Dynamic Stability and Control of a Manipulating Unmanned Aerial … · 2017. 11. 16. · space of a continuous dynamical system [22]. The concept of Lyapunov exponents provides a

Research ArticleDynamic Stability and Control of a Manipulating UnmannedAerial Vehicle

Yunping Liu , Xijie Huang, Yonghong Zhang, and Yukang Zhou

Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET), Nanjing University ofInformation Science & Technology, Nanjing 210044, China

Correspondence should be addressed to Yunping Liu; [email protected]

Received 16 November 2017; Accepted 9 May 2018; Published 12 June 2018

Academic Editor: Paul Williams

Copyright © 2018 Yunping Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper focuses on the dynamic stability analysis of a manipulator mounted on a quadrotor unmanned aerial vehicle, namely, amanipulating unmanned aerial vehicle (MUAV). Manipulator movements and environments interaction will extremely affect thedynamic stability of the MUAV system. So the dynamic stability analysis of the MUAV system is of paramount importance forsafety and satisfactory performance. However, the applications of Lyapunov’s stability theory to the MUAV system have beenextremely limited, due to the lack of a constructive method available for deriving a Lyapunov function. Thus, Lyapunov exponentmethod and impedance control are introduced, and the Lyapunov exponent method can establish the quantitative relationshipsbetween the manipulator movements and the dynamics stability, while impedance control can reduce the impact of environmentalinteraction on system stability. Numerical simulation results have demonstrated the effectiveness of the proposed method.

1. Introduction

Unmanned aerial vehicles (UAVs), especially the quadrotors,have developed rapidly and beenwidely applied inmanyways[1], for example, aerial photography, power line patrol, andsurveillance [2]. However, these applications mentioned allavoid interacting with the environments. For someenvironmental interaction tasks, the active operation ofUAVsendowed with manipulator will greatly expand their range ofapplications, while it brings great challenges in dynamic sta-bility for the whole system. As is well known, UAVs havemany stability problems, such as shaking, out of control, andeven crash, and if endowedwithmanipulators, UAVs stabilitywill additionally be affected by manipulator movements andenvironment contacts. So dynamic stability researches ofUAVs especially MUAVs are very necessary and importantto ensure that the whole system is safe and reliable.

The field of MUAVs is still largely an immature one, but ithas received great concern and some recent attempts have beenmade. Earlier studies focused on the installation of a lightweightgripper at the belly of the aircraft [3–5]. The gripper has lessimpact on the UAV, but the operating ability is very limited,

so the later research mainly focused on changing the gripperdevice into a manipulator [6–9]. Manipulator, especiallyredundant manipulator, has flexible operation [10, 11]. How-ever, manipulator movements and the environment contactwill change the center of gravity position and reduce systemstability [12]. On theoretical research, Euler Lagrange [13–15] and Newton-Euler [16–18] methods have been widelyused to establish the dynamics model. But complex compu-tation always brings great difficulties to model derivation.

As for nonlinear system stability analysis, the most com-monly used method is Lyapunov’s stability theory [19–21]. Itplays a very important role in the system stability proof andthe controller design, and Lyapunov functions are neededwhen using this method. However, there is no constructivemethod available for deriving the functions, so it is quitedifficult to establish the Lyapunov functions especially forthe nonlinear system of MUAV with multiple variables,strong coupling, and underactuation. To solve this problem,Lyapunov exponent method has been introduced and widelyused. Lyapunov exponents are the average exponential ratesof divergence or convergence of nearby orbits in the statespace, and they are related to the expanding or contracting

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 3481328, 13 pageshttps://doi.org/10.1155/2018/3481328

nature of different directions in the n-dimensional phasespace of a continuous dynamical system [22]. The conceptof Lyapunov exponents provides a generalization of thelinear stability analysis for perturbations of steady-statesolutions to time-dependent solutions [23, 24]. Yang andWu and Sun and Wu [25, 26] applied Lyapunov exponentmethod to study the dynamic stability of bipedal robotsduring disturbed standing. Abdulwahab et al. [27] usedLyapunov exponent method to analyze the lateral flightstability of unmanned aerial vehicles. In our previous work[28], quantitative relationship between the structural param-eters of a quadrotor and the dynamic stability was establishedbased on the Lyapunov exponent. By optimizing the differentstructural parameters, we improved the quadrotor dynamicstability. Since MUAVs usually need to interact with theenvironment, it is necessary to take some measures to ensurethe stability of the system and avoid damage to the devices.Bartelds et al. [29] designed a compliant mechanism to allowthe aerial manipulator to remain stable during and afterimpact. Alexis et al. [30] proposed a hybrid model predictivecontrol framework to ensure the quadrotor stable and activeinteraction. In [31], impedance control was introduced todeal with the interaction between the end effector mountedon a ducted fan UAV and a vertical wall.

This paper focuses on the modeling and the dynamicstability analysis of a single DOF manipulator mounted ona quadrotor unmanned aerial vehicle. The Euler-Poincareequations are introduced to derive kinematics and dynamicsmodel of the MUAV system. As for dynamic stability, themanipulator movements and the environmental interactionsare considered separately. Lyapunov exponent method isused to analyze the stability affected by different manipulatormovements and then the Newton-Euler recursive method isused to quantify the coupling between the manipulator andthe quadrotor, after which coupling compensation is consid-ered in the controller design. In order to ensure stability andsafety when the end effector of the manipulator interacts withthe environment, an impedance controller is designed.Numerical simulations illustrate the effectiveness of thecontroller in the force-constrained space as well as the transi-tion from the free space to the force-constrained space.

The rest of this paper is organized as follows. Section 2 pre-sents the overall system model. In Section 3, dynamics stabilityand coupling characteristics of the system due to manipulatormovements are analyzed. In Section 4, the complete interactioncontrol architecture is proposed, and its effectiveness is demon-strated in Section 5 by means of numerical simulation. Finally,some conclusions are presented in Section 6.

2. System Modeling

The system considered in this paper, depicted in Figure 1, iscomposed of a quadrotor vehicle equipped with a single DOFmanipulator. One end of the manipulator is mounted at thegeometric center of the four quadrotors (assuming that thegeometric center coincides with the center of mass ofthe quadrotor).

To facilitate modeling, let ∑ E − XYZ be the inertialframe, and ∑b B − xyz is the UAV body frame with origin

at the vehicle center of mass. ∑m M − x′y′z′ is a manipula-tor frame, whose origin coincides with ∑b, and the z′ axis isin the direction of the reverse extension line of the manipula-tor. The configuration of the MUAV system can be given by

q = ηuT puT aT , 1

where ηu = ψ θ ϕ T denotes the quadrotor attitude, andvector pu = x y z T is the position of the quadrotor withrespect to ∑, and a is the joint angle of the manipulator. It isobvious that q is the generalized coordinate of the MUAVsystem. Relative to q, the pseudovelocity vector of the systemcan be given by

p = ωuT vuT b

T , 2

where vu = u v w T and ωu = p q r T , respectively,denote linear velocity and angular velocity component ofthe quadrotor in ∑b, and b is the joint angular velocity ofthe manipulator.

The position of the end effector in ∑ is represented as

pe = pu + Rupbe, 3

whereRu is the rotationmatrix of the∑b relative to∑. pbe is theposition of the end effector in∑b and pbe = xbe ybe zbe

T .Some symbols and descriptions are as follows:

m1: Mass of the quadrotor vehicle

m2: Mass of the manipulator

g: Local gravity acceleration

L: Length of the quadrotor from the center of mass torotor axis

Rq: Rotor radius

R: Length of the manipulator from the root to the centerof mass

F4

F1

F3

F2x

y

z

�휃

�휓

B

E

Z

XY

L

�휙

x′

y′

z′

M

Figure 1: Quadrotor and manipulator system with thecorresponding frames.

2 International Journal of Aerospace Engineering

S: Length of the manipulator

CQ: Coefficient of the rotor torque

CT: Rotor tensile force coefficient

ρ: Local air density

Ix, Iy, and Iz: Quadrotor moment of inertia of axis x, y,and z

Jy: Moment of inertia of the manipulator link about itscenter of mass

wi: ith rotor speed.

The MUAV system is composed of two single rigidbodies, the quadrotor, and the manipulator. The quadrotorhas 6 degrees of freedom, and the manipulator has 1degree of freedom rotating around the y-axis of ∑b. Thekinematics and dynamics equations of MUAV based onEuler-Poincare equations (5) can be derived with the Mathe-matica software package TSI ProPac [32]. The modelingprocess is as follows:

(i) Define joint data:

r1 = {6}; H1= IdentityMatrix [6];

q1= {ϕ, θ, ψ, x, y, z}; p1= {p, q, r, u, v, w};

r2 = {1}; H2= {{0}, {1}, {0}, {0}, {0}, {0}};

q2= {a}; p2= {b};

JointLst = {{r1, H1, q1, p1}, {r2, H2, q2, p2}};

(ii) Define body data:

com1= {0, 0, 0}; mass1 =m1; out1 = {2, {0, 0, 0}};

Inertia1 = {{Ix, 0, 0}, {0, Iy, 0}, {0, 0, Iz}};

com2= {0, 0, −R}; mass2=m2; out2 = {3, {0, 0, -S}};

Inertia2 = {{0, 0, 0}, {0, Jy, 0}, {0, 0, 0}};

BodyLst = {{com1, {out1}, mass1, Inertia1}, {com2,{out2}, mass2, Inertia2}};

(iii) Define interconnection structure:

TreeLst = {{{1, 1}, {2, 2}}};

(iv) Define potential energy and nonconservative gener-alized input force/torque:

PE=0;

T1=U2; T2=U3; T3=U4;

Q= {T1, T2, T3, 0, 0, U1, T4};

(v) Modeling.

{JV, JX, JH, MM, Cp, Fp, pp, qq} =CreateModel[JointLst, BodyLst, TreeLst, g, PE, Q, JV, JX, JH];

Equations =MakeODEs [pp, qq, JV, MM, Cp, Fp, t]

where U1, U2, U3, and U4 are the control inputs of verticalheight, roll angle, pitching angle, and yawing angle, respec-tively, and

U1 = F1 + F2 + F3 + F4,

U2 = F2 − F4,

U3 = F3 − F1,

U4 = F1 − F2 + F3 − F4,

4

where Fi i = 1,… , 4 is the lift of the ith rotor. If wedefine A = πRq

2, then the thrust of each rotor can be

expressed as Fi = 1/2ρACTRq2wi

2. The main matrix parame-ters can be obtained through the above symbolic calculation.With these matrix parameters, the MUAV system model canbe finally obtained in the form of

q =V q p,M q p +C q, p p + F p, q, u = 0,

5

whereV q ,M q , and C q, p are kinematics matrix, inertiamatrix, and gyroscopic matrix, respectively. F p, q, u is theviscous friction and gravity vector.

3. Dynamic Stability Analysis

The quadrotor, as the base of the manipulator, is differentfrom the traditional fixed base and it is essentially unstable.However, a stable base is really required when the manipula-tor performs some operation. Thus, the dynamic analysis of asystem is quite important. Considering the limitations of theLyapunov’s stability theory for complicated nonlinear sys-tems, in this section, Lyapunov exponent method is appliedto analyze the dynamic stability of the MUAV system inthe case of manipulator movements. As is well known, whenit comes to stability analysis, Lyapunov exponents are usuallycompared with 0, and some relationships between Lyapunovexponents and system stability can be concluded. If allLyapunov exponents are less than zero, the system isexponentially stable about the equilibrium point, whichindicates that the system is dissipative or nonconservative.On the contrary, when at least one Lyapunov exponent isgreater than zero; the system tends to be unstable or chaotic.And if the value of Lyapunov exponents is fixed at zero, thephase trajectory is periodic with time [33]. Lyapunov expo-nent can be applied to establish the quantitative relationshipsbetween the manipulator movements and the MUAVdynamic stability. The Lyapunov exponent calculation isbased on the established mathematical model of the MUAV.Different manipulators movements lead to the kinematicsand dynamics model of the MUAV different, which indicatesthat the corresponding Lyapunov exponent may change.Lyapunov exponent can be calculated as

λ = limn→∞

1n〠n−1

i=0ln

df XdX Xi

, 6

3International Journal of Aerospace Engineering

where df X /dX is Jacoby matrix of the system, and n is thenumber of iterations. In order to compute the Jacoby matrix,(5) needs to be translated into

q =V q p,p = −M−1 C p + F

7

Let

X = q, p T = ϕ, θ, ψ, x, y, z, a, p, q, r, u, v,w, b T , 8

and (7) can be reformulated as

X = f X, u , 9

where u is control input vector. Figure 2 displays the specificcalculation process.

When designing manipulator controller with differentexpected manipulator joint angle, the joint torque T4 in step(iv) of the MUAV dynamic modeling changes, which makesthe dynamic model changes as well. Then, the calculatedLyapunov exponents will also be different based on Figure 2and after n iteration, the trajectory distance between the dis-turbed MUAV system and the original system is as follows:

limn→∞

ε ⋅ enλ x0 = limn→∞

f n x0 + ε − f n x0 10

It can be known that if λ > 0, limn→∞ε ⋅ enλ →∞will tendto infinity, which shows that the disturbed trajectory and theoriginal trajectory of the system is separated with time and thesystem is chaotic, and if λ < 0, limn→∞ε ⋅ enλ → 0 will tend tozero, which illustrates that after disturbed, the systemcan reach the original system state again and the systemis insensitive to the disturbance and stable.

According to the method described above, when thequadrotor is hovering, Lyapunov exponents of the systemare calculated separately with different manipulator move-ments. When setting the number of iterations to 100and the initial state q, p T of the MUAV system to0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 T , the Lyapunov exponentsof the system can be obtained. For a clear analysis, Lyapu-nov exponents of the quadrotor attitudes and manipulatorjoint angle are shown in Figures 3 and 4. When the desiredjoint angle ad of the manipulator is 0, that is, the quadrotorand the manipulator are relative stationary, the Lyapunovexponents spectrum is shown in Figure 3. Similarly, setad = sina, then the Lyapunov exponents spectrum of thequadrotor attitudes and manipulator joint angle are shownin Figure 4.

From Figures 3 and 4, we can see that the Lyapunovexponents of Figure 3 are less than 0 or tend to be 0, whilein Figure 4, there is one exponent that is greater than 200with no tendency to 0. In comparison with these two figures,when there is relative motion between the manipulator andthe quadrotor, the dynamic stability of the whole system isobviously getting worse; in severe cases, the whole systemmay get unstable.

Jacobi matrix computationdf/dX

Initial condition

End

8

�휆 = lim − ln|df (X) / dX|

Dynamics modelequation (3)

Xi

Xi

Next initialcondition

n−1

n→ i=0n

1

Figure 2: Calculation process of the Lyapunov exponent.

20 40 60 80 100

t (s)

−2

−1

LE

�휆 (a)�휆 (�휓)

�휆 (�휃)�휆 (�휙)

1

2

3

4

5

Figure 3: Lyapunov exponents spectrum of the MUAV (ad = 0).

t (s)

LE2

20 40 60 80 100−1

12345

�휆 (�휙)�휆 (�휃)�휆 (�휓)�휆 (a)

40 60 80 10020

50

100

150

200

Figure 4: Lyapunov exponents spectrum of the MUAV (ad =sina).

4 International Journal of Aerospace Engineering

From Figures 3 and 4, it can be seen that the Lyapunovexponents of the corresponding roll angle and yaw angleare almost the same while the Lyapunov exponents of thecorresponding pitch angle and manipulator joint angle differ.For a clearer analysis, the Lyapunov exponents of pitchangles and joint angles are shown in Figures 5 and 6.

Figure 5 shows that the Lyapunov exponent of the pitchangle in the condition of ad = sina is much larger than thatof ad = 0, which indicates that the dynamic stability getsworse when the manipulator movements exist and themanipulator movements will affect the quadrotor pitchingmotion, and even may lead to the instability.

According to Figure 6, the joint angle Lyapunov exponentis less than 0 when ad=0. However, the exponent changeswhen ad= sina, that is, relative motion between the robotarm and the quadrotor presents, which indicates that manip-ulator motion, in turn, is affected by the quadrotor motion.

From Figures 5 and 6, it can be concluded that thequadrotor pitching motion and the manipulator motion arecoupled. This coupling characteristic will have very bad effectson the subsequent control and practical applications, so it is

necessary to analyze and quantify the coupling and then inthe controller design, coupling, and compensation is needed.

Similar to the space manipulator [34], Newton-Eulerrecursive method [35] is used to quantify the coupling.Suppose that the quadrotor is the root of the manipulator,which can be thought link 0 and from the root to the endeffector, the linear velocity, angular velocity, and accelerationof each link can be calculated. With velocity and acceleration,the force and torque on each link are derived backwards.Finally, the force applied on the quadrotor due to themanipulator movements can be calculated, and the deriva-tion process is as follows.

For simplicity, some symbols are defined. Let jωi andjvi denote the angular velocity and the linear velocity oflink i with respect to the link frame ∑j. αi denotes thejoint angle of link i. αi represents the derivative of time.jiR is the rotation matrix of the ∑i relative to ∑j.

jpi is

the position of the ith joint in ∑j, andjhi represents the

axis of rotation of link i in ∑j.From the root to the end effector, the angular velocity and

linear velocity of each link at the origin of link frame can becalculated as follows:

k 1ωk+1 = k 1kRkωk + αk+1

k 1hk+1,k 1vk+1 = k 1

kR kvk + kωk × kpk+1 ,11

and on the time derivative, the corresponding accelerationcan be obtained.

k 1ωk+1 = k 1kRkωk + k 1

kRkωk + αk+1k 1hk+1

= k 1kRkωk + αk+1

k 1hk+1 × k 1k Rkωk

+ αk+1k 1hk+1,

k 1vk+1= k 1

kRkvk + kωk × kpk+1 + kωk × kpk+1

= k 1kR

kvk + kωk × kpk+1 + kωk ×kωk ×

kpk+112

t (s)

ad = 0ad = sina

40 60 80 10020

50

100

150

200

�휆 (�휃

)

Figure 5: Lyapunov exponents of the pitch angles under twoconditions.

20 40 60 80 100

−2

t (s)

ad = 0ad = sina

2

4

6

�휆 (a

)

Figure 6: Lyapunov exponents of the joint angles under twoconditions.

10 20 30 40 50

t

�휃

A�erBefore

−0.0010

−0.0005

0.0005

0.0010

Figure 7: Pitch angle simulation with/without couplingcompensation.

5International Journal of Aerospace Engineering

Let subscript c represent the center of mass of the link.Then, the linear velocity and acceleration of the link at thecenter of mass can be calculated as

k 1vck+1 =k 1vk+1 + k 1ωk+1 × k 1pck+1 ,

k 1vck+1 =k 1vk+1 + k 1ωk+1 × k 1pck+1 +

k 1ωk+1

× k 1ωk+1 × k 1pck+1

13

With the above velocity and acceleration, the general-ized force F and torque N on each link at the center of

mass can be obtained from the end effector to the rootas follows:

k 1Fk+1 =mk+1k 1vck+1 ,

k 1Nk+1 =ck 1Ik 1

k 1ωk+1 + k 1ωk+1 ×ck 1Ik 1

k 1ωk+1 ,

14

where mk+1 is the mass of link k+1 and ck+1Ik+1 is themoment of inertia about the center of mass of link k+1.Then, the generalized force f and torque n on each link atthe origin of link frame can be obtained as

where n is the number of the links.The force and torque acting on the joint 1 can be

obtained by recursive calculation, and they will be transmit-ted to the quadrotor. Thus, considering only the generalizedforce and torque exerted by the manipulator, the couplingbetween the manipulator and the quadrotor can be finallyquantified as

0F0 = 01R1 f1,

0N0 = 0p1 × 01R1 f1 + 0

1R1n116

In order to verify the quantized coupling, firstly, simplePID controllers are designed to control attitude and heightof quadrotor and joint angle of the manipulator. Then, PIDcontrollers with regard to coupling compensation are alsoapplied to control the quadrotor and the manipulator.For the two cases, the pitch angle simulation is presentedin Figure 7.

From Figure 7, the coupling exists both with andwithout feed-forward compensation. However, after com-pensation, the amplitude of the pitch angle oscillation issignificantly decreased. Therefore, quantized coupling, whenapplied to control compensation, can play a certain role incontroller design.

4. Control Scheme

Compared with a conventional quadrotor, the aerial manip-ulator has the advantage of maneuverability. It can interactwith the environments actively and perform some tasks.However, if the interaction force is too large, it is very likelyto damage both the end effector and the objects. Also, thesystem may get unstable during the transition from free

motion to interaction. Thus, it is necessary to control theinteraction force effectively. In this section, impedance con-trol is adopted to solve the problem. Impedance controlshows the dynamic relationship between interaction forceand position [36]. In the Cartesian space, the desiredimpedance model of the end effector can be given by

Md Xd −X + Bd Xd −X +Kd Xd −X = Fe, 17

where Md, Bd, and Kd are the constant matrices of desiredinertia, damping coefficient, and spring stiffness of the endeffector. Vectors X and Xd are the actual and desired endeffector position, respectively, and Fe represents the general-ized force exerted upon the end effector by the environment.By modeling interaction surface of the environment as aspring, the force Fe can be calculated as

Fe =ke Xenv −X if X >Xenv,

0 if X ≤Xenv,18

where ke is the stiffness of the environment while Xenv is theposition of the interaction surface. When the interactionforce arises, the corresponding torque applied to the manip-ulator is shown by

Te = JTFe, 19

where J is the Jacoby matrix of the manipulator. Lete =Xd −X represent the position error of the manipula-tor end effector and then convert (17) into frequencydomain, it can be obtained as

Fe se s

=Mds2 + Bds +Kd 20

k fk =kFk + k

k 1Rk 1 fk+1 k < n ,kFk k = n ,

knk =kNk + k

k 1Rk 1nk+1 + kpck ×kFk + kpk+1 × k

k 1Rk 1 fk+1 k < n

kNk + kpck ×kFk k = n ,

15

6 International Journal of Aerospace Engineering

Equation (20) can be changed into

e s =F s

Mds2 + Bds +Kd, 21

where e is the position correction of the end effector.Summing it and the given reference position, the desiredposition of the end effector can be finally obtained. Inthe joint space, a PID controller applied to obtain the torquefor the manipulator without regard to environmental inter-actions has the form:

Ta = kp ad − a + ki ad − a dt + kd ad − a , 22

where kp, ki, and kd are the proportional, integral, and deriv-ative control coefficients, respectively. ad obtained throughthe inverse kinematics and a are the desired and actual joint

angles of the manipulator. If considering interactions, theactual torque can be given by

T = Ta − Te 23

As is shown in the impedance control block diagram inFigure 8, for the manipulator, a position-based impedancecontrol (PBIC) is introduced and it includes a double controlloop. The inner loop is position control and the outer loop isimpedance control. According to the actual interaction forceand desired impedance parameters, the outer loop generatesposition correction as represented in (21). When giving thereference position per, attitude ηer and joint angle ar of theend effector as well as the feedback roll and pitch angle, thereference position pur, and yaw angle ψr of the quadrotorcan be obtained via a kinematics calculation algorithm pro-posed by [37]. Then with reference and the feedback positionpu and attitude η of the quadrotor, the quadrotor controllercan be designed.

Kinematicsand

dynamicsmodel ofMUAV

ΣJoint angle

controlInverse

kinematics

Forwardkinematics

Impedancemodel

Environmentmodel

Kinematicscalculation

Quadrotorcontrol

JT �훴−

pu �훈

Te T

Taper

ped

pe a

penv

Fe ade

ar per �훈er pur

�휙 �휃

�휓ur U

Figure 8: The impedance control block diagram.

Horizontalposition

controller

Attitudecalculation

Heightcontroller

Liftcalculation

Attitudecontroller

pur

x �휓r

�휙r �휃r

�훈

z

zr U

U1

U1

U1 U3 U4

y

x..

z..

..yxr yr

Figure 9: The quadrotor control block diagram.

7International Journal of Aerospace Engineering

For the quadrotor control shown in Figure 8, we candetail it as Figure 9 and it shows that a double PID controlloop is also introduced. Since a quadrotor is underactuated,the horizontal position cannot be controlled directly withinput forces. However, relationships between the horizontalacceleration and the reference attitude angles can be givenby (24), where x, y denote the acceleration in x and y direc-tions and subscript r represents the meaning of reference.Then, the double PID control loop can be designed withthe inner loop being UAV attitude controller and the outerloop being UAV position controller, which ensures bothposition and attitude of the quadrotor reach expected states.When manipulator moves or makes contact with the envi-ronment, the controller also ensures UAV stability and evenspot hover.

ϕr = arcsinm1 +m2

U1x sin ψr − y cos ψr ,

θr = arcsinm1 +m2/U1 x − sin ϕr sin ψr

cos ϕrcos ψr

24

The impedance control mentioned above enables the endeffector to track the desired trajectory. However, sometimes,the end effector needs to exert a specified force on the envi-ronment, which indicates that the end effector should havethe ability to track the desired force. This can implementbased on (17) and a force error rather than the interactionforce is applied to the impedance model. Then, the desiredimpedance model and the position correction of the endeffector can be reformulated as

Md Xd −X + Bd Xd −X +Kd Xd −X = Fd − Fe,

e s =E s

Mds2 + Bds +Kd,

25

where Fd is the desired force to track and E = Fd − Fe. Similarto the desired trajectory tracking, the desired force controllercan be designed by replacing Fe into E.

5. Simulation

In order to validate the effectiveness of the controller, severalsimulations have been carried out in the MATLAB/Simulinkenvironment. The purpose of the simulation is that when thequadrotor hovers at a specified point, the manipulator makescontact with a surface. Two cases are studied, that is, thedesired trajectory tracking and the desired force tracking.For simplicity, we ignore friction and consider that interac-tion force is applied to only one direction. Reference position

Table 1: MUAV parameters.

Symbols Values

m1 0.875 kg

m2 0.106 kg

g 9.8m/s2

L 0.225m

R 0.1m

S 0.2m

CT 1.0792−005

CQ 1.8992−007

Rq 0.125m

ρ 11.69 kg/m3

Ix 9.5065−003 kg/m2

Iy 1.00−002 kg/m2

Iz 1.658−002 kg/m2

Jy 2.37−003 kg/m2

UAV position

xyz

0.9

0.95

1

1.05

1.1

1.15

Posit

ion

(m)

200 5 10 15Time (sec)

Figure 10: Position of the UAV in the inertial frame.

UAV attitude

�휙�휃�휓

10 15 200 5Time (sec)

−0.2

−0.15

−0.1

−0.05

0

0.05

0.15

0.1

Ang

le (r

ad)

Figure 11: Attitude of the UAV.

8 International Journal of Aerospace Engineering

of the manipulator is set at xer = 0.1m and a surface is locatedat xenv = 0.09m with respect to ∑b. With the given referenceposition of the end effector, reference joint angle ar of themanipulator can be calculated via the inverse kinematicsalgorithm, and the value of ar is π/6. Besides these, the stiff-ness of the environment ke is 100. Md, Bd, and Kd are 1, 50,and 625, respectively. The initial condition of the simulationis set to q= [0, 0, 0, 1, 1, 1, 0]T. Simulation adopts variablestep. The solver is ODE45 and simulation time is 20 seconds.The involved system parameters values are given in Table 1.

5.1. Desired Trajectory Tracking. Simulation results of theend effector during the transition from free motion tointeraction are shown below. The quadrotor UAV firstlyflies to a position of pu= (1, 1, 1) with respect to ∑, afterwhich it hovers over this point and then the manipulatorwith an initial joint angle value of 0 approaches the walland makes contact with it until the end effector arrivesat the desired position.

Simulation results of the UAV are shown in Figures 10–12. When the quadrotor hovers at the point pu, the manipu-lator operates, which results in a quadrotor oscillation. Thequadrotor tends to stabilize after the controller works, butthe coupling caused by the manipulator leads to a slight tiltof the quadrotor in the pitch angle (shown in Figure 11)and the pitch control torque U3 (shown in Figure 12) is nolong 0. The tilt can allow the UAV to generate a force to offsetthe contact force. Figure 10 shows that the UAV translationin the x-axis is obviously affected due to the influence of theUAV pitch motion. The manipulator movements and inter-action between the end effector and the surface affect the

ability of the system, and the control force or torques changeso that the UAV can return to a stable state quickly.

Figures 13–16 shows the simulation of the end effector.From 0 to about 0.4 seconds, the manipulator moves in thefree space, and the interaction force is 0 (shown inFigure 15), which results in a position correction value of 0and the reference position is equal to the desired positionduring this time period (shown in Figure 13). After 0.4

9.3

9.4

9.5

9.6

9.7

9.8

U1 (

N)

5 10 15 200Time (sec)

(a)

−3

−2

−1

0

1

2

3

U2 (

N.m

)

5 10 15 200Time (sec)

(b)

−1.5

−1

−0.5

0

0.5

1

1.5

U3 (

N.m

)

5 10 15 200Time (sec)

(c)

−0.02

−0.01

0

0.01

0.02

0.03

U4 (

N.m

)

5 10 15 200Time (sec)

(d)

Figure 12: Thrust of the UAV.

End effector position

PedPe

0 0.5 10.09

0.095

0.1

0.105

0

0.05

0.1

0.15

Posit

ion

(m)

1 2 3 4 5 60Time (sec)

Figure 13: Desired and actual position of the end effector in thebody frame.

9International Journal of Aerospace Engineering

seconds, the end effector contacts the surface, and the inter-action force is no longer 0. Figure 15 shows that when theend effector makes contact with the surface at a certain speed,the interaction may keep growing due to inertia, then theimpedance controller works so that the interaction forceand the end effector position meet the desired dynamic rela-tionship. Figure 13 shows that the desired position of the endeffector is less than the given reference value 0.1, which canbe explained by the fact that when the desired dynamic rela-tionship mentioned above meets the requirements, the endeffector reaches a proper position to avoid an excessivecontact force. This is why the impedance control can reducethe damage of the end effector and the interactive objects.

5.2. Desired Force Tracking. The desired force Fd is 1N, andother simulation conditions are the same as the desired tra-jectory tracking simulation. Simulation results of the UAVare shown in Figures 17–23. Similar to the desired trajectorytracking, the UAV hovers at a position of pu= (1, 1, 1) withrespect to ∑. Figure 17 shows that the UAV flies in a

neighborhood of point pu and after about 13 seconds, theUAV tends to hovers although there is still a tiny error. Com-pared with Figure 11, Figure 18 reflects a better convergence

Interaction force

−0.5

0

0.5

1

1.5

2

2.5

F e (N

)

1 2 3 4 5 60Time (sec)

Figure 15: Interaction force between the end effector and thesurface.

Manipulator control torque

0 0.5 1−0.5

00.5

1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

T (N

.m)

1 2 3 4 5 60Time (sec)

Figure 16: Joint torque for the manipulator.

UAV position

xyz

0.9

0.95

1

1.05

1.1Po

sitio

n (m

)

5 10 15 200Time (sec)

Figure 17: Position of the UAV in the inertial frame.

UAV attitude

�휙�휃�휓

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Ang

le (r

ad)

5 10 15 200Time (sec)

Figure 18: Attitude of the UAV.

Joint angle

ada

0 0.5 10.5

0.52

0.54

0

0.2

0.4

0.6

0.8

Ang

le (r

ad)

1 2 3 4 5 60Time (sec)

Figure 14: Desired and actual joint angle of the manipulator.

10 International Journal of Aerospace Engineering

9.2

9.4

9.6

9.8

U1 (

N)

5 10 15 200Time (sec)

(a)

−4

−2

0

2

4

U2 (

N.m

)

5 10 15 200Time (sec)

(b)

5 10 15 200Time (sec)

−0.5

0

0.5

1

U3 (

N.m

)

(c)

−0.05

0

0.05

U4 (

N.m

)

5 10 15 200Time (sec)

(d)

Figure 19: Thrust of the UAV.

End effector position

pedpe

0 0.5 10.0960.098

0.10.1020.104

0

0.05

0.1

0.15

Posit

ion

(m)

1 2 3 4 5 60Time (sec)

Figure 20: Desired and actual position of the end effector in thebody frame.

Joint angle

ada

0 0.5 10.5

0.55

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ang

le (r

ad)

1 2 3 4 5 60Time (sec)

Figure 21: Desired and actual joint angle of the manipulator.

Interaction force

FdF

−0.5

0

0.5

1

1.5

2

2.5

3

Forc

e (N

)

1 2 3 4 5 60Time (sec)

Figure 22: Interaction force between the end effector and thesurface.

Manipulator control torque

0 0.5 1−0.5

0

0.5

0

1

2

3

4

T (N

.m)

1 2 3 4 5 60Time (sec)

Figure 23: Joint torque for the manipulator.

11International Journal of Aerospace Engineering

effect especially the case of pitch angle and it reaches 0despite the presence of interaction between the end effectorand the environment. This is mainly because the givendesired force Fd, which may be thought to offset the interac-tion force. Both simulations of the desired trajectory trackingand the force tracking indicate that the manipulator move-ments and the environment contacts have an impact on theUAV stability; however, the height and yaw motion suffera relatively little impact. From Figure 19, due to themanipulator influence, it is can be seen that the UAVpitch control torque U3 is not 0 although the pitch anglecan reach 0 after the controller works.

Figures 20–23 shows the simulation results of the endeffector. From 0 to about 0.4 seconds, the manipulator movesin the free space, and the interaction force is 0, as shown inFigure 22. However, different from Figures 13 and 14,Figures 20 and 21 show that the desired position and jointangle of the manipulator changes during the movements inthe free space. This can be explained by the desired imped-ance model (25), and position correction is not 0 due to thegiven desired force. After about 0.4 seconds, the controllerworks, and it tracks the desired force perfectly and the track-ing error is 0, which results in a position correction value of 0and the reference position is equal to the desired position.The joint control torque slightly increases to make the endeffector reach the surface. Different from Figure 16, the jointtorque decreases as the interaction force increases, as shownin Figure 23 and when the system gets stable, the torqueremains a fixed value.

6. Conclusion

This paper has presented the model and the dynamic stabilityanalysis of a manipulating unmanned aerial vehicle, andnumerical simulation is used to test the work. The systemmodeling based on Euler-Poincare equation is simple andefficient by means of computer symbol derivation. Lyapunovexponent method used to analyze the dynamic stability justconsidering the manipulator movements has shown obviousadvantages. It is constructive and simple for the exponentcalculation when compared with Lyapunov’s stability theory.Simulation results illustrate that the coupling between themanipulator and the quadrotor will affect the dynamic sta-bility of the MUAV, and the coupling compensation canreduce the influence. As for interaction dynamic stability,the proposed impedance controller is shown to be ableto stabilize the system both in the free flight and in thepresence of interaction.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank the anonymous reviewers for helpful andinsightful remarks. Helpful discussions with Professor WuQiong on her guidance in Lyapunov exponent theory aregratefully acknowledged. This research is supported by the

National Natural Science Foundation of China (51405243and 51575283).

References

[1] K. Nonami, “Prospect and recent research & developmentfor civil use autonomous unmanned aircraft as UAV andMAV,” Journal of System Design and Dynamics, vol. 1,no. 2, pp. 120–128, 2007.

[2] K. Herrick, “Development of the unmanned aerial vehiclemarket: forecasts and trends,” Air & Space Europe, vol. 2,no. 2, pp. 25–27, 2000.

[3] V. Ghadiok, J. Goldin, and W. Ren, “Autonomous indooraerial gripping using a quadrotor,” in 2011 IEEE/RSJ Interna-tional Conference on Intelligent Robots and Systems (IROS),pp. 4645–4651, San Francisco, CA, USA, September 2011.

[4] D. Mellinger, Q. Lindsey, M. Shomin, and V. Kumar, “Design,modeling, estimation and control for aerial grasping andmanipulation,” in 2011 IEEE/RSJ International Conference onIntelligent Robots and Systems (IROS), pp. 2668–2673, SanFrancisco, CA, USA, September 2011.

[5] P. E. I. Pounds, D. R. Bersak, and A. M. Dollar, “Grasping fromthe air: hovering capture and load stability,” in 2011 IEEEInternational Conference on Robotics and Automation (ICRA),pp. 2491–2498, Shanghai, China, May 2011.

[6] S. Kim, H. Seo, S. Choi, and H. J. Kim, “Vision-guidedaerial manipulation using a multirotor with a robotic arm,”IEEE/ASME Transactions on Mechatronics, vol. 21, no. 4,pp. 1912–1923, 2016.

[7] R. Mebarki and V. Lippiello, “Image-based control for aerialmanipulation,” Asian Journal of Control, vol. 16, no. 3,pp. 646–656, 2014.

[8] T. W. Danko and P. Y. Oh, “Design and control of a hyper-redundant manipulator for mobile manipulating unmannedaerial vehicles,” Journal of Intelligent & Robotic Systems,vol. 73, no. 1-4, pp. 709–723, 2014.

[9] J. Mendoza-Mendoza, G. Sepulveda-Cervantes, C. Aguilar-Ibanez et al., “Air-arm: a new kind of flying manipulator,” in2015 Workshop on Research, Education and Development ofUnmanned Aerial Systems (RED-UAS), pp. 278–287, Cancun,Mexico, November 2015.

[10] V. Lippiello and F. Ruggiero, “Exploiting redundancy inCartesian impedance control of UAVs equipped with a roboticarm,” in 2012 IEEE/RSJ International Conference on IntelligentRobots and Systems (IROS), pp. 3768–3773, Vilamoura,Portugal, October 2012.

[11] K. Kondak, F. Huber, M. Schwarzbach et al., “Aerialmanipulation robot composed of an autonomous helicop-ter and a 7 degrees of freedom industrial manipulator,”in 2014 IEEE International Conference on Robotics andAutomation (ICRA), pp. 2107–2112, Hong Kong, China,May-June 2014.

[12] P. E. I. Pounds, D. R. Bersak, and A. M. Dollar, “Stability ofsmall-scale UAV helicopters and quadrotors with addedpayload mass under PID control,” Autonomous Robots,vol. 33, no. 1-2, pp. 129–142, 2012.

[13] B. Yang, Y. He, J. Han, and G. Liu, “Modeling and control ofrotor-flying multi-joint manipulator,” IFAC ProceedingsVolumes, vol. 47, no. 3, pp. 11024–11029, 2014.

[14] J. Escareno, M. Rakotondrabe, G. Flores, and R. Lozano,“Rotorcraft MAV having an onboard manipulator:

12 International Journal of Aerospace Engineering

longitudinal modeling and robust control,” in 2013 Euro-pean Control Conference (ECC), pp. 258–3263, Zurich, Swit-zerland, July 2013.

[15] F. Caccavale, G. Giglio, G. Muscio, and F. Pierri, “Adaptivecontrol for UAVs equipped with a robotic arm,” IFACProceedings Volumes, vol. 47, no. 3, pp. 11049–11054, 2014.

[16] M. Fumagalli, R. Naldi, A. Macchelli, R. Carloni, S. Stramigioli,and L. Marconi, “Modeling and control of a flying robot forcontact inspection,” in 2012 IEEE/RSJ International Confer-ence on Intelligent Robots and Systems (IROS), pp. 3532–3537, Vilamoura, Portugal, October 2012.

[17] M. Orsag, C. Korpela, and P. Oh, “Modeling and control ofMM-UAV: mobile manipulating unmanned aerial vehicle,”Journal of Intelligent & Robotic Systems, vol. 69, no. 1–4,pp. 227–240, 2013.

[18] S. Kannan, M. A. Olivares-Mendez, and H. Voos, “Modelingand control of aerial manipulation vehicle with visual sensor,”IFAC Proceedings Volumes, vol. 46, no. 30, pp. 303–309, 2013.

[19] M. Orsag, C. Korpela, S. Bogdan, and P. Oh, “Lyapunov basedmodel reference adaptive control for aerial manipulation,” in2013 International Conference on Unmanned Aircraft Systems(ICUAS), pp. 966–973, Atlanta, GA, USA, May 2013.

[20] P. Xu, R. Huang, D. Lee, and T. Burg, “Dynamics and controlof a novel manipulator on VTOL aircraft (MOVA) system - aplanar case study,” in 2014 American Control Conference(ACC), pp. 3071–3076, Portland, OR, USA, June 2014.

[21] M. Kobilarov, “Nonlinear trajectory control of multi-bodyaerial manipulators,” Journal of Intelligent & Robotic Systems,vol. 73, no. 1–4, pp. 679–692, 2014.

[22] J. Awrejcewicz and G. Kudra, “Stability analysis and Lyapunovexponents of a multi-body mechanical system with rigidunilateral constraints,” Nonlinear Analysis Theory Methods &Applications, vol. 63, no. 5–7, pp. e909–e918, 2005.

[23] Y. Sun, X. Wang, Q. Wu, and N. Sepehri, “On stability analysisvia Lyapunov exponents calculated based on radial basisfunction networks,” International Journal of Control, vol. 84,no. 8, pp. 1326–1341, 2011.

[24] C. Yang, C. Q. Wu, and P. Zhang, “Estimation of Lyapunovexponents from a time series for n -dimensional state spaceusing nonlinear mapping,” Nonlinear Dynamics, vol. 69,no. 4, pp. 1493–1507, 2012.

[25] C. Yang and Q. Wu, “On stabilization of bipedal robots dur-ing disturbed standing using the concept of Lyapunov expo-nents,” in 2006 American Control Conference, pp. 621–624,Minneapolis, MN, USA, June 2006.

[26] Y. Sun and C. Q. Wu, “Stability analysis via the concept ofLyapunov exponents: a case study in optimal controlledbiped standing,” International Journal of Control, vol. 85,no. 12, pp. 1952–1966, 2012.

[27] E. N. Abdulwahab, Q. A. Atiyah, and A. T. A. Alzahra, “Air-craft lateral-directional stability in critical cases via Lyapunovexponent criterion,” Al-Khawarizmi Engineering Journal,vol. 1, no. 9, pp. 29–38, 2013.

[28] Y. Liu, C. Chen, H. Wu, Y. Zhang, and P. Mei, “Structuralstability analysis and optimization of the quadrotor unmannedaerial vehicles via the concept of Lyapunov exponents,” TheInternational Journal of Advanced Manufacturing Technology,vol. 94, no. 9–12, pp. 3217–3227, 2018.

[29] T. Bartelds, A. Capra, S. Hamaza, S. Stramigioli, andM. Fumagalli, “Compliant aerial manipulators: toward a new

generation of aerial robotic workers,” IEEE Robotics andAutomation Letters, vol. 1, no. 1, pp. 477–483, 2016.

[30] K. Alexis, G. Darivianakis, M. Burri, and R. Siegwart, “Aerialrobotic contact-based inspection: planning and control,”Autonomous Robots, vol. 40, no. 4, pp. 631–655, 2016.

[31] F. Forte, R. Naldi, A. Macchelli, and L. Marconi, “Impedancecontrol of an aerial manipulator,” in 2012 American ControlConference (ACC), pp. 3839–3844, Montreal, QC, Canada,June 2012.

[32] H. G. Kwatny and G. L. Blankenship, “Symbolic constructionof models for multibody dynamics,” IEEE Transactions onRobotics and Automation, vol. 11, no. 2, pp. 271–281, 1995.

[33] W. C. Zhang, “Chaotic forecasting of natural circulation flowinstabilities under rolling motion based on Lyapunov expo-nents,” Acta Physica Sinica, vol. 62, no. 6, pp. 675–675, 2013.

[34] F. Aghili, “Optimal control of a space manipulator for detum-bling of a target satellite,” in 2009 IEEE International Confer-ence on Robotics and Automation, pp. 362–367, Kobe, Japan,June 2009.

[35] J. Y. S. Luh, M. W.Walker, and R. P. C. Paul, “On-line compu-tational scheme for mechanical manipulators,” Journal ofDynamic Systems, Measurement, and Control, vol. 102, no. 2,pp. 69–76, 1980.

[36] N. Hogan, “Impedance control of industrial robots,” Roboticsand Computer-Integrated Manufacturing, vol. 1, no. 1,pp. 97–113, 1984.

[37] G. Arleo, F. Caccavale, G. Muscio, and F. Pierri, “Control ofquadrotor aerial vehicles equipped with a robotic arm,” in2013 21st Mediterranean Conference on Control & Automation(MED), pp. 1174–1180, Chania, Greece, June 2013.

13International Journal of Aerospace Engineering

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