Research ArticleModel Predictive Control of Robotic GrindingBased on Deep Belief Network

Shouyan Chen 1 Tie Zhang 2 Yanbiao Zou2 and Meng Xiao2

1Guangzhou University China2South China University of Technology China

Correspondence should be addressed to Shouyan Chen 15989285002sinacn

Received 10 November 2018 Revised 10 February 2019 Accepted 25 February 2019 Published 27 March 2019

Academic Editor Sing Kiong Nguang

Copyright copy 2019 Shouyan Chen et al This 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

Considering the influence of rigid-flexible dynamics on robotic grinding process a model predictive control approach based ondeep belief network (DBN) is proposed to control robotic grinding deformation The rigid-flexible coupling dynamics of roboticgrinding is first established on the basis of which a robotic grinding prediction model is constructed to predict the change ofrobotic grinding status and perform feed-forward control A rolling optimization formula derived from the energy function isalso established to optimize control output in real time and perform feedback control As the accurately model parameters arehard to obtain a deep belief network is constructed to obtain the parameters of robotic grinding predictive model Simulation andexperimental results indicate that the proposed model predictive control approach can predict abrupt change of robotic grindingstatus caused by deformation and perform a feed-forward and feedback based combination control reducing control overflow andsystem oscillation caused by inaccurate feedback control

1 Introduction

The deformation occurs during robotic grinding processhas significant impact on robotic grinding dynamic androbotic grinding performance [1 2] Two ways are mainlypresented in current studies to solve this problem One wayis to optimize mechanical structure of robotic machiningsystem or increase stiffness and stability of robot machiningsystem The other is to adjust machining trajectory by off-line planning or real-time force control according to the robotdynamic model [3]

The real-time force control approaches presented incurrent studies include adaptive control fuzzy control andcontrol based on neural network [4] Mendes et al [5]proposed an adaptive fuzzy control approach which is basedon Hybrid forcemotion control system to cope with contactissues between robot and a given surface Fu et al [6] pro-posed an adaptive fuzzy force control model which includesa speed control loop and a position control loop to controlboth feed rate and position of robot to achieve stable roboticdeburring control Yen [7] proposed an adaptive controlmethod based on recursive fuzzy wavelet neural network to

optimize motion control parameters of three-axis robot inreal time

The above feedback control approaches implement onlywhen trajectory deviations appear which may result inovershoot control overspill and system oscillations In viewof this some scholars attempt to implement model predictivecontrol approach to achieve feed-forward compensationcontrol [8 9] Many nonlinear model predictive controlapproaches are then proposed Wilson [10] discussed theperformances of three model predictive control approachesapplied to robot system control The three approaches arenonlinear model predictive control (nMPC) approach PID-based nonlinear model predictive control (PID nMPC)approach and simplified nonlinear model predictive con-trol (SnMPC) approach The results of discussion indicatedthat the performance of nonlinear model predictive controlapproach is susceptible to systemmodel errors Some scholarstry to improve nonlinear model predictive control approachby using intelligent algorithms such as neural network [11ndash13] Li [14] proposed a nonlinear model control methodbased on neural dynamic network where the neural dynamicnetwork is used to obtain optimal values of the formulated

HindawiComplexityVolume 2019 Article ID 1891365 12 pageshttpsdoiorg10115520191891365

2 Complexity

constrained quadratic programming (QP) problem derivedfrom the cost function of nonlinear model predictive controlmodel Zeng [15] used Gaussian radial basis function (RBF)neural networks to improve the nonlinear model predictivecontrol approach applied in the control of nonlinear mul-tivariable systems Dalamagkidis [16] proposed a nonlinearmodel predictive control approach based on recurrent neuralnetwork to achieve the predictive control of propeller self-rotation process while unmanned aerial vehicle engine isdamaged

In this paper a model predictive control approach basedon a deep belief network (DBN) is proposed to control roboticdeformation and reduce rigid-flexible effect on robotic grind-ing dynamics The following parts are arranged as followsFirstly the dynamic model of robotic grinding is establishedwith the consideration of rigid-flexible coupling effect Basedon this the model predictive controller of robotic grinding isdesigned Since the accurate parameters of robotic grindingdynamics model and model predictive controller are hardto acquire a deep belief network is designed to accessnonlinear predictive model of robotic grinding Simulationand experiments are finally carried out to verify performanceof the proposed approach

2 Rigid-Flexible Coupling Dynamics ofGrinding Robot

Traditional grinding dynamicsmodel can be expressed as [17]

119872119901 + 119862 119909119901 + 119870119889119909119901 = 119865119901 (1)

where 119865119901 is the cutting force119872 is the system mass matrix 119862is the system damping 119870119889 is the system dynamic stiffness119909119901 is the grinding tool position and 119901 119901 are its firstand second derivatives Since the stiffness of CNC is largeand the deformation is small the grinding tool position isapproximate to the planned position However the stiffnessof robot is not sufficient which may lead to large deformationand large deviation between grinding position and plannedposition Therefore the relationship of the grinding position119909119901 the planned position 119909119886 and the deformation 120590 can beexpressed as

119909119901 = 119909119886 minus 120590 (2)

Similarly the relationship of the force acts to robot end-effector 119865 the cutting force 119865119901 and the force caused byrobotic grinding deformation 119865119889 is

119865 = 119865119889 + 119865119901 (3)

21 Robotic Grinding Deformation Robotic grinding defor-mation consists of extrusion deformation and periodic defor-mation The extrusion deformation is caused by relativemotion between grinding tool and workpiece while theperiodic deformation is caused by relative motion betweenblades and workpiece Therefore the grinding deformationcan be expressed as

120590 = 120590119895 (119905119894) + 120590119888 (119905119894) (4)

where 120590119895(119905119894) is the extrusion deformation 120590119888(119905119894) is the peri-odic deformation The generation of extrusion deformationis shown in Figure 1 The grinding tool is driven toward theworkpiece at a feed rate of 119881119908 to perform cutting Accordingto traditional grinding theory there are sliding and extrusionstate before the actual grinding is conducted Based on thisan assumption is made that the deformation generated whenthe grinding tool cut-in workpiece is mostly derived fromextrusion deformation written as

119865119895 (119905119894) = 119870119889120590119895 (119905119894) (5)

where 119870119889 is the dynamic stiffness matrix of robot grindingsystem 120590119895(119905119894) is the extrusion deformation at time 119905119894 Thevalue of extrusion deformation can be regarded as theaccumulation of the difference between the feed rate 119881119908(119905119895)and the removal speed 119881119890(119905119895) from time 1199051 to time 119905119894 writtenas

120590119895 (119905119894) =119894sumℎ=1

(119881119908 (119905ℎ) minus 119881119890 (119905ℎ)) (6)

Similarly the relationship between periodic deformationforce and the periodic deformation 120590119888(119905119894) can be expressedas

119865119888 (119905119894) = 119870119889120590119888 (119905119894) (7)

120590119888 (119905119894) = 120590119888119900 (119905119894) sin (120596119888119905119894 + 120593119888) (8)

where 120596119888 = 2120587119891119888 = 212058711989911988111988860 is the circular frequencyof grinding tool 119899 is the number of blades 119881119888 is the toolrotating speed 120593119888 is the corresponding phase 120590119888119900(119905119894) is thecorrespondingmaximumcutting deformationTherefore theactual grinding position can be obtained by (6) and(8)

119909119901 (119905119894) = 119909119886 (119905119894) minus 120590119895 (119905119894) minus 120590119888 (119905119894) (9)

Substituting (9) and (1) into (3) the dynamic model of robotgrinding can be expressed as

119865 (119905119894) = 119872119901 (119905119894) + 119862119901 (119905119894) + 119870119889119909119901 (119905119894)+ 119870119889 (120590119888 (119905119894) + 120590119895 (119905119894)) (10)

For the convenience of discussion 119865(119905119894) is named as robotgrinding force in the following text

3 Model Predictive Control Based on DeepBelief Network

31 Problem Description During robotic grinding the actualgrinding position 119909119901(119905119894) is expected to be approximate to theplanned position 119909119886(119905119894)

lim119894997888rarrinfin

(119909119901 (119905119894) minus 119909119886 (119905119894)) = 0 (11)

However according to (10) as the grinding robot rigidity isinsufficient and the robotic grinding deformation is consider-able (11) is hard to realize Therefore an ideal deformation is

Complexity 3

Deformationrelease

ap

Feed rate Vw

Figure 1 Generation of extrusion deformation

defined as well as the corresponding ideal trajectory of robotgrinding 119909119901 lowast (119905119894) Equation (11) can be written as

lim119894997888rarrinfin

(119909119901 (119905119894) minus 119909119901 lowast (119905119894)) = 0 (12)

Assuming that there is a corresponding relation between thegrinding position 119909119901(119905119894) and the robotic grinding force 119865(119905119894)when the system parameters (such as feed rate grinding toolrotation speed and material of workpiece) are constant Thecontrol target in this paper can be converted into

lim119894997888rarrinfin

(119865 (119905119894) minus 119865 lowast (119905119894)) = 0 (13)

where 119865lowast(119905119894) is the ideal grinding force corresponding to theideal grinding position 119909119901 lowast (119905119894)32 Model Predictive Control for Robotic Grinding Definerobot grinding force deviation as

119890119889 (119905119894) = 119865 (119905119894) minus 119865 lowast (119905119894) (14)

Assume that robot grinding force deviation at time 119905119894+1 isaffected by robot grinding force deviation 119890119889(119905119894) and systemtransfer function 119866(119881119908(119905119894)) at time 119905119894

119890119889 (119905119894+1) = 119860119890119890119889 (119905119894) + 119866 (119881119908 (119905119894)) (15)

where 119866(119881119908(119905119894)) is the control output which takes feed rate119881119908(119905119894) as variables 119860119890 is the weighting factor of the robotgrinding force deviation Assume 119860119890 = 1 Rewrite (15)according to (14)

119865 (119905119894+1) minus 119865 lowast (119905119894+1) = 119865 (119905119894) minus 119865 lowast (119905119894) + 119866 (119881119908 (119905119894))119865 (119905119894+1) minus 119865 (119905119894) = 119865 lowast (119905119894+1) minus 119865 lowast (119905119894)

+ 119866 (119881119908 (119905119894))Δ119865 (119905119894) = Δ119865 lowast (119905119894) + 119866 (119881119908 (119905119894))

(16)

where

Δ119865 (119905119894) = 119865 (119905119894+1) minus 119865 (119905119894)Δ119865 (119905119894) = 119872Δ 119909119886 (119905119894) + 119862Δ 119909119886 (119905119894) + 119870119889Δ119909119886 (119905119894)

+ 119870119889 (Δ120590119888 (119905119894) + Δ120590119895 (119905119894))(17)

Δ119865 lowast (119905119894) = 119865 lowast (119905119894+1) minus 119865 lowast (119905119894) (18)

Define Δ119881119908(119905119894) = 119881119908(119905119894+1) minus 119881119908(119905119894) 119886(119905119894) = 119881119908(119905119894) Δ119909119886(119905119894) =119881119908(119905119894)119889119905 According to (6) there isΔ120590119895(119905119894) = 120590119895(119905119894+1)minus120590119895(119905119894) =119881119908(119905119894+1) minus 119881119890(119905119894+1) and (17) can be rewritten as

Δ119865 (119905119894) = 119872Δ119886 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894)+ 119870119889119881119908 (119905119894) (119889119905 + 1)+ 119870119889 (Δ120590119888 (119905119894) minus 119881119890 (119905119894+1))

(19)

321 Robotic Grinding Trajectory Prediction Substitute (19)and (15) into (14) to obtain the relationship between roboticgrinding force deviation and feed rate

119890119889 (119905119894+1) = 119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894)+ 119870119889119881119908 (119905119894) (119889119905 + 1) (20)

where1198661015840(119905119894) = 119872Δ119886(119905119894) minus Δ119865 lowast (119905119894) + 119870119889(Δ120590119888(119905119894) minus 119881119890(119905119894+1))Therefore the robotic grinding force deviation matrix 119864119889(119905119894)from time 119905119894+1 to time 119905119894+119901 can be acquired by

119890119889 (119905119894+1 | 119905119894)= 119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894 | 119905119894)

+ 119870119889119881119908 (119905119894) (119889119905 + 1)(21)

119890119889 (119905119894+2 | 119905119894)= 119860119890119890119889 (119905119894+1 | 119905119894) + 1198661015840 (119905119894+1)

+ (119862 + 119870119889) Δ119881119908 (119905119894+1 | 119905119894)+ 119870119889119881119908 (119905119894+1 | 119905119894) (119889119905 + 1)

119864119889 (119905119894)= [119890119889 (119905119894+1 | 119905119894) 119890119889 (119905119894+2 | 119905119894) 119890119889 (119905119894+119901 | 119905119894)]

(22)

119890119889(119905119894+119901 | 119905119894) represents the predictive of robotic grinding forcedeviation at time 119905119894+119901 according to the system information attime 119905119894 Assume that there is a relationship between change offeed rate and input voltage

Δ119881119908 (119905119894) = 119860Δ119881119908 (119905119894minus1) + 119861Δ119906119901 (119905119894) (23)

4 Complexity

where 119860 and 119861 are weight factors The prediction of feed rateincrement from time 119905119894+1 to time 119905119894+119901 can be acquired by

Δ119881119908 (119905119894+1 | 119905119894) = 119860Δ119881119908 (119905119894) + 119861Δ119906119901 (119905119894+1 | 119905119894)Δ119881119908 (119905119894+1 | 119905119894) = 1198602Δ119881119908 (119905119894minus1) + 119860119861Δ119906119901 (119905119894 | 119905119894)

+ 119861Δ119906119901 (119905119894+1 | 119905119894)Δ119881119908119910 (119905119894+119901 | 119905119894) = (Δ119881119908 (119905119894+1 | 119905119894) Δ119881119908 (119905119894+2 | 119905119894)

Δ119881119908 (119905119894+119901 | 119905119894))

(24)

where Δ119881119908(119905119894+1 | 119905119894) Δ119906119901(119905119894+1 | 119905119894) respectively represent thechanges of feed rate and control voltage at time 119905119894 predictedaccording to the system information at time 119905119894+1322 Control Output Optimizing In order to optimize con-trol output voltage an energy function is defined as

119864 (119905119894) =119894+119899sum119894=1

1003817100381710038171003817Γ1 (119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1))10038171003817100381710038172

+ 1003817100381710038171003817Γ2 (119890119889 (119905119894+1 | 119905119894))10038171003817100381710038172(25)

where119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1)= 119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894 | 119905119894)

+ 119870119889119881119908 (119905119894) (119889119905 + 1) minus 119890119889 lowast (119905119894+1)(26)

Substitute (23) into (26)

119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1)= 119860119890119890119889 (119905119894) + 1198661015840 (119905119894)

+ (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1)+ (119862 + 119870119889) 119861Δ119906119901 (119905119894 | 119905119894) )+ 119870119889119881119908 (119905119894minus1) (119889119905 + 1) minus 119890119889 lowast (119905119894+1)

(27)

Define

119864 (119905119894) = 120588T120588120588 = 1198601119885 minus 119863 (28)

where

119863 = [[Γ1 (119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1) + 119870119889119881119908 (119905119894minus1) (119889119905 + 1) minus 119890119889 lowast (119905119894+1))

Γ2 (119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1) + 119870119889119881119908 (119905119894minus1) (119889119905 + 1)) ]]

1198601 = [Γ1 (119862 + 119870119889) 119861Γ2 (119862 + 119870119889) 119861]

119885 = Δ119906119901 (119905119894 | 119905119894)

(29)

According to (26) the extreme value of energy function canbe obtained by min 120588119879120588 Therefore there is extreme valuewhen

Δ119906119901 (119905119894 | 119905119894) = 119885 = (1198601T1198601)minus1 1198601T119863 (30)

Substitute (23) into (30)

Δ119906119901 (119905119894 | 119905119894) = 1198821119890119889 (119905119894) +1198822Δ119881119908 (119905119894minus2)+ 1198823Δ119906119901 (119905119894minus1) +1198824119881119908 (119905119894minus1) + 1198871 (31)

where

1198821 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889) 1198611198601198901198822 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)

sdot 119861 (119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119860

1198823 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)sdot 119861 (119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119861

1198824 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889) 119861119870119889 (119889119905 + 1) 1198871 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)

sdot 119861 (21198661015840 (119905119894) minus 119890119889 lowast (119905119894+1))(32)

323 Stability Analysis According to (22) (23) and (30)there is

119890119889 (119905119894+1 | 119905119894) = 1198825119890119889 (119905119894) + 1198826Δ119881119908 (119905119894minus2)+ 1198827Δ119906119901 (119905119894minus1) + 1198828119881119908 (119905119894minus1) + 1198872 (33)

where

Complexity 5

1198825 = 119860119890 + (119862 + 119870119889) 11986111988211198826 = ((119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119860 + (119862 + 119870119889) 1198611198822) 1198827 = ((119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119861 + (119862 + 119870119889) 1198611198823) 1198828 = ((119862 + 119870119889) 1198611198824 + 119870119889 (119889119905 + 1)) 1198872 = 1198661015840 (119905119894) + (119862 + 119870119889) 1198611198871

(34)

According to the model predictive control principle [18]when the eigenvalues of the matrix1198825 are all within the unitcircle the control system is asymptotically stable

324 Robot Grinding State Prediction Model Merging (33)and (31) there is

119874 = 119868119882119899T + 119861119887 (35)

where

119874 = [Δ119906119901 (119905119894 | 119905119894) 119890119889 (119905119894+1 | 119905119894)] 119882119899 = [11988211198822119882311988241198825119882611988271198828] 119861119887 = [1198871 1198872] 119868 = [119890119889 (119905119894) Δ119881119908 (119905119894minus2) Δ119906119901 (119905119894minus1) 119881119908 (119905119894minus1)]

(36)

119882119899 and 119861119887 represent predictive model parameters Then thedeep belief network is introduced to obtain predictive modelparameters

33 Predictive Model Based on Deep Belief Network Accord-ing to the deep belief network principle

119874 = (((1198681198821198891T + 1198611198891)1198821198892T + 1198611198892) )119882119889119894T + 119861119889119894(119894 = 1 2 119898) (37)

Comparing (37) and (35) it can be found that predictivemodel parameter 119882119899 can be obtained by weights productprod119898119894=1119882119889119894

119882119899T = 119898prod119894=1

119882119889119894T

119861119887 =119898minus1sum119895=1

119861119889119895119898minus1prod119895=1

119882119889119895+1T + 119861119889119898(38)

Therefore an assumption is made that prod119898119894=1119882119889119894 =[11988211198822119882311988241198825119882611988271198828] can be obtained bythe product of Γ1 Γ2 119860119890 119862 119870119889 119861 And subassumptionsare made that 119882119889119894 = [119861 1198612 119860 1198602 119861119889119905 119860119889119905 ]119882119889119894+1 = [119862 1198622 119870119889 1198701198892 ] 119882119889119894+2 = [(1198601T1198601)minus1Γ12(1198601T1198601)minus1Γ22 ] Based on these assumptions thedeep belief network is constructed which is comprisedof three layers including two layers of RestrictedBoltzmann Machine (RBM) and one Back-Propagation(BP) layer Two layers of RBM are used to obtain weights

119882119889119894 = [119861 1198612 119860 1198602 119861119889119905 119860119889119905 ] and 119882119889119894+1 = [119862 1198622 1198701198891198701198892 ] while the BP layer is used to obtain 119882119889119894+2 =[(1198601T1198601)minus1Γ12 (1198601T1198601)minus1Γ22 ] The inputs of DBN aregrinding force deviation 119890119889(119905119894) feed rate 119881119908(119905119894minus1) incrementof feed rate Δ119881119908(119905119894minus2) and increment of control outputvoltage Δ119906119901(119905119894minus1) while the outputs are Δ119906119901(119905119894 | 119905119894) and119890119889(119905119894+1 | 119905119894) The energy function of general RBM is [19]

119864 (V119894 ℎ119894) = minusV119894119879119908119894ℎ119894 minus 119887119894119879V119894 minus 119889119894119879ℎ119894 (119894 = 1 2) (39)

where V119894 ℎ119894 are explicit layer and hidden layer 119908119894 119887119894 119889119894 arerespectively network weight explicit layer bias and hiddenlayer bias Consider119874 as the first visible layer of first RBMandℎ1 as the hidden layer first RBM and (39) can be rewritten as

119864 (119874 ℎ119894) = minus119874119879119908119894ℎ119894 minus 119887119894119879119874 minus 119889119894119879ℎ119894 (40)

The probability that the RBM assigns to a visible vector is

119875 (119874) = 1119885sumℎ119894

119890minus119864(119874ℎ119894) where 119885 = sum119874ℎ119894

119890minus119864(119874ℎ119894) (41)

The contrastive divergence method is then used to obtain theupdate formula of DBN parameters [19]

120597 log119875 (119874)120597119908119894119895 = 119875 (ℎ1119894 = 1 | 119874(0))119874(0)119895

minus 119875 (ℎ1119894 = 1 | 119874(119896))119874(119896)119895120597 log119875 (119874)

1205971198871 = 119874(0) minus 119874(119896)120597 log119875 (119874)

1205971198891 = 119875 (ℎ1119894 = 1 | 119874(0)) minus 119875 (ℎ1119894 = 1 | 119874(119896))

(42)

For the construction of BP network a classical model isused The relevant functions including activation functionenergy function and update formulas are then set based onthe classical model of BP network The structure diagram ofDBN is presented in Figure 2

The overall control flowchart is shown in Figure 3Firstly the function of control output voltage is obtained byminimizing energy function which is the sufficient conditionof optimal control and the MPC model is constructed basedon itTheDBN is designed according toMPCmodel to obtainthe parameters ofMPCThe training of DBN is achieved withthe use of robotic grinding historical data After identificationofMPCparameters theMPC is used to performonline close-loop control with the use of robotic grinding force deviation119890119889(119905119894) and system information Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1)119881119908(119905119894minus1)rdquo4 DBN Training and Simulation

Simulations are conducted to evaluate feasibility and per-formance of the DBN used to obtain the model predictivecontrol parameters of robotic grinding A BP network isconstructed and a comparison between the BP network andthe proposed DBN is conducted

6 Complexity

Table 1 Performance comparison between BP and DBN (mean square error)

Time BP network DBN119890119889 Δ119906119901 119890119889 Δ119906119901119905119894+0 986 lowast 10minus7 686 lowast 10minus7 604 lowast 10minus7 301 lowast 10minus7

119905119894+1 976 lowast 10minus7 896 lowast 10minus7 836 lowast 10minus7 976 lowast 10minus7119905119894+2 0156 0167 0152 0160119905119894+3 0165 0170 0160 0163119905119894+4 0177 0171 0171 0165119905119894+5 0187 0183 0185 0175119905119894+6 0194 0185 0191 0181119905119894+7 0216 0201 0205 0194

Output layer

BP (6 neurons)

RBM 2 (8 neurons)

RBM 1 (8 neurons)

Input layer

Wd3

Wd2

Wd1

ed(ti) ΔVw(ti-2) Δup (ti-1) Vw(ti-1)

ed(ti+1 ti) Δup(ti || ti)

Figure 2 DBN structure diagram

According to (35) and (37) the MPC is used toobtain 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 | 119905119894)] with the use of119868 = [119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] which meansthat the MPC represents the relationship between 119868 =[119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894)119890119889(119905119894+1 | 119905119894)] Therefore the training data is requiredto contain the relationship between 119868 = [119890119889(119905119894)Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 |119905119894)]

The training data of DBN is determined according to thefollowing aspects

(1) The parameters of training data are determinedaccording to influence factors of robotic grinding processincluding grinding depth grinding tool feed rate tool rota-tion speed and control output voltage As there is a functionalrelationship between feed rate and control output voltagewhen a controller is applied this paper uses feed rate changeto represent change of control voltage The ranges of theparameters are grinding depth is from 06mm to 12mminitial feed rate is from 05mms to 12mms feed rate changeis from 01 1198981198982s to 1 1198981198982s tool rotation speed is 4000rmin (2) Since grinding is a material removal process itis difficult and costly to perform dozens of experiments andobtain a large amount of training data the DBN model

used in this paper is only designed for specific case (roboticgrinding) as Oh and Jung did in their research [19] whichcan reduce the quantity demand of data for DBN training butalso reduce the applicability of DBN applying in other roboticoperations Based on this the scale and the size of learningsample are determined The scale of learning sample batch is40 packets and each packet has about 5000lowast4data 34 packetsare randomly selected for training and the remaining 6 areselected for model validation The maximum epoch is 300

The training performances of both networks are shownin Figure 4 For the model training of robotic grindingstatus at time 119905119894+0 the DBN realizes fitting at 31st epochand BP network realizes fitting at 52nd epoch The modeltraining of robotic grinding status from time 119905119894+0 to 119905119894+7 isalso conducted and the training performance of both BPnetwork and DBN is presented in Table 1 In overall thetraining performances of the proposed DBN are approximateto the training performances of the classical BP network Thetraining results are then used to construct

A simulation is conducted to evaluate the predictive per-formance of models obtained respectively by the proposedDBN and the BP network The steps of the simulation are asfollows Firstly a sample is selected randomly and the first10 data of the sample are taken as inputs Then predictionsof robotic grinding state are made according to these inputsAs shown in Figure 5 the prediction results of two modelsare basically consistent with the actual grinding force Theaverage prediction deviation of grinding force is about 05NThemaximum prediction deviation about 2N appears at thepeak of grinding force during cut-into state At 2nd secondthe grinding force predicted by the BP based model changesdramatically while the one obtained by theDBNbasedmodelis smoother Based on the above simulations a preconclusioncan be made that the performance of predictive modelobtained by DBN network is better than the one obtained byBP network

5 Robotic Grinding Control Experiments

51 Robotic Grinding System The robotic machining systemconsists of YASKAWA industrial robot MH24 six-axis forcesensorME-FK6D40 and grinding tool As shown in Figure 6the force sensor and grinding tool are installed on the end-effector of robot while the workpiece is installed on a fixed

Complexity 7

Energy Function

Control Output Voltage

Predictive Control ModelDeep Brief

NetworkModel

Parameters

DBN Structure

Robotic Grinding historical Data

Offline Training

Optimized Process

Offline

Transfer Function G(V)

Actual Force F(t)

Ideal ForceFlowast(t)

Force Deviation e(t)

System Information

+

Online

minus

T

I = [ed(ti) ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

E(ti) = ４

Δup(ti | ti) ed(ti+1 | ti)

Δup(ti | ti) = Z = (A1４A1)

minus1A1４D

ＧＣＨ

Figure 3 Schematic diagram of robot model predictive control process based on DBN

platform The spindle of grinding tool is perpendicular tothe Z direction of industrial robot sixth axis The worldcoordinate system is set according to the robot CartesiancoordinateThe relevant parameters of equipment are the six-axis force sensor is ME-FK6d40 Germany the workpiecematerial isQ235 steel the grinding tool is composed ofmotorhandle and grinding head

The force signals are collected by the force sensor anddelivered to the embedded real-time control system inPC with the use of Ethercat protocol The analogue filterfrequency of force sensor is 2500Hz and the samplingfrequency of embedded real-time control system is 1msThe control output is then calculated by embedded real-time control system and sent to YASKAWA robot controlcabinet tomodify output pulse and adjust the feed rate of end-effector The frequency of output voltage of control system isabout 100ms

52 Robotic Grinding Control Experiments To evaluate theperformance of proposed control approach robotic grind-ing control experiments are conducted The open looprobotic grinding experiment robotic grinding experimentwith fuzzy-PD control and robotic grinding experiment withmodel predictive control are conducted and comparisons

between these experiments are made The robotic grindingcontrol experiments are carried out on the steel plate plane(Q235)The rotation speed of grinding tool is 4000 rmin andthe grinding depth is 12mm the initial feed rate is 1mmsthe material of cutter is D1614M06

The grinding path of the experiment is shown in Figure 7The grinding tool moves from point A to D via point Band C The grinding path AB is perpendicular to workpiecesurface while the path BC is tangential to workpiece surfaceFor the convenience of discussion in subsequent articles ABis regarded as cut-into state while BC is regarded as stablegrinding state

521 Experiment with Open Loop Control and Experimentwith Fuzzy-PD Control

(A) Experiment withOpen LoopControlThe desired grindingforce for the experiment is 14 N The experiment result isshown in Figure 8 The grinding force increases dramaticallyto 29N during cut-into state and then decreases around15N The dramatically increase of grinding force whichis labeled as abrupt change during cut-into state can beillustrated by (10) and (16) When grinding tool cuts intothe workpiece the gap between feed rate and removal rate

8 Complexity

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)Best Training Performance is 98696e-07 at epoch 52

Train

Best

Goal

10 20 30 40 500Epoch

(a)

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 60367e-07 at epoch 31

TrainBestGoal

5 10 15 20 25 300Epoch

(b)

10minus2

100

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 021614 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(c)

10minus2

10minus1

100

101

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 020538 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(d)

Figure 4 Training performance comparison between BP and DBM (Mean Square Error) ((a) BP (time 119905119894+0) (b) DBN (time 119905119894+0) (c) BP(time 119905119894+7) (d) DBN (time 119905119894+7))

results in a deformation and leads to an increase grindingforce After perpendicular move the grinding tool starts tomove in tangential direction and the value of removal ratein perpendicular direction is larger than the one of feed rateresulting in a decrease of grinding force

(B) Experiment with Fuzzy-PD Control To verify theeffectiveness of proposed control approach an exper-iment based on fuzzy-PID control is conducted Theinputs of fuzzy-PD controller are force deviation 119890 andits first-order derivative 119890 and control parameters 119879p

and 119879d Their domains are 119890 = minus6 minus3 0 3 6 119890 =minus6 minus3 0 3 6 119879p = minus06 minus03 minus01 minus005 minus001 and119879d = minus04 minus02 minus01 minus005 minus001 The fuzzy reasoning isldquoIF A1 and B1 Then C1 and D1rdquo while the Gaussian functionis used as fuzzy membership function

The result of grinding experiment with PD control isshown in Figure 9 The grinding force increases considerablyand reaches a peak value of 25 N which is 4 N lower thanthe one of open loop control The corresponding feed rateincreases to 14mms at the first time and then decreaseto 07mms with a valley of 04mms During this process

Complexity 9

Verification SampleDBN based PredictionBP based Prediction

0

5

10

15

20

25

30G

rindi

ng F

orce

F (N

)

2 4 6 80Time t (s)

Figure 5 Prediction performance

5

2

1

43

Figure 6 Architecture of robotic cutting system ((1) YASKAWA 6-DOF industrial robot (2) rotating platform (3) six-axis force sensor(4) cutting tool (5) workpiece)

a fluctuation of feed rate appears at 1st second and thefluctuation amplitude of feed rate is around 01mms This isbecause when the grinding force is close to the target forcethe control parameters swift according to the fuzzy ruleswhich results in a fluctuation of control output This is acommon problem of fuzzy control as fuzzy rules are oftendefined according to the artificial experience which is easyto result in control oscillations caused by switching of controlrate In view of this a model predictive control approach isapplied to robotic grinding control

522 Robotic Grinding Experiment with Model PredictiveControl The robotic grinding experiment with model pre-dictive control is then conducted to reduce the grinding forcedeviation and feed rate fluctuation that occurs in cut-intostate The prediction time of model predictive control is setas 119905119894+10 and the experiment result is shown in Figure 10 Thepeak value of robotic grinding force is 18N which is 7N lowerand the corresponding feed rate curve changes from 14mmsto 07mms with a valley of 03mms The changes of robotic

B

A D

C

Figure 7 Grinding path

0

5

10

15

20

25

30

Grin

ding

For

ce F

(N)

2 4 6 8 100Time t (s)

Figure 8 Robotic grinding experiment with open loop control

grinding force and feed rate shown in Figure 10 are smoothercompared with the ones shown in Figure 9 This is becausethat the model predictive control approach can predict thefuture grinding deviation based on acquired information andresult in a control compensation to reduce the coming upforce deviation and feed rate fluctuation

To further verify the performance of proposed controlapproach another robotic grinding experiment based onmodel predictive control is conducted of which the predictiontime is set as 119905119894+100 As shown in Figure 11 the grinding forceincreases smoothly from 0N to 14N and the fluctuation ofgrinding force is around 2N The abrupt change of grindingforce which occurs during cut-into state as shown in Figures10 and 9 is completely eliminated The corresponding feedrate changes from 1mms to 075mms without an abruptincrease as shown in Figures 9 and 10 This is because whenthe prediction time is adjusted to 119905119894+100 the control systemcan have sufficient time to adjust feed rate smoothly whichcan lead to sufficient control compensation

To explore the flexibility of the proposed controlapproach the grinding depth is changed to 14mm whilethe grinding depth range of study sample is from 06mmto 12mm The experiment result with open loop control isshown in Figure 12(a) while the experiment results withmodel predictive control are shown in Figures 12(b) and12(c) As shown in Figure 12(a) the grinding force increasesdramatically to about 33 N during cut-into state and then

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 3

Deformationrelease

ap

Feed rate Vw

Figure 1 Generation of extrusion deformation

defined as well as the corresponding ideal trajectory of robotgrinding 119909119901 lowast (119905119894) Equation (11) can be written as

lim119894997888rarrinfin

(119909119901 (119905119894) minus 119909119901 lowast (119905119894)) = 0 (12)

Assuming that there is a corresponding relation between thegrinding position 119909119901(119905119894) and the robotic grinding force 119865(119905119894)when the system parameters (such as feed rate grinding toolrotation speed and material of workpiece) are constant Thecontrol target in this paper can be converted into

lim119894997888rarrinfin

(119865 (119905119894) minus 119865 lowast (119905119894)) = 0 (13)

where 119865lowast(119905119894) is the ideal grinding force corresponding to theideal grinding position 119909119901 lowast (119905119894)32 Model Predictive Control for Robotic Grinding Definerobot grinding force deviation as

119890119889 (119905119894) = 119865 (119905119894) minus 119865 lowast (119905119894) (14)

Assume that robot grinding force deviation at time 119905119894+1 isaffected by robot grinding force deviation 119890119889(119905119894) and systemtransfer function 119866(119881119908(119905119894)) at time 119905119894

119890119889 (119905119894+1) = 119860119890119890119889 (119905119894) + 119866 (119881119908 (119905119894)) (15)

where 119866(119881119908(119905119894)) is the control output which takes feed rate119881119908(119905119894) as variables 119860119890 is the weighting factor of the robotgrinding force deviation Assume 119860119890 = 1 Rewrite (15)according to (14)

119865 (119905119894+1) minus 119865 lowast (119905119894+1) = 119865 (119905119894) minus 119865 lowast (119905119894) + 119866 (119881119908 (119905119894))119865 (119905119894+1) minus 119865 (119905119894) = 119865 lowast (119905119894+1) minus 119865 lowast (119905119894)

+ 119866 (119881119908 (119905119894))Δ119865 (119905119894) = Δ119865 lowast (119905119894) + 119866 (119881119908 (119905119894))

(16)

where

Δ119865 (119905119894) = 119865 (119905119894+1) minus 119865 (119905119894)Δ119865 (119905119894) = 119872Δ 119909119886 (119905119894) + 119862Δ 119909119886 (119905119894) + 119870119889Δ119909119886 (119905119894)

+ 119870119889 (Δ120590119888 (119905119894) + Δ120590119895 (119905119894))(17)

Δ119865 lowast (119905119894) = 119865 lowast (119905119894+1) minus 119865 lowast (119905119894) (18)

Define Δ119881119908(119905119894) = 119881119908(119905119894+1) minus 119881119908(119905119894) 119886(119905119894) = 119881119908(119905119894) Δ119909119886(119905119894) =119881119908(119905119894)119889119905 According to (6) there isΔ120590119895(119905119894) = 120590119895(119905119894+1)minus120590119895(119905119894) =119881119908(119905119894+1) minus 119881119890(119905119894+1) and (17) can be rewritten as

Δ119865 (119905119894) = 119872Δ119886 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894)+ 119870119889119881119908 (119905119894) (119889119905 + 1)+ 119870119889 (Δ120590119888 (119905119894) minus 119881119890 (119905119894+1))

(19)

321 Robotic Grinding Trajectory Prediction Substitute (19)and (15) into (14) to obtain the relationship between roboticgrinding force deviation and feed rate

119890119889 (119905119894+1) = 119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894)+ 119870119889119881119908 (119905119894) (119889119905 + 1) (20)

where1198661015840(119905119894) = 119872Δ119886(119905119894) minus Δ119865 lowast (119905119894) + 119870119889(Δ120590119888(119905119894) minus 119881119890(119905119894+1))Therefore the robotic grinding force deviation matrix 119864119889(119905119894)from time 119905119894+1 to time 119905119894+119901 can be acquired by

119890119889 (119905119894+1 | 119905119894)= 119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894 | 119905119894)

+ 119870119889119881119908 (119905119894) (119889119905 + 1)(21)

119890119889 (119905119894+2 | 119905119894)= 119860119890119890119889 (119905119894+1 | 119905119894) + 1198661015840 (119905119894+1)

+ (119862 + 119870119889) Δ119881119908 (119905119894+1 | 119905119894)+ 119870119889119881119908 (119905119894+1 | 119905119894) (119889119905 + 1)

119864119889 (119905119894)= [119890119889 (119905119894+1 | 119905119894) 119890119889 (119905119894+2 | 119905119894) 119890119889 (119905119894+119901 | 119905119894)]

(22)

119890119889(119905119894+119901 | 119905119894) represents the predictive of robotic grinding forcedeviation at time 119905119894+119901 according to the system information attime 119905119894 Assume that there is a relationship between change offeed rate and input voltage

Δ119881119908 (119905119894) = 119860Δ119881119908 (119905119894minus1) + 119861Δ119906119901 (119905119894) (23)

4 Complexity

where 119860 and 119861 are weight factors The prediction of feed rateincrement from time 119905119894+1 to time 119905119894+119901 can be acquired by

Δ119881119908 (119905119894+1 | 119905119894) = 119860Δ119881119908 (119905119894) + 119861Δ119906119901 (119905119894+1 | 119905119894)Δ119881119908 (119905119894+1 | 119905119894) = 1198602Δ119881119908 (119905119894minus1) + 119860119861Δ119906119901 (119905119894 | 119905119894)

+ 119861Δ119906119901 (119905119894+1 | 119905119894)Δ119881119908119910 (119905119894+119901 | 119905119894) = (Δ119881119908 (119905119894+1 | 119905119894) Δ119881119908 (119905119894+2 | 119905119894)

Δ119881119908 (119905119894+119901 | 119905119894))

(24)

where Δ119881119908(119905119894+1 | 119905119894) Δ119906119901(119905119894+1 | 119905119894) respectively represent thechanges of feed rate and control voltage at time 119905119894 predictedaccording to the system information at time 119905119894+1322 Control Output Optimizing In order to optimize con-trol output voltage an energy function is defined as

119864 (119905119894) =119894+119899sum119894=1

1003817100381710038171003817Γ1 (119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1))10038171003817100381710038172

+ 1003817100381710038171003817Γ2 (119890119889 (119905119894+1 | 119905119894))10038171003817100381710038172(25)

where119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1)= 119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894 | 119905119894)

+ 119870119889119881119908 (119905119894) (119889119905 + 1) minus 119890119889 lowast (119905119894+1)(26)

Substitute (23) into (26)

119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1)= 119860119890119890119889 (119905119894) + 1198661015840 (119905119894)

+ (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1)+ (119862 + 119870119889) 119861Δ119906119901 (119905119894 | 119905119894) )+ 119870119889119881119908 (119905119894minus1) (119889119905 + 1) minus 119890119889 lowast (119905119894+1)

(27)

Define

119864 (119905119894) = 120588T120588120588 = 1198601119885 minus 119863 (28)

where

119863 = [[Γ1 (119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1) + 119870119889119881119908 (119905119894minus1) (119889119905 + 1) minus 119890119889 lowast (119905119894+1))

Γ2 (119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1) + 119870119889119881119908 (119905119894minus1) (119889119905 + 1)) ]]

1198601 = [Γ1 (119862 + 119870119889) 119861Γ2 (119862 + 119870119889) 119861]

119885 = Δ119906119901 (119905119894 | 119905119894)

(29)

According to (26) the extreme value of energy function canbe obtained by min 120588119879120588 Therefore there is extreme valuewhen

Δ119906119901 (119905119894 | 119905119894) = 119885 = (1198601T1198601)minus1 1198601T119863 (30)

Substitute (23) into (30)

Δ119906119901 (119905119894 | 119905119894) = 1198821119890119889 (119905119894) +1198822Δ119881119908 (119905119894minus2)+ 1198823Δ119906119901 (119905119894minus1) +1198824119881119908 (119905119894minus1) + 1198871 (31)

where

1198821 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889) 1198611198601198901198822 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)

sdot 119861 (119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119860

1198823 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)sdot 119861 (119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119861

1198824 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889) 119861119870119889 (119889119905 + 1) 1198871 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)

sdot 119861 (21198661015840 (119905119894) minus 119890119889 lowast (119905119894+1))(32)

323 Stability Analysis According to (22) (23) and (30)there is

119890119889 (119905119894+1 | 119905119894) = 1198825119890119889 (119905119894) + 1198826Δ119881119908 (119905119894minus2)+ 1198827Δ119906119901 (119905119894minus1) + 1198828119881119908 (119905119894minus1) + 1198872 (33)

where

Complexity 5

1198825 = 119860119890 + (119862 + 119870119889) 11986111988211198826 = ((119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119860 + (119862 + 119870119889) 1198611198822) 1198827 = ((119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119861 + (119862 + 119870119889) 1198611198823) 1198828 = ((119862 + 119870119889) 1198611198824 + 119870119889 (119889119905 + 1)) 1198872 = 1198661015840 (119905119894) + (119862 + 119870119889) 1198611198871

(34)

According to the model predictive control principle [18]when the eigenvalues of the matrix1198825 are all within the unitcircle the control system is asymptotically stable

324 Robot Grinding State Prediction Model Merging (33)and (31) there is

119874 = 119868119882119899T + 119861119887 (35)

where

119874 = [Δ119906119901 (119905119894 | 119905119894) 119890119889 (119905119894+1 | 119905119894)] 119882119899 = [11988211198822119882311988241198825119882611988271198828] 119861119887 = [1198871 1198872] 119868 = [119890119889 (119905119894) Δ119881119908 (119905119894minus2) Δ119906119901 (119905119894minus1) 119881119908 (119905119894minus1)]

(36)

119882119899 and 119861119887 represent predictive model parameters Then thedeep belief network is introduced to obtain predictive modelparameters

33 Predictive Model Based on Deep Belief Network Accord-ing to the deep belief network principle

119874 = (((1198681198821198891T + 1198611198891)1198821198892T + 1198611198892) )119882119889119894T + 119861119889119894(119894 = 1 2 119898) (37)

Comparing (37) and (35) it can be found that predictivemodel parameter 119882119899 can be obtained by weights productprod119898119894=1119882119889119894

119882119899T = 119898prod119894=1

119882119889119894T

119861119887 =119898minus1sum119895=1

119861119889119895119898minus1prod119895=1

119882119889119895+1T + 119861119889119898(38)

Therefore an assumption is made that prod119898119894=1119882119889119894 =[11988211198822119882311988241198825119882611988271198828] can be obtained bythe product of Γ1 Γ2 119860119890 119862 119870119889 119861 And subassumptionsare made that 119882119889119894 = [119861 1198612 119860 1198602 119861119889119905 119860119889119905 ]119882119889119894+1 = [119862 1198622 119870119889 1198701198892 ] 119882119889119894+2 = [(1198601T1198601)minus1Γ12(1198601T1198601)minus1Γ22 ] Based on these assumptions thedeep belief network is constructed which is comprisedof three layers including two layers of RestrictedBoltzmann Machine (RBM) and one Back-Propagation(BP) layer Two layers of RBM are used to obtain weights

119882119889119894 = [119861 1198612 119860 1198602 119861119889119905 119860119889119905 ] and 119882119889119894+1 = [119862 1198622 1198701198891198701198892 ] while the BP layer is used to obtain 119882119889119894+2 =[(1198601T1198601)minus1Γ12 (1198601T1198601)minus1Γ22 ] The inputs of DBN aregrinding force deviation 119890119889(119905119894) feed rate 119881119908(119905119894minus1) incrementof feed rate Δ119881119908(119905119894minus2) and increment of control outputvoltage Δ119906119901(119905119894minus1) while the outputs are Δ119906119901(119905119894 | 119905119894) and119890119889(119905119894+1 | 119905119894) The energy function of general RBM is [19]

119864 (V119894 ℎ119894) = minusV119894119879119908119894ℎ119894 minus 119887119894119879V119894 minus 119889119894119879ℎ119894 (119894 = 1 2) (39)

where V119894 ℎ119894 are explicit layer and hidden layer 119908119894 119887119894 119889119894 arerespectively network weight explicit layer bias and hiddenlayer bias Consider119874 as the first visible layer of first RBMandℎ1 as the hidden layer first RBM and (39) can be rewritten as

119864 (119874 ℎ119894) = minus119874119879119908119894ℎ119894 minus 119887119894119879119874 minus 119889119894119879ℎ119894 (40)

The probability that the RBM assigns to a visible vector is

119875 (119874) = 1119885sumℎ119894

119890minus119864(119874ℎ119894) where 119885 = sum119874ℎ119894

119890minus119864(119874ℎ119894) (41)

The contrastive divergence method is then used to obtain theupdate formula of DBN parameters [19]

120597 log119875 (119874)120597119908119894119895 = 119875 (ℎ1119894 = 1 | 119874(0))119874(0)119895

minus 119875 (ℎ1119894 = 1 | 119874(119896))119874(119896)119895120597 log119875 (119874)

1205971198871 = 119874(0) minus 119874(119896)120597 log119875 (119874)

1205971198891 = 119875 (ℎ1119894 = 1 | 119874(0)) minus 119875 (ℎ1119894 = 1 | 119874(119896))

(42)

For the construction of BP network a classical model isused The relevant functions including activation functionenergy function and update formulas are then set based onthe classical model of BP network The structure diagram ofDBN is presented in Figure 2

The overall control flowchart is shown in Figure 3Firstly the function of control output voltage is obtained byminimizing energy function which is the sufficient conditionof optimal control and the MPC model is constructed basedon itTheDBN is designed according toMPCmodel to obtainthe parameters ofMPCThe training of DBN is achieved withthe use of robotic grinding historical data After identificationofMPCparameters theMPC is used to performonline close-loop control with the use of robotic grinding force deviation119890119889(119905119894) and system information Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1)119881119908(119905119894minus1)rdquo4 DBN Training and Simulation

Simulations are conducted to evaluate feasibility and per-formance of the DBN used to obtain the model predictivecontrol parameters of robotic grinding A BP network isconstructed and a comparison between the BP network andthe proposed DBN is conducted

6 Complexity

Table 1 Performance comparison between BP and DBN (mean square error)

Time BP network DBN119890119889 Δ119906119901 119890119889 Δ119906119901119905119894+0 986 lowast 10minus7 686 lowast 10minus7 604 lowast 10minus7 301 lowast 10minus7

119905119894+1 976 lowast 10minus7 896 lowast 10minus7 836 lowast 10minus7 976 lowast 10minus7119905119894+2 0156 0167 0152 0160119905119894+3 0165 0170 0160 0163119905119894+4 0177 0171 0171 0165119905119894+5 0187 0183 0185 0175119905119894+6 0194 0185 0191 0181119905119894+7 0216 0201 0205 0194

Output layer

BP (6 neurons)

RBM 2 (8 neurons)

RBM 1 (8 neurons)

Input layer

Wd3

Wd2

Wd1

ed(ti) ΔVw(ti-2) Δup (ti-1) Vw(ti-1)

ed(ti+1 ti) Δup(ti || ti)

Figure 2 DBN structure diagram

According to (35) and (37) the MPC is used toobtain 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 | 119905119894)] with the use of119868 = [119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] which meansthat the MPC represents the relationship between 119868 =[119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894)119890119889(119905119894+1 | 119905119894)] Therefore the training data is requiredto contain the relationship between 119868 = [119890119889(119905119894)Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 |119905119894)]

The training data of DBN is determined according to thefollowing aspects

(1) The parameters of training data are determinedaccording to influence factors of robotic grinding processincluding grinding depth grinding tool feed rate tool rota-tion speed and control output voltage As there is a functionalrelationship between feed rate and control output voltagewhen a controller is applied this paper uses feed rate changeto represent change of control voltage The ranges of theparameters are grinding depth is from 06mm to 12mminitial feed rate is from 05mms to 12mms feed rate changeis from 01 1198981198982s to 1 1198981198982s tool rotation speed is 4000rmin (2) Since grinding is a material removal process itis difficult and costly to perform dozens of experiments andobtain a large amount of training data the DBN model

used in this paper is only designed for specific case (roboticgrinding) as Oh and Jung did in their research [19] whichcan reduce the quantity demand of data for DBN training butalso reduce the applicability of DBN applying in other roboticoperations Based on this the scale and the size of learningsample are determined The scale of learning sample batch is40 packets and each packet has about 5000lowast4data 34 packetsare randomly selected for training and the remaining 6 areselected for model validation The maximum epoch is 300

The training performances of both networks are shownin Figure 4 For the model training of robotic grindingstatus at time 119905119894+0 the DBN realizes fitting at 31st epochand BP network realizes fitting at 52nd epoch The modeltraining of robotic grinding status from time 119905119894+0 to 119905119894+7 isalso conducted and the training performance of both BPnetwork and DBN is presented in Table 1 In overall thetraining performances of the proposed DBN are approximateto the training performances of the classical BP network Thetraining results are then used to construct

A simulation is conducted to evaluate the predictive per-formance of models obtained respectively by the proposedDBN and the BP network The steps of the simulation are asfollows Firstly a sample is selected randomly and the first10 data of the sample are taken as inputs Then predictionsof robotic grinding state are made according to these inputsAs shown in Figure 5 the prediction results of two modelsare basically consistent with the actual grinding force Theaverage prediction deviation of grinding force is about 05NThemaximum prediction deviation about 2N appears at thepeak of grinding force during cut-into state At 2nd secondthe grinding force predicted by the BP based model changesdramatically while the one obtained by theDBNbasedmodelis smoother Based on the above simulations a preconclusioncan be made that the performance of predictive modelobtained by DBN network is better than the one obtained byBP network

5 Robotic Grinding Control Experiments

51 Robotic Grinding System The robotic machining systemconsists of YASKAWA industrial robot MH24 six-axis forcesensorME-FK6D40 and grinding tool As shown in Figure 6the force sensor and grinding tool are installed on the end-effector of robot while the workpiece is installed on a fixed

Complexity 7

Energy Function

Control Output Voltage

Predictive Control ModelDeep Brief

NetworkModel

Parameters

DBN Structure

Robotic Grinding historical Data

Offline Training

Optimized Process

Offline

Transfer Function G(V)

Actual Force F(t)

Ideal ForceFlowast(t)

Force Deviation e(t)

System Information

+

Online

minus

T

I = [ed(ti) ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

E(ti) = ４

Δup(ti | ti) ed(ti+1 | ti)

Δup(ti | ti) = Z = (A1４A1)

minus1A1４D

ＧＣＨ

Figure 3 Schematic diagram of robot model predictive control process based on DBN

platform The spindle of grinding tool is perpendicular tothe Z direction of industrial robot sixth axis The worldcoordinate system is set according to the robot CartesiancoordinateThe relevant parameters of equipment are the six-axis force sensor is ME-FK6d40 Germany the workpiecematerial isQ235 steel the grinding tool is composed ofmotorhandle and grinding head

The force signals are collected by the force sensor anddelivered to the embedded real-time control system inPC with the use of Ethercat protocol The analogue filterfrequency of force sensor is 2500Hz and the samplingfrequency of embedded real-time control system is 1msThe control output is then calculated by embedded real-time control system and sent to YASKAWA robot controlcabinet tomodify output pulse and adjust the feed rate of end-effector The frequency of output voltage of control system isabout 100ms

52 Robotic Grinding Control Experiments To evaluate theperformance of proposed control approach robotic grind-ing control experiments are conducted The open looprobotic grinding experiment robotic grinding experimentwith fuzzy-PD control and robotic grinding experiment withmodel predictive control are conducted and comparisons

between these experiments are made The robotic grindingcontrol experiments are carried out on the steel plate plane(Q235)The rotation speed of grinding tool is 4000 rmin andthe grinding depth is 12mm the initial feed rate is 1mmsthe material of cutter is D1614M06

The grinding path of the experiment is shown in Figure 7The grinding tool moves from point A to D via point Band C The grinding path AB is perpendicular to workpiecesurface while the path BC is tangential to workpiece surfaceFor the convenience of discussion in subsequent articles ABis regarded as cut-into state while BC is regarded as stablegrinding state

521 Experiment with Open Loop Control and Experimentwith Fuzzy-PD Control

(A) Experiment withOpen LoopControlThe desired grindingforce for the experiment is 14 N The experiment result isshown in Figure 8 The grinding force increases dramaticallyto 29N during cut-into state and then decreases around15N The dramatically increase of grinding force whichis labeled as abrupt change during cut-into state can beillustrated by (10) and (16) When grinding tool cuts intothe workpiece the gap between feed rate and removal rate

8 Complexity

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)Best Training Performance is 98696e-07 at epoch 52

Train

Best

Goal

10 20 30 40 500Epoch

(a)

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 60367e-07 at epoch 31

TrainBestGoal

5 10 15 20 25 300Epoch

(b)

10minus2

100

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 021614 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(c)

10minus2

10minus1

100

101

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 020538 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(d)

Figure 4 Training performance comparison between BP and DBM (Mean Square Error) ((a) BP (time 119905119894+0) (b) DBN (time 119905119894+0) (c) BP(time 119905119894+7) (d) DBN (time 119905119894+7))

results in a deformation and leads to an increase grindingforce After perpendicular move the grinding tool starts tomove in tangential direction and the value of removal ratein perpendicular direction is larger than the one of feed rateresulting in a decrease of grinding force

(B) Experiment with Fuzzy-PD Control To verify theeffectiveness of proposed control approach an exper-iment based on fuzzy-PID control is conducted Theinputs of fuzzy-PD controller are force deviation 119890 andits first-order derivative 119890 and control parameters 119879p

and 119879d Their domains are 119890 = minus6 minus3 0 3 6 119890 =minus6 minus3 0 3 6 119879p = minus06 minus03 minus01 minus005 minus001 and119879d = minus04 minus02 minus01 minus005 minus001 The fuzzy reasoning isldquoIF A1 and B1 Then C1 and D1rdquo while the Gaussian functionis used as fuzzy membership function

The result of grinding experiment with PD control isshown in Figure 9 The grinding force increases considerablyand reaches a peak value of 25 N which is 4 N lower thanthe one of open loop control The corresponding feed rateincreases to 14mms at the first time and then decreaseto 07mms with a valley of 04mms During this process

Complexity 9

Verification SampleDBN based PredictionBP based Prediction

0

5

10

15

20

25

30G

rindi

ng F

orce

F (N

)

2 4 6 80Time t (s)

Figure 5 Prediction performance

5

2

1

43

Figure 6 Architecture of robotic cutting system ((1) YASKAWA 6-DOF industrial robot (2) rotating platform (3) six-axis force sensor(4) cutting tool (5) workpiece)

a fluctuation of feed rate appears at 1st second and thefluctuation amplitude of feed rate is around 01mms This isbecause when the grinding force is close to the target forcethe control parameters swift according to the fuzzy ruleswhich results in a fluctuation of control output This is acommon problem of fuzzy control as fuzzy rules are oftendefined according to the artificial experience which is easyto result in control oscillations caused by switching of controlrate In view of this a model predictive control approach isapplied to robotic grinding control

522 Robotic Grinding Experiment with Model PredictiveControl The robotic grinding experiment with model pre-dictive control is then conducted to reduce the grinding forcedeviation and feed rate fluctuation that occurs in cut-intostate The prediction time of model predictive control is setas 119905119894+10 and the experiment result is shown in Figure 10 Thepeak value of robotic grinding force is 18N which is 7N lowerand the corresponding feed rate curve changes from 14mmsto 07mms with a valley of 03mms The changes of robotic

B

A D

C

Figure 7 Grinding path

0

5

10

15

20

25

30

Grin

ding

For

ce F

(N)

2 4 6 8 100Time t (s)

Figure 8 Robotic grinding experiment with open loop control

grinding force and feed rate shown in Figure 10 are smoothercompared with the ones shown in Figure 9 This is becausethat the model predictive control approach can predict thefuture grinding deviation based on acquired information andresult in a control compensation to reduce the coming upforce deviation and feed rate fluctuation

To further verify the performance of proposed controlapproach another robotic grinding experiment based onmodel predictive control is conducted of which the predictiontime is set as 119905119894+100 As shown in Figure 11 the grinding forceincreases smoothly from 0N to 14N and the fluctuation ofgrinding force is around 2N The abrupt change of grindingforce which occurs during cut-into state as shown in Figures10 and 9 is completely eliminated The corresponding feedrate changes from 1mms to 075mms without an abruptincrease as shown in Figures 9 and 10 This is because whenthe prediction time is adjusted to 119905119894+100 the control systemcan have sufficient time to adjust feed rate smoothly whichcan lead to sufficient control compensation

To explore the flexibility of the proposed controlapproach the grinding depth is changed to 14mm whilethe grinding depth range of study sample is from 06mmto 12mm The experiment result with open loop control isshown in Figure 12(a) while the experiment results withmodel predictive control are shown in Figures 12(b) and12(c) As shown in Figure 12(a) the grinding force increasesdramatically to about 33 N during cut-into state and then

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Complexity

where 119860 and 119861 are weight factors The prediction of feed rateincrement from time 119905119894+1 to time 119905119894+119901 can be acquired by

Δ119881119908 (119905119894+1 | 119905119894) = 119860Δ119881119908 (119905119894) + 119861Δ119906119901 (119905119894+1 | 119905119894)Δ119881119908 (119905119894+1 | 119905119894) = 1198602Δ119881119908 (119905119894minus1) + 119860119861Δ119906119901 (119905119894 | 119905119894)

+ 119861Δ119906119901 (119905119894+1 | 119905119894)Δ119881119908119910 (119905119894+119901 | 119905119894) = (Δ119881119908 (119905119894+1 | 119905119894) Δ119881119908 (119905119894+2 | 119905119894)

Δ119881119908 (119905119894+119901 | 119905119894))

(24)

where Δ119881119908(119905119894+1 | 119905119894) Δ119906119901(119905119894+1 | 119905119894) respectively represent thechanges of feed rate and control voltage at time 119905119894 predictedaccording to the system information at time 119905119894+1322 Control Output Optimizing In order to optimize con-trol output voltage an energy function is defined as

119864 (119905119894) =119894+119899sum119894=1

1003817100381710038171003817Γ1 (119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1))10038171003817100381710038172

+ 1003817100381710038171003817Γ2 (119890119889 (119905119894+1 | 119905119894))10038171003817100381710038172(25)

where119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1)= 119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862 + 119870119889) Δ119881119908 (119905119894 | 119905119894)

+ 119870119889119881119908 (119905119894) (119889119905 + 1) minus 119890119889 lowast (119905119894+1)(26)

Substitute (23) into (26)

119890119889 (119905119894+1 | 119905119894) minus 119890119889 lowast (119905119894+1)= 119860119890119890119889 (119905119894) + 1198661015840 (119905119894)

+ (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1)+ (119862 + 119870119889) 119861Δ119906119901 (119905119894 | 119905119894) )+ 119870119889119881119908 (119905119894minus1) (119889119905 + 1) minus 119890119889 lowast (119905119894+1)

(27)

Define

119864 (119905119894) = 120588T120588120588 = 1198601119885 minus 119863 (28)

where

119863 = [[Γ1 (119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1) + 119870119889119881119908 (119905119894minus1) (119889119905 + 1) minus 119890119889 lowast (119905119894+1))

Γ2 (119860119890119890119889 (119905119894) + 1198661015840 (119905119894) + (119862119860 + 119870119889 (119860 + 119889119905 + 1)) Δ119881119908 (119905119894minus1) + 119870119889119881119908 (119905119894minus1) (119889119905 + 1)) ]]

1198601 = [Γ1 (119862 + 119870119889) 119861Γ2 (119862 + 119870119889) 119861]

119885 = Δ119906119901 (119905119894 | 119905119894)

(29)

According to (26) the extreme value of energy function canbe obtained by min 120588119879120588 Therefore there is extreme valuewhen

Δ119906119901 (119905119894 | 119905119894) = 119885 = (1198601T1198601)minus1 1198601T119863 (30)

Substitute (23) into (30)

Δ119906119901 (119905119894 | 119905119894) = 1198821119890119889 (119905119894) +1198822Δ119881119908 (119905119894minus2)+ 1198823Δ119906119901 (119905119894minus1) +1198824119881119908 (119905119894minus1) + 1198871 (31)

where

1198821 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889) 1198611198601198901198822 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)

sdot 119861 (119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119860

1198823 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)sdot 119861 (119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119861

1198824 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889) 119861119870119889 (119889119905 + 1) 1198871 = (1198601T1198601)minus1 (Γ12 + Γ22) (119862 + 119870119889)

sdot 119861 (21198661015840 (119905119894) minus 119890119889 lowast (119905119894+1))(32)

323 Stability Analysis According to (22) (23) and (30)there is

119890119889 (119905119894+1 | 119905119894) = 1198825119890119889 (119905119894) + 1198826Δ119881119908 (119905119894minus2)+ 1198827Δ119906119901 (119905119894minus1) + 1198828119881119908 (119905119894minus1) + 1198872 (33)

where

Complexity 5

1198825 = 119860119890 + (119862 + 119870119889) 11986111988211198826 = ((119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119860 + (119862 + 119870119889) 1198611198822) 1198827 = ((119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119861 + (119862 + 119870119889) 1198611198823) 1198828 = ((119862 + 119870119889) 1198611198824 + 119870119889 (119889119905 + 1)) 1198872 = 1198661015840 (119905119894) + (119862 + 119870119889) 1198611198871

(34)

According to the model predictive control principle [18]when the eigenvalues of the matrix1198825 are all within the unitcircle the control system is asymptotically stable

324 Robot Grinding State Prediction Model Merging (33)and (31) there is

119874 = 119868119882119899T + 119861119887 (35)

where

119874 = [Δ119906119901 (119905119894 | 119905119894) 119890119889 (119905119894+1 | 119905119894)] 119882119899 = [11988211198822119882311988241198825119882611988271198828] 119861119887 = [1198871 1198872] 119868 = [119890119889 (119905119894) Δ119881119908 (119905119894minus2) Δ119906119901 (119905119894minus1) 119881119908 (119905119894minus1)]

(36)

119882119899 and 119861119887 represent predictive model parameters Then thedeep belief network is introduced to obtain predictive modelparameters

33 Predictive Model Based on Deep Belief Network Accord-ing to the deep belief network principle

119874 = (((1198681198821198891T + 1198611198891)1198821198892T + 1198611198892) )119882119889119894T + 119861119889119894(119894 = 1 2 119898) (37)

Comparing (37) and (35) it can be found that predictivemodel parameter 119882119899 can be obtained by weights productprod119898119894=1119882119889119894

119882119899T = 119898prod119894=1

119882119889119894T

119861119887 =119898minus1sum119895=1

119861119889119895119898minus1prod119895=1

119882119889119895+1T + 119861119889119898(38)

Therefore an assumption is made that prod119898119894=1119882119889119894 =[11988211198822119882311988241198825119882611988271198828] can be obtained bythe product of Γ1 Γ2 119860119890 119862 119870119889 119861 And subassumptionsare made that 119882119889119894 = [119861 1198612 119860 1198602 119861119889119905 119860119889119905 ]119882119889119894+1 = [119862 1198622 119870119889 1198701198892 ] 119882119889119894+2 = [(1198601T1198601)minus1Γ12(1198601T1198601)minus1Γ22 ] Based on these assumptions thedeep belief network is constructed which is comprisedof three layers including two layers of RestrictedBoltzmann Machine (RBM) and one Back-Propagation(BP) layer Two layers of RBM are used to obtain weights

119882119889119894 = [119861 1198612 119860 1198602 119861119889119905 119860119889119905 ] and 119882119889119894+1 = [119862 1198622 1198701198891198701198892 ] while the BP layer is used to obtain 119882119889119894+2 =[(1198601T1198601)minus1Γ12 (1198601T1198601)minus1Γ22 ] The inputs of DBN aregrinding force deviation 119890119889(119905119894) feed rate 119881119908(119905119894minus1) incrementof feed rate Δ119881119908(119905119894minus2) and increment of control outputvoltage Δ119906119901(119905119894minus1) while the outputs are Δ119906119901(119905119894 | 119905119894) and119890119889(119905119894+1 | 119905119894) The energy function of general RBM is [19]

119864 (V119894 ℎ119894) = minusV119894119879119908119894ℎ119894 minus 119887119894119879V119894 minus 119889119894119879ℎ119894 (119894 = 1 2) (39)

where V119894 ℎ119894 are explicit layer and hidden layer 119908119894 119887119894 119889119894 arerespectively network weight explicit layer bias and hiddenlayer bias Consider119874 as the first visible layer of first RBMandℎ1 as the hidden layer first RBM and (39) can be rewritten as

119864 (119874 ℎ119894) = minus119874119879119908119894ℎ119894 minus 119887119894119879119874 minus 119889119894119879ℎ119894 (40)

The probability that the RBM assigns to a visible vector is

119875 (119874) = 1119885sumℎ119894

119890minus119864(119874ℎ119894) where 119885 = sum119874ℎ119894

119890minus119864(119874ℎ119894) (41)

The contrastive divergence method is then used to obtain theupdate formula of DBN parameters [19]

120597 log119875 (119874)120597119908119894119895 = 119875 (ℎ1119894 = 1 | 119874(0))119874(0)119895

minus 119875 (ℎ1119894 = 1 | 119874(119896))119874(119896)119895120597 log119875 (119874)

1205971198871 = 119874(0) minus 119874(119896)120597 log119875 (119874)

1205971198891 = 119875 (ℎ1119894 = 1 | 119874(0)) minus 119875 (ℎ1119894 = 1 | 119874(119896))

(42)

For the construction of BP network a classical model isused The relevant functions including activation functionenergy function and update formulas are then set based onthe classical model of BP network The structure diagram ofDBN is presented in Figure 2

The overall control flowchart is shown in Figure 3Firstly the function of control output voltage is obtained byminimizing energy function which is the sufficient conditionof optimal control and the MPC model is constructed basedon itTheDBN is designed according toMPCmodel to obtainthe parameters ofMPCThe training of DBN is achieved withthe use of robotic grinding historical data After identificationofMPCparameters theMPC is used to performonline close-loop control with the use of robotic grinding force deviation119890119889(119905119894) and system information Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1)119881119908(119905119894minus1)rdquo4 DBN Training and Simulation

Simulations are conducted to evaluate feasibility and per-formance of the DBN used to obtain the model predictivecontrol parameters of robotic grinding A BP network isconstructed and a comparison between the BP network andthe proposed DBN is conducted

6 Complexity

Table 1 Performance comparison between BP and DBN (mean square error)

Time BP network DBN119890119889 Δ119906119901 119890119889 Δ119906119901119905119894+0 986 lowast 10minus7 686 lowast 10minus7 604 lowast 10minus7 301 lowast 10minus7

119905119894+1 976 lowast 10minus7 896 lowast 10minus7 836 lowast 10minus7 976 lowast 10minus7119905119894+2 0156 0167 0152 0160119905119894+3 0165 0170 0160 0163119905119894+4 0177 0171 0171 0165119905119894+5 0187 0183 0185 0175119905119894+6 0194 0185 0191 0181119905119894+7 0216 0201 0205 0194

Output layer

BP (6 neurons)

RBM 2 (8 neurons)

RBM 1 (8 neurons)

Input layer

Wd3

Wd2

Wd1

ed(ti) ΔVw(ti-2) Δup (ti-1) Vw(ti-1)

ed(ti+1 ti) Δup(ti || ti)

Figure 2 DBN structure diagram

According to (35) and (37) the MPC is used toobtain 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 | 119905119894)] with the use of119868 = [119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] which meansthat the MPC represents the relationship between 119868 =[119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894)119890119889(119905119894+1 | 119905119894)] Therefore the training data is requiredto contain the relationship between 119868 = [119890119889(119905119894)Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 |119905119894)]

The training data of DBN is determined according to thefollowing aspects

(1) The parameters of training data are determinedaccording to influence factors of robotic grinding processincluding grinding depth grinding tool feed rate tool rota-tion speed and control output voltage As there is a functionalrelationship between feed rate and control output voltagewhen a controller is applied this paper uses feed rate changeto represent change of control voltage The ranges of theparameters are grinding depth is from 06mm to 12mminitial feed rate is from 05mms to 12mms feed rate changeis from 01 1198981198982s to 1 1198981198982s tool rotation speed is 4000rmin (2) Since grinding is a material removal process itis difficult and costly to perform dozens of experiments andobtain a large amount of training data the DBN model

used in this paper is only designed for specific case (roboticgrinding) as Oh and Jung did in their research [19] whichcan reduce the quantity demand of data for DBN training butalso reduce the applicability of DBN applying in other roboticoperations Based on this the scale and the size of learningsample are determined The scale of learning sample batch is40 packets and each packet has about 5000lowast4data 34 packetsare randomly selected for training and the remaining 6 areselected for model validation The maximum epoch is 300

The training performances of both networks are shownin Figure 4 For the model training of robotic grindingstatus at time 119905119894+0 the DBN realizes fitting at 31st epochand BP network realizes fitting at 52nd epoch The modeltraining of robotic grinding status from time 119905119894+0 to 119905119894+7 isalso conducted and the training performance of both BPnetwork and DBN is presented in Table 1 In overall thetraining performances of the proposed DBN are approximateto the training performances of the classical BP network Thetraining results are then used to construct

A simulation is conducted to evaluate the predictive per-formance of models obtained respectively by the proposedDBN and the BP network The steps of the simulation are asfollows Firstly a sample is selected randomly and the first10 data of the sample are taken as inputs Then predictionsof robotic grinding state are made according to these inputsAs shown in Figure 5 the prediction results of two modelsare basically consistent with the actual grinding force Theaverage prediction deviation of grinding force is about 05NThemaximum prediction deviation about 2N appears at thepeak of grinding force during cut-into state At 2nd secondthe grinding force predicted by the BP based model changesdramatically while the one obtained by theDBNbasedmodelis smoother Based on the above simulations a preconclusioncan be made that the performance of predictive modelobtained by DBN network is better than the one obtained byBP network

5 Robotic Grinding Control Experiments

51 Robotic Grinding System The robotic machining systemconsists of YASKAWA industrial robot MH24 six-axis forcesensorME-FK6D40 and grinding tool As shown in Figure 6the force sensor and grinding tool are installed on the end-effector of robot while the workpiece is installed on a fixed

Complexity 7

Energy Function

Control Output Voltage

Predictive Control ModelDeep Brief

NetworkModel

Parameters

DBN Structure

Robotic Grinding historical Data

Offline Training

Optimized Process

Offline

Transfer Function G(V)

Actual Force F(t)

Ideal ForceFlowast(t)

Force Deviation e(t)

System Information

+

Online

minus

T

I = [ed(ti) ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

E(ti) = ４

Δup(ti | ti) ed(ti+1 | ti)

Δup(ti | ti) = Z = (A1４A1)

minus1A1４D

ＧＣＨ

Figure 3 Schematic diagram of robot model predictive control process based on DBN

platform The spindle of grinding tool is perpendicular tothe Z direction of industrial robot sixth axis The worldcoordinate system is set according to the robot CartesiancoordinateThe relevant parameters of equipment are the six-axis force sensor is ME-FK6d40 Germany the workpiecematerial isQ235 steel the grinding tool is composed ofmotorhandle and grinding head

The force signals are collected by the force sensor anddelivered to the embedded real-time control system inPC with the use of Ethercat protocol The analogue filterfrequency of force sensor is 2500Hz and the samplingfrequency of embedded real-time control system is 1msThe control output is then calculated by embedded real-time control system and sent to YASKAWA robot controlcabinet tomodify output pulse and adjust the feed rate of end-effector The frequency of output voltage of control system isabout 100ms

52 Robotic Grinding Control Experiments To evaluate theperformance of proposed control approach robotic grind-ing control experiments are conducted The open looprobotic grinding experiment robotic grinding experimentwith fuzzy-PD control and robotic grinding experiment withmodel predictive control are conducted and comparisons

between these experiments are made The robotic grindingcontrol experiments are carried out on the steel plate plane(Q235)The rotation speed of grinding tool is 4000 rmin andthe grinding depth is 12mm the initial feed rate is 1mmsthe material of cutter is D1614M06

The grinding path of the experiment is shown in Figure 7The grinding tool moves from point A to D via point Band C The grinding path AB is perpendicular to workpiecesurface while the path BC is tangential to workpiece surfaceFor the convenience of discussion in subsequent articles ABis regarded as cut-into state while BC is regarded as stablegrinding state

521 Experiment with Open Loop Control and Experimentwith Fuzzy-PD Control

(A) Experiment withOpen LoopControlThe desired grindingforce for the experiment is 14 N The experiment result isshown in Figure 8 The grinding force increases dramaticallyto 29N during cut-into state and then decreases around15N The dramatically increase of grinding force whichis labeled as abrupt change during cut-into state can beillustrated by (10) and (16) When grinding tool cuts intothe workpiece the gap between feed rate and removal rate

8 Complexity

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)Best Training Performance is 98696e-07 at epoch 52

Train

Best

Goal

10 20 30 40 500Epoch

(a)

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 60367e-07 at epoch 31

TrainBestGoal

5 10 15 20 25 300Epoch

(b)

10minus2

100

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 021614 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(c)

10minus2

10minus1

100

101

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 020538 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(d)

Figure 4 Training performance comparison between BP and DBM (Mean Square Error) ((a) BP (time 119905119894+0) (b) DBN (time 119905119894+0) (c) BP(time 119905119894+7) (d) DBN (time 119905119894+7))

results in a deformation and leads to an increase grindingforce After perpendicular move the grinding tool starts tomove in tangential direction and the value of removal ratein perpendicular direction is larger than the one of feed rateresulting in a decrease of grinding force

(B) Experiment with Fuzzy-PD Control To verify theeffectiveness of proposed control approach an exper-iment based on fuzzy-PID control is conducted Theinputs of fuzzy-PD controller are force deviation 119890 andits first-order derivative 119890 and control parameters 119879p

and 119879d Their domains are 119890 = minus6 minus3 0 3 6 119890 =minus6 minus3 0 3 6 119879p = minus06 minus03 minus01 minus005 minus001 and119879d = minus04 minus02 minus01 minus005 minus001 The fuzzy reasoning isldquoIF A1 and B1 Then C1 and D1rdquo while the Gaussian functionis used as fuzzy membership function

The result of grinding experiment with PD control isshown in Figure 9 The grinding force increases considerablyand reaches a peak value of 25 N which is 4 N lower thanthe one of open loop control The corresponding feed rateincreases to 14mms at the first time and then decreaseto 07mms with a valley of 04mms During this process

Complexity 9

Verification SampleDBN based PredictionBP based Prediction

0

5

10

15

20

25

30G

rindi

ng F

orce

F (N

)

2 4 6 80Time t (s)

Figure 5 Prediction performance

5

2

1

43

Figure 6 Architecture of robotic cutting system ((1) YASKAWA 6-DOF industrial robot (2) rotating platform (3) six-axis force sensor(4) cutting tool (5) workpiece)

a fluctuation of feed rate appears at 1st second and thefluctuation amplitude of feed rate is around 01mms This isbecause when the grinding force is close to the target forcethe control parameters swift according to the fuzzy ruleswhich results in a fluctuation of control output This is acommon problem of fuzzy control as fuzzy rules are oftendefined according to the artificial experience which is easyto result in control oscillations caused by switching of controlrate In view of this a model predictive control approach isapplied to robotic grinding control

522 Robotic Grinding Experiment with Model PredictiveControl The robotic grinding experiment with model pre-dictive control is then conducted to reduce the grinding forcedeviation and feed rate fluctuation that occurs in cut-intostate The prediction time of model predictive control is setas 119905119894+10 and the experiment result is shown in Figure 10 Thepeak value of robotic grinding force is 18N which is 7N lowerand the corresponding feed rate curve changes from 14mmsto 07mms with a valley of 03mms The changes of robotic

B

A D

C

Figure 7 Grinding path

0

5

10

15

20

25

30

Grin

ding

For

ce F

(N)

2 4 6 8 100Time t (s)

Figure 8 Robotic grinding experiment with open loop control

grinding force and feed rate shown in Figure 10 are smoothercompared with the ones shown in Figure 9 This is becausethat the model predictive control approach can predict thefuture grinding deviation based on acquired information andresult in a control compensation to reduce the coming upforce deviation and feed rate fluctuation

To further verify the performance of proposed controlapproach another robotic grinding experiment based onmodel predictive control is conducted of which the predictiontime is set as 119905119894+100 As shown in Figure 11 the grinding forceincreases smoothly from 0N to 14N and the fluctuation ofgrinding force is around 2N The abrupt change of grindingforce which occurs during cut-into state as shown in Figures10 and 9 is completely eliminated The corresponding feedrate changes from 1mms to 075mms without an abruptincrease as shown in Figures 9 and 10 This is because whenthe prediction time is adjusted to 119905119894+100 the control systemcan have sufficient time to adjust feed rate smoothly whichcan lead to sufficient control compensation

To explore the flexibility of the proposed controlapproach the grinding depth is changed to 14mm whilethe grinding depth range of study sample is from 06mmto 12mm The experiment result with open loop control isshown in Figure 12(a) while the experiment results withmodel predictive control are shown in Figures 12(b) and12(c) As shown in Figure 12(a) the grinding force increasesdramatically to about 33 N during cut-into state and then

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 5

1198825 = 119860119890 + (119862 + 119870119889) 11986111988211198826 = ((119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119860 + (119862 + 119870119889) 1198611198822) 1198827 = ((119862119860 + 119870119889 (119860 + 119889119905 + 1)) 119861 + (119862 + 119870119889) 1198611198823) 1198828 = ((119862 + 119870119889) 1198611198824 + 119870119889 (119889119905 + 1)) 1198872 = 1198661015840 (119905119894) + (119862 + 119870119889) 1198611198871

(34)

According to the model predictive control principle [18]when the eigenvalues of the matrix1198825 are all within the unitcircle the control system is asymptotically stable

324 Robot Grinding State Prediction Model Merging (33)and (31) there is

119874 = 119868119882119899T + 119861119887 (35)

where

119874 = [Δ119906119901 (119905119894 | 119905119894) 119890119889 (119905119894+1 | 119905119894)] 119882119899 = [11988211198822119882311988241198825119882611988271198828] 119861119887 = [1198871 1198872] 119868 = [119890119889 (119905119894) Δ119881119908 (119905119894minus2) Δ119906119901 (119905119894minus1) 119881119908 (119905119894minus1)]

(36)

119882119899 and 119861119887 represent predictive model parameters Then thedeep belief network is introduced to obtain predictive modelparameters

33 Predictive Model Based on Deep Belief Network Accord-ing to the deep belief network principle

119874 = (((1198681198821198891T + 1198611198891)1198821198892T + 1198611198892) )119882119889119894T + 119861119889119894(119894 = 1 2 119898) (37)

Comparing (37) and (35) it can be found that predictivemodel parameter 119882119899 can be obtained by weights productprod119898119894=1119882119889119894

119882119899T = 119898prod119894=1

119882119889119894T

119861119887 =119898minus1sum119895=1

119861119889119895119898minus1prod119895=1

119882119889119895+1T + 119861119889119898(38)

Therefore an assumption is made that prod119898119894=1119882119889119894 =[11988211198822119882311988241198825119882611988271198828] can be obtained bythe product of Γ1 Γ2 119860119890 119862 119870119889 119861 And subassumptionsare made that 119882119889119894 = [119861 1198612 119860 1198602 119861119889119905 119860119889119905 ]119882119889119894+1 = [119862 1198622 119870119889 1198701198892 ] 119882119889119894+2 = [(1198601T1198601)minus1Γ12(1198601T1198601)minus1Γ22 ] Based on these assumptions thedeep belief network is constructed which is comprisedof three layers including two layers of RestrictedBoltzmann Machine (RBM) and one Back-Propagation(BP) layer Two layers of RBM are used to obtain weights

119882119889119894 = [119861 1198612 119860 1198602 119861119889119905 119860119889119905 ] and 119882119889119894+1 = [119862 1198622 1198701198891198701198892 ] while the BP layer is used to obtain 119882119889119894+2 =[(1198601T1198601)minus1Γ12 (1198601T1198601)minus1Γ22 ] The inputs of DBN aregrinding force deviation 119890119889(119905119894) feed rate 119881119908(119905119894minus1) incrementof feed rate Δ119881119908(119905119894minus2) and increment of control outputvoltage Δ119906119901(119905119894minus1) while the outputs are Δ119906119901(119905119894 | 119905119894) and119890119889(119905119894+1 | 119905119894) The energy function of general RBM is [19]

119864 (V119894 ℎ119894) = minusV119894119879119908119894ℎ119894 minus 119887119894119879V119894 minus 119889119894119879ℎ119894 (119894 = 1 2) (39)

where V119894 ℎ119894 are explicit layer and hidden layer 119908119894 119887119894 119889119894 arerespectively network weight explicit layer bias and hiddenlayer bias Consider119874 as the first visible layer of first RBMandℎ1 as the hidden layer first RBM and (39) can be rewritten as

119864 (119874 ℎ119894) = minus119874119879119908119894ℎ119894 minus 119887119894119879119874 minus 119889119894119879ℎ119894 (40)

The probability that the RBM assigns to a visible vector is

119875 (119874) = 1119885sumℎ119894

119890minus119864(119874ℎ119894) where 119885 = sum119874ℎ119894

119890minus119864(119874ℎ119894) (41)

The contrastive divergence method is then used to obtain theupdate formula of DBN parameters [19]

120597 log119875 (119874)120597119908119894119895 = 119875 (ℎ1119894 = 1 | 119874(0))119874(0)119895

minus 119875 (ℎ1119894 = 1 | 119874(119896))119874(119896)119895120597 log119875 (119874)

1205971198871 = 119874(0) minus 119874(119896)120597 log119875 (119874)

1205971198891 = 119875 (ℎ1119894 = 1 | 119874(0)) minus 119875 (ℎ1119894 = 1 | 119874(119896))

(42)

For the construction of BP network a classical model isused The relevant functions including activation functionenergy function and update formulas are then set based onthe classical model of BP network The structure diagram ofDBN is presented in Figure 2

The overall control flowchart is shown in Figure 3Firstly the function of control output voltage is obtained byminimizing energy function which is the sufficient conditionof optimal control and the MPC model is constructed basedon itTheDBN is designed according toMPCmodel to obtainthe parameters ofMPCThe training of DBN is achieved withthe use of robotic grinding historical data After identificationofMPCparameters theMPC is used to performonline close-loop control with the use of robotic grinding force deviation119890119889(119905119894) and system information Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1)119881119908(119905119894minus1)rdquo4 DBN Training and Simulation

Simulations are conducted to evaluate feasibility and per-formance of the DBN used to obtain the model predictivecontrol parameters of robotic grinding A BP network isconstructed and a comparison between the BP network andthe proposed DBN is conducted

6 Complexity

Table 1 Performance comparison between BP and DBN (mean square error)

Time BP network DBN119890119889 Δ119906119901 119890119889 Δ119906119901119905119894+0 986 lowast 10minus7 686 lowast 10minus7 604 lowast 10minus7 301 lowast 10minus7

119905119894+1 976 lowast 10minus7 896 lowast 10minus7 836 lowast 10minus7 976 lowast 10minus7119905119894+2 0156 0167 0152 0160119905119894+3 0165 0170 0160 0163119905119894+4 0177 0171 0171 0165119905119894+5 0187 0183 0185 0175119905119894+6 0194 0185 0191 0181119905119894+7 0216 0201 0205 0194

Output layer

BP (6 neurons)

RBM 2 (8 neurons)

RBM 1 (8 neurons)

Input layer

Wd3

Wd2

Wd1

ed(ti) ΔVw(ti-2) Δup (ti-1) Vw(ti-1)

ed(ti+1 ti) Δup(ti || ti)

Figure 2 DBN structure diagram

According to (35) and (37) the MPC is used toobtain 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 | 119905119894)] with the use of119868 = [119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] which meansthat the MPC represents the relationship between 119868 =[119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894)119890119889(119905119894+1 | 119905119894)] Therefore the training data is requiredto contain the relationship between 119868 = [119890119889(119905119894)Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 |119905119894)]

The training data of DBN is determined according to thefollowing aspects

(1) The parameters of training data are determinedaccording to influence factors of robotic grinding processincluding grinding depth grinding tool feed rate tool rota-tion speed and control output voltage As there is a functionalrelationship between feed rate and control output voltagewhen a controller is applied this paper uses feed rate changeto represent change of control voltage The ranges of theparameters are grinding depth is from 06mm to 12mminitial feed rate is from 05mms to 12mms feed rate changeis from 01 1198981198982s to 1 1198981198982s tool rotation speed is 4000rmin (2) Since grinding is a material removal process itis difficult and costly to perform dozens of experiments andobtain a large amount of training data the DBN model

used in this paper is only designed for specific case (roboticgrinding) as Oh and Jung did in their research [19] whichcan reduce the quantity demand of data for DBN training butalso reduce the applicability of DBN applying in other roboticoperations Based on this the scale and the size of learningsample are determined The scale of learning sample batch is40 packets and each packet has about 5000lowast4data 34 packetsare randomly selected for training and the remaining 6 areselected for model validation The maximum epoch is 300

The training performances of both networks are shownin Figure 4 For the model training of robotic grindingstatus at time 119905119894+0 the DBN realizes fitting at 31st epochand BP network realizes fitting at 52nd epoch The modeltraining of robotic grinding status from time 119905119894+0 to 119905119894+7 isalso conducted and the training performance of both BPnetwork and DBN is presented in Table 1 In overall thetraining performances of the proposed DBN are approximateto the training performances of the classical BP network Thetraining results are then used to construct

A simulation is conducted to evaluate the predictive per-formance of models obtained respectively by the proposedDBN and the BP network The steps of the simulation are asfollows Firstly a sample is selected randomly and the first10 data of the sample are taken as inputs Then predictionsof robotic grinding state are made according to these inputsAs shown in Figure 5 the prediction results of two modelsare basically consistent with the actual grinding force Theaverage prediction deviation of grinding force is about 05NThemaximum prediction deviation about 2N appears at thepeak of grinding force during cut-into state At 2nd secondthe grinding force predicted by the BP based model changesdramatically while the one obtained by theDBNbasedmodelis smoother Based on the above simulations a preconclusioncan be made that the performance of predictive modelobtained by DBN network is better than the one obtained byBP network

5 Robotic Grinding Control Experiments

51 Robotic Grinding System The robotic machining systemconsists of YASKAWA industrial robot MH24 six-axis forcesensorME-FK6D40 and grinding tool As shown in Figure 6the force sensor and grinding tool are installed on the end-effector of robot while the workpiece is installed on a fixed

Complexity 7

Energy Function

Control Output Voltage

Predictive Control ModelDeep Brief

NetworkModel

Parameters

DBN Structure

Robotic Grinding historical Data

Offline Training

Optimized Process

Offline

Transfer Function G(V)

Actual Force F(t)

Ideal ForceFlowast(t)

Force Deviation e(t)

System Information

+

Online

minus

T

I = [ed(ti) ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

E(ti) = ４

Δup(ti | ti) ed(ti+1 | ti)

Δup(ti | ti) = Z = (A1４A1)

minus1A1４D

ＧＣＨ

Figure 3 Schematic diagram of robot model predictive control process based on DBN

platform The spindle of grinding tool is perpendicular tothe Z direction of industrial robot sixth axis The worldcoordinate system is set according to the robot CartesiancoordinateThe relevant parameters of equipment are the six-axis force sensor is ME-FK6d40 Germany the workpiecematerial isQ235 steel the grinding tool is composed ofmotorhandle and grinding head

The force signals are collected by the force sensor anddelivered to the embedded real-time control system inPC with the use of Ethercat protocol The analogue filterfrequency of force sensor is 2500Hz and the samplingfrequency of embedded real-time control system is 1msThe control output is then calculated by embedded real-time control system and sent to YASKAWA robot controlcabinet tomodify output pulse and adjust the feed rate of end-effector The frequency of output voltage of control system isabout 100ms

52 Robotic Grinding Control Experiments To evaluate theperformance of proposed control approach robotic grind-ing control experiments are conducted The open looprobotic grinding experiment robotic grinding experimentwith fuzzy-PD control and robotic grinding experiment withmodel predictive control are conducted and comparisons

between these experiments are made The robotic grindingcontrol experiments are carried out on the steel plate plane(Q235)The rotation speed of grinding tool is 4000 rmin andthe grinding depth is 12mm the initial feed rate is 1mmsthe material of cutter is D1614M06

The grinding path of the experiment is shown in Figure 7The grinding tool moves from point A to D via point Band C The grinding path AB is perpendicular to workpiecesurface while the path BC is tangential to workpiece surfaceFor the convenience of discussion in subsequent articles ABis regarded as cut-into state while BC is regarded as stablegrinding state

521 Experiment with Open Loop Control and Experimentwith Fuzzy-PD Control

(A) Experiment withOpen LoopControlThe desired grindingforce for the experiment is 14 N The experiment result isshown in Figure 8 The grinding force increases dramaticallyto 29N during cut-into state and then decreases around15N The dramatically increase of grinding force whichis labeled as abrupt change during cut-into state can beillustrated by (10) and (16) When grinding tool cuts intothe workpiece the gap between feed rate and removal rate

8 Complexity

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)Best Training Performance is 98696e-07 at epoch 52

Train

Best

Goal

10 20 30 40 500Epoch

(a)

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 60367e-07 at epoch 31

TrainBestGoal

5 10 15 20 25 300Epoch

(b)

10minus2

100

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 021614 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(c)

10minus2

10minus1

100

101

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 020538 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(d)

Figure 4 Training performance comparison between BP and DBM (Mean Square Error) ((a) BP (time 119905119894+0) (b) DBN (time 119905119894+0) (c) BP(time 119905119894+7) (d) DBN (time 119905119894+7))

results in a deformation and leads to an increase grindingforce After perpendicular move the grinding tool starts tomove in tangential direction and the value of removal ratein perpendicular direction is larger than the one of feed rateresulting in a decrease of grinding force

(B) Experiment with Fuzzy-PD Control To verify theeffectiveness of proposed control approach an exper-iment based on fuzzy-PID control is conducted Theinputs of fuzzy-PD controller are force deviation 119890 andits first-order derivative 119890 and control parameters 119879p

and 119879d Their domains are 119890 = minus6 minus3 0 3 6 119890 =minus6 minus3 0 3 6 119879p = minus06 minus03 minus01 minus005 minus001 and119879d = minus04 minus02 minus01 minus005 minus001 The fuzzy reasoning isldquoIF A1 and B1 Then C1 and D1rdquo while the Gaussian functionis used as fuzzy membership function

The result of grinding experiment with PD control isshown in Figure 9 The grinding force increases considerablyand reaches a peak value of 25 N which is 4 N lower thanthe one of open loop control The corresponding feed rateincreases to 14mms at the first time and then decreaseto 07mms with a valley of 04mms During this process

Complexity 9

Verification SampleDBN based PredictionBP based Prediction

0

5

10

15

20

25

30G

rindi

ng F

orce

F (N

)

2 4 6 80Time t (s)

Figure 5 Prediction performance

5

2

1

43

Figure 6 Architecture of robotic cutting system ((1) YASKAWA 6-DOF industrial robot (2) rotating platform (3) six-axis force sensor(4) cutting tool (5) workpiece)

a fluctuation of feed rate appears at 1st second and thefluctuation amplitude of feed rate is around 01mms This isbecause when the grinding force is close to the target forcethe control parameters swift according to the fuzzy ruleswhich results in a fluctuation of control output This is acommon problem of fuzzy control as fuzzy rules are oftendefined according to the artificial experience which is easyto result in control oscillations caused by switching of controlrate In view of this a model predictive control approach isapplied to robotic grinding control

522 Robotic Grinding Experiment with Model PredictiveControl The robotic grinding experiment with model pre-dictive control is then conducted to reduce the grinding forcedeviation and feed rate fluctuation that occurs in cut-intostate The prediction time of model predictive control is setas 119905119894+10 and the experiment result is shown in Figure 10 Thepeak value of robotic grinding force is 18N which is 7N lowerand the corresponding feed rate curve changes from 14mmsto 07mms with a valley of 03mms The changes of robotic

B

A D

C

Figure 7 Grinding path

0

5

10

15

20

25

30

Grin

ding

For

ce F

(N)

2 4 6 8 100Time t (s)

Figure 8 Robotic grinding experiment with open loop control

grinding force and feed rate shown in Figure 10 are smoothercompared with the ones shown in Figure 9 This is becausethat the model predictive control approach can predict thefuture grinding deviation based on acquired information andresult in a control compensation to reduce the coming upforce deviation and feed rate fluctuation

To further verify the performance of proposed controlapproach another robotic grinding experiment based onmodel predictive control is conducted of which the predictiontime is set as 119905119894+100 As shown in Figure 11 the grinding forceincreases smoothly from 0N to 14N and the fluctuation ofgrinding force is around 2N The abrupt change of grindingforce which occurs during cut-into state as shown in Figures10 and 9 is completely eliminated The corresponding feedrate changes from 1mms to 075mms without an abruptincrease as shown in Figures 9 and 10 This is because whenthe prediction time is adjusted to 119905119894+100 the control systemcan have sufficient time to adjust feed rate smoothly whichcan lead to sufficient control compensation

To explore the flexibility of the proposed controlapproach the grinding depth is changed to 14mm whilethe grinding depth range of study sample is from 06mmto 12mm The experiment result with open loop control isshown in Figure 12(a) while the experiment results withmodel predictive control are shown in Figures 12(b) and12(c) As shown in Figure 12(a) the grinding force increasesdramatically to about 33 N during cut-into state and then

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Complexity

Table 1 Performance comparison between BP and DBN (mean square error)

Time BP network DBN119890119889 Δ119906119901 119890119889 Δ119906119901119905119894+0 986 lowast 10minus7 686 lowast 10minus7 604 lowast 10minus7 301 lowast 10minus7

119905119894+1 976 lowast 10minus7 896 lowast 10minus7 836 lowast 10minus7 976 lowast 10minus7119905119894+2 0156 0167 0152 0160119905119894+3 0165 0170 0160 0163119905119894+4 0177 0171 0171 0165119905119894+5 0187 0183 0185 0175119905119894+6 0194 0185 0191 0181119905119894+7 0216 0201 0205 0194

Output layer

BP (6 neurons)

RBM 2 (8 neurons)

RBM 1 (8 neurons)

Input layer

Wd3

Wd2

Wd1

ed(ti) ΔVw(ti-2) Δup (ti-1) Vw(ti-1)

ed(ti+1 ti) Δup(ti || ti)

Figure 2 DBN structure diagram

According to (35) and (37) the MPC is used toobtain 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 | 119905119894)] with the use of119868 = [119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] which meansthat the MPC represents the relationship between 119868 =[119890119889(119905119894) Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894)119890119889(119905119894+1 | 119905119894)] Therefore the training data is requiredto contain the relationship between 119868 = [119890119889(119905119894)Δ119881119908(119905119894minus2) Δ119906119901(119905119894minus1) 119881119908(119905119894minus1)] and 119874 = [Δ119906119901(119905119894 | 119905119894) 119890119889(119905119894+1 |119905119894)]

The training data of DBN is determined according to thefollowing aspects

(1) The parameters of training data are determinedaccording to influence factors of robotic grinding processincluding grinding depth grinding tool feed rate tool rota-tion speed and control output voltage As there is a functionalrelationship between feed rate and control output voltagewhen a controller is applied this paper uses feed rate changeto represent change of control voltage The ranges of theparameters are grinding depth is from 06mm to 12mminitial feed rate is from 05mms to 12mms feed rate changeis from 01 1198981198982s to 1 1198981198982s tool rotation speed is 4000rmin (2) Since grinding is a material removal process itis difficult and costly to perform dozens of experiments andobtain a large amount of training data the DBN model

used in this paper is only designed for specific case (roboticgrinding) as Oh and Jung did in their research [19] whichcan reduce the quantity demand of data for DBN training butalso reduce the applicability of DBN applying in other roboticoperations Based on this the scale and the size of learningsample are determined The scale of learning sample batch is40 packets and each packet has about 5000lowast4data 34 packetsare randomly selected for training and the remaining 6 areselected for model validation The maximum epoch is 300

The training performances of both networks are shownin Figure 4 For the model training of robotic grindingstatus at time 119905119894+0 the DBN realizes fitting at 31st epochand BP network realizes fitting at 52nd epoch The modeltraining of robotic grinding status from time 119905119894+0 to 119905119894+7 isalso conducted and the training performance of both BPnetwork and DBN is presented in Table 1 In overall thetraining performances of the proposed DBN are approximateto the training performances of the classical BP network Thetraining results are then used to construct

A simulation is conducted to evaluate the predictive per-formance of models obtained respectively by the proposedDBN and the BP network The steps of the simulation are asfollows Firstly a sample is selected randomly and the first10 data of the sample are taken as inputs Then predictionsof robotic grinding state are made according to these inputsAs shown in Figure 5 the prediction results of two modelsare basically consistent with the actual grinding force Theaverage prediction deviation of grinding force is about 05NThemaximum prediction deviation about 2N appears at thepeak of grinding force during cut-into state At 2nd secondthe grinding force predicted by the BP based model changesdramatically while the one obtained by theDBNbasedmodelis smoother Based on the above simulations a preconclusioncan be made that the performance of predictive modelobtained by DBN network is better than the one obtained byBP network

5 Robotic Grinding Control Experiments

51 Robotic Grinding System The robotic machining systemconsists of YASKAWA industrial robot MH24 six-axis forcesensorME-FK6D40 and grinding tool As shown in Figure 6the force sensor and grinding tool are installed on the end-effector of robot while the workpiece is installed on a fixed

Complexity 7

Energy Function

Control Output Voltage

Predictive Control ModelDeep Brief

NetworkModel

Parameters

DBN Structure

Robotic Grinding historical Data

Offline Training

Optimized Process

Offline

Transfer Function G(V)

Actual Force F(t)

Ideal ForceFlowast(t)

Force Deviation e(t)

System Information

+

Online

minus

T

I = [ed(ti) ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

E(ti) = ４

Δup(ti | ti) ed(ti+1 | ti)

Δup(ti | ti) = Z = (A1４A1)

minus1A1４D

ＧＣＨ

Figure 3 Schematic diagram of robot model predictive control process based on DBN

platform The spindle of grinding tool is perpendicular tothe Z direction of industrial robot sixth axis The worldcoordinate system is set according to the robot CartesiancoordinateThe relevant parameters of equipment are the six-axis force sensor is ME-FK6d40 Germany the workpiecematerial isQ235 steel the grinding tool is composed ofmotorhandle and grinding head

The force signals are collected by the force sensor anddelivered to the embedded real-time control system inPC with the use of Ethercat protocol The analogue filterfrequency of force sensor is 2500Hz and the samplingfrequency of embedded real-time control system is 1msThe control output is then calculated by embedded real-time control system and sent to YASKAWA robot controlcabinet tomodify output pulse and adjust the feed rate of end-effector The frequency of output voltage of control system isabout 100ms

52 Robotic Grinding Control Experiments To evaluate theperformance of proposed control approach robotic grind-ing control experiments are conducted The open looprobotic grinding experiment robotic grinding experimentwith fuzzy-PD control and robotic grinding experiment withmodel predictive control are conducted and comparisons

between these experiments are made The robotic grindingcontrol experiments are carried out on the steel plate plane(Q235)The rotation speed of grinding tool is 4000 rmin andthe grinding depth is 12mm the initial feed rate is 1mmsthe material of cutter is D1614M06

The grinding path of the experiment is shown in Figure 7The grinding tool moves from point A to D via point Band C The grinding path AB is perpendicular to workpiecesurface while the path BC is tangential to workpiece surfaceFor the convenience of discussion in subsequent articles ABis regarded as cut-into state while BC is regarded as stablegrinding state

521 Experiment with Open Loop Control and Experimentwith Fuzzy-PD Control

(A) Experiment withOpen LoopControlThe desired grindingforce for the experiment is 14 N The experiment result isshown in Figure 8 The grinding force increases dramaticallyto 29N during cut-into state and then decreases around15N The dramatically increase of grinding force whichis labeled as abrupt change during cut-into state can beillustrated by (10) and (16) When grinding tool cuts intothe workpiece the gap between feed rate and removal rate

8 Complexity

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)Best Training Performance is 98696e-07 at epoch 52

Train

Best

Goal

10 20 30 40 500Epoch

(a)

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 60367e-07 at epoch 31

TrainBestGoal

5 10 15 20 25 300Epoch

(b)

10minus2

100

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 021614 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(c)

10minus2

10minus1

100

101

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 020538 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(d)

Figure 4 Training performance comparison between BP and DBM (Mean Square Error) ((a) BP (time 119905119894+0) (b) DBN (time 119905119894+0) (c) BP(time 119905119894+7) (d) DBN (time 119905119894+7))

results in a deformation and leads to an increase grindingforce After perpendicular move the grinding tool starts tomove in tangential direction and the value of removal ratein perpendicular direction is larger than the one of feed rateresulting in a decrease of grinding force

(B) Experiment with Fuzzy-PD Control To verify theeffectiveness of proposed control approach an exper-iment based on fuzzy-PID control is conducted Theinputs of fuzzy-PD controller are force deviation 119890 andits first-order derivative 119890 and control parameters 119879p

and 119879d Their domains are 119890 = minus6 minus3 0 3 6 119890 =minus6 minus3 0 3 6 119879p = minus06 minus03 minus01 minus005 minus001 and119879d = minus04 minus02 minus01 minus005 minus001 The fuzzy reasoning isldquoIF A1 and B1 Then C1 and D1rdquo while the Gaussian functionis used as fuzzy membership function

The result of grinding experiment with PD control isshown in Figure 9 The grinding force increases considerablyand reaches a peak value of 25 N which is 4 N lower thanthe one of open loop control The corresponding feed rateincreases to 14mms at the first time and then decreaseto 07mms with a valley of 04mms During this process

Complexity 9

Verification SampleDBN based PredictionBP based Prediction

0

5

10

15

20

25

30G

rindi

ng F

orce

F (N

)

2 4 6 80Time t (s)

Figure 5 Prediction performance

5

2

1

43

Figure 6 Architecture of robotic cutting system ((1) YASKAWA 6-DOF industrial robot (2) rotating platform (3) six-axis force sensor(4) cutting tool (5) workpiece)

a fluctuation of feed rate appears at 1st second and thefluctuation amplitude of feed rate is around 01mms This isbecause when the grinding force is close to the target forcethe control parameters swift according to the fuzzy ruleswhich results in a fluctuation of control output This is acommon problem of fuzzy control as fuzzy rules are oftendefined according to the artificial experience which is easyto result in control oscillations caused by switching of controlrate In view of this a model predictive control approach isapplied to robotic grinding control

522 Robotic Grinding Experiment with Model PredictiveControl The robotic grinding experiment with model pre-dictive control is then conducted to reduce the grinding forcedeviation and feed rate fluctuation that occurs in cut-intostate The prediction time of model predictive control is setas 119905119894+10 and the experiment result is shown in Figure 10 Thepeak value of robotic grinding force is 18N which is 7N lowerand the corresponding feed rate curve changes from 14mmsto 07mms with a valley of 03mms The changes of robotic

B

A D

C

Figure 7 Grinding path

0

5

10

15

20

25

30

Grin

ding

For

ce F

(N)

2 4 6 8 100Time t (s)

Figure 8 Robotic grinding experiment with open loop control

grinding force and feed rate shown in Figure 10 are smoothercompared with the ones shown in Figure 9 This is becausethat the model predictive control approach can predict thefuture grinding deviation based on acquired information andresult in a control compensation to reduce the coming upforce deviation and feed rate fluctuation

To further verify the performance of proposed controlapproach another robotic grinding experiment based onmodel predictive control is conducted of which the predictiontime is set as 119905119894+100 As shown in Figure 11 the grinding forceincreases smoothly from 0N to 14N and the fluctuation ofgrinding force is around 2N The abrupt change of grindingforce which occurs during cut-into state as shown in Figures10 and 9 is completely eliminated The corresponding feedrate changes from 1mms to 075mms without an abruptincrease as shown in Figures 9 and 10 This is because whenthe prediction time is adjusted to 119905119894+100 the control systemcan have sufficient time to adjust feed rate smoothly whichcan lead to sufficient control compensation

To explore the flexibility of the proposed controlapproach the grinding depth is changed to 14mm whilethe grinding depth range of study sample is from 06mmto 12mm The experiment result with open loop control isshown in Figure 12(a) while the experiment results withmodel predictive control are shown in Figures 12(b) and12(c) As shown in Figure 12(a) the grinding force increasesdramatically to about 33 N during cut-into state and then

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 7

Energy Function

Control Output Voltage

Predictive Control ModelDeep Brief

NetworkModel

Parameters

DBN Structure

Robotic Grinding historical Data

Offline Training

Optimized Process

Offline

Transfer Function G(V)

Actual Force F(t)

Ideal ForceFlowast(t)

Force Deviation e(t)

System Information

+

Online

minus

T

I = [ed(ti) ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

ΔVw(ti-2) Δup(ti-1) Vw(ti-1)]

E(ti) = ４

Δup(ti | ti) ed(ti+1 | ti)

Δup(ti | ti) = Z = (A1４A1)

minus1A1４D

ＧＣＨ

Figure 3 Schematic diagram of robot model predictive control process based on DBN

platform The spindle of grinding tool is perpendicular tothe Z direction of industrial robot sixth axis The worldcoordinate system is set according to the robot CartesiancoordinateThe relevant parameters of equipment are the six-axis force sensor is ME-FK6d40 Germany the workpiecematerial isQ235 steel the grinding tool is composed ofmotorhandle and grinding head

The force signals are collected by the force sensor anddelivered to the embedded real-time control system inPC with the use of Ethercat protocol The analogue filterfrequency of force sensor is 2500Hz and the samplingfrequency of embedded real-time control system is 1msThe control output is then calculated by embedded real-time control system and sent to YASKAWA robot controlcabinet tomodify output pulse and adjust the feed rate of end-effector The frequency of output voltage of control system isabout 100ms

52 Robotic Grinding Control Experiments To evaluate theperformance of proposed control approach robotic grind-ing control experiments are conducted The open looprobotic grinding experiment robotic grinding experimentwith fuzzy-PD control and robotic grinding experiment withmodel predictive control are conducted and comparisons

between these experiments are made The robotic grindingcontrol experiments are carried out on the steel plate plane(Q235)The rotation speed of grinding tool is 4000 rmin andthe grinding depth is 12mm the initial feed rate is 1mmsthe material of cutter is D1614M06

The grinding path of the experiment is shown in Figure 7The grinding tool moves from point A to D via point Band C The grinding path AB is perpendicular to workpiecesurface while the path BC is tangential to workpiece surfaceFor the convenience of discussion in subsequent articles ABis regarded as cut-into state while BC is regarded as stablegrinding state

521 Experiment with Open Loop Control and Experimentwith Fuzzy-PD Control

(A) Experiment withOpen LoopControlThe desired grindingforce for the experiment is 14 N The experiment result isshown in Figure 8 The grinding force increases dramaticallyto 29N during cut-into state and then decreases around15N The dramatically increase of grinding force whichis labeled as abrupt change during cut-into state can beillustrated by (10) and (16) When grinding tool cuts intothe workpiece the gap between feed rate and removal rate

8 Complexity

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)Best Training Performance is 98696e-07 at epoch 52

Train

Best

Goal

10 20 30 40 500Epoch

(a)

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 60367e-07 at epoch 31

TrainBestGoal

5 10 15 20 25 300Epoch

(b)

10minus2

100

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 021614 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(c)

10minus2

10minus1

100

101

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 020538 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(d)

Figure 4 Training performance comparison between BP and DBM (Mean Square Error) ((a) BP (time 119905119894+0) (b) DBN (time 119905119894+0) (c) BP(time 119905119894+7) (d) DBN (time 119905119894+7))

results in a deformation and leads to an increase grindingforce After perpendicular move the grinding tool starts tomove in tangential direction and the value of removal ratein perpendicular direction is larger than the one of feed rateresulting in a decrease of grinding force

(B) Experiment with Fuzzy-PD Control To verify theeffectiveness of proposed control approach an exper-iment based on fuzzy-PID control is conducted Theinputs of fuzzy-PD controller are force deviation 119890 andits first-order derivative 119890 and control parameters 119879p

and 119879d Their domains are 119890 = minus6 minus3 0 3 6 119890 =minus6 minus3 0 3 6 119879p = minus06 minus03 minus01 minus005 minus001 and119879d = minus04 minus02 minus01 minus005 minus001 The fuzzy reasoning isldquoIF A1 and B1 Then C1 and D1rdquo while the Gaussian functionis used as fuzzy membership function

The result of grinding experiment with PD control isshown in Figure 9 The grinding force increases considerablyand reaches a peak value of 25 N which is 4 N lower thanthe one of open loop control The corresponding feed rateincreases to 14mms at the first time and then decreaseto 07mms with a valley of 04mms During this process

Complexity 9

Verification SampleDBN based PredictionBP based Prediction

0

5

10

15

20

25

30G

rindi

ng F

orce

F (N

)

2 4 6 80Time t (s)

Figure 5 Prediction performance

5

2

1

43

Figure 6 Architecture of robotic cutting system ((1) YASKAWA 6-DOF industrial robot (2) rotating platform (3) six-axis force sensor(4) cutting tool (5) workpiece)

a fluctuation of feed rate appears at 1st second and thefluctuation amplitude of feed rate is around 01mms This isbecause when the grinding force is close to the target forcethe control parameters swift according to the fuzzy ruleswhich results in a fluctuation of control output This is acommon problem of fuzzy control as fuzzy rules are oftendefined according to the artificial experience which is easyto result in control oscillations caused by switching of controlrate In view of this a model predictive control approach isapplied to robotic grinding control

522 Robotic Grinding Experiment with Model PredictiveControl The robotic grinding experiment with model pre-dictive control is then conducted to reduce the grinding forcedeviation and feed rate fluctuation that occurs in cut-intostate The prediction time of model predictive control is setas 119905119894+10 and the experiment result is shown in Figure 10 Thepeak value of robotic grinding force is 18N which is 7N lowerand the corresponding feed rate curve changes from 14mmsto 07mms with a valley of 03mms The changes of robotic

B

A D

C

Figure 7 Grinding path

0

5

10

15

20

25

30

Grin

ding

For

ce F

(N)

2 4 6 8 100Time t (s)

Figure 8 Robotic grinding experiment with open loop control

grinding force and feed rate shown in Figure 10 are smoothercompared with the ones shown in Figure 9 This is becausethat the model predictive control approach can predict thefuture grinding deviation based on acquired information andresult in a control compensation to reduce the coming upforce deviation and feed rate fluctuation

To further verify the performance of proposed controlapproach another robotic grinding experiment based onmodel predictive control is conducted of which the predictiontime is set as 119905119894+100 As shown in Figure 11 the grinding forceincreases smoothly from 0N to 14N and the fluctuation ofgrinding force is around 2N The abrupt change of grindingforce which occurs during cut-into state as shown in Figures10 and 9 is completely eliminated The corresponding feedrate changes from 1mms to 075mms without an abruptincrease as shown in Figures 9 and 10 This is because whenthe prediction time is adjusted to 119905119894+100 the control systemcan have sufficient time to adjust feed rate smoothly whichcan lead to sufficient control compensation

To explore the flexibility of the proposed controlapproach the grinding depth is changed to 14mm whilethe grinding depth range of study sample is from 06mmto 12mm The experiment result with open loop control isshown in Figure 12(a) while the experiment results withmodel predictive control are shown in Figures 12(b) and12(c) As shown in Figure 12(a) the grinding force increasesdramatically to about 33 N during cut-into state and then

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Complexity

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)Best Training Performance is 98696e-07 at epoch 52

Train

Best

Goal

10 20 30 40 500Epoch

(a)

10minus5

100

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 60367e-07 at epoch 31

TrainBestGoal

5 10 15 20 25 300Epoch

(b)

10minus2

100

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 021614 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(c)

10minus2

10minus1

100

101

102

Mea

n Sq

uare

d Er

ror (

mse

)

Best Training Performance is 020538 at epoch 300

TrainBestGoal

50 100 150 200 250 3000Epoch

(d)

Figure 4 Training performance comparison between BP and DBM (Mean Square Error) ((a) BP (time 119905119894+0) (b) DBN (time 119905119894+0) (c) BP(time 119905119894+7) (d) DBN (time 119905119894+7))

results in a deformation and leads to an increase grindingforce After perpendicular move the grinding tool starts tomove in tangential direction and the value of removal ratein perpendicular direction is larger than the one of feed rateresulting in a decrease of grinding force

(B) Experiment with Fuzzy-PD Control To verify theeffectiveness of proposed control approach an exper-iment based on fuzzy-PID control is conducted Theinputs of fuzzy-PD controller are force deviation 119890 andits first-order derivative 119890 and control parameters 119879p

and 119879d Their domains are 119890 = minus6 minus3 0 3 6 119890 =minus6 minus3 0 3 6 119879p = minus06 minus03 minus01 minus005 minus001 and119879d = minus04 minus02 minus01 minus005 minus001 The fuzzy reasoning isldquoIF A1 and B1 Then C1 and D1rdquo while the Gaussian functionis used as fuzzy membership function

The result of grinding experiment with PD control isshown in Figure 9 The grinding force increases considerablyand reaches a peak value of 25 N which is 4 N lower thanthe one of open loop control The corresponding feed rateincreases to 14mms at the first time and then decreaseto 07mms with a valley of 04mms During this process

Complexity 9

Verification SampleDBN based PredictionBP based Prediction

0

5

10

15

20

25

30G

rindi

ng F

orce

F (N

)

2 4 6 80Time t (s)

Figure 5 Prediction performance

5

2

1

43

Figure 6 Architecture of robotic cutting system ((1) YASKAWA 6-DOF industrial robot (2) rotating platform (3) six-axis force sensor(4) cutting tool (5) workpiece)

a fluctuation of feed rate appears at 1st second and thefluctuation amplitude of feed rate is around 01mms This isbecause when the grinding force is close to the target forcethe control parameters swift according to the fuzzy ruleswhich results in a fluctuation of control output This is acommon problem of fuzzy control as fuzzy rules are oftendefined according to the artificial experience which is easyto result in control oscillations caused by switching of controlrate In view of this a model predictive control approach isapplied to robotic grinding control

522 Robotic Grinding Experiment with Model PredictiveControl The robotic grinding experiment with model pre-dictive control is then conducted to reduce the grinding forcedeviation and feed rate fluctuation that occurs in cut-intostate The prediction time of model predictive control is setas 119905119894+10 and the experiment result is shown in Figure 10 Thepeak value of robotic grinding force is 18N which is 7N lowerand the corresponding feed rate curve changes from 14mmsto 07mms with a valley of 03mms The changes of robotic

B

A D

C

Figure 7 Grinding path

0

5

10

15

20

25

30

Grin

ding

For

ce F

(N)

2 4 6 8 100Time t (s)

Figure 8 Robotic grinding experiment with open loop control

grinding force and feed rate shown in Figure 10 are smoothercompared with the ones shown in Figure 9 This is becausethat the model predictive control approach can predict thefuture grinding deviation based on acquired information andresult in a control compensation to reduce the coming upforce deviation and feed rate fluctuation

To further verify the performance of proposed controlapproach another robotic grinding experiment based onmodel predictive control is conducted of which the predictiontime is set as 119905119894+100 As shown in Figure 11 the grinding forceincreases smoothly from 0N to 14N and the fluctuation ofgrinding force is around 2N The abrupt change of grindingforce which occurs during cut-into state as shown in Figures10 and 9 is completely eliminated The corresponding feedrate changes from 1mms to 075mms without an abruptincrease as shown in Figures 9 and 10 This is because whenthe prediction time is adjusted to 119905119894+100 the control systemcan have sufficient time to adjust feed rate smoothly whichcan lead to sufficient control compensation

To explore the flexibility of the proposed controlapproach the grinding depth is changed to 14mm whilethe grinding depth range of study sample is from 06mmto 12mm The experiment result with open loop control isshown in Figure 12(a) while the experiment results withmodel predictive control are shown in Figures 12(b) and12(c) As shown in Figure 12(a) the grinding force increasesdramatically to about 33 N during cut-into state and then

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 9

Verification SampleDBN based PredictionBP based Prediction

0

5

10

15

20

25

30G

rindi

ng F

orce

F (N

)

2 4 6 80Time t (s)

Figure 5 Prediction performance

5

2

1

43

Figure 6 Architecture of robotic cutting system ((1) YASKAWA 6-DOF industrial robot (2) rotating platform (3) six-axis force sensor(4) cutting tool (5) workpiece)

a fluctuation of feed rate appears at 1st second and thefluctuation amplitude of feed rate is around 01mms This isbecause when the grinding force is close to the target forcethe control parameters swift according to the fuzzy ruleswhich results in a fluctuation of control output This is acommon problem of fuzzy control as fuzzy rules are oftendefined according to the artificial experience which is easyto result in control oscillations caused by switching of controlrate In view of this a model predictive control approach isapplied to robotic grinding control

522 Robotic Grinding Experiment with Model PredictiveControl The robotic grinding experiment with model pre-dictive control is then conducted to reduce the grinding forcedeviation and feed rate fluctuation that occurs in cut-intostate The prediction time of model predictive control is setas 119905119894+10 and the experiment result is shown in Figure 10 Thepeak value of robotic grinding force is 18N which is 7N lowerand the corresponding feed rate curve changes from 14mmsto 07mms with a valley of 03mms The changes of robotic

B

A D

C

Figure 7 Grinding path

0

5

10

15

20

25

30

Grin

ding

For

ce F

(N)

2 4 6 8 100Time t (s)

Figure 8 Robotic grinding experiment with open loop control

grinding force and feed rate shown in Figure 10 are smoothercompared with the ones shown in Figure 9 This is becausethat the model predictive control approach can predict thefuture grinding deviation based on acquired information andresult in a control compensation to reduce the coming upforce deviation and feed rate fluctuation

To further verify the performance of proposed controlapproach another robotic grinding experiment based onmodel predictive control is conducted of which the predictiontime is set as 119905119894+100 As shown in Figure 11 the grinding forceincreases smoothly from 0N to 14N and the fluctuation ofgrinding force is around 2N The abrupt change of grindingforce which occurs during cut-into state as shown in Figures10 and 9 is completely eliminated The corresponding feedrate changes from 1mms to 075mms without an abruptincrease as shown in Figures 9 and 10 This is because whenthe prediction time is adjusted to 119905119894+100 the control systemcan have sufficient time to adjust feed rate smoothly whichcan lead to sufficient control compensation

To explore the flexibility of the proposed controlapproach the grinding depth is changed to 14mm whilethe grinding depth range of study sample is from 06mmto 12mm The experiment result with open loop control isshown in Figure 12(a) while the experiment results withmodel predictive control are shown in Figures 12(b) and12(c) As shown in Figure 12(a) the grinding force increasesdramatically to about 33 N during cut-into state and then

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Complexity

03040506070809

11112131415

Feed

rate

(mm

s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

0

5

10

15

20

25G

rindi

ng F

orce

F (N

)

Figure 9 Robotic grinding experiment with fuzzy-PD control

0 2 4 6 8 10Time (s)

03040506070809

11112131415

Feed

Rat

e (m

ms

)

0

5

10

15

20

Grin

ding

For

ce F

(N)

2 4 6 8 100Time (s)

Figure 10 Robotic grinding experiment with model predictive control (119905119894+10)

2 4 6 80Time (s)

07

08

09

1

11

12

13

14

Feed

rate

(mm

s)

2 4 6 80Time (s)

0

5

10

15

20

Grin

ding

For

ce F

(N)

Figure 11 Robotic grinding experiment with model predictive control (119905119894+100 depth 12mm)

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 11

0

10

20

30

Grin

ding

For

ce F

(N)

2 4 6 80Time (s)

(a)

2 4 6 80Time (s)

0

5

10

15

20

25

Grin

ding

For

ce F

(N)

(b)

0 2 4 6 8 10Time (s)

06

07

08

09

1

11

12

Grin

ding

For

ce F

(N)

(c)

Figure 12 Robotic grinding experiment with model predictive control (119905119894+100 depth 14mm)

decreases to around 19N When model predictive con-trol approach is implemented the grinding force increasessmoothly from 0N to 19N without an abrupt change duringcut-into state The corresponding feed rate curve shown inFigure 12(c) is also smooth which decreases from 1mms to07mms with a small valley of 065mms at 4th second

Comparedwith fuzzy-PD control approach the proposedMPC approach can avoid mutation of grinding force andreduce the grinding force deviation when prediction intervalis large enough as shown in Figure 12 Also the feed ratechange of robotic grinding based on MPC is smoothercompared with the one based on fuzzy-PD control Theconclusion is the same while the comparison is made withthe experiments based on adaptive PID approach which pre-sented in a previous research of writers [20]The comparisonshows that adaptive PID control can reduce the mutation ofgrinding force but not able to avoid it as the adaptive PIDcontrol reacts only when the mutation comes out

6 Conclusion

In this paper a model predictive control approach based ondeep belief network is proposed to control the deviation ofgrinding force caused by robotic deformation The roboticgrinding dynamic impacted by rigid-flexible coupling effectis analyzed and the deep belief network is used to study therobotic grinding dynamic to acquire predict model Based onthis the model predictive control approach can predict thechange of grinding force deviation and achieve a combinationof feed-forward and feedback control

The experimental results show that the prediction modelof robot grinding based on the deep belief network canaccurately predict the change of robot grinding state in time119905119894+10 and in time 119905119894+100 Compared with open loop control andfuzzy-PD control the model predictive control approach canperform a compensate control to beforehand reduce the forcedeviation The elimination of the abrupt change of grindingforce caused by deformation as shown in Figures 11 and 12verifies the performance of proposed control approach

Since current researches are mostly focusing on thefeedback control of robot machining while feed-forward

compensation control is rarely applied to robotic machiningthe contributions of this paper are as follows (1) the modelpredictive control approach is adjusted according to roboticgrinding model within the consideration of coupling effectbetween rigid-flexible system and control system The MPCis used to predict the mutation of grinding force and per-form forward control to avoid grinding force mutation Theproposed approach has both the advantages of feed-forwardcontrol and feedback control which can reduce the controloscillations and overflow caused by system delay and feed-back control (2) A deep belief network is designed accordingto the MPC model of robotic grinding control system Sincethe MPC model is nonlinear and the parameters are difficultto obtain the use of deep belief network can benefit theobtainment of MPC model parameters Therefore this studyprovidesmodel reference and data support for the research ofnonlinear control method of robot machining process basedon intelligent model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] S Mousavi V Gagnol B C Bouzgarrou and P Ray ldquoSta-bility optimization in robotic milling through the control offunctional redundanciesrdquo Robotics and Computer-IntegratedManufacturing vol 50 pp 181ndash192 2018

[2] M Cordes andW Hintze ldquoOffline simulation of path deviationdue to joint compliance and hysteresis for robot machiningrdquoe International Journal of Advanced Manufacturing Technol-ogy vol 90 no 1-4 pp 1075ndash1083 2017

[3] A Brunete E Gambao J Koskinen et al ldquoHardmaterial small-batch industrial machining robotrdquo Robotics and Computer-Integrated Manufacturing vol 54 pp 185ndash199 2018

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Complexity

[4] R Mei and C Yu ldquoAdaptive neural output feedback control foruncertain robot manipulators with input saturationrdquo Complex-ity vol 2017 Article ID 7413642 12 pages 2017

[5] N Mendes and P Neto ldquoIndirect adaptive fuzzy control forindustrial robots a solution for contact applicationsrdquo ExpertSystems with Applications vol 42 no 22 pp 8929ndash8935 2015

[6] F-Y Hsu and L-C Fu ldquoIntelligent robot deburring using adap-tive fuzzy hybrid positionforce controlrdquo IEEE Transactions onRobotics and Automation vol 16 no 4 pp 325ndash335 2000

[7] V T Yen W Y Nan and P van Cuong ldquoRecurrent fuzzywavelet neural networks based on robust adaptive sliding modecontrol for industrial robot manipulatorsrdquo Neural Computingand Applications vol 18 pp 1ndash14 2018

[8] T Faulwasser T Weber P Zometa and R Findeisen ldquoImple-mentation of nonlinear model predictive path-following con-trol for an industrial robotrdquo IEEE Transactions on ControlSystems Technology vol 25 no 4 pp 1505ndash1511 2017

[9] Y G Xi D L Li and S Lin ldquoModel predictive control mdash statusand challengesrdquo Acta Automatica Sinica vol 39 no 3 pp 222ndash236 2013

[10] J Wilson M Charest and R Dubay ldquoNon-linear modelpredictive control schemes with application on a 2 link verticalrobot manipulatorrdquo Robotics and Computer-Integrated Manu-facturing vol 41 no C pp 23ndash30 2016

[11] H Peng JWuG Inoussa QDeng andK Nakano ldquoNonlinearsystem modeling and predictive control using the RBF nets-based quasi-linear ARX modelrdquo Control Engineering Practicevol 17 no 1 pp 59ndash66 2009

[12] H Peng K Nakano and H Shioya ldquoNonlinear predic-tive control using neural nets-based local linearization arxmodelmdashstability and industrial applicationrdquo IEEE Transactionson Control Systems Technology vol 15 no 1 pp 130ndash143 2007

[13] NNikdel PNikdelMA Badamchizadeh and IHassanzadehldquoUsing neural network model predictive control for controllingshape memory alloy-based manipulatorrdquo IEEE Transactions onIndustrial Electronics vol 61 no 3 pp 1394ndash1401 2014

[14] Z Li W Yuan Y Chen F Ke X Chu and C L P ChenldquoNeural-dynamic optimization-based model predictive controlfor tracking and formation of nonholonomic multirobot sys-temsrdquo IEEE Transactions on Neural Networks and LearningSystems pp 1ndash10 2018

[15] X Zeng H Peng and J Wu ldquoAn improved multivariable RBF-ARXmodel-basednonlinearmodel predictive control approachand applicationrdquo Journal of Central South University (Scienceand Technology) vol 46 no 10 pp 3710ndash3717 2015

[16] K Dalamagkidis K P Valavanis and L A Piegl ldquoNonlinearmodel predictive control with neural network optimizationfor autonomous autorotation of small unmanned helicoptersrdquoIEEE Transactions on Control Systems Technology vol 19 no 4pp 818ndash831 2011

[17] C Zhang Machinery Dynamics Higher Education Press Bei-jing China 2nd edition 2007

[18] H Chen Model Predictive Control Science Press BeijingChina 1st edition 2013

[19] H Oh J H Jung B C Jeon and B D Youn ldquoScalable andunsupervised feature engineering using vibration-imaging anddeep learning for rotor system diagnosisrdquo IEEE Transactions onIndustrial Electronics vol 65 no 4 pp 3539ndash3549 2018

[20] S Chen and T Zhang ldquoForce control approaches research forrobotic machining based on particle swarm optimization andadaptive iteration algorithmsrdquo Industrial Robot An Interna-tional Journal vol 45 no 1 pp 141ndash151 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom