Hybrid Adaptive Control for Aerial Manipulation

J Intell Robot Syst (2014) 73:693–707DOI 10.1007/s10846-013-9936-1

Hybrid Adaptive Control for Aerial Manipulation

Matko Orsag ·Christopher Michael Korpela ·Stjepan Bogdan ·Paul Yu Oh

Received: 1 September 2013 / Accepted: 12 September 2013 / Published online: 30 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract This paper presents a control scheme toachieve dynamic stability in a mobile manipulat-ing unmanned aerial vehicle (MM-UAV) usinga combination of Gain scheduling and Lyapunovbased model reference adaptive control (MRAC).

This material is based on research sponsored by theAir Force Research Laboratory, under agreementnumber FA8655-13-1-3055. The U.S. Government isauthorized to reproduce and distribute reprints forGovernmental purposes notwithstanding anycopyright notation thereon. The views and conclusionscontained herein are those of the authors and shouldnot be interpreted as necessarily representing theofficial policies or endorsements, either expressed orimplied, of the Air Force Research Laboratory or theU.S. Government.

M. Orsag (B) · S. BogdanLaboratory for Robotics and IntelligentControl Systems, Faculty of Electrical Engineeringand Computing, University of Zagreb,10000 Zagreb, Croatiae-mail: [email protected]

S. Bogdane-mail: [email protected]

C. M. Korpela · P. Y. OhDrexel Autonomous Systems Lab, Drexel University,Philadelphia, PA 19104, USA

C. M. Korpelae-mail: [email protected]

P. Y. Ohe-mail: [email protected]

Our test flight results indicate that we can accu-rately model and control our aerial vehicle whenboth moving the manipulators and interactingwith target objects. Using the Lyapunov stabilitytheory, the controller is proven to be stable. Thesimulation results showed how the MRAC is ca-pable of stabilizing the oscillations produced fromthe unstable PI-D attitude control loop. Finallya high level control system based on a switchingautomaton is proposed in order to ensure thesaftey of the aerial manipulation missions.

Keywords Adaptive control ·Dexterous manipulation ·Aerial robots

1 Introduction

Historically, unmanned aerial vehicle (UAV) re-search has been focused on avoiding interactionwith the environment. Recently, the focus hasshifted towards aerial manipulation and bringingthe unmanned aerial systems in contact with theirsurroundings. The ability to manipulate objectswill, ultimately, greatly expand the use of un-manned aerial systems. Although some researchgroups tested various forms of adaptive controlon quadrotor UAVs [1, 2], its full potential canbe exposed through aerial manipulation missions,where the moment of inertia and the center of

694 J Intell Robot Syst (2014) 73:693–707

mass constantly fluctuate. Model reference adap-tive control (MRAC) concept [3], proposed in thispaper has been used in numerous applications [4–6]. Together with the original PID controller ofthe aircraft it is used to assure aircraft stabilitythroughout the manipulation process.

Current State of the art in aerial manipulationhas focused mostly on single degree of freedom(DOF) gripper manipulators and slung load trans-port. A lot of researchers that contributed to thisfield of aerial robotics have so far focused on3 key aspects: Contact inspection (Pose/Wrenchcontrol), Slung load transport, Single Degree offreedom grippers.

1.1 Pose/Wrench Control

In [7], authors introduced wrench control to givethe quadrotor an ability of stable motion while incontact. The authors used a hybrid Pose/Wrenchframework capable of switching between purePose and Pose/Wrench control but the operatoris required to make the switch. Instead of usingadditional force/torque sensors, authors utilized awrench estimator using quadrotor inputs and posemeasurements. Another hybrid Force/Positioncontrol concept based on state feedback is intro-duced in [8, 9]. In this work, a ducted-fan aerialvehicle is used to achieve contact inspection tasks.The authors presented a mechanical design tocope with inherent zero dynamics of the system.

1.2 Slung Load

Authors in [10] explored the possibility of us-ing single and multiple UAVs to assist in searchand rescue (SAR) missions. They tested forma-tions of up to three small size petrol poweredhelicopters that cooperatively transport a slungload. The authors were capable of transportinga video camera with three small size helicoptersin adverse weather conditions that exhibited highwind speeds of up to 35 km/h. A group of authorsin [11] implemented an additional vision systemthat measured the position of the slung load. Theyproposed an adaptive controller that reduced theswing in the load. In order to solve a similar prob-lem with quadrotors carrying a slung load, authors

in [12] proposed a technique based on dynamicprogramming which ensured swing free trajectorytracking.

In [13], a tethered helicopter configuration ismodeled and tested. This configuration proved tobe more stable in presence of disturbances (i.e.wind gusts) then a non-tethered helicopter. Tetheralso introduced coupled dynamics, adding to thecomplexity of the overall controller. Nevertheless,tether is a very interesting concept that couldpotentially be used in twofold manner: It couldprovide unlimited supply of electricity; It couldalso be used as additional pose and position mea-surement system.

1.3 Single DOF Grippers

The authors in [14–16] analyzed the stability ofa helicopter and a quadrotor with added payloadmass. Using Routh-Hourwitz criteria, the authorsderived a connection between the mass off theadded payload, its offset from UAV center ofmass and the stability of the UAV. Plotting offsetvs. mass stability regions, they clearly showed howa bigger mass tightens the stability region of theaircraft. Namely, the bigger the mass, the smallerthe available offset region, so that the vehicle hasto grab the additional payload as close to thecenter of gravity as possible. Quadrotor stabilityin presence of unknown payload disturbances wasdiscussed in [17]. Here the authors look into thepossibility of estimating disturbance parameters(i.e. mass and moments of inertia). Using hovermode to estimate the parameters, they effectivelyeliminated the Euler angles and the derivativesof the position from the equations. This researchgroup also contributed by doing an experimen-tal study with teams of quadrotors cooperativelygrasping, stabilizing, and transporting payloadsalong desired three-dimensional trajectories [18].They went a step further and showed the exper-imental results of team of quadrotors performingautomated assembly of Special Cubic type Struc-tures [19]. They used their gripping tool to pickup the simple structural nodes and used magneticendings on the structures to piece them together.

The aim of this research is to extend the currentstate of the art by introducing multiple degree offreedom manipulators. To the extent of authors

J Intell Robot Syst (2014) 73:693–707 695

knowledge, little or no attempt has been made toimplement such an aerial vehicle so far. Afore-mentioned state of the art is limited to simple ma-nipulation problems (i.e. pick and place, contactinspection, painting, etc.). Multiple—DOFmanip-ulators expand the capabilities of aerial robots bygiving them the ability to: perch and manipulate,twist valves on or off, assemble objects, removeobstacles and many other.

Introducing multiple DOF adds to the com-plexity of the control problem. Therefore, the firstgoal of this research is to achieve and sustain astable flight while moving manipulator arms. Alot of research in mobile robotics as well as theresearch in aerial robots with 1 DOF grippers canbe utilized to solve this problem. After this, nextresearch step would be to implement current stateof the art in contact stability and load estimationwhich would yield a fully dexterous aerial robot.The following sections present a first step towardsa fully dexterous MM-UAV (Fig. 1). First, andextensive mathematical model of the aerial ro-bot system is presented, followed by a stabilityanalysis of low level attitude controllers. Next,the two nonlinear adaptive control techniques arepresented (i.e. gain scheduling and model refer-ence adaptive control). Finally, nonlinear controltechniques are tested and verified in simulations.These simulations and experiments serve to showthe efficacy and performance of the proposed hy-brid nonlinear control concept.

2 Mathematical Model

The proposed mobile manipulating unmannedaerial system, featuring a quadrotor UAV andtwo multi DOF manipulators, is shown in Fig. 3,and a table of dynamic parameters is given inTable 1. Using the recursive Newton-Euler ap-proach andDenavit-Hartenberg parameterizationfor forward kinematics, each arm is modeled as aserial chain RRRR manipulator (Fig. 2) [20]. Theconnection between the quadrotor body frameand the first joint of each arm is represented withstatic revolute joint with a constant angular offsetfor each MM-UAV arm (Link B-0). Applying theresults from [21] quadrotor dynamics are intro-duced to the aerial platform of the robot.

Fig. 1 MM-UAV carrying a 40 cm long cylinder rod

2.1 Manipulator Model

Denavit-Hartenberg (DH) parameters of the ma-nipulator arms shown in Fig. 2 are given inTable 2, and Fig. 3 depicts the overall UAV-manipulator system. Parameters θ , d, a, and α arein standard DH convention and q1

i , q2i , q

3i , and q4

iare joint variables of each manipulator arm i =[A, B]. Since the whole aircraft is symmetrical,the general kinematic structure is identical forthe right and left arms, the coordinate frames arethe same for each arm, and only the link B-0is different for the two arms. Reference framesare shown in Fig. 2 which relate the fist joint L1

to the end effector frame E. To make the DHparameters consistent, an additional, virtual frameLT is set in the origin of frame L4.

With Denavit-Hartenberg parameterization,joint frames are set and direct kinematic equationsfor each serial chain are derived. This procedure

Table 1 Mobile manipulating unmanned aerial vehicle(MM-UAV) parameters

Quadrotor

Jxx Jyy Jzz1.41 kgcm2 1.42 kgcm2 2.79 kgcm2

mQ

957.597 g

Manipulator arms

Jxx Jyy Jzz49.43 gcm2 39.85 gcm2 28.95 gcm2

mA,B a37 g 4.25 cm


Fig. 2 Reference frames for manipulator arms. See Fig. 3 for detail on overall UAV-manipulator system

is repeated for both manipulator arms. Individuallinks that form the arm are consistent in size,mass and shape (i.e. their mass mL, kinematicparameter a and tensor of inertia JL are identical).Under a reasonable assumption, that the links aresymmetrical with respect to the chosen coordi-nate systems, the following simplification holdsfor their respective tensors of inertia:

JL =⎡⎣jxx 0 00 jyy 00 0 jzz

⎤⎦ (1)

Once the arms start moving, the tensors of iner-tia change according to the joint angle changes. Tocalculate the overall tensor of inertia of the arms,first, each link’s tensor of inertia needs to be trans-formed to the body origin frame LB. Transforma-tion matrices, necessary for the calculation of theoverall tensor of inertia can easily be devised fromthe DH parameters. Judging from the patterns inTable 2, it is clear that the transformation matricesfor the first three joints Ti

i−1, where i ∈ 〈1, 2, 3〉are the same. Virtual joint LT , that has no size,mass or moment of inertia; and the end effector

Table 2 Denavit-Hartenberg parameters for manipulators[cm]

Link θ d a α

1 q1A − π

20 3.75 −π

2

2 q2A 0 3.75

π

2

3 q3A 0 3.75 −π

2

4 q4A + π

20 0

π

2T-E 0 0 3.75 0

Showing arm a only for clarity

joint LE, with a constant angle q5 = 0 have theirspecific transformation matrices:

Tii−1 =

⎡⎢⎢⎣

Ci 0 −Si aCi

Si 0 Ci aSi0 −1 0 00 0 0 1

⎤⎥⎥⎦ TT

3 =

⎡⎢⎢⎣

C4 0 −S4 0S4 0 C4 00 −1 0 00 0 0 1

⎤⎥⎥⎦

TET =

⎡⎢⎢⎣

1 0 0 00 1 0 00 0 1 a0 0 0 1

⎤⎥⎥⎦ (2)

Using a Recursive Newton-Euler algorithm [22,23] and neglecting friction forces, one can de-rive force/torque equations produced from all thejoints:

[FA(δqA)

τ A(δqA)

]= D(qA)qA +C(qA, qA)+H(qA)

(3)

with D as a generalized inertia tensor, C(q, q) isthe generalized Coriolis and Centrifugal force ma-trix equation, H(q) is a generalized gravity forceand δqA = [

qA, qA, qA

]denotes a triplet of joint

angles, angular speeds and acceleration. The sameexpression could, of course, be written in the samemanner for arm B.

2.2 Quadrotor Model

The end goal of the mathematical model is tocombine both manipulator dynamics and the dy-namics of the quadrotor. Due to the resultingcomplexity of the model and keeping in mindspecific mission requirements, the quadrotor dy-namics considered in this paper do not account forvarious aerodynamic effects (i.e. blade flapping,


Fig. 3 Complete modelof the mobilemanipulating unmannedaerial system

Center of Quadrotor Construction

A

1A

A2A

A

3A

A

4A

A

TA

A

EA

CMQ

c1A

ground effect, etc.) experienced during highly dy-namic flying maneuvers. Most of the missions thatinclude some form of interaction with the environ-ment require stable hovering maneuvers, whichjustify a simplified mathematical model withoutaccounting for the previously mentioned aerody-namic effects.

As the manipulator dynamics are introducedthrough the recursive Newton-Euler method, it ispossible to separately model the quadrotor mo-tion based on Newton-Euler equations for rigidbody translation and rotation [24]. The two mod-els are then combined to yield a complete modelof the proposed mobile manipulating unmannedaerial system.

In the first aerodynamic approximation, thethrust and torque that come from the rotors, TandQ respectively, are proportional to the square

of rotor speed �. The rotation speed is propor-tional to the voltage � ∝ U[V] applied to the DCmotor. In the presence of the added payload andits inertia, as well as the mission requirementsfor air robots, the dynamics of the motors has animportant impact in the overall aircraft stabilityand cannot be omitted from theMM-UAVmodel.Off the shelf electronic speed controllers are usedto power and control the motors. The underlyingdynamics of the speed controllers can be modeledas a simplified first order PT1 transfer function.

� = Km

1 + TmsU (4)

Considering a general quadrotor construction,each propeller produces thrust force Ti and torqueQi in body frame z-axis. Both torque and thrustare complex nonlinear functions which depend on


multiple aerodynamic conditions, such as: verticaland horizontal speed of the quadrotor, air density,wind etc. In these simplified aerodynamic condi-tions, first approximation is used, where the re-sulting thrust forces and torques depend solely onrotor speed, which in turn is linearly dependant onthe applied voltage. Summing all forces togethergives the total aircraft thrust. Total torque on theother hand depends not only on the speed andthrust of each rotor, but on the position of theMM-UAVs centroid as well. Therefore, propellertorque τ i has two components, one coming fromthe actual propeller drag, and the other due tothe displacement of the propeller from the cen-ter of mass. Of course, in a mobile manipulatingunmanned aerial system, the center of mass shiftsas each joint of the manipulator moves and thetorque becomes a nonlinear function of manipu-lator joint angles.

Fq(u) =4∑

i=1

F(u)i (5a)

τq(u, q j

A, qjB

)=4∑

i=1

Q(u)i+�oiCM

(q jA, q

jB

) × F(u)i

(5b)

2.3 Mobile Manipulating Unmanned AerialVehicle Dynamics

Combining the dynamic equations of motion ofan aerial vehicle together with the mathematicalmodel of the attached manipulators yields a com-plete mathematical model of the system:

[Fq(u)

τ q(u, q j

A, qjB

)]+

[FA

(δq j

A

) + FB(δq j

B

)τ A

(δq j

A

) + τ B(δq j

B

)]

=[mI 00 J

(q jA, q

jB

)] [

v

ω

]+

[0

ω × Jω

](6)

The resulting equation is highly nonlinear. Theexpression on the right hand side, is a vector formon Euler equation of rigid body motion, wherem stands for total system mass, I a 3 × 3 identitymatrix, J(q j

A, qjB) is a nonlinear equation of total

moment of inertia, with v and ω as system’s lin-ear and angular speed respectively. The left hand

side consists of quadrotor propulsion system andmanipulator arm dynamics.

Previous work in this field has concentratedonly on load mass stability problem, ignoringthe coupled manipulator dynamics [15, 17]. Fora 1 DOF gripper tools, this assumption is wel-come, but for 4-DOF arms that introduce a sig-nificant increase in payload and moment of inertiathis simplification cannot be applied. Manipulatordynamics that have been discussed in detail inSection 2.1 is introduced to system dynamicsthrough the second part of left hand side of Eq. 6.It can be treated as a disturbance to the quadrotorcontrol system, and is thus the focus of stabilitydiscussion. On the other hand, two underlyingeffects that are identified to directly influencequadrotor control are the change in the overallcenter of mass CM and moment of inertia J.

Building upon the results from [25, 26] wepresent a complete mathematical model to estab-lish a stability criteria for this mobile manipulatingunmanned aerial system.

2.4 Center of Mass and Momentof Inertia Distribution

As shown in [26], one can easily calculate thevarying center of mass CM by summing all thebody part distances from the center of quadrotorconstruction, weighted by their respective mass:

CM(q jA, q

jB

)

=QcmmQ +mL

∑4i=1

[cAi

(q jA, q

jB

) + cBi(q jA, q

jB

)]

mQ + 4mL

(7)

Recalling from Eq. 5b the quadrotor propulsionsystem torque τq(u, q

jA, q

jB) is a nonlinear func-

tion of systems centroid CM(q jA, q

jB). As the cen-

troid shifts due to joint angle changes, the dis-tance between each propeller and the centroid�oiCM(q j

A, qjB) varies:

�oiCM

(q jA, q

jB

)

=[Cos

(π4+(i− 1)

π

2

)×Sin

(π4+ (i − 1)

π

2

)0]T

+CM(q jA, q

jB

)(8)


where i denotes each propeller which are, due tothe chosen X configuration, placed in a mannerthat each propeller closes a 45◦ angle between itsclosest coordinate system axis. The total torqueapplied to the aerial robot thus becomes:

τq(u, q j

A, qjB

) =4∑

i=1

Q(u)i + CM(q jA, q

jB

) × F(u)i

(9)

if we assume perfect construction of the quad-rotor, resulting torques zero out the first term[Cos( π4 + (i − 1) π2 )Sin(

π4 + (i − 1) π2 )0

T] × F(u)i

in the sum of all rotors.The overall moment of inertia changes are a bit

more tedious to devise, but still manageable in twoseparate steps:

– Transforming each body part moment of in-ertia from its own principal axis to the bodyorigin coordinate system.

RiBJ

A,iL Ri

BT

where RiB is a 3 × 3 rotation part of transfor-

mation matrix TiB

– After the coordinate systems are aligned itis possible to apply the parallel axis theorem

and yield a complete expression for systemsmoment of inertia.

JA,iL |B +mL

[(cA,i · cA,i) I− cA,i ⊗ cA,i]

where c is a vector distance from the body partcenter of mass to the center of construction,I again denotes 3 × 3 identity matrix, and ⊗represent the outer product of the two vectors.

Putting it all together yields the equation for mo-ment of inertia variations with respect to manipu-lator joint angle changes:

J(q jA, q

jB) =

4∑i=1

∑j=A,B

RiBJ

j,iLR

iBT

+mL[(c j,i · c j,i) I− c j,i ⊗ c j,i] (10)

3 Angle Control Stability

Control of the MM-UAV body is achieved with astandard PI-D control loop shown in Fig. 4. Thistype of PID implementation eliminates the po-tential damages to the actuators, normally experi-enced when driving the control difference directlythrough the derivation channel [27]. The majorfactor that affects the stability of the aircraft is the

Fig. 4 Attitude modelreference adaptive PI-Dcontrol


variable moment of inertia J = Ixx, Iyy discussedin Section 2.4.

The PI-D angle control with the additionaladaptive control loop is shown in Fig. 4. Its trans-fer function yields a 4th order characteristic poly-nomial a4s4 + a3s3 + a2s2 + a1s+ a0, where the4th order dynamic system includes both thedynamics of the aircraft and the motor dynam-ics (11).

GαCL =KiKDKm

Tm J

(Kp

Kis+ 1

)

s4 + 1Tm

s3 + KDKmTm J s2 + KDKmKp

Tm J s+ KiKDKmTmJ

(11)

Coefficients KD, Kp and Ki are PI-D respectivegains and Km and Tm represent propulsion systemgain and motor time constant. After applying theRouth-Hurwitz stability criteria, the analytical so-lutions for stability conditions, which require thatall coefficients be positive and that inequalities inEq. 12 hold, are derived.

℘ = KDKmKp

Ki

(1 − TmKp

)> J(qA, qB) (12a)

KDKm(1 − TmKp

)> 0 (12b)

The stabilty criterion can be visualized in Fig. 5.Criterion (12) shows how stability could be

maintained through adaptation of three parame-ters, KD,Ki and Kp respectively. Of course, thestability criterion has different sensitivity for eachvariable which adaptation algorithm needs to take

Fig. 5 Visualization of the stability criterion

into account. For KD, the stability criteria ℘ is alinear function

δ℘

δKD= KmKp

Ki

(1 − TmKp

)(13)

which for all stable parameter values is greaterthan zero, making the ℘ a monotonically risingfunction of KD. Thus increasing KD should in-crease the stability of the system, making KD

an optimal choice for Model Reference AdaptiveControl. On the other hand, Kp has a more com-plex affect on stability:

δ℘

δKp= KmKD

Ki

(1 − TmKp

)(14)

where ℘ is rising for Kp <1

2Tmand falling for

12Tm

> KP < 1Tm

. Due to Eq. 12b, the system be-comes unstable for all KP > 1

Tm. Increasing Kp for

the first range keeps the system stable. Increasingit too much, reverses the process and ultimatelydrives the system unstable. Finally, the integralpart of PID controller, Ki, has a reverse, nonlin-ear affect on system stability ℘:

δ℘

δKi= −KDKm

Ki2 (1 − TmKp). (15)

Previous equation shows that ℘ is a monotonicallyfalling nonlinear function of Ki. Both Ki and Kp

have nonlinear affects on system stability ℘ andare not suitable for direct adaptive control. There-fore, we propose using them in a separate adaptivecontroller based on gain scheduling technique.

4 Hybrid Adaptive Control

Transfer function (11) shows that there exist fourparameters in the attitude control loop of a mobilemanipualting quadrotor that change during flightand manipulation:

– Km - Propulsion system gain changes drasti-cally through time, especially in load manipu-lation missions.

– Tm - Propulsion system dynamics changes dueto various effects as the mission progressesthrough time.


– β - The aerodynamic conditions constantlychange during the flight.

– J - As previously discussed (2.4) the momentof inertia changes depending on the load andarm pose.

Aerial manipulation missions mostly requiresteady flight conditions, for which the changes inthe aerodynamic conditions as well as the aerody-namic coefficient β can be neglected. On the otherhand, if the battery power supply is kept constantthroughout the mission, the variations in the Tm

can be minimized. The two remaining parameters,J and Km diverge the most during aerial manip-ulation. The variations in the moment of inertiahave been previously discussed. The propulsionsystem gain changes are mostly caused from thevariations in the load mass, which changes thepiecewise linearization of the quadratic relation-ship between the propeller thrust and the appliedvoltage. Apart from that, the variations in temper-ature and the battery depletion also change thelinearized motor gain throughout the mission. Inorder to adapt to parameter changes, AdaptiveControl mechanisms should be applied.

4.1 Gain Scheduling Adaptive Controller

As shown in the previous paragraph, it is pos-sible to predict the changes in the dynamics ofthe proposed aerial manipulator when it is not incontact with the environment. In these situations,changes in the overall dynamics come from armmovement, which is a known nonlinearity. There-fore, the controller parameters could be modifiedsimply by monitoring the position of the manip-ulator joints and relate the controller parame-ters to these auxiliary variables (i.e. manipulatorjoints). Gain scheduling has been proposed andverified on different aerial platforms [28, 29] andcan be regarded as a common adaptive controlmechanism in flight control systems [30]. Becausethere is no explicit method for gain schedulingcontroller synthesis [3], this paper discusses twopossible ways to implement such a controller onthe proposed aerial robot.

The first approach is straighforward: adaptingthe KD gain with respect to joint angle changes qA

and qB respectively, while maintaining the speedcontrol loop quality parameters:

ωn =√

KDKm

J(qA, qB)Tm(16a)

ζ = 1

2

√J(qA, qB)

KDKmTm(16b)

Even though there exists only one DOF for para-meter adaptation, it is possible to satisfy both nat-ural frequency ωn (16a), and dampening ζ (16b)conditions. Thus rewriting the equations fromEq. 16 yields two versions of gain scheduling func-tion for KD:

KD(qA, qB) = ωn

2 J(qA, qB

)Tm

Km(17a)

KD(qA, qB) = J

(qA, qB

)

4ζ 2KmTm(17b)

Because both equations have linear relation tovariations in moment of inertia, setting KD

proportional with respect to J(qA, qB) cancelsout both variations of natural frequency anddampening.

On the other hand, in this control loop thereexist other parameters that vary with time. For in-stance, aforementioned Km and Tm change as thebattery depletes. Even more, linear relationshipbetween KD and the stability condition ℘ makes itideal for model reference adaptation. Therefore,in order to use both gain scheduling and MRACtogether, gain scheduling should be applied toother controller parameters.

To that end, a different parameter adaptationlaw is proposed: adapting the Kp and Ki gainswith respect to joint angle changes, qA and qB re-spectively, while maintaining the stability condition℘ > J(qA, qB).

The design approach for this gain schedulingadaptation controller is straightforward. First wechose a nominal design pose. The pose can bechosen arbitrary, but for practical reasons thispose should be chosen in a way that simplifiesparameter tuning. Once the pose is chosen, thesystem is tuned according to the controller designgoals, which yields a stable, nominal value ℘N >


Fig. 6 Visualization of the gain scheduling algorithm basedon maintaining the stability condition ℘ > J(qA,qB)

JN . Gain scheduling adaptation algorithm is thenapplied in the following form:

Ki(qA, qB) ∼ 1

J(qA, qB

) → ℘(qA, qB)

−℘N = J(qA, qB) − JN . (18)

This adaptation algorithm successfully creates anenvelope above the varying moment of inertia,graphically shown in Fig. 6. Wrapping the stabilitycriterion around the moment of inertia keeps thesystem stable at all times.

4.2 Lyapunov Based MRAC Control

Discussions from the previous chapters show thatKD is an optimal choice for model reference adap-tation. In this paragraph, the stability and feasabil-ity of such an adaptation controller is devised.

4.2.1 Stability Analysis

Separating the highly inconsistent parameters Km

and J on one side, and fairly constant Tm andβ on the other, allows us to write the open looprotation speed transfer function as a class of IPT1,y = kG(s):

�(s) ={Km

J

} {1

1 + Tms1

s

}u(s) (19)

The PI-D controller is then modified so that itsoutput is multiplied by the adaptive gain ζ and theadjustment rule for it is chosen according to:

dζdt

= −γuTpi-de (20)

where e is the error difference between the modeland the actual orientation dynamics

y =⎡⎣

0 1

−1 − 1

Tm

⎤⎦ y+ Km

JTmu (21)

Using the Lyapunov stability theory, it can beshown that such a system is uniformly asymp-totically stable. Candidate Lyapunov function ischosen:

V (e, ζ ) = γ

2eTe+ k

2(ζ − ζ0)

2 (22)

with ζ0 as a steady state value of ζ and arbitrarychosen gains γ and k so that its derivative can be:

dV (e, ζ )dt

= γ eTe+ k (ζ − ζ0)

(dζdt

+ γueTKm

TmJ

)

(23)

For the chosen adaptation rule (20), it can beshown that the Lyapunov Candidate (22) is neg-ative semi-definite if and only if Tm > 0, whichis always true, and γ > 0 which can be arbitrarychosen. Although, the Lyapunov stability analysissets no upper bound for the correction factor γ ,it is still necessary to choose its appropriate valueand to that end, the approach in [6] is chosen. Apractical implementation requires that the upperand the lower bound for the adaptation gain ζ ,ζmax and ζmin are set. According to criteria (8),the range of KD for which the system is stablecan be determined once the range of changes ofthe moment of inertia J is known. In our caseKDmax = 2KD0 and KDmin = KD0/2 , with KD0 asthe nominal value of the control parameter KD.Since the adaptation mechanism influences thesystem through multiplication KDζ (Fig. 6), deter-mination of ζ maximum and minimum is straight-forward, i.e. in our case ζmax = 2 and ζmin = 1/2.Now, one is able to estimate the range of thecorrection factor γ . Rewriting Eq. 10 gives:

ζ(t) = −γ

∫uPI−D(t)e(t)dt (24)

During the adaptation phase (Fig. 7), a set ofpulses is generated by PI-D controller in order


Fig. 7 Matlab Simulation(Take off with armsstowed, Oscillationssettled; Deploying armsmove): Roll and pitchangles

0 5 10 15 20 25 300

2

4

Z [

m]

0 5 10 15 20 25 30

−2

0

2

Θ [° ]

0 5 10 15 20 25 30

−2

0

2Ψ

[° ]

time [s]

to perturb the system so that new value of theadaptation parameter can be determined, thus,

ζ(t) = −γ

∫δ(t)

[y(t)− ym(t)

]dt (25)

As we already wrote, J and Km are two parame-ters that are mostly influenced by aerial manipula-tion. Hence, including inverse Laplace transformof the system and the model (neglecting influenceof Tm) in Eq. 25, one gets

ζ(t) = −γ

(Km

J− KM

JM

)t + ζ0 (26)

Since dynamics of the adaptation loop must beslower (usually 5 to 10 times) than the systemdynamics, in case of large change of parame-ters, the adaptation parameter should attain maxi-mum/minimum value at t ≈ 5 · (5Tm), which gives

ζmax ≈ −γ

(Km

Jmax− KM

JM

)· 25Tm + ζ0 (27)

ζmin ≈ −γ

(Km

Jmin− KM

JM

)· 25Tm + ζ0 (28)

Finally, lower of two values of the correction fac-tor γ , calculated from Eqs. 17a, 17b and 18, shouldbe included in the MRAC.

Because model reference adaptation is highlysusceptible to disturbances, one has to take intoaccount the static and dynamic torque distur-bances produced from the arm movement. Thatis why the disturbance torque estimator is intro-duced to the MRAC control schematics. Dynamicdisturbances are cancelled out by using a lowpass filter for the adaptation rule. Static torqueshowever, cannot be bypassed with a filter. Statictorque is caused from the shift in the center ofmass of the aircraft and the gravity that affectsits unbalanced body. Learning from the results in[17], one can find the unknown center of massoffset, in a least square minimization sense, simplyas an average over collected data:

ˆCMx = u3

u1(29)

Where u1 and u3 are simply controller outputs forpitch and height control respectively. The estima-tion results are then fed to the MRAC model,thus minimizing the controller vulnerabilities todisturbances. Static toreque estimation works wellfor steady state estimation, but fails to accuratelyestimate the dynamic changes in the gravity torqe.Therefore, we propose adding a dead zone tothe adaptation rule, in order to cancel out theestimation errors.


Fig. 8 Left to right,MM-UAV arm transitionfrom stowed to fullydeployed

Fig. 9 The adaptive gainζ changes as theoscillations occur, andbrings the system back inthe stability region

0 5 10 15 20 25 300.5

1

1.5

ζ

time [s]

Fig. 10 MatlabSimulation (Take off witharms stowed, Oscillationssettled; Deploying armsmove): Propulsion systemthrust and torque values

0 5 10 15 20 25 30−20

−10

0

Fz [

N]

0 5 10 15 20 25 30

−0.2

0

0.2

Mx, M

y [N

m2 ]

time [s]

M

x

My


4.2.2 Simulation verif ication

In our previous work, we have showed how witha poorly designed PID controller aerial robot be-comes unstable during manipulation tasks, eventhough it is perfectly stable during the flight [25,26]. In this paragraph, we put the adaptive controlto the test, trying to stabilize the same systemfrom our previous work. Matlab was used for sim-ulations and a recursive Newton-Euler dynamicsmodel of the manipulators was implemented usingthe Robotics Toolbox [31]. Figures 7 and 10 showthe results of one of the performed tests wherethe quadrotors roll controller was tuned closeto the stability boundary. The aircraft takes offwith arms tucked and stowed. After the vehiclessettles to a hover, the arms are deployed downand fully extended (Fig. 8), thus increasing themoments of inertia. This change in the momentof inertia tries to destabilize the system and thusproduces undesired oscillations in the roll anglecontrol loop. The oscillations trigger the MRACthat changes the overall control loop gain, andtherefore stabilizes the system. According to thestability criteria (8), the adaptive gain ζ needs toincrease the derivative gain KD to account forthe rise in J. Figure 9 shows how the adaptivegain ζ changes throughout the simulation, andFig. 10 shows the system response and the pro-duced forces and torques.

4.3 Hybrid Automaton

To better utilize the MRAC and Gain Schedulingadaptive capabilities, a hybrid system based on aswitching automaton shown in Fig. 11 is proposed.The hybrid system defines four phases of aerialmanipulation missions: Flight phase, Arm De-ployment phase, Manipulation phase and Adap-tation phase. The system starts in the flight phase,where the MM-UAV flies to the designated point.Once it arrives to the set point (i.e. ‖x− xref‖ ≤�x), and before switching to the manipulationstage, the aircraft repositions the arms for themanipulation. This is called the arm deploymentphase. During this phase, gain scheduling algo-rithm is crucial for keeping the system stable.After the arms are deployed, the UAV switchesto the adaptation phase and starts the self induced

AdaptationPhase

Flight Phase

ManipulationPhase

DESTINATION REACHED

OBJECT UNREACHABLE

UNDESIRED OSCILLATIONS

ArmDeployment

Phase

MANIPULATION PENDING

ARMSDEPLOYED

ADAPTATION DONE

MANIPULATION DONE

MANIPULATION DONE

READY TO FLY

Fig. 11 Hybrid system automaton

oscillations in order to fine tune the controllerusing model reference adaptation. Once ready,the MM-UAV starts the manipulation phase ofthe mission. The aircraft can leave the adaptationphase once

∥∥ δζ

δt

∥∥ ≤ �ζ .During the manipulation phase, the shift in the

center of mass or the change of moment of inertiaproduced from interaction with the environmentor from picking up an object, could drive the ve-hicle unstable. Therefore, the vehicle is allowed toswitch back to Adaptation phase if such a problemoccurs. After successfully picking up an object andbefore flying back to the base, the air robot onceagain changes to the Adaptation phase. Becausethe load changed the center of mass and momentof inertia of the body, it is necessary to fine tunethe controller once again, before flying off. Oncetuning is complete, the aircraft switches back toflight phase and flies back to the base.

5 Conclusions

This paper analyzes the potential use of adaptivecontrol techniques (i.e. Model Reference Adap-tive Control and Gain Scheduling) to controla multi-arm manipulating aerial vehicle imple-mented on a small, off-the-shelf quadrotor. Pre-vious experimental results proved that such a ve-hicle is not necessarily stable for all the possiblechanges in the moment of inertia. Therefore, the


additional adaptive control loops for attitude sta-bilization are proposed. Multiple gain schedulingcontrol algorithms are proposed and analyticallyverified. Model Reference Adaptive Controller isproven to be stable using the Lyapunov stabil-ity theory. Furthermore, the controller is verifiedin the simulation environment, where the resultsproved that the MRAC is capable of stabilizingthe oscillations produced from the unstable PI-Dattitude control loop.

Finally a high level control system is proposedin order to ensure the safety of the aerial ma-nipulation missions. It is based on a switchingautomaton with four distinct mission phases. Ineach phase, single or multiple adaptive controlalgorithms are supposed to maintain the overallaircraft stability.

In the future, the MRAC controller will beimplemented and tested on the experimental plat-form. Also, other adaptive and robust controltechniques should be tested both in simulationand experiment. Using adaptive and robust con-trol greater flight stability should be achieved,which would enable fully dexterous manipulation.

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Hybrid Adaptive Control for Aerial Manipulation