research article dynamic model identification for 6...

Research ArticleDynamic Model Identification for 6-DOF Industrial Robots

Li Ding1 Hongtao Wu1 Yu Yao2 and Yuxuan Yang1

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

Correspondence should be addressed to Hongtao Wu mehtwu126com

Received 25 August 2015 Accepted 12 October 2015

Academic Editor Keigo Watanabe

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

A complete and systematic procedure for the dynamical parameters identification of industrial robot manipulator is presentedThesystemmodel of robot including joint frictionmodel is linear with respect to the dynamical parameters Identification experimentsare carried out for a 6-degree-of-freedom (DOF) ER-16 robot Relevant data is sampled while the robot is tracking optimaltrajectories that excite the system The artificial bee colony algorithm is introduced to estimate the unknown parameters And wevalidate the dynamical model according to torque prediction accuracy All the results are presented to demonstrate the efficiencyof our proposed identification algorithm and the accuracy of the identified robot model

1 Introduction

In recent years industrial robots have been greatly used asorienting devices in industry especially in the shipbuildingautomotive and aerospace manufacturing industries [1 2]Advanced control techniques for robots have become moreandmore affordable thanks to increasing power of computingresources and their dramatic cost reduction However thedynamical model of robot contains uncertainties in someparameters and many control methods are sensitive to theirvalues especially in high speed operations Hence dynamicalparameters identification approach has importance for devel-oping model based controllers

In terms of academic research a standard robot iden-tification procedure consists of dynamic modeling exci-tation trajectory design data collection signal preprocessparameter identification and model validation [3] Theparameter identification has attracted considerable attentionfrom numerous researchers Atkeson et al [4] proposed theleast square method to realize the estimation of dynamicalparameters Grotjahn et al [5] used the two-step approachto perform the identification of robot dynamics Gautierand Poignet [6] obtained a dynamical model of SCARArobot from experimental data with weighted least squaresmethod Behzad et al [7] applied fractional subspace method

to identify a robot model in simulation field Recently someintelligence computation algorithms have been reported as auseful tool in robotmodel identificationA traditional geneticalgorithm (GA) was proposed to identify the autonomousunderwater robot in [8] Liu et al [9] introduced theimproved genetic algorithm to obtain the space robot modelHowever while dealing with complex and large-scale param-eters identification problems the GA algorithm would bestuck on local optimum

Artificial bee colony algorithm (ABC) was first pro-posed by Karaboga in 2005 [10] and successfully appliedto parameters identification of aerial robot [11] The ABCalgorithm has been proved to possess a better performancein function optimization problems compared with differen-tial evolution algorithm (DE) particle swarm optimizationalgorithm (PSO) and GA algorithm [12] As we know usualoptimization algorithms conduct only one search operationin one iteration but ABC algorithm can conduct both localsearch and global search in each iteration and as a result theprobability of finding the optimal parameters is significantlyincreased which efficiently avoids local optimum to a largeextent In this paper the ABC algorithm was introduced toconquer the parameters identification problem of the indus-trial robots The identification experiment was implementedon 6-DOF ER-16 robot manipulator

Hindawi Publishing CorporationJournal of RoboticsVolume 2015 Article ID 471478 9 pageshttpdxdoiorg1011552015471478

2 Journal of Robotics

The outline of this paper is organized as follows Firstlythe linear robot dynamical model is given in Section 2Then Section 3 presents the identification process of thelinear model based on the ABC algorithm where excitationdesign data collection and signal preprocess are describedLater on the experimental platform identified results andmodel validation are presented in Section 4 Finally the mainconclusions are given in Section 5

2 Dynamic Modeling

Since the 119899-DOF industrial robot is represented by a kine-matic chain of rigid bodies the exhaustive description for itsmotion can be found in [13]The dynamicmodel of industrialrobot is derived by the Newton-Euler or Lagrangian method

120591dyn = M (119902) 119902 + C (119902 119902) + G (119902) (1)

where 120591dyn is the 119899-vector of actuator torques as well as thejoint positions 119902 velocities 119902 and accelerations 119902M(119902) is the119899 times 119899 inertia matrix C(119902 119902) denotes the 119899-vector includingCoriolis and centrifugal forces and G(119902) is the 119899-vector ofgravity

According to themodifiedNewton-Euler parameters [14]or the barycentric parameters [15] (1) can be rewritten as alinear form

120591dyn = Φdyn (119902 119902 119902) 120579dyn (2)

whereΦdyn denotes the 119899 times 10119899 observation or identificationmatrix which depends only on the motion data 120579dyn isthe barycentric parameter vector This property considerablysimplifies the parameters identification

Dynamic model of robots also contains the torquescaused by joint frictions and inertias of actuator rotors apartfrom the effects of dynamic parameters in (2) The inertias ofactuator rotors are generally provided by producers and cor-responding torques should be compensated for the dynamicequations In fact joint friction is a complex nonlinearmodelespecially during motion reversal In order to simplify themodel the friction model consisting of only Coulomb andviscous friction [16] is given by

120591fric = 119891119888sign ( 119902) + 119891V 119902 (3)

where 120591fric is the friction torques and 119891119888 119891V respectively

mean the Coulomb and viscous friction parametersThe integrated dynamicmodel of robots can be written as

120591119904= Φ119904(119902 119902 119902) 120579

119904 (4)

where 120591119904is actuator torques including 120591dyn and 120591fricΦ119904 is the

119899 times 12119899 observation matrix and 120579119904is 12119899-vector of unknown

dynamic parameters In addition the dynamic parameters oflink 119894 are governed by the form

120579119894

119904= [119868119909119909119894 119868119909119910119894 119868119909119911119894 119868119910119910119894 119868119910119911119894 119868119911119911119894 119898119894119903119909119894 119898119894119903119910119894 119898119894119903119911119894 119898119894 119891119888119894

119891V119894]119879

(5)

where 119868120577120577119894

(120577 = 119909 119910 119911) is the inertial tensor of link 119894 Similarly119898119894119903120577119894denotes the first-order mass moment and119898

119894is the mass

of link 119894In general the observation matrix Φ

119904in (4) is not

a full rank that is not all dynamic parameters have aninfluence on the dynamic model In order to obtain a setof minimum parameters a case-by-case analysis method isadopted [17] Consequently the dynamic model based on thebasic dynamic parameters can be rewritten as

120591 = Φ (119902 119902 119902) 120579 (6)

whereΦ is the 119899 times (119901 + 2119899) observation matrix 120579 is (119901 + 2119899)-vector of dynamic parameters including the basic parametersand the friction parameters 119901 denotes the number of theminimum dynamical parameters 2119899 denotes the number ofthe friction parameters

3 Parameters Identification Procedures

31 Basic Principles of Identification Algorithm In order tointroduce the search mechanism of ABC algorithm weshould define three essential components employed beesunemployed bees and food source [11] And the unemployedbees are divided into following bees and scout beesThepopu-lation of the colony bees is119873

119904 the number of employed bees is

119873119890 and the number of unemployed bees is119873

119906 which satisfies

the relation119873119904= 2119873119890= 2119873119906 We also define119863 as the dimen-

sion of solution vector that is the number of the unknownparameters ABC algorithm treats each unknown parameteras a food source The detailed procedure of executing theproposed algorithm is described as follows

Step 1 Randomly initialize a set of possible solutions(1199091 119909

119873119904

) and the particular solution 119909119894can be governed

by

119909119895

119894= 119909119895

min + rand (0 1) (119909119895max minus 119909119895

min) (7)

where 119895 isin 1 119863 denotes the 119895th dimension of thesolution vector 119909119895min and 119909

119895

max mean the lower and upperbounds respectively

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

119894according to the following equations and select the

top119873119890best solutions as the number of the employed bees

fit119894=

1

(1 + 119865119894)

(8)

119865119894=

1

119873

119873

sum

119894=1

(1205751

100381710038171003817100381710038171205911119894minus 1205911199011119894

10038171003817100381710038171003817+ 1205752

100381710038171003817100381710038171205912119894minus 1205911199012119894

10038171003817100381710038171003817

+ 1205753

100381710038171003817100381710038171205913119894minus 1205911199013119894

10038171003817100381710038171003817)

(9)

where fit119894is the fitness function 119865

119894is the objective function

119873 is the data length and 120591120585119894(120585 = 1 2 3) is the vector of

the actual torques data from the first three joints Similarly120591119901120585119894

(120585 = 1 2 3) is the vector of the predicted data fromthe identified model 120575

120585(120585 = 1 2 3) is a weight coefficient

between 0 and 1

Journal of Robotics 3

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

V119895119894= 119909119895

119894+ 120582119895

119894(119909119895

119894minus 119909119895

119896) (10)

where 119896 isin 1 119863 119896 = 119894 both 119896 and 119895 are randomlygenerated and 120582

119895

119894is a random parameter in the range from

minus1 to 1 Next we apply the greedy selection equation (11) tochoose the better solution between V119895

119894and 119909

119895

119894into the next

generation

119909119895

119894=

V119895119894 fit (V119895

119894) gt fit (119909119895

119894)

119909119895

119894 fit (V119895

119894) le fit (119909119895

119894)

(11)

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

119901119894=

fit119894

sum119873119890

119894=1fit119894

(12)

Step 5 The following bee searches in the neighborhood ofthe 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 by scout bees according to the following equation

119909119894(119899 + 1)

=

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

119909119894(119899) trial le limit

(13)

Step 7 Output the best solution parameters achieved at thepresent time and go back to Step 3 until termination criterion119879max is met

The detailed procedure of ABC algorithm for parametersidentification can be also depicted in Figure 1

32 Excitation Trajectory When designing an identificationexperiment for the robot it is necessary to design properexcitation trajectories to ensure the accuracy of estimation inpresence of disturbances [18] In this work a finite Fourierseries is adopted as excitation trajectories that is a finite sumof harmonic sine and cosine functions The trajectories forjoint 119894 of a robot are designed as

119902119894(119905) = 119902

119894119900+

119873

sum

119896=1

119886119894119896sin (119896120596

119891119905) +

119873

sum

119896=1

119887119894119896cos (119896120596

119891119905) (14)

where 119902119894119900

is the offset term and 120596119891is the fundamental

pulsation of the Fourier series This Fourier series specifiesa periodic function with period 119879

119891= 2120587120596

119891 Each Fourier

Start

T = 1

Employed bees search new

Following bees search

Yes

Reinitialize the parametersand scout bees search

Output

No

T = T + 1

Initialization Ns Tmax limit

solution and generate ji

Trial gt limit

T gt Tmax

Figure 1 Sketch of the identification algorithm

series contains 2119873 + 1 parameters and 119886119894119896 119887119894119896

are theamplitudes of the sine and cosine functions

The noise immunity and convergence rate of an identi-fication experiment depend directly upon the constraints ofthe excitation trajectories It is important to emphasize thatthe configurations for which measurements are taken mustcorrespond to awell-conditioned reduced observationmatrixsince the constraints represent some limits for inputoutputIn the literature the constraints of the excitation trajectoriescan be described as

min cond (Φ)

119902min le 119902 (120573) le 119902max

1003816100381610038161003816119902 (120573)

1003816100381610038161003816le 119902max

1003816100381610038161003816119902 (120573)

1003816100381610038161003816le 119902max

119908 (119902 (120573)) sub 119882119900

120591min le Φ (119902 (120573) 119902 (120573) 119902 (120573)) 120579 le 120591max

(15)

where 119902min and 119902max are the lower and upper of the jointpositions 119902max and 119902max are the upper of velocities andaccelerations 120573 is optimal trajectory parameters 119882

119900is the

available workspace of robot and 120591max is the maximum jointtorque


33 Preprocessing of Measured Data The measured torquesare obtained through collecting the data of motor currentwhich is described as follows

120591 = 119870119868 (16)

where 119868 is motor current119870 is just coefficientSince there are measurement noises in experiments it is

necessary to preprocess the collection data before identifica-tion In order to remove outliers and attenuate the effect ofinterference signal a five-spot triple smoothing method isadopted to smooth the raw data according to the followingequations

1199101=

1

70

[691199101+ 4 (119910

2+ 1199104) minus 6119910

3minus 1199105]

1199102=

1

35

[2 (1199101+ 1199105) + 27119910

2+ 12119910

3minus 81199104]

119910119894=

1

35

[minus3 (119910119894minus2

+ 119910119894+2) + 12 (119910

119894minus1+ 119910119894+1) + 17119910

119894]

119910119898minus1

=

1

35

[2 (119910119898minus4

+ 119910119898) minus 8119910

119898minus3+ 12119910

119898minus2+ 27119910

119898minus1]

119910119898

=

1

70

[minus119910119898minus4

+ 4 (119910119898minus3

+ 119910119898minus1

) minus 6119910119898minus2

+ 69119910119898]

(17)

where 119894 = 3 119898 minus 2 Y = [1199101 1199102 119910

119898] is the measured

raw data and Y = [1199101 119910

119898]119879 is the data for identification

after preprocessing The more the number of using (17) isthe smoother the curves will be It should be noted thatexcessively using (17) to smooth the raw data can lead to theerror of the parameters identification increasing

In addition the velocities and accelerations of jointscannot be measured directly However these pieces of infor-mation are usually obtained by joint positions and numericaldifferentiation for joint positions can amplify the measure-ment noise and decrease accuracy of the velocities andaccelerations An analytical approach is adopted to overcomethe aforementioned difficulty which was proposed in [19]and used successfully in [20] The average joint positions areapproximated as finite Fourier series through the linear leastsquare technique and the joint velocities and accelerationscan be estimated by the derivatives of the obtained finiteFourier series

4 Experiment Results

41 Parameters Identification An experiment is conductedto test the proposed identification algorithm The ER-16shown in Figure 2 is a 6-DOF industrial robot manipulatorwithout payload And the link frame of the robot is depictedin Figure 3 The geometric parameters of the ER-16 robotare given in Table 1 Only the first joints are consideredhere A fundamental pulsation of 005Hz is selected forthe excitation trajectories resulting in a period of 25 s Asshown in Figure 4 the commanded trajectories are five-term

Figure 2 ER-16 6-DOF robot

z5z6

x4 x5 x6

z4

z3

x3

z1 z2

x1 x2

z0

x0

Figure 3 The link frame of ER-16 robot

0 5 10 15 20 25minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3

Figure 4 Optimized robot excitation trajectories


Table 1 DH parameters of ER-16 robot

Link 119894 120572119894minus1

(rad) 119886119894minus1

(m) 119889119894(m) 120579

119894(rad)

1 120587 0 0 1205791

2 1205872 016 0 1205792minus 1205872

3 0 068 0 1205793

4 1205872 013 minus075 1205794

5 minus1205872 0 0 1205795

6 1205872 0 0 1205796

minus10

12

minus2

minus1

0

1minus1

0

1

x (m)

y (m)

z (m

)

Figure 5 3D visualization of the optimized trajectory

Fourier series involving 11 optimal trajectory parameters foreach joint which are listed in the Appendices and a 025Hzbandwidth The 3D visualization of this optimized trajectoryin the workspace of the robot is shown in Figure 5 Thetotal measured time is 25 s corresponding to 1 period of theexcitation trajectory The data is sampled with 1 kHz

Identification procedures are carried out with ABC algo-rithm in Matlab 2013b programming environment on anIntel Core i7-3770 PC running Windows 7 No commercialtools are used According to [12] the performance of ABCalgorithm is relative to the population size of colony bees Asthe population size increases the algorithm produces betterresults However after a sufficient value for colony size anyincrement in the value does not improve the performanceof ABC algorithm And the control parameter limit is basedon location vector reinitialized frequency As the value oflimit approaches infinity the total number of location vectorsreinitialized goes to zero After many trials in this paperwe set the parameters of ABC algorithm as follows 119873

119904=

30 limit = 15 and 119879max = 50 It should be notedthat when the predetermined iterations exceed 50 the ABCalgorithm converged and the objective value could not beobviously improved And the search scope of the unknownparameters is also listed in the Appendices The objectivevalue of optimization process for the parameters is shownin Figure 6 The result shows that the convergence speed ofour algorithm is fast and the final objective value which iscalculated by (9) is 03182

The robot dynamic model for the first joints contains 21parameters 15 base parameters and 6 friction parameters

0 10 20 30 40 500315

032

0325

033

0335

034

0345

Obj

ectiv

e val

ues

Iteration

Figure 6 Evolutionary curves of identification algorithm

Table 2 Identification dynamic parameters

Parameter Value1198681199111199111

(kgsdotm2) 5044381198681199091199092

(kgsdotm2) 3775031198681199091199102

(kgsdotm2) 1616931198681199091199112

(kgsdotm2) minus2300751198681199101199112

(kgsdotm2) 320471198681199111199112

(kgsdotm2) 1551211989821199031199092(kgsdotm) 59390

11989821199031199102(kgsdotm) 468898

1198681199091199093

(kgsdotm2) 3093191198681199091199103

(kgsdotm2) minus075151198681199101199113

(kgsdotm2) 6329971198681199111199113

(kgsdotm2) minus0406911989831199031199093(kgsdotm) 23608

11989831199031199103(kgsdotm) 14471

1198911198881(Nsdotm) minus00448

119891V1 (Nmsdotsrad) 011441198911198882(Nsdotm) 110195


119891V3 (Nmsdotsrad) 2734751198681199091199113

(kgsdotm2) 23092

The parameters identified by our proposed algorithm arelisted in Table 2 It should be noted that the values ofthe dynamic parameters of the first three joints are muchbigger than those of the other three joints It is reasonableto ignore the effect of the torques caused by the 4 5 and6 joints Figure 7 compares the measured torques for theexcitation trajectories with the predicted torques based onthe identified dynamical model Although the results showthat the predicted error is slightly big during velocity reversalthe predicted torques have the same trend as the measured


0 5 10 15 20 25minus200

minus100

0

100

200

300

Time (s)

Measured torquePredicted torque

τ 1(N

m)

(a) Joint 1

0 5 10 15 20 25minus800

minus600

minus400

minus200

0

200

400

600

Time (s)


2(N

m)

τ(b) Joint 2

0 5 10 15 20 25minus300

minus200

minus100

0

100

200

300

Time (s)


τ 3(N

m)

(c) Joint 3

Figure 7 Comparison of the measured torques and predicted torques

torques It indicates that ABC algorithm has a strong abilityto find the optimal parameters

To verify the precision of the identified model by ABCalgorithm the correlation coefficient between the measuredtorques 120591

119894and predicted torques 120591

119901119894 defined as the nor-

malized cross-covariance function is applied to estimatehow well the identified model can reproduce the measuredtorques and the function is defined as

120588 =

sum119873

119894=1(120591119894minus 120591) (120591

119898119894minus 120591119898)

radicsum119873

119894=1(120591119894minus 120591)2

sum119873

119894=1(120591119898119894minus 120591119898)2

(18)

where 120591 = (1119873)sum119873

119894=1120591119894and 120591

119898= (1119873)sum

119873

119894=1120591119898119894 The

closer the correlation coefficient is to unity the better theidentified model is While the coefficient is close to zero

the identified model is poor As a result the correlationcoefficients of the predicted first joints are 09533 09856 and09801 respectively It indicates that the identified parametershave satisfactory precision

42 Model Validation Since our aim for current stage isto investigate the validity of the model calculated by ourproposed method we focus on the validation test Obviouslythe appropriate validation test is to use the identified modelin application and evaluate its success As shown in Figure 8three-term Fourier series are chosen as the excitation trajec-tories the same as the aforementioned excitation trajectoriesCorresponding optimal trajectory parameters for each jointare listed in the Appendices And the comparison of themeasured and predicted torques is shown in Figure 9 It


0 5 10 15 20minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3

Figure 8 Robot model validation trajectory

0 5 10 15 20minus300

minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20minus600

minus400

minus200

0

200

400

600

Time (s)


τ 2(N

m)

(b) Joint 2

0 5 10 15 20minus300

minus200

minus100

0

100

200

Time (s)


τ 3(N

m)

(c) Joint 3Figure 9 Comparison of the measured torques and predicted torques


Table 3 Search scope of dynamical parameters

Parameter Scope1198681199111199111

(kgsdotm2) [minus60 60]

1198681199091199092


1198681199091199102


1198681199091199112


1198681199101199112


1198681199111199112


11989821199031199092(kgsdotm) [minus10 10]

11989821199031199102(kgsdotm) [minus80 80]

1198681199091199093


1198681199091199103


1198681199101199113


1198681199111199113


11989831199031199093(kgsdotm) [minus10 10]

11989831199031199103(kgsdotm) [minus10 10]

1198911198881(Nsdotm) [minus10 10]

119891V1 (Nmsdotsrad) [minus10 10]

1198911198882(Nsdotm) [minus10 10]


1198911198883(Nsdotm) [minus30 30]


1198681199091199113


indicates that the model we obtain is capable of accu-rately predicting the actuator torque data In addition the

correlation coefficients of the predicted first joints are 0927209534 and 09606 according to (18) The validation test notonly shows very good results but also demonstrates that ourproposed identification method is reliable enough

5 Conclusion

In this paper a systematic procedure for the dynamicalparameters identification of a 6-DOF industrial robot hasbeen presented We design optimal periodic excitation tra-jectories to integrate the identification experiment data col-lection and signal preprocess All the unknown parametersare well identified by ABC algorithm When comparing themeasured torques and the predicting torques we concludethat our proposed method can accurately estimate the robotdynamical parameters Further a model validation has beencarried out to verify the validity of the identified modelThe results of this paper are useful for researchers andmanufactures of industrial robots

Appendices

A Optimal Parameters ofExcitation Trajectories

The optimal trajectory parameters for the five-term Fourierseries are listed as follows

1205731=

[

[

[

[

[

minus03857 minus01911 minus02842 minus02289 minus02455 minus05705 03359 04129 05744 04842 00730

00011 03597 minus00179 minus00117 00443 04355 02551 00386 00411 00057 minus00065

minus06641 minus01638 05347 minus01448 01029 00826 04426 03359 06126 minus06714 minus05792

]

]

]

]

]

(A1)

The optimal trajectory parameters for the three-term Fourierseries are listed as follows

1205732=

[

[

[

[

[

03016 04006 06921 minus04210 minus00423 06066 minus09719

minus05026 minus00051 00492 08074 04578 minus02521 06288

minus01276 minus00624 03805 04006 03016 03531 01524

]

]

]

]

]

(A2)

B Search Scope of Dynamical Parameters

After many trials the probable suboptimal or optimal searchscope of the unknown parameters is listed in Table 3

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 BE2013010-2)

References

[1] J A Persson X Feng D Wappling and J Olvander ldquoAframework for multidisciplinary optimization of a balancing


mechanism for an industrial robotrdquo Journal of Robotics vol2015 Article ID 389769 8 pages 2015

[2] A Paoli and A V Razionale ldquoLarge yacht hull measurementby integrating optical scanning withmechanical tracking-basedmethodologiesrdquoRobotics andComputer-IntegratedManufactur-ing vol 28 no 5 pp 592ndash601 2012

[3] J Wu J Wang and Z You ldquoAn overview of dynamic parameteridentification of robotsrdquo Robotics and Computer-IntegratedManufacturing vol 26 no 5 pp 414ndash419 2010

[4] C G Atkeson C H An and J M Hollerbach ldquoEstimationof inertial parameters of manipulator loads and linksrdquo Interna-tional Journal of Robotics Research vol 5 no 3 pp 101ndash119 1986

[5] M Grotjahn M Daemi and B Heimann ldquoFriction and rigidbody identification of robot dynamicsrdquo International Journal ofSolids and Structures vol 38 no 10 pp 1889ndash1902 2001

[6] M Gautier and P Poignet ldquoExtended Kalman filtering andweighted least squares dynamic identification of robotrdquo ControlEngineering Practice vol 9 no 12 pp 1361ndash1372 2001

[7] H Behzad H T Shandiz A Noori and T Abrishami ldquoRobotidentification using fractional subspacemethodrdquo in Proceedingsof the 2nd International Conference on Control Instrumentationand Automation (ICCIA rsquo11) pp 1193ndash1199 Shiraz Iran Decem-ber 2011

[8] J Xuan J G Liu and S F Zhu ldquoIdentification method ofhydrodynamic parameters of autonomous underwater vehiclebased on genetic algorithmrdquo Journal of Mechanical Engineeringvol 46 no 11 pp 96ndash100 2010

[9] Y Liu G-X Li D Xia and W-F Xu ldquoIdentifying dynamicparameters of a space robot based on improved genetic algo-rithmrdquo Journal of Harbin Institute of Technology vol 42 no 11pp 1734ndash1739 2010

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

[11] L Ding H T Wu and Y Yao ldquoChaotic artificial bee colonyalgorithm for system identification of a small-scale unmannedhelicopterrdquo International Journal of Aerospace Engineering vol2015 Article ID 801874 11 pages 2015

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

[13] J Swevers W Verdonck and J De Schutter ldquoDynamic modelidentification for industrial robotsrdquo IEEE Control Systems vol27 no 5 pp 58ndash71 2007

[14] A Calanca L M Capisani A Ferrara and L MagnanildquoMIMO closed loop identification of an industrial robotrdquo IEEETransactions on Control Systems Technology vol 19 no 5 pp1214ndash1224 2011

[15] N D Vuong and M H Ang Jr ldquoDynamic model identificationfor industrial robotsrdquoActa Polytechnica Hungarica vol 6 no 5pp 51ndash68 2009

[16] M Grotjahn M Daemi and B Heimann ldquoFriction and rigidbody identification of robot dynamicsrdquo International Journal ofSolids and Structures vol 38 no 10ndash13 pp 1889ndash1902 2001

[17] M Gautier andW Khalil ldquoA direct determination of minimuminertial parameters of robotsrdquo in Proceedings of the IEEE Inter-national Conference on Robotics and Automation pp 1682ndash1687 1988

[18] W-XWu S-Q Zhu and X-L Jin ldquoDynamic identification forrobotmanipulators based onmodified fourier seriesrdquo Journal ofZhejiang University Engineering Science vol 47 no 2 pp 231ndash237 2013

[19] C Ganseman J Swevers J De Schutter et al ldquoExperimentalrobot identification using optimized periodic trajectoriesrdquo inProceedings of the International Conference on Noise and Vibra-tion Engineering pp 585ndash595 1994

[20] J Swevers C Ganseman D Bilgin Tukel J de Schutter and Hvan Brussel ldquoOptimal robot excitation and identificationrdquo IEEETransactions on Robotics and Automation vol 13 no 5 pp 730ndash740 1997

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



The outline of this paper is organized as follows Firstlythe linear robot dynamical model is given in Section 2Then Section 3 presents the identification process of thelinear model based on the ABC algorithm where excitationdesign data collection and signal preprocess are describedLater on the experimental platform identified results andmodel validation are presented in Section 4 Finally the mainconclusions are given in Section 5

2 Dynamic Modeling

Since the 119899-DOF industrial robot is represented by a kine-matic chain of rigid bodies the exhaustive description for itsmotion can be found in [13]The dynamicmodel of industrialrobot is derived by the Newton-Euler or Lagrangian method

120591dyn = M (119902) 119902 + C (119902 119902) + G (119902) (1)

where 120591dyn is the 119899-vector of actuator torques as well as thejoint positions 119902 velocities 119902 and accelerations 119902M(119902) is the119899 times 119899 inertia matrix C(119902 119902) denotes the 119899-vector includingCoriolis and centrifugal forces and G(119902) is the 119899-vector ofgravity

According to themodifiedNewton-Euler parameters [14]or the barycentric parameters [15] (1) can be rewritten as alinear form

120591dyn = Φdyn (119902 119902 119902) 120579dyn (2)

whereΦdyn denotes the 119899 times 10119899 observation or identificationmatrix which depends only on the motion data 120579dyn isthe barycentric parameter vector This property considerablysimplifies the parameters identification

Dynamic model of robots also contains the torquescaused by joint frictions and inertias of actuator rotors apartfrom the effects of dynamic parameters in (2) The inertias ofactuator rotors are generally provided by producers and cor-responding torques should be compensated for the dynamicequations In fact joint friction is a complex nonlinearmodelespecially during motion reversal In order to simplify themodel the friction model consisting of only Coulomb andviscous friction [16] is given by

120591fric = 119891119888sign ( 119902) + 119891V 119902 (3)

where 120591fric is the friction torques and 119891119888 119891V respectively

mean the Coulomb and viscous friction parametersThe integrated dynamicmodel of robots can be written as

120591119904= Φ119904(119902 119902 119902) 120579

119904 (4)

where 120591119904is actuator torques including 120591dyn and 120591fricΦ119904 is the

119899 times 12119899 observation matrix and 120579119904is 12119899-vector of unknown

dynamic parameters In addition the dynamic parameters oflink 119894 are governed by the form

120579119894

119904= [119868119909119909119894 119868119909119910119894 119868119909119911119894 119868119910119910119894 119868119910119911119894 119868119911119911119894 119898119894119903119909119894 119898119894119903119910119894 119898119894119903119911119894 119898119894 119891119888119894

119891V119894]119879

(5)

where 119868120577120577119894

(120577 = 119909 119910 119911) is the inertial tensor of link 119894 Similarly119898119894119903120577119894denotes the first-order mass moment and119898

119894is the mass

of link 119894In general the observation matrix Φ

119904in (4) is not

a full rank that is not all dynamic parameters have aninfluence on the dynamic model In order to obtain a setof minimum parameters a case-by-case analysis method isadopted [17] Consequently the dynamic model based on thebasic dynamic parameters can be rewritten as

120591 = Φ (119902 119902 119902) 120579 (6)

whereΦ is the 119899 times (119901 + 2119899) observation matrix 120579 is (119901 + 2119899)-vector of dynamic parameters including the basic parametersand the friction parameters 119901 denotes the number of theminimum dynamical parameters 2119899 denotes the number ofthe friction parameters

3 Parameters Identification Procedures

31 Basic Principles of Identification Algorithm In order tointroduce the search mechanism of ABC algorithm weshould define three essential components employed beesunemployed bees and food source [11] And the unemployedbees are divided into following bees and scout beesThepopu-lation of the colony bees is119873

119904 the number of employed bees is

119873119890 and the number of unemployed bees is119873

119906 which satisfies

the relation119873119904= 2119873119890= 2119873119906 We also define119863 as the dimen-

sion of solution vector that is the number of the unknownparameters ABC algorithm treats each unknown parameteras a food source The detailed procedure of executing theproposed algorithm is described as follows

Step 1 Randomly initialize a set of possible solutions(1199091 119909

119873119904

) and the particular solution 119909119894can be governed

by

119909119895

119894= 119909119895

min + rand (0 1) (119909119895max minus 119909119895

min) (7)

where 119895 isin 1 119863 denotes the 119895th dimension of thesolution vector 119909119895min and 119909

119895

max mean the lower and upperbounds respectively

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

119894according to the following equations and select the

top119873119890best solutions as the number of the employed bees

fit119894=

1

(1 + 119865119894)

(8)

119865119894=

1

119873

119873

sum

119894=1

(1205751

100381710038171003817100381710038171205911119894minus 1205911199011119894

10038171003817100381710038171003817+ 1205752

100381710038171003817100381710038171205912119894minus 1205911199012119894

10038171003817100381710038171003817

+ 1205753

100381710038171003817100381710038171205913119894minus 1205911199013119894

10038171003817100381710038171003817)

(9)

where fit119894is the fitness function 119865

119894is the objective function

119873 is the data length and 120591120585119894(120585 = 1 2 3) is the vector of

the actual torques data from the first three joints Similarly120591119901120585119894

(120585 = 1 2 3) is the vector of the predicted data fromthe identified model 120575

120585(120585 = 1 2 3) is a weight coefficient

between 0 and 1



V119895119894= 119909119895

119894+ 120582119895

119894(119909119895

119894minus 119909119895

119896) (10)


119895



119894and 119909

119895

119894into the next

generation

119909119895

119894=

V119895119894 fit (V119895

119894) gt fit (119909119895

119894)

119909119895

119894 fit (V119895

119894) le fit (119909119895

119894)

(11)


119901119894=

fit119894

sum119873119890

119894=1fit119894

(12)



119909119894(119899 + 1)

=


119909119894(119899) trial le limit

(13)




119902119894(119905) = 119902

119894119900+

119873

sum

119896=1

119886119894119896sin (119896120596

119891119905) +

119873

sum

119896=1

119887119894119896cos (119896120596

119891119905) (14)

where 119902119894119900



119891= 2120587120596

119891 Each Fourier

Start

T = 1



Yes


Output

No

T = T + 1



Trial gt limit

T gt Tmax





min cond (Φ)

119902min le 119902 (120573) le 119902max

1003816100381610038161003816119902 (120573)

1003816100381610038161003816le 119902max

1003816100381610038161003816119902 (120573)

1003816100381610038161003816le 119902max

119908 (119902 (120573)) sub 119882119900

120591min le Φ (119902 (120573) 119902 (120573) 119902 (120573)) 120579 le 120591max

(15)


119900is the




120591 = 119870119868 (16)



1199101=

1

70

[691199101+ 4 (119910

2+ 1199104) minus 6119910

3minus 1199105]

1199102=

1

35

[2 (1199101+ 1199105) + 27119910

2+ 12119910

3minus 81199104]

119910119894=

1

35

[minus3 (119910119894minus2

+ 119910119894+2) + 12 (119910

119894minus1+ 119910119894+1) + 17119910

119894]

119910119898minus1

=

1

35

[2 (119910119898minus4

+ 119910119898) minus 8119910

119898minus3+ 12119910

119898minus2+ 27119910

119898minus1]

119910119898

=

1

70


+ 4 (119910119898minus3

+ 119910119898minus1

) minus 6119910119898minus2

+ 69119910119898]

(17)

where 119894 = 3 119898 minus 2 Y = [1199101 1199102 119910


raw data and Y = [1199101 119910







z5z6

x4 x5 x6

z4

z3

x3

z1 z2

x1 x2

z0

x0


0 5 10 15 20 25minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3




Link 119894 120572119894minus1

(rad) 119886119894minus1

(m) 119889119894(m) 120579

119894(rad)

1 120587 0 0 1205791

2 1205872 016 0 1205792minus 1205872

3 0 068 0 1205793

4 1205872 013 minus075 1205794

5 minus1205872 0 0 1205795

6 1205872 0 0 1205796

minus10

12

minus2

minus1

0

1minus1

0

1

x (m)

y (m)

z (m

)




119904=



0 10 20 30 40 500315

032

0325

033

0335

034

0345

Obj

ectiv

e val

ues

Iteration




(kgsdotm2) 5044381198681199091199092

(kgsdotm2) 3775031198681199091199102

(kgsdotm2) 1616931198681199091199112

(kgsdotm2) minus2300751198681199101199112

(kgsdotm2) 320471198681199111199112

(kgsdotm2) 1551211989821199031199092(kgsdotm) 59390

11989821199031199102(kgsdotm) 468898

1198681199091199093

(kgsdotm2) 3093191198681199091199103

(kgsdotm2) minus075151198681199101199113

(kgsdotm2) 6329971198681199111199113


11989831199031199103(kgsdotm) 14471

1198911198881(Nsdotm) minus00448



119891V3 (Nmsdotsrad) 2734751198681199091199113

(kgsdotm2) 23092



0 5 10 15 20 25minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20 25minus800

minus600

minus400

minus200

0

200

400

600

Time (s)


2(N

m)

τ(b) Joint 2

0 5 10 15 20 25minus300

minus200

minus100

0

100

200

300

Time (s)


τ 3(N

m)

(c) Joint 3







120588 =

sum119873

119894=1(120591119894minus 120591) (120591

119898119894minus 120591119898)

radicsum119873

119894=1(120591119894minus 120591)2

sum119873

119894=1(120591119898119894minus 120591119898)2

(18)

where 120591 = (1119873)sum119873

119894=1120591119894and 120591

119898= (1119873)sum

119873

119894=1120591119898119894 The





0 5 10 15 20minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3


0 5 10 15 20minus300

minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20minus600

minus400

minus200

0

200

400

600

Time (s)


τ 2(N

m)

(b) Joint 2

0 5 10 15 20minus300

minus200

minus100

0

100

200

Time (s)


τ 3(N

m)






1198681199091199092


1198681199091199102


1198681199091199112


1198681199101199112


1198681199111199112


11989821199031199092(kgsdotm) [minus10 10]

11989821199031199102(kgsdotm) [minus80 80]

1198681199091199093


1198681199091199103


1198681199101199113


1198681199111199113


11989831199031199093(kgsdotm) [minus10 10]

11989831199031199103(kgsdotm) [minus10 10]

1198911198881(Nsdotm) [minus10 10]


1198911198882(Nsdotm) [minus10 10]


1198911198883(Nsdotm) [minus30 30]


1198681199091199113




5 Conclusion


Appendices



1205731=

[

[

[

[

[




]

]

]

]

]

(A1)


1205732=

[

[

[

[

[



minus01276 minus00624 03805 04006 03016 03531 01524

]

]

]

]

]

(A2)





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











V119895119894= 119909119895

119894+ 120582119895

119894(119909119895

119894minus 119909119895

119896) (10)


119895



119894and 119909

119895

119894into the next

generation

119909119895

119894=

V119895119894 fit (V119895

119894) gt fit (119909119895

119894)

119909119895

119894 fit (V119895

119894) le fit (119909119895

119894)

(11)


119901119894=

fit119894

sum119873119890

119894=1fit119894

(12)



119909119894(119899 + 1)

=


119909119894(119899) trial le limit

(13)




119902119894(119905) = 119902

119894119900+

119873

sum

119896=1

119886119894119896sin (119896120596

119891119905) +

119873

sum

119896=1

119887119894119896cos (119896120596

119891119905) (14)

where 119902119894119900



119891= 2120587120596

119891 Each Fourier

Start

T = 1



Yes


Output

No

T = T + 1



Trial gt limit

T gt Tmax





min cond (Φ)

119902min le 119902 (120573) le 119902max

1003816100381610038161003816119902 (120573)

1003816100381610038161003816le 119902max

1003816100381610038161003816119902 (120573)

1003816100381610038161003816le 119902max

119908 (119902 (120573)) sub 119882119900

120591min le Φ (119902 (120573) 119902 (120573) 119902 (120573)) 120579 le 120591max

(15)


119900is the




120591 = 119870119868 (16)



1199101=

1

70

[691199101+ 4 (119910

2+ 1199104) minus 6119910

3minus 1199105]

1199102=

1

35

[2 (1199101+ 1199105) + 27119910

2+ 12119910

3minus 81199104]

119910119894=

1

35

[minus3 (119910119894minus2

+ 119910119894+2) + 12 (119910

119894minus1+ 119910119894+1) + 17119910

119894]

119910119898minus1

=

1

35

[2 (119910119898minus4

+ 119910119898) minus 8119910

119898minus3+ 12119910

119898minus2+ 27119910

119898minus1]

119910119898

=

1

70


+ 4 (119910119898minus3

+ 119910119898minus1

) minus 6119910119898minus2

+ 69119910119898]

(17)

where 119894 = 3 119898 minus 2 Y = [1199101 1199102 119910


raw data and Y = [1199101 119910







z5z6

x4 x5 x6

z4

z3

x3

z1 z2

x1 x2

z0

x0


0 5 10 15 20 25minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3




Link 119894 120572119894minus1

(rad) 119886119894minus1

(m) 119889119894(m) 120579

119894(rad)

1 120587 0 0 1205791

2 1205872 016 0 1205792minus 1205872

3 0 068 0 1205793

4 1205872 013 minus075 1205794

5 minus1205872 0 0 1205795

6 1205872 0 0 1205796

minus10

12

minus2

minus1

0

1minus1

0

1

x (m)

y (m)

z (m

)




119904=



0 10 20 30 40 500315

032

0325

033

0335

034

0345

Obj

ectiv

e val

ues

Iteration




(kgsdotm2) 5044381198681199091199092

(kgsdotm2) 3775031198681199091199102

(kgsdotm2) 1616931198681199091199112

(kgsdotm2) minus2300751198681199101199112

(kgsdotm2) 320471198681199111199112

(kgsdotm2) 1551211989821199031199092(kgsdotm) 59390

11989821199031199102(kgsdotm) 468898

1198681199091199093

(kgsdotm2) 3093191198681199091199103

(kgsdotm2) minus075151198681199101199113

(kgsdotm2) 6329971198681199111199113


11989831199031199103(kgsdotm) 14471

1198911198881(Nsdotm) minus00448



119891V3 (Nmsdotsrad) 2734751198681199091199113

(kgsdotm2) 23092



0 5 10 15 20 25minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20 25minus800

minus600

minus400

minus200

0

200

400

600

Time (s)


2(N

m)

τ(b) Joint 2

0 5 10 15 20 25minus300

minus200

minus100

0

100

200

300

Time (s)


τ 3(N

m)

(c) Joint 3







120588 =

sum119873

119894=1(120591119894minus 120591) (120591

119898119894minus 120591119898)

radicsum119873

119894=1(120591119894minus 120591)2

sum119873

119894=1(120591119898119894minus 120591119898)2

(18)

where 120591 = (1119873)sum119873

119894=1120591119894and 120591

119898= (1119873)sum

119873

119894=1120591119898119894 The





0 5 10 15 20minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3


0 5 10 15 20minus300

minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20minus600

minus400

minus200

0

200

400

600

Time (s)


τ 2(N

m)

(b) Joint 2

0 5 10 15 20minus300

minus200

minus100

0

100

200

Time (s)


τ 3(N

m)






1198681199091199092


1198681199091199102


1198681199091199112


1198681199101199112


1198681199111199112


11989821199031199092(kgsdotm) [minus10 10]

11989821199031199102(kgsdotm) [minus80 80]

1198681199091199093


1198681199091199103


1198681199101199113


1198681199111199113


11989831199031199093(kgsdotm) [minus10 10]

11989831199031199103(kgsdotm) [minus10 10]

1198911198881(Nsdotm) [minus10 10]


1198911198882(Nsdotm) [minus10 10]


1198911198883(Nsdotm) [minus30 30]


1198681199091199113




5 Conclusion


Appendices



1205731=

[

[

[

[

[




]

]

]

]

]

(A1)


1205732=

[

[

[

[

[



minus01276 minus00624 03805 04006 03016 03531 01524

]

]

]

]

]

(A2)





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120591 = 119870119868 (16)



1199101=

1

70

[691199101+ 4 (119910

2+ 1199104) minus 6119910

3minus 1199105]

1199102=

1

35

[2 (1199101+ 1199105) + 27119910

2+ 12119910

3minus 81199104]

119910119894=

1

35

[minus3 (119910119894minus2

+ 119910119894+2) + 12 (119910

119894minus1+ 119910119894+1) + 17119910

119894]

119910119898minus1

=

1

35

[2 (119910119898minus4

+ 119910119898) minus 8119910

119898minus3+ 12119910

119898minus2+ 27119910

119898minus1]

119910119898

=

1

70


+ 4 (119910119898minus3

+ 119910119898minus1

) minus 6119910119898minus2

+ 69119910119898]

(17)

where 119894 = 3 119898 minus 2 Y = [1199101 1199102 119910


raw data and Y = [1199101 119910







z5z6

x4 x5 x6

z4

z3

x3

z1 z2

x1 x2

z0

x0


0 5 10 15 20 25minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3




Link 119894 120572119894minus1

(rad) 119886119894minus1

(m) 119889119894(m) 120579

119894(rad)

1 120587 0 0 1205791

2 1205872 016 0 1205792minus 1205872

3 0 068 0 1205793

4 1205872 013 minus075 1205794

5 minus1205872 0 0 1205795

6 1205872 0 0 1205796

minus10

12

minus2

minus1

0

1minus1

0

1

x (m)

y (m)

z (m

)




119904=



0 10 20 30 40 500315

032

0325

033

0335

034

0345

Obj

ectiv

e val

ues

Iteration




(kgsdotm2) 5044381198681199091199092

(kgsdotm2) 3775031198681199091199102

(kgsdotm2) 1616931198681199091199112

(kgsdotm2) minus2300751198681199101199112

(kgsdotm2) 320471198681199111199112

(kgsdotm2) 1551211989821199031199092(kgsdotm) 59390

11989821199031199102(kgsdotm) 468898

1198681199091199093

(kgsdotm2) 3093191198681199091199103

(kgsdotm2) minus075151198681199101199113

(kgsdotm2) 6329971198681199111199113


11989831199031199103(kgsdotm) 14471

1198911198881(Nsdotm) minus00448



119891V3 (Nmsdotsrad) 2734751198681199091199113

(kgsdotm2) 23092



0 5 10 15 20 25minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20 25minus800

minus600

minus400

minus200

0

200

400

600

Time (s)


2(N

m)

τ(b) Joint 2

0 5 10 15 20 25minus300

minus200

minus100

0

100

200

300

Time (s)


τ 3(N

m)

(c) Joint 3







120588 =

sum119873

119894=1(120591119894minus 120591) (120591

119898119894minus 120591119898)

radicsum119873

119894=1(120591119894minus 120591)2

sum119873

119894=1(120591119898119894minus 120591119898)2

(18)

where 120591 = (1119873)sum119873

119894=1120591119894and 120591

119898= (1119873)sum

119873

119894=1120591119898119894 The





0 5 10 15 20minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3


0 5 10 15 20minus300

minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20minus600

minus400

minus200

0

200

400

600

Time (s)


τ 2(N

m)

(b) Joint 2

0 5 10 15 20minus300

minus200

minus100

0

100

200

Time (s)


τ 3(N

m)






1198681199091199092


1198681199091199102


1198681199091199112


1198681199101199112


1198681199111199112


11989821199031199092(kgsdotm) [minus10 10]

11989821199031199102(kgsdotm) [minus80 80]

1198681199091199093


1198681199091199103


1198681199101199113


1198681199111199113


11989831199031199093(kgsdotm) [minus10 10]

11989831199031199103(kgsdotm) [minus10 10]

1198911198881(Nsdotm) [minus10 10]


1198911198882(Nsdotm) [minus10 10]


1198911198883(Nsdotm) [minus30 30]


1198681199091199113




5 Conclusion


Appendices



1205731=

[

[

[

[

[




]

]

]

]

]

(A1)


1205732=

[

[

[

[

[



minus01276 minus00624 03805 04006 03016 03531 01524

]

]

]

]

]

(A2)





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Link 119894 120572119894minus1

(rad) 119886119894minus1

(m) 119889119894(m) 120579

119894(rad)

1 120587 0 0 1205791

2 1205872 016 0 1205792minus 1205872

3 0 068 0 1205793

4 1205872 013 minus075 1205794

5 minus1205872 0 0 1205795

6 1205872 0 0 1205796

minus10

12

minus2

minus1

0

1minus1

0

1

x (m)

y (m)

z (m

)




119904=



0 10 20 30 40 500315

032

0325

033

0335

034

0345

Obj

ectiv

e val

ues

Iteration




(kgsdotm2) 5044381198681199091199092

(kgsdotm2) 3775031198681199091199102

(kgsdotm2) 1616931198681199091199112

(kgsdotm2) minus2300751198681199101199112

(kgsdotm2) 320471198681199111199112

(kgsdotm2) 1551211989821199031199092(kgsdotm) 59390

11989821199031199102(kgsdotm) 468898

1198681199091199093

(kgsdotm2) 3093191198681199091199103

(kgsdotm2) minus075151198681199101199113

(kgsdotm2) 6329971198681199111199113


11989831199031199103(kgsdotm) 14471

1198911198881(Nsdotm) minus00448



119891V3 (Nmsdotsrad) 2734751198681199091199113

(kgsdotm2) 23092



0 5 10 15 20 25minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20 25minus800

minus600

minus400

minus200

0

200

400

600

Time (s)


2(N

m)

τ(b) Joint 2

0 5 10 15 20 25minus300

minus200

minus100

0

100

200

300

Time (s)


τ 3(N

m)

(c) Joint 3







120588 =

sum119873

119894=1(120591119894minus 120591) (120591

119898119894minus 120591119898)

radicsum119873

119894=1(120591119894minus 120591)2

sum119873

119894=1(120591119898119894minus 120591119898)2

(18)

where 120591 = (1119873)sum119873

119894=1120591119894and 120591

119898= (1119873)sum

119873

119894=1120591119898119894 The





0 5 10 15 20minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3


0 5 10 15 20minus300

minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20minus600

minus400

minus200

0

200

400

600

Time (s)


τ 2(N

m)

(b) Joint 2

0 5 10 15 20minus300

minus200

minus100

0

100

200

Time (s)


τ 3(N

m)






1198681199091199092


1198681199091199102


1198681199091199112


1198681199101199112


1198681199111199112


11989821199031199092(kgsdotm) [minus10 10]

11989821199031199102(kgsdotm) [minus80 80]

1198681199091199093


1198681199091199103


1198681199101199113


1198681199111199113


11989831199031199093(kgsdotm) [minus10 10]

11989831199031199103(kgsdotm) [minus10 10]

1198911198881(Nsdotm) [minus10 10]


1198911198882(Nsdotm) [minus10 10]


1198911198883(Nsdotm) [minus30 30]


1198681199091199113




5 Conclusion


Appendices



1205731=

[

[

[

[

[




]

]

]

]

]

(A1)


1205732=

[

[

[

[

[



minus01276 minus00624 03805 04006 03016 03531 01524

]

]

]

]

]

(A2)





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20 25minus800

minus600

minus400

minus200

0

200

400

600

Time (s)


2(N

m)

τ(b) Joint 2

0 5 10 15 20 25minus300

minus200

minus100

0

100

200

300

Time (s)


τ 3(N

m)

(c) Joint 3







120588 =

sum119873

119894=1(120591119894minus 120591) (120591

119898119894minus 120591119898)

radicsum119873

119894=1(120591119894minus 120591)2

sum119873

119894=1(120591119898119894minus 120591119898)2

(18)

where 120591 = (1119873)sum119873

119894=1120591119894and 120591

119898= (1119873)sum

119873

119894=1120591119898119894 The





0 5 10 15 20minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3


0 5 10 15 20minus300

minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20minus600

minus400

minus200

0

200

400

600

Time (s)


τ 2(N

m)

(b) Joint 2

0 5 10 15 20minus300

minus200

minus100

0

100

200

Time (s)


τ 3(N

m)






1198681199091199092


1198681199091199102


1198681199091199112


1198681199101199112


1198681199111199112


11989821199031199092(kgsdotm) [minus10 10]

11989821199031199102(kgsdotm) [minus80 80]

1198681199091199093


1198681199091199103


1198681199101199113


1198681199111199113


11989831199031199093(kgsdotm) [minus10 10]

11989831199031199103(kgsdotm) [minus10 10]

1198911198881(Nsdotm) [minus10 10]


1198911198882(Nsdotm) [minus10 10]


1198911198883(Nsdotm) [minus30 30]


1198681199091199113




5 Conclusion


Appendices



1205731=

[

[

[

[

[




]

]

]

]

]

(A1)


1205732=

[

[

[

[

[



minus01276 minus00624 03805 04006 03016 03531 01524

]

]

]

]

]

(A2)





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20minus3

minus2

minus1

0

1

2

Time (s)

Join

t pos

ition

(rad

)

Joint 1Joint 2

Joint 3


0 5 10 15 20minus300

minus200

minus100

0

100

200

300

Time (s)


τ 1(N

m)

(a) Joint 1

0 5 10 15 20minus600

minus400

minus200

0

200

400

600

Time (s)


τ 2(N

m)

(b) Joint 2

0 5 10 15 20minus300

minus200

minus100

0

100

200

Time (s)


τ 3(N

m)






1198681199091199092


1198681199091199102


1198681199091199112


1198681199101199112


1198681199111199112


11989821199031199092(kgsdotm) [minus10 10]

11989821199031199102(kgsdotm) [minus80 80]

1198681199091199093


1198681199091199103


1198681199101199113


1198681199111199113


11989831199031199093(kgsdotm) [minus10 10]

11989831199031199103(kgsdotm) [minus10 10]

1198911198881(Nsdotm) [minus10 10]


1198911198882(Nsdotm) [minus10 10]


1198911198883(Nsdotm) [minus30 30]


1198681199091199113




5 Conclusion


Appendices



1205731=

[

[

[

[

[




]

]

]

]

]

(A1)


1205732=

[

[

[

[

[



minus01276 minus00624 03805 04006 03016 03531 01524

]

]

]

]

]

(A2)





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1198681199091199092


1198681199091199102


1198681199091199112


1198681199101199112


1198681199111199112


11989821199031199092(kgsdotm) [minus10 10]

11989821199031199102(kgsdotm) [minus80 80]

1198681199091199093


1198681199091199103


1198681199101199113


1198681199111199113


11989831199031199093(kgsdotm) [minus10 10]

11989831199031199103(kgsdotm) [minus10 10]

1198911198881(Nsdotm) [minus10 10]


1198911198882(Nsdotm) [minus10 10]


1198911198883(Nsdotm) [minus30 30]


1198681199091199113




5 Conclusion


Appendices



1205731=

[

[

[

[

[




]

]

]

]

]

(A1)


1205732=

[

[

[

[

[



minus01276 minus00624 03805 04006 03016 03531 01524

]

]

]

]

]

(A2)





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article dynamic model identification for 6...

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