Research ArticleKinematic Calibration of Industrial Robots Based on DistanceInformation Using a Hybrid Identification Method

Guanbin Gao Yuan Li Fei Liu and Shichang Han

Faculty of Mechanical and Electrical Engineering Kunming University of Science and Technology Kunming 650500 China

Correspondence should be addressed to Guanbin Gao gbgao163com

Received 16 August 2020 Revised 1 September 2020 Accepted 17 March 2021 Published 26 March 2021

Academic Editor Jing Na

Copyright copy 2021 Guanbin Gao et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To improve the positioning accuracy of industrial robots and avoid using the coordinates of the end effector a novel kinematiccalibration method based on the distance information is proposed (e kinematic model of an industrial robot is established (erelationship between the moving distance of the end effector and the kinematic parameters is analyzed Based on the results of theanalysis and the kinematic model of the robot the error model with displacements as the reference is built which is linearized forthe convenience of the following identification (e singular value decomposition (SVD) is used to eliminate the redundantparameters of the error model To solve the problem that traditional optimization algorithms are easily affected by data noise inhigh dimension identification a novel extended Kalman filter (EKF) and regularized particle filter (RPF) hybrid identificationmethod is presented EKF is used in the preidentification of the linearized error model With the preidentification results as theinitial parameters RPF is used to identify the kinematic parameters of the linearized error model Simulations are carried out tovalidate the effectiveness of the proposed method which shows that the method can identify the error of the parameters and aftercompensation the accuracy of the robot is improved

1 Introduction

With high generality and industrial flexibility robots havebeen widely used in modern manufacturing which play animportant role in the fields of automobile manufacturinglogistics machinery manufacture and so forth Tradition-ally industrial robots are used in the way of teachingprogramming for example in spot welding handling andother similar repetitive works In recent years the positionaccuracy of industrial robots is required to be higher andhigher in the fields of mechanical manufacturing carvingmeasuring testing and so forth

About 90 position errors are caused by the inaccuracyof the kinematic parameters in the controller [1] (e mainway to improve the position accuracy of industrial robots iskinematic calibration by which the structural parameterserrors due to the tolerance in machining and assembly ofrobots can be identified and compensated [2] (ere are twolevels for kinematic calibration (e first level is to identifyand compensate the errors between the transducer readingof joints and the actual joint angle (e second level is to

identify and compensate all the kinematic parameters ofrobots Generally the kinematic calibration of robots can beclassified into four steps [3] modelling measurement pa-rameter identification and compensation (e kinematicparameter identification of industrial robots is a highnonlinear problem To obtain precision identified parame-ters the kinematic model is required with the features ofcompleteness continuity and minimality [4] (e standardDenavitndashHartenberg (DH) model [5] is the most commonlyused model in kinematic modelling of serial open-chainrobots However there will be singularity when two adjacentjoints are completely or almost parallel Hayati [6] proposeda modified DH (MDH) method which introduces a rotationparameter to solve the singularity problem S-model was alsoproposed [7] to solve the singularity problem by Stone AComplete and Parametrically Continuous (CPC) kinematicmodel was proposed by [8] to provide a kinematic modellingmethod with completeness and continuity

In the step of measurement a group of poses and po-sitions of the robot are acquired for the following parameteridentification usually by laser trackers [9ndash12] or other

HindawiComplexityVolume 2021 Article ID 8874226 10 pageshttpsdoiorg10115520218874226

coordinate measuring instruments [13ndash15] Nadia Schillre[16] reported a calibration method with a laser tracker toacquire data (e maximum position error of the robot isreduced from 04883mm to 01801mm Albert Nubiola [17]presented a calibration case for ABB IRB 1600 using lasertracker in which the mean position error of the robot wasreduced from 0968mm to 0696mm With the advantagesof high precision dynamic tracking and noncontact mea-suring laser trackers become the main measurement in-strument in the calibration field of robots But they are soexpensive that ordinary users cannot afford them Exceptlaser trackers coordinate measuringmachines (CMMs) [18]measuring arms [19] ballbars [20] and so forth are also usedin calibrating robots Actually the coordinates of the endeffector of the robots are acquired by instruments in theprevious researches [21ndash23] which are needed in the fol-lowing identification (ese coordinates are acquired in themeasurement coordinate system of the instruments whichhave to be transformed into the base coordinate system ofthe robot and this transformation will bring error andincrease the difficulty of identification Unlike coordinatesdistance is a relative value which is always the same indifferent coordinate systems without any transformation

(e third step identification actually is an optimizationproblem in which the least square (LS) algorithm [24ndash27] isthe most commonly used way But LS is easily affected by thenoise of data which can lead to failure in identificationExtended Kalman filter (EKF) algorithm was more suitablefor the identification of the robot the working principle ofwhich is similar to that of Kalman filter [28] (e expectedEKF deals with nonlinear systems that retain the Taylorexpansion first-order terms of the nonlinear function andignore the higher-order terms But EKF cannot deal withhigh nonlinear system and only suit Gaussian noise system[29] Actually the kinematic model of the robot is a highlynonlinear system

Referring to ISO-9258 [30 31] the distance accuracy ofindustrial robot can be defined as the deviation between themeasured distance and the command distance between thecontinuous moving points in workspace To avoid usingcoordinate measuring instruments and the transformationbetween the measurement coordinate system and the basecoordinate system of the robot we propose a kinematic

calibration method based on distance error modelling andthe definition of distance accuracy to improve the positionaccuracy of the robot A regularized Particle Filter (RPF)[32] is used to improve the ability of identification for thekinematic model of the robot in the light of the excellentperformance of sample filtering method in solving highlynonlinear problem in identification Since RPF is sensitive toits initial value EKF is used in preidentification of the ki-nematic parameters of the robot (e results of the pre-identification are used as the initial value of RPF

(e paper is organized as follows In Section 2 we in-troduce the kinematic modelling and distance error mod-elling In Section 3 the identification of the robot isdiscussed and the redundant parameters in the identificationmodel are analyzed by numerical method based on distanceerror Comparative simulations are conducted in Section 4and some conclusions are given in Section 5

2 Kinematic Modelling and DistanceError Modelling

21 KinematicModelling (e controller of industrial robotsis essentially a semi-closed-loop system which sends controlinstructions to the joint servo motors and acquires thepositions of them (e positions of the joints are calculatedfrom the positions of the servo motors which are used todetermine the poses of the industrial robots by kinematicmodel

An MDH method [33] is used to build the kinematicmodel of the industrial robot in which the relationshipbetween two adjacent rotation joints is described by a ho-mogeneous transformation matrix (e kinematic modelconsists of the controller which consists of four groups ofparameters link length a link offset d link twist angle α andjoint angle θ [33] First the coordinate systems of a 6-degree-of-freedom (DOF) industrial robot named ER20-C10 areestablished according to the MDH method as shown inFigure 1 (en the nominal values of kinematic parametersof the robot are obtained as shown in Table 1

According to MDH method the transformation matrixTiiminus1 of two arbitrary adjacent rotation links iminus1 and i isas follows

Timinus1i Rot ximinus1 αiminus1( 1113857Trans aiminus1 0 0( 1113857Rot zi θi( 1113857Trans 0 0 di( 1113857

cos θi minussin θi 0 aiminus1

sin θi cos αiminus1 cos θi cos αiminus1 minussin αiminus1 minusdi sin αiminus1

sin θi sin αiminus1 cos θi sin αiminus1 cos αiminus1 di cos αiminus1

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

For the generic industrial robot with Nt-DOF thetransformation matrixes from the base frame 0 to the endeffector can be written as

T0E 1113944

Nt

i1Timinus1i

⎛⎝ ⎞⎠ middot TKE (2)

2 Complexity

where TKE 0 0 lz 11113858 1113859T and lz is the end effector (EE)

offset along the axis of jointNt (e first three rows of the lastcolumn of T0E denote the position of EE that is P(middot)

22 Distance Error Modelling According to ISO-9258[30 31] between the continuous moving points during themovement of robot in workspace the distance accuracy ofindustrial robot refers to the deviation between the mea-sured distance and the command distance illustrated asFigure 2 (e distance of the two adjacent points (i and i+ 1)in workspace can be expanded as

Δl(i i + 1) lt(i i + 1) minus lm(i i + 1)

Pt(i + 1) minus Pt(i)1113858 1113859 minus Pm(i + 1) minus Pm(i)1113858 1113859

Pt(i + 1) minus Pm(i + 1)1113858 1113859 minus Pt(i) minus Pm(i)1113858 1113859

(3)

(e position error of the ith point of the acquisition datais defined as

Pe(i) Pt(i) minus Pm(i) JX (4)

where J is the Jacobeanmatrix andX is the error of kinematicparameters According to equations (3) and (4) one canobtain the distance error formula shown as follows

lm(i i + 1) middot Δl(i i + 1) Pm(i + 1) minus Pm(i)1113858 1113859

middot Pe(i + 1) minus Pe(i)(5)

Equation (5) indicates that the distance error can beexpressed by partial pose information Furthermore equa-tion (5) can be rewritten as [34]

Δl(i i + 1) Pm(i + 1) minus Pm(i)

lm(i i + 1)1113890 1113891 middot Pe(i + 1) minus Pe(i)

(6)

According to equation (6) the measured distance can becalculated by two corresponding adjacent points [34](erefore at every acquisition position one has to measuretwice to obtain the distance used which is time-consumingand labor-consuming In this paper we proposed a modifieddistance model which is more convenient and less time-consuming than the general one

(e original position of the device is set as P0(xp0yp0zp0)in the frame of the measurement device which is fixed P0 isconsidered to be error-free which is also set as one of the twoadjacent points in equation (6) Hence equation (6) can berewritten as

Δl(i) Pm(i) minus P0

lm(i)1113890 1113891 middot Pe(i) (7)

where lm(i) is the ith distance from EE to P0 which can bedirectly measured by the measurement system P0 is knownas the fixed point of the measurement system (e originalerror of the measurement system can be neglectable for thedistance is a relative value With P0 being the fixed referencepoint to get the distance one only needs to measure onetime at a position (us the measuring process can besimplified with proposed strategy Equation (7) describes therelationship between the distance error model and the

Y4Y6

Y5

X6 X5

X4

Z6 Z5Z2

Z2

Y3

Y2

Y1

Y0

O2

O1

Z1

Z0

O0

X1

X0

O3

X3

X2

Figure 1 (e kinematic model of ER20-C10 robot

Table 1 Nominal values of robotrsquos MDH model parameters

No of joints αiminus1(deg) aiminus1(mm) θi(deg) di (mm)1 0 0 θ1 5042 90 166605 minus90 + θ2 03 0 minus78227 θ3 04 minus90 minus138826 θ4 761355 90 0 θ5 06 minus90 0 θ6 125

O

Pt(i) Pt(i + 1)Pe(i + 1)

Pm(i + 1)Pe(i)

Pm(i)

lt

lm

Figure 2 (e definition of distance accuracy

Complexity 3

position error Assuming that the deviation of the param-eters is small (7) can be linearized as the first-order term ofits Taylor expansion around the nominal values of the ki-nematic parameters as shown in the following equation

Δl(i) 1113957JX (8)

where 1113957J isin R3ntimesN n is the number of calibration distancesand N is the number of parameters (e Jacobian matrix 1113957J indistance error model is

1113957Ji Pm(i) minus P0

lm(i)1113890 1113891 middot Ji (9)

To define the problem of identification with filteringalgorithm the errors for kinematic parameters X and themeasured data Y are described in the state space which canbe rewritten as follows

X Δα1 middot middot middot Δα6 Δa1 middot middot middot Δa6 Δθ1 middot middot middot Δθ6 Δd1 middot middot middot Δd61113858 1113859T

Xk+1 Xk

Yk h Xk( 1113857 + uk

(10)

where k is the number of iterations and Xk represents theparametersrsquo errors Yk is the corresponding calibrationdistance measured by the measurement device uk is theprocess noise of measurement and h (middot) is the measurementfunction

3 Kinematic Parameter Identification

(e effect of gradual error accumulation of links decreasesthe absolute position accuracy significantly for the open-chain structure of industrial robots Kinematic parameteridentification is an important way to improve the absoluteposition accuracy

In the kinematic parameter identification the data of thejoint angles and the corresponding distances are required(e forward kinematic solutions of the robot can be ob-tained by combining a group of joint angles with which thedistances from EE of the robot to P0 can be calculatedUnlike the kinematic parameter identification using lasertracker to acquire the coordinates of EE the proposedmethod in this paper only needs the distances acquired bypull wire sensors or another distance measurement system

31 Preidentification Based on EKF (e solution for func-tion (8) can be obtained commonly using a nonlinear LSapproach Although nonlinear LS approach is a fast andefficient algorithm that can be used to solve nonlinearequations it is very sensitive to noise To obtain more ac-curate and stable solutions EKF algorithm is used in pre-liminary identification

(e EKF algorithm consists of two steps prediction andupdate [29] (e parametersrsquo error X can be regarded as thestate vector to be preliminarily identified First the priorestimate of state vector 1113954Xk+1 and the covariance matrix arepredicted by the former state vector Xk and Pk

1113954Xk+1 Xk

1113954Pk+1 Pk + Qk(11)

where k denotes the number of iterations Xk is N-statevector consisting of parametersrsquo deviation at kth iteration

1113954Xk+1 is the prior estimation of state vector Pk is the co-variance matrix of the estimation state and Qk is an NtimesNcovariance matrix of the system process noise

With the prior estimate of state vector the optimalKalman gain can be calculated with (12) and the covariancematrix of state vector is obtained as shown in (14)(en thestate vector can be updated by the measurement data with(13)

Kk+1 1113954Pk+11113957J

T

k+11113957Jk+1

1113954Pk+11113957J

T

k+1 + Rk+11113874 1113875minus1

(12)

Xk+1 1113954Xk+1 + K Yk+1 minus 1113957JT

k+11113954Xk+11113874 1113875 (13)

Pk+1 1113954Pk+1 minus Kk+11113957Jk+1

1113954Pk+1 (14)

where Kk+ 1 is the Kalman gain at kth iteration and Rk+ 1denotes the covariancematrix of measurement noiseYk+ 1 isthe distance error measured by the measurement system 1113957J isthe Jacobian matrix Xk+ 1 is the posterior estimate of statevector (e details of the EKF algorithm are illustrated inFigure 3

324eRPFAlgorithm When the noise of the acquired datais non-Gaussian it is difficult for the EKF algorithm to get areliable identified result (e RPF algorithm does not needthe model to be linear or compellent hypothesis for Gaussiannoise [35] the basis of which is particle filter (e pre-identification result of EKF is applied as the initial value ofthe RPF which can identify the state of nonlinear and non-Gaussian noise system With a certain amount of randomstate samples (ie particles) the RPF algorithm can ap-proximate the posterior probability density function of theidentifying model It provides a suboptimal solution with afinite number of samples All state samples are transferred indynamic systems through importance sampling Subse-quently it updates the posterior distribution sequentially

Apart from the resampling step the RPF is similar togeneric sampling importance resampling (SIR) filter [36]Considering the robot that is described by the state space

4 Complexity

model the RPF algorithm operates Np particles to ap-proximate the posterior density

(e RPF algorithm also consists of two steps predictionand update similar to EKF But it is described statisticallywith the state transition probability density p(xk|xkminus1) andthe observation likelihood probability density of the statevector p(xk|y1k)

In the prediction step the predicted state vector is es-timated by the last time state Xk-1 and its measurement dataY1kminus 1

p Xk|Y1kminus1( 1113857 1113946 p Xk|Ykminus1( 1113857p Xkminus1|Y1kminus1( 1113857dXkminus1 (15)

With the latest measurement data Y1k the predictedstate can be updated then its approximated discrete pos-terior density can be obtained

p Xk|Y1k( 1113857 p Yk|Xk( 1113857p Xk|Y1kminus1( 1113857

p Yk|Y1kminus1( 1113857 (16)

where p(Yk|Y1kminus1) is defined as the normalization constantas shown in (20)

p Yk|Y1kminus1( 1113857 1113946 p Yk|Xk( 1113857p Xk|Y1kminus1( 1113857dXk (17)

With Monte Carlo method [37] the approximateddiscrete posterior density can be simplified as shown in thefollowing equation

p Xk|Y1k( 1113857 asymp 1113944

Np

i1ωi

kδ Xk minus Xik1113872 1113873 (18)

where δ(middot) is the Dirac delta function Np is the number ofparticles and ωi

k is the weight of the ith particle at kth it-eration p(Xk|Y1k) is the approximated discrete posteriordensity According to Bayesian estimation [37] all theparticles need to sample from the target posterior densitywith the knowledge of observation Y1k but this is unreal-istic Importance sampling [37] is introduced in the RPF

algorithm as well as the importance samples from a knownposterior density (ie importance density) In the identifi-cation the weight of the particles and the normalized weightcan be defined as (19) and (20) respectively

ωik

12π|R|

radic exp minus12

edk1113872 1113873

TR

minus 1e

dk1113874 1113875 (19)

1113957ωik

ωik

1113936Np

i1 ωik

(20)

where edk is the distance error of the ith particle According to

Bayesian estimation [37] the minimum mean square(MMS) estimate 1113954X

i

k can be described as the expectation ofidentified state vector combining all particles [38] as shownin the following equation

1113954Xi

k asymp E 1113954Xi

k1113874 1113875 asymp 1113944

Np

i11113957ωi

kXik (21)

In the identification model with distance error theweight of each particle is evaluated using its distance error(e particles with lower distance error Δl have higherweight that is

Δl lm minus lr

(22)

After a number of iterations the weight will occupy onone particle (almost close to one) and the weights of otherparticles are so small that they will be negligible due to thedegeneracy of particles [37] (e degeneracy problem in-dicates that abundant computation is applied to updatingparticles whose weight is almost zero (e negligible par-ticles with tiny weight make no contribution to approxi-mation of the target posterior density Resampling [39]should be applied to avoid this problem Note that aneffective sample size Neff is introduced to measure thedegeneracy of particles [39] If Neff is too small resamplewill be triggered

Measurementsystem

Statepredictor

Stateupdater

Covarianceupdater

Covariancepredictor Kalman gain

Pk

XkXk+1ˆ

Pk+1ˆ Kk+1

Xk+1

Pk+1

Yk+1

Figure 3 (e flow chart of the EKF algorithm

Complexity 5

Neff 1

1113936Np

i1 ωik1113872 1113873

2 (23)

Resampling is an approach to prevent particles from de-generacy which will introduce the decay of diversity of par-ticles in turn (is problem leads to poor approximation of theposterior density One way to solve this problem is modifyingthe resampling in which the particles are resampled from thecontinuous approximation of the posterior density [40]

(e approximated continuous posterior density in theRPF [40] is

p X0k|Y1k( 1113857 asymp 1113944

Np

i1ωi

kKh X0k minus Xi0k1113872 1113873 (24)

where

kh(x) 1

hnx

kx

h1113874 1113875 (25)

where kh(middot) is theKernel probability density function hgt0 is theKernel bandwidth and n is the dimension of the estimated statevector X kh will be used for resampling particles on continuousapproximated density With appropriate function kh and h themean integrated square error (MISE) between real and regu-larized empirical posterior densities in (24) can be minimized[38](e details of the RPF algorithm are illustrated in Figure 4

It is assumed that the coordinate of P0 is constant whichcan be regarded as one of identified parameters (e offset ofthe adapter of the measurement device should also be con-sidered In general a complete kinematic model includes 28unknown kinematic parameters which need to be identifiedin our framework (e distances used in identification shouldbe more than 10mm to increase the certainty of identifiedresults Figure 5 shows the flow chart of the identification

33 Determination of Redundant Parameters Singularityanalysis of the identified matrix J is very important inidentification calculation which can lead to nonconvergence

or inaccurate results Generally speaking the maximumnumber of parameters is 4r1 + 2p1 + 6 in a generic serial-linkrobot with r1 rotation axes and p1 translation axes [41] Toobtain reliable results with the identification algorithmredundant parameters must be determined and excludedfrom the error model before identification (e conditionnumber of the identified matrix is

κ(J) J Jminus1

(26)

where κ(middot) is the measure of sensitivity or stability of theidentified matrix with regard to a small change in the inputargument In high dimension full-parameter identificationwith a larger condition number the identified matrix J isoften nonfull rank In this paper the singular value de-composition (SVD) [42] is applied to reduce the conditionnumber and avoid singularity problem Considering totalJacobian matrix H [J1 J2 Jn] it can be written as

H U 1113944 VT (27)

where U isin R3ntimes3n V isin R25times25 and they are orthogonalmatrices H isin R3ntimesN Σ diag(σ1 σ2 σ3 σr) r is therank of H n is the number of calibration distances N is thenumber of parameters and in this case N is 25 Hence thenumber of the parameters that are redundant can be ob-tained as 25minus r

After SVD calculation r can be determined that isr 20(erefore there are 5 redundant parameters that needto be eliminated from the process of identificationAccording to the property of distance error the originalcoordinates of P0 are determined in the base frame FromFigure 1 it can be found that the four parameter errors injoint 1 may cause the overall change of the global coordinatesystem Hence P0 will change accordingly which makes itidentifiable (erefore there are 19 parameters including P0(xp0 yp0 zp0) which need to be identified and they can bewritten in the format of a vector

X Δα1 middot middot middot Δα5 Δa1 middot middot middot Δa4 Δθ2 middot middot middot Δθ5 Δd2 Δd4 Δd61113858 1113859T (28)

4 Verification

To verify the proposed kinematic parameter identificationmethod a 6-DOF industrial robot ER20-C10 was usedwhose maximum load and repeatability are 20 kg and008mm respectively (e nominal values of parameters areshown in Table 1 (e settings of the error for each pa-rameter are shown in Table 2 It should be noted that theerror of the redundant parameters is set as zero since theseparameters are not independent (e two terminationconditions of the iteration are as follows the maximumnumber of iterations is 10000 and the minimum objective

function is less than 10minus6 Both of the covariance matrices Qand P were initialized as 10minus4 times I3lowast3 in EKF and the noise ofthe covariance matrices of measurement R is set as10minus4

times I3lowast3 in the simulation (e bandwidth h is set as 042A random and bounded measurement disturbance[minus002mm 002mm] is added to each distance to simulatethe measuring error in the actual device

To verify the effectiveness of the proposed algorithm agroup of 100 poses were generated randomly for identifi-cation and verification Considering the workspace of therobot the six joint angles were chosen randomly from[minus30deg90deg] [minus50deg50deg] [minus45deg90deg] [minus100deg100deg] [minus90deg90deg] and

6 Complexity

[minus360deg360deg] respectively (e former 50 poses were used foridentification and the other 50 poses were used forvalidation

To compare the effectiveness three different algorithms(ie LS EKF and EKF+RPF) were used to identify thekinematic parameters that were compensated to the controlmodel of the robot(emaximum distance error was used asthe index to evaluate the results of compensation Figure 6 isthe distance error distribution chart of the robot aftercompensation by LS which shows that all the distance errorsare reduced greatly and the average error is from 04827mmto 01120mm by a reduction rate of 76 Figure 7 is the

distance error distribution chart of the robot after com-pensation by EKF which shows that the average error isreduced from 04827mm to 01352mm by a reduction rate

Xk

Yk

Xki

Xki

Xk = Xk + hDk τii i Draw τ k ~ kh fromepanechnikov kernel

iDraw Xk fromq (Xk | Xkndash1 Y1k)

ii˜

Calculate ωk andtotal weight ωk

i ii ˜ωk = (ωk)ndash1 ωk

Measurementsystem

Normalize

Yes

No

Calculate effectivesample size Neff

Calculate the empiricalcovariance matrix Sk

Neff lt threshold

Resample Compute Dk such that Dk Dk = SkT

Xk = ωk Xki˜ i

i = 1

M

Figure 4 (e flow chart of the RPF algorithm

Initialvalue

Robot(ER20-C10)

A set of jointangle and P0

Δ l = ĴΔxlmisin

N

Set of distancemeasurements

eisinΩ contains 19parameters error

EKF

RPF

Kinematicsparameters error

Compensate tonominal parameters

Calculate distanceaccuracy

Figure 5 (e flow chart of the proposed algorithm

Table 2 Settings of the kinematic parametersrsquo error

No of joints Δαiminus1 (deg) Δaiminus1 (mm) Δθi(deg) Δdi (mm)1 0 0 0 02 002 10 003 minus033 minus001 05 004 04 0015 minus04 006 0855 minus001 minus03 005 06 minus006 0 0 minus04

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 6(e distance error of the robot after compensation by LS

Complexity 7

of 72 Figure 8 is the distance error distribution chart of therobot after compensation by EKF+RPF which shows thatthe average error is reduced from 04827mm to 00780mmby a reduction rate of 84

Table 3 shows the statistics of the maximum mean andRMS values of the distance errors before and after com-pensation by the three algorithms By comparison one canfind that the effectiveness of RPF is the best of the three

5 Conclusion

(is paper proposed a novel kinematic calibration methodbased on the distance information for the industrial robot inwhich an EKF and RPF hybrid identification method wasadopted To simplify the data acquisition procedure and alsoavoid using expensive coordinate measuring instrumentsthe moving displacements of the end effector were used asthe reference to build the kinematic parameter error iden-tification model (en the error model was linearized bydifferentiating of the coefficient matrix By SVD of thecoefficient matrix the redundant parameters were deter-mined and removed from the list of parameters to beidentified (e preidentification of the parameters wasconducted with EKF by which we obtained a group of initialparameters of the kinematic model RPF was used to identifythe kinematic parameters of the linearized error model withthe initial parameters Comparative simulations were con-ducted based on the linearized kinematic error model (ecompensation results of the three algorithms are all effectivein improving the accuracy of the industrial robot but thehybrid algorithm proposed in this paper is more precise andfast in iteration calculation In the future work precisiondistance measuring device will be developed and effec-tiveness of the device and the proposed algorithm will befurther verified by experimental tests

Data Availability

(e data were curated by the authors and are available uponrequest

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Natural ScienceFoundation of China (Grant no 51865020) and the ScientificResearch Fund of Yunnan Education Department (Grant no2019J0046)

References

[1] Z Roth B Mooring and B Ravani ldquoAn overview of robotcalibrationrdquo IEEE Journal on Robotics and Automation vol 3no 5 pp 377ndash385 1987

[2] B Mooring M Driels and Z Roth Fundamentals of Ma-nipulator Calibration Wiley Hoboken NJ USA 1991

[3] W K Veitschegger and C-H Wu ldquoRobot calibration andcompensationrdquo IEEE Journal on Robotics and Automationvol 4 no 6 pp 643ndash656 1988

[4] K Schroeer S L Albright and M Grethlein ldquoCompleteminimal and model-continuous kinematic models for robotcalibrationrdquo Robotics and Computer-Integrated Manufactur-ing vol 13 pp 73ndash85 1997

[5] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanism based on matricesrdquo Transof theAsmeJournal of AppliedMechanics vol 22 pp 215ndash221 1955

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 7 (e distance error of the robot after compensation byEKF

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 8 (e distance error of the robot after compensation byEKF+RPF

Table 3 (e distance error of robot before and after compensationby 3 different algorithms

Algorithm Max(mm)

Mean(mm)

RMS(mm)

Before compensation mdash 12612 04827 05791

After compensationLS 03236 01120 01376EKF 03255 01352 01653

EKF+RPF 02404 00780 00936

8 Complexity

[6] S A Hayati ldquoRobot arm geometric link parameter estima-tionrdquo in Proceedings of the 22nd IEEE Conference on Decisionand Control San Antonio TX USA November 1983

[7] H W Stone and A C Sanderson ldquoStatistical performanceevaluation of the S-model arm signature identificationtechniquerdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Philadelphia PA USAAugust 1988

[8] H Zhuang Z S Roth and F Hamano ldquoA complete andparametrically continuous kinematic model for robot ma-nipulatorsrdquo IEEE Transactions on Robotics amp Automationvol 8 pp 451ndash463 1990

[9] T Sun Y Zhai Y Song and J Zhang ldquoKinematic calibrationof a 3-DoF rotational parallel manipulator using lasertrackerrdquo Robotics and Computer-Integrated Manufacturingvol 41 pp 78ndash91 2016

[10] P Yuan D Chen T Wang S Cao Y Cai and L Xue ldquoAcompensation method based on extreme learning machine toenhance absolute position accuracy for aviation drilling ro-botrdquo Advances in Mechanical Engineering vol 10 2018

[11] G Gao F Liu H San XWu andWWang ldquoHybrid optimalkinematic parameter identification for an industrial robotbased on BPNN-PSOrdquo Complexity vol 2018 Article ID4258676 2018

[12] Y Bai H Zhuang and Z S Roth ldquoExperiment study ofPUMA robot calibration using a laser tracking systemrdquo inProceeding of the IEEE International Workshop on SoftComputing in Industrial Applications Binghamton NY USASeptember 2003

[13] X Zhang Y Song Y Yang and H Pan ldquoStereo vision basedautonomous robot calibrationrdquo Robotics and AutonomousSystems vol 93 pp 43ndash51 2017

[14] T Messay R Ordontildeez and E Marcil ldquoComputationallyefficient and robust kinematic calibration methodologies andtheir application to industrial robotsrdquo Robotics and Com-puter-Integrated Manufacturing vol 37 pp 33ndash48 2016

[15] G Gatti and G Danieli ldquoA practical approach to compensatefor geometric errors in measuring arms application to a six-degree-of-freedom kinematic structurerdquo Measurement Sci-ence and Technology vol 19 Article ID 015107 2008

[16] N Schillreff M Nykolaychuk and F Ortmeier ldquoTowardshigh accuracy robot-assisted surgerylowastlowast this work was partlyfunded by the federal ministry of education and Research inGermany within the research campus STIMULATE undergrant No 13GW0095Ardquo IFAC-PapersOnLine vol 50 no 1pp 5666ndash5671 2017

[17] A Nubiola and I A Bonev ldquoAbsolute calibration of an ABBIRB 1600 robot using a laser trackerrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 236ndash245 2013

[18] A Nubiola M Slamani A Joubair and I A BonevldquoComparison of two calibrationmethods for a small industrialrobot based on an optical CMM and a laser trackerrdquo Roboticavol 32 no 3 pp 447ndash466 2014

[19] G Gao H Zhang H San X Wu and W Wang ldquoModelingand error compensation of robotic articulated arm coordinatemeasuring machines using BP neural networkrdquo Complexityvol 2017 Article ID 5156264 2017

[20] A Nubiola M Slamani and I A Bonev ldquoA new method formeasuring a large set of poses with a single telescopingballbarrdquo Precision Engineering vol 37 no 2 pp 451ndash4602013

[21] H-N Nguyen J Zhou andH-J Kang ldquoA calibrationmethodfor enhancing robot accuracy through integration of an

extended Kalman filter algorithm and an artificial neuralnetworkrdquo Neurocomputing vol 151 pp 996ndash1005 2015

[22] J Zhang X Wang K Wen Y Zhou Y Yue and J Yang ldquoAsimple and rapid calibration methodology for industrial robotbased on geometric constraint and two-step errorrdquo IndustrialRobot An International Journal vol 45 no 6 pp 715ndash7212018

[23] G Zhao P Zhang GMa andW Xiao ldquoSystem identificationof the nonlinear residual errors of an industrial robot usingmassive measurementsrdquo Robotics and Computer-IntegratedManufacturing vol 59 pp 104ndash114 2019

[24] K Wang G Liu Q Tao and M Zhai ldquoEfficient parametersestimation method for the separable nonlinear least squaresproblemrdquo Complexity vol 2020 Article ID 9619427 17 pages2020

[25] I Dattner H Ship and E O Voit ldquoSeparable nonlinear least-squares parameter estimation for complex dynamic systemsrdquoComplexity vol 2020 Article ID 6403641 18 pages 2020

[26] J Ding ldquo(e hierarchical iterative identification algorithm formulti-input-output-error systems with autoregressive noiserdquoComplexity vol 2017 Article ID 5292894 2017

[27] A Omodei G Legnani and R Adamini ldquo(ree methodol-ogies for the calibration of industrial manipulators experi-mental results on a SCARA robotrdquo Journal of Robotic Systemsvol 17 no 6 pp 291ndash307 2000

[28] Y Ho and R Lee ldquoA Bayesian approach to problems instochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 333ndash339 1964

[29] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processingvol 50 no 2 pp 174ndash188 2002

[30] K Schroer Precision and Calibration John Wiley amp SonsHoboken NJ USA 2007

[31] B Greenway ldquoRobot accuracyrdquo Industrial Robot An Inter-national Journal vol 27 no 4 pp 257ndash265 2000

[32] C Musso N Oudjane and F Le Gland ldquoImproving regu-larised particle filtersrdquo in Sequential Monte Carlo Methods inPractice A Doucet N de Freitas and N Gordon Edspp 247ndash271 Springer Berlin Germany 2001

[33] J J Craig Introduction to Robotics Mechanics and ControlPearson Education India Uttar Pradesh India 3rd edition2009

[34] Z Wang H Xu G Chen R Sun and L Sun ldquoA distanceerror based industrial robot kinematic calibration methodrdquoIndustrial Robot An International Journal vol 41 pp 439ndash446 2014

[35] K Senne ldquoStochastic processes and filtering theoryrdquo IEEETransactions on Automatic Control vol 17 no 5 pp 752-7531972

[36] J Carpenter P Clifford and P Fearnhead ldquoImproved particlefilter for nonlinear problemsrdquo IEE Proceedings-Radar Sonarand Navigation vol 146 no 1 pp 2ndash7 1999

[37] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[38] S Godsill and T Clapp Improvement Strategies for MonteCarlo Particle Filters Springer Berlin Germany 2001

[39] S Liu and R Chen ldquoSequential Monte Carlo methods fordynamic systemsrdquo Journal of the American Statistical Asso-ciation vol 93 1998

[40] C Musso N Oudjane and F L Gland ldquoImproving regu-larised particle filtersrdquo Statistics for Engineering amp Informa-tion Science vol 28 2001

Complexity 9

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity

where TKE 0 0 lz 11113858 1113859T and lz is the end effector (EE)

offset along the axis of jointNt (e first three rows of the lastcolumn of T0E denote the position of EE that is P(middot)

22 Distance Error Modelling According to ISO-9258[30 31] between the continuous moving points during themovement of robot in workspace the distance accuracy ofindustrial robot refers to the deviation between the mea-sured distance and the command distance illustrated asFigure 2 (e distance of the two adjacent points (i and i+ 1)in workspace can be expanded as

Δl(i i + 1) lt(i i + 1) minus lm(i i + 1)

Pt(i + 1) minus Pt(i)1113858 1113859 minus Pm(i + 1) minus Pm(i)1113858 1113859

Pt(i + 1) minus Pm(i + 1)1113858 1113859 minus Pt(i) minus Pm(i)1113858 1113859

(3)

(e position error of the ith point of the acquisition datais defined as

Pe(i) Pt(i) minus Pm(i) JX (4)

where J is the Jacobeanmatrix andX is the error of kinematicparameters According to equations (3) and (4) one canobtain the distance error formula shown as follows

lm(i i + 1) middot Δl(i i + 1) Pm(i + 1) minus Pm(i)1113858 1113859

middot Pe(i + 1) minus Pe(i)(5)

Equation (5) indicates that the distance error can beexpressed by partial pose information Furthermore equa-tion (5) can be rewritten as [34]

Δl(i i + 1) Pm(i + 1) minus Pm(i)

lm(i i + 1)1113890 1113891 middot Pe(i + 1) minus Pe(i)

(6)

According to equation (6) the measured distance can becalculated by two corresponding adjacent points [34](erefore at every acquisition position one has to measuretwice to obtain the distance used which is time-consumingand labor-consuming In this paper we proposed a modifieddistance model which is more convenient and less time-consuming than the general one

(e original position of the device is set as P0(xp0yp0zp0)in the frame of the measurement device which is fixed P0 isconsidered to be error-free which is also set as one of the twoadjacent points in equation (6) Hence equation (6) can berewritten as

Δl(i) Pm(i) minus P0

lm(i)1113890 1113891 middot Pe(i) (7)

where lm(i) is the ith distance from EE to P0 which can bedirectly measured by the measurement system P0 is knownas the fixed point of the measurement system (e originalerror of the measurement system can be neglectable for thedistance is a relative value With P0 being the fixed referencepoint to get the distance one only needs to measure onetime at a position (us the measuring process can besimplified with proposed strategy Equation (7) describes therelationship between the distance error model and the

Y4Y6

Y5

X6 X5

X4

Z6 Z5Z2

Z2

Y3

Y2

Y1

Y0

O2

O1

Z1

Z0

O0

X1

X0

O3

X3

X2

Figure 1 (e kinematic model of ER20-C10 robot

Table 1 Nominal values of robotrsquos MDH model parameters

No of joints αiminus1(deg) aiminus1(mm) θi(deg) di (mm)1 0 0 θ1 5042 90 166605 minus90 + θ2 03 0 minus78227 θ3 04 minus90 minus138826 θ4 761355 90 0 θ5 06 minus90 0 θ6 125

O

Pt(i) Pt(i + 1)Pe(i + 1)

Pm(i + 1)Pe(i)

Pm(i)

lt

lm

Figure 2 (e definition of distance accuracy

Complexity 3

position error Assuming that the deviation of the param-eters is small (7) can be linearized as the first-order term ofits Taylor expansion around the nominal values of the ki-nematic parameters as shown in the following equation

Δl(i) 1113957JX (8)

where 1113957J isin R3ntimesN n is the number of calibration distancesand N is the number of parameters (e Jacobian matrix 1113957J indistance error model is

1113957Ji Pm(i) minus P0

lm(i)1113890 1113891 middot Ji (9)

To define the problem of identification with filteringalgorithm the errors for kinematic parameters X and themeasured data Y are described in the state space which canbe rewritten as follows

X Δα1 middot middot middot Δα6 Δa1 middot middot middot Δa6 Δθ1 middot middot middot Δθ6 Δd1 middot middot middot Δd61113858 1113859T

Xk+1 Xk

Yk h Xk( 1113857 + uk

(10)

where k is the number of iterations and Xk represents theparametersrsquo errors Yk is the corresponding calibrationdistance measured by the measurement device uk is theprocess noise of measurement and h (middot) is the measurementfunction

3 Kinematic Parameter Identification

(e effect of gradual error accumulation of links decreasesthe absolute position accuracy significantly for the open-chain structure of industrial robots Kinematic parameteridentification is an important way to improve the absoluteposition accuracy

In the kinematic parameter identification the data of thejoint angles and the corresponding distances are required(e forward kinematic solutions of the robot can be ob-tained by combining a group of joint angles with which thedistances from EE of the robot to P0 can be calculatedUnlike the kinematic parameter identification using lasertracker to acquire the coordinates of EE the proposedmethod in this paper only needs the distances acquired bypull wire sensors or another distance measurement system

31 Preidentification Based on EKF (e solution for func-tion (8) can be obtained commonly using a nonlinear LSapproach Although nonlinear LS approach is a fast andefficient algorithm that can be used to solve nonlinearequations it is very sensitive to noise To obtain more ac-curate and stable solutions EKF algorithm is used in pre-liminary identification

(e EKF algorithm consists of two steps prediction andupdate [29] (e parametersrsquo error X can be regarded as thestate vector to be preliminarily identified First the priorestimate of state vector 1113954Xk+1 and the covariance matrix arepredicted by the former state vector Xk and Pk

1113954Xk+1 Xk

1113954Pk+1 Pk + Qk(11)

where k denotes the number of iterations Xk is N-statevector consisting of parametersrsquo deviation at kth iteration

1113954Xk+1 is the prior estimation of state vector Pk is the co-variance matrix of the estimation state and Qk is an NtimesNcovariance matrix of the system process noise

With the prior estimate of state vector the optimalKalman gain can be calculated with (12) and the covariancematrix of state vector is obtained as shown in (14)(en thestate vector can be updated by the measurement data with(13)

Kk+1 1113954Pk+11113957J

T

k+11113957Jk+1

1113954Pk+11113957J

T

k+1 + Rk+11113874 1113875minus1

(12)

Xk+1 1113954Xk+1 + K Yk+1 minus 1113957JT

k+11113954Xk+11113874 1113875 (13)

Pk+1 1113954Pk+1 minus Kk+11113957Jk+1

1113954Pk+1 (14)

where Kk+ 1 is the Kalman gain at kth iteration and Rk+ 1denotes the covariancematrix of measurement noiseYk+ 1 isthe distance error measured by the measurement system 1113957J isthe Jacobian matrix Xk+ 1 is the posterior estimate of statevector (e details of the EKF algorithm are illustrated inFigure 3

324eRPFAlgorithm When the noise of the acquired datais non-Gaussian it is difficult for the EKF algorithm to get areliable identified result (e RPF algorithm does not needthe model to be linear or compellent hypothesis for Gaussiannoise [35] the basis of which is particle filter (e pre-identification result of EKF is applied as the initial value ofthe RPF which can identify the state of nonlinear and non-Gaussian noise system With a certain amount of randomstate samples (ie particles) the RPF algorithm can ap-proximate the posterior probability density function of theidentifying model It provides a suboptimal solution with afinite number of samples All state samples are transferred indynamic systems through importance sampling Subse-quently it updates the posterior distribution sequentially

Apart from the resampling step the RPF is similar togeneric sampling importance resampling (SIR) filter [36]Considering the robot that is described by the state space

4 Complexity

model the RPF algorithm operates Np particles to ap-proximate the posterior density

(e RPF algorithm also consists of two steps predictionand update similar to EKF But it is described statisticallywith the state transition probability density p(xk|xkminus1) andthe observation likelihood probability density of the statevector p(xk|y1k)

In the prediction step the predicted state vector is es-timated by the last time state Xk-1 and its measurement dataY1kminus 1

p Xk|Y1kminus1( 1113857 1113946 p Xk|Ykminus1( 1113857p Xkminus1|Y1kminus1( 1113857dXkminus1 (15)

With the latest measurement data Y1k the predictedstate can be updated then its approximated discrete pos-terior density can be obtained

p Xk|Y1k( 1113857 p Yk|Xk( 1113857p Xk|Y1kminus1( 1113857

p Yk|Y1kminus1( 1113857 (16)

where p(Yk|Y1kminus1) is defined as the normalization constantas shown in (20)

p Yk|Y1kminus1( 1113857 1113946 p Yk|Xk( 1113857p Xk|Y1kminus1( 1113857dXk (17)

With Monte Carlo method [37] the approximateddiscrete posterior density can be simplified as shown in thefollowing equation

p Xk|Y1k( 1113857 asymp 1113944

Np

i1ωi

kδ Xk minus Xik1113872 1113873 (18)

where δ(middot) is the Dirac delta function Np is the number ofparticles and ωi

k is the weight of the ith particle at kth it-eration p(Xk|Y1k) is the approximated discrete posteriordensity According to Bayesian estimation [37] all theparticles need to sample from the target posterior densitywith the knowledge of observation Y1k but this is unreal-istic Importance sampling [37] is introduced in the RPF

algorithm as well as the importance samples from a knownposterior density (ie importance density) In the identifi-cation the weight of the particles and the normalized weightcan be defined as (19) and (20) respectively

ωik

12π|R|

radic exp minus12

edk1113872 1113873

TR

minus 1e

dk1113874 1113875 (19)

1113957ωik

ωik

1113936Np

i1 ωik

(20)

where edk is the distance error of the ith particle According to

Bayesian estimation [37] the minimum mean square(MMS) estimate 1113954X

i

k can be described as the expectation ofidentified state vector combining all particles [38] as shownin the following equation

1113954Xi

k asymp E 1113954Xi

k1113874 1113875 asymp 1113944

Np

i11113957ωi

kXik (21)

In the identification model with distance error theweight of each particle is evaluated using its distance error(e particles with lower distance error Δl have higherweight that is

Δl lm minus lr

(22)

After a number of iterations the weight will occupy onone particle (almost close to one) and the weights of otherparticles are so small that they will be negligible due to thedegeneracy of particles [37] (e degeneracy problem in-dicates that abundant computation is applied to updatingparticles whose weight is almost zero (e negligible par-ticles with tiny weight make no contribution to approxi-mation of the target posterior density Resampling [39]should be applied to avoid this problem Note that aneffective sample size Neff is introduced to measure thedegeneracy of particles [39] If Neff is too small resamplewill be triggered

Measurementsystem

Statepredictor

Stateupdater

Covarianceupdater

Covariancepredictor Kalman gain

Pk

XkXk+1ˆ

Pk+1ˆ Kk+1

Xk+1

Pk+1

Yk+1

Figure 3 (e flow chart of the EKF algorithm

Complexity 5

Neff 1

1113936Np

i1 ωik1113872 1113873

2 (23)

Resampling is an approach to prevent particles from de-generacy which will introduce the decay of diversity of par-ticles in turn (is problem leads to poor approximation of theposterior density One way to solve this problem is modifyingthe resampling in which the particles are resampled from thecontinuous approximation of the posterior density [40]

(e approximated continuous posterior density in theRPF [40] is

p X0k|Y1k( 1113857 asymp 1113944

Np

i1ωi

kKh X0k minus Xi0k1113872 1113873 (24)

where

kh(x) 1

hnx

kx

h1113874 1113875 (25)

where kh(middot) is theKernel probability density function hgt0 is theKernel bandwidth and n is the dimension of the estimated statevector X kh will be used for resampling particles on continuousapproximated density With appropriate function kh and h themean integrated square error (MISE) between real and regu-larized empirical posterior densities in (24) can be minimized[38](e details of the RPF algorithm are illustrated in Figure 4

It is assumed that the coordinate of P0 is constant whichcan be regarded as one of identified parameters (e offset ofthe adapter of the measurement device should also be con-sidered In general a complete kinematic model includes 28unknown kinematic parameters which need to be identifiedin our framework (e distances used in identification shouldbe more than 10mm to increase the certainty of identifiedresults Figure 5 shows the flow chart of the identification

33 Determination of Redundant Parameters Singularityanalysis of the identified matrix J is very important inidentification calculation which can lead to nonconvergence

or inaccurate results Generally speaking the maximumnumber of parameters is 4r1 + 2p1 + 6 in a generic serial-linkrobot with r1 rotation axes and p1 translation axes [41] Toobtain reliable results with the identification algorithmredundant parameters must be determined and excludedfrom the error model before identification (e conditionnumber of the identified matrix is

κ(J) J Jminus1

(26)

where κ(middot) is the measure of sensitivity or stability of theidentified matrix with regard to a small change in the inputargument In high dimension full-parameter identificationwith a larger condition number the identified matrix J isoften nonfull rank In this paper the singular value de-composition (SVD) [42] is applied to reduce the conditionnumber and avoid singularity problem Considering totalJacobian matrix H [J1 J2 Jn] it can be written as

H U 1113944 VT (27)

where U isin R3ntimes3n V isin R25times25 and they are orthogonalmatrices H isin R3ntimesN Σ diag(σ1 σ2 σ3 σr) r is therank of H n is the number of calibration distances N is thenumber of parameters and in this case N is 25 Hence thenumber of the parameters that are redundant can be ob-tained as 25minus r

After SVD calculation r can be determined that isr 20(erefore there are 5 redundant parameters that needto be eliminated from the process of identificationAccording to the property of distance error the originalcoordinates of P0 are determined in the base frame FromFigure 1 it can be found that the four parameter errors injoint 1 may cause the overall change of the global coordinatesystem Hence P0 will change accordingly which makes itidentifiable (erefore there are 19 parameters including P0(xp0 yp0 zp0) which need to be identified and they can bewritten in the format of a vector

X Δα1 middot middot middot Δα5 Δa1 middot middot middot Δa4 Δθ2 middot middot middot Δθ5 Δd2 Δd4 Δd61113858 1113859T (28)

4 Verification

To verify the proposed kinematic parameter identificationmethod a 6-DOF industrial robot ER20-C10 was usedwhose maximum load and repeatability are 20 kg and008mm respectively (e nominal values of parameters areshown in Table 1 (e settings of the error for each pa-rameter are shown in Table 2 It should be noted that theerror of the redundant parameters is set as zero since theseparameters are not independent (e two terminationconditions of the iteration are as follows the maximumnumber of iterations is 10000 and the minimum objective

function is less than 10minus6 Both of the covariance matrices Qand P were initialized as 10minus4 times I3lowast3 in EKF and the noise ofthe covariance matrices of measurement R is set as10minus4

times I3lowast3 in the simulation (e bandwidth h is set as 042A random and bounded measurement disturbance[minus002mm 002mm] is added to each distance to simulatethe measuring error in the actual device

To verify the effectiveness of the proposed algorithm agroup of 100 poses were generated randomly for identifi-cation and verification Considering the workspace of therobot the six joint angles were chosen randomly from[minus30deg90deg] [minus50deg50deg] [minus45deg90deg] [minus100deg100deg] [minus90deg90deg] and

6 Complexity

[minus360deg360deg] respectively (e former 50 poses were used foridentification and the other 50 poses were used forvalidation

To compare the effectiveness three different algorithms(ie LS EKF and EKF+RPF) were used to identify thekinematic parameters that were compensated to the controlmodel of the robot(emaximum distance error was used asthe index to evaluate the results of compensation Figure 6 isthe distance error distribution chart of the robot aftercompensation by LS which shows that all the distance errorsare reduced greatly and the average error is from 04827mmto 01120mm by a reduction rate of 76 Figure 7 is the

distance error distribution chart of the robot after com-pensation by EKF which shows that the average error isreduced from 04827mm to 01352mm by a reduction rate

Xk

Yk

Xki

Xki

Xk = Xk + hDk τii i Draw τ k ~ kh fromepanechnikov kernel

iDraw Xk fromq (Xk | Xkndash1 Y1k)

ii˜

Calculate ωk andtotal weight ωk

i ii ˜ωk = (ωk)ndash1 ωk

Measurementsystem

Normalize

Yes

No

Calculate effectivesample size Neff

Calculate the empiricalcovariance matrix Sk

Neff lt threshold

Resample Compute Dk such that Dk Dk = SkT

Xk = ωk Xki˜ i

i = 1

M

Figure 4 (e flow chart of the RPF algorithm

Initialvalue

Robot(ER20-C10)

A set of jointangle and P0

Δ l = ĴΔxlmisin

N

Set of distancemeasurements

eisinΩ contains 19parameters error

EKF

RPF

Kinematicsparameters error

Compensate tonominal parameters

Calculate distanceaccuracy

Figure 5 (e flow chart of the proposed algorithm

Table 2 Settings of the kinematic parametersrsquo error

No of joints Δαiminus1 (deg) Δaiminus1 (mm) Δθi(deg) Δdi (mm)1 0 0 0 02 002 10 003 minus033 minus001 05 004 04 0015 minus04 006 0855 minus001 minus03 005 06 minus006 0 0 minus04

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 6(e distance error of the robot after compensation by LS

Complexity 7

of 72 Figure 8 is the distance error distribution chart of therobot after compensation by EKF+RPF which shows thatthe average error is reduced from 04827mm to 00780mmby a reduction rate of 84

Table 3 shows the statistics of the maximum mean andRMS values of the distance errors before and after com-pensation by the three algorithms By comparison one canfind that the effectiveness of RPF is the best of the three

5 Conclusion

(is paper proposed a novel kinematic calibration methodbased on the distance information for the industrial robot inwhich an EKF and RPF hybrid identification method wasadopted To simplify the data acquisition procedure and alsoavoid using expensive coordinate measuring instrumentsthe moving displacements of the end effector were used asthe reference to build the kinematic parameter error iden-tification model (en the error model was linearized bydifferentiating of the coefficient matrix By SVD of thecoefficient matrix the redundant parameters were deter-mined and removed from the list of parameters to beidentified (e preidentification of the parameters wasconducted with EKF by which we obtained a group of initialparameters of the kinematic model RPF was used to identifythe kinematic parameters of the linearized error model withthe initial parameters Comparative simulations were con-ducted based on the linearized kinematic error model (ecompensation results of the three algorithms are all effectivein improving the accuracy of the industrial robot but thehybrid algorithm proposed in this paper is more precise andfast in iteration calculation In the future work precisiondistance measuring device will be developed and effec-tiveness of the device and the proposed algorithm will befurther verified by experimental tests

Data Availability

(e data were curated by the authors and are available uponrequest

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Natural ScienceFoundation of China (Grant no 51865020) and the ScientificResearch Fund of Yunnan Education Department (Grant no2019J0046)

References

[1] Z Roth B Mooring and B Ravani ldquoAn overview of robotcalibrationrdquo IEEE Journal on Robotics and Automation vol 3no 5 pp 377ndash385 1987

[2] B Mooring M Driels and Z Roth Fundamentals of Ma-nipulator Calibration Wiley Hoboken NJ USA 1991

[3] W K Veitschegger and C-H Wu ldquoRobot calibration andcompensationrdquo IEEE Journal on Robotics and Automationvol 4 no 6 pp 643ndash656 1988

[4] K Schroeer S L Albright and M Grethlein ldquoCompleteminimal and model-continuous kinematic models for robotcalibrationrdquo Robotics and Computer-Integrated Manufactur-ing vol 13 pp 73ndash85 1997

[5] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanism based on matricesrdquo Transof theAsmeJournal of AppliedMechanics vol 22 pp 215ndash221 1955

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 7 (e distance error of the robot after compensation byEKF

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 8 (e distance error of the robot after compensation byEKF+RPF

Table 3 (e distance error of robot before and after compensationby 3 different algorithms

Algorithm Max(mm)

Mean(mm)

RMS(mm)

Before compensation mdash 12612 04827 05791

After compensationLS 03236 01120 01376EKF 03255 01352 01653

EKF+RPF 02404 00780 00936

8 Complexity

[6] S A Hayati ldquoRobot arm geometric link parameter estima-tionrdquo in Proceedings of the 22nd IEEE Conference on Decisionand Control San Antonio TX USA November 1983

[7] H W Stone and A C Sanderson ldquoStatistical performanceevaluation of the S-model arm signature identificationtechniquerdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Philadelphia PA USAAugust 1988

[8] H Zhuang Z S Roth and F Hamano ldquoA complete andparametrically continuous kinematic model for robot ma-nipulatorsrdquo IEEE Transactions on Robotics amp Automationvol 8 pp 451ndash463 1990

[9] T Sun Y Zhai Y Song and J Zhang ldquoKinematic calibrationof a 3-DoF rotational parallel manipulator using lasertrackerrdquo Robotics and Computer-Integrated Manufacturingvol 41 pp 78ndash91 2016

[10] P Yuan D Chen T Wang S Cao Y Cai and L Xue ldquoAcompensation method based on extreme learning machine toenhance absolute position accuracy for aviation drilling ro-botrdquo Advances in Mechanical Engineering vol 10 2018

[11] G Gao F Liu H San XWu andWWang ldquoHybrid optimalkinematic parameter identification for an industrial robotbased on BPNN-PSOrdquo Complexity vol 2018 Article ID4258676 2018

[12] Y Bai H Zhuang and Z S Roth ldquoExperiment study ofPUMA robot calibration using a laser tracking systemrdquo inProceeding of the IEEE International Workshop on SoftComputing in Industrial Applications Binghamton NY USASeptember 2003

[13] X Zhang Y Song Y Yang and H Pan ldquoStereo vision basedautonomous robot calibrationrdquo Robotics and AutonomousSystems vol 93 pp 43ndash51 2017

[14] T Messay R Ordontildeez and E Marcil ldquoComputationallyefficient and robust kinematic calibration methodologies andtheir application to industrial robotsrdquo Robotics and Com-puter-Integrated Manufacturing vol 37 pp 33ndash48 2016

[15] G Gatti and G Danieli ldquoA practical approach to compensatefor geometric errors in measuring arms application to a six-degree-of-freedom kinematic structurerdquo Measurement Sci-ence and Technology vol 19 Article ID 015107 2008

[16] N Schillreff M Nykolaychuk and F Ortmeier ldquoTowardshigh accuracy robot-assisted surgerylowastlowast this work was partlyfunded by the federal ministry of education and Research inGermany within the research campus STIMULATE undergrant No 13GW0095Ardquo IFAC-PapersOnLine vol 50 no 1pp 5666ndash5671 2017

[17] A Nubiola and I A Bonev ldquoAbsolute calibration of an ABBIRB 1600 robot using a laser trackerrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 236ndash245 2013

[18] A Nubiola M Slamani A Joubair and I A BonevldquoComparison of two calibrationmethods for a small industrialrobot based on an optical CMM and a laser trackerrdquo Roboticavol 32 no 3 pp 447ndash466 2014

[19] G Gao H Zhang H San X Wu and W Wang ldquoModelingand error compensation of robotic articulated arm coordinatemeasuring machines using BP neural networkrdquo Complexityvol 2017 Article ID 5156264 2017

[20] A Nubiola M Slamani and I A Bonev ldquoA new method formeasuring a large set of poses with a single telescopingballbarrdquo Precision Engineering vol 37 no 2 pp 451ndash4602013

[21] H-N Nguyen J Zhou andH-J Kang ldquoA calibrationmethodfor enhancing robot accuracy through integration of an

extended Kalman filter algorithm and an artificial neuralnetworkrdquo Neurocomputing vol 151 pp 996ndash1005 2015

[22] J Zhang X Wang K Wen Y Zhou Y Yue and J Yang ldquoAsimple and rapid calibration methodology for industrial robotbased on geometric constraint and two-step errorrdquo IndustrialRobot An International Journal vol 45 no 6 pp 715ndash7212018

[23] G Zhao P Zhang GMa andW Xiao ldquoSystem identificationof the nonlinear residual errors of an industrial robot usingmassive measurementsrdquo Robotics and Computer-IntegratedManufacturing vol 59 pp 104ndash114 2019

[24] K Wang G Liu Q Tao and M Zhai ldquoEfficient parametersestimation method for the separable nonlinear least squaresproblemrdquo Complexity vol 2020 Article ID 9619427 17 pages2020

[25] I Dattner H Ship and E O Voit ldquoSeparable nonlinear least-squares parameter estimation for complex dynamic systemsrdquoComplexity vol 2020 Article ID 6403641 18 pages 2020

[26] J Ding ldquo(e hierarchical iterative identification algorithm formulti-input-output-error systems with autoregressive noiserdquoComplexity vol 2017 Article ID 5292894 2017

[27] A Omodei G Legnani and R Adamini ldquo(ree methodol-ogies for the calibration of industrial manipulators experi-mental results on a SCARA robotrdquo Journal of Robotic Systemsvol 17 no 6 pp 291ndash307 2000

[28] Y Ho and R Lee ldquoA Bayesian approach to problems instochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 333ndash339 1964

[29] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processingvol 50 no 2 pp 174ndash188 2002

[30] K Schroer Precision and Calibration John Wiley amp SonsHoboken NJ USA 2007

[31] B Greenway ldquoRobot accuracyrdquo Industrial Robot An Inter-national Journal vol 27 no 4 pp 257ndash265 2000

[32] C Musso N Oudjane and F Le Gland ldquoImproving regu-larised particle filtersrdquo in Sequential Monte Carlo Methods inPractice A Doucet N de Freitas and N Gordon Edspp 247ndash271 Springer Berlin Germany 2001

[33] J J Craig Introduction to Robotics Mechanics and ControlPearson Education India Uttar Pradesh India 3rd edition2009

[34] Z Wang H Xu G Chen R Sun and L Sun ldquoA distanceerror based industrial robot kinematic calibration methodrdquoIndustrial Robot An International Journal vol 41 pp 439ndash446 2014

[35] K Senne ldquoStochastic processes and filtering theoryrdquo IEEETransactions on Automatic Control vol 17 no 5 pp 752-7531972

[36] J Carpenter P Clifford and P Fearnhead ldquoImproved particlefilter for nonlinear problemsrdquo IEE Proceedings-Radar Sonarand Navigation vol 146 no 1 pp 2ndash7 1999

[37] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[38] S Godsill and T Clapp Improvement Strategies for MonteCarlo Particle Filters Springer Berlin Germany 2001

[39] S Liu and R Chen ldquoSequential Monte Carlo methods fordynamic systemsrdquo Journal of the American Statistical Asso-ciation vol 93 1998

[40] C Musso N Oudjane and F L Gland ldquoImproving regu-larised particle filtersrdquo Statistics for Engineering amp Informa-tion Science vol 28 2001

Complexity 9

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity

position error Assuming that the deviation of the param-eters is small (7) can be linearized as the first-order term ofits Taylor expansion around the nominal values of the ki-nematic parameters as shown in the following equation

Δl(i) 1113957JX (8)

where 1113957J isin R3ntimesN n is the number of calibration distancesand N is the number of parameters (e Jacobian matrix 1113957J indistance error model is

1113957Ji Pm(i) minus P0

lm(i)1113890 1113891 middot Ji (9)

To define the problem of identification with filteringalgorithm the errors for kinematic parameters X and themeasured data Y are described in the state space which canbe rewritten as follows

X Δα1 middot middot middot Δα6 Δa1 middot middot middot Δa6 Δθ1 middot middot middot Δθ6 Δd1 middot middot middot Δd61113858 1113859T

Xk+1 Xk

Yk h Xk( 1113857 + uk

(10)

where k is the number of iterations and Xk represents theparametersrsquo errors Yk is the corresponding calibrationdistance measured by the measurement device uk is theprocess noise of measurement and h (middot) is the measurementfunction

3 Kinematic Parameter Identification

(e effect of gradual error accumulation of links decreasesthe absolute position accuracy significantly for the open-chain structure of industrial robots Kinematic parameteridentification is an important way to improve the absoluteposition accuracy

In the kinematic parameter identification the data of thejoint angles and the corresponding distances are required(e forward kinematic solutions of the robot can be ob-tained by combining a group of joint angles with which thedistances from EE of the robot to P0 can be calculatedUnlike the kinematic parameter identification using lasertracker to acquire the coordinates of EE the proposedmethod in this paper only needs the distances acquired bypull wire sensors or another distance measurement system

31 Preidentification Based on EKF (e solution for func-tion (8) can be obtained commonly using a nonlinear LSapproach Although nonlinear LS approach is a fast andefficient algorithm that can be used to solve nonlinearequations it is very sensitive to noise To obtain more ac-curate and stable solutions EKF algorithm is used in pre-liminary identification

(e EKF algorithm consists of two steps prediction andupdate [29] (e parametersrsquo error X can be regarded as thestate vector to be preliminarily identified First the priorestimate of state vector 1113954Xk+1 and the covariance matrix arepredicted by the former state vector Xk and Pk

1113954Xk+1 Xk

1113954Pk+1 Pk + Qk(11)

where k denotes the number of iterations Xk is N-statevector consisting of parametersrsquo deviation at kth iteration

1113954Xk+1 is the prior estimation of state vector Pk is the co-variance matrix of the estimation state and Qk is an NtimesNcovariance matrix of the system process noise

With the prior estimate of state vector the optimalKalman gain can be calculated with (12) and the covariancematrix of state vector is obtained as shown in (14)(en thestate vector can be updated by the measurement data with(13)

Kk+1 1113954Pk+11113957J

T

k+11113957Jk+1

1113954Pk+11113957J

T

k+1 + Rk+11113874 1113875minus1

(12)

Xk+1 1113954Xk+1 + K Yk+1 minus 1113957JT

k+11113954Xk+11113874 1113875 (13)

Pk+1 1113954Pk+1 minus Kk+11113957Jk+1

1113954Pk+1 (14)

where Kk+ 1 is the Kalman gain at kth iteration and Rk+ 1denotes the covariancematrix of measurement noiseYk+ 1 isthe distance error measured by the measurement system 1113957J isthe Jacobian matrix Xk+ 1 is the posterior estimate of statevector (e details of the EKF algorithm are illustrated inFigure 3

324eRPFAlgorithm When the noise of the acquired datais non-Gaussian it is difficult for the EKF algorithm to get areliable identified result (e RPF algorithm does not needthe model to be linear or compellent hypothesis for Gaussiannoise [35] the basis of which is particle filter (e pre-identification result of EKF is applied as the initial value ofthe RPF which can identify the state of nonlinear and non-Gaussian noise system With a certain amount of randomstate samples (ie particles) the RPF algorithm can ap-proximate the posterior probability density function of theidentifying model It provides a suboptimal solution with afinite number of samples All state samples are transferred indynamic systems through importance sampling Subse-quently it updates the posterior distribution sequentially

Apart from the resampling step the RPF is similar togeneric sampling importance resampling (SIR) filter [36]Considering the robot that is described by the state space

4 Complexity

model the RPF algorithm operates Np particles to ap-proximate the posterior density

(e RPF algorithm also consists of two steps predictionand update similar to EKF But it is described statisticallywith the state transition probability density p(xk|xkminus1) andthe observation likelihood probability density of the statevector p(xk|y1k)

In the prediction step the predicted state vector is es-timated by the last time state Xk-1 and its measurement dataY1kminus 1

p Xk|Y1kminus1( 1113857 1113946 p Xk|Ykminus1( 1113857p Xkminus1|Y1kminus1( 1113857dXkminus1 (15)

With the latest measurement data Y1k the predictedstate can be updated then its approximated discrete pos-terior density can be obtained

p Xk|Y1k( 1113857 p Yk|Xk( 1113857p Xk|Y1kminus1( 1113857

p Yk|Y1kminus1( 1113857 (16)

where p(Yk|Y1kminus1) is defined as the normalization constantas shown in (20)

p Yk|Y1kminus1( 1113857 1113946 p Yk|Xk( 1113857p Xk|Y1kminus1( 1113857dXk (17)

With Monte Carlo method [37] the approximateddiscrete posterior density can be simplified as shown in thefollowing equation

p Xk|Y1k( 1113857 asymp 1113944

Np

i1ωi

kδ Xk minus Xik1113872 1113873 (18)

where δ(middot) is the Dirac delta function Np is the number ofparticles and ωi

k is the weight of the ith particle at kth it-eration p(Xk|Y1k) is the approximated discrete posteriordensity According to Bayesian estimation [37] all theparticles need to sample from the target posterior densitywith the knowledge of observation Y1k but this is unreal-istic Importance sampling [37] is introduced in the RPF

algorithm as well as the importance samples from a knownposterior density (ie importance density) In the identifi-cation the weight of the particles and the normalized weightcan be defined as (19) and (20) respectively

ωik

12π|R|

radic exp minus12

edk1113872 1113873

TR

minus 1e

dk1113874 1113875 (19)

1113957ωik

ωik

1113936Np

i1 ωik

(20)

where edk is the distance error of the ith particle According to

Bayesian estimation [37] the minimum mean square(MMS) estimate 1113954X

i

k can be described as the expectation ofidentified state vector combining all particles [38] as shownin the following equation

1113954Xi

k asymp E 1113954Xi

k1113874 1113875 asymp 1113944

Np

i11113957ωi

kXik (21)

In the identification model with distance error theweight of each particle is evaluated using its distance error(e particles with lower distance error Δl have higherweight that is

Δl lm minus lr

(22)

After a number of iterations the weight will occupy onone particle (almost close to one) and the weights of otherparticles are so small that they will be negligible due to thedegeneracy of particles [37] (e degeneracy problem in-dicates that abundant computation is applied to updatingparticles whose weight is almost zero (e negligible par-ticles with tiny weight make no contribution to approxi-mation of the target posterior density Resampling [39]should be applied to avoid this problem Note that aneffective sample size Neff is introduced to measure thedegeneracy of particles [39] If Neff is too small resamplewill be triggered

Measurementsystem

Statepredictor

Stateupdater

Covarianceupdater

Covariancepredictor Kalman gain

Pk

XkXk+1ˆ

Pk+1ˆ Kk+1

Xk+1

Pk+1

Yk+1

Figure 3 (e flow chart of the EKF algorithm

Complexity 5

Neff 1

1113936Np

i1 ωik1113872 1113873

2 (23)

Resampling is an approach to prevent particles from de-generacy which will introduce the decay of diversity of par-ticles in turn (is problem leads to poor approximation of theposterior density One way to solve this problem is modifyingthe resampling in which the particles are resampled from thecontinuous approximation of the posterior density [40]

(e approximated continuous posterior density in theRPF [40] is

p X0k|Y1k( 1113857 asymp 1113944

Np

i1ωi

kKh X0k minus Xi0k1113872 1113873 (24)

where

kh(x) 1

hnx

kx

h1113874 1113875 (25)

where kh(middot) is theKernel probability density function hgt0 is theKernel bandwidth and n is the dimension of the estimated statevector X kh will be used for resampling particles on continuousapproximated density With appropriate function kh and h themean integrated square error (MISE) between real and regu-larized empirical posterior densities in (24) can be minimized[38](e details of the RPF algorithm are illustrated in Figure 4

It is assumed that the coordinate of P0 is constant whichcan be regarded as one of identified parameters (e offset ofthe adapter of the measurement device should also be con-sidered In general a complete kinematic model includes 28unknown kinematic parameters which need to be identifiedin our framework (e distances used in identification shouldbe more than 10mm to increase the certainty of identifiedresults Figure 5 shows the flow chart of the identification

33 Determination of Redundant Parameters Singularityanalysis of the identified matrix J is very important inidentification calculation which can lead to nonconvergence

or inaccurate results Generally speaking the maximumnumber of parameters is 4r1 + 2p1 + 6 in a generic serial-linkrobot with r1 rotation axes and p1 translation axes [41] Toobtain reliable results with the identification algorithmredundant parameters must be determined and excludedfrom the error model before identification (e conditionnumber of the identified matrix is

κ(J) J Jminus1

(26)

where κ(middot) is the measure of sensitivity or stability of theidentified matrix with regard to a small change in the inputargument In high dimension full-parameter identificationwith a larger condition number the identified matrix J isoften nonfull rank In this paper the singular value de-composition (SVD) [42] is applied to reduce the conditionnumber and avoid singularity problem Considering totalJacobian matrix H [J1 J2 Jn] it can be written as

H U 1113944 VT (27)

where U isin R3ntimes3n V isin R25times25 and they are orthogonalmatrices H isin R3ntimesN Σ diag(σ1 σ2 σ3 σr) r is therank of H n is the number of calibration distances N is thenumber of parameters and in this case N is 25 Hence thenumber of the parameters that are redundant can be ob-tained as 25minus r

After SVD calculation r can be determined that isr 20(erefore there are 5 redundant parameters that needto be eliminated from the process of identificationAccording to the property of distance error the originalcoordinates of P0 are determined in the base frame FromFigure 1 it can be found that the four parameter errors injoint 1 may cause the overall change of the global coordinatesystem Hence P0 will change accordingly which makes itidentifiable (erefore there are 19 parameters including P0(xp0 yp0 zp0) which need to be identified and they can bewritten in the format of a vector

X Δα1 middot middot middot Δα5 Δa1 middot middot middot Δa4 Δθ2 middot middot middot Δθ5 Δd2 Δd4 Δd61113858 1113859T (28)

4 Verification

To verify the proposed kinematic parameter identificationmethod a 6-DOF industrial robot ER20-C10 was usedwhose maximum load and repeatability are 20 kg and008mm respectively (e nominal values of parameters areshown in Table 1 (e settings of the error for each pa-rameter are shown in Table 2 It should be noted that theerror of the redundant parameters is set as zero since theseparameters are not independent (e two terminationconditions of the iteration are as follows the maximumnumber of iterations is 10000 and the minimum objective

function is less than 10minus6 Both of the covariance matrices Qand P were initialized as 10minus4 times I3lowast3 in EKF and the noise ofthe covariance matrices of measurement R is set as10minus4

times I3lowast3 in the simulation (e bandwidth h is set as 042A random and bounded measurement disturbance[minus002mm 002mm] is added to each distance to simulatethe measuring error in the actual device

To verify the effectiveness of the proposed algorithm agroup of 100 poses were generated randomly for identifi-cation and verification Considering the workspace of therobot the six joint angles were chosen randomly from[minus30deg90deg] [minus50deg50deg] [minus45deg90deg] [minus100deg100deg] [minus90deg90deg] and

6 Complexity

[minus360deg360deg] respectively (e former 50 poses were used foridentification and the other 50 poses were used forvalidation

To compare the effectiveness three different algorithms(ie LS EKF and EKF+RPF) were used to identify thekinematic parameters that were compensated to the controlmodel of the robot(emaximum distance error was used asthe index to evaluate the results of compensation Figure 6 isthe distance error distribution chart of the robot aftercompensation by LS which shows that all the distance errorsare reduced greatly and the average error is from 04827mmto 01120mm by a reduction rate of 76 Figure 7 is the

distance error distribution chart of the robot after com-pensation by EKF which shows that the average error isreduced from 04827mm to 01352mm by a reduction rate

Xk

Yk

Xki

Xki

Xk = Xk + hDk τii i Draw τ k ~ kh fromepanechnikov kernel

iDraw Xk fromq (Xk | Xkndash1 Y1k)

ii˜

Calculate ωk andtotal weight ωk

i ii ˜ωk = (ωk)ndash1 ωk

Measurementsystem

Normalize

Yes

No

Calculate effectivesample size Neff

Calculate the empiricalcovariance matrix Sk

Neff lt threshold

Resample Compute Dk such that Dk Dk = SkT

Xk = ωk Xki˜ i

i = 1

M

Figure 4 (e flow chart of the RPF algorithm

Initialvalue

Robot(ER20-C10)

A set of jointangle and P0

Δ l = ĴΔxlmisin

N

Set of distancemeasurements

eisinΩ contains 19parameters error

EKF

RPF

Kinematicsparameters error

Compensate tonominal parameters

Calculate distanceaccuracy

Figure 5 (e flow chart of the proposed algorithm

Table 2 Settings of the kinematic parametersrsquo error

No of joints Δαiminus1 (deg) Δaiminus1 (mm) Δθi(deg) Δdi (mm)1 0 0 0 02 002 10 003 minus033 minus001 05 004 04 0015 minus04 006 0855 minus001 minus03 005 06 minus006 0 0 minus04

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 6(e distance error of the robot after compensation by LS

Complexity 7

of 72 Figure 8 is the distance error distribution chart of therobot after compensation by EKF+RPF which shows thatthe average error is reduced from 04827mm to 00780mmby a reduction rate of 84

Table 3 shows the statistics of the maximum mean andRMS values of the distance errors before and after com-pensation by the three algorithms By comparison one canfind that the effectiveness of RPF is the best of the three

5 Conclusion

(is paper proposed a novel kinematic calibration methodbased on the distance information for the industrial robot inwhich an EKF and RPF hybrid identification method wasadopted To simplify the data acquisition procedure and alsoavoid using expensive coordinate measuring instrumentsthe moving displacements of the end effector were used asthe reference to build the kinematic parameter error iden-tification model (en the error model was linearized bydifferentiating of the coefficient matrix By SVD of thecoefficient matrix the redundant parameters were deter-mined and removed from the list of parameters to beidentified (e preidentification of the parameters wasconducted with EKF by which we obtained a group of initialparameters of the kinematic model RPF was used to identifythe kinematic parameters of the linearized error model withthe initial parameters Comparative simulations were con-ducted based on the linearized kinematic error model (ecompensation results of the three algorithms are all effectivein improving the accuracy of the industrial robot but thehybrid algorithm proposed in this paper is more precise andfast in iteration calculation In the future work precisiondistance measuring device will be developed and effec-tiveness of the device and the proposed algorithm will befurther verified by experimental tests

Data Availability

(e data were curated by the authors and are available uponrequest

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Natural ScienceFoundation of China (Grant no 51865020) and the ScientificResearch Fund of Yunnan Education Department (Grant no2019J0046)

References

[1] Z Roth B Mooring and B Ravani ldquoAn overview of robotcalibrationrdquo IEEE Journal on Robotics and Automation vol 3no 5 pp 377ndash385 1987

[2] B Mooring M Driels and Z Roth Fundamentals of Ma-nipulator Calibration Wiley Hoboken NJ USA 1991

[3] W K Veitschegger and C-H Wu ldquoRobot calibration andcompensationrdquo IEEE Journal on Robotics and Automationvol 4 no 6 pp 643ndash656 1988

[4] K Schroeer S L Albright and M Grethlein ldquoCompleteminimal and model-continuous kinematic models for robotcalibrationrdquo Robotics and Computer-Integrated Manufactur-ing vol 13 pp 73ndash85 1997

[5] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanism based on matricesrdquo Transof theAsmeJournal of AppliedMechanics vol 22 pp 215ndash221 1955

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 7 (e distance error of the robot after compensation byEKF

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 8 (e distance error of the robot after compensation byEKF+RPF

Table 3 (e distance error of robot before and after compensationby 3 different algorithms

Algorithm Max(mm)

Mean(mm)

RMS(mm)

Before compensation mdash 12612 04827 05791

After compensationLS 03236 01120 01376EKF 03255 01352 01653

EKF+RPF 02404 00780 00936

8 Complexity

[6] S A Hayati ldquoRobot arm geometric link parameter estima-tionrdquo in Proceedings of the 22nd IEEE Conference on Decisionand Control San Antonio TX USA November 1983

[7] H W Stone and A C Sanderson ldquoStatistical performanceevaluation of the S-model arm signature identificationtechniquerdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Philadelphia PA USAAugust 1988

[8] H Zhuang Z S Roth and F Hamano ldquoA complete andparametrically continuous kinematic model for robot ma-nipulatorsrdquo IEEE Transactions on Robotics amp Automationvol 8 pp 451ndash463 1990

[9] T Sun Y Zhai Y Song and J Zhang ldquoKinematic calibrationof a 3-DoF rotational parallel manipulator using lasertrackerrdquo Robotics and Computer-Integrated Manufacturingvol 41 pp 78ndash91 2016

[10] P Yuan D Chen T Wang S Cao Y Cai and L Xue ldquoAcompensation method based on extreme learning machine toenhance absolute position accuracy for aviation drilling ro-botrdquo Advances in Mechanical Engineering vol 10 2018

[11] G Gao F Liu H San XWu andWWang ldquoHybrid optimalkinematic parameter identification for an industrial robotbased on BPNN-PSOrdquo Complexity vol 2018 Article ID4258676 2018

[12] Y Bai H Zhuang and Z S Roth ldquoExperiment study ofPUMA robot calibration using a laser tracking systemrdquo inProceeding of the IEEE International Workshop on SoftComputing in Industrial Applications Binghamton NY USASeptember 2003

[13] X Zhang Y Song Y Yang and H Pan ldquoStereo vision basedautonomous robot calibrationrdquo Robotics and AutonomousSystems vol 93 pp 43ndash51 2017

[14] T Messay R Ordontildeez and E Marcil ldquoComputationallyefficient and robust kinematic calibration methodologies andtheir application to industrial robotsrdquo Robotics and Com-puter-Integrated Manufacturing vol 37 pp 33ndash48 2016

[15] G Gatti and G Danieli ldquoA practical approach to compensatefor geometric errors in measuring arms application to a six-degree-of-freedom kinematic structurerdquo Measurement Sci-ence and Technology vol 19 Article ID 015107 2008

[16] N Schillreff M Nykolaychuk and F Ortmeier ldquoTowardshigh accuracy robot-assisted surgerylowastlowast this work was partlyfunded by the federal ministry of education and Research inGermany within the research campus STIMULATE undergrant No 13GW0095Ardquo IFAC-PapersOnLine vol 50 no 1pp 5666ndash5671 2017

[17] A Nubiola and I A Bonev ldquoAbsolute calibration of an ABBIRB 1600 robot using a laser trackerrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 236ndash245 2013

[18] A Nubiola M Slamani A Joubair and I A BonevldquoComparison of two calibrationmethods for a small industrialrobot based on an optical CMM and a laser trackerrdquo Roboticavol 32 no 3 pp 447ndash466 2014

[19] G Gao H Zhang H San X Wu and W Wang ldquoModelingand error compensation of robotic articulated arm coordinatemeasuring machines using BP neural networkrdquo Complexityvol 2017 Article ID 5156264 2017

[20] A Nubiola M Slamani and I A Bonev ldquoA new method formeasuring a large set of poses with a single telescopingballbarrdquo Precision Engineering vol 37 no 2 pp 451ndash4602013

[21] H-N Nguyen J Zhou andH-J Kang ldquoA calibrationmethodfor enhancing robot accuracy through integration of an

extended Kalman filter algorithm and an artificial neuralnetworkrdquo Neurocomputing vol 151 pp 996ndash1005 2015

[22] J Zhang X Wang K Wen Y Zhou Y Yue and J Yang ldquoAsimple and rapid calibration methodology for industrial robotbased on geometric constraint and two-step errorrdquo IndustrialRobot An International Journal vol 45 no 6 pp 715ndash7212018

[23] G Zhao P Zhang GMa andW Xiao ldquoSystem identificationof the nonlinear residual errors of an industrial robot usingmassive measurementsrdquo Robotics and Computer-IntegratedManufacturing vol 59 pp 104ndash114 2019

[24] K Wang G Liu Q Tao and M Zhai ldquoEfficient parametersestimation method for the separable nonlinear least squaresproblemrdquo Complexity vol 2020 Article ID 9619427 17 pages2020

[25] I Dattner H Ship and E O Voit ldquoSeparable nonlinear least-squares parameter estimation for complex dynamic systemsrdquoComplexity vol 2020 Article ID 6403641 18 pages 2020

[26] J Ding ldquo(e hierarchical iterative identification algorithm formulti-input-output-error systems with autoregressive noiserdquoComplexity vol 2017 Article ID 5292894 2017

[27] A Omodei G Legnani and R Adamini ldquo(ree methodol-ogies for the calibration of industrial manipulators experi-mental results on a SCARA robotrdquo Journal of Robotic Systemsvol 17 no 6 pp 291ndash307 2000

[28] Y Ho and R Lee ldquoA Bayesian approach to problems instochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 333ndash339 1964

[29] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processingvol 50 no 2 pp 174ndash188 2002

[30] K Schroer Precision and Calibration John Wiley amp SonsHoboken NJ USA 2007

[31] B Greenway ldquoRobot accuracyrdquo Industrial Robot An Inter-national Journal vol 27 no 4 pp 257ndash265 2000

[32] C Musso N Oudjane and F Le Gland ldquoImproving regu-larised particle filtersrdquo in Sequential Monte Carlo Methods inPractice A Doucet N de Freitas and N Gordon Edspp 247ndash271 Springer Berlin Germany 2001

[33] J J Craig Introduction to Robotics Mechanics and ControlPearson Education India Uttar Pradesh India 3rd edition2009

[34] Z Wang H Xu G Chen R Sun and L Sun ldquoA distanceerror based industrial robot kinematic calibration methodrdquoIndustrial Robot An International Journal vol 41 pp 439ndash446 2014

[35] K Senne ldquoStochastic processes and filtering theoryrdquo IEEETransactions on Automatic Control vol 17 no 5 pp 752-7531972

[36] J Carpenter P Clifford and P Fearnhead ldquoImproved particlefilter for nonlinear problemsrdquo IEE Proceedings-Radar Sonarand Navigation vol 146 no 1 pp 2ndash7 1999

[37] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[38] S Godsill and T Clapp Improvement Strategies for MonteCarlo Particle Filters Springer Berlin Germany 2001

[39] S Liu and R Chen ldquoSequential Monte Carlo methods fordynamic systemsrdquo Journal of the American Statistical Asso-ciation vol 93 1998

[40] C Musso N Oudjane and F L Gland ldquoImproving regu-larised particle filtersrdquo Statistics for Engineering amp Informa-tion Science vol 28 2001

Complexity 9

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity

model the RPF algorithm operates Np particles to ap-proximate the posterior density

(e RPF algorithm also consists of two steps predictionand update similar to EKF But it is described statisticallywith the state transition probability density p(xk|xkminus1) andthe observation likelihood probability density of the statevector p(xk|y1k)

In the prediction step the predicted state vector is es-timated by the last time state Xk-1 and its measurement dataY1kminus 1

p Xk|Y1kminus1( 1113857 1113946 p Xk|Ykminus1( 1113857p Xkminus1|Y1kminus1( 1113857dXkminus1 (15)

With the latest measurement data Y1k the predictedstate can be updated then its approximated discrete pos-terior density can be obtained

p Xk|Y1k( 1113857 p Yk|Xk( 1113857p Xk|Y1kminus1( 1113857

p Yk|Y1kminus1( 1113857 (16)

where p(Yk|Y1kminus1) is defined as the normalization constantas shown in (20)

p Yk|Y1kminus1( 1113857 1113946 p Yk|Xk( 1113857p Xk|Y1kminus1( 1113857dXk (17)

With Monte Carlo method [37] the approximateddiscrete posterior density can be simplified as shown in thefollowing equation

p Xk|Y1k( 1113857 asymp 1113944

Np

i1ωi

kδ Xk minus Xik1113872 1113873 (18)

where δ(middot) is the Dirac delta function Np is the number ofparticles and ωi

k is the weight of the ith particle at kth it-eration p(Xk|Y1k) is the approximated discrete posteriordensity According to Bayesian estimation [37] all theparticles need to sample from the target posterior densitywith the knowledge of observation Y1k but this is unreal-istic Importance sampling [37] is introduced in the RPF

algorithm as well as the importance samples from a knownposterior density (ie importance density) In the identifi-cation the weight of the particles and the normalized weightcan be defined as (19) and (20) respectively

ωik

12π|R|

radic exp minus12

edk1113872 1113873

TR

minus 1e

dk1113874 1113875 (19)

1113957ωik

ωik

1113936Np

i1 ωik

(20)

where edk is the distance error of the ith particle According to

Bayesian estimation [37] the minimum mean square(MMS) estimate 1113954X

i

k can be described as the expectation ofidentified state vector combining all particles [38] as shownin the following equation

1113954Xi

k asymp E 1113954Xi

k1113874 1113875 asymp 1113944

Np

i11113957ωi

kXik (21)

In the identification model with distance error theweight of each particle is evaluated using its distance error(e particles with lower distance error Δl have higherweight that is

Δl lm minus lr

(22)

After a number of iterations the weight will occupy onone particle (almost close to one) and the weights of otherparticles are so small that they will be negligible due to thedegeneracy of particles [37] (e degeneracy problem in-dicates that abundant computation is applied to updatingparticles whose weight is almost zero (e negligible par-ticles with tiny weight make no contribution to approxi-mation of the target posterior density Resampling [39]should be applied to avoid this problem Note that aneffective sample size Neff is introduced to measure thedegeneracy of particles [39] If Neff is too small resamplewill be triggered

Measurementsystem

Statepredictor

Stateupdater

Covarianceupdater

Covariancepredictor Kalman gain

Pk

XkXk+1ˆ

Pk+1ˆ Kk+1

Xk+1

Pk+1

Yk+1

Figure 3 (e flow chart of the EKF algorithm

Complexity 5

Neff 1

1113936Np

i1 ωik1113872 1113873

2 (23)

Resampling is an approach to prevent particles from de-generacy which will introduce the decay of diversity of par-ticles in turn (is problem leads to poor approximation of theposterior density One way to solve this problem is modifyingthe resampling in which the particles are resampled from thecontinuous approximation of the posterior density [40]

(e approximated continuous posterior density in theRPF [40] is

p X0k|Y1k( 1113857 asymp 1113944

Np

i1ωi

kKh X0k minus Xi0k1113872 1113873 (24)

where

kh(x) 1

hnx

kx

h1113874 1113875 (25)

where kh(middot) is theKernel probability density function hgt0 is theKernel bandwidth and n is the dimension of the estimated statevector X kh will be used for resampling particles on continuousapproximated density With appropriate function kh and h themean integrated square error (MISE) between real and regu-larized empirical posterior densities in (24) can be minimized[38](e details of the RPF algorithm are illustrated in Figure 4

It is assumed that the coordinate of P0 is constant whichcan be regarded as one of identified parameters (e offset ofthe adapter of the measurement device should also be con-sidered In general a complete kinematic model includes 28unknown kinematic parameters which need to be identifiedin our framework (e distances used in identification shouldbe more than 10mm to increase the certainty of identifiedresults Figure 5 shows the flow chart of the identification

33 Determination of Redundant Parameters Singularityanalysis of the identified matrix J is very important inidentification calculation which can lead to nonconvergence

or inaccurate results Generally speaking the maximumnumber of parameters is 4r1 + 2p1 + 6 in a generic serial-linkrobot with r1 rotation axes and p1 translation axes [41] Toobtain reliable results with the identification algorithmredundant parameters must be determined and excludedfrom the error model before identification (e conditionnumber of the identified matrix is

κ(J) J Jminus1

(26)

where κ(middot) is the measure of sensitivity or stability of theidentified matrix with regard to a small change in the inputargument In high dimension full-parameter identificationwith a larger condition number the identified matrix J isoften nonfull rank In this paper the singular value de-composition (SVD) [42] is applied to reduce the conditionnumber and avoid singularity problem Considering totalJacobian matrix H [J1 J2 Jn] it can be written as

H U 1113944 VT (27)

where U isin R3ntimes3n V isin R25times25 and they are orthogonalmatrices H isin R3ntimesN Σ diag(σ1 σ2 σ3 σr) r is therank of H n is the number of calibration distances N is thenumber of parameters and in this case N is 25 Hence thenumber of the parameters that are redundant can be ob-tained as 25minus r

After SVD calculation r can be determined that isr 20(erefore there are 5 redundant parameters that needto be eliminated from the process of identificationAccording to the property of distance error the originalcoordinates of P0 are determined in the base frame FromFigure 1 it can be found that the four parameter errors injoint 1 may cause the overall change of the global coordinatesystem Hence P0 will change accordingly which makes itidentifiable (erefore there are 19 parameters including P0(xp0 yp0 zp0) which need to be identified and they can bewritten in the format of a vector

X Δα1 middot middot middot Δα5 Δa1 middot middot middot Δa4 Δθ2 middot middot middot Δθ5 Δd2 Δd4 Δd61113858 1113859T (28)

4 Verification

To verify the proposed kinematic parameter identificationmethod a 6-DOF industrial robot ER20-C10 was usedwhose maximum load and repeatability are 20 kg and008mm respectively (e nominal values of parameters areshown in Table 1 (e settings of the error for each pa-rameter are shown in Table 2 It should be noted that theerror of the redundant parameters is set as zero since theseparameters are not independent (e two terminationconditions of the iteration are as follows the maximumnumber of iterations is 10000 and the minimum objective

function is less than 10minus6 Both of the covariance matrices Qand P were initialized as 10minus4 times I3lowast3 in EKF and the noise ofthe covariance matrices of measurement R is set as10minus4

times I3lowast3 in the simulation (e bandwidth h is set as 042A random and bounded measurement disturbance[minus002mm 002mm] is added to each distance to simulatethe measuring error in the actual device

To verify the effectiveness of the proposed algorithm agroup of 100 poses were generated randomly for identifi-cation and verification Considering the workspace of therobot the six joint angles were chosen randomly from[minus30deg90deg] [minus50deg50deg] [minus45deg90deg] [minus100deg100deg] [minus90deg90deg] and

6 Complexity

[minus360deg360deg] respectively (e former 50 poses were used foridentification and the other 50 poses were used forvalidation

To compare the effectiveness three different algorithms(ie LS EKF and EKF+RPF) were used to identify thekinematic parameters that were compensated to the controlmodel of the robot(emaximum distance error was used asthe index to evaluate the results of compensation Figure 6 isthe distance error distribution chart of the robot aftercompensation by LS which shows that all the distance errorsare reduced greatly and the average error is from 04827mmto 01120mm by a reduction rate of 76 Figure 7 is the

distance error distribution chart of the robot after com-pensation by EKF which shows that the average error isreduced from 04827mm to 01352mm by a reduction rate

Xk

Yk

Xki

Xki

Xk = Xk + hDk τii i Draw τ k ~ kh fromepanechnikov kernel

iDraw Xk fromq (Xk | Xkndash1 Y1k)

ii˜

Calculate ωk andtotal weight ωk

i ii ˜ωk = (ωk)ndash1 ωk

Measurementsystem

Normalize

Yes

No

Calculate effectivesample size Neff

Calculate the empiricalcovariance matrix Sk

Neff lt threshold

Resample Compute Dk such that Dk Dk = SkT

Xk = ωk Xki˜ i

i = 1

M

Figure 4 (e flow chart of the RPF algorithm

Initialvalue

Robot(ER20-C10)

A set of jointangle and P0

Δ l = ĴΔxlmisin

N

Set of distancemeasurements

eisinΩ contains 19parameters error

EKF

RPF

Kinematicsparameters error

Compensate tonominal parameters

Calculate distanceaccuracy

Figure 5 (e flow chart of the proposed algorithm

Table 2 Settings of the kinematic parametersrsquo error

No of joints Δαiminus1 (deg) Δaiminus1 (mm) Δθi(deg) Δdi (mm)1 0 0 0 02 002 10 003 minus033 minus001 05 004 04 0015 minus04 006 0855 minus001 minus03 005 06 minus006 0 0 minus04

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 6(e distance error of the robot after compensation by LS

Complexity 7

of 72 Figure 8 is the distance error distribution chart of therobot after compensation by EKF+RPF which shows thatthe average error is reduced from 04827mm to 00780mmby a reduction rate of 84

Table 3 shows the statistics of the maximum mean andRMS values of the distance errors before and after com-pensation by the three algorithms By comparison one canfind that the effectiveness of RPF is the best of the three

5 Conclusion

(is paper proposed a novel kinematic calibration methodbased on the distance information for the industrial robot inwhich an EKF and RPF hybrid identification method wasadopted To simplify the data acquisition procedure and alsoavoid using expensive coordinate measuring instrumentsthe moving displacements of the end effector were used asthe reference to build the kinematic parameter error iden-tification model (en the error model was linearized bydifferentiating of the coefficient matrix By SVD of thecoefficient matrix the redundant parameters were deter-mined and removed from the list of parameters to beidentified (e preidentification of the parameters wasconducted with EKF by which we obtained a group of initialparameters of the kinematic model RPF was used to identifythe kinematic parameters of the linearized error model withthe initial parameters Comparative simulations were con-ducted based on the linearized kinematic error model (ecompensation results of the three algorithms are all effectivein improving the accuracy of the industrial robot but thehybrid algorithm proposed in this paper is more precise andfast in iteration calculation In the future work precisiondistance measuring device will be developed and effec-tiveness of the device and the proposed algorithm will befurther verified by experimental tests

Data Availability

(e data were curated by the authors and are available uponrequest

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Natural ScienceFoundation of China (Grant no 51865020) and the ScientificResearch Fund of Yunnan Education Department (Grant no2019J0046)

References

[1] Z Roth B Mooring and B Ravani ldquoAn overview of robotcalibrationrdquo IEEE Journal on Robotics and Automation vol 3no 5 pp 377ndash385 1987

[2] B Mooring M Driels and Z Roth Fundamentals of Ma-nipulator Calibration Wiley Hoboken NJ USA 1991

[3] W K Veitschegger and C-H Wu ldquoRobot calibration andcompensationrdquo IEEE Journal on Robotics and Automationvol 4 no 6 pp 643ndash656 1988

[4] K Schroeer S L Albright and M Grethlein ldquoCompleteminimal and model-continuous kinematic models for robotcalibrationrdquo Robotics and Computer-Integrated Manufactur-ing vol 13 pp 73ndash85 1997

[5] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanism based on matricesrdquo Transof theAsmeJournal of AppliedMechanics vol 22 pp 215ndash221 1955

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 7 (e distance error of the robot after compensation byEKF

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 8 (e distance error of the robot after compensation byEKF+RPF

Table 3 (e distance error of robot before and after compensationby 3 different algorithms

Algorithm Max(mm)

Mean(mm)

RMS(mm)

Before compensation mdash 12612 04827 05791

After compensationLS 03236 01120 01376EKF 03255 01352 01653

EKF+RPF 02404 00780 00936

8 Complexity

[6] S A Hayati ldquoRobot arm geometric link parameter estima-tionrdquo in Proceedings of the 22nd IEEE Conference on Decisionand Control San Antonio TX USA November 1983

[7] H W Stone and A C Sanderson ldquoStatistical performanceevaluation of the S-model arm signature identificationtechniquerdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Philadelphia PA USAAugust 1988

[8] H Zhuang Z S Roth and F Hamano ldquoA complete andparametrically continuous kinematic model for robot ma-nipulatorsrdquo IEEE Transactions on Robotics amp Automationvol 8 pp 451ndash463 1990

[9] T Sun Y Zhai Y Song and J Zhang ldquoKinematic calibrationof a 3-DoF rotational parallel manipulator using lasertrackerrdquo Robotics and Computer-Integrated Manufacturingvol 41 pp 78ndash91 2016

[10] P Yuan D Chen T Wang S Cao Y Cai and L Xue ldquoAcompensation method based on extreme learning machine toenhance absolute position accuracy for aviation drilling ro-botrdquo Advances in Mechanical Engineering vol 10 2018

[11] G Gao F Liu H San XWu andWWang ldquoHybrid optimalkinematic parameter identification for an industrial robotbased on BPNN-PSOrdquo Complexity vol 2018 Article ID4258676 2018

[12] Y Bai H Zhuang and Z S Roth ldquoExperiment study ofPUMA robot calibration using a laser tracking systemrdquo inProceeding of the IEEE International Workshop on SoftComputing in Industrial Applications Binghamton NY USASeptember 2003

[13] X Zhang Y Song Y Yang and H Pan ldquoStereo vision basedautonomous robot calibrationrdquo Robotics and AutonomousSystems vol 93 pp 43ndash51 2017

[14] T Messay R Ordontildeez and E Marcil ldquoComputationallyefficient and robust kinematic calibration methodologies andtheir application to industrial robotsrdquo Robotics and Com-puter-Integrated Manufacturing vol 37 pp 33ndash48 2016

[15] G Gatti and G Danieli ldquoA practical approach to compensatefor geometric errors in measuring arms application to a six-degree-of-freedom kinematic structurerdquo Measurement Sci-ence and Technology vol 19 Article ID 015107 2008

[16] N Schillreff M Nykolaychuk and F Ortmeier ldquoTowardshigh accuracy robot-assisted surgerylowastlowast this work was partlyfunded by the federal ministry of education and Research inGermany within the research campus STIMULATE undergrant No 13GW0095Ardquo IFAC-PapersOnLine vol 50 no 1pp 5666ndash5671 2017

[17] A Nubiola and I A Bonev ldquoAbsolute calibration of an ABBIRB 1600 robot using a laser trackerrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 236ndash245 2013

[18] A Nubiola M Slamani A Joubair and I A BonevldquoComparison of two calibrationmethods for a small industrialrobot based on an optical CMM and a laser trackerrdquo Roboticavol 32 no 3 pp 447ndash466 2014

[19] G Gao H Zhang H San X Wu and W Wang ldquoModelingand error compensation of robotic articulated arm coordinatemeasuring machines using BP neural networkrdquo Complexityvol 2017 Article ID 5156264 2017

[20] A Nubiola M Slamani and I A Bonev ldquoA new method formeasuring a large set of poses with a single telescopingballbarrdquo Precision Engineering vol 37 no 2 pp 451ndash4602013

[21] H-N Nguyen J Zhou andH-J Kang ldquoA calibrationmethodfor enhancing robot accuracy through integration of an

extended Kalman filter algorithm and an artificial neuralnetworkrdquo Neurocomputing vol 151 pp 996ndash1005 2015

[22] J Zhang X Wang K Wen Y Zhou Y Yue and J Yang ldquoAsimple and rapid calibration methodology for industrial robotbased on geometric constraint and two-step errorrdquo IndustrialRobot An International Journal vol 45 no 6 pp 715ndash7212018

[23] G Zhao P Zhang GMa andW Xiao ldquoSystem identificationof the nonlinear residual errors of an industrial robot usingmassive measurementsrdquo Robotics and Computer-IntegratedManufacturing vol 59 pp 104ndash114 2019

[24] K Wang G Liu Q Tao and M Zhai ldquoEfficient parametersestimation method for the separable nonlinear least squaresproblemrdquo Complexity vol 2020 Article ID 9619427 17 pages2020

[25] I Dattner H Ship and E O Voit ldquoSeparable nonlinear least-squares parameter estimation for complex dynamic systemsrdquoComplexity vol 2020 Article ID 6403641 18 pages 2020

[26] J Ding ldquo(e hierarchical iterative identification algorithm formulti-input-output-error systems with autoregressive noiserdquoComplexity vol 2017 Article ID 5292894 2017

[27] A Omodei G Legnani and R Adamini ldquo(ree methodol-ogies for the calibration of industrial manipulators experi-mental results on a SCARA robotrdquo Journal of Robotic Systemsvol 17 no 6 pp 291ndash307 2000

[28] Y Ho and R Lee ldquoA Bayesian approach to problems instochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 333ndash339 1964

[29] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processingvol 50 no 2 pp 174ndash188 2002

[30] K Schroer Precision and Calibration John Wiley amp SonsHoboken NJ USA 2007

[31] B Greenway ldquoRobot accuracyrdquo Industrial Robot An Inter-national Journal vol 27 no 4 pp 257ndash265 2000

[32] C Musso N Oudjane and F Le Gland ldquoImproving regu-larised particle filtersrdquo in Sequential Monte Carlo Methods inPractice A Doucet N de Freitas and N Gordon Edspp 247ndash271 Springer Berlin Germany 2001

[33] J J Craig Introduction to Robotics Mechanics and ControlPearson Education India Uttar Pradesh India 3rd edition2009

[34] Z Wang H Xu G Chen R Sun and L Sun ldquoA distanceerror based industrial robot kinematic calibration methodrdquoIndustrial Robot An International Journal vol 41 pp 439ndash446 2014

[35] K Senne ldquoStochastic processes and filtering theoryrdquo IEEETransactions on Automatic Control vol 17 no 5 pp 752-7531972

[36] J Carpenter P Clifford and P Fearnhead ldquoImproved particlefilter for nonlinear problemsrdquo IEE Proceedings-Radar Sonarand Navigation vol 146 no 1 pp 2ndash7 1999

[37] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[38] S Godsill and T Clapp Improvement Strategies for MonteCarlo Particle Filters Springer Berlin Germany 2001

[39] S Liu and R Chen ldquoSequential Monte Carlo methods fordynamic systemsrdquo Journal of the American Statistical Asso-ciation vol 93 1998

[40] C Musso N Oudjane and F L Gland ldquoImproving regu-larised particle filtersrdquo Statistics for Engineering amp Informa-tion Science vol 28 2001

Complexity 9

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity

Neff 1

1113936Np

i1 ωik1113872 1113873

2 (23)

Resampling is an approach to prevent particles from de-generacy which will introduce the decay of diversity of par-ticles in turn (is problem leads to poor approximation of theposterior density One way to solve this problem is modifyingthe resampling in which the particles are resampled from thecontinuous approximation of the posterior density [40]

(e approximated continuous posterior density in theRPF [40] is

p X0k|Y1k( 1113857 asymp 1113944

Np

i1ωi

kKh X0k minus Xi0k1113872 1113873 (24)

where

kh(x) 1

hnx

kx

h1113874 1113875 (25)

where kh(middot) is theKernel probability density function hgt0 is theKernel bandwidth and n is the dimension of the estimated statevector X kh will be used for resampling particles on continuousapproximated density With appropriate function kh and h themean integrated square error (MISE) between real and regu-larized empirical posterior densities in (24) can be minimized[38](e details of the RPF algorithm are illustrated in Figure 4

It is assumed that the coordinate of P0 is constant whichcan be regarded as one of identified parameters (e offset ofthe adapter of the measurement device should also be con-sidered In general a complete kinematic model includes 28unknown kinematic parameters which need to be identifiedin our framework (e distances used in identification shouldbe more than 10mm to increase the certainty of identifiedresults Figure 5 shows the flow chart of the identification

33 Determination of Redundant Parameters Singularityanalysis of the identified matrix J is very important inidentification calculation which can lead to nonconvergence

or inaccurate results Generally speaking the maximumnumber of parameters is 4r1 + 2p1 + 6 in a generic serial-linkrobot with r1 rotation axes and p1 translation axes [41] Toobtain reliable results with the identification algorithmredundant parameters must be determined and excludedfrom the error model before identification (e conditionnumber of the identified matrix is

κ(J) J Jminus1

(26)

where κ(middot) is the measure of sensitivity or stability of theidentified matrix with regard to a small change in the inputargument In high dimension full-parameter identificationwith a larger condition number the identified matrix J isoften nonfull rank In this paper the singular value de-composition (SVD) [42] is applied to reduce the conditionnumber and avoid singularity problem Considering totalJacobian matrix H [J1 J2 Jn] it can be written as

H U 1113944 VT (27)

where U isin R3ntimes3n V isin R25times25 and they are orthogonalmatrices H isin R3ntimesN Σ diag(σ1 σ2 σ3 σr) r is therank of H n is the number of calibration distances N is thenumber of parameters and in this case N is 25 Hence thenumber of the parameters that are redundant can be ob-tained as 25minus r

After SVD calculation r can be determined that isr 20(erefore there are 5 redundant parameters that needto be eliminated from the process of identificationAccording to the property of distance error the originalcoordinates of P0 are determined in the base frame FromFigure 1 it can be found that the four parameter errors injoint 1 may cause the overall change of the global coordinatesystem Hence P0 will change accordingly which makes itidentifiable (erefore there are 19 parameters including P0(xp0 yp0 zp0) which need to be identified and they can bewritten in the format of a vector

X Δα1 middot middot middot Δα5 Δa1 middot middot middot Δa4 Δθ2 middot middot middot Δθ5 Δd2 Δd4 Δd61113858 1113859T (28)

4 Verification

To verify the proposed kinematic parameter identificationmethod a 6-DOF industrial robot ER20-C10 was usedwhose maximum load and repeatability are 20 kg and008mm respectively (e nominal values of parameters areshown in Table 1 (e settings of the error for each pa-rameter are shown in Table 2 It should be noted that theerror of the redundant parameters is set as zero since theseparameters are not independent (e two terminationconditions of the iteration are as follows the maximumnumber of iterations is 10000 and the minimum objective

function is less than 10minus6 Both of the covariance matrices Qand P were initialized as 10minus4 times I3lowast3 in EKF and the noise ofthe covariance matrices of measurement R is set as10minus4

times I3lowast3 in the simulation (e bandwidth h is set as 042A random and bounded measurement disturbance[minus002mm 002mm] is added to each distance to simulatethe measuring error in the actual device

To verify the effectiveness of the proposed algorithm agroup of 100 poses were generated randomly for identifi-cation and verification Considering the workspace of therobot the six joint angles were chosen randomly from[minus30deg90deg] [minus50deg50deg] [minus45deg90deg] [minus100deg100deg] [minus90deg90deg] and

6 Complexity

[minus360deg360deg] respectively (e former 50 poses were used foridentification and the other 50 poses were used forvalidation

To compare the effectiveness three different algorithms(ie LS EKF and EKF+RPF) were used to identify thekinematic parameters that were compensated to the controlmodel of the robot(emaximum distance error was used asthe index to evaluate the results of compensation Figure 6 isthe distance error distribution chart of the robot aftercompensation by LS which shows that all the distance errorsare reduced greatly and the average error is from 04827mmto 01120mm by a reduction rate of 76 Figure 7 is the

distance error distribution chart of the robot after com-pensation by EKF which shows that the average error isreduced from 04827mm to 01352mm by a reduction rate

Xk

Yk

Xki

Xki

Xk = Xk + hDk τii i Draw τ k ~ kh fromepanechnikov kernel

iDraw Xk fromq (Xk | Xkndash1 Y1k)

ii˜

Calculate ωk andtotal weight ωk

i ii ˜ωk = (ωk)ndash1 ωk

Measurementsystem

Normalize

Yes

No

Calculate effectivesample size Neff

Calculate the empiricalcovariance matrix Sk

Neff lt threshold

Resample Compute Dk such that Dk Dk = SkT

Xk = ωk Xki˜ i

i = 1

M

Figure 4 (e flow chart of the RPF algorithm

Initialvalue

Robot(ER20-C10)

A set of jointangle and P0

Δ l = ĴΔxlmisin

N

Set of distancemeasurements

eisinΩ contains 19parameters error

EKF

RPF

Kinematicsparameters error

Compensate tonominal parameters

Calculate distanceaccuracy

Figure 5 (e flow chart of the proposed algorithm

Table 2 Settings of the kinematic parametersrsquo error

No of joints Δαiminus1 (deg) Δaiminus1 (mm) Δθi(deg) Δdi (mm)1 0 0 0 02 002 10 003 minus033 minus001 05 004 04 0015 minus04 006 0855 minus001 minus03 005 06 minus006 0 0 minus04

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 6(e distance error of the robot after compensation by LS

Complexity 7

of 72 Figure 8 is the distance error distribution chart of therobot after compensation by EKF+RPF which shows thatthe average error is reduced from 04827mm to 00780mmby a reduction rate of 84

Table 3 shows the statistics of the maximum mean andRMS values of the distance errors before and after com-pensation by the three algorithms By comparison one canfind that the effectiveness of RPF is the best of the three

5 Conclusion

(is paper proposed a novel kinematic calibration methodbased on the distance information for the industrial robot inwhich an EKF and RPF hybrid identification method wasadopted To simplify the data acquisition procedure and alsoavoid using expensive coordinate measuring instrumentsthe moving displacements of the end effector were used asthe reference to build the kinematic parameter error iden-tification model (en the error model was linearized bydifferentiating of the coefficient matrix By SVD of thecoefficient matrix the redundant parameters were deter-mined and removed from the list of parameters to beidentified (e preidentification of the parameters wasconducted with EKF by which we obtained a group of initialparameters of the kinematic model RPF was used to identifythe kinematic parameters of the linearized error model withthe initial parameters Comparative simulations were con-ducted based on the linearized kinematic error model (ecompensation results of the three algorithms are all effectivein improving the accuracy of the industrial robot but thehybrid algorithm proposed in this paper is more precise andfast in iteration calculation In the future work precisiondistance measuring device will be developed and effec-tiveness of the device and the proposed algorithm will befurther verified by experimental tests

Data Availability

(e data were curated by the authors and are available uponrequest

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Natural ScienceFoundation of China (Grant no 51865020) and the ScientificResearch Fund of Yunnan Education Department (Grant no2019J0046)

References

[1] Z Roth B Mooring and B Ravani ldquoAn overview of robotcalibrationrdquo IEEE Journal on Robotics and Automation vol 3no 5 pp 377ndash385 1987

[2] B Mooring M Driels and Z Roth Fundamentals of Ma-nipulator Calibration Wiley Hoboken NJ USA 1991

[3] W K Veitschegger and C-H Wu ldquoRobot calibration andcompensationrdquo IEEE Journal on Robotics and Automationvol 4 no 6 pp 643ndash656 1988

[4] K Schroeer S L Albright and M Grethlein ldquoCompleteminimal and model-continuous kinematic models for robotcalibrationrdquo Robotics and Computer-Integrated Manufactur-ing vol 13 pp 73ndash85 1997

[5] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanism based on matricesrdquo Transof theAsmeJournal of AppliedMechanics vol 22 pp 215ndash221 1955

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 7 (e distance error of the robot after compensation byEKF

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 8 (e distance error of the robot after compensation byEKF+RPF

Table 3 (e distance error of robot before and after compensationby 3 different algorithms

Algorithm Max(mm)

Mean(mm)

RMS(mm)

Before compensation mdash 12612 04827 05791

After compensationLS 03236 01120 01376EKF 03255 01352 01653

EKF+RPF 02404 00780 00936

8 Complexity

[6] S A Hayati ldquoRobot arm geometric link parameter estima-tionrdquo in Proceedings of the 22nd IEEE Conference on Decisionand Control San Antonio TX USA November 1983

[7] H W Stone and A C Sanderson ldquoStatistical performanceevaluation of the S-model arm signature identificationtechniquerdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Philadelphia PA USAAugust 1988

[8] H Zhuang Z S Roth and F Hamano ldquoA complete andparametrically continuous kinematic model for robot ma-nipulatorsrdquo IEEE Transactions on Robotics amp Automationvol 8 pp 451ndash463 1990

[9] T Sun Y Zhai Y Song and J Zhang ldquoKinematic calibrationof a 3-DoF rotational parallel manipulator using lasertrackerrdquo Robotics and Computer-Integrated Manufacturingvol 41 pp 78ndash91 2016

[10] P Yuan D Chen T Wang S Cao Y Cai and L Xue ldquoAcompensation method based on extreme learning machine toenhance absolute position accuracy for aviation drilling ro-botrdquo Advances in Mechanical Engineering vol 10 2018

[11] G Gao F Liu H San XWu andWWang ldquoHybrid optimalkinematic parameter identification for an industrial robotbased on BPNN-PSOrdquo Complexity vol 2018 Article ID4258676 2018

[12] Y Bai H Zhuang and Z S Roth ldquoExperiment study ofPUMA robot calibration using a laser tracking systemrdquo inProceeding of the IEEE International Workshop on SoftComputing in Industrial Applications Binghamton NY USASeptember 2003

[13] X Zhang Y Song Y Yang and H Pan ldquoStereo vision basedautonomous robot calibrationrdquo Robotics and AutonomousSystems vol 93 pp 43ndash51 2017

[14] T Messay R Ordontildeez and E Marcil ldquoComputationallyefficient and robust kinematic calibration methodologies andtheir application to industrial robotsrdquo Robotics and Com-puter-Integrated Manufacturing vol 37 pp 33ndash48 2016

[15] G Gatti and G Danieli ldquoA practical approach to compensatefor geometric errors in measuring arms application to a six-degree-of-freedom kinematic structurerdquo Measurement Sci-ence and Technology vol 19 Article ID 015107 2008

[16] N Schillreff M Nykolaychuk and F Ortmeier ldquoTowardshigh accuracy robot-assisted surgerylowastlowast this work was partlyfunded by the federal ministry of education and Research inGermany within the research campus STIMULATE undergrant No 13GW0095Ardquo IFAC-PapersOnLine vol 50 no 1pp 5666ndash5671 2017

[17] A Nubiola and I A Bonev ldquoAbsolute calibration of an ABBIRB 1600 robot using a laser trackerrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 236ndash245 2013

[18] A Nubiola M Slamani A Joubair and I A BonevldquoComparison of two calibrationmethods for a small industrialrobot based on an optical CMM and a laser trackerrdquo Roboticavol 32 no 3 pp 447ndash466 2014

[19] G Gao H Zhang H San X Wu and W Wang ldquoModelingand error compensation of robotic articulated arm coordinatemeasuring machines using BP neural networkrdquo Complexityvol 2017 Article ID 5156264 2017

[20] A Nubiola M Slamani and I A Bonev ldquoA new method formeasuring a large set of poses with a single telescopingballbarrdquo Precision Engineering vol 37 no 2 pp 451ndash4602013

[21] H-N Nguyen J Zhou andH-J Kang ldquoA calibrationmethodfor enhancing robot accuracy through integration of an

extended Kalman filter algorithm and an artificial neuralnetworkrdquo Neurocomputing vol 151 pp 996ndash1005 2015

[22] J Zhang X Wang K Wen Y Zhou Y Yue and J Yang ldquoAsimple and rapid calibration methodology for industrial robotbased on geometric constraint and two-step errorrdquo IndustrialRobot An International Journal vol 45 no 6 pp 715ndash7212018

[23] G Zhao P Zhang GMa andW Xiao ldquoSystem identificationof the nonlinear residual errors of an industrial robot usingmassive measurementsrdquo Robotics and Computer-IntegratedManufacturing vol 59 pp 104ndash114 2019

[24] K Wang G Liu Q Tao and M Zhai ldquoEfficient parametersestimation method for the separable nonlinear least squaresproblemrdquo Complexity vol 2020 Article ID 9619427 17 pages2020

[25] I Dattner H Ship and E O Voit ldquoSeparable nonlinear least-squares parameter estimation for complex dynamic systemsrdquoComplexity vol 2020 Article ID 6403641 18 pages 2020

[26] J Ding ldquo(e hierarchical iterative identification algorithm formulti-input-output-error systems with autoregressive noiserdquoComplexity vol 2017 Article ID 5292894 2017

[27] A Omodei G Legnani and R Adamini ldquo(ree methodol-ogies for the calibration of industrial manipulators experi-mental results on a SCARA robotrdquo Journal of Robotic Systemsvol 17 no 6 pp 291ndash307 2000

[28] Y Ho and R Lee ldquoA Bayesian approach to problems instochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 333ndash339 1964

[29] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processingvol 50 no 2 pp 174ndash188 2002

[30] K Schroer Precision and Calibration John Wiley amp SonsHoboken NJ USA 2007

[31] B Greenway ldquoRobot accuracyrdquo Industrial Robot An Inter-national Journal vol 27 no 4 pp 257ndash265 2000

[32] C Musso N Oudjane and F Le Gland ldquoImproving regu-larised particle filtersrdquo in Sequential Monte Carlo Methods inPractice A Doucet N de Freitas and N Gordon Edspp 247ndash271 Springer Berlin Germany 2001

[33] J J Craig Introduction to Robotics Mechanics and ControlPearson Education India Uttar Pradesh India 3rd edition2009

[34] Z Wang H Xu G Chen R Sun and L Sun ldquoA distanceerror based industrial robot kinematic calibration methodrdquoIndustrial Robot An International Journal vol 41 pp 439ndash446 2014

[35] K Senne ldquoStochastic processes and filtering theoryrdquo IEEETransactions on Automatic Control vol 17 no 5 pp 752-7531972

[36] J Carpenter P Clifford and P Fearnhead ldquoImproved particlefilter for nonlinear problemsrdquo IEE Proceedings-Radar Sonarand Navigation vol 146 no 1 pp 2ndash7 1999

[37] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[38] S Godsill and T Clapp Improvement Strategies for MonteCarlo Particle Filters Springer Berlin Germany 2001

[39] S Liu and R Chen ldquoSequential Monte Carlo methods fordynamic systemsrdquo Journal of the American Statistical Asso-ciation vol 93 1998

[40] C Musso N Oudjane and F L Gland ldquoImproving regu-larised particle filtersrdquo Statistics for Engineering amp Informa-tion Science vol 28 2001

Complexity 9

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity

[minus360deg360deg] respectively (e former 50 poses were used foridentification and the other 50 poses were used forvalidation

To compare the effectiveness three different algorithms(ie LS EKF and EKF+RPF) were used to identify thekinematic parameters that were compensated to the controlmodel of the robot(emaximum distance error was used asthe index to evaluate the results of compensation Figure 6 isthe distance error distribution chart of the robot aftercompensation by LS which shows that all the distance errorsare reduced greatly and the average error is from 04827mmto 01120mm by a reduction rate of 76 Figure 7 is the

distance error distribution chart of the robot after com-pensation by EKF which shows that the average error isreduced from 04827mm to 01352mm by a reduction rate

Xk

Yk

Xki

Xki

Xk = Xk + hDk τii i Draw τ k ~ kh fromepanechnikov kernel

iDraw Xk fromq (Xk | Xkndash1 Y1k)

ii˜

Calculate ωk andtotal weight ωk

i ii ˜ωk = (ωk)ndash1 ωk

Measurementsystem

Normalize

Yes

No

Calculate effectivesample size Neff

Calculate the empiricalcovariance matrix Sk

Neff lt threshold

Resample Compute Dk such that Dk Dk = SkT

Xk = ωk Xki˜ i

i = 1

M

Figure 4 (e flow chart of the RPF algorithm

Initialvalue

Robot(ER20-C10)

A set of jointangle and P0

Δ l = ĴΔxlmisin

N

Set of distancemeasurements

eisinΩ contains 19parameters error

EKF

RPF

Kinematicsparameters error

Compensate tonominal parameters

Calculate distanceaccuracy

Figure 5 (e flow chart of the proposed algorithm

Table 2 Settings of the kinematic parametersrsquo error

No of joints Δαiminus1 (deg) Δaiminus1 (mm) Δθi(deg) Δdi (mm)1 0 0 0 02 002 10 003 minus033 minus001 05 004 04 0015 minus04 006 0855 minus001 minus03 005 06 minus006 0 0 minus04

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 6(e distance error of the robot after compensation by LS

Complexity 7

of 72 Figure 8 is the distance error distribution chart of therobot after compensation by EKF+RPF which shows thatthe average error is reduced from 04827mm to 00780mmby a reduction rate of 84

Table 3 shows the statistics of the maximum mean andRMS values of the distance errors before and after com-pensation by the three algorithms By comparison one canfind that the effectiveness of RPF is the best of the three

5 Conclusion

(is paper proposed a novel kinematic calibration methodbased on the distance information for the industrial robot inwhich an EKF and RPF hybrid identification method wasadopted To simplify the data acquisition procedure and alsoavoid using expensive coordinate measuring instrumentsthe moving displacements of the end effector were used asthe reference to build the kinematic parameter error iden-tification model (en the error model was linearized bydifferentiating of the coefficient matrix By SVD of thecoefficient matrix the redundant parameters were deter-mined and removed from the list of parameters to beidentified (e preidentification of the parameters wasconducted with EKF by which we obtained a group of initialparameters of the kinematic model RPF was used to identifythe kinematic parameters of the linearized error model withthe initial parameters Comparative simulations were con-ducted based on the linearized kinematic error model (ecompensation results of the three algorithms are all effectivein improving the accuracy of the industrial robot but thehybrid algorithm proposed in this paper is more precise andfast in iteration calculation In the future work precisiondistance measuring device will be developed and effec-tiveness of the device and the proposed algorithm will befurther verified by experimental tests

Data Availability

(e data were curated by the authors and are available uponrequest

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Natural ScienceFoundation of China (Grant no 51865020) and the ScientificResearch Fund of Yunnan Education Department (Grant no2019J0046)

References

[1] Z Roth B Mooring and B Ravani ldquoAn overview of robotcalibrationrdquo IEEE Journal on Robotics and Automation vol 3no 5 pp 377ndash385 1987

[2] B Mooring M Driels and Z Roth Fundamentals of Ma-nipulator Calibration Wiley Hoboken NJ USA 1991

[3] W K Veitschegger and C-H Wu ldquoRobot calibration andcompensationrdquo IEEE Journal on Robotics and Automationvol 4 no 6 pp 643ndash656 1988

[4] K Schroeer S L Albright and M Grethlein ldquoCompleteminimal and model-continuous kinematic models for robotcalibrationrdquo Robotics and Computer-Integrated Manufactur-ing vol 13 pp 73ndash85 1997

[5] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanism based on matricesrdquo Transof theAsmeJournal of AppliedMechanics vol 22 pp 215ndash221 1955

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 7 (e distance error of the robot after compensation byEKF

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 8 (e distance error of the robot after compensation byEKF+RPF

Table 3 (e distance error of robot before and after compensationby 3 different algorithms

Algorithm Max(mm)

Mean(mm)

RMS(mm)

Before compensation mdash 12612 04827 05791

After compensationLS 03236 01120 01376EKF 03255 01352 01653

EKF+RPF 02404 00780 00936

8 Complexity

[6] S A Hayati ldquoRobot arm geometric link parameter estima-tionrdquo in Proceedings of the 22nd IEEE Conference on Decisionand Control San Antonio TX USA November 1983

[7] H W Stone and A C Sanderson ldquoStatistical performanceevaluation of the S-model arm signature identificationtechniquerdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Philadelphia PA USAAugust 1988

[8] H Zhuang Z S Roth and F Hamano ldquoA complete andparametrically continuous kinematic model for robot ma-nipulatorsrdquo IEEE Transactions on Robotics amp Automationvol 8 pp 451ndash463 1990

[9] T Sun Y Zhai Y Song and J Zhang ldquoKinematic calibrationof a 3-DoF rotational parallel manipulator using lasertrackerrdquo Robotics and Computer-Integrated Manufacturingvol 41 pp 78ndash91 2016

[10] P Yuan D Chen T Wang S Cao Y Cai and L Xue ldquoAcompensation method based on extreme learning machine toenhance absolute position accuracy for aviation drilling ro-botrdquo Advances in Mechanical Engineering vol 10 2018

[11] G Gao F Liu H San XWu andWWang ldquoHybrid optimalkinematic parameter identification for an industrial robotbased on BPNN-PSOrdquo Complexity vol 2018 Article ID4258676 2018

[12] Y Bai H Zhuang and Z S Roth ldquoExperiment study ofPUMA robot calibration using a laser tracking systemrdquo inProceeding of the IEEE International Workshop on SoftComputing in Industrial Applications Binghamton NY USASeptember 2003

[13] X Zhang Y Song Y Yang and H Pan ldquoStereo vision basedautonomous robot calibrationrdquo Robotics and AutonomousSystems vol 93 pp 43ndash51 2017

[14] T Messay R Ordontildeez and E Marcil ldquoComputationallyefficient and robust kinematic calibration methodologies andtheir application to industrial robotsrdquo Robotics and Com-puter-Integrated Manufacturing vol 37 pp 33ndash48 2016

[15] G Gatti and G Danieli ldquoA practical approach to compensatefor geometric errors in measuring arms application to a six-degree-of-freedom kinematic structurerdquo Measurement Sci-ence and Technology vol 19 Article ID 015107 2008

[16] N Schillreff M Nykolaychuk and F Ortmeier ldquoTowardshigh accuracy robot-assisted surgerylowastlowast this work was partlyfunded by the federal ministry of education and Research inGermany within the research campus STIMULATE undergrant No 13GW0095Ardquo IFAC-PapersOnLine vol 50 no 1pp 5666ndash5671 2017

[17] A Nubiola and I A Bonev ldquoAbsolute calibration of an ABBIRB 1600 robot using a laser trackerrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 236ndash245 2013

[18] A Nubiola M Slamani A Joubair and I A BonevldquoComparison of two calibrationmethods for a small industrialrobot based on an optical CMM and a laser trackerrdquo Roboticavol 32 no 3 pp 447ndash466 2014

[19] G Gao H Zhang H San X Wu and W Wang ldquoModelingand error compensation of robotic articulated arm coordinatemeasuring machines using BP neural networkrdquo Complexityvol 2017 Article ID 5156264 2017

[20] A Nubiola M Slamani and I A Bonev ldquoA new method formeasuring a large set of poses with a single telescopingballbarrdquo Precision Engineering vol 37 no 2 pp 451ndash4602013

[21] H-N Nguyen J Zhou andH-J Kang ldquoA calibrationmethodfor enhancing robot accuracy through integration of an

extended Kalman filter algorithm and an artificial neuralnetworkrdquo Neurocomputing vol 151 pp 996ndash1005 2015

[22] J Zhang X Wang K Wen Y Zhou Y Yue and J Yang ldquoAsimple and rapid calibration methodology for industrial robotbased on geometric constraint and two-step errorrdquo IndustrialRobot An International Journal vol 45 no 6 pp 715ndash7212018

[23] G Zhao P Zhang GMa andW Xiao ldquoSystem identificationof the nonlinear residual errors of an industrial robot usingmassive measurementsrdquo Robotics and Computer-IntegratedManufacturing vol 59 pp 104ndash114 2019

[24] K Wang G Liu Q Tao and M Zhai ldquoEfficient parametersestimation method for the separable nonlinear least squaresproblemrdquo Complexity vol 2020 Article ID 9619427 17 pages2020

[25] I Dattner H Ship and E O Voit ldquoSeparable nonlinear least-squares parameter estimation for complex dynamic systemsrdquoComplexity vol 2020 Article ID 6403641 18 pages 2020

[26] J Ding ldquo(e hierarchical iterative identification algorithm formulti-input-output-error systems with autoregressive noiserdquoComplexity vol 2017 Article ID 5292894 2017

[27] A Omodei G Legnani and R Adamini ldquo(ree methodol-ogies for the calibration of industrial manipulators experi-mental results on a SCARA robotrdquo Journal of Robotic Systemsvol 17 no 6 pp 291ndash307 2000

[28] Y Ho and R Lee ldquoA Bayesian approach to problems instochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 333ndash339 1964

[29] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processingvol 50 no 2 pp 174ndash188 2002

[30] K Schroer Precision and Calibration John Wiley amp SonsHoboken NJ USA 2007

[31] B Greenway ldquoRobot accuracyrdquo Industrial Robot An Inter-national Journal vol 27 no 4 pp 257ndash265 2000

[32] C Musso N Oudjane and F Le Gland ldquoImproving regu-larised particle filtersrdquo in Sequential Monte Carlo Methods inPractice A Doucet N de Freitas and N Gordon Edspp 247ndash271 Springer Berlin Germany 2001

[33] J J Craig Introduction to Robotics Mechanics and ControlPearson Education India Uttar Pradesh India 3rd edition2009

[34] Z Wang H Xu G Chen R Sun and L Sun ldquoA distanceerror based industrial robot kinematic calibration methodrdquoIndustrial Robot An International Journal vol 41 pp 439ndash446 2014

[35] K Senne ldquoStochastic processes and filtering theoryrdquo IEEETransactions on Automatic Control vol 17 no 5 pp 752-7531972

[36] J Carpenter P Clifford and P Fearnhead ldquoImproved particlefilter for nonlinear problemsrdquo IEE Proceedings-Radar Sonarand Navigation vol 146 no 1 pp 2ndash7 1999

[37] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[38] S Godsill and T Clapp Improvement Strategies for MonteCarlo Particle Filters Springer Berlin Germany 2001

[39] S Liu and R Chen ldquoSequential Monte Carlo methods fordynamic systemsrdquo Journal of the American Statistical Asso-ciation vol 93 1998

[40] C Musso N Oudjane and F L Gland ldquoImproving regu-larised particle filtersrdquo Statistics for Engineering amp Informa-tion Science vol 28 2001

Complexity 9

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity

of 72 Figure 8 is the distance error distribution chart of therobot after compensation by EKF+RPF which shows thatthe average error is reduced from 04827mm to 00780mmby a reduction rate of 84

Table 3 shows the statistics of the maximum mean andRMS values of the distance errors before and after com-pensation by the three algorithms By comparison one canfind that the effectiveness of RPF is the best of the three

5 Conclusion

(is paper proposed a novel kinematic calibration methodbased on the distance information for the industrial robot inwhich an EKF and RPF hybrid identification method wasadopted To simplify the data acquisition procedure and alsoavoid using expensive coordinate measuring instrumentsthe moving displacements of the end effector were used asthe reference to build the kinematic parameter error iden-tification model (en the error model was linearized bydifferentiating of the coefficient matrix By SVD of thecoefficient matrix the redundant parameters were deter-mined and removed from the list of parameters to beidentified (e preidentification of the parameters wasconducted with EKF by which we obtained a group of initialparameters of the kinematic model RPF was used to identifythe kinematic parameters of the linearized error model withthe initial parameters Comparative simulations were con-ducted based on the linearized kinematic error model (ecompensation results of the three algorithms are all effectivein improving the accuracy of the industrial robot but thehybrid algorithm proposed in this paper is more precise andfast in iteration calculation In the future work precisiondistance measuring device will be developed and effec-tiveness of the device and the proposed algorithm will befurther verified by experimental tests

Data Availability

(e data were curated by the authors and are available uponrequest

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Natural ScienceFoundation of China (Grant no 51865020) and the ScientificResearch Fund of Yunnan Education Department (Grant no2019J0046)

References

[1] Z Roth B Mooring and B Ravani ldquoAn overview of robotcalibrationrdquo IEEE Journal on Robotics and Automation vol 3no 5 pp 377ndash385 1987

[2] B Mooring M Driels and Z Roth Fundamentals of Ma-nipulator Calibration Wiley Hoboken NJ USA 1991

[3] W K Veitschegger and C-H Wu ldquoRobot calibration andcompensationrdquo IEEE Journal on Robotics and Automationvol 4 no 6 pp 643ndash656 1988

[4] K Schroeer S L Albright and M Grethlein ldquoCompleteminimal and model-continuous kinematic models for robotcalibrationrdquo Robotics and Computer-Integrated Manufactur-ing vol 13 pp 73ndash85 1997

[5] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanism based on matricesrdquo Transof theAsmeJournal of AppliedMechanics vol 22 pp 215ndash221 1955

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 7 (e distance error of the robot after compensation byEKF

15

1

05

0

ndash05

ndash1

Dist

ance

erro

r (m

m)

5 10 15 20 25 30 35 40 45 50e number of samples

Before calibrationAer calibration

Figure 8 (e distance error of the robot after compensation byEKF+RPF

Table 3 (e distance error of robot before and after compensationby 3 different algorithms

Algorithm Max(mm)

Mean(mm)

RMS(mm)

Before compensation mdash 12612 04827 05791

After compensationLS 03236 01120 01376EKF 03255 01352 01653

EKF+RPF 02404 00780 00936

8 Complexity

[6] S A Hayati ldquoRobot arm geometric link parameter estima-tionrdquo in Proceedings of the 22nd IEEE Conference on Decisionand Control San Antonio TX USA November 1983

[7] H W Stone and A C Sanderson ldquoStatistical performanceevaluation of the S-model arm signature identificationtechniquerdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Philadelphia PA USAAugust 1988

[8] H Zhuang Z S Roth and F Hamano ldquoA complete andparametrically continuous kinematic model for robot ma-nipulatorsrdquo IEEE Transactions on Robotics amp Automationvol 8 pp 451ndash463 1990

[9] T Sun Y Zhai Y Song and J Zhang ldquoKinematic calibrationof a 3-DoF rotational parallel manipulator using lasertrackerrdquo Robotics and Computer-Integrated Manufacturingvol 41 pp 78ndash91 2016

[10] P Yuan D Chen T Wang S Cao Y Cai and L Xue ldquoAcompensation method based on extreme learning machine toenhance absolute position accuracy for aviation drilling ro-botrdquo Advances in Mechanical Engineering vol 10 2018

[11] G Gao F Liu H San XWu andWWang ldquoHybrid optimalkinematic parameter identification for an industrial robotbased on BPNN-PSOrdquo Complexity vol 2018 Article ID4258676 2018

[12] Y Bai H Zhuang and Z S Roth ldquoExperiment study ofPUMA robot calibration using a laser tracking systemrdquo inProceeding of the IEEE International Workshop on SoftComputing in Industrial Applications Binghamton NY USASeptember 2003

[13] X Zhang Y Song Y Yang and H Pan ldquoStereo vision basedautonomous robot calibrationrdquo Robotics and AutonomousSystems vol 93 pp 43ndash51 2017

[14] T Messay R Ordontildeez and E Marcil ldquoComputationallyefficient and robust kinematic calibration methodologies andtheir application to industrial robotsrdquo Robotics and Com-puter-Integrated Manufacturing vol 37 pp 33ndash48 2016

[15] G Gatti and G Danieli ldquoA practical approach to compensatefor geometric errors in measuring arms application to a six-degree-of-freedom kinematic structurerdquo Measurement Sci-ence and Technology vol 19 Article ID 015107 2008

[16] N Schillreff M Nykolaychuk and F Ortmeier ldquoTowardshigh accuracy robot-assisted surgerylowastlowast this work was partlyfunded by the federal ministry of education and Research inGermany within the research campus STIMULATE undergrant No 13GW0095Ardquo IFAC-PapersOnLine vol 50 no 1pp 5666ndash5671 2017

[17] A Nubiola and I A Bonev ldquoAbsolute calibration of an ABBIRB 1600 robot using a laser trackerrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 236ndash245 2013

[18] A Nubiola M Slamani A Joubair and I A BonevldquoComparison of two calibrationmethods for a small industrialrobot based on an optical CMM and a laser trackerrdquo Roboticavol 32 no 3 pp 447ndash466 2014

[19] G Gao H Zhang H San X Wu and W Wang ldquoModelingand error compensation of robotic articulated arm coordinatemeasuring machines using BP neural networkrdquo Complexityvol 2017 Article ID 5156264 2017

[20] A Nubiola M Slamani and I A Bonev ldquoA new method formeasuring a large set of poses with a single telescopingballbarrdquo Precision Engineering vol 37 no 2 pp 451ndash4602013

[21] H-N Nguyen J Zhou andH-J Kang ldquoA calibrationmethodfor enhancing robot accuracy through integration of an

extended Kalman filter algorithm and an artificial neuralnetworkrdquo Neurocomputing vol 151 pp 996ndash1005 2015

[22] J Zhang X Wang K Wen Y Zhou Y Yue and J Yang ldquoAsimple and rapid calibration methodology for industrial robotbased on geometric constraint and two-step errorrdquo IndustrialRobot An International Journal vol 45 no 6 pp 715ndash7212018

[23] G Zhao P Zhang GMa andW Xiao ldquoSystem identificationof the nonlinear residual errors of an industrial robot usingmassive measurementsrdquo Robotics and Computer-IntegratedManufacturing vol 59 pp 104ndash114 2019

[24] K Wang G Liu Q Tao and M Zhai ldquoEfficient parametersestimation method for the separable nonlinear least squaresproblemrdquo Complexity vol 2020 Article ID 9619427 17 pages2020

[25] I Dattner H Ship and E O Voit ldquoSeparable nonlinear least-squares parameter estimation for complex dynamic systemsrdquoComplexity vol 2020 Article ID 6403641 18 pages 2020

[26] J Ding ldquo(e hierarchical iterative identification algorithm formulti-input-output-error systems with autoregressive noiserdquoComplexity vol 2017 Article ID 5292894 2017

[27] A Omodei G Legnani and R Adamini ldquo(ree methodol-ogies for the calibration of industrial manipulators experi-mental results on a SCARA robotrdquo Journal of Robotic Systemsvol 17 no 6 pp 291ndash307 2000

[28] Y Ho and R Lee ldquoA Bayesian approach to problems instochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 333ndash339 1964

[29] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processingvol 50 no 2 pp 174ndash188 2002

[30] K Schroer Precision and Calibration John Wiley amp SonsHoboken NJ USA 2007

[31] B Greenway ldquoRobot accuracyrdquo Industrial Robot An Inter-national Journal vol 27 no 4 pp 257ndash265 2000

[32] C Musso N Oudjane and F Le Gland ldquoImproving regu-larised particle filtersrdquo in Sequential Monte Carlo Methods inPractice A Doucet N de Freitas and N Gordon Edspp 247ndash271 Springer Berlin Germany 2001

[33] J J Craig Introduction to Robotics Mechanics and ControlPearson Education India Uttar Pradesh India 3rd edition2009

[34] Z Wang H Xu G Chen R Sun and L Sun ldquoA distanceerror based industrial robot kinematic calibration methodrdquoIndustrial Robot An International Journal vol 41 pp 439ndash446 2014

[35] K Senne ldquoStochastic processes and filtering theoryrdquo IEEETransactions on Automatic Control vol 17 no 5 pp 752-7531972

[36] J Carpenter P Clifford and P Fearnhead ldquoImproved particlefilter for nonlinear problemsrdquo IEE Proceedings-Radar Sonarand Navigation vol 146 no 1 pp 2ndash7 1999

[37] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[38] S Godsill and T Clapp Improvement Strategies for MonteCarlo Particle Filters Springer Berlin Germany 2001

[39] S Liu and R Chen ldquoSequential Monte Carlo methods fordynamic systemsrdquo Journal of the American Statistical Asso-ciation vol 93 1998

[40] C Musso N Oudjane and F L Gland ldquoImproving regu-larised particle filtersrdquo Statistics for Engineering amp Informa-tion Science vol 28 2001

Complexity 9

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity

[6] S A Hayati ldquoRobot arm geometric link parameter estima-tionrdquo in Proceedings of the 22nd IEEE Conference on Decisionand Control San Antonio TX USA November 1983

[7] H W Stone and A C Sanderson ldquoStatistical performanceevaluation of the S-model arm signature identificationtechniquerdquo in Proceedings of the IEEE International Confer-ence on Robotics and Automation Philadelphia PA USAAugust 1988

[8] H Zhuang Z S Roth and F Hamano ldquoA complete andparametrically continuous kinematic model for robot ma-nipulatorsrdquo IEEE Transactions on Robotics amp Automationvol 8 pp 451ndash463 1990

[9] T Sun Y Zhai Y Song and J Zhang ldquoKinematic calibrationof a 3-DoF rotational parallel manipulator using lasertrackerrdquo Robotics and Computer-Integrated Manufacturingvol 41 pp 78ndash91 2016

[10] P Yuan D Chen T Wang S Cao Y Cai and L Xue ldquoAcompensation method based on extreme learning machine toenhance absolute position accuracy for aviation drilling ro-botrdquo Advances in Mechanical Engineering vol 10 2018

[11] G Gao F Liu H San XWu andWWang ldquoHybrid optimalkinematic parameter identification for an industrial robotbased on BPNN-PSOrdquo Complexity vol 2018 Article ID4258676 2018

[12] Y Bai H Zhuang and Z S Roth ldquoExperiment study ofPUMA robot calibration using a laser tracking systemrdquo inProceeding of the IEEE International Workshop on SoftComputing in Industrial Applications Binghamton NY USASeptember 2003

[13] X Zhang Y Song Y Yang and H Pan ldquoStereo vision basedautonomous robot calibrationrdquo Robotics and AutonomousSystems vol 93 pp 43ndash51 2017

[14] T Messay R Ordontildeez and E Marcil ldquoComputationallyefficient and robust kinematic calibration methodologies andtheir application to industrial robotsrdquo Robotics and Com-puter-Integrated Manufacturing vol 37 pp 33ndash48 2016

[15] G Gatti and G Danieli ldquoA practical approach to compensatefor geometric errors in measuring arms application to a six-degree-of-freedom kinematic structurerdquo Measurement Sci-ence and Technology vol 19 Article ID 015107 2008

[16] N Schillreff M Nykolaychuk and F Ortmeier ldquoTowardshigh accuracy robot-assisted surgerylowastlowast this work was partlyfunded by the federal ministry of education and Research inGermany within the research campus STIMULATE undergrant No 13GW0095Ardquo IFAC-PapersOnLine vol 50 no 1pp 5666ndash5671 2017

[17] A Nubiola and I A Bonev ldquoAbsolute calibration of an ABBIRB 1600 robot using a laser trackerrdquo Robotics and Computer-Integrated Manufacturing vol 29 no 1 pp 236ndash245 2013

[18] A Nubiola M Slamani A Joubair and I A BonevldquoComparison of two calibrationmethods for a small industrialrobot based on an optical CMM and a laser trackerrdquo Roboticavol 32 no 3 pp 447ndash466 2014

[19] G Gao H Zhang H San X Wu and W Wang ldquoModelingand error compensation of robotic articulated arm coordinatemeasuring machines using BP neural networkrdquo Complexityvol 2017 Article ID 5156264 2017

[20] A Nubiola M Slamani and I A Bonev ldquoA new method formeasuring a large set of poses with a single telescopingballbarrdquo Precision Engineering vol 37 no 2 pp 451ndash4602013

[21] H-N Nguyen J Zhou andH-J Kang ldquoA calibrationmethodfor enhancing robot accuracy through integration of an

extended Kalman filter algorithm and an artificial neuralnetworkrdquo Neurocomputing vol 151 pp 996ndash1005 2015

[22] J Zhang X Wang K Wen Y Zhou Y Yue and J Yang ldquoAsimple and rapid calibration methodology for industrial robotbased on geometric constraint and two-step errorrdquo IndustrialRobot An International Journal vol 45 no 6 pp 715ndash7212018

[23] G Zhao P Zhang GMa andW Xiao ldquoSystem identificationof the nonlinear residual errors of an industrial robot usingmassive measurementsrdquo Robotics and Computer-IntegratedManufacturing vol 59 pp 104ndash114 2019

[24] K Wang G Liu Q Tao and M Zhai ldquoEfficient parametersestimation method for the separable nonlinear least squaresproblemrdquo Complexity vol 2020 Article ID 9619427 17 pages2020

[25] I Dattner H Ship and E O Voit ldquoSeparable nonlinear least-squares parameter estimation for complex dynamic systemsrdquoComplexity vol 2020 Article ID 6403641 18 pages 2020

[26] J Ding ldquo(e hierarchical iterative identification algorithm formulti-input-output-error systems with autoregressive noiserdquoComplexity vol 2017 Article ID 5292894 2017

[27] A Omodei G Legnani and R Adamini ldquo(ree methodol-ogies for the calibration of industrial manipulators experi-mental results on a SCARA robotrdquo Journal of Robotic Systemsvol 17 no 6 pp 291ndash307 2000

[28] Y Ho and R Lee ldquoA Bayesian approach to problems instochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 333ndash339 1964

[29] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processingvol 50 no 2 pp 174ndash188 2002

[30] K Schroer Precision and Calibration John Wiley amp SonsHoboken NJ USA 2007

[31] B Greenway ldquoRobot accuracyrdquo Industrial Robot An Inter-national Journal vol 27 no 4 pp 257ndash265 2000

[32] C Musso N Oudjane and F Le Gland ldquoImproving regu-larised particle filtersrdquo in Sequential Monte Carlo Methods inPractice A Doucet N de Freitas and N Gordon Edspp 247ndash271 Springer Berlin Germany 2001

[33] J J Craig Introduction to Robotics Mechanics and ControlPearson Education India Uttar Pradesh India 3rd edition2009

[34] Z Wang H Xu G Chen R Sun and L Sun ldquoA distanceerror based industrial robot kinematic calibration methodrdquoIndustrial Robot An International Journal vol 41 pp 439ndash446 2014

[35] K Senne ldquoStochastic processes and filtering theoryrdquo IEEETransactions on Automatic Control vol 17 no 5 pp 752-7531972

[36] J Carpenter P Clifford and P Fearnhead ldquoImproved particlefilter for nonlinear problemsrdquo IEE Proceedings-Radar Sonarand Navigation vol 146 no 1 pp 2ndash7 1999

[37] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[38] S Godsill and T Clapp Improvement Strategies for MonteCarlo Particle Filters Springer Berlin Germany 2001

[39] S Liu and R Chen ldquoSequential Monte Carlo methods fordynamic systemsrdquo Journal of the American Statistical Asso-ciation vol 93 1998

[40] C Musso N Oudjane and F L Gland ldquoImproving regu-larised particle filtersrdquo Statistics for Engineering amp Informa-tion Science vol 28 2001

Complexity 9

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity

[41] A Goswami A Quaid andM Peshkin ldquoComplete parameteridentification of a robot from partial pose informationrdquo inProceedings IEEE International Conference on Robotics andAutomation pp 168ndash173 Paris France May 1993

[42] Y Tian C Lu Z Wang and Z Wang ldquoFault diagnosis basedon lmd-SVD and information-geometric support vectormachine for hydraulic pumpsrdquo Transactions of the CanadianSociety for Mechanical Engineering vol 39 no 3 pp 569ndash5802015

10 Complexity