mohamed fanni and ahmed khalifa - unal.edu.co

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 22, NO. 3, JUNE 2017 1315

A New 6-DOF Quadrotor Manipulation System:Design, Kinematics, Dynamics, and Control

Mohamed Fanni and Ahmed Khalifa

Abstract—The research on aerial manipulation hasincreased rapidly in recent years. In the previous work,a manipulator or a gripper is attached to the bottom of aquadrotor to facilitate the interaction with the environment.However, the previously introduced systems suffer fromeither limited end-effector degrees of freedom (DOF) orsmall payload capacity. In this paper, a quadrotor with a2-DOF manipulator that has a unique topology to enablethe end-effector to track a desired 6-DOF trajectory withminimum possible actuators is investigated. The proposedsystem is designed and modeled. However, such a systemproduces complexity in its inverse kinematics and control.A novel solution to the inverse kinematics problem ispresented, which requires a solution of complicatedalgebraic/differential equations. Its accuracy is verified vianumerical results. In order to solve the control problem, thesystem nonholonomic constraints are utilized with a model-free and low computation cost robust control technique.Moreover, the system stability proof under the proposedcontroller is carried out. A prototype of the proposed systemis built and its design is verified via a flight test. In addition,the system feasibility and efficiency are enlightened.

Index Terms—Aerial manipulation, disturbance observer,dynamic modeling, inverse kinematics, quadrotor.

NOMENCLATURE

Σb Robot body frame.Σ World-fixed inertial reference frame.pb Position of body frame with respect to Σ.Rb Rotation matrix from body frame to world frame.Φb The triple ZYX yaw-pitch-roll angles.Σe Frame attached to the end-effector.Θ Vector of joint angles of the manipulator.pbeb Position of Σe with respect to Σb expressed in Σb .pe The position of Σe with respect to Σ.χe The 6-degrees of freedom (DOF) operational coordi-

nates.

Manuscript received July 18, 2016; revised October 29, 2016 andDecember 27, 2016; accepted March 4, 2017. Date of publication March10, 2017; date of current version June 14, 2017. Recommended byTechnical Editor M. O. Efe.

M. Fanni is with the Department of Mechatronics and Robotics Engi-neering, Egypt-Japan University of Science and Technology, New BorgEl Arab 21934, Egypt, on leave from the Department of Production Engi-neering and Mechanical Design, Mansoura University, Mansoura 35516,Egypt (e-mail: [email protected]).

A. Khalifa is with the Department of Industrial Electronics and Con-trol Engineering, Faculty of Electronic Engineering, Menoufia University,Al Minufiyah 32952, Egypt (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2017.2681179

q The quadrotor/joint space coordinates.Φe Euler angles of the end-effector.ωb Angular velocity of the quadrotor expressed in Σ.ωe The angular velocity of Σe .ωbeb The angular velocity of the end-effector relative to Σb

and is expressed in Σb .vbeb Generalized velocity of end-effector w.r.t Σb .Jbeb The manipulator Jacobian.ve The generalized end-effector velocity.Tb Transformation matrix between ωb and Φb .J The system Jacobian.ζ Independent coordinate.σb The dependent coordinates.χb Quadrotor body coordinates.τmi

Torque acting on joint i.Fbm,q Manipulator forces on the quadrotor in Σb .Mb

m,q Interaction moments.Fm,q Interaction forces expressed in the inertial frame.Mi , Ni Nonlinear terms of the manipulator dynamics.m The mass of the quadrotor.τai Input moments about the three body axes.Ωj Angular velocity of rotor j; j = 1, 2, 3, 4.Fj/Mj Thrust force/Drag moment of rotor j.Kfj Thrust coefficient of rotor j.Kmj

Drag coefficient of rotor j.T The total thrust applied to the quadrotor.d Length of the quadrotor arm.Ω Collective speed of the speed of the four rotor.Ir Rotor inertia.Ii Inertia of the vehicle around its body-frame.M Inertia matrix of the combined system.C Matrix of Coriolis and centrifugal terms.G Vector of gravity terms.dex Vector of external disturbances.u Vector of the actuator inputs.B,N Input and Control matrices.H Transforms the body input forces to Σ.χe,r Reference trajectory for the end-effector pose.rij The ijth element of the rotation matrix Re .τ Vector of the input generalized forces.Mn System nominal inertia matrix.τdes Desired input to the robot.gi Low-pass filter bandwidth of the ith variable of q.P Vector of gis.Q(s) Matrix of the low-pass filter of disturbance observer

(DOb).

1083-4435 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

1316 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 22, NO. 3, JUNE 2017

τdis System extended disturbances.τdis System estimated disturbances.Mna Nominal inertia for the translational coordinates.Mnv Nominal inertia for the rotational coordinates.qdes Output from the outer loop controller.ev Error of the DOb internal loop.Lp Lp space; p = [1, ∞).HPD Exponentially stable transfer function.

I. INTRODUCTION

QUADROTOR system is one of the unmanned aerial vehi-cles (UAVs), which offer possibilities of speed and access

to regions that are otherwise inaccessible to ground robotic ve-hicles. However, most research on UAVs has typically beenlimited to monitoring and surveillance applications, where theobjectives are limited to “look” and “search” but “do not touch.”Due to their superior mobility, much interest is given to utilizethem for aerial manipulation. Previous research on aerial ma-nipulation can be divided into three approaches.

In the first approach, a gripper or a tool is installed at thebottom of a UAV to transport a payload or interact with existingstructures. In [1]–[3], a quadrotor with a gripper is used for trans-porting blocks and to build structures. In [4], a force sensor isused to apply a normal force to a surface. Accordingly, not onlythe attitude of the payload/tool is restricted to the attitude of theUAV, but also the accessible range of the end-effector is confineddue to the fixed configuration of the gripper/tool with respect tothe UAV body and blades. Consequently, the resulting aerial sys-tem has four DOF, three translational DOF, and one rotationalDOF (Yaw), i.e., the gripper/tool cannot possess pitch or rollrotation without moving horizontally. The second approach isto suspend a payload with cables. In [5], an adaptive controller ispresented to avoid swing excitation of a payload. In [6]–[8], spe-cific attitude and position of a payload is achieved using cablesconnected to one or three quadrotors. However, this approachhas a drawback that the movement of the payload cannot be al-ways regulated directly because manipulation is achieved usinga cable which cannot always drive the motion of the payload asdesired.

To overcome these limitations, a third approach is developedin which an aerial vehicle is equipped with a robotic manipulatorthat can actively interact with the environment. For example, in[9], a test bed including two 4-DOF robot arms and a crane em-ulating an aerial robot is proposed. By combining the mobilityof the aerial vehicle with the versatility of a robotic manip-ulator, the utility of mobile manipulation can be maximized.When employing the robotic manipulator, the dynamics of therobotic manipulator are highly coupled with that of the aerialvehicle, which should be carefully considered in the controllerdesign. Also, an aerial robot needs to tolerate the reaction forcesfrom the interactions with the object or external environment.These reaction forces may affect the stability of an aerial ve-hicle significantly. Very few reports that exist in the literatureinvestigate the combination of aerial vehicle with the roboticmanipulator. Kinematic and dynamic models of the quadrotorcombined with arbitrary multi-DOF robot arm are derived us-

ing the Euler–Lagrangian formalism in [10]. Based on that,simulation results using the Cartesian impedance control forthe combined system with 3-DOF robotic arm is presented in[11]. In addition, the effects of manipulators, two 4-DOF arms,on the quadrotor are simulated based on the dynamic modelwhich considers a quadrotor and robotic arms separately, treat-ing the arms as the disturbance to the quadrotor control loop. In[12], a quadrotor with lightweight manipulators, three 2-DOFarms, are tested, although the movement of the manipulators isnot explicitly considered during the design of the proportional–integral–derivative controller. In [13], an aerial manipulationusing a quadrotor with a 2-DOF robotic arm is presented butwith certain topology that disable the system from making arbi-trary position and orientation of the end-effector. In this system,the axes of the manipulator joints are parallel to each other andparallel to one in-plane axis of the quadrotor. Thus, the systemcannot achieve orientation around the second in-plane axis ofthe quadrotor without moving horizontally.

From the above discussion, the systems that use a grippersuffer from the limited allowable DOF of the end-effector. Theother systems have a manipulator with either two DOF but incertain topology that disables the end-effector to track arbi-trary 6-DOF trajectory, or more than two DOF which decreasesgreatly the possible payload carried by the system.

In [14], [15], Khalifa et al. propose a new aerial manipu-lation system that consists of two-link manipulator, with tworevolute joints whose axes are perpendicular to each other andthe axis of the first joint is parallel to one in-plane axis of thequadrotor. Thus, the end-effector is able to reach arbitrary posi-tion and orientation without moving horizontally. However, theproof that this system can perform any desired trajectory in thetask space has not been achieved yet due to the resulted com-plications in the inverse kinematics and control. This system issimilar to a serial manipulator with eight joints, two of them arepassive.

This system opens new application area for robotics. Suchapplications are inspection, maintenance, firefighting, servicerobot in crowded cities to deliver light stuff such as post mailsor quick meals, rescue operation, surveillance, demining, per-forming tasks in dangerous places, or transportation in remoteplaces.

The main contributions of this paper are as follows.1) New quadrotor-based aerial manipulator, which utilizes

a two-link robotic arm with unique topology, such thatthe end-effector can track a desired 6-DOF trajectory,is designed. This system is nonredundant and use min-imum possible number of actuators and links such thatthe payload carried by the system can be maximized andconsequently, the practical use can be enhanced.

2) Novel inverse kinematics analysis, which is based oncomplex set of nonlinear differential and algebraic equa-tions, is presented and utilized to prove the end-effectorability to track an arbitrary 6-DOF trajectory that layswithin the capability of the used actuators.

3) Design of a robust motion control scheme, with rigorousstability analysis, is carried out to track the desired 6-DOFtrajectory.

FANNI AND KHALIFA: NEW 6-DOF QUADROTOR MANIPULATION SYSTEM: DESIGN, KINEMATICS, DYNAMICS, AND CONTROL 1317

Fig. 1. Model of the proposed aerial manipulator. (a) Three-dimensional CAD model. (b) Schematic with relevant frames.

Fig. 2. Prototype of the proposed aerial manipulator.

II. DESCRIPTION AND DESIGN OF THE PROPOSED SYSTEM

Three-dimensional (3-D) computer-aided design (CAD)model of the proposed system is shown in Fig. 1(a). The systemconsists mainly of two parts; the quadrotor and the manipu-lator. Fig. 1(b) presents a sketch of the proposed system withthe relevant frames which indicates the unique topology thatpermits the end-effector to achieve arbitrary pose. The framesare assumed to satisfy the Denavit–Hartenberg convention. Themanipulator has two revolute joints. The axis of the first revo-lute joint (z0), that is fixed to the quadrotor, is parallel to thebody x-axis of the quadrotor [see Fig. 1(b)]. The axis of thesecond joint (z1) is perpendicular to the axis of the first jointand will be parallel to the body y-axis of the quadrotor at home(extended) configuration. Thus, the pitching and rolling rotationof the end-effector is now possible independently from the hor-izontal motion of the quadrotor. Hence, with this new system,the capability of manipulating objects with arbitrary locationand orientation is achieved. By this nonredundant system, theend-effector can achieve 6-DOF motion with minimum numberof actuators/links which is an important factor in flight. The pro-posed system is distinguished from all other previous systemsin the literature by having maximum mobility with minimumweight. The resulted complexity of the inverse kinematics andcontrol are handled later to prove the capability of the end-effector to track the reference 6-DOF trajectory.

The prototype of the proposed system is presented in Fig. 2.The quadrotor is selected such that it can carry an additionalweight equals 500 g (larger than the total arm weight and the

Fig. 3. Manipulator’s structure analysis. (a) Deflection analysis.(b) Stress analysis.

maximum payload). Asctec pelican quadrotor [16] is used as thequadrotor platform. It has a flight control unit (FCU) as well as amodular carbon fiber structure which enables us to easily mountdifferent components like computer boards, position sensors,and the robotic arm with its electronics. The FCU has an inertialmeasurement unit (IMU) as well as two 32-Bit, 60 MHz ARM-7microcontrollers. One of these microcontrollers, the low-levelprocessor (LLP), is responsible for the hardware managementand IMU sensor data fusion. The other microcontroller, high-level processor, is dedicated for custom code such as a newcontrol algorithm. The IMU provides the accelerations and theangular velocities. It contains also a magnetic compass. It hasa global positioning system receiver. A 11.1 V Li-Po batteryis used as a power supply to the whole system. The angularposition and angular velocity are available from the FCU via theIMU data fusion algorithm implemented in the LLP. The linearposition and linear velocity can be obtained via data fusion ofan on-board IMU and either monocular vision system [17], [18]or range finders (laser or ultrasonic) [19].

A lightweight manipulator that can carry a payload of 200 gand has a maximum reach of 22 cm is designed. The arm com-ponents are selected, purchased, and assembled. Its weight is200 g. The arm components are as follows. Three dc motors;HS-422 (Max torque = 0.4 N·m) for the gripper, HS-5485HB(Max torque = 0.7 N·m) for joint 1, and HS-422 (Max torque =0.4 N·m) for joint 2. Motor’s Driver (SSC) is used as an interfacebetween the main control unit and the motors. Wireless PS2 R/Cis used to send commands to manipulator’s motors remotely. Themanipulator structure accessories are aluminum tubing—1.50in diameter, aluminum multipurpose servo bracket, Aluminumtubing connector hub, and aluminum long “C’ servo bracketwith ball bearings [20]. An Arduino board (Mega 2560) is uti-lized as an interface between the low-level peripherals (such asultrasonic sensor, PS2 wireless receiver, and motor driver), andthe on-board computer.

The safety of this design, with respect to the strength andrigidity, is checked through finite element analysis using AN-SYS software (see Fig. 3). A 250 g force is applied at the gripperend, while the arm end (i.e., point at which the arm is connectedto the body center of the quadrotor) is fixed. From these figures,the maximum deflection is about 0.6 mm which is smaller thanthe allowable value which equals 1 mm. In addition, the maxi-mum stress of the structure is 113 MPa which is smaller than theyield strength of aluminum alloy which is 270 MPa. Also, thebearings and gripper are selected to sustain the loads. Therefore,


this design is safe. A prototype of the proposed system is builtand its design is verified through flight test.1

III. FORWARD KINEMATICS

In this section, position and velocity kinematics analysis arepresented. Let Σb , Ob - xb yb zb denote the vehicle body-fixedreference frame with origin at the quadrotor center of mass [seeFig. 1(b)]. Its position with respect to the world-fixed inertialreference frame Σ, O- x y z is given by the (3 × 1) vectorpb = [x y z]T , while its orientation is given by the rotationmatrix Rb

Rb =

⎡⎢⎣CψCθ SφSθCψ − SψCφ SψSφ + CψSθCφ

SψCθ CψCφ + SψSθSφ SψSθCφ − CψSφ

−Sθ CθSφ CθCφ

⎤⎥⎦

(1)where Φb = [ψ θ φ]T is the triple ZY X yaw-pitch-roll angles.Note that C. and S. are short notations for cos(.) and sin(.),respectively. Let us consider the frame Σe ,O2- x2 y2 z2 attachedto the end-effector of the manipulator [see Fig. 1(b)].

Thus, the position of Σe with respect to Σ is given by

pe = pb +Rbpbeb (2)

where the vector pbeb describes the position of Σe with respectto Σb expressed in Σb . The orientation of Σe can be defined bythe rotation matrix

Re = RbRbe (3)

where Rbe describes the orientation of Σe w.r.t Σb . The forward

kinematics problem consists of determining the operational co-ordinates χe = [xe ye ze ψe θe φe ]T , as a function of the vehi-cle/joint space coordinates q = [x y z ψ θ φ θ1 θ2 ]T . For solvingthe forward kinematics, the inputs are eight variables, q, and theoutput are six variables, χe , obtained from six algebraic equa-tions. The end-effector position can be found from (2). Eulerangles of the end-effector Φe can be computed from the rotationmatrix of Re , from (3).

In the rest of this section, the velocity kinematics analysis isdiscussed. The linear velocity pe of Σe in the world-fixed frameis obtained by the differentiation of (2) as

pe = pb − Skew(Rbpbeb)ωb +Rbp

beb , (4)

where Skew(.) is the (3 × 3) skew-symmetric matrix operator,while ωb is the angular velocity of the quadrotor expressed inΣ. The angular velocity ωe of Σe is expressed as

ωe = ωb +Rbωbeb (5)

where ωbeb is the angular velocity of the end-effector relative toΣb and is expressed in Σb .

Let Θ = [θ1 θ2 ]T be the (2 × 1) vector of joint angles of themanipulator. The (6 × 1) vector of the generalized velocity ofthe end-effector with respect to Σb , vbeb = [pbTeb ω

bTeb ]T , can be

expressed in terms of the joint velocities Θ via the manipulator

1https://www.dropbox.com/s/g9psmcbici3mphk/Prototype_Test.avi?dl=0

Jacobian, Jbeb , such that

vbeb = JbebΘ. (6)

From (4) and (5), the generalized end-effector velocity, ve =[pTe ω

Te ]T , can be expressed as

ve = Jbvb + JebΘ (7)

where vb = [pTb ωTb ]T , Jb = [ I3O3

−Skew(Rb pbe b )

I3], Jeb =

[Rb

O3

O3Rb

]Jbeb where Im and Om denote (m×m) identityand (m×m) null matrices, respectively. If the attitude of thevehicle is expressed in terms of yaw-pitch-roll angles, then (7)will be

ve = JbQbχb + JebΘ (8)

with χb = [ pbΦ b], Qb = [ I3

O3

O3Tb

], where Tb describes the trans-formation matrix between the angular velocity ωb and the timederivative of Euler angles Φb , and it is given as

Tb(Φb) =

⎡⎢⎣

0 −Sψ CψCθ

0 Cψ SψCθ

1 0 −Sθ

⎤⎥⎦ . (9)

Since the vehicle is an underactuated system, i.e., only fourindependent control inputs are available for the 6-DOF system,the position and the yaw angle are usually the controlled vari-ables. Hence, it is worth to define ζ = [x y z ψ θ1 θ2 ]T asthe controlled variables and σb = [θ φ]T as the intermediatevariables.

IV. DYNAMIC MODEL

The equations of motion of the proposed robot have beenderived by us in details in [14]. Applying the Newton Euler al-gorithm to the manipulator, one can get the equations of motionof the manipulator as well as the interaction forces and momentsbetween the manipulator and the quadrotor.

For the system structure, we assume the following.Assumption 1: The quadrotor body is rigid and symmetric.

The manipulator links are rigid.The equations of motion of the manipulator are

M1(q)θ1 +N1(q, q, χb) = τm 1 , (10)

M2(q)θ2 +N2(q, q, χb) = τm 2 (11)

where τm 1 and τm 2 are the manipulator actuators’ torques.M1(q), M2(q), N1(q, q, χb), and N2(q, q, χb) are nonlinearterms and they are functions of the system states (q, q) andaccelerations (χb ).

The Newton Euler method are used to find the equations ofmotion of the quadrotor after adding the forces/moments fromthe manipulator. They are given by the following equations:

mx = T (CψSθCφ + SψSφ) + Fm,qx (12)

my = T (SψSθCφ − CψSφ) + Fm,qy (13)

mzp = −mg + TCθCφ + Fm,qz (14)

Ixφ = θφ(Iy − Iz ) − Ir θΩ + Ta1 +Mbm,qφ

(15)


Iy θ = ψφ(Iz − Ix) + Ir φΩ + Ta2 +Mbm,qθ

(16)

Iz ψ = θφ(Ix − Iy ) + Ta3 +Mbm,qψ

(17)

where Fm,qx , Fm,qy , and Fm,qz are the interaction forces fromthe manipulator to the quadrotor in x, y, and z directions ex-pressed in the inertial frame andMb

m,qφ,Mb

m,qθ, andMb

m,qψare

the interaction moments from the manipulator to the quadrotoraround xb , yb , and zb directions.

The variables in (12)–(17) are defined as follows: m is themass of the quadrotor. Each rotor j has angular velocity Ωj

and it produces thrust force Fj and drag moment Mj which aregiven by

Fj = Kfj Ω2j , Mj = Kmj

Ω2j (18)

whereKfj andKmjare the thrust and drag coefficients, respec-

tively.T is the total thrust applied to the quadrotor from all four

rotors. Ta1 , Ta2 , and Ta3 are the three input moments about thethree body axes, xb , yb , and zb , respectively. d is the distancebetween the quadrotor center of mass and rotor rotational axis.Ω is given by

Ω = Ω1 − Ω2 + Ω3 − Ω4 . (19)

Ir is the rotor inertia. If is the inertia matrix of the vehiclearound its body-frame assuming that the vehicle is symmetricabout xb−, yb−, and zb−axis.

The dynamical model of the quadrotor-manipulator systemcan be written as follows:

M(q)q + C(q, q)q +G(q) + dex = Bu (20)

where M ∈ R8×8 represents the symmetric and positive def-inite inertia matrix of the combined system, C ∈ R8×8 is thematrix of Coriolis and centrifugal terms, G ∈ R8 is the vectorof gravity terms, dex ∈ R8 is vector of the external disturbances,u = [F1 F2 F3 F4 τm 1 τm 2 ]

T ∈ R6 is vector of the actua-tor inputs, and B = H(q) N(Kfj ,Kmj

, d) is the input matrixwhich is used to generate the body forces and moments fromthe actuator inputs. The control matrix N is given by

N =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 00 0 0 0 0 01 1 1 1 0 0γ1 −γ2 γ3 −γ4 0 0−d 0 d 0 0 00 −d 0 d 0 00 0 0 0 1 00 0 0 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

where γj = Kmj/Kfj , and H ∈ R8×8 is the matrix that trans-

forms body input forces to be expressed in Σ and is given by

H =

⎡⎢⎣

Rb O3 O2

O3 TTb Rb O2

O2×3 O2×3 I2

⎤⎥⎦ . (22)

Fig. 4. Verification of the inverse kinematic algorithm: Block diagram.

From the equations of the translation dynamics part of(12)–(14), one can extract the expressions of the second-ordernonholonomic constraints as

sin (φ) − xf Sψ − yf Cψ√x2f + y2

f + z2f

= 0 (23)

tan (θ) − xf Cψ + yf Sψzf

= 0 (24)

where xf = x− Fm , q x

m , yf = y − Fm , q y

m , and zf = z + g −Fm , q z

m .It is to be noted that the force terms in the above equations are

also function of the system states and their derivatives. Equations(23) and (24) can be solved for the desired trajectories of φ andθ through substituting by the desired trajectories of the othervariables. These nonholonomic constraints will be utilized laterto solve the inverse kinematics problem.

V. INVERSE KINEMATICS

A task for the quadrotor manipulation system is usually speci-fied in terms of a desired trajectory for the end-effector position,pe,r (t), and orientation, Φe,r (t). In this section, six algebraickinematic equations relating χe with q are derived.

However, to find the eight variables of q from the given sixvariables of χe , we need additional two equations. These twoequations are the nonholonomic constraints (23) and (24) whichare differential equations. We solve these eight equation sym-bolically and reduce them to two differential equations inσb . It isfound that the exact solution of them has very high computationcost which prevents its practical implementation.

For the desired task space trajectories, we assume the follow-ing.

Assumption 2: The desired trajectories for the end-effectorare bounded.

Thus, we propose an algorithm to get an approximate solutionto these eight algebraic/differential equations, see Fig. 4, asfollows.

1) Specify the desired 6-DOF trajectory in the task space,χe,r (t).

2) Put i = 1, t1 = 0, and Δt = very small positive number(1 ms).

3) Put σb(t0) = σb(t0) = σb(t0) = [0 0]T .4) At time ti , obtain the values of χe,r (ti) and put σb(ti) =σb(ti−1), σb(ti) = σb(ti−1), and σb(ti) = σb(ti−1).


5) Use the six inverse kinematics algebraic equations andget the unknowns ζr (ti), ζr (ti), ζr (ti) which with σb(ti),σb(ti), and σb(ti) constitute q(ti), q(ti), and q(ti).

6) In the equations of the nonholonomic constraints, weseparate sin (φ) and tan (θ) in one side (left side) and theother terms in the other side (right side).

7) We substitute by q(ti), q(ti), q(ti) values in the right-hand side of the nonholonomic constraints obtained fromstep 6, and obtain the new σb(ti) from left-hand side andby numerical differentiation, one can get the new σb(ti)and σb(ti).

8) Put i = i+ 1, ti = ti−1 + Δt, and repeat steps 4 to 8 toobtain the reference trajectories of the controlled vari-ables ζr (ti).

The validation of this algorithm will be checked at the end ofthis section. The six algebraic inverse kinematic equations arederived next.

Define the general form for the rotation matrix Re as a func-tion of end-effector variables χe , as

Re =

⎡⎢⎣r11 r12 r13

r21 r22 r23

r31 r32 r33

⎤⎥⎦ . (25)

Equating (3) and (25), an expression for the elements ofRe , rij ;i, j = 1, 2, 3, can be found. According to the structure of Re

from (3), the inverse orientation is carried out first followed byinverse position. The inverse orientation has three cases basedon the value of θ1 which can be calculated from the elementr33 as

CφCθCθ1 − SφCθSθ1 = r33 . (26)

By rearranging (26) and solving the resulting equation for θ1 ,then

θ1 = 2 atan2(−2b1 ±√

(2b1)2 − 4(−a1 − r33)(a1 − r33),

2(−a1 − r33)) (27)

where a1 = CφCθ and b1 = −SφCθ .The three cases are derived as follows.Case 1: θ1 �= 0 and θ1 �= πInspecting r13 and r23 , one can find the value of ψ as

r13 = a2Sψ + b2Cψ , (28)

r23 = c2Sψ + d2Cψ (29)

where a2 = CφSθ1 + SφCθ1 , b2 = CφSθCθ1 − SφSθSθ1 , c2 =b2 , and d2 = −a2 .

Solving (28) and (29) will give

Sψ =r23b2 − r13d2

−d2a2 + b2c2, (30)

Cψ =r23a2 − r13c2d2a2 − b2c2

, (31)

ψ = atan2(Sψ ,Cψ ). (32)

By inspecting r32 and r31 , θ2 can be found as

r32 = a3Sθ2 + b3Cθ2 , (33)

r31 = c3Sθ2 + d3Cθ2 (34)

where a3 = CφCθSθ1 + CθSφCθ1 , b3 = −Sθ , c3 = b3 , andd3 = −a3 .

Solving (33) and (34), then

Sθ2 =r31b3 − r32d3

−d3a3 + b3c3, (35)

Cθ2 =r31a3 − r32c3d3a3 − b3c3

, (36)

θ2 = atan2(Sθ2 , Cθ2 ). (37)

Case 2: θ1 = 0If θ1 = 0, then the sum θ2 + ψ can be determined. One can

assume any value for ψ and get θ2 . Therefore, there are infinityof solutions. By puttingψ= 0, the value of θ2 can be determinedas follows.

Inspecting r11 and r12 , θ2 can be found as

r11 = a4Sθ2 + b4Cθ2 , (38)

r12 = c4Sθ2 + d4Cθ2 (39)

where a4 = Cθ , b4 = −SφSθ , c4 = −b4 , and d4 = a4 .Solving (38) and (39), then

Sθ2 =r12b4 − r11d4

−d4a4 + b4c4, (40)

Cθ2 =r12a4 − r11c4d4a4 − b4c4

, (41)

θ2 = atan2(Sθ2 , Cθ2 ). (42)

Case 3: θ1 = πSince Sθ1 = 0, this is similar to case 2. However, θ2 − ψ can

be determined and by choosing ψ = 0, an expression for θ2 canbe determined as follows.

Inspecting r11 and r12 , θ2 can be found as

r11 = a5Sθ2 + b5Cθ2 , (43)

r12 = c5Sθ2 + d5Cθ2 (44)

where a5 = a4 , b5 = −b4 , c5 = −b5 , and d5 = a5 .Solving (43) and (44), then

Sθ2 =r12b5 − r11d5

−d5a5 + b5c5, (45)

Cθ2 =r12a5 − r11c5d5a5 − b5c5

, (46)

θ2 = atan2(Sθ2 , Cθ2 ). (47)

As shown above, there are four possible solutions for the rota-tional inverse kinematics problem provided that we put ψ = 0 incases 2 and 3. When we start applying the above-described algo-rithm in real time, we select the solution of θ1 which coincideswith the given configuration of the quadrotor manipulation sys-tem. After that, we continue with the one of the four solutionswhich produces this θ1 .


Fig. 5. Verification of the inverse kinematic algorithm: End-effector 3-Dtrajectory. (a) Slower trajectory. (b) Faster trajectory.

Finally, the inverse position is determined from (2) as

pb = pe −Rbpbeb . (48)

To validate the proposed approximate solution of the inversekinematics, the desired task space trajectories are chosen tomake a circular helix in position and quintic polynomial [21]for the orientation.

After obtaining the joint space variables from the proposedalgorithm, as shown in Fig. 4, we apply the forward kinematicsto find the actual task space trajectory.

The comparison between the actual and desired task spacetrajectories is shown in Fig. 5(a) from which one can recognizethat the actual and desired trajectories coincide. In Fig. 5(b),another case study is investigated to show the capability of theproposed algorithm to deal with higher speed desired trajecto-ries. In this study, the desired trajectories are chosen to be fasterthan that in Fig. 5(a) by ten times. These figures show that the ac-tual and desired trajectories coincide under both slower or fastertrajectories, which ensures the validity of the proposed inversekinematic algorithm, and proves that the proposed system hasthe ability to track arbitrary 6-DOF task space trajectory. Withthis result in mind, we propose a control algorithm in the nextsection to track the desired 6-DOF trajectories in the task space.

VI. CONTROL DESIGN

A. Control Objectives

We target the design of the controller output, τ , in order tosatisfy the following objectives.

1) Control Objective 1 (Robust Stability): The consideredrobotic system is stable and robust in the presence of theexternal disturbances and noises.

2) Control Objective 2: Minimizing the error of the 6-DOFtask space trajectory tracking.

To achieve these control objectives, we utilize a control tech-nique based on both the DOb method and the above inversekinematics analysis.

B. DOb-Based Controller

DOb-based controller is one of the most popular methodsin the field of robust motion control due to its simplicity and

Fig. 6. Block diagram of DOb-based controller.

computational efficiency. Kim and Chung [22], and Li et al.[23] present the principles of DOb-based control system. InDOb-based robust motion control systems, internal and exter-nal disturbances are observed by DOb, and the robustness issimply achieved by feeding-back the estimated disturbances inan inner loop. Another controller is designed in an outer loop sothat the performance goals are achieved without considering in-ternal and external disturbances. In [24]–[26], DOb-based con-trol technique has been applied to robotic systems and showedefficient performance.

A block diagram of the DOb controller is shown in Fig. 6 (theplant dynamics are highlighted by yellow box, while the dobloops are highlighted by green box) for robotic system whichwill be utilized later to design robust control for the proposedsystem. It is well known that the linear accelerations and angularrates of the vehicle can be measured directly from IMU. Inaddition, the angular velocities of the manipulator joints can bemeasured via an encoder. Therefore, two different DOb loopsare used. One is based on the measured acceleration, while theother is based on the measured velocity.

In Fig. 6, Mn = [Mn aO5×3

O3×5Mn v

] ∈ R8×8 is the system nominal

inertia matrix with Mna ∈ R3×3 represents the nominal inertiafor the translational coordinates pb , while Mnv ∈ R5×5 repre-sents the nominal inertia for the rotational coordinates, Φb andΘ. τdes is the robot desired input. τ = [τTa τTv ]T is the robotgeneralized input. Q(s) = [ Qa

O5×3

O3×5Qv

] ∈ R8×8 is the matrix ofthe low-pass filter of DOb, Qa(s) = diag([ g1

s+g1. . . g3

s+g3]),

and Qv (s) = diag([ g4s+g4

. . . g8s+g8

]). P = [ PaO5×3

O3×5Pv

] withPa = diag([g1 g2 g3 ]) for the translational coordinates andPv = diag([g4 . . . gi . . . g8 ]) for the rotational coordinates,where gi is the bandwidth of the ith variable of q. τdis repre-sents the system disturbances including the Coriolis, centrifu-gal and gravitational terms in addition to external disturbance.τdis = [τdisT

a τdisTv ]T (collected by the block called Mux, like

multiplexer symbol in MATLAB) represents the system esti-mated disturbances. qn and qn represent the measurement noiseof the velocity and acceleration, respectively. The block named“Demux” is a demultiplexer which is used to select some signalsfrom bus of signals. It is used, for example, to select the linearaccelerations pb , from the accelerations vector of the wholesystem, q.


Fig. 7. Detailed block diagram of the control system.

The system disturbance τdis can be assumed as

τdis = (M(q) −Mn )q + τd ,

τ d = C(q, q)q +G(q) + dex. (49)

The control input, τ , in Fig. 6 can be calculated as

τ = Mnqdes + τdis (50)

where

τdis = Q(τ −Mnq) (51)

and qdes is the output of the outer loop controller K(s), and itis given as

qdes = qr +KP (qr − q) +KD (qr − q) (52)

where KP and KD ∈ R8×8 are the proportional and derivativegains of the PD controller, respectively. qr , qr , and qr are thereferences for linear/angular positions, velocities, and accelera-tions, respectively.

Thus, the control input, τ , can be calculated from(49)–(51) as

τ = Mnqdes +Kvev (53)

where Kv = PMn = kv I8 with kv is a positive constant,and ev = qdes − q with qdes is the output from the outer loopcontroller.

Fig. 7 presents a block diagram of the proposed motion con-trol system based on the inverse kinematics analysis and onquadrotor/joint space-based control. The desired trajectories forthe end-effector’s position and orientation χe,r (t) are fed to theinverse kinematics algorithm together with σb,r (t) from a sim-plified version of the nonholonomic constraints (54) such thatthe desired vehicle/joint space trajectories ζr (t) are obtained.After that, the controller block receives the desired trajecto-ries and the feedback signals from the system and provides thecontrol signal, τ = Bu.

Since the quadrotor position, manipulator joints angles, andthe yaw angle are usually the controlled variables (i.e., inde-pendent coordinates, ζ), while pitch and roll angles are usedas intermediate control inputs (i.e., dependent coordinates, σb )for horizontal positions control, the proposed control system

consists of two DOb-based controllers; one for ζ (with Kζ (s),Mna ,Mnv1 ,Pv1 ,Qv1 , andQa ) and the other forσb (withKσ (s),Mnσ , Pv2 ,Qv2 ). The subscript v1 refers to the rotational motionin the independent coordinates which are ψ, θ1 , and θ2 , whilethe subscript v2 refers to the rotational motion in the dependentcoordinates which are σb = [θ φ]T .

The desired independent quadrotor/joint space trajectoriesgenerated from the inverse kinematics algorithm ζr are fed tothe PD controller Kζ (s) that will produce the desired accelera-tion for the independent coordinates ζdes . This desired accelera-tion ζdes is converted to a desired torque τdes

ζ via multiplicationby the nominal inertia matrix of the independent coordinatesMnζ . The desired torque is added to the estimated disturbances,

τdisζ = [τdisT

ζ a τdisTζ v ]T , from the DOb of the independent coordi-

nates ζ, such that the required input torques in the independentcoordinates τζ are produced. Inside the lower DOb block thereare two DObs. The first one is for the translation independentcoordinates pb . It is an acceleration-based DOb, with inputs assystem linear accelerations and the input torques for the transla-tional motion τζa = τζ (1 : 3). It uses the corresponding part ofthe low-pass filterQa and the corresponding part of the nominalinertia matrix Mna . This block produces the estimated distur-bances for the translational motion τdis

ζ a . The second DOb insidethe lower DOb block is for the rotational independent coordi-nates, ψ, θ1 , and θ2 , and it is a velocity-based DOb, with inputsas rates of yaw and manipulator joints angles and the inputtorques for the independent rotational motion τζv = τζ (4 : 6).It uses the corresponding part of the low-pass filter Qv1 thecorresponding part of Pv , Pv1 and the corresponding part of thenominal inertia matrixMnv1 . This block produces the estimateddisturbances for the rotational motion τdis

ζ v . After that, τdisζ a and

τdisζ v are gathered to get the final estimated disturbances for the

independent coordinates τdisζ .

The desired values for the intermediate controller σb,r areobtained from the output of independent coordinates’ controllerτζ through the relation

σb,r =1

τζ (3)

[Cψ Sψ

Sψ −Cψ

][τζ (1)τζ (2)

](54)

which can be derived from (1) based on small angleapproximation.

The desired value of the pitching and rolling σb,r is then fedback to both the inverse kinematics algorithm and the outer loopcontroller for the dependent coordinates Kσ (s). This controllerproduces the desired acceleration for the dependent coordinatesσdesb that is converted to desired torque τdes

σ via the nominalinertia matrix of the rotational dependent coordinatesMnσ . Thedesired torque is added to the estimated disturbances τdis

σ fromthe DOb of the dependent coordinates σb , such that the requiredinput torques for the dependent coordinates τσ are produced.The upper DOb block is a velocity-based DOb, with inputs asrate of σb and the input torque for the dependent rotationalmotion τσ . It uses the corresponding part of the low-pass filterQv2 the corresponding part of Pv , Pv2 , and the correspondingpart of the nominal inertia matrixMnσ . This block produces theestimated disturbances for the dependent rotational motion τdis

σ .


After that, the output of the two controllers τζ and τσ aremixed to generate the final control vector τ which is convertedto the forces/torques applied to quadrotor/manipulator by

u = B−16

⎡⎢⎣τζ (3, 4)τσ

τζ (5, 6)

⎤⎥⎦ (55)

where B6 ∈ R6×6 is the part of B matrix and it is given byB6 = B(3 : 8, 1 : 6).

Theorem 1: Consider the control system shown in Fig. 7,and described in (20), (50), and (52) with Assumptions 1–2.Then, as long as the following conditions; Kv > 0, KP > 0,and KD > 0, are satisfied, the system is Lp input/output stablewith respect to the pair (δ, ev ), and e ∈ Lp and e ∈ Lp for allp ∈ [1,∞) such that

‖ev‖p ≤ 1γ

+√

2λmin

(2pγ

) 1p √

V (0, ev (0))‖δ‖p . (56)

Proof: By substituting from the control law (52) and (53) intothe system dynamics equation (20), one can get the followingerror dynamics:

M(q)ev + C(q, q)ev +Kvev = δ (57)

and

e+KDe+KP

∫e dt = ev (58)

where

δ = ΔM(q)qdes + C(q, q)qdes +G(q) + dex (59)

and ΔM(q) = M(q) −Mn .Assume the following Lyapunov function:

V =12eTv M(q)ev . (60)

The time derivative of this function is

V = eTv M(q)ev +12eTv M(q)ev . (61)

Substituting from (57), then (61) becomes

V = eTv δ − eTv Kvev +12eTv (M(q) − 2C(q, q))ev . (62)

The dynamic equation of motion (20) posses several well-knownproperties [27].

Property 1:

λmin‖ν‖2 ≤ νT M(q)ν ≤ λmax‖ν‖2 . (63)

Property 2:

νT (M(q) − 2C(q, q))ν = 0 (64)

where M(q) − 2C(q, q) is a skew-symmetric matrix, ν ∈ R8

represents a 8-D vector, and λmin and λmax are positive realconstants.

Substituting from (64) into (62), then (62) will be

V = eTv δ − eTv Kvev . (65)

From the robot dynamic property (63), one can find

V ≤ −γV +√

2Vλmin

|δ| (66)

where γ = 2kvλm a x

. From the analysis presented in [28] and [29],we complete the proof as the following. Dividing (66) byV 0.5 �=0 will give

d

dt(V 0.5) + 0.5γV 0.5 ≤

√2

λmin|δ|. (67)

By multiplying (67) by e−0.5γ and integrating, then we obtain

V 0.5 ≤ e−0.5γ tV 0.5(0, ev (0)) + Vc (68)

with

Vc =1√

2λmin

∫ t

0e−0.5γ (t−ι) |δ(ι)|dι. (69)

We can rewrite Vc as

Vc =1√

2λmin(e−0.5γ t ∗ |δ(t)|). (70)

Taking the ‖.‖p , gives

‖Vc‖p =1√

2λmin‖e‖−0.5γ t ∗ ‖δ(t)|p . (71)

From Lemma 3 in [29], then (71) is

‖Vc‖p =1√

2λmin‖e‖−0.5γt1‖δ(t)‖p (72)

but ‖e‖−0.5γ t1 = 0.5γ, so

‖Vc‖p = 0.5γ1√

2λmin‖δ(t)‖p . (73)

Thus, (68) is

‖V 0.5‖p ≤ ‖e‖−0.5γ tV 0.5(0, ev (0))p + 0.5γ1√

2λmin‖δ(t)‖p .

(74)This expression (74) can be simplified to reach (56).

Lemma 1 ([30]): Let

e = HPD (s)ev (75)

where HPD (s) is an strictly proper and exponentially stabletransfer function. Then, ev ∈ Lp implies that e ∈ Lp and e ∈Lp .

Form (58), the transfer function between e and ev is properexponentially stable as long as the parameters KP and KD arepositive definite matrices. Thus, Lemma 1 implies that both eand e ∈ Lp . �

Remark 1: Equation (66) indicates that the convergence rateincreases proportionally withKv . However, we have to take intoconsiderations the constraints on gi due to the sampling time.

Remark 2: It is noted from the quadrotor dynamics [31] thatthe response of the intermediate σ controller must be faster thanthat of the position control loop. However, as stated in the aboveanalysis, the convergence rate of errors dynamics (ev and e) isdirectly proportional to the Kv , KP , KD . Therefore, we have


TABLE ISYSTEM PARAMETERS

Par. Value Unit Par. Value Unit

m 1 kg L2 85 × 10−3 md 223.5 × 10−3 m m 0 30 × 10−3 kgIx 13.215 × 10−3 N·m.s2 m 1 55 × 10−3 kgIy 12.522 × 10−3 N·m.s2 m 2 112 × 10−3 kgIz 23.527 × 10−3 N·m.s2 Ir 33.216 × 10−6 N·m.s2

L0 30 × 10−3 m L1 70 × 10−3 mKF 1 1.667 × 10−5 kg.m.rad2 KF 2 1.285 × 10−5 kg.m.rad2

KF 3 1.711 × 10−5 kg.m.rad2 KF 4 1.556 × 10−5 kg.m.rad2

KM 1 3.965 × 10−7 kg.m2 .rad2 KM 2 2.847 × 10−7 kg.m2 .rad2

KM 3 4.404 × 10−7 kg.m2 .rad2 KM 4 3.170 × 10−7 kg.m2 .rad2

TABLE IIDOB-BASED CONTROLLER PARAMETERS

Par. Value Par. Value

Mn ζdiag{2, 2, 2, 0.5, 0.5, 0.5} Mn σ diag{0.5, 0.5}

KP ζdiag{3, 3, 10, 3, 3, 3} KP σ diag{10, 10}

KD ζdiag{1.5, 1.5, 5, 3, 3, 3} KD σ diag{7, 7}

P [10 10 10 100 100 100 100 100]T

to tune Kv , KP , and KD , such that the response speed of σb ismuch faster than that of ζ.

VII. SIMULATION STUDY

In this section, the previously proposed control strategy issimulated in MATLAB/SIMULINK program for the consideredaerial manipulation system.

A. Simulation Environment

In order to make the simulation quite realistic, the followingsetup and assumptions have been made.

1) The model has been identified on the basis of real datathrough experimental tests. The identified parameters aregiven in Table I.

2) Linear and angular position and velocity of the quadrotoras well as the position and velocity of the manipulatorjoints are available at rate of 1 KHz.

3) A normally distributed measurement noise, with mean of10−3 and standard deviation of 5 × 10−3 , has been addedto the measured signals.

4) The controller outputs are computed at a rate of 1 KHz.5) In order to test the robustness to the model uncertainties,

a step disturbance is introduced, at instant 15 s, in thecontrol matrix N (Actuators’ losses), whose elementsare assumed to be equal to 0.9 times their true values(i.e., 10% error). In addition, the end-effector has to picka payload of value 200 g at instant 20 s and release it at40 s.

B. Results and Discussion

Table II presents the controller parameter for the proposedcontrol technique. Tuning of the controller parameters is carried

Fig. 8. Actual response of the end-effector pose (a) xe , (b) ye , (c) ze ,(d) φe , (e) θe , and (f) ψe .

out to maximize the speed of system response and minimize thetracking error. The reference trajectories for the end-effectorare chosen such that the end-effector moves on a circular he-lix, while its orientation is fixed in a case and follows quinticpolynomial trajectories in another case.

The simulation results are presented in Figs. 8 and 9(a) (themarker represents the end-effector orientation; green, blue, andred for x-, y-, and z-axis). From these figures, it is possible to rec-ognize that there are small oscillations after starting the systemoperation due to the time need to estimate the nonlinearities anddisturbances. This time is about 4 s. After that period, the con-troller can quickly recover this error and provides good trackingof the desired trajectories. Moreover, it is clear that the capabil-ity of the controller to recover, with fast response (about 1 s),the tracking error due to the presence of the payload and the un-certainty in system parameters. Furthermore, Fig. 9(b) indicatesthat the required actuators efforts, which are the required thrustforce from each rotor and the torque from each manipulator’s


Fig. 9. Three-dimensional trajectory and controller effort. (a) 3-D tra-jectory of end-effector with fixed orientation. (b) The required controlefforts.

motor u are in the allowable limit. The maximum thrust forcefor each rotor is 6 N as obtained from the identification process,while the allowable input torque for the motor of joint 1 is 0.7and 0.4 N·m for that of joint 2 as stated in the motors’ datasheet. Therefore, one can contend that the control objectives areachieved by using this motion control scheme.

VIII. CONCLUSION

A new 6-DOF quadrotor-based aerial manipulator with min-imum number of actuators is investigated. It provides solutionsto the limitations found on the currently developed aerial manip-ulation systems by having maximum mobility with minimumweight. Description and design of the proposed system are in-troduced. Mathematical modeling are carried out. Proof thatthis system can perform any desired end-effector trajectory isdone through analysis and numerical simulation of the complexinverse kinematics with nonholonomic constraints. A robustmotion control scheme is designed and tested based on both theproposed inverse kinematics and the DOb technique in order toachieve 6-DOb task space trajectory tracking. Stability of thecontroller is analyzed. Simulation results enlighten the feasi-bility of the proposed system and the efficiency of the motioncontrol scheme. As a future work, the proposed system will betested experimentally.

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Mohamed Fanni received the B.E. degree inmechanical engineering from the Faculty of En-gineering, Cairo University, Giza, Egypt, in 1981;the M.Sc. degree in mechanical engineeringfrom Mansoura University, Mansoura, Egypt,in 1986; and the Ph.D. degree in engineeringfrom Karlsruhe University, Karlsruhe, Germany,in 1993.

He is an Associate Professor with the Egypt-Japan University of Science and Technology,New Borg El Arab, Egypt, on leave from the

Production Engineering and Mechanical Design Department, Facultyof Engineering, Mansoura University. His major research interests in-clude robotics engineering, automatic control, and mechanical design.His current research focuses on design and control of mechatronic sys-tems, surgical manipulators, and flying/walking robots.

Ahmed Khalifa received the B.Sc. degree in in-dustrial electronics and control engineering fromMenoufia University, Al Minufya, Egypt, in 2009,and the M.Sc. and Ph.D. degrees in mechatron-ics and robotics engineering from Egypt-JapanUniversity of Science and Technology, New BorgEl Arab, Egypt, in 2013 and 2016, respectively.

He is currently with the Faculty of ElectronicEngineering, Menofia University, as an Assis-tant Professor. His current research focuses onflying robots, multirobot systems, model-free ro-

bust and optimal control, and decentralized/cooperative control and stateestimation.

mohamed fanni and ahmed khalifa - unal.edu.co

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