ∈ TRdSO(3)

cannot be directly compared since they lie in differenttangent spaces. We transform Rd into a vector in TRSO(3),

and we compare it with R as follows:

R− Rd(RTdR) = R(Ω−RTRdΩd)∧,

where Ωd = RTd Rd and we use equation (A.5) in AppendixA. This motivates our choice of the tracking error for theangular velocity eΩ as follows:

eΩ = Ω−RTRdΩd. (9)

We show that eΩ is the angular velocity of the relativerotation matrix RTdR, represented in the body-fixed frame,since:

d

dt(RTdR) = RTdR(Ω−RTRdΩd)∧ = (RTdR) eΩ. (10)

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

M = −kReR − kΩeΩ + Ω× JΩ

− J(ΩRTRdΩd −RTRdΩd), (11)

where kR, kΩ are positive constants and Rd(t) ∈ SO(3) isthe specified attitude command for this attitude controlledflight mode. The control moment vector depends of feed-back of the attitude error and angular velocity error andit depends on the commanded attitude, angular velocityand angular acceleration.

In this attitude controlled mode, it is possible to ignore thetranslational motion of the quadrotor UAV; consequentlythe reduced model for the attitude dynamics are given byequations (4), (5), using the controller expression (11).

We now show that the reduced closed loop dynamics havethe property that (eR, eΩ) = (0, 0) is an equilibrium thatis exponentially stable.

Proposition 1. (Exponential Stability of Attitude Con-trolled Flight Mode) Consider the control moment Mdefined in (11) for any positive constants kR, kΩ. Supposethat the initial conditions satisfy

Ψ(R(0), Rd(0)) < 2, (12)

‖eΩ(0)‖2 < 2

λmax(J)kR(2−Ψ(R(0), Rd(0))), (13)

where λmax(J) denotes the maximum eigenvalue of theinertia matrix J . Then, the zero equilibrium of the closedloop tracking error (eR, eΩ) = (0, 0) is exponentiallystable. Furthermore, there exist constants α2, β2 > 0 suchthat

Ψ(R(t), Rd(t)) ≤ min

2, α2e−β2t

. (14)

In this proposition, equations (12), (13) describe a re-gion of attraction for the reduced closed loop dynam-ics. An estimate of the domain of attraction is obtainedfor which the quadrotor attitude lies in the sublevel setL2 = R ∈ SO(3) |Ψ(R,Rd) < 2. This requires that theinitial attitude error should be less than 180, in termsof the rotation angle about the eigenaxis between R andRd. Therefore, in Proposition 1, exponential stability isguaranteed for almost all initial attitude errors. The atti-tude error function defined in (6) has the following criticalpoints or equilibrium solutions: the identity matrix, androtation matrices that can be written as exp(πv) for anyv ∈ S2. These non-identity critical points of the attitudeerror function lie outside of the region of attraction. Sincethese critical points are a two-dimensional manifold inthe three-dimensional SO(3), we claim that the presentedcontroller exhibits almost global properties in SO(3). It isimpossible to construct a smooth controller on SO(3) thathas global asymptotic stability. The region of attractionfor the angular velocity can be increased by choosing alarge controller gain kR in (13).

Asymptotic tracking of the quadrotor attitude does notrequire specification of the total thrust force. As an auxil-iary problem, the total thrust force can be chosen in manydifferent ways to achieve a 1-D translational motion ob-jective. As one illustration of a specific selection approach,we assume that the objective is to asymptotically track aquadrotor altitude command. It is straightfoward to obtainthe following corollary of Proposition 1.

Proposition 2. (Exponential Stability of Attitude Con-trolled Flight Mode with Altitude Tracking) Consider the

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 18th IFAC World Congress. Received October 18, 2010.

control moment M defined in (11) satisfying the assump-tions of Proposition 1. In addition, the total thrust forceis given by

f =kx(x3 − x3d

) + kv(x3 − x3d) +mg −mx3d

e3 ·Re3, (15)

where kx, kv are positive constants, x3d(t) is the quadrotoraltitude command, and we assume that

e3 ·Re3 6= 0. (16)

The conclusions of Proposition 1 hold and in addition thequadrotor altitude x3(t) asymptotically tracks the altitudecommand x3d

(t).

Since the translational motion of the quadrotor UAVcan be only partially controlled; this flight mode is mostsuitable only for short time periods where an attitudemaneuver is to be completed. The translational equationsof motion of the quadrotor UAV, during an attitudeflight mode, are given by equations (2), (3), and whatevertotal thrust controller, e.g. equation (15), is selected.These equations can be analyzed to determine the fulltranslational motion of the quadrotor UAV during anattitude controlled flight mode.

5. POSITION CONTROLLED FLIGHT MODE

We now introduce a nonlinear controller for the posi-tion controlled flight mode. We show that this controllerachieves almost global asymptotic position tracking, thatis the output position vector of the quadrotor UAV asymp-totically tracks the commanded position. This flight moderequires analysis of the coupled translational and rota-tional equations of motion; hence, we make use of thenotation and analysis in the prior section to describe theproperties of the closed loop system in this flight mode.

An arbitrary position tracking command t → xd(t) ∈ R3

is given. The position tracking errors for the position andthe velocity are given by:

ex = x− xd, (17)

ev = v − xd. (18)

The nonlinear controller for the position controlled flightmode, described by control expressions for the total thrustmagnitude and the total moment vector, are:

f = (kxex + kvev +mge3 −mxd) ·Re3, (19)

M = −kReR − kΩeΩ + Ω× JΩ

− J(ΩRTRcΩc −RTRcΩc), (20)

where kx, kv, kR, kΩ are positive constants. Following theprior definition of the attitude error and the angularvelocity error

eR =1

2(RTc R−RTRc)∨, eΩ = Ω−RTRcΩc, (21)

where Rc(t) ∈ SO(3) and Ωc ∈ R3 are constructed as:

Rc = [b1c ; b3c × b1c ; b3c ], Ωc = RTc Rc, (22)

where b3c∈ S2 is defined by

b3c = − −kxex − kvev −mge3 +mxd‖−kxex − kvev −mge3 +mxd‖

. (23)

and b1c ∈ S2 is selected to be orthogonal to b3c, therebyguaranteeing that Rc ∈ SO(3). We assume that

‖−kxex − kvev −mge3 +mxd‖ 6= 0, (24)

Forcecontroller

Momentcontroller

--

-

- -Quadrotor

Dynamics-

f

M

b3cxd

(b1d )

x, v,R,Ω

6 -q qController

Fig. 2. Controller structure for position controlled flightmode

and

‖ −mge3 +mxd‖ < B (25)

for a given positive constant B.

The control thrust magnitude and the control momentvector depend on feedback of the position error and trans-lational velocity error and they depend on the commandedposition, translational velocity and translational accelera-tion. The control moment vector has a form that is similarto that for the attitude controlled flight mode. However,the attitude error and angular velocity error are definedwith respect to a computed attitude, angular velocity andangular acceleration, that are constructed according to theindicated procedure. This construction has the propertythat the direction of the thrust vector, namely −Re3, issuch that the thrust magnitude (19) achieves the desiredposition tracking objectives.

In short, this control system is designed to achieve asymp-totic tracking of the complete dynamics. The closed loopsystem for this position controlled flight mode is illustratedin Fig. 2. The corresponding closed loop control systemis described by equations (2), (3), (4), (5), using thecontroller expressions (19) and (20).

We now show that the closed loop dynamics have theproperty that (ex, ev, eR, eΩ) = (0, 0, 0, 0) is an equilibriumthat is exponentially stable.

Proposition 3. (Exponential Stability of Position Con-trolled Flight Mode) Consider the thrust magnitude fand moment vector M defined in expressions (19), (20).Suppose that the initial conditions satisfy

Ψ(R(0), Rc(0)) < 1. (26)

Define W1,W12,W2 ∈ R2×2 to be

W1 =

c1kxm

−c1kv2m

(1 + α)

−c1kv2m

(1 + α) kv(1− α)− c1

, (27)

W12 =

[kxevmax +

c1mB 0

B 0

], (28)

W2 =

c2kR

λmax(J)− c2kΩ

2λmin(J)

− c2kΩ

2λmin(J)kΩ − c2

, (29)

where ψ1 < Ψ(R(0), Rc(0)) < 1, α =√ψ1(2− ψ1),

evmax = max‖ev(0)‖, Bkv(1−α). For positive constants

kx, kv, we choose positive constants c1, c2, kR, kΩ such that

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 18th IFAC World Congress. Received October 18, 2010.

c1 < min

kv(1− α),

4mkxkv(1− α)

k2v(1 + α)2 + 4mkx

,√kxm

,

(30)

c2 < min

kΩ,

4kΩkRλmin(J)2

k2Ωλmax(J) + 4kRλmin(J)2

,√kRλmin(J)

,

(31)

λmin(W2) >4‖W12‖2

λmin(W1). (32)

Then, the zero equilibrium of the closed loop trackingerrors (ex, ev, eR, eΩ) = (0, 0, 0, 0) is exponentially stable.A region of attraction is characterized by (26) and

‖eΩ(0)‖2 < 2

λmax(J)kR(ψ1 −Ψ(R(0), Rc(0))). (33)

Note that the attitude error defined above is based onthe computed attitude Rc ∈ SO(3), which depends onfeedback according to the above specifications. Note thatthe construction of Rc is not completely determined butinvolves an orthogonality selection process; this construc-tion freedom arises as a consequence of the fact that Rcis constructed to define the direction of the thrust vectorwhich is invariant for rotations about that direction.

Proposition 3 requires that the initial attitude error isless than 90 to achieve exponential stability for thisflight mode. Suppose that this is not satisfied, i.e. 1 ≤Ψ(R(0), Rc(0)) < 2. We can apply the conclusions ofProposition 1 to obtain that the attitude error functionΨ exponentially decreases, and therefore, it enters theregion of attraction of Proposition 3 in a finite time.Therefore, by combining the results of Proposition 1 and3, we can show almost global exponential attractivenesswhen Ψ(R(0), Rc(0)) < 2.

Definition 1. (Exponential Attractiveness Qu [1998]) Anequilibrium point z = 0 of a dynamic systems is ex-ponentially attractive if, for some δ > 0, there exists aconstant α(δ) > 0 and β > 0 such that ‖z(0)‖ < δ implies‖z(t)‖ ≤ α(δ)e−βt for all t > 0.

This should be distinguished from the stronger notion ofexponential stability, in which the constant α(δ) in theabove bound is replaced by α(δ) ‖z(0)‖.Proposition 4. (Almost Global Exponential Attractive-ness of the Position Controlled Flight Mode) Consider thethrust force f and moment vector M defined in expressions(19), (20). Suppose that the initial conditions satisfy

1 ≤ Ψ(R(0), Rc(0)) < 2, (34)

‖eΩ(0)‖2 < 2

λmax(J)kR(2−Ψ(R(0), Rc(0))). (35)

Then, the zero equilibrium of the closed loop trackingerrors (ex, ev, eR, eΩ) = (0, 0, 0, 0) is exponentially attrac-tive.

In Proposition 4, exponential attractiveness is guaranteedfor almost all initial attitude errors. Since the criticalpoints of the attitude error function are a two-dimensionalmanifold in the three-dimensional SO(3), as discussed inSection 4, we claim that the presented controller exhibitsalmost global properties in SO(3).

As described above, the construction of the orthogonalmatrix Rc involves definition of its third column b3c by

normalization of a feedback function and selection of itsfirst column b1c

to be orthogonal to this third column. Thetwo-dimensional unit vector b1c can be arbitrarily chosenin the plane normal to b3c . This reflects the fact that therestill remains one degree of freedom in control inputs, sincefour control inputs of the quadrotor UAV is designed tofollow a three-dimensional position command.

The rotation matrix Rc represents the attitude of thequadrotor UAV required to follow a given position com-mand, and according to the structures of the presentedposition controlled flight mode, the attitude of the quadro-tor UAV asymptotically converges to Rc, i.e. R → Rc ast→∞. Therefore, by choosing b1c

properly, we can controlthe direction of the first body fixed axis. However, wecannot control the two-dimensional direction of the firstbody fixed axis completely arbitrarily, since there existsonly one additional degree of freedom in control inputs.More explicitly, there is a constraint that b1c

should lie inthe plane normal to b3c to guarantee Rc ∈ SO(3). Here,the additional one dimensional control input is designedto control the projection of the first body fixed axis on tothe plane normal to b3c .

There are several ways to specify the desired direction ofthe first body fixed axis projected on to the plane normalto b3c

. Here, we choose a desired direction b1d∈ S2, that is

not parallel to b3c, and b1c

is selected to be the projectionof b1d

onto the plane normal to b3c, i.e. b1c

= Proj[b1d],

where Proj[·] denotes the normalized projection onto theplane perpendicular to b3c

. In this case, the first body fixedaxis does not converge to b1d

exactly, but it converges tothe projection of b1d

, i.e. b1 → b1c= Proj[b1d

] as t → ∞.In other words, the first body fixed axis converges to theintersection of two planes, namely the plane normal tob3c

and the plane spanned by b3cand b1d

(see Fig. 3).From (23), we observe that b3c

asymptotically convergesto the direction of ge3 − xd. In short, the additional onedimensional input is used to guarantee that the first bodyfixed axis asymptotically lies in the plane spanned by b1d

and ge3 − xd.Suppose that xd = 0, then the third body fixed axisconverges to the gravity direction e3. In this case, wecan choose b1d

arbitrarily in the horizontal plane, whichfollows that b1c

= Proj[b1d] = b1d

as t → ∞. Therefore,the first body fixed axis b1 asymptotically converges tob1d

, which can be used to specify the heading directionof the quadrotor UAV in the horizontal plane. These aresummarized as follows.

Proposition 5. (Almost Global Exponential Attractive-ness of Position Controlled Flight Mode with AsymptoticDirection of First Body-Fixed Axis) Consider the momentvector M defined in (20) and the total thrust f defined in(19) satisfying the assumptions of Propositions 3 and 4.

In addition, the first column of Rc, namely b1cis con-

structed as follows. We choose t → b1d(t) ∈ S2, and we

assume that it is not parallel to b3c. The unit vector b1c

is constructed by projecting b1donto the plane normal to

b3c:

b1c= Proj[b1d

] = − 1

‖b3c× b1d

‖(b3c× (b3c

× b1d)). (36)

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 18th IFAC World Congress. Received October 18, 2010.

b3c

b1d

b2c = b3c × b1c

b1c = Proj[b1d ]Planenormal to b3c

Plane spanned by

b1d and b3c

Fig. 3. Convergence property of the first body fixed axis:b3c

is determined by (23). Then, there still remainsone additional input degree of freedom, that corre-sponds to the direction of b1c

in the plane normal tob3c

. We choose an arbitrary b1dthat is not parallel to

b3c, and project it on to the plane normal to b3c

toobtain b1c

. This guarantees that the first body fixedaxis asymptotically lies in the plane spanned by b1d

and b3c, which converges to the direction of ge3 − xd

as t→∞.

Then, the conclusions of Propositions 3 and 4 hold, andthe first body fixed axis asymptotically lies in the planecomposed of b1d

and ge3 − xd.In a special case when xd = 0. We choose b1d

on the hori-zontal plane. Then, the first body fixed axis asymptoticallyconverges to b1d

.

These additional properties of the closed loop can beinterpreted as characterizing the asymptotic direction ofthe first body-fixed axis and the asymptotic direction ofthe third body-fixed axis as it depends on the commandedvehicle acceleration. These physical properties may be ofimportance in some flight maneuvers.

The control input of the position controlled flight mode de-pends on Ωc and Ωc as shown at (20). The expressions for

Ωc and Ωc corresponding to Proposition 5 are summarizedat Appendix.

6. VELOCITY CONTROLLED FLIGHT MODE

We now introduce a nonlinear controller for the veloc-ity controlled flight mode. We show that this controllerachieves almost global asymptotic velocity tracking, thatis the output velocity vector of the quadrotor UAV asymp-totically tracks the commanded velocity.

An arbitrary velocity tracking command t→ vd(t) ∈ R3 isgiven. The velocity tracking error given by:

ev = v − vd. (37)

The nonlinear controller for the velocity controlled flightmode, described by control expressions for the total thrustmagnitude and the total moment vector, are:

f = (kvev +mge3 −mvd) ·Re3, (38)

M = −kReR − kΩeΩ + Ω× JΩ

− J(ΩRTRcΩc −RTRcΩc), (39)

where kv, kR, kΩ are positive constants, and following theprior definition of the attitude error and the angularvelocity error

eR =1

2(RTc R−RTRc)∨, eΩ = Ω−RTRcΩc, (40)

and Rc(t) ∈ SO(3) and Ωc ∈ R3 are constructed as:

Rc = [b1c; b3c

× b1c; b3c

], Ωc = RTc Rc, (41)

where b3c∈ S2 is defined by

b3c= − −kvev −mge3 +mvd‖−kvev −mge3 +mvd‖

. (42)

and b1c ∈ S2 is selected to be orthogonal to b3c, therebyguaranteeing that Rc ∈ SO(3). We assume that

‖−kvev −mge3 +mxd‖ 6= 0, (43)

and

‖ −mge3 +mxd‖ < B (44)

for a given positive constant B.

The overall controller structure is similar to the positioncontrolled flight mode. The control thrust magnitude andthe control moment vector depend on feedback of thetranslational velocity error and they depend on the com-manded translational velocity and translational accelera-tion. The control moment vector has a form that is similarto that for the attitude controlled flight mode. However,the attitude error and angular velocity error are definedwith respect to a computed attitude, angular velocity andangular acceleration, that are constructed according to theindicated procedure. This construction has the propertythat the direction of the thrust vector, namely −Re3, issuch that the thrust magnitude (19) achieves the desiredvelocity tracking objectives. This is again verified in theproof.

We now show that the closed loop dynamics have theproperty that (ev, eR, eΩ) = (0, 0, 0) is an equilibrium thatis exponentially stable.

Proposition 6. (Exponential Stability of Velocity Con-trolled Flight Mode) Consider the thrust magnitude fand moment vector M defined in expressions (38), (39).Suppose that the initial conditions satisfy

Ψ(R(0), Rc(0)) < 1. (45)

Define W2 ∈ R2×2 to be

W2 =

c2kR

λmax(J)− c2kΩ

2λmin(J)

− c2kΩ

2λmin(J)kΩ − c2

, (46)

For positive constants kv, we choose positive constantsc2, kR, kΩ such that

c2 < min

kΩ,

4kΩkRλmin(J)2

k2Ωλmax(J) + 4kRλmin(J)2

,√kRλmin(J)

,

(47)

λmin(W2) >4B2

kv(1− α), (48)

where ψ1 < Ψ(R(0), Rc(0)) < 1, α =√ψ1(2− ψ1). Then,

the zero equilibrium of the closed loop tracking errors(ev, eR, eΩ) = (0, 0, 0) is exponentially stable. A region ofattraction is characterized by (45) and

‖eΩ(0)‖2 < 2

λmax(J)kR(ψ1 −Ψ(R(0), Rc(0))). (49)

Proposition 6 requires that the initial attitude error is lessthan 90 to achieve exponential stability for this flightmode. Similar to Proposition 4, we can show almost globalexponential attractiveness when Ψ(R(0), Rc(0)) < 2.

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 18th IFAC World Congress. Received October 18, 2010.

Proposition 7. (Almost Global Exponential Attractive-ness of Velocity Controlled Flight Mode) Consider thethrust force f and moment vector M defined in expressions(38), (39). Suppose that the initial conditions satisfy

1 ≤ Ψ(R(0), Rc(0)) < 2, (50)

‖eΩ(0)‖2 < 2

λmax(J)kR(2−Ψ(R(0), Rc(0))). (51)

Then, the zero equilibrium of the closed loop trackingerrors (ev, eR, eΩ) = (0, 0, 0) is exponentially attractive.

As described in Section 5, there is one-dimensional freedomin constructing Rc ∈ SO(3), namely the direction of thequadrotor UAV in the plane normal to b3c

. This can bechosen as discussed in Proposition 5.

Proposition 8. (Almost Global Exponential Attractive-ness of Velocity Controlled Flight Mode with AsymptoticDirection of First Body-Fixed Axis) Consider the momentvector M defined in (39) and the total thrust f defined in(38) satisfying the assumptions of Propositions 6 and 7.

In addition, the first column of Rc, namely b1c is con-structed as follows. We choose t → b1d

(t) ∈ S2, and weassume that it is not parallel to b3c . The unit vector b1c

is constructed by projecting b1donto the plane normal to

b3c :

b1c = − 1

‖b3c× b1d

‖(b3c × (b3c × b1d

)). (52)

Then, the conclusions of Propositions 6 and 7 hold, andthe first body fixed axis asymptotically lies in the planecomposed of b1d

and ge3 − vd.In a special case when vd = 0. We choose b1d

on the hori-zontal plane. Then, the first body fixed axis asymptoticallyconverges to b1d

.

7. NUMERICAL EXAMPLE

Numerical results are presented to illustrate the priorapproach to determining complex flight maneuvers for atypical quadrotor UAV. The parameters of the quadrotorUAV are chosen according to a quadrotor UAV developedin Pounds et al. [2010].

J = [0.0820, 0.0845, 0.1377] kg −m2, m = 4.34 kg

d = 0.315 m, cτf = 8.004× 10−3 m.

The controller parameters are chosen as follows:

kx = 16m, kv = 5.6m, kR = 8.81, kΩ = 2.54.

We consider a complex flight maneuver involving transi-tions between five flight modes (see Fig. 4). The trackingcommand for each flight mode is given by

(a) Velocity controlled flight mode (t ∈ [0, 4))

vd(t) = [1 + 0.5t, 0.2 sin(2πt), −0.1], b1d(t) = [1, 0, 0].

(b) Attitude controlled flight mode (t ∈ [4, 6)): rotationabout e2 by 720

Rd(t) = exp(2π(t− 4)e2).

(c) Position controlled flight mode (t ∈ [6, 8))

xd(t) = [14− t, 0, 0], b1d(t) = [1, 0, 0].

(d) Attitude controlled flight mode (t ∈ [8, 9)): rotationabout e1 by 360

Rd(t) = exp(2π(t− 8)e1).

(e) Position controlled flight mode (t ∈ [9, 12])

xd(t) = [20− 5

3t, 0, 0], b1d

(t) = [0, 1, 0].

x(0) = [0, 0, 0], v(0) = [0, 0, 0],

R(0) = I, Ω(0) = [0, 0, 0].

Simulation results for these three flight maneuvers areillustrated at Figure 5. It begins with a velocity controlledflight mode. As the initial attitude error function is lessthan 1, according to Proposition 6, the velocity trackingerror exponentially converges as shown at Fig. 5(d). FromProposition 8, the first body fixed axis asymptotically liesin the plane spanned by b1d

and ge3 − vd. Since ‖vd‖ < g,it stays close to the plane composed of e1 and e3, asillustrated at Fig. 5(e).

This is followed by an attitude tracking mode to rotate thequadrotor by 720 about e2. During this time period, theattitude error converges according to Proposition 1, butthe quadrotor altitude decreases by 1 meter. But, a posi-tion tracking mode is again engaged, and the quadrotorUAV soon follows a straight line again. Another attitudetracking mode and a position tracking mode are repeatedto rotate the quadrotor by 360 about the direction of thevelocity vector. For the position tracking modes (c) and(e), we have xd = 0, and b1d

lies in the horizontal plane.Therefore, according to Proposition 5, the first body fixedb1 asymptotically converges to b1d

, as shown at Fig. 5(e).For example, at the last position tracking mode (e), thefirst body fixed axis points the left side of the flight pathsince b1d

is specified to be e2. These illustrate that byswitching between an attitude mode and a position andheading flight mode, the quadrotor UAV can perform theindicated complex acrobatic maneuver.

Appendix A. PROPERTIES OF THE HAT MAP

The hat map · : R3 → so(3) is defined such that xy = x×yfor any x, y ∈ R3. This identifies the Lie algebra so(3) withR3 using the vector cross product in R3. The inverse ofthe hat map is referred to as the vee map, ∨ : so(3)→ R3.Several properties are summarized as follows.

xy = x× y = −y × x = −yx, (A.1)

−1

2tr[xy] = xT y, (A.2)

= tr[Ax] =1

2tr[x(A−AT )

]= −xT (A−AT )∨, (A.3)

xA+AT x = (tr[A] I3×3 −Ax)∧, (A.4)

RxRT = (Rx)∧, (A.5)

for any x, y ∈ R3, A ∈ R3×3, and R ∈ SO(3).

REFERENCES

A. M. Bloch. Nonholonomic Mechanics and Control,volume 24 of Interdisciplinary Applied Mathematics.Springer-Verlag, 2003.

S. Bouabdalla and R. Siegward. Backstepping and sliding-mode techniques applied to an indoor micro quadrotor.In Proceedings of the IEEE International Conference onRobotics and Automation, pages 2259–2264, 2005.

S. Bouabdalla, P. Murrieri, and R. Siegward. Towardsautonomous indoor micro VTOL. Autonomous Robots,18(2):171–183, 2005.

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 18th IFAC World Congress. Received October 18, 2010.

initial/terminal

position

(a) velocity tracking

(b) attitude tracking

(c) position

tracking

(d) attitude tracking(e) position

trackinge1

e2e3

b1

b1

b1

Fig. 4. Complex maneuver of a quadrotor UAV involving a rotation by 720 about e2 (b), and a rotation by 360

about e1 (d), with transitions between several flight modes (an animation illustrating this maneuver is availableat http://my.fit.edu/˜taeyoung).

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

t

Ψ

(a) Attitude error function Ψ

0 2 4 6 8 10 120

5

10

0 2 4 6 8 10 12−0.5

0

0.5

x,x

d

0 2 4 6 8 10 12−1

0

1

t

(b) Position (x:solid, xd:dotted,(m))

0 2 4 6 8 10 12−10

0

10

0 2 4 6 8 10 12−10

0

10

Ω,Ω

d

0 2 4 6 8 10 12−5

0

5

t

(c) Angular velocity (Ω:solid,Ωd:dotted, (rad/sec))

0 2 4 6 8 10 12−5

0

5

0 2 4 6 8 10 12−2

0

2

v,v

d

0 2 4 6 8 10 12−5

0

5

t

(d) Velocity (v:solid, vd:dotted,(m/s))

0 2 4 6 8 10 12−1

0

1

0 2 4 6 8 10 12−1

0

1

b 1,b

1d

0 2 4 6 8 10 12−1

0

1

t

(e) First body fixed axis (b1:solid,b1d :dotted)

Fig. 5. Simulation results

F. Bullo and A. D. Lewis. Geometric control of mechanicalsystems, volume 49 of Texts in Applied Mathematics.Springer-Verlag, New York, 2005. Modeling, analysis,and design for simple mechanical control systems.

D. Cabecinhas, R. Cunha, and C. Silvestre. Output-feedback control for almost global stabilization of fully-acuated rigid bodies. In 3583-3588, editor, Proceedingsof IEEE Conference on Decision and Control, 2008.

P. Castillo, R. Lozano, and A. Dzul. Stabilization of amini rotorcraft with four rotors. IEEE Control System

Magazine, pages 45–55, 2005.N. A. Chaturvedi, N. H. McClamroch, and D. Bernstein.

Asymptotic smooth stabilization of the inverted 3-Dpendulum. IEEE Transactions on Automatic Control,54(6):1204–1215, 2009.

R. Ghosh and C. Tomlin. Nonlinear inverse dynamiccontrol for mode-based flight. In Proceedings of theAIAA Guidance, Navigation and Control Conference,2000.

N. Guenard, T. Hamel, and V. Moreau. Dynamic mod-eling and intuitive control strategy for an X4-flyer. InProceedings of the IEEE International Conference onControl and Application, 2005.

G. Hoffmann, H. Huang, S. Waslander, and C. Tomlin.Quadrotor helicopter flight dynamics and control: The-ory and experiment. In Proceedings of the AIAA Guid-ance, Navigation, and Control Conference, 2007. AIAA2007-6461.

T. Lee, M. Leok, and N. H. McClamroch. Geomet-ric tracking control of a quadrotor UAV on SE(3).arXiv:1003.2005v1, 2010a. URL http://arxiv.org/abs/1003.2005v1.

T. Lee, M. Leok, and N. H. McClamroch. Geometrictracking control of a quadrotor UAV on SE(3). InProceedings of the IEEE Conference on Decision andControl, 2010b. accepted.

E. Nice. Design of a four rotor hovering vehicle. Master’sthesis, Cornell University, 2004.

M. Oishi and C. Tomlin. Switched nonlinear control ofa vstol aircraft. In Proceedings of IEEE Conference onDecision and Control, pages 2685–2690, 1999.

M. Oishi and C. Tomlin. Switching in nonlinear minimumphase systems: Applications to a vstol aircraft. InProceedings of American Control Conference, 2002.

P. Pounds, R. Mahony, and P. Corke. Modeling andcontrol of a large quadrotor robot. Control EngineeringPractice, 18:691–699, 2010.

Zhihua Qu. Robust Control of Nonlinear Uncertain Sys-tems. John Wiley & Sons, Inc., New York, NY, USA,1998. ISBN 0471115894.

M Valenti, B. Bethke, G. Fiore, and J. How. Indoormulti-vehicle flight testbed for fault detection, indoormulti-vehicle flight testbed for fault detection, isolation,and recovery. In Proceedings of the AIAA Guidance,Navigation and Control Conference, 2006.

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 18th IFAC World Congress. Received October 18, 2010.