Research Article Chaotic Artificial Bee Colony Algorithm for...

Research ArticleChaotic Artificial Bee Colony Algorithm for SystemIdentification of a Small-Scale Unmanned Helicopter

Li Ding1 Hongtao Wu1 and Yu Yao2

1College of Mechanical and Electrical Engineering Nanjing University of Aeronautics and AstronauticsNo 29 Yudao Street Nanjing 210016 China2College of Aerospace Engineering Nanjing University of Aeronautics and Astronautics No 29 Yudao StreetNanjing 210016 China

Correspondence should be addressed to Hongtao Wu mehtwu126com

Received 12 April 2015 Revised 18 June 2015 Accepted 24 June 2015

Academic Editor Kenneth M Sobel

Copyright copy 2015 Li Ding 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

The purpose of this paper is devoted to developing a chaotic artificial bee colony algorithm (CABC) for the system identificationof a small-scale unmanned helicopter state-space model in hover condition In order to avoid the premature of traditional artificialbee colony algorithm (ABC) which is stuck in local optimum and can not reach the global optimum a novel chaotic operator withthe characteristics of ergodicity and irregularity was introduced to enhance its performanceWith input-output data collected fromactual flight experiments the identification results showed the superiority of CABC over the ABC and the genetic algorithm (GA)Simulations are presented to demonstrate the effectiveness of our proposed algorithm and the accuracy of the identified helicoptermodel

1 Introduction

Small-scale unmanned helicopter is a typical aerial robotthat can carry cameras sensors or other payloads whichis identified as an important part of the Unmanned AerialVehicles (UAVs) and will be used for some challenging mis-sions such as information gathering accurate measurementborder patrol and forest protection to name several aerialrobotic applications [1ndash3] Compared to fixed-wing aircraftsthe helicopters can take off and land vertically and they havethe ability of hovering and flying at low altitudes Moreovera small-scale helicopter not only is extremely sensitive tocontrol inputs and disturbances but also is a complex systemwith high unstable nonlinear multiple-inputmultiple output(MIMO) and high degree of coupling characteristics Hencethese factors make the unmanned helicopter attract muchattention and effort of researchers to design a satisfactoryMIMO linear controller And an accurate mathematicalmodel is the basis for the design of a feasible controller

In terms of academic research a small-scale unmannedhelicopter modeling approaches can be generally categorizedinto two approaches [4] first principle modeling and

modeling through system identification The first principleapproach to helicopter modeling applies the fundamentallaws of mechanics and aerodynamics to derive from amathematicalmodel with a large number of parametersmostof which cannot be directly measured [5] This modelingapproach is unsuitable for helicopter system because itrequires detailed knowledge and numerous measurementsof the helicopter flight mechanics Hence the first prin-ciple modeling approach may be tedious expensive andinefficient Conversely system identification is a convenientmethod based on integration of a mathematical model withreal flight data in determining the model parameters

In the realm of the unmanned helicopters a numberof system identification approaches have been reported inthe literature US army and NASA developed a frequencydomain method named Comprehensive Identification fromFrequency Responses (CIFER) and Mettler et al [6 7]used this software to identify the linear model for CarnegieMellonrsquos Yamaha R50 helicopter This model structure wasalso used successfully by other researchers [8 9] Howeverit is difficult for the frequency domain method to getthe suitable frequency spectrum through lots of frequency

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 801874 11 pageshttpdxdoiorg1011552015801874

2 International Journal of Aerospace Engineering

sweep experiments The use of prediction-error modeling(PEM) method has also been reported as a useful tool inautonomous helicopter model estimate Cai et al [10] appliedthe PEM method to identify the model of the yaw channelof a UAV helicopter Dharmayanda et al [11] obtained alinear model of the RUAV helicopter from flight data withPEM Nonetheless the PEM method is highly sensitive toparameter initialization and this increases the difficulty ofthe system identification for helicopters Raptis et al [12]used the Recursive least squares algorithm (RLS) to identify anonlinear helicoptermodel inX-Plane simulation Accordingto [13] the RLS algorithm was used to determine thehelicopterrsquos parameters as the entire model was divided intosix decoupled SISO subsystems Despite the wide applicationof the RLS algorithm the method may be trapped into localoptimum and ignore the global optimum

In recent years some intelligence computation algo-rithms have been employed in system identification forhelicopters In order to get the model of a UAV helicopter animmune particle swarm optimization algorithm (PSO) wasproposed to identify a linear state-space model in [14] Zhi-gang and Tiansheng [15] introduced the traditional geneticalgorithm (GA) to obtain the helicopter model and Lei andDu [16] used the adaptive GA to estimate the parametersof two decoupled linear helicopter models However whiledealing with complex and large-scale parameters identifi-cation problems the flaw that the premature convergencecan make the aforementioned methods stuck in a localoptimum

The artificial bee colony algorithm (ABC) was first pro-posed by Karaboga in 2005 [17] and successfully applied tomany fields like UAV path planning [18] function optimiza-tion problem [19] and clustering analysis [20] The ABC canbe also trapped into local optimum as well as other intelligentalgorithms In this paper considering the outstanding per-formance of chaotic operator in jumping out of stagnation anovel chaotic artificial bee colony algorithm (CABC) whichcombines chaotic optimization method with the ABC wasintroduced to conquer the system identification problem ofthe helicopters The identification was implemented on aTREX 600 Radio Controlled (RC) model helicopter with afull set of avionics system The comparison of three differentidentification methods GA ABC and CABC illustrated thesuperiority of our proposed algorithm

The outline of this paper is organized as follows Firstlythe helicopter dynamical model is given in Section 2 whereboth the nonlinear and linear models are described ThenSection 3 presents the identification process of the linearhelicopter model using the chaotic artificial bee colonyalgorithm Later on the experimental platform flight datapreprocess and identified results are presented in Section 4Finally the main conclusions and further research directionare given in Section 5

2 Helicopter Dynamical Model

An experimental TREX 600 helicopter is adopted asthe unmanned helicopter platform Since the small-scale

as

bs

OEYE

XE

ZE

OB

YB 120579 q

XB u 120601 p

ZB w 120595 r

Figure 1 TREX 600 states in body-fixed coordinate frame system

helicopter is treated as a rigid body it possesses 6 Degreesof Freedom (DOF) and the exhaustive description for forcesmoments and physical parameters for it can be foundin [21] All the force and moment components primarilyconsist of the contributions from the fuselage main rotorand tail rotor As shown in Figure 1 the standard rigidbody dynamical equations of the helicopter that describethe helicopterrsquos translational and rotational motion in thebody-fixed coordinate frame (BF) are given by Newton-Eulerequations

[[[[

V

]]]]= 1119898119865minus

[[[[

119901119902119903

]]]]sdot[[[[

119906V

119908

]]]]

[[[[

119902119903

]]]]= 119868minus1(119872minus[[[

[

119901119902119903

]]]]sdot 119868[[[[

119901119902119903

]]]])

(1)

where [119906 V 119908]119879 and [119901 119902 119903]119879 are the linear velocity and theangular velocity in BF 119865 and 119872 denote the external forcematrix and moment matrix acting on the center of mass ofthe helicopter respectively 119898 is the helicopter mass 119868 is theinertial moment matrix about the reference axes

The rotation matrix 119877(Θ) is defined by the yaw-pitch-roll Euler angles and maps vectors from BF to Earth-fixedcoordinate frame (EF) 119877(Θ) is governed by the equation

119877 (Θ) = [[[[

119888120579119888120595 119904120601119904120579119888120595 minus 119888120601119904120595 119888120601119904120579119888120595 + 119904120601119904120595119888120579119904120595 119904120601119904120579119904120595 + 119888120601119888120595 119888120601119904120579119904120595 minus 119904120601119888120595minus119904120579 119904120601119888120579 119888120601119888120579

]]]] (2)

where 119904120572 and 119888120572 are the abbreviations for cos120572 and sin120572The parameters 120601 120579 and 120595 represent the roll pitch andyaw angles of the helicopter in BF respectively The angular

International Journal of Aerospace Engineering 3

velocity of the helicopter is related to the time rate of changeof the Euler angles through the following relation

[[[[

120601120579

]]]]=[[[[[[

1119904120601119904120579119888120579

119888120601119904120579119888120579

0 119888120601 minus119904120601

0119904120601119888120579

119888120601119888120579

]]]]]]

[[[

119901119902119903]]] (3)

There are four control commands associated with heli-copter piloting The control inputs are defined as 119906 =[119906lat 119906lon 119906col 119906ped]119879 where 119906lat and 119906lon are the lateral andlongitudinal cyclic rotor control inputs 119906col is the collectivepitch control input and 119906ped is the tail rotor pedal controlinput

The simplified rotor dynamics which is common to allsmall-scale helicopters can be derived from the first-orderflapping dynamic equations [7]

[119886119904

119904

] = [[[

minus119886119904120591119904

+ 1198961119901 minus 119902 + 119860 lat119906latminus119887119904120591119904

minus 1198961119902 minus 119901 + 119861lon119906lon]]] (4)

where 119886119904

and 119887119904

are the longitudinal and lateral main rotorflapping angles 120591

119904

is the effective rotor time constant and119860 lat 119861lon and 1198961 are just gains

In the real control of the TREX 600 a feedback yawrate gyro installed on the helicopter is used to reduce theeffect of the antitorque fluctuation on the yaw response Hereaccording to [4] the additional yaw-rate gyro feedback term119903fb is added to account for the effect of the tail gyro

119903fb119903 = 119896

119903

119904 + 119896fb (5)

where 119896119903

and 119896fb are the parameters to be identifiedThe above nonlinear differential equations represent the

motion and orientation of the helicopter They can be rewrit-ten in a compact form as

= 119891 (119909 119906) (6)

where 119909 is a vector of 12 states written as 119909 =[119906 V 120601 120579 120595 119902 119901 119886

119904

119887119904

119908 119903 119903fb]119879However the nonlinear model is unfit for system iden-

tification for the helicopter Hence a linearized state-spacemodel of the helicopter system is derived from the nonlinearequations by using the small perturbation theory throughcapturing the actual behavior of the system near the trimcondition (eg hoveringcruise) [22] In that case120595 has littlecouple relationship with other variables and it is true that asymp 119903 Additionally it is difficult and impractical to estimateall the parameters in one go Hence the helicopter dynamicscan be separated into two interconnected subsystems thatis the horizontal and vertical motions In particular thesubsystems are given by (7) which were proposed in [23] andused successfully in [24]

120575hor = 119860hor120575119909hor +119861hor120575119906hor120575ver = 119860ver120575119909ver +119861ver120575119906ver

(7)

where 120575 is the perturbation from the trim condition and therelated matrices of the above models can be found as follows

119883hor = [119906 V 120579 120601 119902 119901 119886119904 119887119904]119879

119906hor = [119906lat 119906lon]119879

119883ver = [119908 119903 119903fb]119879

119906ver = [119906col 119906ped]119879

119860hor =

[[[[[[[[[[[[[[[[[[[[[[[

119883119906

0 minus119892 0 0 0 minus119892 0

0 119884V 0 119892 0 0 0 1198920 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

119872119906

119872V 0 0 119872119902

0 119872119886

119872119887

119871119906

119871V 0 0 0 119871119902

119871119886

119871119887

0 0 0 0 minus1 1198961 minus 1120591119904

0

0 0 0 0 minus1198961 minus1 0 minus 1120591119904

]]]]]]]]]]]]]]]]]]]]]]]

119861hor =

[[[[[[[[[[[[[[[[[[

119883lat 00 119884lon0 00 0119872lat 119872lon

119871 lat 119871 lon

119860 lat 00 119861lon

]]]]]]]]]]]]]]]]]]

119860ver = [[[

119885119908

119885119903

0119873119908

119873119903

119873fb

0 119896119903

minus119896fb

]]]

119861ver = [[[

119885col 119885ped

119873col 119873ped

0 0

]]]

(8)

In above matrices there are more than 30 unknownparameters to be identified Moreover some parameters areunable to be directly identified from the actual flight dataAccording to [7] the assumptions that 119873fb = minus119873ped and119896fb = minus2119873119903 are accepted in this paper

3 Identification of the Model Based onCABC Algorithm

31 Introduction to Chaos Theory Chaos theory is epito-mized by the so-called ldquobutteryrdquo detailed by Lorenz [25]


1 2 3 40

02

04

06

08

1

Iteration

Valu

e

02505075

Figure 2 Loss of chaotic property at three specific initial points

he discovered that tiny changes in initial conditions willmake radically disparate final results rendering long-termprediction impossible in general Chaotic map can bedescribed as a bounded nonlinear system with deterministicdynamic behavior that has ergodic and stochastic propertiesMoreover it has a very sensitive dependence upon initialconditions and parameters

Some common chaotic maps are logistic map tent mapsinusmap andGaussmap [26] in this paper the well-knownlogistic map is chosen to generate initialize condition Thelogistic map equation is defined as

119909119898+1 = 120583119909119898 (1minus119909119898) (9)

In the equation 119898 is the iteration number and the controlparameter 120583 isin [0 4] The behavior of the system (9) is greatlychanged with the variation of 120583 When 120583 = 4 the systemexhibits chaotic dynamics in which very small change in theinitial value of 119909 would give rise to very large differences inits long-term behavior [27] In this case the variable 119909 iscalled chaotic variable and the series 1199091 119909119898 is calledchaotic sequence With the characteristics of ergodicitypseudorandomness and irregularity the chaotic variable cantravel ergodically over the whole search space and neverrepeat a value having appeared already Suppose that 0 le119909119898

le 1 (expect 025 05 and 075 to be the fixed point)Figure 2 shows that the chaotic sequence is stuck in theendless loopwhen initial points are 025 or 075 since 4times025times(1minus025) = 075 and 4times075times(1minus075) = 075 Analogouslywhen initial point becomes 05 the chaotic sequence can alsobe stuck in the endless loop Consequently these three pointswill lead chaotic sequence lose chaotic property

32 Basic Principles of CABC As a highly robust and effi-cient method the real-coded chaotic artificial bee colony

algorithm (CABC) is applied to identify the TREX 600helicopter model In CABC the position of each food sourcerepresents a possible solution to the appropriate identifiedparameter In order to introduce the self-organization modelof bee selection that leads to the emergence of collectiveintelligence of honey bee swarms we define the populationof the colony bees as 119873

119904

the number of employed beesas 119873119890

and the number of unemployed bees as 119873119906

whichsatisfies the relation 119873

119904

= 2119873119890

= 2119873119906

We also define 119863 asthe dimension of solution vector that is the number of theunknown parameters The detailed procedure of executingthe proposed algorithm is expressed as follows

Step 1 (initialization) Randomly initialize a set of possiblesolutions (1199091 119909119873

119904

) and the particular solution 119909119894

can begoverned by

119909119895119894

= 119909119895min + rand (0 1) (119909119895max minus119909119895min) (10)

where 119895 isin 1 119863 donates the 119895th dimension of thesolution vector 119909119895min and 119909119895max mean the lower and upperbounds respectively

Step 2 Apply a specific function to calculate the fitness of thesolution119909

119894

according to the following equations and select thetop119873

119890

best solutions as the number of the employed bees

119865119894

=119873

sum119894=1

1003817100381710038171003817119910119894 minus 11991011989811989410038171003817100381710038171003817100381710038171003817119910119894 minus 1199101003817100381710038171003817 (11)

fit119894

= 1(1 + 119865

119894

) (12)

where fit119894

is the fitness function119865119894

is the objective function119873is the simulation length 119910

119894

is the actual flight data vector and119910 is themean of119910

119894

Similarly119910119898119894

is the vector of the simulateddata from the identified model

Step 3 Each employed bee searches new solution in theneighborhood of the current position vector in the 119899thiteration as follows

V119895119894

= 119909119895119894

+Φ119895119894

(119909119895119894

minus119909119895119896

) (13)

where 119896 isin 1 119863 119896 = 119894 both 119896 and 119895 are randomlygenerated andΦ119895

119894

is a randomparameter in the range fromminus1to 1 In order to ensure that the algorithm evolves to the globaloptimal we apply the following greedy selection equation tochoose the better solution between V119895

119894

and 119909119895119894

into the nextgeneration

119909119895119894

=

V119895119894

fit (V119895119894

) gt fit (119909119895119894

)119909119895119894

fit (V119895119894

) le fit (119909119895119894

) (14)

Step 4 Eachunemployed bee selects an employed bee to traceaccording to the parameter of probability value The formulaof the probability method is described as

119901119894

= fit119894

sum119873119890119894=1 fit119894

(15)


Step 5 The unemployed bee searches in the neighborhoodof the selected employed beersquos position to find new solutionsUpdate the current solution according to their fitness

Step 6 If the search time trial is larger than the pre-determined threshold limit and the optimal value cannotbe improved then the location vector can be reinitializedrandomly according to the following equation

119909119894

(119899 + 1)

=

119909min + rand (0 1) (119909max minus 119909min) trial gt limit

119909119894

(119899) trial le limit(16)

Step 7 Store the best solution and conduct the chaotic searcharound the best solution Among the engendered series ofsolutions the best one can be selected to replace a randomemployed bee The chaotic operator is written as follows

119909119898

= 119909119894

+119877119894

(2119909119898

minus 1) (17)

where chaotic sequence 119909119898

is mapped into the optimizationvector 119909

119898

And 119909119898

is located in the circle with center of 119909119894

and radius of 119877119894

Step 8 Output the best solution parameters achieved at thepresent time and go back to the Step 3 until terminationcriterion 119879max is met

The detailed procedure of CABC algorithm for systemidentification can be depicted in Figure 3

4 Model Validation and Analysis

41 Flight Data Collection and Processing The entire experi-ments are implemented on a TREX 600 RCmodel helicopterplatform equipped with flight control system GPS cameraand so forth (Figure 4) and the general physical parametersare shown in Table 1

To get the data of attitude angles and angular velocitiesin experiment an instrument IMU (inertial measurementunit) consisting of a three-axis accelerometer a three-axisgyroscope and a three-axis magnetometer was installed inthe flight control system module A brief photo of the flightcontrol system components is shown in Figure 5 And thespecifications of the components are given in Table 2 Itshould be noted that the flight control module is only usedfor obtaining flight data

In addition a low-cost GPS module installed on thealuminium board is kept on the bottom of the helicopter dueto the short cable whichmay result in poor satellite receptionand inaccurate data The appearance of the GPS module isshown in Figure 6 Some key specifications of the module arelisted in Table 3

Due to the structural vibrations frommain rotor a shockabsorber is installed below the flight control system moduleas shown in Figure 7 And the specifications of the shockabsorber are shown in Table 4

Start

employed bees

Employed bees search withgreedy selection

Unemployed bees search

solution

Reinitialize the parameters

Keep the best solution andconduct chaotic search

T = T + 1

Output

No

No

Yes

Yes

T = 1

Initialization Ns Tmax limit

with pi

T gt Tmax

Calculate fiti and select

Calculate fiti and update the

trial gt limit

Figure 3 The procedure of CABC method

Table 1 General physical parameters of TREX 600 helicopter

Physical quantities ValueRotor speed 2200 rpmFull length of fuselage 116mMain rotor diameter 135mTotal weight 64 kg

In the flight experiment five servos are used to controlthe longitudinal cyclic input lateral cyclic input collectivepedal and throttle respectively The pilot excites each of thesystem channels (roll pitch yaw and collective) with a seriesof inputs whilemaking the helicopter fly in hoverThe controlinputs and relative raw outputs were sampled with 50Hzfrequency and recorded in a 1 GB SD card


LiPo batteries

CameraGyro

Wireless transmissionFlight control system

GPS

Figure 4 TREX 600 helicopter and flight experiment

Figure 5 Flight control system in our helicopter

Table 2 Specifications of the flight control system components

Specification Sending rangeAcceleration range plusmn196ms2

Angular rate range plusmn200 degsMagnetometer range plusmn075GRollpitch accuracy 0005 radYaw accuracy 0012 radSize 65mm times 42mm times 13mmWeight 58 g

Since there are various errors in flight tests such as winddisturbance installation error andmotor error it is necessary

Figure 6 The appearance of GPS module

Figure 7 Shock absorber

Table 3 Specifications of GPS module

Specification GPS modulePosition accuracy 1mUpdate rate 18HzSensitivity minus167 dBmTime to first time 1 s (Hot)26 s (cold)Size 12060154mm times 15mmWeight 30 g

Table 4 Specifications of damping plate

Specification Shock absorberMaterial Glass fiberSize of lower plate 100mm times 69mm times 15mmSize of upper plate 91mm times 61mm times 15mmNet weight 22 gPackage weight 70 g

to preprocess the flight data before identification In orderto remove outliers and attenuate the effect of interferencesignal a five-point average FIR filter is adopted to smooth


the raw data 119884 = [1199101 119910119898]119879 according to the followingequation

1199101 =170

[691199101 + 4 (1199102 +1199104) minus 61199103 minus1199105]

1199102 =135

[2 (1199101 +1199105) + 271199102 + 121199103 minus 81199104]

119910119894

= 135

[minus3 (119910119894minus2 +119910119894+2) + 12 (119910119894minus1 +119910119894+1) + 17119910119894]

119910119898minus1

= 135

[2 (119910119898minus4 +119910119898) minus 8119910119898minus3 + 12119910119898minus2 + 27119910119898minus1]

119910119898

= 170

[minus119910119898minus4 + 4 (119910119898minus3 +119910119898minus1) minus 6119910119898minus2 + 69119910119898]

(18)

where 119894 = 3 119898 minus 2 and 119884 = [1199101 119910119898]119879 isthe data for identification after preprocessing through theaforementioned FIR filter The principle of the five-pointaverage FIR filter is the use of the least square method tosmooth the discrete data In the flight experiment the size ofthe flight data collected is far more than 5 and two adjacentpoints before and after each data point are approximated bya three-order polynomial with the least square method Themore the number of using (18) is the smoother the curveswill be It should be noted that excessive use of (18) to smooththe raw data can lead to the error of the system identificationincreasing

42 Experiment and Simulation System identification pro-cedures are carried out with CABC in Matlab 2012b pro-gramming environment on an Intel Core i7-3770 PC runningWindows 7 No commercial tools are used

According to [19] the performance of algorithm is basedon the population size of colony bees As the population sizeincreases the algorithm produces better results Howeverafter a sufficient value for colony size any increment in thevalue does not improve the performance of the algorithmsignificantly The control parameter limit is related to thelocal vector reinitialized frequency As the value of limitapproaches to infinity the total number of local vectorreinitialized goes to zero After many trials in this paperwe set the parameters of traditional ABC and CABC asfollows 119873

119904

= 20 Limit = 5 Moreover a GA algo-rithm [16] is also used to identify the helicopter modelsand the parameters selection is similar to ABC algorithmThe optimal parameters that is population size crossoverprobability and mutation probability are chosen as 20 08and 02 All the above algorithms are run 20 times theperformance comparison of the three different identificationmethods is presented as the three-axis linear velocities three-axis angular velocities and Euler angles illustrated in Figures8ndash10 The results show that the estimated data by applyingthe three proposed algorithms have the same trend as

Table 5 The values of identified parameters

Parameter Value119883119906

minus01064119884V minus002873119872119906

minus01215119872V 03012119872119902

02456119872119886

minus12648119872119887

minus231095119871119906

10106119871V minus09456119871119901

minus80143119871119886

625780119871119887

7831821198961 00510120591119904

minus72351119883lat minus58015119884lon 72462119872lat minus40126119872lon minus85145119871 lat 92156119871 lon minus56877119860 lat 64578119861lon minus52111119885119908

minus71048119885119903

20663119873119908

minus13975119873119903

04905119873fb minus07884119896119903

45578119896fb minus09810119885col minus80137119885ped 70868119873col 115558119873ped 07884

the measured data Nonetheless it turns out that the sim-ulation results generated by CABC approximate the actualtest data best The parameters identified by our proposedalgorithm are listed in Table 5

To further prove the performance of the CABC againstthe other algorithms the match degree 120588 between themeasured data 120582 and estimated data is governed by thefollowing equation and the results are listed in Table 6

120588 = 1minus10038171003817100381710038171003817 minus 120582

10038171003817100381710038171003817120582 times 100 (19)


0 2 4 6 8 10minus2

minus15

minus1

minus05

0

05

1

15

Time (s)

Actual dataGA

ABCCABC

u(m

s)

(a) Longitudinal velocity response

0 2 4 6 8 10Time (s)

Actual dataGA

ABCCABC

minus15

minus1

minus05

0

05

1

15

(m

s)

(b) Lateral velocity response

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2minus15

minus1minus05

005

151

2

w(m

s)

(c) Heave velocity response

Figure 8 Comparison of the actual and estimated velocities responses

Table 6 Comparison of math degrees

Parameter GA ABC CABC119906 7464 7715 7891V 7236 7541 7967119908 6586 6384 6880119902 5297 6726 7502119901 6612 5637 6988119903 4023 5397 6235120579 6778 6565 7206120601 7573 7814 7962

From the Table 6 we can clearly see that the matchdegree of the identification results produced by our proposedalgorithm is better than those of other algorithms Forexample the match degree of pitch rate by applying theCABC can be improved 8 percent and 22 percent comparedto applying ABC and GA methods respectively It should benoted that our proposed algorithm has a great ability to findthe global optimum with high accuracy

Figure 11 shows the evolution curves of CABC ABCand GA regarding (12) The figure demonstrates that thefitness function increases as the generation iterates with timegradually converging to an optimal result Compared withGA and ABC CABC achieves a better result with higherfitness value after 20 iterations It can be seen from the resultthat the chaotic operator cannot only avoid the traditionalABC being trapped in a local optimum but also improve itsrobustness and efficiency

5 Conclusion

In this paper the small-scale unmanned helicopter nonlinearmodel development and extraction of linearized model havebeen presented Based on the flight data collected from flightexperiment we use a novel identification algorithm CABCincluding ABC method and chaotic operator to identify theunknown parameters of the two decoupled linear modelsFurthermore the evolutionary curves show that CABC canget a better fitness value when compared with GA and ABCSimulation results verify the effectiveness feasibility androbustness of our proposed algorithm

In the future we will continue to study the design ofcontrol law based on our identified model


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)

(a) Pitch rate response

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1

(b) Roll rate response

r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1

(c) Yaw rate response


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16minus03minus02minus01

00102030405

Time (s)

120579(r

ad)

(a) Pitch angle response

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)

(b) Roll angle response

Figure 10 Comparison of the actual and estimated angles responses


0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA

Figure 11 Evolutionary curves of CABC ABC and GA

Conflict of Interests

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

Acknowledgments

This work was partially supported by the National NaturalSciences Foundation of China (51375230) and the project ofScience and Technology Support Plan of Jiangsu province(BE2013003-1 and BE2013010-2)

References

[1] I A Raptis K P Valavanis and G J Vachtsevanos ldquoLineartracking control for small-scale unmanned helicoptersrdquo IEEETransactions on Control Systems Technology vol 20 no 4 pp995ndash1010 2012

[2] I B Tijani R Akmeliawati A Legowo A Budiyono and A GAbdul Muthalif ldquoHybrid DE-PEM algorithm for identificationof UAV helicopterrdquo Aircraft Engineering and Aerospace Technol-ogy vol 86 no 5 pp 385ndash405 2014

[3] D Song and J Qi ldquoModeling and control for unmannedhelicopter based aerial manipulatorrdquo in Proceedings of the6th International Conference on Modelling Identification andControl (ICMIC rsquo14) pp 259ndash264 IEEE December 2014

[4] S Tang X Lu and Z Zheng ldquoPlatform and state estimationdesign of a small-scale UAV helicopter systemrdquo InternationalJournal of Aerospace Engineering vol 2013 Article ID 52485613 pages 2013

[5] K Kondak M Bernard N Losse et al ldquoElaborated modelingand control for autonomous small size helicoptersrdquo VDI-Berichte no 1956 pp 207ndash219 2006

[6] B Mettler T Kanade and M B Tischler System IdentificationModeling of a Model-Scale Helicopter Carnegie Mellon Univer-sity The Robotics Institute 2000

[7] V Gavrilets B Mettler and E Feron ldquoNonlinear model fora small-size acrobatic helicopterrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and Exhibit pp6ndash9 August 2001

[8] D H Shim H J Kim and S Sastry ldquoControl system design forrotorcraft-based unmanned aerial vehicles using time-domain

system identificationrdquo in Proceedings of the IEEE InternationalConference on Control Applications pp 808ndash813 IEEE 2000

[9] B Kim Y Chang and M H Lee ldquoSystem identificationand 6-DOF hovering controller design of unmanned modelhelicopterrdquo JSME International Journal Series C MechanicalSystems Machine Elements and Manufacturing vol 49 no 4pp 1048ndash1057 2007

[10] G Cai BM Chen K PengM Dong and T H Lee ldquoModelingand control of the yaw channel of a UAV helicopterrdquo IEEETransactions on Industrial Electronics vol 55 no 9 pp 3426ndash3434 2008

[11] H R Dharmayanda A Budiyono and T Kang ldquoState spaceidentification and implementation of 119867

infin

control design forsmall-scale helicopterrdquoAircraft Engineering andAerospace Tech-nology vol 82 no 6 pp 340ndash352 2010

[12] I A Raptis K P Valavanis and W A Moreno ldquoSystem identi-fication and discrete nonlinear control of miniature helicoptersusing backsteppingrdquo Journal of Intelligent and Robotic SystemsTheory and Applications vol 55 no 2-3 pp 223ndash243 2009

[13] S K Kim and D M Tilbury ldquoMathematical modeling andexperimental identification of a model helicopterrdquo in Pro-ceedings of the AIAA Modeling and Simulation TechnologiesConference and Exhibit pp 203ndash213 Boston Mass USA 1998

[14] T T Yang and A J Li ldquoAn unmanned helicopter modelidentification method based on the immune particle swarmoptimization algorithmrdquo Applied Mechanics and Materials vol347ndash350 pp 3890ndash3893 2013

[15] Z Zhigang and L Tiansheng ldquoGA-based evolutionary iden-tification of model structure for small-scale robot helicoptersystemrdquo in Embedded SystemsmdashModeling Technology andApplications pp 149ndash158 Springer Netherlands 2006

[16] X Lei and Y Du ldquoA linear domain system identification forsmall unmanned aerial rotorcraft based on adaptive geneticalgorithmrdquo Journal of Bionic Engineering vol 7 no 2 pp 142ndash149 2010

[17] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep tr06 Computer Engineering Depart-ment Engineering Faculty Erciyes University 2005

[18] C Xu H Duan and F Liu ldquoChaotic artificial bee colonyapproach to Uninhabited Combat Air Vehicle (UCAV) pathplanningrdquo Aerospace Science and Technology vol 14 no 8 pp535ndash541 2010

[19] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008

[20] C Zhang D Ouyang and J Ning ldquoAn artificial bee colonyapproach for clusteringrdquo Expert Systems with Applications vol37 no 7 pp 4761ndash4767 2010

[21] RW ProutyHelicopter Performance Stability andControl REKrieger Publishing 1995

[22] S Bhandari and R Colgren ldquo14-DOF linear parameter varyingmodel of a UAV helicopter using analytical techniquesrdquo inProceedings of the American Institute of Aeronautics and Astro-nautics Modeling and Simulation Conference and Exhibit 2008

[23] I Raptis and K Valavanis Linear and Nonlinear Control ofSmall-Scale Unmanned Helicopters Springer 2010

[24] T Wang Y Chen J Liang Y Wu C Wang and Y ZhangldquoChaos-genetic algorithm for the system identification of asmall unmanned helicopterrdquo Journal of Intelligent and RoboticSystems Theory and Applications vol 67 no 3-4 pp 323ndash3382012


[25] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[26] S Talatahari B F Azar R Sheikholeslami and A H GandomildquoImperialist competitive algorithm combined with chaos forglobal optimizationrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 3 pp 1312ndash1319 2012

[27] Y Wu and S Zhang ldquoA self-adaptive chaos particle swarmoptimization algorithmrdquo TELKOMNIKA (TelecommunicationComputing Electronics and Control) vol 13 no 1 pp 331ndash3402015

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



sweep experiments The use of prediction-error modeling(PEM) method has also been reported as a useful tool inautonomous helicopter model estimate Cai et al [10] appliedthe PEM method to identify the model of the yaw channelof a UAV helicopter Dharmayanda et al [11] obtained alinear model of the RUAV helicopter from flight data withPEM Nonetheless the PEM method is highly sensitive toparameter initialization and this increases the difficulty ofthe system identification for helicopters Raptis et al [12]used the Recursive least squares algorithm (RLS) to identify anonlinear helicoptermodel inX-Plane simulation Accordingto [13] the RLS algorithm was used to determine thehelicopterrsquos parameters as the entire model was divided intosix decoupled SISO subsystems Despite the wide applicationof the RLS algorithm the method may be trapped into localoptimum and ignore the global optimum

In recent years some intelligence computation algo-rithms have been employed in system identification forhelicopters In order to get the model of a UAV helicopter animmune particle swarm optimization algorithm (PSO) wasproposed to identify a linear state-space model in [14] Zhi-gang and Tiansheng [15] introduced the traditional geneticalgorithm (GA) to obtain the helicopter model and Lei andDu [16] used the adaptive GA to estimate the parametersof two decoupled linear helicopter models However whiledealing with complex and large-scale parameters identifi-cation problems the flaw that the premature convergencecan make the aforementioned methods stuck in a localoptimum

The artificial bee colony algorithm (ABC) was first pro-posed by Karaboga in 2005 [17] and successfully applied tomany fields like UAV path planning [18] function optimiza-tion problem [19] and clustering analysis [20] The ABC canbe also trapped into local optimum as well as other intelligentalgorithms In this paper considering the outstanding per-formance of chaotic operator in jumping out of stagnation anovel chaotic artificial bee colony algorithm (CABC) whichcombines chaotic optimization method with the ABC wasintroduced to conquer the system identification problem ofthe helicopters The identification was implemented on aTREX 600 Radio Controlled (RC) model helicopter with afull set of avionics system The comparison of three differentidentification methods GA ABC and CABC illustrated thesuperiority of our proposed algorithm

The outline of this paper is organized as follows Firstlythe helicopter dynamical model is given in Section 2 whereboth the nonlinear and linear models are described ThenSection 3 presents the identification process of the linearhelicopter model using the chaotic artificial bee colonyalgorithm Later on the experimental platform flight datapreprocess and identified results are presented in Section 4Finally the main conclusions and further research directionare given in Section 5

2 Helicopter Dynamical Model

An experimental TREX 600 helicopter is adopted asthe unmanned helicopter platform Since the small-scale

as

bs

OEYE

XE

ZE

OB

YB 120579 q

XB u 120601 p

ZB w 120595 r

Figure 1 TREX 600 states in body-fixed coordinate frame system

helicopter is treated as a rigid body it possesses 6 Degreesof Freedom (DOF) and the exhaustive description for forcesmoments and physical parameters for it can be foundin [21] All the force and moment components primarilyconsist of the contributions from the fuselage main rotorand tail rotor As shown in Figure 1 the standard rigidbody dynamical equations of the helicopter that describethe helicopterrsquos translational and rotational motion in thebody-fixed coordinate frame (BF) are given by Newton-Eulerequations

[[[[

V

]]]]= 1119898119865minus

[[[[

119901119902119903

]]]]sdot[[[[

119906V

119908

]]]]

[[[[

119902119903

]]]]= 119868minus1(119872minus[[[

[

119901119902119903

]]]]sdot 119868[[[[

119901119902119903

]]]])

(1)

where [119906 V 119908]119879 and [119901 119902 119903]119879 are the linear velocity and theangular velocity in BF 119865 and 119872 denote the external forcematrix and moment matrix acting on the center of mass ofthe helicopter respectively 119898 is the helicopter mass 119868 is theinertial moment matrix about the reference axes

The rotation matrix 119877(Θ) is defined by the yaw-pitch-roll Euler angles and maps vectors from BF to Earth-fixedcoordinate frame (EF) 119877(Θ) is governed by the equation

119877 (Θ) = [[[[

119888120579119888120595 119904120601119904120579119888120595 minus 119888120601119904120595 119888120601119904120579119888120595 + 119904120601119904120595119888120579119904120595 119904120601119904120579119904120595 + 119888120601119888120595 119888120601119904120579119904120595 minus 119904120601119888120595minus119904120579 119904120601119888120579 119888120601119888120579

]]]] (2)

where 119904120572 and 119888120572 are the abbreviations for cos120572 and sin120572The parameters 120601 120579 and 120595 represent the roll pitch andyaw angles of the helicopter in BF respectively The angular



[[[[

120601120579

]]]]=[[[[[[

1119904120601119904120579119888120579

119888120601119904120579119888120579

0 119888120601 minus119904120601

0119904120601119888120579

119888120601119888120579

]]]]]]

[[[

119901119902119903]]] (3)



[119886119904

119904

] = [[[

minus119886119904120591119904



where 119886119904

and 119887119904


119904



119903fb119903 = 119896

119903

119904 + 119896fb (5)

where 119896119903



= 119891 (119909 119906) (6)


119904

119887119904




(7)


119883hor = [119906 V 120579 120601 119902 119901 119886119904 119887119904]119879

119906hor = [119906lat 119906lon]119879

119883ver = [119908 119903 119903fb]119879

119906ver = [119906col 119906ped]119879

119860hor =

[[[[[[[[[[[[[[[[[[[[[[[

119883119906

0 minus119892 0 0 0 minus119892 0

0 119884V 0 119892 0 0 0 1198920 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

119872119906

119872V 0 0 119872119902

0 119872119886

119872119887

119871119906

119871V 0 0 0 119871119902

119871119886

119871119887

0 0 0 0 minus1 1198961 minus 1120591119904

0


]]]]]]]]]]]]]]]]]]]]]]]

119861hor =

[[[[[[[[[[[[[[[[[[

119883lat 00 119884lon0 00 0119872lat 119872lon

119871 lat 119871 lon

119860 lat 00 119861lon

]]]]]]]]]]]]]]]]]]

119860ver = [[[

119885119908

119885119903

0119873119908

119873119903

119873fb

0 119896119903

minus119896fb

]]]

119861ver = [[[

119885col 119885ped

119873col 119873ped

0 0

]]]

(8)





1 2 3 40

02

04

06

08

1

Iteration

Valu

e

02505075




119909119898+1 = 120583119909119898 (1minus119909119898) (9)





119904




119904

= 2119873119890

= 2119873119906



119904


can begoverned by

119909119895119894




119894


119890


119865119894

=119873

sum119894=1

1003817100381710038171003817119910119894 minus 11991011989811989410038171003817100381710038171003817100381710038171003817119910119894 minus 1199101003817100381710038171003817 (11)

fit119894

= 1(1 + 119865

119894

) (12)

where fit119894



119894


119894

Similarly119910119898119894



V119895119894

= 119909119895119894

+Φ119895119894

(119909119895119894

minus119909119895119896

) (13)


119894


119894

and 119909119895119894


119909119895119894

=

V119895119894

fit (V119895119894

) gt fit (119909119895119894

)119909119895119894

fit (V119895119894

) le fit (119909119895119894

) (14)


119901119894

= fit119894

sum119873119890119894=1 fit119894

(15)




119909119894

(119899 + 1)

=


119909119894



119909119898

= 119909119894

+119877119894

(2119909119898

minus 1) (17)



119898

And 119909119898










Start

employed bees



solution



T = T + 1

Output

No

No

Yes

Yes

T = 1


with pi

T gt Tmax



trial gt limit






LiPo batteries

CameraGyro


GPS
















1199101 =170

[691199101 + 4 (1199102 +1199104) minus 61199103 minus1199105]

1199102 =135

[2 (1199101 +1199105) + 271199102 + 121199103 minus 81199104]

119910119894

= 135


119910119898minus1

= 135


119910119898

= 170


(18)




119904




minus01064119884V minus002873119872119906

minus01215119872V 03012119872119902

02456119872119886

minus12648119872119887

minus231095119871119906

10106119871V minus09456119871119901

minus80143119871119886

625780119871119887

7831821198961 00510120591119904


minus71048119885119903

20663119873119908

minus13975119873119903

04905119873fb minus07884119896119903




120588 = 1minus10038171003817100381710038171003817 minus 120582

10038171003817100381710038171003817120582 times 100 (19)


0 2 4 6 8 10minus2

minus15

minus1

minus05

0

05

1

15

Time (s)

Actual dataGA

ABCCABC

u(m

s)


0 2 4 6 8 10Time (s)

Actual dataGA

ABCCABC

minus15

minus1

minus05

0

05

1

15

(m

s)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2minus15

minus1minus05

005

151

2

w(m

s)







5 Conclusion




Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1


r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1



Actual dataGA

ABCCABC


00102030405

Time (s)

120579(r

ad)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)




0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[[[[

120601120579

]]]]=[[[[[[

1119904120601119904120579119888120579

119888120601119904120579119888120579

0 119888120601 minus119904120601

0119904120601119888120579

119888120601119888120579

]]]]]]

[[[

119901119902119903]]] (3)



[119886119904

119904

] = [[[

minus119886119904120591119904



where 119886119904

and 119887119904


119904



119903fb119903 = 119896

119903

119904 + 119896fb (5)

where 119896119903



= 119891 (119909 119906) (6)


119904

119887119904




(7)


119883hor = [119906 V 120579 120601 119902 119901 119886119904 119887119904]119879

119906hor = [119906lat 119906lon]119879

119883ver = [119908 119903 119903fb]119879

119906ver = [119906col 119906ped]119879

119860hor =

[[[[[[[[[[[[[[[[[[[[[[[

119883119906

0 minus119892 0 0 0 minus119892 0

0 119884V 0 119892 0 0 0 1198920 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

119872119906

119872V 0 0 119872119902

0 119872119886

119872119887

119871119906

119871V 0 0 0 119871119902

119871119886

119871119887

0 0 0 0 minus1 1198961 minus 1120591119904

0


]]]]]]]]]]]]]]]]]]]]]]]

119861hor =

[[[[[[[[[[[[[[[[[[

119883lat 00 119884lon0 00 0119872lat 119872lon

119871 lat 119871 lon

119860 lat 00 119861lon

]]]]]]]]]]]]]]]]]]

119860ver = [[[

119885119908

119885119903

0119873119908

119873119903

119873fb

0 119896119903

minus119896fb

]]]

119861ver = [[[

119885col 119885ped

119873col 119873ped

0 0

]]]

(8)





1 2 3 40

02

04

06

08

1

Iteration

Valu

e

02505075




119909119898+1 = 120583119909119898 (1minus119909119898) (9)





119904




119904

= 2119873119890

= 2119873119906



119904


can begoverned by

119909119895119894




119894


119890


119865119894

=119873

sum119894=1

1003817100381710038171003817119910119894 minus 11991011989811989410038171003817100381710038171003817100381710038171003817119910119894 minus 1199101003817100381710038171003817 (11)

fit119894

= 1(1 + 119865

119894

) (12)

where fit119894



119894


119894

Similarly119910119898119894



V119895119894

= 119909119895119894

+Φ119895119894

(119909119895119894

minus119909119895119896

) (13)


119894


119894

and 119909119895119894


119909119895119894

=

V119895119894

fit (V119895119894

) gt fit (119909119895119894

)119909119895119894

fit (V119895119894

) le fit (119909119895119894

) (14)


119901119894

= fit119894

sum119873119890119894=1 fit119894

(15)




119909119894

(119899 + 1)

=


119909119894



119909119898

= 119909119894

+119877119894

(2119909119898

minus 1) (17)



119898

And 119909119898










Start

employed bees



solution



T = T + 1

Output

No

No

Yes

Yes

T = 1


with pi

T gt Tmax



trial gt limit






LiPo batteries

CameraGyro


GPS
















1199101 =170

[691199101 + 4 (1199102 +1199104) minus 61199103 minus1199105]

1199102 =135

[2 (1199101 +1199105) + 271199102 + 121199103 minus 81199104]

119910119894

= 135


119910119898minus1

= 135


119910119898

= 170


(18)




119904




minus01064119884V minus002873119872119906

minus01215119872V 03012119872119902

02456119872119886

minus12648119872119887

minus231095119871119906

10106119871V minus09456119871119901

minus80143119871119886

625780119871119887

7831821198961 00510120591119904


minus71048119885119903

20663119873119908

minus13975119873119903

04905119873fb minus07884119896119903




120588 = 1minus10038171003817100381710038171003817 minus 120582

10038171003817100381710038171003817120582 times 100 (19)


0 2 4 6 8 10minus2

minus15

minus1

minus05

0

05

1

15

Time (s)

Actual dataGA

ABCCABC

u(m

s)


0 2 4 6 8 10Time (s)

Actual dataGA

ABCCABC

minus15

minus1

minus05

0

05

1

15

(m

s)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2minus15

minus1minus05

005

151

2

w(m

s)







5 Conclusion




Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1


r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1



Actual dataGA

ABCCABC


00102030405

Time (s)

120579(r

ad)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)




0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 40

02

04

06

08

1

Iteration

Valu

e

02505075




119909119898+1 = 120583119909119898 (1minus119909119898) (9)





119904




119904

= 2119873119890

= 2119873119906



119904


can begoverned by

119909119895119894




119894


119890


119865119894

=119873

sum119894=1

1003817100381710038171003817119910119894 minus 11991011989811989410038171003817100381710038171003817100381710038171003817119910119894 minus 1199101003817100381710038171003817 (11)

fit119894

= 1(1 + 119865

119894

) (12)

where fit119894



119894


119894

Similarly119910119898119894



V119895119894

= 119909119895119894

+Φ119895119894

(119909119895119894

minus119909119895119896

) (13)


119894


119894

and 119909119895119894


119909119895119894

=

V119895119894

fit (V119895119894

) gt fit (119909119895119894

)119909119895119894

fit (V119895119894

) le fit (119909119895119894

) (14)


119901119894

= fit119894

sum119873119890119894=1 fit119894

(15)




119909119894

(119899 + 1)

=


119909119894



119909119898

= 119909119894

+119877119894

(2119909119898

minus 1) (17)



119898

And 119909119898










Start

employed bees



solution



T = T + 1

Output

No

No

Yes

Yes

T = 1


with pi

T gt Tmax



trial gt limit






LiPo batteries

CameraGyro


GPS
















1199101 =170

[691199101 + 4 (1199102 +1199104) minus 61199103 minus1199105]

1199102 =135

[2 (1199101 +1199105) + 271199102 + 121199103 minus 81199104]

119910119894

= 135


119910119898minus1

= 135


119910119898

= 170


(18)




119904




minus01064119884V minus002873119872119906

minus01215119872V 03012119872119902

02456119872119886

minus12648119872119887

minus231095119871119906

10106119871V minus09456119871119901

minus80143119871119886

625780119871119887

7831821198961 00510120591119904


minus71048119885119903

20663119873119908

minus13975119873119903

04905119873fb minus07884119896119903




120588 = 1minus10038171003817100381710038171003817 minus 120582

10038171003817100381710038171003817120582 times 100 (19)


0 2 4 6 8 10minus2

minus15

minus1

minus05

0

05

1

15

Time (s)

Actual dataGA

ABCCABC

u(m

s)


0 2 4 6 8 10Time (s)

Actual dataGA

ABCCABC

minus15

minus1

minus05

0

05

1

15

(m

s)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2minus15

minus1minus05

005

151

2

w(m

s)







5 Conclusion




Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1


r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1



Actual dataGA

ABCCABC


00102030405

Time (s)

120579(r

ad)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)




0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119909119894

(119899 + 1)

=


119909119894



119909119898

= 119909119894

+119877119894

(2119909119898

minus 1) (17)



119898

And 119909119898










Start

employed bees



solution



T = T + 1

Output

No

No

Yes

Yes

T = 1


with pi

T gt Tmax



trial gt limit






LiPo batteries

CameraGyro


GPS
















1199101 =170

[691199101 + 4 (1199102 +1199104) minus 61199103 minus1199105]

1199102 =135

[2 (1199101 +1199105) + 271199102 + 121199103 minus 81199104]

119910119894

= 135


119910119898minus1

= 135


119910119898

= 170


(18)




119904




minus01064119884V minus002873119872119906

minus01215119872V 03012119872119902

02456119872119886

minus12648119872119887

minus231095119871119906

10106119871V minus09456119871119901

minus80143119871119886

625780119871119887

7831821198961 00510120591119904


minus71048119885119903

20663119873119908

minus13975119873119903

04905119873fb minus07884119896119903




120588 = 1minus10038171003817100381710038171003817 minus 120582

10038171003817100381710038171003817120582 times 100 (19)


0 2 4 6 8 10minus2

minus15

minus1

minus05

0

05

1

15

Time (s)

Actual dataGA

ABCCABC

u(m

s)


0 2 4 6 8 10Time (s)

Actual dataGA

ABCCABC

minus15

minus1

minus05

0

05

1

15

(m

s)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2minus15

minus1minus05

005

151

2

w(m

s)







5 Conclusion




Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1


r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1



Actual dataGA

ABCCABC


00102030405

Time (s)

120579(r

ad)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)




0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










LiPo batteries

CameraGyro


GPS
















1199101 =170

[691199101 + 4 (1199102 +1199104) minus 61199103 minus1199105]

1199102 =135

[2 (1199101 +1199105) + 271199102 + 121199103 minus 81199104]

119910119894

= 135


119910119898minus1

= 135


119910119898

= 170


(18)




119904




minus01064119884V minus002873119872119906

minus01215119872V 03012119872119902

02456119872119886

minus12648119872119887

minus231095119871119906

10106119871V minus09456119871119901

minus80143119871119886

625780119871119887

7831821198961 00510120591119904


minus71048119885119903

20663119873119908

minus13975119873119903

04905119873fb minus07884119896119903




120588 = 1minus10038171003817100381710038171003817 minus 120582

10038171003817100381710038171003817120582 times 100 (19)


0 2 4 6 8 10minus2

minus15

minus1

minus05

0

05

1

15

Time (s)

Actual dataGA

ABCCABC

u(m

s)


0 2 4 6 8 10Time (s)

Actual dataGA

ABCCABC

minus15

minus1

minus05

0

05

1

15

(m

s)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2minus15

minus1minus05

005

151

2

w(m

s)







5 Conclusion




Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1


r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1



Actual dataGA

ABCCABC


00102030405

Time (s)

120579(r

ad)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)




0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1199101 =170

[691199101 + 4 (1199102 +1199104) minus 61199103 minus1199105]

1199102 =135

[2 (1199101 +1199105) + 271199102 + 121199103 minus 81199104]

119910119894

= 135


119910119898minus1

= 135


119910119898

= 170


(18)




119904




minus01064119884V minus002873119872119906

minus01215119872V 03012119872119902

02456119872119886

minus12648119872119887

minus231095119871119906

10106119871V minus09456119871119901

minus80143119871119886

625780119871119887

7831821198961 00510120591119904


minus71048119885119903

20663119873119908

minus13975119873119903

04905119873fb minus07884119896119903




120588 = 1minus10038171003817100381710038171003817 minus 120582

10038171003817100381710038171003817120582 times 100 (19)


0 2 4 6 8 10minus2

minus15

minus1

minus05

0

05

1

15

Time (s)

Actual dataGA

ABCCABC

u(m

s)


0 2 4 6 8 10Time (s)

Actual dataGA

ABCCABC

minus15

minus1

minus05

0

05

1

15

(m

s)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2minus15

minus1minus05

005

151

2

w(m

s)







5 Conclusion




Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1


r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1



Actual dataGA

ABCCABC


00102030405

Time (s)

120579(r

ad)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)




0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10minus2

minus15

minus1

minus05

0

05

1

15

Time (s)

Actual dataGA

ABCCABC

u(m

s)


0 2 4 6 8 10Time (s)

Actual dataGA

ABCCABC

minus15

minus1

minus05

0

05

1

15

(m

s)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2minus15

minus1minus05

005

151

2

w(m

s)







5 Conclusion




Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1


r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1



Actual dataGA

ABCCABC


00102030405

Time (s)

120579(r

ad)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)




0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus1

minus05

0

05

1

q(r

ads

)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

p(r

ads

)

minus2

minus15

minus1

minus05

0

05

1


r(r

ads

)

Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus2

minus1

0

1



Actual dataGA

ABCCABC


00102030405

Time (s)

120579(r

ad)


Actual dataGA

ABCCABC

0 2 4 6 8 10 12 14 16Time (s)

minus05

minus025

0

025

05

120601(r

ad)




0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20

010203040506070809

Iteration

Fitn

ess

CABCABCGA




Acknowledgments


References













infin





















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









Research Article Chaotic Artificial Bee Colony Algorithm for...

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