research article dynamics modeling and control of a...

Research ArticleDynamics Modeling and Control of a Quadrotor withSwing Load

S Sadr1 S Ali A Moosavian1 and P Zarafshan2

1 Center of Excellence inRobotics andControl AdvancedRobotics andAutomated Systems LabDepartment ofMechanical EngineeringK N Toosi University of Technology Tehran 19991 43344 Iran

2Department of Agro-Technology College of Aburaihan University of Tehran Pakdasht Tehran 113654117 Iran

Correspondence should be addressed to P Zarafshan pzarafshanutacir

Received 19 May 2014 Revised 15 September 2014 Accepted 21 October 2014 Published 17 November 2014

Academic Editor Bijan Shirinzadeh

Copyright copy 2014 S Sadr et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Nowadays aerial robots or Unmanned Aerial Vehicles (UAV) have many applications in civilian and military fields For exampleof these applications is aerial monitoring picking loads and moving them by different grippers In this research a quadrotor witha cable-suspended load with eight degrees of freedom is considered The purpose is to control the position and attitude of thequadrotor on a desired trajectory in order to move the considered load with constant length of cable So the purpose of thisresearch is proposing and designing an antiswing control algorithm for the suspended load To this end control and stabilizationof the quadrotor are necessary for designing the antiswing controller Furthermore this paper is divided into two parts In thefirst part dynamics model is developed using Newton-Euler formulation and obtained equations are verified in comparison withLagrange approach Consequently a nonlinear control strategy based on dynamic model is used in order to control the positionand attitude of the quadrotor The performance of this proposed controller is evaluated by nonlinear simulations and finally theresults demonstrate the effectiveness of the control strategy for the quadrotor with suspended load in various maneuvers

1 Introduction

Quadrotor is a rotorcraft whose flight is based on rotationof two pairs of rotors that rotate opposite to each otherAs shown in Figure 1 the different movement of quadrotoris created by a difference in the velocity of rotors If thevelocity of rotor 1 (or 2) decreases and the velocity of rotor3 (or 4) increases then the roll (or pitch) motion is createdand the quadrotor moves along the 119910-axis (or the 119909-axis)Moreover a quadrotor is an aerial robot which has thepotential to hover and take off fly and land in small areas Inaddition this robot has applications in different fields amongwhich are safety natural risk management environmentalprotection infrastructures management agriculture andfilm protection Moreover a quadrotor is an underactuatedsystem since it has six degrees of freedom and only fourinputs However a quadrotor is inherently unstable and it canbe difficult to flyThus the control of this nonlinear system isa problem for both practical and theoretical interest Manycontrol algorithms are tested and implemented on this aerial

robot in order to stabilize andmove in different tasks Amongthese algorithms are classic control linear and nonlinearstate feedback control sliding mode control back steppingcontrol and fuzzy and neural network control In 2010Vazquez and Valenzuela designed a nonlinear control systemfor the position and attitude control based on the classiccontrol PID indeed the quadrotor altitude is controlled bya PI-action controller [1] In 2012 Lee et al implemented aLinear Quadratic Regulator (LQR) controller for the positioncontrol of the quadrotor [2] In 2004 Hoffmann [3] proposeda slidingmodemethod for the altitude control and an optimalcontrol method for the attitude control But many difficultiesoccurred because of motor vibrations in the high thrustand the chattering phenomena Also for realizing the robustcontrol of the quadrotor a back stepping control algorithm isproposed in [4] This algorithm could estimate disturbancesonline and so they could improve the robustness of systemErginer andAltug in 2012 performed dynamicsmodeling andcontrol of a quadrotor They obtained the dynamic modelof the quadrotor by Newton-Euler method and controlled

Hindawi Publishing CorporationJournal of RoboticsVolume 2014 Article ID 265897 12 pageshttpdxdoiorg1011552014265897

2 Journal of Robotics

1

2

3

4x

y

Figure 1 Scheme of quadrotorrsquos rotor rotation

the quadrotor using a hybrid fuzzy-PD control algorithm [5]In 2008 Raffo et al implemented a nonlinear H

infinalgorithm

to control and stabilize the angular motion of the quadrotorThe simulation results show that this nonlinear algorithm caneliminate disturbances and stabilize the rotation motion ofthe quadrotor [6] de Vries and Subbarao in 2010 designed aback stepping multiloop controller for the hover flight [7]

Recently with introduction of inexpensive micro-Unmanned Aerial Vehicles (UAV) and advanced sensorscontrollers have been designed to enable these systems formany tasks for example offensive maneuvers [8] balancinga flying inverted pendulum [9] Also the progress in sensorsand controllers leads very well to use of UAV for differentapplications One of these applications is transportationof external loads Different grippers for grasping andtransporting of a load are designed [10ndash14] Another one iscable-suspended loads which have been studied in recentyears [15ndash17] Cable-suspended systems are underactuatedsystems Therefore canceling or reducing oscillation of thesuspended load is very necessary since oscillations of loadin an industrial environment and other areas can result indamage [18ndash22] So different control methods have beenproposed to control these robots since the suspended loadsignificantly alters the flight characteristics of the quadrotorThese methods are divided into feedback and feed-forwardapproaches Feedback control methods use measurementsand estimations of system states to reduce the vibrationwhile feed-forward approaches change actuator commandsfor reducing the oscillation of system The feed-forwardcontroller can often improve the performance of feedbackcontrollerThus proposing feed-forward algorithms can leadto more practical and accurate control of these systems Oneeffective feed-forward method is the input shaping theorywhich has proven to be a practical and effective approachof reducing vibrations [23 24] Also several methods areproposed in order to minimize the residual vibration Smithproposed the Posicast control of the damped oscillatorysystems which is a technique to generate a nonoscillatoryresponse from a damped system to a step input This methodbreaks a step of a certain magnitude into two smaller stepsone of which is delayed in time [25] Swigert proposedshaped torques techniques which consider the sensitivity ofterminal states to variation in the model parameters [26]

x

y

z

zb

T1

T2

T3

T4

Q1

Q2Q3

Q4

120573

ybxb

120574

Fcablemg

mpg

Figure 2 Free body diagram of quadrotor slung load system

Recently in the control of overhead cranes Mita and Kanaisolved a minimum time control problem for swing freevelocity profiles which resulted in an open loop control [27]Also Yu proposed a nonlinear control based on the singularperturbation method [28] In addition Lee designed a highperformance control based on the loop shaping and rootlocus methods [29] Also in 2012 Adams et al designedinput shaping control of a micro-coaxial radio-controlledhelicopter carrying a suspended load [30] Zain et al in2006 proposed hybrid learning control schemes with aninput shaping of a flexible manipulator system [31]

In this paper the problem of the quadrotor flying with asuspended load is addressedwhich is widely used for differentkinds of a cargo transport The paper is organized in twoparts In the first part a nonlinear model of an underactu-ated eight-degree-of-freedom quadrotor slung load systemis derived on the basis of the Newton-Euler formulationNext this dynamic model is verified in comparison withLagrange method Then a nonlinear model based controlalgorithm is designed for the position and attitude controlof the quadrotor with the suspended load In next part thedescription of the input shaping algorithm is presented andthen this method is implemented to the quadrotor with asuspended load Finally simulation results are studied todamp the oscillation of the suspended load

2 Dynamics Modeling

The quadrotor slung load system is shown in Figure 2 Itis considered to be a system consisting of two rigid bodiesconnected by massless straight-line links which support onlyforces along the linkThe system is characterized bymass andinertia parameters of rigid bodies and suspensionrsquos attach-ment point locations In this section dynamics equations of

Journal of Robotics 3

the quadrotor slung load system are presented by Newton-Euler method The following assumptions are made formodeling the quadrotor with a swinging load

(i) Elastic deformation and shock of the quadrotor areignored

(ii) Inertia matrix is time-invariant

(iii) Mass distribution of the quadrotor is symmetrical inthe 119909-119910 plane

(iv) Drag factor and thrust factor of the quadrotor areconstant

(v) Air density around of the quadrotor is constant

(vi) Thrust force and drag moment of each propellers areproportional to the square of the propeller speed

(vii) Both bodies are assumed to be rigidThis assumptionexcludes an elastic quadrotor and rotormodes such asflapping and nonrigid loads

(viii) The cable mass and aerodynamic effects on the loadare neglected

(ix) The cable is considered to be inelastic

These assumptions are considered to be sufficient for therealistic representation of the quadrotor with a swinging loadsystem which is used for a nonaggressive trajectory tracking

21 Aerodynamics of Rotor and Propeller The aerodynamicforce andmoment are obtained by combining themomentumtheory of the blade element [8 9] A quadrotor has fourmotors with propellers The power applied to each motor 119875generates a torque on the rotor shaft 119876 and a force 119879 Thesetorques and forces are generated by each rotor-propeller andthey are proportional to the square of the propeller speed as

119879 = 119862119879

41205881198774

12058721205962

119898

119876 = 119862119876

41205881198775

12058731205962

119898

(1)

where 120596119898

is the rotor velocity 120588 is the air density 119877 isthe propeller radius 119862

119879is the thrust factor and 119862

119876is the

momentum factor [9 10]

22 Dynamics Equations of Motion

221 Kinematics Equation of Quadrotor As shown inFigure 1 the quadrotor has four rotors which can generateidentical thrusts and moments denoted by 119879

119894and 119876

119894 for 119894 =

1 2 3 4 respectively Let IF represent a right-hand inertiaframe with the 119911-axis being the vertical direction to the earthThe body fixed frame is denoted by BF that center of thisframe is located on the mass center of the quadrotor TheEuclidean position of the quadrotor with respect to IF isrepresented by the 119909 119910 and 119911 Also the Euler angle of thequadrotorwith respect to IF is represented by the120593 120579 and120595

Thus the rotation matrix from BF to IF can be representedby R as

R = (119888120579119888120595 119904120593119904120579119888120595 minus 119888120593119904120595 119888120593119904120579119888120595 + 119904120593119904120595

119888120579119904120595 119904120593119904120579119904120595 + 119888120593119888120595 119888120593119904120579119904120595 minus 119904120593119888120595

minus119904120579 119888120579119904120593 119888120579119888120595

) (2)

where 119888 and 119904 refer to cos and sin function respectively Alsothe translational and the rotational kinematics equationswithrespect to the inertial frame IF can be yielded as

X = RV

= P120596(3)

where V and 120596 denote the linear velocity and the angularvelocity of the quadrotor with respect to the inertial frameIF expressed in the body fixed frame BF So the rotationvelocity transfer matrix P can be given as

P = (1 119904120593119905120579 119888120593119905120579

0 119888120593 minus119904120593

0119904120593

119888120579

119888120593

119888120579

) (4)

222 Newton-Euler Equation of Quadrotor As the free bodydiagram of the quadrotor slung load system shown inFigure 2 the Newton-Euler equations for quadrotor in theinertia frame can be obtained as

mX = minusmG minus RKtR119879X + RT minus RFcable

J = minus P (P119879 times JP119879) minus PKrP119879

minus PJ(120597P119879

120597120593 +

120597P119879

120597120579) + P120591

(5)

where m is the mass matrix of the quadrotor J is the inertiamatrix of the quadrotor G = [0 0 119892]

119879 is the gravity matrixFcable is the cable force and Kt and Kr are the linear andangular aerodynamic friction factor respectively AlsoT and120591matrices are given as

T = 0 0 119862119879

41205881198774

1205872

4

sum

119894=1

1205962

119898119894

119879

120591 =

119897119862119879

41205881198774

1205872(minus1205962

1198982

+ 1205962

1198984

)

119897119862119879

41205881198774

1205872(minus1205962

1198983

+ 1205962

1198981

)

119862119901

41205881198775

1205873

4

sum

119894=1

(minus1)119894

1205962

119898119894

(6)

However actuator forces and moments are summarizedas

1198801= 1198791+ 1198792+ 1198793+ 1198794

1198802= 119880120593= 119897 (1198794minus 1198792)

1198803= 119880120579= 119897 (1198793minus 1198791)

1198804= 119880120595= (minus119876

1+ 1198762minus 1198763+ 1198764)

(7)


where 119897 is distance of two rotors opposite to each other 1198791to

1198794are thrust forces which are generated by rotors 1 to 4 and

1198761to 1198764are moments which are generated by rotors 1 to 4

So 1198801results in the motion along the 119911

119887axis Also 119880

2 1198803

and1198804create the roll pitch and yaw motion respectively In

this system Fcable is considered systemrsquos input (or the actuatorforce for changing the cable length) which can be representedin the body frame as

Fcable =1003816100381610038161003816Fcable

1003816100381610038161003816[

[

sin120573 cos 120574sin120573 sin 120574cos120573

]

]

(8)

where |Fcable| is cablersquos force magnitude Also the relationbetween Fcable and the cable length can be stated as

1003816100381610038161003816Fcable1003816100381610038161003816 = 119898119901 119903 (9)

where 119903 is the cable length 119903 is the acceleration of massrelative to the quadrotor in the body coordinate and 119898

119901is

the load mass So motionrsquos equation of the load in the inertiaframe can be obtained as

119898119901(X + R (A + B + 120596 times (A)))

minus 119898119901R (120596 times Rb) + 119898119901G = RFcable

(10)

where

A = [

[

119904120573119888120574 minus119903119904120573119904120574 119903119888120573119888120574

119904120573119904120574 119903119904120573119888120574 119903119888120573119904120574

119888120573 0 minus119903119904120573

]

]

(11a)

120572 =

119903

120574

120573

B =

(2 119903 120573119888120573119888120574 minus 2 119903 120574119904120573119904120574 minus 2119903 120573 120574119888120573119904120574

minus119903 1205732

119904120573119888120574 minus 119903 1205742

119904120573119888120574)

(2 119903 120573119888120573119904120574 + 2 119903 120574119904120573119888120574 + 2119903 120573 120574119888120573119888120574

minus119903 1205732

119904120573119904120574 minus 119903 1205742

119904120573119904120574)

(minus2 119903 120573119904120573 minus 119903 1205732

119888120573)

(11b)

And Rb is the load velocity with respect to the quadrotorwhile it is expressed in the body frame Equations (5) and (10)are motion equations of system with generalized coordinatesas follows

q = [119909 119910 119911 120593 120579 120595 119903 120574 120573]119879

(12)

By considering the constant length of the cable thesystem has eight degrees of freedom

223 Lagrange Equation of Quadrotor To obtain the dy-namic equations of motion by Lagrange method generalizedcoordinates are defined as

q = [119909 119910 119911 120593 120579 120595 119903 120574 120573]119879

(13)

So the kinetic energy for the quadrotor is

119870119902=1

2XTM119902X + 1

2119879Ji (14)

and the swinging loadrsquos kinetic energy can be attained as

Kp =1

2X119879pMpXp =

1

2(X + Rs)

119879

Mq (X + Rs) (15)

whereMqMp are themass diagonal matrix and the swingingload of the quadrotor respectively Also Rs is the swingingload velocity with respect to the quadrotor inertia frame(IF) Moreover Ji is the inertia moment matrix of thequadrotor in IF which can be calculated as

Ji = RJR119879 (16)

where J is the inertia moment matrix of the quadrotor inthe body frame (BF) Finally the closed form equation ofmotion can be obtained as

Mq + V (q q) + G (q) = Q (17)

whereas by considering (9)M9times9

is the mass matrix V9times1

isthe nonlinear velocity matrix and G

9times1is the gravity matrix

AlsoQ is the generalized force

224 Model Verification In this section in order to verifythe obtained dynamics model since the motion equations byNewton-Euler method are inhomogeneous unlike Lagrangeequations a desired path for the quadrotor flight is definedThen by solving Lagrange equations rotors input are com-puted for tracking this desired path (solving the inversedynamics) Next these forces are exerted to Newton-Eulerequations as inputs and these equations are solved (solvingthe forward dynamics) This procedure shows that responsesof Newton-Euler and Lagrange equations are the same as thedesired path To this end in simulations a specified pathfor the quadrotor is considered and the inverse dynamic issolved to obtain the desired forces for the considered path (forLagrange equations) By defining these forces as inputs for theNewton-Euler equations and solving these equations to findthe tracked path by the quadrotor the dynamics model canbe verified So a hover flight is defined in 119911 = 2m and thesimulation result is shown in Figure 3 As shown form thisfigure both paths are the same In next scenario a verticaltake-off flight is defined Therefore the desired path for thisflight is

119911 = (minus00021199053

) + (0031199052

) + 2 (18)

The simulation result for this path is shown in Figure 4It is shown that both responses of these dynamic models arethe same and are reasonable

3 Controller Design

31 Position andAttitude Control of Quadrotor Themain aimof this section is to design a model based control scheme


0 1 2 3 4 5 6 7 8 9 1019999

2

20001

Time (s)

Position of quadrotorDesired position of quadrotor

zan

dz d

(m)

Figure 3 Position of quadrotor in hover flight

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

Time

Erro

r

times10minus6

Z(quadrotor) minus Z(desired path)

Figure 4 Error of responses in vertical take-off flight

for a full control of a quadrotor This control method isa compensation of nonlinear terms based on the accurateknowledge of the dynamics systemThus dynamic equationsof motion based on the accurate knowledge of the dynamicssystem can be stated as

120591 = M (q) q + V (q q) + G (q) (19)

where M V and G are the mass matrix the nonlinearvelocity matrix and the gravity matrix respectively Thesematrices are obtained based on the physical knowledge andthe geometrical dimensions In Figure 5 the block diagram

qd(t)

qd(t)

qd(t)

+

+ +

+

minus

+ minus

e

e

Kp

Kd

V + G

MMVG

q

qMq + V + G = Fτ

Figure 5 Model based control algorithm scheme

of this control algorithm is shown According to this diagramthe control law can be calculated as

120591cc = V (q q) + G (q) +M qd + Kpe + Kde (20)

Also this control torque can be applied to the belowdynamics equations of the considered system as

120591 = M (q) q + V (q q) + G (q) (21)

Moreover by considering what is well known about thedynamic parameters of the system it can be concluded that

M ≜ M V ≜ V G ≜ G (22)

So by substituting (20) into (21) and considering theassumption in (22) it yields

M e + Kpe + Kde = 0 (23)

AsM is the positive definition matrix so it can be writtenas

e + Kpe + Kde = 0 (24)

which confirms the error convergence by choosing the propercontroller gains Kp and Kd

Using this algorithm and by choosing optimal gains forthe designed controller the position and attitude of thequadrotor are controlled So the model can be viewed as twoindependent subsystems which are the transitional move-ment subsystem and the angular movement subsystem Thetransitional motion does not affect the angular motion butthe angular motion affects the transitional motion Howeverfor designing a control algorithm in order to take full controlof degrees of freedom quadrotorrsquos dynamics equations mustbe divided into two subsystems the transitional subsystemand the rotation subsystem Therefore the control algorithmis designed in two parts the position controller and theattitude controller Figure 6 shows this control algorithmscheme

Based on quadrotorrsquos operation in different flights itis obvious that the quadrotor does not have any actuatorforce which directly creates a movement along the 119909- andthe 119910-axis Thus this robot can fly in the 119909 and the 119910direction by creation of the pitch and the roll motion Based


Desired path

Position controller

controllerAttitude

Position dynamic

Attitude dynamic

U1

U2U3U4

(x y z r 120574 120573)

(120593 120579 120595)

120579ctrl120593ctrl

120593d120579d120595d

xdydzdrd

Figure 6 Block diagram of the considered control system

on this principle control forces can be obtained using thetransitional subsystem as

(119888120593119904120579119888120595 + 119904120593119904120595)1198801= 119865119909

(119888120593119904120579119904120595 minus 119904120593119888120595)1198801= 119865119910

1198881205791198881205931198801= 119865119911

(25)

where 119865119909 119865119910 and 119865

119911are created forces motion along the 119909-

119910- and 119911-axis which are calculated according to the desiredtrajectory Also the vector of control forcesFcc = [Fx Fy Fz]

119879

can be calculated according to the transitional subsystem(transitional motion equation of the quadrotor and swingload) as

Fcc = Mt (qd + Kpte + Kdte) + Vt (q q) + Gt (q) (26)

where Mt(6 times 6) Vt(6 times 1) and Gt(6 times 1) are the massmatrix the nonlinear velocity matrix and the gravity matrixof the transitional dynamic subsystem respectively Alsoqd is the desired acceleration and e = qd minus q (whereq = [119909 119910 119911 119903 120574 120573]) is the position error Moreover Kpt andKdt are controller gains By solving three equations of (25)simultaneously control outputs 119880

1-ctrl 120593ctrl and 120579ctrl whichcreate motion along the 119909- 119910- and 119911-axis are calculated as

1198801-ctrl =

119865119911

119888120579119888120593

120593ctrl = arcsin 11198801

(119865119909119904120595119889minus 119865119910119888120595119889)

120579ctrl = arcsin 1

1198801119888120593ctrl

(119865119909119888120595119889+ 119865119910119904120595119889)

(27)

where 120595119889is the desired value of the yaw angle In the

same way the attitude controller is designed according tothe rotational subsystem Therefore to perform this controlpropose the torque control vector or 120591cc = [U2U3U4]

119879 canbe considered

120591cc = Mr (qd + Kpre + Kdre) + Vr (q q) + Gr (q) (28)

where Mr Vr and Gr are the mass matrix the nonlinearvelocity matrix and the gravity matrix of the rotationaldynamic subsystem (rotational motion of the quadrotor)respectively Also q = [120593 120579 120595] and Kpr Kdr are rotationcontrollerrsquos gains

Table 1 MBA and PID controller gains values

Controllers gains Value119870119901119905= 119870119901119903

100119870119889119905= 119870119889119903

80119870119901119909

= 119870119901119910

50119870119889119909

= 119870119889119910

8119870119894119909= 119870119894119910

8119870119901119911

40119870119889119911

14119870119894119911

20119870119901120593

= 119870119901120579

= 119870119901120595

80119870119894120593= 119870119894120579= 119870119894120595

10119870119889120593

= 119870119889120579

8119870119889120595

6

Table 2 Physical parameters of the system

Parameter Value119898 (Kg) 065

119898119901(Kg) 03

119897 (m) 0232

119903 (m) 1

119869119909(Kgm2) 75 times 10

minus3

119869119910(Kgm2) 75 times 10

minus3

119869119911(Kgm2) 13 times 10

minus2

119862119879

007428

119862119876

010724

119870119903= 119870119905

10 times 10minus15

32 Simulation Results of Designed Position and AttitudeController To verify the effectiveness and the applicationeffect of the proposed control method the simulation studyhas been carried on the quadrotor with a swinging load Inthis section simulation results of theModel Based Algorithmcontroller (MBA controller) are compared to a PID controllerin different maneuvers It should be noted that the gains ofPID controller are calculated by try and error method Socontroller gains and the system parameters of simulations aredescribed in Tables 1 and 2 In the first case study a linear pathin the 119909-119910 plane at 119911 = 3m is definedThe equation of desiredpath is defined as follows

119911 = 3 119909 = 02119905 + 1 119910 = 02119905 + 1 (29)

Simulation results of the designed controller are shownin Figures 7ndash9 As shown in Figure 7 the controller perfor-mance in comparison with a PID controller in this maneuveris good As seen from Figure 8 the quadrotor finds thedesired path by the MBA controller earlier than the PIDcontroller Also Figure 9 shows the maximum error of the 119909and the 119910 position so that the consideredMBA controller hasthe better performance than the PID controller


05 1 15 2 25 3 350

24

14

16

18

2

22

24

26

28

3

32

x

y

z

Desired pathReal path of quadrotor (MBA controller)Real path of slung load (MBA controller)Real path of quadrotor (PID controller)Real path of slung load (PID controller)

Figure 7 Linear path of quadrotor slung load system

0 1 2 3 4 5 6 7 8 9 1025

26

27

28

29

3

31

32

Altitude of quadrotor (MBA controller) Altitude of quadrotor (PID controller)

Time

Figure 8 Altitude of quadrotor in linear flight

The last case study is a circular path for the quadrotorflight (Figure 10) In this scenario the superior performanceof the proposed MBA controller is compared to the PIDcontroller (Figures 11 and 12) The position error of thequadrotor is converged to the zero after a few seconds by thedesigned model based control algorithm while by the PIDcontroller this error has oscillations about zero and it is notzero

33 Antiswing Control of Quadrotor In the first step of theantiswing controller design procedure it should be better tofind commands that move systems without vibration So it ishelpful to start with the simplest command It is known that

Erro

r

Error of x (MBA controller)Error of y (MBA controller)

Error of x (PID controller)Error of y (PID controller)

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

times10minus3

minus6

minus4

minus2

0

2

4

6

8

10

12

Figure 9 Position error of quadrotor in linear flight

02

40

24

0

05

1

15

xy

z


minus4

minus2

minus4minus2

minus05

Figure 10 Circle path flight of quadrotor slung load system

by giving an impulse to the system it will cause vibrationHowever if a second impulse is applied to the system in theright moment the vibration included by the first impulse iscancelled [23] This concept is shown in Figure 13 Subse-quently input shaping theory is implemented by convolvingthe reference command with a sequence of impulses Thisprocess is illustrated in Figure 14The impulse amplitudes andtime locations of their sequence are calculated in order toobtain a stair-like command to reduce the detrimental effectsof system oscillations The amplitude and time locations ofimpulses are calculated by estimation of systemrsquos naturalfrequency and the damping ratio [30]


0 1 2 3 4 5 6Time

Erro

r

Error of xError of y

Error of z

minus01

0

01

02

03

04

05

06

Figure 11 Position error of quadrotor in circle flight with MBAcontroller

0 1 2 3 4 5 6Time


Error of z

Erro

r

minus01

0

01

02

03

04

05

06

Figure 12 Position error of quadrotor in circle flight with PIDcontroller

If estimations of systemrsquos natural frequency 120596119899 and the

damping ratio 120577 for vibration modes of the system whichmust be canceled are known then the residual vibration thatresults from the sequence of impulses can be described as

] (120596119899 120577) = 119890

minus120577120596119899119905119873radic119862 (120596119899 120577)2

+ 119878 (120596119899 120577)2

(30)

where

119862 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 cos (120596

119889119905119894)

119878 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 sin (120596

119889119905119894)

(31)

where 119860119894and 119905119894are the amplitude and the time locations

of impulses 119873 is the number of impulses in the impulse

0

05

Am

plitu

deResponse to both impulses

Response to first impulsesResponse to second impulses

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

minus05

0

05

Am

plitu

de

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

minus05

Figure 13 Response of dynamic system to two sequent impulsesA

mpl

itude

Time Time

Am

plitu

de

lowastA1

A2

0 t2 0 t2

Figure 14 Input shaping process with ZV shaper

sequence and 120596119889= 120596119899radic(1 minus 1205772) To avoid the trivial solution

of all zero valued impulses and to obtain a normalized resultit is necessary to satisfy the below equation

119873

sum

119894=1

119860119894= 1 (32)

In themodeling of a suspended load the load damping or120577 can be neglected and the frequency of the linear model canbe considered

120596119899= radic

119892

119897 (33)

where 119897 is length of the cable When the amplitude ofimpulses is considered positive and the estimation of systemrsquosparameters is accurate the simplest shaper or the ZV shapercan be proposed with two impulses in two times 119905

1and 1199052

Convolving this shaper to the step input is shown in Figure 14The ZV shaper can cancel the oscillation of systems whosenatural frequency and damping ratio are specific and knownIn this case this assumption can be considered to drivedynamic equations The cable is considered inelastic and theaerodynamic force on the suspended load is neglected So theexact estimation of natural frequency and the damping ratiodoes not exist In 1993 Bohlke proposed amethod to increasethe robustness of the ZV shaper against errors of systemrsquos


parameter estimation [23] In this proposed method thederivation of the residual vibration to the natural frequencyand the damping ratio must be zero as

119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)

These two equations can be written as

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)

Due to adding these two constraints two variables arerequired to be added by an impulse with the 119860

3amplitude

at the time 1199053to the ZV shaper The result of this addition is

the ZVD shaper with 3 impulses and 5 unknown 1198601 1198602 1198603

1199052 and 119905

3(with the assumption 119905

1= 0) for this shaper Thus

the following equations can be obtained

1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)

By solving these four equations unknown parameters ofthe input shaper are calculated as

1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)

For the quadrotor with the suspended load the cablelength is assumed as 119903 = 1m By estimating the naturalfrequency and the damping ratio input shaperrsquos parametersare obtained as

1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)

However the transfer function of the designed ZVDshaper is

119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)

For the small oscillation suspended loadrsquos motion iscaused by the horizontal motion of the quadrotor So it isnecessary that the shaper is implemented on the part of thedesired path If the part of the desired path for the shaper isdefined as

Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)

consequently the shaped path for the quadrotor is

X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)

where ΓZVD is the transfer functionmatrix of the ZVD shaperin the time domain By considering the shaped path for thequadrotor controller the controller performance is improvedand the oscillation of the swing load is reduced In nextsection simulation results of the implementing shaper oninput commands are shown and discussed

34 Simulation Results of Designed Antiswing Controller Toverify the effectiveness and the application effect of thecontrol method for canceling the oscillation of the suspendedload the simulation routine has been carried on the quadro-tor with the suspended load In addition for estimatingthe natural frequency and the damping ratio this systemis compared to a cart-pendulum system So the naturalfrequency of system is assumed as (33) Also the dampingratio of oscillation mode of the swing load is related to theaerodynamics drag on the load and suspension system andso the damping ratio is zero according to the small scaleof system consideration Moreover the system parametersused in the simulation are listed in Table 2 Moreover thedesired path is defined as the line-curve-line and simulationresults are shown in Figure 15 By comparing Figure 15 withFigure 16 it can be concluded that the designed feed-forwardcontroller can effectively cancel the oscillation of the sus-pended load In addition Figure 17 shows the position errorof the quadrotor both with ZVD shaper and without ZVDshaper As seen from this figure the error is converged to zeroafter a few seconds when input shaping is implemented ondesired command and the performance of the controller is


02

46

8

02

460

1

2

3

4

xy

z

Desired pathReal path of quadrotor

Real path of slung load

minus2 minus2

Figure 15 3D flight of quadrotor with input shaping controller

05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2

Figure 16 3D flight of quadrotor without input shaping controller

better Also it is clear that by implementing input shapingon a desired path the required time for tracking the desiredpath is increased So the quadrotor requires more time fortracking the desired path

In the second senario the square path is designed for thequadrotor flight In Figures 18 and 19 simulation results areshown both with ZVD shaper and without it As can be seenfrom these figures the feed-forward controller can cancel theoscillation Also Figure 20 shows the position error of thequadrotor with input shaping As shown from this figureerror is zero after few seconds

0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)

Figure 17 Position error of quadrotor (a) without input shaper(b) with input shaper

4 Conclusions

In this paper the problem of the quadrotor flying is addressedwith a suspended load which is widely used for differentkinds of cargo transportation The suspended load is alsoknown as either the slung load or the sling load Alsoflying with a suspended load can be a very challengingand sometimes hazardous task because the suspended loadsignificantly alters the flight characteristics of the quadrotorSo many different control algorithms have been proposed tocontrol these systems To this end dynamic model of thissystem was obtained and verified by comparing two Newton-Euler and Lagrange methods Next a control algorithm was


005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1



Figure 18 Square path of quadrotor with input shaping controller

4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05

Figure 19 Square path of quadrotor without input shaping con-troller

presented for the position and attitude of the quadrotor Inthis algorithm swinging objectrsquos oscillationmay cause dangerin the work space and it canmake instability in the quadrotorflight Using comprehensive simulation routine it was shownthat this designed controller could control the robot motionon the desired path but could not reduce the load oscillationin noncontinuous and nondifferentiable paths To deal withthis issue a feed-forwarded control algorithmwas introducedfor reducing or canceling swinging loadrsquos oscillation Thiscontroller was designed by implementing the input shap-ing theory which convolves the reference command with

0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02

Figure 20 Position Error of quadrotor with input shaping

a sequence of impulses Finally it was shown that the feed-forward controller could actively improve the performanceof the feedback controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S G Vazquez and J M Valenzuela ldquoA new nonlinear PIPIDcontroller for quadrotor posture regulationrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference pp 642ndash647 October 2010

[2] G Lee D Y Jeong N Khoi and T Kang ldquoAttitude controlsystem design for a quadrotor flying robotrdquo in Proceedings of the8th International conference on Ubiquitous Robots and AmbientIntelligence (URAI rsquo12) Incheon Republic of Korea 2012

[3] G Hoffmann ldquoThe Stanford testbed of autonomous rotorcraftfor multi agent control (STARMAC)rdquo in Proceedings of the 23rdDigital Avionics Systems Conference (DASC rsquo04) vol 2 October2004

[4] Z Fang and W Gao ldquoAdaptive integral backstepping controlof a Micro-Quadrotorrdquo in Proceedings of the 2nd InternationalConference on Intelligent Control and Information Processing(ICICIP rsquo11) pp 910ndash915 Harbin China July 2011

[5] B Erginer and E Altug ldquoDesign and implementation of ahybrid fuzzy logic controller for a quadrotor VTOL vehiclerdquoInternational Journal of Control Automation and Systems vol10 no 1 pp 61ndash70 2012

[6] G V Raffo M G Ortega and F R Rubio ldquoBackstep-pingnonlinear 119867

infincontrol for path tracking of a quadrotor

unmanned aerial vehiclerdquo in Proceedings of the AmericanControl Conference (ACC rsquo08) pp 3356ndash3361 Seattle WashUSA June 2008

[7] E de Vries and K Subbarao ldquoBackstepping based nested multi-loop control laws for a quadrotorrdquo in Proceedings of the 11th


International Conference on Control Automation Robotics andVision pp 1911ndash1916 Singapore December 2010

[8] D Mellinger N Michael M Shomin and V Kumar ldquoRecentadvances in quadrotor capabilitiesrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation (ICRArsquo11) pp 2964ndash2965 Shanghai China May 2011

[9] M Hehn and R DrsquoAndrea ldquoA flying inverted pendulumrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo11) pp 763ndash770 May 2011

[10] V Ghadiok J Goldin and W Ren ldquoAutonomous indoor aerialgripping using a quadrotorrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems 2011

[11] E Doyle J Bird T A Isom et al ldquoAn Avian-inspired passivemechanism for quadrotor perchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems San Francisco Calif USA 2011

[12] A Mattio ldquoDevelopment of autonomous cargo transport for anunmanned aerial vehicle using visual servoingrdquo in Proceedingsof the ASMEDynamic Systems and Control Conference pp 407ndash414 October 2008

[13] P E I Pounds D R Bersak and A M Dollar ldquoGrasping fromthe air hovering capture and load stabilityrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo11) pp 2491ndash2498 Shanghai China May 2011

[14] M Korpela W Danko and Y OhDesigning a system for mobilemanipulation from an unmanned aerial vehicle [Master thesis]2005

[15] K Sreenath N Michael and V Kumar ldquoTrajectory generationand control of a quadrotor with a cable-suspended loadmdashadifferentially-flat hybrid systemrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation (ICRArsquo13) pp 4888ndash4895 May 2013

[16] I Palunko R Fierro and P Cruz ldquoTrajectory generation forswing-free maneuvers of a quadrotor with suspended payloada dynamic programming approachrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation (ICRArsquo12) pp 2691ndash2697 2012

[17] N Michael J Fink and V Kumar ldquoCooperative manipulationand transportation with aerial robotsrdquoAutonomous Robots vol30 no 1 pp 73ndash86 2011

[18] J Thalapil ldquoInput shaping for sway control in gantry cranesrdquoIOSR Journal of Mechanical and Civil Engineering vol 14 no 4pp 36ndash46 2012

[19] P Benes and M Valasek ldquoInput shaping control with reentrycommands of prescribed durationrdquo Applied and ComputationalMechanics vol 2 pp 227ndash234 2008

[20] P Benes O Marek and M Valasek ldquoInput shaping control ofelectronic cams with adjusted input profilerdquo Bulletin of AppliedMechanics vol 8 no 29 pp 10ndash14 2011

[21] M Kenison and W Singhose ldquoInput shaper design for double-pendulum planar gantry cranesrdquo in Proceedings of the IEEEInternational Conference on Control Applications vol 1 pp 539ndash544 Kohala Coast Hawaii USA August 1999

[22] M A Ahmad Z Mohamed and Z H Ismail ldquoHybrid inputshaping and PID control of flexible robot manipulatorrdquo Journalof the Institution of Engineers vol 72 no 3 pp 56ndash62 2009

[23] K A BohlkeUsing input shaping to minimize residual vibrationin flexible space structures [MS thesis] Princeton University1993

[24] D BlackburnW Singhose J Kitchen et al ldquoCommand shapingfor nonlinear crane dynamicsrdquo Journal of Vibration and Controlvol 16 no 4 pp 477ndash501 2010

[25] O JM Smith ldquoPosicast control of damped oscillatory systemsrdquoProceedings of the IRE vol 45 no 9 pp 1249ndash1255 1957

[26] C J Swigert ldquoShaped Torques Techniquesrdquo Journal of Guidanceand Control vol 3 no 5 pp 460ndash467 1980

[27] TMita andT Kanai ldquoOptimal control of the crane systemusingthemaximum speed of the trolleyrdquo Transactions of the Society ofInstrument and Control Engineers vol 15 pp 833ndash838 1979

[28] J Yu ldquoNonlinear feedback control of a gantry cranerdquo inProceedings of the American Control Conference pp 4310ndash4314Seattle Wash USA June 1995

[29] H Lee ldquoModeling and control of a three-dimensional overheadcranerdquo ASME Transaction on Journal of Dynamic SystemsMeasurement and Control vol 120 no 4 pp 471ndash476 1998

[30] C Adams J Potter and W Singhose ldquoModeling and inputshaping control of a micro coaxial radio-controlled helicoptercarrying a suspended loadrdquo in Proceedings of the 12th Interna-tional Conference on Control Automation and Systems (ICCASrsquo12) pp 645ndash650 October 2012

[31] M ZM ZainM O Tokhi and ZMohamed ldquoHybrid learningcontrol schemes with input shaping of a flexible manipulatorsystemrdquoMechatronics vol 16 no 3-4 pp 209ndash219 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



1

2

3

4x

y

Figure 1 Scheme of quadrotorrsquos rotor rotation

the quadrotor using a hybrid fuzzy-PD control algorithm [5]In 2008 Raffo et al implemented a nonlinear H

infinalgorithm

to control and stabilize the angular motion of the quadrotorThe simulation results show that this nonlinear algorithm caneliminate disturbances and stabilize the rotation motion ofthe quadrotor [6] de Vries and Subbarao in 2010 designed aback stepping multiloop controller for the hover flight [7]

Recently with introduction of inexpensive micro-Unmanned Aerial Vehicles (UAV) and advanced sensorscontrollers have been designed to enable these systems formany tasks for example offensive maneuvers [8] balancinga flying inverted pendulum [9] Also the progress in sensorsand controllers leads very well to use of UAV for differentapplications One of these applications is transportationof external loads Different grippers for grasping andtransporting of a load are designed [10ndash14] Another one iscable-suspended loads which have been studied in recentyears [15ndash17] Cable-suspended systems are underactuatedsystems Therefore canceling or reducing oscillation of thesuspended load is very necessary since oscillations of loadin an industrial environment and other areas can result indamage [18ndash22] So different control methods have beenproposed to control these robots since the suspended loadsignificantly alters the flight characteristics of the quadrotorThese methods are divided into feedback and feed-forwardapproaches Feedback control methods use measurementsand estimations of system states to reduce the vibrationwhile feed-forward approaches change actuator commandsfor reducing the oscillation of system The feed-forwardcontroller can often improve the performance of feedbackcontrollerThus proposing feed-forward algorithms can leadto more practical and accurate control of these systems Oneeffective feed-forward method is the input shaping theorywhich has proven to be a practical and effective approachof reducing vibrations [23 24] Also several methods areproposed in order to minimize the residual vibration Smithproposed the Posicast control of the damped oscillatorysystems which is a technique to generate a nonoscillatoryresponse from a damped system to a step input This methodbreaks a step of a certain magnitude into two smaller stepsone of which is delayed in time [25] Swigert proposedshaped torques techniques which consider the sensitivity ofterminal states to variation in the model parameters [26]

x

y

z

zb

T1

T2

T3

T4

Q1

Q2Q3

Q4

120573

ybxb

120574

Fcablemg

mpg

Figure 2 Free body diagram of quadrotor slung load system

Recently in the control of overhead cranes Mita and Kanaisolved a minimum time control problem for swing freevelocity profiles which resulted in an open loop control [27]Also Yu proposed a nonlinear control based on the singularperturbation method [28] In addition Lee designed a highperformance control based on the loop shaping and rootlocus methods [29] Also in 2012 Adams et al designedinput shaping control of a micro-coaxial radio-controlledhelicopter carrying a suspended load [30] Zain et al in2006 proposed hybrid learning control schemes with aninput shaping of a flexible manipulator system [31]

In this paper the problem of the quadrotor flying with asuspended load is addressedwhich is widely used for differentkinds of a cargo transport The paper is organized in twoparts In the first part a nonlinear model of an underactu-ated eight-degree-of-freedom quadrotor slung load systemis derived on the basis of the Newton-Euler formulationNext this dynamic model is verified in comparison withLagrange method Then a nonlinear model based controlalgorithm is designed for the position and attitude controlof the quadrotor with the suspended load In next part thedescription of the input shaping algorithm is presented andthen this method is implemented to the quadrotor with asuspended load Finally simulation results are studied todamp the oscillation of the suspended load

2 Dynamics Modeling

The quadrotor slung load system is shown in Figure 2 Itis considered to be a system consisting of two rigid bodiesconnected by massless straight-line links which support onlyforces along the linkThe system is characterized bymass andinertia parameters of rigid bodies and suspensionrsquos attach-ment point locations In this section dynamics equations of














119879 = 119862119879

41205881198774

12058721205962

119898

119876 = 119862119876

41205881198775

12058731205962

119898

(1)

where 120596119898



119876is the




119894and 119876

119894 for 119894 =



R = (119888120579119888120595 119904120593119904120579119888120595 minus 119888120593119904120595 119888120593119904120579119888120595 + 119904120593119904120595

119888120579119904120595 119904120593119904120579119904120595 + 119888120593119888120595 119888120593119904120579119904120595 minus 119904120593119888120595

minus119904120579 119888120579119904120593 119888120579119888120595

) (2)


X = RV

= P120596(3)


P = (1 119904120593119905120579 119888120593119905120579

0 119888120593 minus119904120593

0119904120593

119888120579

119888120593

119888120579

) (4)




minus PJ(120597P119879

120597120593 +

120597P119879

120597120579) + P120591

(5)



T = 0 0 119862119879

41205881198774

1205872

4

sum

119894=1

1205962

119898119894

119879

120591 =

119897119862119879

41205881198774

1205872(minus1205962

1198982

+ 1205962

1198984

)

119897119862119879

41205881198774

1205872(minus1205962

1198983

+ 1205962

1198981

)

119862119901

41205881198775

1205873

4

sum

119894=1

(minus1)119894

1205962

119898119894

(6)


1198801= 1198791+ 1198792+ 1198793+ 1198794

1198802= 119880120593= 119897 (1198794minus 1198792)

1198803= 119880120579= 119897 (1198793minus 1198791)

1198804= 119880120595= (minus119876

1+ 1198762minus 1198763+ 1198764)

(7)






119887axis Also 119880

2 1198803



Fcable =1003816100381610038161003816Fcable

1003816100381610038161003816[

[


]

]

(8)


1003816100381610038161003816Fcable1003816100381610038161003816 = 119898119901 119903 (9)


119901is


119898119901(X + R (A + B + 120596 times (A)))


(10)

where

A = [

[

119904120573119888120574 minus119903119904120573119904120574 119903119888120573119888120574

119904120573119904120574 119903119904120573119888120574 119903119888120573119904120574

119888120573 0 minus119903119904120573

]

]

(11a)

120572 =

119903

120574

120573

B =

(2 119903 120573119888120573119888120574 minus 2 119903 120574119904120573119904120574 minus 2119903 120573 120574119888120573119904120574

minus119903 1205732

119904120573119888120574 minus 119903 1205742

119904120573119888120574)

(2 119903 120573119888120573119904120574 + 2 119903 120574119904120573119888120574 + 2119903 120573 120574119888120573119888120574

minus119903 1205732

119904120573119904120574 minus 119903 1205742

119904120573119904120574)

(minus2 119903 120573119904120573 minus 119903 1205732

119888120573)

(11b)


q = [119909 119910 119911 120593 120579 120595 119903 120574 120573]119879

(12)



q = [119909 119910 119911 120593 120579 120595 119903 120574 120573]119879

(13)


119870119902=1

2XTM119902X + 1

2119879Ji (14)


Kp =1

2X119879pMpXp =

1

2(X + Rs)

119879

Mq (X + Rs) (15)


Ji = RJR119879 (16)


Mq + V (q q) + G (q) = Q (17)







119911 = (minus00021199053

) + (0031199052

) + 2 (18)


3 Controller Design



0 1 2 3 4 5 6 7 8 9 1019999

2

20001

Time (s)


zan

dz d

(m)


0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

Time

Erro

r

times10minus6




120591 = M (q) q + V (q q) + G (q) (19)


qd(t)

qd(t)

qd(t)

+

+ +

+

minus

+ minus

e

e

Kp

Kd

V + G

MMVG

q

qMq + V + G = Fτ





120591 = M (q) q + V (q q) + G (q) (21)


M ≜ M V ≜ V G ≜ G (22)


M e + Kpe + Kde = 0 (23)


e + Kpe + Kde = 0 (24)





Desired path

Position controller

controllerAttitude

Position dynamic

Attitude dynamic

U1

U2U3U4

(x y z r 120574 120573)

(120593 120579 120595)


120593d120579d120595d

xdydzdrd



(119888120593119904120579119888120595 + 119904120593119904120595)1198801= 119865119909

(119888120593119904120579119904120595 minus 119904120593119888120595)1198801= 119865119910

1198881205791198881205931198801= 119865119911

(25)

where 119865119909 119865119910 and 119865



119879





1198801-ctrl =

119865119911

119888120579119888120593

120593ctrl = arcsin 11198801

(119865119909119904120595119889minus 119865119910119888120595119889)


1198801119888120593ctrl

(119865119909119888120595119889+ 119865119910119904120595119889)

(27)








100119870119889119905= 119870119889119903

80119870119901119909

= 119870119901119910

50119870119889119909

= 119870119889119910

8119870119894119909= 119870119894119910

8119870119901119911

40119870119889119911

14119870119894119911

20119870119901120593

= 119870119901120579

= 119870119901120595

80119870119894120593= 119870119894120579= 119870119894120595

10119870119889120593

= 119870119889120579

8119870119889120595

6



119898119901(Kg) 03

119897 (m) 0232

119903 (m) 1

119869119909(Kgm2) 75 times 10

minus3

119869119910(Kgm2) 75 times 10

minus3

119869119911(Kgm2) 13 times 10

minus2

119862119879

007428

119862119876

010724

119870119903= 119870119905

10 times 10minus15


119911 = 3 119909 = 02119905 + 1 119910 = 02119905 + 1 (29)



05 1 15 2 25 3 350

24

14

16

18

2

22

24

26

28

3

32

x

y

z



0 1 2 3 4 5 6 7 8 9 1025

26

27

28

29

3

31

32


Time




Erro

r



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

times10minus3

minus6

minus4

minus2

0

2

4

6

8

10

12


02

40

24

0

05

1

15

xy

z


minus4

minus2

minus4minus2

minus05




0 1 2 3 4 5 6Time

Erro

r


Error of z

minus01

0

01

02

03

04

05

06


0 1 2 3 4 5 6Time


Error of z

Erro

r

minus01

0

01

02

03

04

05

06




] (120596119899 120577) = 119890

minus120577120596119899119905119873radic119862 (120596119899 120577)2

+ 119878 (120596119899 120577)2

(30)

where

119862 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 cos (120596

119889119905119894)

119878 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 sin (120596

119889119905119894)

(31)



0

05

Am

plitu



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

minus05

0

05

Am

plitu

de

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

minus05


mpl

itude

Time Time

Am

plitu

de

lowastA1

A2

0 t2 0 t2




119873

sum

119894=1

119860119894= 1 (32)


120596119899= radic

119892

119897 (33)


1and 1199052




119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)


119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)


3amplitude



1199052 and 119905




1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)


1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)


1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)


119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)


Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)


X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)




02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















119879 = 119862119879

41205881198774

12058721205962

119898

119876 = 119862119876

41205881198775

12058731205962

119898

(1)

where 120596119898



119876is the




119894and 119876

119894 for 119894 =



R = (119888120579119888120595 119904120593119904120579119888120595 minus 119888120593119904120595 119888120593119904120579119888120595 + 119904120593119904120595

119888120579119904120595 119904120593119904120579119904120595 + 119888120593119888120595 119888120593119904120579119904120595 minus 119904120593119888120595

minus119904120579 119888120579119904120593 119888120579119888120595

) (2)


X = RV

= P120596(3)


P = (1 119904120593119905120579 119888120593119905120579

0 119888120593 minus119904120593

0119904120593

119888120579

119888120593

119888120579

) (4)




minus PJ(120597P119879

120597120593 +

120597P119879

120597120579) + P120591

(5)



T = 0 0 119862119879

41205881198774

1205872

4

sum

119894=1

1205962

119898119894

119879

120591 =

119897119862119879

41205881198774

1205872(minus1205962

1198982

+ 1205962

1198984

)

119897119862119879

41205881198774

1205872(minus1205962

1198983

+ 1205962

1198981

)

119862119901

41205881198775

1205873

4

sum

119894=1

(minus1)119894

1205962

119898119894

(6)


1198801= 1198791+ 1198792+ 1198793+ 1198794

1198802= 119880120593= 119897 (1198794minus 1198792)

1198803= 119880120579= 119897 (1198793minus 1198791)

1198804= 119880120595= (minus119876

1+ 1198762minus 1198763+ 1198764)

(7)






119887axis Also 119880

2 1198803



Fcable =1003816100381610038161003816Fcable

1003816100381610038161003816[

[


]

]

(8)


1003816100381610038161003816Fcable1003816100381610038161003816 = 119898119901 119903 (9)


119901is


119898119901(X + R (A + B + 120596 times (A)))


(10)

where

A = [

[

119904120573119888120574 minus119903119904120573119904120574 119903119888120573119888120574

119904120573119904120574 119903119904120573119888120574 119903119888120573119904120574

119888120573 0 minus119903119904120573

]

]

(11a)

120572 =

119903

120574

120573

B =

(2 119903 120573119888120573119888120574 minus 2 119903 120574119904120573119904120574 minus 2119903 120573 120574119888120573119904120574

minus119903 1205732

119904120573119888120574 minus 119903 1205742

119904120573119888120574)

(2 119903 120573119888120573119904120574 + 2 119903 120574119904120573119888120574 + 2119903 120573 120574119888120573119888120574

minus119903 1205732

119904120573119904120574 minus 119903 1205742

119904120573119904120574)

(minus2 119903 120573119904120573 minus 119903 1205732

119888120573)

(11b)


q = [119909 119910 119911 120593 120579 120595 119903 120574 120573]119879

(12)



q = [119909 119910 119911 120593 120579 120595 119903 120574 120573]119879

(13)


119870119902=1

2XTM119902X + 1

2119879Ji (14)


Kp =1

2X119879pMpXp =

1

2(X + Rs)

119879

Mq (X + Rs) (15)


Ji = RJR119879 (16)


Mq + V (q q) + G (q) = Q (17)







119911 = (minus00021199053

) + (0031199052

) + 2 (18)


3 Controller Design



0 1 2 3 4 5 6 7 8 9 1019999

2

20001

Time (s)


zan

dz d

(m)


0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

Time

Erro

r

times10minus6




120591 = M (q) q + V (q q) + G (q) (19)


qd(t)

qd(t)

qd(t)

+

+ +

+

minus

+ minus

e

e

Kp

Kd

V + G

MMVG

q

qMq + V + G = Fτ





120591 = M (q) q + V (q q) + G (q) (21)


M ≜ M V ≜ V G ≜ G (22)


M e + Kpe + Kde = 0 (23)


e + Kpe + Kde = 0 (24)





Desired path

Position controller

controllerAttitude

Position dynamic

Attitude dynamic

U1

U2U3U4

(x y z r 120574 120573)

(120593 120579 120595)


120593d120579d120595d

xdydzdrd



(119888120593119904120579119888120595 + 119904120593119904120595)1198801= 119865119909

(119888120593119904120579119904120595 minus 119904120593119888120595)1198801= 119865119910

1198881205791198881205931198801= 119865119911

(25)

where 119865119909 119865119910 and 119865



119879





1198801-ctrl =

119865119911

119888120579119888120593

120593ctrl = arcsin 11198801

(119865119909119904120595119889minus 119865119910119888120595119889)


1198801119888120593ctrl

(119865119909119888120595119889+ 119865119910119904120595119889)

(27)








100119870119889119905= 119870119889119903

80119870119901119909

= 119870119901119910

50119870119889119909

= 119870119889119910

8119870119894119909= 119870119894119910

8119870119901119911

40119870119889119911

14119870119894119911

20119870119901120593

= 119870119901120579

= 119870119901120595

80119870119894120593= 119870119894120579= 119870119894120595

10119870119889120593

= 119870119889120579

8119870119889120595

6



119898119901(Kg) 03

119897 (m) 0232

119903 (m) 1

119869119909(Kgm2) 75 times 10

minus3

119869119910(Kgm2) 75 times 10

minus3

119869119911(Kgm2) 13 times 10

minus2

119862119879

007428

119862119876

010724

119870119903= 119870119905

10 times 10minus15


119911 = 3 119909 = 02119905 + 1 119910 = 02119905 + 1 (29)



05 1 15 2 25 3 350

24

14

16

18

2

22

24

26

28

3

32

x

y

z



0 1 2 3 4 5 6 7 8 9 1025

26

27

28

29

3

31

32


Time




Erro

r



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

times10minus3

minus6

minus4

minus2

0

2

4

6

8

10

12


02

40

24

0

05

1

15

xy

z


minus4

minus2

minus4minus2

minus05




0 1 2 3 4 5 6Time

Erro

r


Error of z

minus01

0

01

02

03

04

05

06


0 1 2 3 4 5 6Time


Error of z

Erro

r

minus01

0

01

02

03

04

05

06




] (120596119899 120577) = 119890

minus120577120596119899119905119873radic119862 (120596119899 120577)2

+ 119878 (120596119899 120577)2

(30)

where

119862 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 cos (120596

119889119905119894)

119878 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 sin (120596

119889119905119894)

(31)



0

05

Am

plitu



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

minus05

0

05

Am

plitu

de

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

minus05


mpl

itude

Time Time

Am

plitu

de

lowastA1

A2

0 t2 0 t2




119873

sum

119894=1

119860119894= 1 (32)


120596119899= radic

119892

119897 (33)


1and 1199052




119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)


119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)


3amplitude



1199052 and 119905




1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)


1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)


1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)


119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)


Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)


X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)




02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119887axis Also 119880

2 1198803



Fcable =1003816100381610038161003816Fcable

1003816100381610038161003816[

[


]

]

(8)


1003816100381610038161003816Fcable1003816100381610038161003816 = 119898119901 119903 (9)


119901is


119898119901(X + R (A + B + 120596 times (A)))


(10)

where

A = [

[

119904120573119888120574 minus119903119904120573119904120574 119903119888120573119888120574

119904120573119904120574 119903119904120573119888120574 119903119888120573119904120574

119888120573 0 minus119903119904120573

]

]

(11a)

120572 =

119903

120574

120573

B =

(2 119903 120573119888120573119888120574 minus 2 119903 120574119904120573119904120574 minus 2119903 120573 120574119888120573119904120574

minus119903 1205732

119904120573119888120574 minus 119903 1205742

119904120573119888120574)

(2 119903 120573119888120573119904120574 + 2 119903 120574119904120573119888120574 + 2119903 120573 120574119888120573119888120574

minus119903 1205732

119904120573119904120574 minus 119903 1205742

119904120573119904120574)

(minus2 119903 120573119904120573 minus 119903 1205732

119888120573)

(11b)


q = [119909 119910 119911 120593 120579 120595 119903 120574 120573]119879

(12)



q = [119909 119910 119911 120593 120579 120595 119903 120574 120573]119879

(13)


119870119902=1

2XTM119902X + 1

2119879Ji (14)


Kp =1

2X119879pMpXp =

1

2(X + Rs)

119879

Mq (X + Rs) (15)


Ji = RJR119879 (16)


Mq + V (q q) + G (q) = Q (17)







119911 = (minus00021199053

) + (0031199052

) + 2 (18)


3 Controller Design



0 1 2 3 4 5 6 7 8 9 1019999

2

20001

Time (s)


zan

dz d

(m)


0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

Time

Erro

r

times10minus6




120591 = M (q) q + V (q q) + G (q) (19)


qd(t)

qd(t)

qd(t)

+

+ +

+

minus

+ minus

e

e

Kp

Kd

V + G

MMVG

q

qMq + V + G = Fτ





120591 = M (q) q + V (q q) + G (q) (21)


M ≜ M V ≜ V G ≜ G (22)


M e + Kpe + Kde = 0 (23)


e + Kpe + Kde = 0 (24)





Desired path

Position controller

controllerAttitude

Position dynamic

Attitude dynamic

U1

U2U3U4

(x y z r 120574 120573)

(120593 120579 120595)


120593d120579d120595d

xdydzdrd



(119888120593119904120579119888120595 + 119904120593119904120595)1198801= 119865119909

(119888120593119904120579119904120595 minus 119904120593119888120595)1198801= 119865119910

1198881205791198881205931198801= 119865119911

(25)

where 119865119909 119865119910 and 119865



119879





1198801-ctrl =

119865119911

119888120579119888120593

120593ctrl = arcsin 11198801

(119865119909119904120595119889minus 119865119910119888120595119889)


1198801119888120593ctrl

(119865119909119888120595119889+ 119865119910119904120595119889)

(27)








100119870119889119905= 119870119889119903

80119870119901119909

= 119870119901119910

50119870119889119909

= 119870119889119910

8119870119894119909= 119870119894119910

8119870119901119911

40119870119889119911

14119870119894119911

20119870119901120593

= 119870119901120579

= 119870119901120595

80119870119894120593= 119870119894120579= 119870119894120595

10119870119889120593

= 119870119889120579

8119870119889120595

6



119898119901(Kg) 03

119897 (m) 0232

119903 (m) 1

119869119909(Kgm2) 75 times 10

minus3

119869119910(Kgm2) 75 times 10

minus3

119869119911(Kgm2) 13 times 10

minus2

119862119879

007428

119862119876

010724

119870119903= 119870119905

10 times 10minus15


119911 = 3 119909 = 02119905 + 1 119910 = 02119905 + 1 (29)



05 1 15 2 25 3 350

24

14

16

18

2

22

24

26

28

3

32

x

y

z



0 1 2 3 4 5 6 7 8 9 1025

26

27

28

29

3

31

32


Time




Erro

r



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

times10minus3

minus6

minus4

minus2

0

2

4

6

8

10

12


02

40

24

0

05

1

15

xy

z


minus4

minus2

minus4minus2

minus05




0 1 2 3 4 5 6Time

Erro

r


Error of z

minus01

0

01

02

03

04

05

06


0 1 2 3 4 5 6Time


Error of z

Erro

r

minus01

0

01

02

03

04

05

06




] (120596119899 120577) = 119890

minus120577120596119899119905119873radic119862 (120596119899 120577)2

+ 119878 (120596119899 120577)2

(30)

where

119862 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 cos (120596

119889119905119894)

119878 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 sin (120596

119889119905119894)

(31)



0

05

Am

plitu



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

minus05

0

05

Am

plitu

de

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

minus05


mpl

itude

Time Time

Am

plitu

de

lowastA1

A2

0 t2 0 t2




119873

sum

119894=1

119860119894= 1 (32)


120596119899= radic

119892

119897 (33)


1and 1199052




119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)


119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)


3amplitude



1199052 and 119905




1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)


1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)


1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)


119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)


Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)


X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)




02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5 6 7 8 9 1019999

2

20001

Time (s)


zan

dz d

(m)


0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

Time

Erro

r

times10minus6




120591 = M (q) q + V (q q) + G (q) (19)


qd(t)

qd(t)

qd(t)

+

+ +

+

minus

+ minus

e

e

Kp

Kd

V + G

MMVG

q

qMq + V + G = Fτ





120591 = M (q) q + V (q q) + G (q) (21)


M ≜ M V ≜ V G ≜ G (22)


M e + Kpe + Kde = 0 (23)


e + Kpe + Kde = 0 (24)





Desired path

Position controller

controllerAttitude

Position dynamic

Attitude dynamic

U1

U2U3U4

(x y z r 120574 120573)

(120593 120579 120595)


120593d120579d120595d

xdydzdrd



(119888120593119904120579119888120595 + 119904120593119904120595)1198801= 119865119909

(119888120593119904120579119904120595 minus 119904120593119888120595)1198801= 119865119910

1198881205791198881205931198801= 119865119911

(25)

where 119865119909 119865119910 and 119865



119879





1198801-ctrl =

119865119911

119888120579119888120593

120593ctrl = arcsin 11198801

(119865119909119904120595119889minus 119865119910119888120595119889)


1198801119888120593ctrl

(119865119909119888120595119889+ 119865119910119904120595119889)

(27)








100119870119889119905= 119870119889119903

80119870119901119909

= 119870119901119910

50119870119889119909

= 119870119889119910

8119870119894119909= 119870119894119910

8119870119901119911

40119870119889119911

14119870119894119911

20119870119901120593

= 119870119901120579

= 119870119901120595

80119870119894120593= 119870119894120579= 119870119894120595

10119870119889120593

= 119870119889120579

8119870119889120595

6



119898119901(Kg) 03

119897 (m) 0232

119903 (m) 1

119869119909(Kgm2) 75 times 10

minus3

119869119910(Kgm2) 75 times 10

minus3

119869119911(Kgm2) 13 times 10

minus2

119862119879

007428

119862119876

010724

119870119903= 119870119905

10 times 10minus15


119911 = 3 119909 = 02119905 + 1 119910 = 02119905 + 1 (29)



05 1 15 2 25 3 350

24

14

16

18

2

22

24

26

28

3

32

x

y

z



0 1 2 3 4 5 6 7 8 9 1025

26

27

28

29

3

31

32


Time




Erro

r



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

times10minus3

minus6

minus4

minus2

0

2

4

6

8

10

12


02

40

24

0

05

1

15

xy

z


minus4

minus2

minus4minus2

minus05




0 1 2 3 4 5 6Time

Erro

r


Error of z

minus01

0

01

02

03

04

05

06


0 1 2 3 4 5 6Time


Error of z

Erro

r

minus01

0

01

02

03

04

05

06




] (120596119899 120577) = 119890

minus120577120596119899119905119873radic119862 (120596119899 120577)2

+ 119878 (120596119899 120577)2

(30)

where

119862 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 cos (120596

119889119905119894)

119878 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 sin (120596

119889119905119894)

(31)



0

05

Am

plitu



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

minus05

0

05

Am

plitu

de

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

minus05


mpl

itude

Time Time

Am

plitu

de

lowastA1

A2

0 t2 0 t2




119873

sum

119894=1

119860119894= 1 (32)


120596119899= radic

119892

119897 (33)


1and 1199052




119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)


119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)


3amplitude



1199052 and 119905




1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)


1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)


1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)


119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)


Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)


X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)




02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Desired path

Position controller

controllerAttitude

Position dynamic

Attitude dynamic

U1

U2U3U4

(x y z r 120574 120573)

(120593 120579 120595)


120593d120579d120595d

xdydzdrd



(119888120593119904120579119888120595 + 119904120593119904120595)1198801= 119865119909

(119888120593119904120579119904120595 minus 119904120593119888120595)1198801= 119865119910

1198881205791198881205931198801= 119865119911

(25)

where 119865119909 119865119910 and 119865



119879





1198801-ctrl =

119865119911

119888120579119888120593

120593ctrl = arcsin 11198801

(119865119909119904120595119889minus 119865119910119888120595119889)


1198801119888120593ctrl

(119865119909119888120595119889+ 119865119910119904120595119889)

(27)








100119870119889119905= 119870119889119903

80119870119901119909

= 119870119901119910

50119870119889119909

= 119870119889119910

8119870119894119909= 119870119894119910

8119870119901119911

40119870119889119911

14119870119894119911

20119870119901120593

= 119870119901120579

= 119870119901120595

80119870119894120593= 119870119894120579= 119870119894120595

10119870119889120593

= 119870119889120579

8119870119889120595

6



119898119901(Kg) 03

119897 (m) 0232

119903 (m) 1

119869119909(Kgm2) 75 times 10

minus3

119869119910(Kgm2) 75 times 10

minus3

119869119911(Kgm2) 13 times 10

minus2

119862119879

007428

119862119876

010724

119870119903= 119870119905

10 times 10minus15


119911 = 3 119909 = 02119905 + 1 119910 = 02119905 + 1 (29)



05 1 15 2 25 3 350

24

14

16

18

2

22

24

26

28

3

32

x

y

z



0 1 2 3 4 5 6 7 8 9 1025

26

27

28

29

3

31

32


Time




Erro

r



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

times10minus3

minus6

minus4

minus2

0

2

4

6

8

10

12


02

40

24

0

05

1

15

xy

z


minus4

minus2

minus4minus2

minus05




0 1 2 3 4 5 6Time

Erro

r


Error of z

minus01

0

01

02

03

04

05

06


0 1 2 3 4 5 6Time


Error of z

Erro

r

minus01

0

01

02

03

04

05

06




] (120596119899 120577) = 119890

minus120577120596119899119905119873radic119862 (120596119899 120577)2

+ 119878 (120596119899 120577)2

(30)

where

119862 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 cos (120596

119889119905119894)

119878 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 sin (120596

119889119905119894)

(31)



0

05

Am

plitu



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

minus05

0

05

Am

plitu

de

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

minus05


mpl

itude

Time Time

Am

plitu

de

lowastA1

A2

0 t2 0 t2




119873

sum

119894=1

119860119894= 1 (32)


120596119899= radic

119892

119897 (33)


1and 1199052




119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)


119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)


3amplitude



1199052 and 119905




1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)


1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)


1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)


119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)


Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)


X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)




02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










05 1 15 2 25 3 350

24

14

16

18

2

22

24

26

28

3

32

x

y

z



0 1 2 3 4 5 6 7 8 9 1025

26

27

28

29

3

31

32


Time




Erro

r



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

times10minus3

minus6

minus4

minus2

0

2

4

6

8

10

12


02

40

24

0

05

1

15

xy

z


minus4

minus2

minus4minus2

minus05




0 1 2 3 4 5 6Time

Erro

r


Error of z

minus01

0

01

02

03

04

05

06


0 1 2 3 4 5 6Time


Error of z

Erro

r

minus01

0

01

02

03

04

05

06




] (120596119899 120577) = 119890

minus120577120596119899119905119873radic119862 (120596119899 120577)2

+ 119878 (120596119899 120577)2

(30)

where

119862 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 cos (120596

119889119905119894)

119878 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 sin (120596

119889119905119894)

(31)



0

05

Am

plitu



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

minus05

0

05

Am

plitu

de

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

minus05


mpl

itude

Time Time

Am

plitu

de

lowastA1

A2

0 t2 0 t2




119873

sum

119894=1

119860119894= 1 (32)


120596119899= radic

119892

119897 (33)


1and 1199052




119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)


119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)


3amplitude



1199052 and 119905




1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)


1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)


1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)


119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)


Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)


X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)




02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5 6Time

Erro

r


Error of z

minus01

0

01

02

03

04

05

06


0 1 2 3 4 5 6Time


Error of z

Erro

r

minus01

0

01

02

03

04

05

06




] (120596119899 120577) = 119890

minus120577120596119899119905119873radic119862 (120596119899 120577)2

+ 119878 (120596119899 120577)2

(30)

where

119862 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 cos (120596

119889119905119894)

119878 (120596119899 120577) =

119873

sum

119894=1

119860119894119890120577120596119899119905119894 sin (120596

119889119905119894)

(31)



0

05

Am

plitu



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

minus05

0

05

Am

plitu

de

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

minus05


mpl

itude

Time Time

Am

plitu

de

lowastA1

A2

0 t2 0 t2




119873

sum

119894=1

119860119894= 1 (32)


120596119899= radic

119892

119897 (33)


1and 1199052




119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)


119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)


3amplitude



1199052 and 119905




1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)


1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)


1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)


119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)


Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)


X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)




02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119889

119889120596119899

] (120596119899 120577) = 0

119889

119889120577] (120596119899 120577) = 0

(34)


119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 cos (120596

119889119905119894) = 0

119873

sum

119894=1

119860119894119905119894119890120577120596119899119905119894 sin (120596

119889119905119894) = 0

(35)


3amplitude



1199052 and 119905




1198601+ 11986021198901205771205961198991199052 cos (120596

1198891199052) + 11986031198901205771205961198991199053 cos (120596

1198891199053) = 0

11986021198901205771205961198991199052 sin (120596

1198891199052) + 11986031198901205771205961198991199053 sin (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 cos (120596

1198891199052) + 119860311990531198901205771205961198991199053 cos (120596

1198891199053) = 0

119860211990521198901205771205961198991199052 sin (120596

1198891199052) + 119860311990531198901205771205961198991199053 sin (120596

1198891199053) = 0

(36)


1199052=119879119889

2 119905

3= 119879119889

1198601=

1

1 + 2119870 + 1198702

1198602=

2119870

1 + 2119870 + 1198702

1198603=

1198702

1 + 2119870 + 1198702

(37)

where

119870 = exp(minus 120577120587

radic1 minus 1205772) (38)


1199052= 100322 119905

3= 200644

1198601=1

4 119860

2=1

2 119860

3=1

4

(39)


119866119894119904= 1198601119890minus1199051119904 + 119860

2119890minus1199052119904 + 119860

3119890minus1199053119904 (40)


Xd = [119909119889 119910119889 120595119889 V119909 119889 V119910 119889 119889 119886119909 119889 119886119910 119889 119889]119879

(41)


X119889

lowast

(119905) = ΓZVD sdot X119889 (119905) (42)




02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02

46

8

02

460

1

2

3

4

xy

z



minus2 minus2


05

10



minus5

0

2

4

60

1

2

3

4

xy

z

minus2




0 5 10 15 20 25

0

01

02

03

04

05

06

07

08

Time (s)

Erro

r (m

)


Error of z

minus01

(a)

0 5 10 15 20 25Time (s)


Error of z

0

02

04

06

08

1

12

14Er

ror (

m)

(b)


4 Conclusions



005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










005

115

0

05

1

15

0

1

2

3

x

y

z

minus05 minus05

minus1




4



0

1

2

3

z

0

05

1

15

y

minus050

051

15

xminus05



0 5 10 15 20 25Time


Error of z

0

02

04

06

08

1

12

Erro

r

minus02





References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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