Nonlinear Control of a Dual-Quadrotor AssemblyAnthony Dzaba and Eugenio Schuster

Abstract—In response to the need for accurate position sens-

ing in unmanned aerial vehicles (UAVs), we propose a concept

aircraft in the form of a mobile global vision system. The dual-

quad prototype resembles a miniature quadrotor nesting within

a larger quadrotor. Inspired by the gaze stabilization abilities

of hunting birds, the inner quadrotor is used as an actively

stabilized sensor deck, while the outer quadrotor is used as a

position-control flight deck. The sensor deck is stabilized to

impose motion constraints on the visual odometry problem.

This reduces the computational expense of implementing a

vision system on a UAV where resources are limited. We

propose a combination of feedback linearization, nonlinear

transformation, and backstepping control. Results for aggressive

flight maneuvers are demonstrated in simulation.

I. INTRODUCTION

Advanced motion control techniques capable of producingaggressive flight maneuvers, generally require non-portable,high-precision pose measurement systems such as the Viconcamera array used to track the agile micro quadrotors featuredin [1]. In order to duplicate these results outside of thelaboratory setting, one would be faced with the difficulttask of integrating capable on-board vision systems on microaerial vehicles (MAVs) where resources are at a premium. Forthis reason it may prove more expedient to avoid giving eachMAV “eyes” and instead give a single global vision system“wings”. To this end we envision a mobile global visionsystem. The concept of a dedicated airborne tracking assetbecomes familiar when we consider the airborne warningand control system (AWACS). To reduce the computationalexpense of implementing vision-based tracking on a UAV-size AWACS, we look to introduce motion constraints.Specifically, fixing the rotation of the vision system (withrespect to the scene under observation) reduces the visualodometry problem to a simpler 3D correlation task.

We borrow from a familiar biological solution in which abird of prey is able to isolate the chaotic motion of its body(flight frame) from the stabilized motion of its head (sensorframe), which continuously tracks a target of interest [2].This type of attitude decoupling would prove difficult shouldour AWACS UAV assume the form of a quadrotor, which byreason of underactuation, must change its attitude in orderto change its position. We address this issue by designing anovel aircraft featuring an additional set of actuators in theform of a second nested quadrotor (increasing the system’sdegrees of freedom from six to nine). The ramifications ofthis design choice are discussed in the Conclusions section.

From the vast body of work dedicated to quadrotor back-stepping, we site the following examples. A classical three-

A. Dzaba (anthony@lehigh.edu) and E. Schuster are with the Departmentof Mechanical Engineering, Lehigh University, Bethlehem, PA 18015 USA.

step backstepping controller is proposed in [3]. A simplifiedbackstepping approach with a nested saturation controller isproposed in [4]. A combination of backstepping and adaptivesliding mode control is proposed in [5]. A combination ofbackstepping and robust control, designed to address modelparameter uncertainty, is proposed in [6]. Lastly, a Lagrange-form backstepping controller with neural network estimationof the attitude dynamics is proposed in [7].

In this paper, we propose a combination of feedback lin-earization, nonlinear transformation, and backstepping con-trol to stabilize the seven outputs, which are the 3D sensordeck attitude, the scalar flight deck yaw heading, and the3D position. We begin by simplifying the dynamics of thesensor deck and flight deck attitude states using feedbacklinearization. Next, we use PD controllers to stabilize thealtitude and flight deck yaw heading. Next, we decouple thehorizontal position states using a nonlinear transformation.And last, we stabilize the horizontal position via the flightdeck’s pitch and roll states using a backstepping controller.

This paper is organized as follows. In Section II, we derivethe equations of motion for the dual-quad using the Euler-Lagrange equation. In Section III, we develop the back-stepping framework. In Section IV, we present Simulationresults. Finally, in Section V, we discuss conclusions, designconsiderations, and future work.

II. EQUATIONS OF MOTION

The dual-quad is actuated by eight brushless DC motorsoutfitted with orthogonal sets of counter-rotating propellers.The outer quadrotor is designated as the flight deck whilethe inner quadrotor is designated as the sensor deck. The twodecks are connected via a ball joint assembly, as illustratedin Fig. 1. The equations of motion are derived in the inertialframe using the approach proposed in [4]. The inertial axesare separated from the Earth-fixed axes by a 3D translationvector and a scalar yaw rotation denoted

f

. The dual-quadposition is given by the 3D vector ⇠ = [x, y, z]

T . The dual-quad attitude is defined by two sets of body-fixed axes. Theattitude of the flight deck is given by the Tait-Bryan angles⌘

f

= [

f

, ✓

f

,�

f

]

T , while the attitude of the sensor deck isgiven by the analogous angles ⌘

s

= [

s

, ✓

s

,�

s

]

T . The modelassumes perfect knowledge of the states.

With q = [⇠, ⌘

f

, ⌘

s

]

T , we define the Lagrangian

L(q, q) = T

trans

+ T

rot

+ U, (1)

T

trans

=

m

2

˙

⇠

T

˙

⇠, T

rot

=

1

2

⌦

T

f

If

⌦

f

+

1

2

⌦

T

s

Is

⌦

s

, (2)

U = �mgz, ⌦

f

= W

v

(⌘

f

)⌘

f

, ⌦

s

= W

v

(⌘

s

)⌘

s

, (3)

yf

xf

zf

ys

xs

zsȥf

ijf

șf

șs ȥs

ijs

f1

f4

f3

f2f5

f6

f7f8

Figure 1. By design, the orientation of the flight deck (purple axes) isindependent of the orientation of the sensor deck (green axes). The twodecks are connected via the stem which originates at the anterior apex ofthe dome and terminates at the ball joint in the center of the sensor deck.

W

v

(⌘) =

2

4� sin ✓ 0 1

cos ✓ sin cos 0

cos ✓ cos � sin 0

3

5, (4)

where If

and Is

are the moment of inertia tensors for theflight deck and sensor deck, respectively, m is the vehiclemass, and g is the acceleration due to gravity. The vectors⌦

f

and ⌦

s

are the 3D angular velocities of the flight deckand the sensor deck in their respective frames.

In the vehicle-fixed frames, the external forces are

F

f

=

2

40

0

�F

f

z

3

5 ,

2

40

0

(f3 + f1) + (f2 + f4)

3

5, (5)

F

s

=

2

40

0

F

s

z

3

5 ,

2

40

0

(f7 + f5) + (f6 + f8)

3

5, (6)

⌧

f

=

2

4⌧

f

⌧

f

✓

⌧

f

�

3

5 ,

2

4(f3 + f1)bf � (f2 + f4)bf

(f2 � f4)lf

(f3 � f1)lf

3

5, (7)

⌧

s

=

2

4⌧

s

⌧

s

✓

⌧

s

�

3

5 ,

2

4(f7 + f5)bs � (f6 + f8)bs

(f6 � f8)ls

(f7 � f5)ls

3

5, (8)

where F

f

and F

s

are the cumulative thrusts, and ⌧

f

and⌧

s

are the 3D moments on the flight deck and sensor deck,respectively. The scalars f

i=1:4 2⇥0 f

f

max

⇤are the

four (non-negative) flight deck rotor thrusts, and f

i=5:8 2⇥�f

s

max

f

s

max

⇤are the four sensor deck rotor thrusts.

The parameters b

f

and b

s

are positive electromechanicalconstants, and l

f

and l

s

are the moment arms betweenopposing motors on the flight and sensor decks, respectively.Note that the sums and differences on the right-hand side of(5)-(8) are eight independent equations in the eight inputs.

We map F

f

and F

s

to the inertial frame using

F

⇠

=R(⌘

f

)F

f

+R(⌘

s

)F

s

, (9)

R(⌘)=

2

4c

c

✓

c

✓

s

�s

✓

c

s

✓

s

�

� c

�

s

c

c

�

+ s

s

✓

s

�

c

✓

s

�

s

s

�

+ c

c

�

s

✓

c

�

s

s

✓

� c

s

�

c

�

c

✓

3

5, (10)

where R is the rotation matrix from the flight or sensor frameto the inertial frame. The functions sin ✓ and cos� have beenabbreviated as s

✓

and c

�

. Evaluating the Lagrangian,d

dt

@L@q

� @L@q

=

⇥F

⇠

⌧

f

⌧

s

⇤T

, (11)

we obtain the equations of motion

m

¨

⇠ +

2

40

0

�mg

3

5= R(⌘

f

)F

f

+R(⌘

s

)F

s

, (12)

J(⌘f

, If

)⌘

f

+ C(⌘

f

, ⌘

f

)⌘

f

= ⌧

f

, (13)J(⌘

s

, Is

)⌘

s

+ C(⌘

s

, ⌘

s

)⌘

s

= ⌧

s

, (14)J(⌘, I) = W

T

v

(⌘)IWv

(⌘), (15)

C(⌘, ⌘) =

˙J� 1

2

@

@⌘

(⌘

T J), (16)2

4x

y

z

3

5=

1

m

2

4�F

f

z

s

✓

f

� F

s

z

s

✓

s

F

f

z

c

✓

f

s

�

f

+ F

s

z

c

✓

s

s

�

s

�F

f

z

c

✓

f

c

�

f

� F

s

z

c

✓

s

c

�

s

+mg

3

5, (17)

2

4¨

f

¨

✓

f

¨

�

f

3

5= J�1

(⌘

f

, If

)

0

@

2

4⌧

f

⌧

f

✓

⌧

f

�

3

5�C(⌘

f

, ⌘

f

)

2

4˙

f

˙

✓

f

˙

�

f

3

5

1

A, (18)

2

4¨

s

¨

✓

s

¨

�

s

3

5= J�1

(⌘

s

, Is

)

0

@

2

4⌧

s

⌧

s

✓

⌧

s

�

3

5�C(⌘

s

, ⌘

s

)

2

4˙

s

˙

✓

s

˙

�

s

3

5

1

A, (19)

with inputs F

f

z

2 R, Fs

z

2 R, ⌧f

2 R

3, and ⌧s

2 R

3.

III. CONTROL SCHEME

In Section III-A, we simplify the system dynamics viafeedback. In Section III-B, we use a set of PD controllersto stabilize the sensor deck attitude. In Section III-C we usea similar set of controllers to stabilize the vehicle altitude andthe flight deck yaw heading. In Section III-D, we present abackstepping control strategy for stabilizing the horizontalposition using the flight deck pitch and roll states. A blockdiagram of the control system is given in Fig. 2. The top tworows show the feedback linearization control of z and

f

.The bottom row illustrates the combined state feedback andbackstepping control processes for the x and y states.A. Feedback Linearization

First, we simplify the dynamics of the six attitude statesby substituting the feedback linearization control laws

⌧

f

, C(⌘

f

, ⌘

f

)⌘

f

+ Jf

(⌘

f

)

⇥r

f

r

✓

f

r

�

f

⇤T

, (20)⌧

s

, C(⌘

s

, ⌘

s

)⌘

s

+ Js

(⌘

s

)

⇥r

s

r

✓

s

r

�

s

⇤T

, (21)

into (18) and (19) to obtain the linear subsystems⇥

¨

f

¨

✓

f

¨

�

f

⇤T

=

⇥r

f

r

✓

f

r

�

f

⇤T

, (22)⇥

¨

s

¨

✓

s

¨

�

s

⇤T

=

⇥r

s

r

✓

s

r

�

s

⇤T

, (23)

(50)(51)

(73)(74)

(17)(18)(44) (20)

> @Tx y

T

f fT Iª º¬ ¼

T

f fT Iª º¬ ¼

f f

T

r rT Iª º¬ ¼

T

x yr rª º¬ ¼

> @1 2

TX X

(36)

(37)f

r\

zr(25)

Dz

Df\

fW

T

f f\ \ª º¬ ¼

> @Tz zzfF

> @Tx y

PLANTT

D Dx yª º¬ ¼

Figure 2. Each block in the diagram represents an equation in the text. The control design linearizes the system from the references⇥x

D

y

D

z

D

f

D

⇤T , to the outputs

⇥x y z

f

⇤T . Note that the decoupled states z and

f

(as well as the sensor deck states s

, ✓s

,and �

s

, not pictured) are controlled outside of the backstepping framework.

where r

f

, r

✓

f

, and r

�

f

are the new control inputs tothe flight deck subsystem, and r

s

, r

✓

s

, and r

�

s

are thenew control inputs to the sensor deck subsystem. Next, wedecouple the forces on the sensor deck from those of theattitude deck by setting F

s

z

, 0, so that (17) simplifies to2

4x

y

z

3

5=

1

m

2

4�F

f

z

s

✓

f

F

f

z

c

✓

f

s

�

f

�F

f

z

c

✓

f

c

�

f

+mg

3

5. (24)

We then use the feedback linearization control law

F

f

z

, m

r

z

+ g

cos ✓

f

cos�

f

, (25)

to simplify the altitude state in (24), which becomes

2

4x

y

z

3

5=

2

4�(r

z

+ g) sec�

f

tan ✓

f

(r

z

+ g) tan�

f

�r

z

3

5, (26)

where the position control inputs are now ✓

f

, �f

, and r

z

.

B. Stabilization of the Sensor Deck Attitude

First, we convert the second-order system (23), into theequivalent first-order system

� =

2

6666664

�1

�2

�3

�4

�5

�6

3

7777775=

2

6666664

s

✓

s

�

s

˙

s

˙

✓

s

˙

�

s

3

7777775,

2

6666664

�1

�2

�3

�4

�5

�6

3

7777775=

2

6666664

�4

�5

�6

r

s

r

✓

s

r

�

s

3

7777775, (27)

where r

f

, r✓

f

, and r

�

f

are the inputs, and �1, �2, and �3,are the outputs. We define the control laws

r

s

, �2k

�1�4 � k

2�1�1, (28)

r

✓

s

, �2k

�2�5 � k

2�2�2, (29)

r

�

s

, �2k

�3�6 � k

2�3�3, (30)

where k�1 , k

�2 , and k

�3 are the positive-definite control gains.Substituting (28)-(30) into (27), we find the linear subsystems�1

�3

�= A

�1

�1

�4

�, A

�1 =

0 1

�k

2�1

�2k

�1

�, (31)

�2

�4

�= A

�2

�2

�5

�, A

�2 =

0 1

�k

2�2

�2k

�2

�, (32)

�3

�6

�= A

�3

�3

�6

�, A

�3 =

0 1

�k

2�3

�2k

�3

�, (33)

where eig(A

�

i

) =

⇥�k

�

i

�k

�

i

⇤T

, 8i = 1 : 3, makingthe closed-loop �-subsystem globally, exponentially stable.

C. Stabilization of the Flight Deck Altitude and YawBy stabilizing the error signals2

664

z

e

(t)

e

(t)

z

e

(t)

˙

e

(t)

3

775 =

2

664

z(t)� z

D

(t)

(t)�

f

D

(t)

z(t)� z

D

(t)

˙

(t)� ˙

f

D

(t)

3

775 , (34)

we can track the altitude reference, z

D

(t), and the yawreference,

f

D

(t). As before, we first convert the second-order systems (22) and (26), into the first-order system

↵=

2

664

↵1

↵2

↵3

↵4

3

775=

2

664

z

e

f

e

z

e

˙

f

e

3

775,

2

664

↵1

↵2

↵3

↵4

3

775=

2

664

↵3

↵4

�r

z

� z

D

(t)

r

f

� ¨

f

D

(t)

3

775, (35)

where r

z

and r

f

are the inputs, and ↵1 and ↵2 are theoutputs. We define the control laws

�r

z

, �2k

↵1↵3 � k

2↵1↵1 + z

D

(t), (36)

r

f

, �2k

↵2↵4 � k

2↵2↵2 +

¨

f

D

(t), (37)

where k

↵1 and k

↵2 are the positive-definite control gains.Substituting (36) and (37) into (35), we obtain the globally,exponentially stable subsystems↵1

↵3

�= A

↵1

↵1

↵3

�, A

↵1 =

0 1

�k

2↵1

�2k

↵1

�, (38)

↵2

↵4

�= A

↵2

↵2

↵4

�, A

↵2 =

0 1

�k

2↵2

�2k

↵2

�. (39)

D. Stabilization of the Flight Deck Horizontal PositionBy stabilizing the error signals

x

e

(t)

y

e

(t)

�=

x(t)� x

D

(t)

y(t)� y

D

(t)

�, (40)

we can track the position references x

D

(t) and y

D

(t). Webegin by converting (22) and (26) into the equivalent system

�=

2

664

�1

�2

�3

�4

3

775=

2

664

x

e

y

e

x

e

y

e

3

775, �=

2

664

�1

�2

�3

�4

3

775=

2

664

✓

f

�

f

˙

✓

f

˙

�

f

3

775, (41)

2

664

˙

�1˙

�2˙

�3˙

�4

3

775=

2

664

�3

�4

�(r

z

+ g) sec �2 tan �1 � x

D

(t)

(r

z

+ g) tan �2 � y

D

(t)

3

775, (42)

⇥˙

�1˙

�2˙

�3˙

�4

⇤T

=

⇥�3 �4 r

✓

f

r

�

f

⇤T

, (43)with inputs r

✓

f

and r

�

f

, and outputs �1 and �2.Using the nonlinear transformation

� =

2

664

�1

�2

�3

�4

3

775 ,

2

664

� sec �2 tan �1 � r

x

r

z

+g

tan �2 � r

y

r

z

+g

�1

�2

3

775 , (44)

we convert the �-� system (42)-(43), to the �-� system

˙

�1

˙

�3

�=

�3

�(r

z

+ g)�1 + r

x

� x

D

(t)

�,

�1

�3

�=

�3

µ1

�, (45)

˙

�2

˙

�4

�=

�4

(r

z

+ g)�2 + r

y

� y

D

(t)

�,

�2

�4

�=

�4

µ2

�, (46)

µ1 , (cos

2�2(2 sin �1�

23 � sin �1�

24 cos

2�1 + r

✓

f

cos �1)

+ cos �2(r�f

sin �1 sin �2 cos2�1 + 2�3�4 sin �2 cos �1)

+2�

24 cos

2�1 sin �1) sec

3�1 sec

3�2

� d

2

dt

2

2k

�1�3 + k

2�1�1 � x

D

(t)

r

z

+ g

!, (47)

µ2 , sec

2�2(2�

24 tan �2 + r

�

f

)

� d

2

dt

2

�2k

�2�4 � k

2�2�2 + y

D

(t)

r

z

+ g

!, (48)

�1

�2

�=

2

4� arctan

⇣⇣�1 +

r

x

r

z

+g

⌘cos �2

⌘

arctan

⇣�2 +

r

y

r

z

+g

⌘

3

5, (49)

r

x

, �2k

�1�3 � k

2�1�1 + x

D

(t) , (50)r

y

, �2k

�2�4 � k

2�2�2 + y

D

(t) , (51)

where k

�1 and k

�2 are positive-definite control gains. Thenew variables, �1 and �2, are the virtual control inputs to(45) and (46). The control functions, r

✓

f

and r

�

f

, are thereal control inputs to (45) and (46).

Note that when �1 and �2 are both equal to zero, (45) and(46) become the exponentially stable subsystems

˙

�1˙

�3

�= A

�1

�1

�3

�, A

�1 =

0 1

�k

2�1

�2k

�1

�, (52)

˙

�2˙

�4

�= A

�3

�2

�4

�, A

�3 =

0 1

�k

2�2

�2k

�2

�. (53)

!10 !5 0 5 10

!10

!5

0

5

10

�2

�

4

Figure 3. Phase portrait of the unforced response of the abridged model(53) for various initial conditions and control gain k

�2 = 2. Notice thetwo regions subtended by rotating the line �4 = �k

�2�2 (cyan) clockwisetoward the line �4 = k

�1�2�2 (magenta). Trajectories originating in these

regions will remain in the ball R�

(green) while exhibiting some overshootbefore converging to zero. Also notice the two regions subtended by rotatingthe line �4 = k

�1�2�2 (magenta) clockwise toward the line �4 = �k

�2�2

(cyan). Trajectories originating in these regions will escape the ball R

�

but remain in the larger ball R

✏

before converging to the origin with noovershoot. As t ! 1, all trajectories tend to the line �4 = �k

�2�2.Trajectories starting on �4 = �k

�2�2 will remain thereon while proceedingdirectly to the origin (dotted traces). The time t = ⌧

�4occurs at the

supremum (vertical crest or trough) of �4(t) where each trace intersectsthe (purple dashed) line �4 = � 1

2k�2�2 (i.e. ˙

�4 = 0).

In order to drive � to zero, we set

µ1 , �2k

�1�3 � k

2�1�1, (54)

µ2 , �2k

�2�4 � k

2�2�2, (55)

where k

�1 and k

�2 are the positive-definite control gains.Substituting (54) and (55) into (45) and (46), we find theglobally, exponentially stable subsystem�1

�3

�= A

�1

�1

�3

�, A

�1 =

0 1

�k

2�1

�2k

�1

�,

�2

�4

�= A

�2

�2

�4

�, A

�2 =

0 1

�k

2�2

�2k

�2

�. (56)

Having demonstrated that the unforced � subsystems, (52)and (53), are globally exponentially stable, and taking intoaccount that r

z

+g is bounded, we can prove that the cascade�-� system, (45)-(46), is globally asymptotically stable byshowing that the forced � subsystems are input-to-state stable(ISS). Examining the subsystem

˙

�2˙

�4

�=

�4

(r

z

+ g)�2 � 2k

�2�4 � k

2�2�2

�, (57)

�2

�4

�=

�4

�2k

�2�4 � k

2�2�2

�, (58)

we proceed with the ISS-Lyapunov function

V =

1

2

�

T

24P�24, �24 ,⇥�2 �4

⇤T

, (59)

P =

2k

�2 + k

2�2k

f

1

1 k

f

�, k

f

,k

2�2

+ 1

2k

�2

� 1 8k�2 , (60)

V =

1

2

�2k

�2 + k

2�2k

f

��

22 +

1

2

k

f

�

24 + �2�4, (61)

a1 V a2, (62)a1 = �

min

(P )||�24||2, a2 = �

max

(P )||�24||2, (63)

where �

min

and �

max

are the minimum and maximumeigenvalues. Taking the derivative of V , we have

˙

V =

�2k

�2 + k

2�2k

f

��2�4 + k

f

�4˙

�4 + �

24 + �2

˙

�4, (64)˙

V = �k

2�2

��

22 + �

24

�+ (�2 + k

f

�4) (rz + g) �2. (65)

Making use of the triangle inequality, we assert

˙

V �k

2�2(�

22 + �

24) + k

f

(|�2|+ |�4|)|rz + g||�2|, (66)˙

V �k

2�2(�

22 + �

24) + k

f

(|�24|+ |�24|)|rz + g||�2|, (67)˙

V �k

2�2||�24||2 + 2k

f

||�24|||rz + g||�2|. (68)

We can use the term �k

2�2||�24||2 to dominate the term

2k

f

||�24|||rz + g||�2| by rewriting (68) as

˙

V

�(1� ⇣)k

2�2||�24||2 � ⇣k

2�2||�24||2+

+2k

f

||�24||max |rz

+ g||�2|,(69)

where 0 < ⇣ < 1. We can then say

˙

V �W3(�24) 8||�24||2 � ⇢(|�2|) > 0, (70)W3(�24) = (1� ⇣)k

2�2||�24||2, (71)

⇢(�2) =k

2�2

+ 1

⇣k

3�2

max |rz

+ g||�2|, (72)

with ultimate bound � = a

�11 (a2(⇢)). The foregoing Lya-

punov analysis proves that the � subsystems are in fact ISS.We therefore conclude that the closed loop �-� system isasymptotically stable (in the region where the Euler anglesare valid) [8, Theorem 4.19].

In summary, our seven outputs, xe

, ye

, ze

, f

e

, s

e

, ✓s

e

,and �

s

e

, are stabilized using five linear control laws, (36),(37), (28), (29), and (30), and two nonlinear control laws

r

✓

f

, cos

2�1 cos �2

��2k

�1�3 � k

2�1�1

�� 2 tan �1

� sec �2(r�f

cos �1 sin �1 sin �2 + 2�3�4 sin �2)

�2�

24 cos �1 sin �1 sec

2�2 + cos �1 sin �1�

24�

23

+cos

2�1cos �2

d

2

dt

2

2k

�1�3 + k

2�1�1 � x

D

(t)

r

z

+ g

!,(73)

r

�

f

, cos

2�2

��2k

�2�4 � k

2�2�2

�� 2�

24 tan �2

+cos

2�2

d

2

dt

2

�2k

�2�4 � k

2�2�2 + y

D

(t)

r

z

+ g

!, (74)

where (73) and (74) are found by inverting (54) and (55).

!1

!0.5

0

0.5

1

!1

!0.5

0

0.5

1

0

1

2

3

4

5

6

7

8

9

10

x (m)

y (m)

z(m)

⇠ (t )

⇠

D

(t )

(a)

0 1 2 3 4 5 6 7 8 9 10!2

0

2

x (m

)

0 1 2 3 4 5 6 7 8 9 10!5

0

5

y (m

)

0 1 2 3 4 5 6 7 8 9 100

5

10

z (m

)

t (s)

(b)Figure 4. Sub-figures (a) and (b) give alternate views of the system’sperformance while tracking a reference signal (green dashed traces) of theform x

D

(t) = exp(�t) cos(t), yD

(t) = exp(�t) sin(t), zD

(t) = �t.

IV. SIMULATION

Model simulations are carried out in Simulink. Fig. 3shows a phase portrait of the unforced subsystem (53), forvarious initial conditions. Fig. 4 shows the full closed-loopsystem tracking an aggressive 3D position trajectory withzero overshoot. The corresponding control function inputsare plotted in Fig. 5. Lastly, Fig. 6 shows how the controlfunctions are used to steer the flight deck attitude states whilethe sensor deck attitude states are independently stabilized.

V. CONCLUSIONS

The findings of this paper suggest that the proposed controlstrategy will be suitable for tracking aggressive trajectoriesin both the position and attitude states of the dual-quadprototype pictured in Fig. 7. The presence of significantunmodeled dynamics could, however, render the obtainedresults invalid. Unmodeled dynamics are most likely to resultfrom unintended coupling between the sensor and flightdecks. To address this concern, the geometry of the designis such that aerodynamic interference between the decksis minimal. Similarly, mechanical interference between thedecks is made negligible by virtue of a low-friction ball joint.

0 1 2 3 4 5 6 7 8 9 10!4

!2

0

2

r z

0 1 2 3 4 5 6 7 8 9 10!0.6

!0.4

!0.2

0

0.2

r !

0 1 2 3 4 5 6 7 8 9 10!100

0

100

200

r "

0 1 2 3 4 5 6 7 8 9 10!200

!100

0

100

200

r #

t (s)

Figure 5. The top two subplots show linear trajectory tracking in the altitudeand flight deck yaw inputs, while the bottom two subplots show nonlineartrajectory tracking in the flight deck pitch and roll inputs.

The decision to actuate the sensor deck with propellerswas not made trivially. While a servo assembly would havebeen simpler to implement, adding servos would increase thevehicle’s overall weight without also increasing the thrustavailable to lift said weight. A nested quadrotor, on theother hand, offers attitude control as well as increased thrust.Additionally, the inclusion of servos would introduce newdynamics to the system while the inclusion of a secondquadrotor merely duplicates the existing dynamics.

Perhaps the greatest motivation for choosing the nestedquadrotor design is the ability to apply a tugboat conceptwhere a larger vessel outsources certain motion control tasksto a smaller vessel. In [9], we find a framework for robustperching and landing, while in [10], we find a framework foraerial grasping and lifting. Together, these suggest that it isfeasible to use a quadrotor MAV to tow a larger quadrotorUAV. In this case, the larger UAV would serve as the sensordeck, using large-diameter propellers to carry the visionsystem payload more efficiently. In order to allow the UAVsensor deck to maintain its motion constraint while carryingout the vision tracking task for an MAV swarm, one of theMAVs would dock with and tow the UAV as needed. As such,the dual-quad prototype is a physical model of the dockingmaneuver between two rotorcraft in a heterogeneous swarm.

REFERENCES

[1] A. Kushleyev, D. Mellinger, and V. Kumar, “Towards a swarm of agilemicro quadrotors,” Robotics: Science and Systems, July 2012.

[2] A. Haque and J. D. Dickman, “Vestibular gaze stabilization: differentbehavioral strategies for arboreal and terrestrial avians,” J. Neurophys-iology, vol. 93, no. 3, pp. 1165–1173, 2005.

[3] T. Madani and A. Benallegue, “Control of a quadrotor mini-helicoptervia full state backstepping technique,” in Decision and Control, 200645th IEEE Conference on, dec. 2006, pp. 1515 –1520.

[4] P. Castillo, R. Lozano, and A. Dzul, “Stabilization of a mini rotorcraftwith four rotors,” Control Systems, IEEE, vol. 25, no. 6, pp. 45 – 55,dec 2005.

0 1 2 3 4 5 6 7 8 9 100

5

10

!f (

deg.)

Flight Deck Attitude

0 1 2 3 4 5 6 7 8 9 10!100

0

100

" f (deg.)

0 1 2 3 4 5 6 7 8 9 10!100

0

100

# f (deg.)

0 1 2 3 4 5 6 7 8 9 100

5

!s (

deg.)

Sensor Deck Attitude

0 1 2 3 4 5 6 7 8 9 10!10

!5

0

" s (deg.)

0 1 2 3 4 5 6 7 8 9 100

10

20

# s (deg.)

t (s)

Figure 6. The top three subplots show the flight deck tracking a periodictrajectory (green dashed traces) while the bottom three subplots show thesensor deck independently stabilized at the origin.

Figure 7. A small inner quadrotor (sensor deck) carrying a vision system,is pictured nesting inside a large outer quadrotor (flight deck).

[5] S. Seghour, M. Bouchoucha, and H. Osmani, “From integralbackstepping to integral sliding mode attitude stabilization of aquadrotor system: Real time implementation on an embedded controlsystem based on a dspic c,” Istanbul, Turkey, 2011, pp. 154 – 161.[Online]. Available: http://dx.doi.org/10.1109/ICMECH.2011.5971273

[6] G. Raffo, M. Ortega, and F. Rubio, “Backstepping/nonlinear h-infinitycontrol for path tracking of a quadrotor unmanned aerial vehicle,” inAmerican Control Conference, 2008, june 2008, pp. 3356 –3361.

[7] A. Das, F. Lewis, and K. Subbarao, “Backstepping approach forcontrolling a quadrotor using lagrange form dynamics,” Journalof Intelligent and Robotic Systems: Theory and Applications,vol. 56, no. 1-2, pp. 127 – 151, 2009. [Online]. Available:http://dx.doi.org/10.1007/s10846-009-9331-0

[8] H. Khalil, Nonlinear Systems. Prentice Hall, Jan. 2002. [Online].Available: http://www.worldcat.org/isbn/0131227408

[9] D. Mellinger, M. Shomin, and V. Kumar, “Control of quadrotorsfor robust perching and landing,” in Proceedings of the InternationalPowered Lift Conference, Oct 2010.

[10] D. Mellinger, Q. Lindsey, M. Shomin, and V. Kumar, “Design, mod-eling, estimation and control for aerial grasping and manipulation,”in Proceedings of the IEEE International Conference on IntelligentRobots and Systems (IROS), Sept 2011.