A Robust Control for an Aerial Robot Quadrotor under Wind …2 + − = 3 + zz − = 4 + − where , ,and arethemomentofinertia, isthe mass,and isthegravitationalacceleration.And ˘andˇ˘

Research ArticleA Robust Control for an Aerial Robot Quadrotor underWind Gusts

Li Ding 12 and Zhenwei Wang1

1College of Mechanical Engineering Jiangsu University of Technology Changzhou 213001 China2Department of Industrial and System Engineering The Hong Kong Polytechnic University 999077 Hung Hom Hong Kong

Correspondence should be addressed to Li Ding nuaadli163com

Received 3 May 2018 Accepted 25 July 2018 Published 1 August 2018

Academic Editor L Fortuna

Copyright copy 2018 Li Ding and Zhenwei Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A robust flight controller based on linear active disturbance rejection control (LADRC) is proposed for stability control of an aerialrobot quadrotor under wind gusts The nonlinear dynamical model of the quadrotor considering the wind disturbance is firstlyestablished through Newton-Euler method Subsequently a robust LADRC technique is proposed to design the controllers for theinner loop and outer loop of the aircraft In this control scheme the linear extended state observer (LESO) serves as a compensatorwhich can effectively reject the wind gusts Then a method of parameter tuning is introduced to obtain the optimized controlperformance Finally the effectiveness and advantages of the proposed controller are demonstrated through series of simulationcase

1 Introduction

Quadrotor is the most popular configuration of multirotorunmanned aerial robot extensively for civilian applications[1 2] In recent years significant growth of interest is wit-nessed towards the academic research of the quadrotor Thisis possibly attributed to their certain features such as simplemechanical structure hovering and agile maneuverabilityWith the increasing requirement of autonomous flight underdifferent conditions control of the quadrotor is an importantchallenge Complex rotation and translation operations arealso commonly encountered in the mathematical modelresulting in complex equations that have to be taken intoaccount when designing control laws On the other handthe position control and attitude control of the quadrotorare extremely sensitive to external disturbance such aswind gusts Particularly robustness issues may be criticalfor the control of quadrotor since they are subjected tothe complicated dynamics and external disturbances Severalproper control methodologies for the quadrotor have beenreported in the literatures such as proportion integrationdifferentiation (PID) control [3] backstepping control [4]sliding mode control [5] and model predictive control [6]

Nevertheless there are a few researches about the robustcontrol of the quadrotor especially under the wind gusts

In order to weaken the effect of wind gusts some authorshave proposed complicated control techniques to achievestability In the literatures [7 8] adaptive controllers ofdifferent structures were verified by simulations and exper-iments on test bench For control of a multirotor aircraftSalazar et al [9] proposed a novel mathematical model of thewind gusts and designed a simple nonlinear control law toresist the wind disturbance The effect of wind disturbancewas discussed towards the issue of (unmanned air vehicle)UAV path planning in the literatures [10 11] From theanalysis compensation control for the aircraft may be aneffective method Dong et al [12] designed a flight controllerwith disturbance observer (DOB) to guarantee the high-performance trajectory tracking of a quadrotor The DOB-based controller is robust to wind gusts without the use ofhigh control gain or extensive computational power

Our team has proposed in previous articles like [13] theuse of LADRC to control a small-scale unmanned helicopterThis control strategy is rooted in active disturbance rejectioncontrol (ADRC) which was firstly proposed by Han [14]LADRC is a simplified implementation of the ADRC in

HindawiJournal of RoboticsVolume 2018 Article ID 5607362 8 pageshttpsdoiorg10115520185607362

2 Journal of Robotics

which bandwidth is the only adjustable parameter of thecontrol performance A previous contribution from our teampresented the parameter tuning towards the LADRCbased onthe artificial bee colony algorithm [15] Detailed backgroundand principles on the ABC can be found in the literatures[16 17] and the authors analyzed the performance of theABCon optimization problem

The content is organized in the following manner Sec-tion 2 presents the mathematical model of the quadrotorunder the wind gusts In Section 3 the robust controlleris described using the LADRC strategy And the proof ofthe stability analysis for the proposed controller is givenin Section 4 A method of parameter tuning is applied inSection 5 to optimize the performance of our proposedcontroller The control law with wind gusts is simulatedand illustrated in Section 6 Finally our conclusions andcontributions are discussed in Section 7

2 Mathematical Model

The free body diagram and coordinate systems for thequadrotor are given in Figure 1 It is controlled by the angularspeeds of four electric motors Each motor produces a thrustand a torque119898119894 (119894 = 1 2 3 4) whose combination generatesthe main thrust input 1198801 pitch input 1198802 roll input 1198803 andyaw input 1198804 Two reference frames are subjected to thequadrotor namely the earth-fixed frame 119874119864119883119864119884119864119885119864 andthe body-fixed frame 119874119861119883119861119884119861119885119861 The position (119909 119910 119911)attitude (pitch 120579 roll 120601 and yaw 120595) and motion can beachieved through the two frames According to Newton-Euler method The six-degrees-of-freedom dynamics of thequadrotor are modeled as [18]

= 1198801 (c120595s120579c120601 + s120595s120601)119898

119910 = 1198801 (s120595s120579c120601 minus c120595s120601)119898

= 1198801c120601c120579119898 minus 119892120601 = 1198802119868119909119909 +

120579 (119868119910119910 minus 119868119911119911)119868119909119909120579 = 1198803119868119910119910 +

120601 (119868zz minus 119868119909119909)119868119910119910 = 1198804119868119911119911 +

120601 120579 (119868119909119909 minus 119868119910119910)119868119911119911

(1)

where 119868119909119909 119868119910119910 and 119868119911119911 are the moment of inertia 119898 is themass and 119892 is the gravitational acceleration And 119904120572 and 119888120572are the abbreviations for cos120572 and sin120572

The thrust of each propeller is calculated by 119891119898119894 =1198961199051205962119894 (119894 = 1 2 3 4) with 119896119905 being the thrust coefficient and 120596119894

fm4

lm4

fm1

ZE

XE

OE

YE

m1

p

mgXB

ZBr

OB q

YB

m3

fm2

fm3

m2

Figure 1 Free body diagram of quadrotor

Vp Vw

fm

fw

V

Figure 2 Analysis of the main and lateral thrusts

being the angular speeds The relationship between the fourinputs and angular speed is expressed by

[[[[[[

1198801119880211988031198804

]]]]]]= [[[[[[

119896119905 119896119905 119896119905 1198961199050 minus119896119905119897 0 119896119905119897minus119896119905119897 0 119896119905119897 0minus119896119898 119896119898 minus119896119898 119896119898

]]]]]]

[[[[[[[

12059621120596221205962312059624

]]]]]]]

(2)

where 119897 denotes the distance from the center of mass tothe center of the propeller and 119896119898 is termed as antitorquecoefficient

In the outdoor flight the quadrotor is generally exposedto lateral wind gustsThe aircraft will be yawed or overturnedif it is subjected to the lateral wind gusts Stating precisely thisleads to additional lateral air flow acting on the propeller asshown in Figure 2 Hence the total thrust 119891119879119894 = 119891119898119894 +119891119908119894 (119894 =1 2 3 4) could be described as [19]

119891119879119894 = 2120588119860119881119901 (3)

Journal of Robotics 3

where 120588 is the air density and119860 is the propeller area119881119901 is theinduced wind speed of the propeller and is the total windinduced speed of the rotor The relationship between them isgiven by

= [(119881119908 cos120572 + 119881119901)2 + (119881119908 sin120572)2]12 (4)

where 120572 represents the angle between the propeller axis andthe lateral wind gusts Since the latter is perpendicular to theformer 120572 = 90∘ At this moment the additional lateral forcesare expressed by

119891119908119894 = 21205881198601198812119901 (1 + 11988121199081198812119901 )12 minus 119891119898119894 (5)

Moreover the aerodynamics drag is defined as 119898119889119903119886119892 =12058811986011988121199082 = 1198961198891199031198861198921198812119908 where 119896119889119903119886119892 gt 0 is a constant dependingon the parameter 120588 shape of the blade and other factorsWhen the aircraft is disturbed by the lateral wind gusts theadditional torques acting on the propellers are represented by

[[[[

119898119908120601119898119908120579119898119908120595]]]]=[[[[[[[

(1198911199084 minus 1198911199082) 119897(1198911199083 minus 1198911199081) 1198974sum119894=1

119898119889119903119886119892119894

]]]]]]]

(6)

Using (1) (5) and (6) the integrated quadrotor dynamicalmodel can be rewritten as

= 1198801 (c120595s120579c120601 + s120595s120601)119898 + 4sum

119894=1

119891119908119894119898119910 = 1198801 (s120595s120579c120601 minus c120595s120601)

119898 + 4sum119894=1

119891119908119894119898 = 1198801c120601c120579119898 minus 119892 + 4sum

119894=1

119891119908119894119898120601 = 1198802119868119909119909 +

120579 (119868119910119910 minus 119868119911119911)119868119909119909 + 119898119908120601119868119909119909120579 = 1198803119868119910119910 +

120601 (119868zz minus 119868119909119909)119868119910119910 + 119898119908120579119868119910119910 = 1198804119868119911119911 +

120601 120579 (119868119909119909 minus 119868119910119910)119868119911119911 + 119898119908120595119868119911119911

(7)

3 Robust Controller Design

According to (7) the input and output of each control channelpresent second-order derivative relation Thus we use thesecond-order system in the following form for describing theLADRC

119904 = 119891119904 (119909119904 119904 119908119904 119905) + 119887 (119905) 119906119910119904 = 119909119904 (8)

where 119906 and 119910119904 denote input and output respectively 119891119904 isthe external disturbance 119909119904 is state variables and 119887(119905) is thecontrol function

Suppose 1198870 is the approximate value of 119887(119905) and (8) iswritten equivalently as

119904 = 119891119904 (119909119904 119904 119908119904 119905) + 1198870119906 = 119891119904 + 1198870119906119910119904 = 119909119904 (9)

By rewriting the above equation in the form of differentialequation one has

1199041 = 11990911990421199042 = 1199091199043 + 11988701199061199043 = ℎ119910119904 = 1199091199041

(10)

with 1199091199043 = 119891119904 added as an augmented state and ℎ = 119891119904 asunknown disturbance Now 119891119904 can be estimated through astate observer based on a state space model

x = Ax + Bu + Eh

y = Cx(11)

where

A = [[[0 1 00 0 10 0 0

]]]

119861 = [[[011988700]]]

C = [[[100]]]

E = [[[001]]]

(12)

The LESO of (11) can be constructed as

z = Az + Bu + L (y minus y)y = Cz

(13)

where L is the observer gain It is calculated as [17]

L = [[[119897111989721198973]]]= [[[[

12057211205961199001205722120596119900212057231205961199003

]]]]

(14)


where 120596119900 gt 0 is the observer bandwidth It results in thecharacteristic polynomial of (13) being

120582 (119904) = 1199043 + 12057211199042 + 1205722119904 + 1205723 = (119904 + 120596119900)3 (15)

From the polynomial one obtains

1205721 = 31205722 = 31205723 = 1

(16)

The observer can track the state variables with well-tunedobserver bandwidth and yield 1199111(119905) 997888rarr 119910(119905) 1199112(119905) 997888rarr 119910(119905)and 1199113(119905) 997888rarr 119891119904

With the estimate of119891119904 the LADRC can actively compen-sate for effect of the external disturbance in real time as

119906 = 1199060 minus 11991131198870 (17)

The closed-loop control system is converted into a unitdouble integral plant And the control law is [20 21]

119910 = 119891 minus 1199113 + 1199060 asymp 1199060 = 1205821 (119903119890 minus 1199111) minus 12058221199112 (18)

where 119903119890 is the reference signalThe transfer function of the second-order plant is given

as

1198662 = 12058211199042 + 1205822119904 + 1205821 (19)

Here the gains can be calculated as [17]

1205821 = 12059611988821205822 = 2120596119888 (20)

4 Stability Analysis

Define the error as

e = x minus z (21)

Subtracting (13) from (11) the error equation is obtainedas

e = A119890e + Eh (22)

where

A119890 = A minus LC = [[[minus1198971 1 0minus1198972 0 1minus1198973 0 0

]]]

(23)

Obviously the LESO is bounded-input bounded-output(BIBO) stable if the roots of the characteristic polynomialnamely (15) are all in the left half plane and h is bounded

Theorem 1 The design of LADRC from (13) to (18) yields aBIBO stable closed-loop system if the LESO and the control lawfor the unit double integral plant are stable

ReferenceTrajectories

PositionController Attitude

Controller

Motors Auadrotor

Prr U1

U2

U3

U4

rrr

xy z

Figure 3 Control system architecture of the quadrotor

Proof According to (17) and (18) one has

119906 = 11198870 [minus1205821 minus1205822 minus1][[[1199111 minus 11990311989011991121199113

]]]= Qz minus 11198870

[[[11990311989000]]]

= Qz minus G

(24)

where

Q = 11198870 [minus1205821 minus1205822 minus1] (25)

The closed-loop control system is rewritten as

x = Ax + Bu + Eh = Ax + BQz minus BG + Eh (26)

z = Az + Bu + L (y minus y)= LCx + (A minus LC + BQ) z minus BG

(27)

Using the state space equation to represent the closed-loop control system one gets

[xz] = [ A BQ

LC A minus LC + BQ][x

z] + [minusB E

minusB 0][G

h] (28)

Similarly the control system is BIBO stable if the eigen-values of (28) are in the left half plane Since the referencesignal 119903119890 is always bounded with the condition that thedisturbance 119891119904 satisfies differentiable h is bounded

5 Parameter Tuning

According to the previous analysis the control structureof the quadrotor is divided into two loops ie inner loopand outer loop The block diagram of such a structure isschematically shown in Figure 3 In the outer loop theposition vector P119903 and yaw angle 120595119903 are chosen as thereference signals In the inner loop the reference signal(120579119903 120601119903 120595119903) comes from the outer loop Moreover the fourinputs generated by the two loops are converted into theangular speeds of motors

The evaluation function is crucial for the parameter tun-ing Consequently an improved integral of time multipliedby absolute error (IITAE) is employed as performance index


10 20 30 40 50 60 70 80 90 100Iteration

0

100

200

300

400

500

600

700

800

900

1000Pa

ram

eter

s

Ｉ4

＜04

＝4

Ｉ5

＜05

＝5

Ｉ6

＜06

＝6

Figure 4 Evolving curves of the control parameters

0 1 2 3 4 5 6Time (s)

0

02

04

06

08

1

12

Attit

ude (

rad)

2671 2673

099981

10002

Reference

Figure 5 Response of the attitude

The evaluation function of the performance index is definedas

119865 (119905) = 1205721 intinfin0119905 |119890 (119905)| 119889119905 + 1205722 intinfin

0(119906 (119905))2 119889119905 + 1205723119905119904

+ 1205724119900(29)

where 119890(119905) is the error signal in the time domain 119905119904 is thesettling time 119900119904 is the overshoot and 1205721 1205722 1205723 and 1205724 arethe weight coefficients Note that the parameter tuning ofthe LADRC is conducted by the ABC algorithm and the

20 40 60 80 100Iteration

0

100

200

300

400

500

600

700

800

900

1000

Para

met

ers

Ｉ1

＜01

＝1

Ｉ2

＜02

＝2

Ｉ3

＜03

＝3

Figure 6 Tuned parameters of outer loop controller

Table 1 Physical parameters

Item Quantity119871 (m) 02053119898 (kg) 1923119868119909119909 (kgsdotm2) 0094119868119910119910 (kgsdotm2) 0094119868119911119911 (kgsdotm2) 0086

exhaustive process is reported in the literature [13] and hererepeating is unnecessary

As taking the inner loop as an instance the referencesignal is set to 1 rad 1 rad 1 rad There are three second-order LADRC in the inner loop ie 120579-LADRC 120601-LADRCand 120595-LADRC The physical parameters of the quadrotorare listed in Table 1 We use the ABC algorithm to tunethe unknown control parameters under a unit step responsecase The number of the iterations is 100 In addition thesimulation parameters of the wind gusts are given as 120588 =1293 gL 119860 = 00016m2 119881119875 = 2ms and 119881119908 = 6msAnd a white noise with the amplitude of 02 rad is added tothe measured portsThe simulation time lasts for 6s Figure 4displays the evolving curves of the control parameters whichindicates the process of the parameter tuning by ABC Theattitude response is shown in Figure 5 It is obvious that theLADRC has a satisfactory performance of wind resistanceand the steady state error is less than 2 Furthermore theresponses of the three attitude channels are fast and accurate


ReferenceLADRCPID

0

1

2

3

4

5

6

7

z (m

)

10 150 5Time (s)

(a) 119881119908 = 4ms

0 5 10 15Time (s)

0

1

2

3

4

5

6

7

z (m

)

ReferenceLADRCPID

(b) 119881119908 = 10ms

Figure 7 Position response under different wind gusts

(a) Time=0s

10m

(b) Time=6s

Figure 8 Hovering control in Simscape environment

6 Simulation Results

In order to validate the robustness and wind resistance ofthe proposed control scheme for stabilizing the quadrotor attrajectory tracking the simulation is conducted in MATLAB2017a programming environment on an Intel Core i7-4720PCrunningWindows 10 Similarly randomwind gusts are addedto the input of the outer loop

As pointed out in Section 5 the control parameters of theouter loop controller are also tuned by the ABC algorithmThe tuned parameters are shown in Figure 6 using a unit stepresponse case

Moreover we introduce a PID controller from the lit-erature [22] as a comparison to test the robustness ofour proposed controller In the comparative simulation thequadrotor follows a referenced trajectory of z direction as

shown in Figure 7 Meanwhile lateral wind gusts are appliedwith different wind speeds at a specific time period 8 s-12s The PID and LADRC are applied to stabilize the positionof the aircraft respectively From the result the response ofLADRC is slightly faster than that of PID As the wind speedincreases the tracking error becomes bigger Nevertheless itis obvious that the steady state error of LADRC is smaller thanthat of PID due to the LESO In conclusion the LADRC hasa strong ability of wind disturbance compensation

Lastly a visual simulation is run in the MATALBSimscape environment to display the trajectory tracking ofthe quadrotor Note that the Simscape link utility creates aphysical modeling XML file that represents the quadrotorrsquosparts as bodies and maps the constraints between the partsinto joints [23] In this case the task requires that thequadrotor takes off from the ground and flies in hover The


simulation time lasts for 8s From the result in Figure 8the quadrotor has a stable flight performance based on theproposed controller

7 Conclusion

A new LADRC control scheme is given for the stabilitycontrol of an aerial robot quadrotor under wind gusts in thisarticle The proposed controller roughly has two parts theLESO part applied to properly estimate and compensate thewind disturbance leading to an attractive model-free featureand a PD part applied to ensure satisfactory control perfor-mance Moreover corresponding stability of the closed-loopcontrol plant is analyzed and an ABC algorithm is applied forparameter tuning of the LADRC Finally all the simulationresults show the proposed controller has a better ability ofresisting wind gusts comparing to the traditional PID

In the future more advanced control strategies such asthat reported in the literature [24] will be studied to controlthe movement of the quadrotor under wind gusts

Data Availability

Figure 4 displays the variation of the tuning parametersThere is no need to give the detailed parameters searched byABC In addition the data of quadrotor is listed in Table 1

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Helpful discussions with Professor Wu Hongtao from Nan-jing University of Aeronautics and Astronautics on his guid-ance in dynamical modeling of aerial robots are gratefullyacknowledged This work was partially supported by theFoundation Research Project of Jiangsu Province (The Natu-ral Science Fund no BK20170315) and Changzhou SciampTechProgram of China (Grant no CJ20179017)

References

[1] R Amin L Aijun and S Shamshirband ldquoA review of quadrotorUAV Control methodologies and performance evaluationrdquoInternational Journal of Automation and Control Engineeringvol 10 no 2 pp 87ndash103 2016

[2] M A Tofigh M J Mahjoob andM Ayati ldquoDynamic modelingand nonlinear tracking control of a novel modified quadrotorrdquoInternational Journal of Robust and Nonlinear Control vol 28no 2 pp 552ndash567 2018

[3] S An S Yuan and H Li ldquoSelf-tuning of PID controllers designby adaptive interaction for quadrotorUAVrdquo in Proceedings of the7th IEEE Chinese Guidance Navigation and Control ConferenceCGNCC 2016 pp 1547ndash1552 IEEE Xiamen China August2016

[4] A Das F Lewis and K Subbarao ldquoBackstepping approach forcontrolling a quadrotor using lagrange form dynamicsrdquo Journalof IntelligentampRobotic Systems vol 56 no 1-2 pp 127ndash151 2009

[5] D Lee H J Kim and S Sastry ldquoFeedback linearization vsadaptive sliding mode control for a quadrotor helicopterrdquoInternational Journal of Control Automation and Systems vol7 no 3 pp 419ndash428 2009

[6] A Chamseddine Y Zhang C-A Rabbath C Fulford and JApkarian ldquoModel reference adaptive fault tolerant control of aquadrotor UAVrdquo in Infotech Aerospace St Louis MO USAMarch 2011

[7] F Chen F Lu B Jiang and G Tao ldquoAdaptive compensationcontrol of the quadrotor helicopter using quantum informationtechnology and disturbance observerrdquo Journal of The FranklinInstitute vol 351 no 1 pp 442ndash455 2014

[8] S Salazar H Romero R Lozano et al ldquoModeling and real-time stabilization of an aircraft having eight rotorsrdquo Journal ofIntelligent and Robotic Systems vol 54 no 1-3 pp 455ndash4702008

[9] L Ding R Ma H Wu C Feng and Q Li ldquoYaw controlof an unmanned aerial vehicle helicopter using linear activedisturbance rejection controlrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 231 no 6 pp 427ndash435 2017

[10] S Jackson J Tisdale M Kamgarpour B Basso and J KHedrick ldquoTracking controllers for small UAVs with winddisturbances theory and flight resultsrdquo in Proceedings of the47th IEEE Conference on Decision and Control (CDC rsquo08) pp564ndash569 IEEE Cancun Mexico December 2008

[11] A L Jennings R Ordonez and N Ceccarelli ldquoAn ant colonyoptimization using training data applied to UAV way pointpath planning in windrdquo in Proceedings of the 2008 IEEE SwarmIntelligence Symposium SIS 2008 USA September 2008

[12] W Dong G-Y Gu X Zhu and H Ding ldquoHigh-performancetrajectory tracking control of a quadrotor with disturbanceobserverrdquo Sensors and Actuators A Physical vol 211 pp 67ndash772014

[13] J Song Z Gan and J Han ldquoStudy of active disturbancerejection controller on filteringrdquo Control and Decision vol 18no 1 pp 110ndash112 2003

[14] DMa Y Xia T Li and K Chang ldquoActive disturbance rejectionand predictive control strategy for a quadrotor helicopterrdquo IETControl Theory amp Applications vol 10 no 17 pp 2213ndash22222016

[15] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Technical report-tr06 Erciyes university engi-neering faculty computer engineering department KayseriTurkey 2005

[16] L Ding H Wu Y Yao and Y Yang ldquoDynamic model identi-fication for 6-DOF industrial robotsrdquo Journal of Robotics vol2015 Article ID 471478 pp 1ndash9 2015

[17] N Michael D Mellinger Q Lindsey and V Kumar ldquoTheGRASP multiple micro-UAV testbedrdquo IEEE Robotics andAutomation Magazine vol 17 no 3 pp 56ndash65 2010

[18] Y Guo B Jiang and Y Zhang ldquoA novel robust attitude controlfor quadrotor aircraft subject to actuator faults and wind gustsrdquoIEEECAA Journal of Automatica Sinica vol 5 no 1 pp 292ndash300 2018

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence pp 4989ndash4996 Denver Colo USA June 2003

[20] Z Gao ldquoActive disturbance rejection control a paradigmshift in feedback control system designrdquo in Proceedings ofthe American Control Conference pp 2399ndash2405 MinneapolisMinn USA June 2006


[21] Y Liu J Liu and S Zhou ldquoLinear active disturbance rejectioncontrol for pressurized water reactor powerrdquo Annals of NuclearEnergy vol 111 pp 22ndash30 2018

[22] W Dong G Gu Y X Zhu et al ldquoModeling and control of aquadrotor UAV with aerodynamic conceptsrdquo in Proceedings ofWorld Academy of Science Engineering and Technology vol 77p 437World Academy of Science Engineering and Technology(WASET) 2013

[23] httpsww2mathworkscnhelpphysmodsimscapeindexhtmls tid=srchtitle

[24] L Fortuna and G Muscato ldquoA roll stabilization system for amonohull ship modeling identification and adaptive controlrdquoIEEE Transactions on Control Systems Technology vol 4 no 1pp 18ndash28 1996

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which bandwidth is the only adjustable parameter of thecontrol performance A previous contribution from our teampresented the parameter tuning towards the LADRCbased onthe artificial bee colony algorithm [15] Detailed backgroundand principles on the ABC can be found in the literatures[16 17] and the authors analyzed the performance of theABCon optimization problem

The content is organized in the following manner Sec-tion 2 presents the mathematical model of the quadrotorunder the wind gusts In Section 3 the robust controlleris described using the LADRC strategy And the proof ofthe stability analysis for the proposed controller is givenin Section 4 A method of parameter tuning is applied inSection 5 to optimize the performance of our proposedcontroller The control law with wind gusts is simulatedand illustrated in Section 6 Finally our conclusions andcontributions are discussed in Section 7

2 Mathematical Model

The free body diagram and coordinate systems for thequadrotor are given in Figure 1 It is controlled by the angularspeeds of four electric motors Each motor produces a thrustand a torque119898119894 (119894 = 1 2 3 4) whose combination generatesthe main thrust input 1198801 pitch input 1198802 roll input 1198803 andyaw input 1198804 Two reference frames are subjected to thequadrotor namely the earth-fixed frame 119874119864119883119864119884119864119885119864 andthe body-fixed frame 119874119861119883119861119884119861119885119861 The position (119909 119910 119911)attitude (pitch 120579 roll 120601 and yaw 120595) and motion can beachieved through the two frames According to Newton-Euler method The six-degrees-of-freedom dynamics of thequadrotor are modeled as [18]

= 1198801 (c120595s120579c120601 + s120595s120601)119898

119910 = 1198801 (s120595s120579c120601 minus c120595s120601)119898

= 1198801c120601c120579119898 minus 119892120601 = 1198802119868119909119909 +

120579 (119868119910119910 minus 119868119911119911)119868119909119909120579 = 1198803119868119910119910 +

120601 (119868zz minus 119868119909119909)119868119910119910 = 1198804119868119911119911 +

120601 120579 (119868119909119909 minus 119868119910119910)119868119911119911

(1)

where 119868119909119909 119868119910119910 and 119868119911119911 are the moment of inertia 119898 is themass and 119892 is the gravitational acceleration And 119904120572 and 119888120572are the abbreviations for cos120572 and sin120572

The thrust of each propeller is calculated by 119891119898119894 =1198961199051205962119894 (119894 = 1 2 3 4) with 119896119905 being the thrust coefficient and 120596119894

fm4

lm4

fm1

ZE

XE

OE

YE

m1

p

mgXB

ZBr

OB q

YB

m3

fm2

fm3

m2

Figure 1 Free body diagram of quadrotor

Vp Vw

fm

fw

V

Figure 2 Analysis of the main and lateral thrusts

being the angular speeds The relationship between the fourinputs and angular speed is expressed by

[[[[[[

1198801119880211988031198804

]]]]]]= [[[[[[

119896119905 119896119905 119896119905 1198961199050 minus119896119905119897 0 119896119905119897minus119896119905119897 0 119896119905119897 0minus119896119898 119896119898 minus119896119898 119896119898

]]]]]]

[[[[[[[

12059621120596221205962312059624

]]]]]]]

(2)

where 119897 denotes the distance from the center of mass tothe center of the propeller and 119896119898 is termed as antitorquecoefficient

In the outdoor flight the quadrotor is generally exposedto lateral wind gustsThe aircraft will be yawed or overturnedif it is subjected to the lateral wind gusts Stating precisely thisleads to additional lateral air flow acting on the propeller asshown in Figure 2 Hence the total thrust 119891119879119894 = 119891119898119894 +119891119908119894 (119894 =1 2 3 4) could be described as [19]

119891119879119894 = 2120588119860119881119901 (3)



= [(119881119908 cos120572 + 119881119901)2 + (119881119908 sin120572)2]12 (4)


119891119908119894 = 21205881198601198812119901 (1 + 11988121199081198812119901 )12 minus 119891119898119894 (5)


[[[[

119898119908120601119898119908120579119898119908120595]]]]=[[[[[[[

(1198911199084 minus 1198911199082) 119897(1198911199083 minus 1198911199081) 1198974sum119894=1

119898119889119903119886119892119894

]]]]]]]

(6)


= 1198801 (c120595s120579c120601 + s120595s120601)119898 + 4sum

119894=1

119891119908119894119898119910 = 1198801 (s120595s120579c120601 minus c120595s120601)

119898 + 4sum119894=1

119891119908119894119898 = 1198801c120601c120579119898 minus 119892 + 4sum

119894=1

119891119908119894119898120601 = 1198802119868119909119909 +

120579 (119868119910119910 minus 119868119911119911)119868119909119909 + 119898119908120601119868119909119909120579 = 1198803119868119910119910 +

120601 (119868zz minus 119868119909119909)119868119910119910 + 119898119908120579119868119910119910 = 1198804119868119911119911 +

120601 120579 (119868119909119909 minus 119868119910119910)119868119911119911 + 119898119908120595119868119911119911

(7)



119904 = 119891119904 (119909119904 119904 119908119904 119905) + 119887 (119905) 119906119910119904 = 119909119904 (8)



119904 = 119891119904 (119909119904 119904 119908119904 119905) + 1198870119906 = 119891119904 + 1198870119906119910119904 = 119909119904 (9)


1199041 = 11990911990421199042 = 1199091199043 + 11988701199061199043 = ℎ119910119904 = 1199091199041

(10)


x = Ax + Bu + Eh

y = Cx(11)

where

A = [[[0 1 00 0 10 0 0

]]]

119861 = [[[011988700]]]

C = [[[100]]]

E = [[[001]]]

(12)



(13)


L = [[[119897111989721198973]]]= [[[[

12057211205961199001205722120596119900212057231205961199003

]]]]

(14)



120582 (119904) = 1199043 + 12057211199042 + 1205722119904 + 1205723 = (119904 + 120596119900)3 (15)


1205721 = 31205722 = 31205723 = 1

(16)



119906 = 1199060 minus 11991131198870 (17)




as

1198662 = 12058211199042 + 1205822119904 + 1205821 (19)


1205821 = 12059611988821205822 = 2120596119888 (20)


Define the error as

e = x minus z (21)


e = A119890e + Eh (22)

where


]]]

(23)





Controller

Motors Auadrotor

Prr U1

U2

U3

U4

rrr

xy z




]]]= Qz minus 11198870

[[[11990311989000]]]

= Qz minus G

(24)

where





(27)


[xz] = [ A BQ


z] + [minusB E

minusB 0][G

h] (28)


5 Parameter Tuning




10 20 30 40 50 60 70 80 90 100Iteration

0

100

200

300

400

500

600

700

800

900

1000Pa

ram

eter

s

Ｉ4

＜04

＝4

Ｉ5

＜05

＝5

Ｉ6

＜06

＝6


0 1 2 3 4 5 6Time (s)

0

02

04

06

08

1

12

Attit

ude (

rad)

2671 2673

099981

10002

Reference




0(119906 (119905))2 119889119905 + 1205723119905119904

+ 1205724119900(29)


20 40 60 80 100Iteration

0

100

200

300

400

500

600

700

800

900

1000

Para

met

ers

Ｉ1

＜01

＝1

Ｉ2

＜02

＝2

Ｉ3

＜03

＝3







ReferenceLADRCPID

0

1

2

3

4

5

6

7

z (m

)

10 150 5Time (s)

(a) 119881119908 = 4ms

0 5 10 15Time (s)

0

1

2

3

4

5

6

7

z (m

)

ReferenceLADRCPID

(b) 119881119908 = 10ms


(a) Time=0s

10m

(b) Time=6s










7 Conclusion



Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




= [(119881119908 cos120572 + 119881119901)2 + (119881119908 sin120572)2]12 (4)


119891119908119894 = 21205881198601198812119901 (1 + 11988121199081198812119901 )12 minus 119891119898119894 (5)


[[[[

119898119908120601119898119908120579119898119908120595]]]]=[[[[[[[

(1198911199084 minus 1198911199082) 119897(1198911199083 minus 1198911199081) 1198974sum119894=1

119898119889119903119886119892119894

]]]]]]]

(6)


= 1198801 (c120595s120579c120601 + s120595s120601)119898 + 4sum

119894=1

119891119908119894119898119910 = 1198801 (s120595s120579c120601 minus c120595s120601)

119898 + 4sum119894=1

119891119908119894119898 = 1198801c120601c120579119898 minus 119892 + 4sum

119894=1

119891119908119894119898120601 = 1198802119868119909119909 +

120579 (119868119910119910 minus 119868119911119911)119868119909119909 + 119898119908120601119868119909119909120579 = 1198803119868119910119910 +

120601 (119868zz minus 119868119909119909)119868119910119910 + 119898119908120579119868119910119910 = 1198804119868119911119911 +

120601 120579 (119868119909119909 minus 119868119910119910)119868119911119911 + 119898119908120595119868119911119911

(7)



119904 = 119891119904 (119909119904 119904 119908119904 119905) + 119887 (119905) 119906119910119904 = 119909119904 (8)



119904 = 119891119904 (119909119904 119904 119908119904 119905) + 1198870119906 = 119891119904 + 1198870119906119910119904 = 119909119904 (9)


1199041 = 11990911990421199042 = 1199091199043 + 11988701199061199043 = ℎ119910119904 = 1199091199041

(10)


x = Ax + Bu + Eh

y = Cx(11)

where

A = [[[0 1 00 0 10 0 0

]]]

119861 = [[[011988700]]]

C = [[[100]]]

E = [[[001]]]

(12)



(13)


L = [[[119897111989721198973]]]= [[[[

12057211205961199001205722120596119900212057231205961199003

]]]]

(14)



120582 (119904) = 1199043 + 12057211199042 + 1205722119904 + 1205723 = (119904 + 120596119900)3 (15)


1205721 = 31205722 = 31205723 = 1

(16)



119906 = 1199060 minus 11991131198870 (17)




as

1198662 = 12058211199042 + 1205822119904 + 1205821 (19)


1205821 = 12059611988821205822 = 2120596119888 (20)


Define the error as

e = x minus z (21)


e = A119890e + Eh (22)

where


]]]

(23)





Controller

Motors Auadrotor

Prr U1

U2

U3

U4

rrr

xy z




]]]= Qz minus 11198870

[[[11990311989000]]]

= Qz minus G

(24)

where





(27)


[xz] = [ A BQ


z] + [minusB E

minusB 0][G

h] (28)


5 Parameter Tuning




10 20 30 40 50 60 70 80 90 100Iteration

0

100

200

300

400

500

600

700

800

900

1000Pa

ram

eter

s

Ｉ4

＜04

＝4

Ｉ5

＜05

＝5

Ｉ6

＜06

＝6


0 1 2 3 4 5 6Time (s)

0

02

04

06

08

1

12

Attit

ude (

rad)

2671 2673

099981

10002

Reference




0(119906 (119905))2 119889119905 + 1205723119905119904

+ 1205724119900(29)


20 40 60 80 100Iteration

0

100

200

300

400

500

600

700

800

900

1000

Para

met

ers

Ｉ1

＜01

＝1

Ｉ2

＜02

＝2

Ｉ3

＜03

＝3







ReferenceLADRCPID

0

1

2

3

4

5

6

7

z (m

)

10 150 5Time (s)

(a) 119881119908 = 4ms

0 5 10 15Time (s)

0

1

2

3

4

5

6

7

z (m

)

ReferenceLADRCPID

(b) 119881119908 = 10ms


(a) Time=0s

10m

(b) Time=6s










7 Conclusion



Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




120582 (119904) = 1199043 + 12057211199042 + 1205722119904 + 1205723 = (119904 + 120596119900)3 (15)


1205721 = 31205722 = 31205723 = 1

(16)



119906 = 1199060 minus 11991131198870 (17)




as

1198662 = 12058211199042 + 1205822119904 + 1205821 (19)


1205821 = 12059611988821205822 = 2120596119888 (20)


Define the error as

e = x minus z (21)


e = A119890e + Eh (22)

where


]]]

(23)





Controller

Motors Auadrotor

Prr U1

U2

U3

U4

rrr

xy z




]]]= Qz minus 11198870

[[[11990311989000]]]

= Qz minus G

(24)

where





(27)


[xz] = [ A BQ


z] + [minusB E

minusB 0][G

h] (28)


5 Parameter Tuning




10 20 30 40 50 60 70 80 90 100Iteration

0

100

200

300

400

500

600

700

800

900

1000Pa

ram

eter

s

Ｉ4

＜04

＝4

Ｉ5

＜05

＝5

Ｉ6

＜06

＝6


0 1 2 3 4 5 6Time (s)

0

02

04

06

08

1

12

Attit

ude (

rad)

2671 2673

099981

10002

Reference




0(119906 (119905))2 119889119905 + 1205723119905119904

+ 1205724119900(29)


20 40 60 80 100Iteration

0

100

200

300

400

500

600

700

800

900

1000

Para

met

ers

Ｉ1

＜01

＝1

Ｉ2

＜02

＝2

Ｉ3

＜03

＝3







ReferenceLADRCPID

0

1

2

3

4

5

6

7

z (m

)

10 150 5Time (s)

(a) 119881119908 = 4ms

0 5 10 15Time (s)

0

1

2

3

4

5

6

7

z (m

)

ReferenceLADRCPID

(b) 119881119908 = 10ms


(a) Time=0s

10m

(b) Time=6s










7 Conclusion



Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



10 20 30 40 50 60 70 80 90 100Iteration

0

100

200

300

400

500

600

700

800

900

1000Pa

ram

eter

s

Ｉ4

＜04

＝4

Ｉ5

＜05

＝5

Ｉ6

＜06

＝6


0 1 2 3 4 5 6Time (s)

0

02

04

06

08

1

12

Attit

ude (

rad)

2671 2673

099981

10002

Reference




0(119906 (119905))2 119889119905 + 1205723119905119904

+ 1205724119900(29)


20 40 60 80 100Iteration

0

100

200

300

400

500

600

700

800

900

1000

Para

met

ers

Ｉ1

＜01

＝1

Ｉ2

＜02

＝2

Ｉ3

＜03

＝3







ReferenceLADRCPID

0

1

2

3

4

5

6

7

z (m

)

10 150 5Time (s)

(a) 119881119908 = 4ms

0 5 10 15Time (s)

0

1

2

3

4

5

6

7

z (m

)

ReferenceLADRCPID

(b) 119881119908 = 10ms


(a) Time=0s

10m

(b) Time=6s










7 Conclusion



Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ReferenceLADRCPID

0

1

2

3

4

5

6

7

z (m

)

10 150 5Time (s)

(a) 119881119908 = 4ms

0 5 10 15Time (s)

0

1

2

3

4

5

6

7

z (m

)

ReferenceLADRCPID

(b) 119881119908 = 10ms


(a) Time=0s

10m

(b) Time=6s










7 Conclusion



Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




7 Conclusion



Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

A Robust Control for an Aerial Robot Quadrotor under Wind …2 + − = 3 + zz − = 4 + − where , ,and arethemomentofinertia, isthe mass,and isthegravitationalacceleration.And ˘andˇ˘