constantforcepidcontrolforroboticmanipulatorbasedon...

Research ArticleConstant Force PID Control for Robotic Manipulator Based onFuzzy Neural Network Algorithm

Zhu Dachang 1 Du Baolin 1 Zhu Puchen 2 and Chen Shouyan 1

1School of Mechanical and Electrical Engineering Guangzhou University Guangzhou 510006 China2School of Automation Guangdong University of Technology Guangzhou 510006 China

Correspondence should be addressed to Zhu Dachang zdc98998gzhueducn

Received 13 April 2020 Accepted 14 May 2020 Published 17 July 2020

Academic Editor Shuping He

Copyright copy 2020 Zhu Dachang et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

e increased demand for robotic manipulator has driven the development of industrial manufacturing In particular thetrajectory tracking and contact constant force control of the robotic manipulator for the working environment under contactcondition has become popular because of its high precision and quality operation However the two factors are opposite that is tosay to maintain constant force control it is necessary to make limited adjustment to the trajectory It is difficult for the traditionalPID controller because of the complexity parameters and nonlinear characteristics In order to overcome this issue a PIDcontroller based on fuzzy neural network algorithm is developed in this paper for tracking the trajectory and contact constantforce simultaneously Firstly the kinetic and potential energy is calculated and the Lagrange function is constructed for a two-linkrobotic manipulator Furthermore a precise dynamic model is built for analyzing Secondly fuzzy neural network algorithm isproposed and two kinds of turning parameters are derived for trajectory tracking and contact constant force control Finallynumerical simulation results are reported to demonstrate the effectiveness of the proposed method

1 Introduction

Nowadays the increased involvement of robotic manipu-lators in various industrial manufacturing facilities such asprecision assembly and polishing in the field of ceramicsanitary ware or aircraft engine precision surgery andautomobile manufacturing lead to new and innovative de-velopments in the domain of dynamic modeling of mech-anisms and control [1ndash3] According to the workingconditions of the robotic manipulators there are two typescontract and noncontact e vital features of the contactworking condition are highly precise trajectory trackingcapabilities at the same time it should have the ability toadjust the contact forcetwist flexible [4 5] Along with thecomplexity and nonlinearity issues the robotic manipulatorsuffered from various uncertainties external disturbancespayload variations and parameter variations during theiroperations [6 7] erefore it is not possible for the tra-ditional proportional-integral-derivative (PID) controllers

to provide effective control for trajectory tracking andconstant forcetwist control simultaneously

According to the given value r(t) and the actual outputvalue c(t) the traditional PID control combines the devi-ation e(t) r(t) minus c(t) by proportion (P) integration (I)and differentiation (D) to form the control quantity econtrol law is given by

u(t) Kp e(t) +1Ti

1113946t

0e(t)dt + Td

de(t)

dt1113890 1113891

Kpe(t) + Ki 1113946t

0e(t)dt + Kd

de(t)

dt

(1)

where Kp is proportionality coefficient Ti is integral timeconstant Td is differential time constant andKi KpTiisintegral coefficient and Kd Kp middot Td is differential coeffi-cient e parameter selection of a traditional PID controllermust take into account the dynamic and static performancerequirements By using the method of closed-loop response

HindawiComplexityVolume 2020 Article ID 3491845 11 pageshttpsdoiorg10115520203491845

a heuristic tuning method of PID parameters is proposed byZiegler and Nichols in the 1940s

In recent times an overwhelming development in theterrain of advantage PID control schemes for parameterstuning has encouraged various control engineers to work inthis developing field Differential evolution (DE) geneticalgorithm (GA) and particle swarm optimization (PSO)were used to determine the optimal gains of the positiondomain PID controller and three distinct fitness functionswere also used to quantify the contour tracking performanceof each solution set in [8] In order to expand the robustnessand adaptive capabilities of conventional PID controller aneural network- (NN-) based PID controller which is tunedwhen the controller is operating in an online mode for high-performance magnet synchronous motor position controlwas proposed by Kumar et al [9] and the training algorithmfor the PID controller gain initialization based on theminimum norm least square solution was also proposed Forthe last few years several works have been cited on theintelligent algorithm as well as online policy iterative ap-proach adaptive optimal control sliding mode control andneural network control for different processes or plants suchas uncertain nonlinear systems [10 11] autonomous underwater vehicle (AUV) [12] mobile robot [13] permanentmagnet synchronous motor drivers [14] and many others[15ndash19]

Recently real-time trajectory optimization of roboticmanipulators has received a lot of attention and research dueto the increasing demand to improve movement speedaccuracy and lower energy consumption when executingtasks Several authors have been working toward intelligentPID control techniques of trajectory tracking for roboticmanipulators such as global asymptotic saturated PIDcontrol [20] fuzzy PID for enhanced position control insurgery robot [21] mobile robot [22ndash25] and flexible jointrobot [26] However trajectory tracking is not the onlyfactor to ensure the accuracy of the operation during thecontact working conditions of the robotic manipulator suchas high precision flexible assembly surface polishing andlocation of clamp in minimally invasive surgery (shown inFigure 1)

With the changes of the pressure temperature frictionand wear of the contact surface the contact forcetwistbetween the robot and the work piece will also changeAlthough the robotic manipulator can run accuratelyaccording to the present trajectory the machining quality orassembly quality on the contact surface will decline By usinga modified hybrid computed torque method based on theprinciple of orthogonalization Sanchez presented an im-proving force tracking control structure to reduce thenumber of sensors with a velocity observer [27] Cortesaoproposed double active observer architecture to tackleprecise force control in the presence of heart motion andone controls the desired interaction force and the other oneis responsible for compensating heart motion autonomously[28] An adaptive fuzzy backstepping position trackingcontrol scheme was proposed for multirobot manipulatorsystems and extra terms were also added to the controlsignals to consider the force tracking problem by utilizing

the properties of internal forces in [29] Accurate and robustforcetwist control is still a great challenge for robot-envi-ronment contact applications such as in drilling operations[30] polishing [31ndash33] welding [34] and surgical tasks [35]

In general the traditional PID controller helps ineliminating the steady state error However this controllerdoes not provide effective control in presence of nonline-arities and uncertainties Moreover motivated by the re-search studies carried out in [36 37] We proposed in thispaper fuzzy neural network proportion-integration-differ-entiation (FNN-PID) constant forcetwist control of therobotic manipulator for contact working conditions emain contributions of the current paper are summarized asfollows

(1) By using Lagrange energy function the precise dy-namic model of two-link robotic manipulator isbuilt Compared with the literature [38] it com-pletely represents the relationship between actuatedtorque and displacement and velocity and acceler-ation in joint space

(2) For the contact working condition the change ofcontact force will reduce the machining quality so itis important to keep the contact constant forceduring manufacturing process However constantcontact force and trajectory tracking are opposite toeach other In order to tune PID controller pa-rameters quickly and effectively a fuzzy neuralnetwork algorithm is proposed for two kindsfunctions of PID controller that is trajectorytracking and limited change online while contactconstant force control

is paper is organized in the following manner Section2 presents the precise dynamic modeling method and theprecise dynamic model of two types of planar robotic ma-nipulator with RR (revolute joint) and RP (prismatic joint)are derived Section 3 describes the PID controller with fuzzyneural network and its stability is analyzed Section 4 dis-cusses the relationship between the contact forcetwist withthe trajectory according to the contact working conditionsand numerial simulation results are reported to demonstratethe effectiveness of the proposed method Finally conclu-sions are given in Section 5

2 Dynamic Modeling of Two-Link Robot

e dynamic mathematical model for a rigid planar roboticmanipulator having two links and a contact surface and theexternal force acted on the surface as shown in Figure 2According to the coordinate system o minus xy1113864 1113865 it consists oftwo links having link length l1 and l2 with their center ofmass m1 and m2 lying at the middle of links respectivelye length of center of mass are p1 and p2 respectively

Different from the literature [39] the Lagrange methodis used to build the precise dynamic model of two-linkrobotic manipulator with their nominal values as listed inTable 1

e location coordinates and the square of velocity of 1center of mass can be derived as follows

2 Complexity

x1 p1 sin θ1

y1 p1 cos θ1

_x21 + _y

21 p1

_θ11113872 11138732

(2)

e location coordinates and the square of velocity of 1center of mass can also be derived as follows

x2 l1 sin θ1 + p2 sin θ1 + θ2( 1113857

y2 l1 cos θ1 + p2 cos θ1 + θ2( 1113857

_x22 + _y

22 l

21_θ21 + p

22

_θ1 + _θ21113872 11138732

+ 2l1p2_θ21 + _θ1 _θ21113874 1113875cos θ2

(3)

e total kinetic energy of the two-link robotic ma-nipulator is

Ek Ek1 + Ek2 12m1p

21_θ21 +

12m2l

21_θ21 +

12m2p

22

_θ1 + _θ21113872 11138732

+ m2l1p2_θ21 + _θ1 _θ21113874 1113875cos θ2

(4)

e total potential energy of the two-link robotic ma-nipulator is

Ep Ep1 + Ep2 m1gp1 cos θ1+ m2g l1 cos θ1 + p2 cos θ1 + θ2( 1113857( 1113857

(5)

y Contact force he

θ2

θ1

m2 I2

x

m1 I1

τ2

τ1

l 1

p1

p 2

l 2

Figure 2 Two-link robotic manipulator plant with contact force at tip

Table 1 Variables description of two-link robotic manipulator [39]

Description Nominal value Description Nominal valueCenter of mass link 1 m1 (kg) 01 Center of mass link 2 m2 (kg) 01Length of link 1 l1 (m) 08 Length of link 2 l2 (m) 04Length of center of mass link 1 p1 (m) 04 Length of center of mass link 2 p2 (m) 02Centroid inertia of link 1 I1 (kgm2) 0064 Centroid inertia of link 2 I2 (kgm2) 0016Joint variable of link 1 θ1 (rad) mdash Joint variable of link 2 θ2 (rad) mdashTorque of link 1 τ1 (Nm) mdash Torque of link 2 τ2 (Nm) mdash

(a) (b) (c)

Figure 1 Constant working conditions (a) High precision flexible assembly (b) minimally invasive surgery (c) complexity surfacepolishing

Complexity 3

e Lagrange function is constructed as followsL Ek minus Ep (6)

e dynamic equation of the two-link robotic manip-ulator can be calculated by

τi d

dt

zL

z _θi

minuszL

zθi

i 1 2 (7)

e torque of joint 1 τ1 can be derived as followszL

z _θ1 m1p

21 + m2l

211113872 1113873 _θ1 + m2l1p2 2 _θ1 + _θ21113872 1113873c2

+ m2p22

_θ1 + _θ21113872 1113873

zL

zθ1 minusm1gp1s1 minus m2g l1s1 + p2s12( 1113857

τ1 ddt

zL

z _θ1minus

zL

zθ1 D11

euroθ1 + D12euroθ2 + D112

_θ1 _θ2

+D122_θ22 + D1

(8)

where s1 sin θ1 c1 cos θ1 s12 sin(θ1 + θ2) and

D11 m1p21 + m2p

22 + m2l

21 + 2m2l1p2c2

D12 m2p22 + m2l1p2c2

D112 minus2m2l1p2s2

D122 minusm2l1p2s2

D1 m1p1 + m2l1( 1113857gs1 + m2p2gs12

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Meanwhile the torque of joint 2 τ2 can be also derived as

τ2 ddt

zL

z _θ2minus

zL

zθ2 D21


_θ1 _θ2 + D211_θ22 + D2

(10)where

D21 m1p22 + m2l1p2c2

D22 m2p22

D212 0

D211 m2l1p2s2

D2 m2p2gs12

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

Combined (8) with (10) the robotic plant can be re-written with the following mathematical model

D11 D12

D21 D221113890 1113891

euroθ1euroθ2

⎡⎣ ⎤⎦ +D112

D2121113890 1113891

_θ1 _θ2_θ2 _θ1

⎡⎣ ⎤⎦ +D211

D1221113890 1113891

_θ22

_θ21

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

+D1

D21113890 1113891

τ1τ2

1113890 1113891

(12)

Comparing with the literature [39] (12) completelyrepresents the relationship between actuated torque anddisplacement and velocity and acceleration in joint space

e issues with D11 and D22 represent the moment of inertiacaused by the acceleration of joint 1 and joint 2 respectivelye issues with D12 and D21 represent the moment of inertiaof the acceleration coupling between two joints e issueswith D122 and D211 represent the coupling moment term ofthe centripetal force caused by the velocity between twojoints e issues with D112 and D212 represent the couplingmoment term of the Coriolis force between two joints D1and D2 represent the gravity moment term Considering theeffect of centroid inertia (12) can be modified by

D11prime I1 + I2 + D11

D12prime I2 + D12

D21prime I2 + D21

D22prime I2 + D22

(13)

Furthermore to consider the effect of contact force acton the end-effector right side of (12) can be replaced by

τprime τ minus JT(q)he (14)

where he is the force and moment vector applied by the end-effector in the working environment and JT(q) is the ve-locity Jacobian matrix which yields to

J(q) minusl1s1 minus l2s12 minusl2s12

l1c1 + l2c12 l2c121113890 1113891 (15)

3 PID Controller with Fuzzy NeuralNetwork Algorithm

In order to design a fuzzy neural network PID (FNN-PID)controller for trajectory tracking and constant force controlof the robotic manipulator it is essential to derive the ab-solute error and the rate change of the external force act onthe end-effector as the inputs for the fuzzy controller whichare calculated as

Δe1 r(t) minus y(t)

Δe2 c(t) minus τ(t)(16)

where Δe1 is the error of trajectory tracking r(t) is the givenplanning trajectory y(t) is output trajectory at time t Δe2 isthe error of contact force c(t) is the given contact constantforce and τ(t) is the output contact force A block diagramillustrating the proposed FNN-PID controller is shown inFigure 3

e fuzzy control algorithm comprises the followingthree units fuzzification fuzzy rules and defuzzification

(1) Fuzzification it takes e and ec as input languagevariables and Kp KI and KD as output languagevariables e fuzzy subset of input and outputvariables are expressed as negative big (NB) negativemiddle (NM) negative small (NS) zero (ZO)positive small (PS) positive middle (PM) andpositive big (PB) e domain of e and ec is minus3 31113864 1113865and the domain of Kp KI and KD are minus03 031113864 1113865minus006 0061113864 1113865 and 3 31113864 1113865 respectively e

4 Complexity

membership function for the proposed fuzzy algo-rithm is shown in Figure 4

(2) Fuzzy rules fuzzy rules were designed for tuning thegain control parameters of the PID controller (thesame rules for trajectory tracking and constant force)as follows

(i) When the error is large in order to maintain arapid response and make the absolute or errorsreduce with the maximum speed Kp should bebigger and KI and KD should be smaller As theerror decreases to prevent an excessively largeovershoot KD should be added and Kp and KI

should be decreased uniformly(ii) When the system is over the steady state and the

error is increasing in order to decrease theovershoot KD should be bigger and KI and KD

should be smaller(iii) When the system is tending toward a steady

state KP should be assigned a bigger value inorder to enhance the response speed and accessthe steady state quickly KD should be added todecrease the overshoot and KI should be re-duced to avoid the oscillations caused by inte-gral overshoot

(iv) When the overshoot of the system is negativeand the error is increasing KD should beassigned a bigger value When the error reachesthe maximum and the system tends toward asteady state KD should be decreased and KP

and KI should be increasedForty-nine fuzzy rules are obtained for theproposed fuzzy algorithm as followsRule one if (e is NB) and (ec is NB) then (KPisPB) (KIis NB) and (KDis PS)Rule two if (e is NB) and (ec is NM) then (KPisPB) (KIis NB) and (KDis NS)

Rule three if (e is NB) and (ec is NS) then (KPisPM) (KIis NM) and (KDis NB)middot middot middot middot middot middot middot middot middot

Fuzzy rule setting for Kp KI and KD is given byTable 2

(3) Defuzzification defuzzification was performed toextract the values from all of the rules given abovewhere the number of rules was transformed into thenumber of variables by using a membership func-tion A center of gravity method is employed fordefuzzification in this paper [40]

Furthermore in order to accelerate the trajectorytracking and constant force control by the fuzzy PID con-troller the NN is utilized in the proposed methodpreviously

Neural network function solves nonlinear problemswhich change into linear ones by mapping the low-di-mension original space to the high-dimension feature spaceand approximating any continuous function with theweighted sum of multiple basis functions e neural net-work structure is shown in Figure 5

In the network structure X [r(t) c(t)]T is the inputY [y(t) τ(t)]T is the output H [h1 h2 middot middot middot hm]T is thevector of radial basis and hj is usually the Gauss function

hj exp minusX minus Cj

2

2b2j

⎛⎜⎜⎝ ⎞⎟⎟⎠ X [r(t) c(t)]T (17)

where Cj [cj1 cj2 middot middot middot cjn]T is the center vector of the nthnode and B [b1 b2 middot middot middot bm]T is the width of the basisvector the weight vector of network is selected asW [w1 w2 wm]T

e output of the neural network is given by

Ym(t) w1h1 + w2h2 + middot middot middot + wmhm (18)

c(t) = JT he

e2

e1

τ i

+

+ndash

ndash

+ ndash

Fuzzification DefuzzificationInferenceengine

Rule base Neural network

NN output

PIDcontroller Control

variable

Feedback linearization Model of roboticmanipulator

τ(t)

r(t)

y(t)

Figure 3 Block diagram of the proposed fuzzy PID with NN

Complexity 5

e optimization objective function is given by

F 12

Y(t) minus Ym(t)( 11138572

12(u(t))

2 (19)

According to the gradient descent method the iterationalgorithm of the neural network parameters is satisfied asfollows

Membership function plots

10

05

00

NMNB

ndash3 ndash2 ndash1 0 1 2 3

NS ZO

Input variable lsquolsquoersquorsquo

PS PM PB

Plots points 181

(a)


10

05

00

NMNB


NS ZO

Input variable lsquolsquoecrsquorsquo

PS PM PB

Plots points 181

(b)


10

05

00

NMNB


NS ZO

Input variable lsquolsquokprsquorsquo

PS PM PB

Plots points 181

(c)


10

05

00

NMNB


NS ZO

Input variable lsquolsquokirsquorsquo

PS PM PB

Plots points 181

(d)


10

05

00

NMNB


NS ZO

Input variable lsquolsquokdrsquorsquo

PS PM PB

Plots points 181

(e)

Figure 4 Fuzzy membership function

6 Complexity

HiddenlayerInput

layerOutput

layer

Model of two-linkrobotic manipulator

NN algorithm

y(t)r(t)

c(t)τ(t)

Figure 5 Structure diagram of NN algorithm for two-link robotic manipulator

Table 2 Fuzzy rule setting for Kp KI and KD

e ec NB NM NS ZO PS PM PBNB PBNBPS PBNBNS PMNMNB PMNMNB PSNSNB ZOZONM ZOZOPSNM PBNBPS PBNBNS PMNMNB PSNSNM PSNSNM ZOZONS NSZOZONS PMNMZO PMNMNS PSNSNM PSNSNM ZOZONS NSPSNS NSPSZOZO PMNMZO PMNMNS PSNSNS ZOZONS NSPSNS NMPMNS NMPMZOPS PSNMZO PSNSZO ZOZOZO NSPSZO NSPSZO NMPMZO NMPBZOPM PSZOPB ZOZONS NSPSPS NMPSPS NMPMPS NMPBPS NBPBPBPB ZOZOPB ZOZOPM NMPSPM NMPMPM NMPMPS NBPBPS NBPBPB

02

ndash023 3

ndash3 ndash32 2

eec ndash2 ndash21 1ndash1 ndash10 0

015

ndash015

01

ndash01

005

ndash0050k p

(a)

004

ndash0043 3

ndash3 ndash32 2


003

ndash003

002

ndash002

001

ndash0010k i

(b)

3 3

ndash3 ndash32 2


2

ndash2

15

ndash15

1

ndash1

05

ndash050k d

(c)

Figure 6 Surface schematic representations for KP KI and KD

Complexity 7

wj(k) wj(k minus 1) + ηu(t)hj + α wj(k minus 1) minus wj(k minus 2)1113872 1113873

Δbj u(t)wjhj

X minus Cj

2

b3j

bj(k) bj(k minus 1) + ηΔbj + α bj(k minus 1) minus bj(k minus 2)1113872 1113873

Δcji u(t)wj

xj minus cji

b2j

cji(k) cji(k minus 1) + ηΔcji + α cji(k minus 1) minus cji(k minus 2)1113872 1113873

(20)

where η is the learning rate and α is the momentum factorRegarding the sensitivity of the output to input changes

the algorithm is given as follows

zY

zu 1113944

m

j1wjhj

cji minus Δub2j

(21)

4 Simulation and Discussion

In this section numerical simulation is conducted to dem-onstrate the performance of the proposed controller e

initialization parameters are determined as η 025 andα 004e initial value of the weight vector the node centerand the width value of the basis function are shown as follows

W [minus06132 minus09216 01835 06783 minus02210 minus03275]T

B [15039 03736 26521 24361 02505 15362]T

C

05862 06302 minus08965 minus20464 minus18435 16532

minus10356 minus02987 10536 05626 23748 05032

minus21452 minus02098 18364 26869 10592 21782

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(22)

Surface schematic representations of the gain controlparameters comprising KP KI and KD are shown inFigure 6

e contact constant force is loaded as 10N and theresponse curves of two control strategies are shown inFigure 7 e constant force traction error in the stabilityinterval (15ndash10 s) is shown in Figure 8 As shown in Fig-ures 7 and 8 the response time of the complete trackingprocess in 15 s and the maximum traction error is 022Ne parameters of the FNN-PID controller are self-adjustedtherefore the response speed improved and the constantforce tracking process is more stable

While maintaining constant force control simulationcomparisons of the displacement trajectory tracking

e desired tracking value (constant force)e response curve of traditional PIDe response curve of FNN-PID

2 4 6 8 100Time (s)

0

2

4

6

8

10

12

14

Con

stant

forc

e tra

ckin

g (N

)

Figure 7 Comparison of response curves of two control strategies

ndash03

ndash02

ndash01

0

01

02

Forc

e tra

ckin

g er

ror (

N)

4 6 8 102Time (s)

Figure 8 e constant force tracking error

8 Complexity

performance are derived Displacement of the directions of x

and y of the end-effector is shown in Figures 9(a) and 9(b)respectively and the trajectory tracking errors of the end-effector are derived and shown as Figure 10 It shows that themaximum error is 003mm and the error is close to balanceposition and keeps stabilities after 08 s

In order to test the dynamic characteristics of theproposed system step function is used as an external dis-turbance at 5 s and the displacement tracking of directions x

and y is shown in Figure 11It shows that the tracking performance is convergence

behind 085 s for x direction and 025 s for y direction

Planning trajectoryFNN-PIDTraditional PID

0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)

Figure 9 Displacement tracking of (a) x direction (b) y direction

ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)

Figure 10 Position tracking errors of the end-effector of two-link robotic manipulator

ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)

Figure 11 Displacement tracking (a) x direction and (b) y direction with external disturbance

Complexity 9

respectively Furthermore the constant force tracking is alsoadjusted convergence with external disturbance as shown inFigure 12

5 Conclusions

e framework of the fuzzy-neural-network PID (FNN-PID) control of the robotic manipulator has been presentedin this article By using Lagrange energy function a precisedynamic model of two-link robotic manipulator is built andthe relationship among actuated torque displacement ve-locity and acceleration in joint space is presented In orderto tune PID controller parameters quickly and effectively afuzzy neural network algorithm is proposed for two kinds ofPID controller that is trajectory tracking and limitedchange online while in contact with constant force controlNumerical simulations are conducted and analyzed to il-lustrate the effectiveness of the proposed method

Data Availability

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

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no51165009) and Innovation School Project of EducationDepartment of Guangdong Province China (Grant nos2017KZDXM060 and 2018KCXTD023)

References

[1] I Rodriguez K Nottensteiner D Leidner M DurnerF Stulp and A Albu-Schaffer ldquoPattern recognition forknowledge transfer in robotic assembly sequence planningrdquoIEEE Robotics and Automation Letters vol 5 no 2pp 3666ndash3673 2020

[2] K Numakura Y Muto M Saito S Narita T Inoue andT Habuchi ldquoRobot-assisted laparoscopic pyeloplasty forureteropelvic junction obstruction with duplex systemrdquoUrology Case Reports vol 30 Article ID 101138 2020

[3] Y F Dong T Y Ren K Hu D Wu and K Chen ldquoContactforce detection and control for robotic polishing based onjoint torque sensorsrdquo International Journal of AdvancedManufacturing Technology vol 107 no 5-6 pp 2745ndash27562020

[4] T Ozaki T Sujuki T Furuhashi S Okuma and Y UchikawaldquoTrajectory control of robotic manipulators using neuralnetworksrdquo IEEE Transaction on Industrial Electronics vol 38no 3 pp 195ndash202 1999

[5] V A Mut J F Postigo R O Carelli and B Kuchen ldquoRobusthybrid motion-force control algorithm for robot manipula-torsrdquo International Journal of Engineering vol 13 no 3pp 233ndash242 2000

[6] T-H S Li and Y-C Huang ldquoMIMO adaptive fuzzy terminalsliding-mode controller for robotic manipulatorsrdquo Informa-tion Sciences vol 180 no 23 pp 4641ndash4660 2010

[7] M-D Tran and H-J Kang ldquoAdaptive terminal sliding modecontrol of uncertain robotic manipulators based on localapproximation of a dynamic systemrdquo Neurocomputingvol 228 pp 231ndash240 2017

[8] P Ouyang and V Pano ldquoComparative study of DE PSO andGA for position domain PID controller tuningrdquo Algorithmsvol 8 no 3 pp 697ndash711 2015

[9] V Kumar P Gaur and A P Mittal ldquoANN based self tunedPID like adaptive controller design for high performancePMSM position controlrdquo Expert Systems with Applicationsvol 41 no 17 pp 7995ndash8002 2014

[10] S He H Fang M Zhang F Liu X Luan and Z DingldquoOnline policy iterative-based Hinfin optimization algorithmfor a class of nonlinear systemsrdquo Information Sciencesvol 495 pp 1ndash13 2019

[11] S He H Fang M Zhang F Liu and Z Ding ldquoAdaptiveoptimal control for a class of nonlinear systems the onlinepolicy iteration approachrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 31 no 2 pp 549ndash5582020

[12] M H Khodayari and S Balochian ldquoModeling and control ofautonomous underwater vehicle (AUV) in heading and depthattitude via self-adaptive fuzzy PID controllerrdquo Journal ofMarine Science and Technology vol 20 no 3 pp 559ndash5782015

[13] F G Rossomando and CM Soria ldquoIdentification and controlof nonlinear dynamics of a mobile robot in discrete time usingan adaptive technique based on neural PIDrdquo Neural Com-puting and Applications vol 26 no 5 pp 1179ndash1191 2015

[14] J -W Jung V Q Leu T D Do E -K Kim and H H ChoildquoAdaptive PID speed control design for permanent magnetsynchronous motor drivesrdquo IEEE Transactions on PowerElectronics vol 30 no 2 pp 900ndash908 2015

[15] X Zhou C Yang and T Cai ldquoA model reference adaptivecontrolPID compound scheme on disturbance rejection foran aerial initially stabilized platformrdquo Journal of Sensorsvol 2016 Article ID 7964727 11 pages 2016

[16] A S W P Annal and S Kanthalakshmi ldquoAn adaptive PIDcontrol algorithm for nonlinear process with uncertain dy-namicsrdquo International Journal of Automation and Controlvol 11 no 3 pp 262ndash273 2017

[17] C C Ren and S P He ldquoFinite-time stabilization for positiveMarkovian jumping neural networksrdquo UNSP vol 365Article ID 124631 2020

0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)

Figure 12 Constant force tracking with external disturbance

10 Complexity

[18] P Pei Z C Pei Z Y Tang and H Gu ldquoPosition trackingcontrol o PMSM based on fuzzy PID-variable structureadaptive controlrdquo Mathematical Problems in Engineeringvol 2018 Article ID 5794067 15 pages 2018

[19] Z Pan D Li K Yang and H Deng ldquoMulti-robot obstacleavoidance based on the improved artificial potential field andPID adaptive tracking control algorithmrdquo Robotica vol 37no 11 pp 1883ndash1903 2019

[20] Y X Su P C Muller and C H Zheng ldquoGlobal asymptoticsaturated PID control for robot manipulator (vol 18 pg 12802010)rdquo IEEE Transactions on Control Systems Technologyvol 23 no 1 p 412 2015

[21] S J Song Y Moon D H Lee C B Ahn Y Jo and J ChoildquoComparative study of fuzzy PID control algorithms forenhanced position control in laparoscopic surgery robotrdquoJournal of Medical and Biological Engineering vol 35 no 1pp 34ndash44 2015

[22] J M Seok ldquoType-2 fuzzy self-tuning PID controller designand steering angle control for mobile robot turningrdquo 6eJournal of Korea Navigation Institute vol 20 no 3pp 226ndash231 2016

[23] K S Chia and X Y Yap ldquoA portable PID control learning toolby means of a mobile robotrdquo International Journal of OnlineEngineering (iJOE) vol 12 no 6 pp 54ndash57 2016

[24] S Wang X Yin P Li M Zhang and X Wang ldquoTrajectorytracking control for mobile robots using reinforcementlearning and PIDrdquo Iranian Journal of Science and TechnologyTransactions of Electrical Engineering vol 44 p 1031 2019

[25] A K khalaji ldquoPID-based target tracking control of a tractor-trailer mobile robotrdquo Proceedings of the Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 233 no 13 pp 4776ndash4787 2019

[26] J Ju Y Zhao C Zhang and Y Liu ldquoVibration suppression ofa flexible-joint robot based on parameter identification andfuzzy PID controlrdquo Algorithms vol 11 no 11 p 189 2018

[27] P Sanchez-Sanchez and M A Arteaga-Perez ldquoImprovingforce tracking control performance in cooperative robotsrdquoInternational Journal of Advanced Robotic Systems vol 14no 4 Article ID 1729881417708969 2017

[28] R Cortesao and M Dominici ldquoRobot force control on abeating heartrdquo IEEEASME Transactions on Mechatronicsvol 22 no 4 pp 1736ndash1743 2017

[29] B Baigzadehnoe Z Rahmani A Khosravi and B Rezaie ldquoOnpositionforce tracking control problem of cooperative robotmanipulators using adaptive fuzzy backstepping approachrdquoISA Transactions vol 70 pp 432ndash446 2017

[30] D G G Rosa J F S Feiteira A M Lopes andP A F de Abreu ldquoAnalysis and implementation of a forcecontrol strategy for drilling operations with an industrialrobotrdquo Journal of the Brazilian Society of Mechanical Sciencesand Engineering vol 39 no 11 pp 4749ndash4756 2017

[31] A E K Mohammad J Hong and D Wang ldquoDesign of aforce-controlled end-effector with low-inertia effect for ro-botic polishing using macro-mini robot approachrdquo Roboticsand Computer-Integrated Manufacturing vol 49 pp 54ndash652018

[32] J E Solanes L Gracia P Munoz-Benavent J V MiroC Perez-Vidal and J Tornero ldquoRobust hybrid position-forcecontrol for robotic surface polishingrdquo Journal ofManufacturing Science and Engineering-Transactions of theASME vol 141 no 1 Article ID 011013 2019

[33] Y Ding X Min W Fu and Z Liang ldquoResearch and ap-plication on force control of industrial robot polishing con-cave curved surfacesrdquo Proceedings of the Institution of

Mechanical Engineers Part B Journal of Engineering Man-ufacture vol 233 no 6 pp 1674ndash1686 2019

[34] Y Gan J Duan M Chen and X Dai ldquoMulti-robot trajectoryplanning and positionforce coordination control in complexwelding tasksrdquo Applied Sciences vol 9 no 5 p 924 2019

[35] T Osa N Sugita and M Mitsuishi ldquoOnline trajectoryplanning and force control for automation of surgical tasksrdquoIEEE Transactions on Automation Science and Engineeringvol 15 no 2 pp 675ndash691 2018

[36] T Zhang M Xiao Y B Zou and J D Xiao ldquoRoboticconstant-force grinding control with a press-and-releasemodel and model-based reinforcement learningrdquo Interna-tional Journal of Advanced Manufacturing Technologyvol 106 no 1-2 pp 589ndash602 2020

[37] T Zhang and X Liang ldquoDisturbance observer-based robotend constant contact force-tracking controlrdquo Complexityvol 2019 Article ID 5802453 20 pages 2019

[38] S Richa B Shubhendu G Prerna and J Deepak ldquoAswitching-based collaborative fractional order fuzzy logiccontrollers for robotic manipulatorsrdquo Applied MathematicalModelling vol 73 pp 228ndash246 2019

[39] M Vijay C Himanshu R Asha and S Vijander ldquoAn expert2DOF fractional order fuzz PID controller for nonlinearsystemsrdquo Neural Computing and Applications vol 31pp 4253ndash4270 2019

[40] H Sadeghi H Motameni A Ebrahimnejad and J VahidildquoMorphology of composition functions in Persian sentencesthrough a newly proposed classified fuzzy method and centerof gravity defuzzification methodrdquo Journal of Intelligent ampFuzzy Systems vol 36 no 6 pp 5463ndash5473 2019

Complexity 11

a heuristic tuning method of PID parameters is proposed byZiegler and Nichols in the 1940s

In recent times an overwhelming development in theterrain of advantage PID control schemes for parameterstuning has encouraged various control engineers to work inthis developing field Differential evolution (DE) geneticalgorithm (GA) and particle swarm optimization (PSO)were used to determine the optimal gains of the positiondomain PID controller and three distinct fitness functionswere also used to quantify the contour tracking performanceof each solution set in [8] In order to expand the robustnessand adaptive capabilities of conventional PID controller aneural network- (NN-) based PID controller which is tunedwhen the controller is operating in an online mode for high-performance magnet synchronous motor position controlwas proposed by Kumar et al [9] and the training algorithmfor the PID controller gain initialization based on theminimum norm least square solution was also proposed Forthe last few years several works have been cited on theintelligent algorithm as well as online policy iterative ap-proach adaptive optimal control sliding mode control andneural network control for different processes or plants suchas uncertain nonlinear systems [10 11] autonomous underwater vehicle (AUV) [12] mobile robot [13] permanentmagnet synchronous motor drivers [14] and many others[15ndash19]

Recently real-time trajectory optimization of roboticmanipulators has received a lot of attention and research dueto the increasing demand to improve movement speedaccuracy and lower energy consumption when executingtasks Several authors have been working toward intelligentPID control techniques of trajectory tracking for roboticmanipulators such as global asymptotic saturated PIDcontrol [20] fuzzy PID for enhanced position control insurgery robot [21] mobile robot [22ndash25] and flexible jointrobot [26] However trajectory tracking is not the onlyfactor to ensure the accuracy of the operation during thecontact working conditions of the robotic manipulator suchas high precision flexible assembly surface polishing andlocation of clamp in minimally invasive surgery (shown inFigure 1)

With the changes of the pressure temperature frictionand wear of the contact surface the contact forcetwistbetween the robot and the work piece will also changeAlthough the robotic manipulator can run accuratelyaccording to the present trajectory the machining quality orassembly quality on the contact surface will decline By usinga modified hybrid computed torque method based on theprinciple of orthogonalization Sanchez presented an im-proving force tracking control structure to reduce thenumber of sensors with a velocity observer [27] Cortesaoproposed double active observer architecture to tackleprecise force control in the presence of heart motion andone controls the desired interaction force and the other oneis responsible for compensating heart motion autonomously[28] An adaptive fuzzy backstepping position trackingcontrol scheme was proposed for multirobot manipulatorsystems and extra terms were also added to the controlsignals to consider the force tracking problem by utilizing

the properties of internal forces in [29] Accurate and robustforcetwist control is still a great challenge for robot-envi-ronment contact applications such as in drilling operations[30] polishing [31ndash33] welding [34] and surgical tasks [35]

In general the traditional PID controller helps ineliminating the steady state error However this controllerdoes not provide effective control in presence of nonline-arities and uncertainties Moreover motivated by the re-search studies carried out in [36 37] We proposed in thispaper fuzzy neural network proportion-integration-differ-entiation (FNN-PID) constant forcetwist control of therobotic manipulator for contact working conditions emain contributions of the current paper are summarized asfollows

(1) By using Lagrange energy function the precise dy-namic model of two-link robotic manipulator isbuilt Compared with the literature [38] it com-pletely represents the relationship between actuatedtorque and displacement and velocity and acceler-ation in joint space

(2) For the contact working condition the change ofcontact force will reduce the machining quality so itis important to keep the contact constant forceduring manufacturing process However constantcontact force and trajectory tracking are opposite toeach other In order to tune PID controller pa-rameters quickly and effectively a fuzzy neuralnetwork algorithm is proposed for two kindsfunctions of PID controller that is trajectorytracking and limited change online while contactconstant force control

is paper is organized in the following manner Section2 presents the precise dynamic modeling method and theprecise dynamic model of two types of planar robotic ma-nipulator with RR (revolute joint) and RP (prismatic joint)are derived Section 3 describes the PID controller with fuzzyneural network and its stability is analyzed Section 4 dis-cusses the relationship between the contact forcetwist withthe trajectory according to the contact working conditionsand numerial simulation results are reported to demonstratethe effectiveness of the proposed method Finally conclu-sions are given in Section 5

2 Dynamic Modeling of Two-Link Robot

e dynamic mathematical model for a rigid planar roboticmanipulator having two links and a contact surface and theexternal force acted on the surface as shown in Figure 2According to the coordinate system o minus xy1113864 1113865 it consists oftwo links having link length l1 and l2 with their center ofmass m1 and m2 lying at the middle of links respectivelye length of center of mass are p1 and p2 respectively

Different from the literature [39] the Lagrange methodis used to build the precise dynamic model of two-linkrobotic manipulator with their nominal values as listed inTable 1

e location coordinates and the square of velocity of 1center of mass can be derived as follows

2 Complexity

x1 p1 sin θ1

y1 p1 cos θ1

_x21 + _y

21 p1

_θ11113872 11138732

(2)


x2 l1 sin θ1 + p2 sin θ1 + θ2( 1113857

y2 l1 cos θ1 + p2 cos θ1 + θ2( 1113857

_x22 + _y

22 l

21_θ21 + p

22

_θ1 + _θ21113872 11138732

+ 2l1p2_θ21 + _θ1 _θ21113874 1113875cos θ2

(3)


Ek Ek1 + Ek2 12m1p

21_θ21 +

12m2l

21_θ21 +

12m2p

22

_θ1 + _θ21113872 11138732

+ m2l1p2_θ21 + _θ1 _θ21113874 1113875cos θ2

(4)



(5)

y Contact force he

θ2

θ1

m2 I2

x

m1 I1

τ2

τ1

l 1

p1

p 2

l 2




(a) (b) (c)


Complexity 3



τi d

dt

zL

z _θi

minuszL

zθi

i 1 2 (7)


z _θ1 m1p

21 + m2l

211113872 1113873 _θ1 + m2l1p2 2 _θ1 + _θ21113872 1113873c2

+ m2p22

_θ1 + _θ21113872 1113873

zL


τ1 ddt

zL

z _θ1minus

zL

zθ1 D11


_θ1 _θ2

+D122_θ22 + D1

(8)


D11 m1p21 + m2p

22 + m2l

21 + 2m2l1p2c2

D12 m2p22 + m2l1p2c2

D112 minus2m2l1p2s2

D122 minusm2l1p2s2

D1 m1p1 + m2l1( 1113857gs1 + m2p2gs12

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)


τ2 ddt

zL

z _θ2minus

zL

zθ2 D21


_θ1 _θ2 + D211_θ22 + D2

(10)where

D21 m1p22 + m2l1p2c2

D22 m2p22

D212 0

D211 m2l1p2s2

D2 m2p2gs12

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)


D11 D12

D21 D221113890 1113891

euroθ1euroθ2

⎡⎣ ⎤⎦ +D112

D2121113890 1113891

_θ1 _θ2_θ2 _θ1

⎡⎣ ⎤⎦ +D211

D1221113890 1113891

_θ22

_θ21

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

+D1

D21113890 1113891

τ1τ2

1113890 1113891

(12)




D12prime I2 + D12

D21prime I2 + D21

D22prime I2 + D22

(13)





l1c1 + l2c12 l2c121113890 1113891 (15)








4 Complexity

















2

2b2j

⎛⎜⎜⎝ ⎞⎟⎟⎠ X [r(t) c(t)]T (17)




c(t) = JT he

e2

e1

τ i

+

+ndash

ndash

+ ndash



NN output


variable


τ(t)

r(t)

y(t)


Complexity 5


F 12

Y(t) minus Ym(t)( 11138572

12(u(t))

2 (19)



10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(a)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(b)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(c)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(d)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(e)


6 Complexity

HiddenlayerInput

layerOutput

layer


NN algorithm

y(t)r(t)

c(t)τ(t)




02

ndash023 3

ndash3 ndash32 2


015

ndash015

01

ndash01

005

ndash0050k p

(a)

004

ndash0043 3

ndash3 ndash32 2


003

ndash003

002

ndash002

001

ndash0010k i

(b)

3 3

ndash3 ndash32 2


2

ndash2

15

ndash15

1

ndash1

05

ndash050k d

(c)


Complexity 7


Δbj u(t)wjhj

X minus Cj

2

b3j


Δcji u(t)wj

xj minus cji

b2j


(20)



zY

zu 1113944

m

j1wjhj

cji minus Δub2j

(21)





B [15039 03736 26521 24361 02505 15362]T

C


minus10356 minus02987 10536 05626 23748 05032

minus21452 minus02098 18364 26869 10592 21782

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(22)





2 4 6 8 100Time (s)

0

2

4

6

8

10

12

14

Con

stant

forc

e tra

ckin

g (N

)


ndash03

ndash02

ndash01

0

01

02

Forc

e tra

ckin

g er

ror (

N)

4 6 8 102Time (s)


8 Complexity







0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)


ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)


ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)


Complexity 9


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11

x1 p1 sin θ1

y1 p1 cos θ1

_x21 + _y

21 p1

_θ11113872 11138732

(2)


x2 l1 sin θ1 + p2 sin θ1 + θ2( 1113857

y2 l1 cos θ1 + p2 cos θ1 + θ2( 1113857

_x22 + _y

22 l

21_θ21 + p

22

_θ1 + _θ21113872 11138732

+ 2l1p2_θ21 + _θ1 _θ21113874 1113875cos θ2

(3)


Ek Ek1 + Ek2 12m1p

21_θ21 +

12m2l

21_θ21 +

12m2p

22

_θ1 + _θ21113872 11138732

+ m2l1p2_θ21 + _θ1 _θ21113874 1113875cos θ2

(4)



(5)

y Contact force he

θ2

θ1

m2 I2

x

m1 I1

τ2

τ1

l 1

p1

p 2

l 2




(a) (b) (c)


Complexity 3



τi d

dt

zL

z _θi

minuszL

zθi

i 1 2 (7)


z _θ1 m1p

21 + m2l

211113872 1113873 _θ1 + m2l1p2 2 _θ1 + _θ21113872 1113873c2

+ m2p22

_θ1 + _θ21113872 1113873

zL


τ1 ddt

zL

z _θ1minus

zL

zθ1 D11


_θ1 _θ2

+D122_θ22 + D1

(8)


D11 m1p21 + m2p

22 + m2l

21 + 2m2l1p2c2

D12 m2p22 + m2l1p2c2

D112 minus2m2l1p2s2

D122 minusm2l1p2s2

D1 m1p1 + m2l1( 1113857gs1 + m2p2gs12

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)


τ2 ddt

zL

z _θ2minus

zL

zθ2 D21


_θ1 _θ2 + D211_θ22 + D2

(10)where

D21 m1p22 + m2l1p2c2

D22 m2p22

D212 0

D211 m2l1p2s2

D2 m2p2gs12

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)


D11 D12

D21 D221113890 1113891

euroθ1euroθ2

⎡⎣ ⎤⎦ +D112

D2121113890 1113891

_θ1 _θ2_θ2 _θ1

⎡⎣ ⎤⎦ +D211

D1221113890 1113891

_θ22

_θ21

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

+D1

D21113890 1113891

τ1τ2

1113890 1113891

(12)




D12prime I2 + D12

D21prime I2 + D21

D22prime I2 + D22

(13)





l1c1 + l2c12 l2c121113890 1113891 (15)








4 Complexity

















2

2b2j

⎛⎜⎜⎝ ⎞⎟⎟⎠ X [r(t) c(t)]T (17)




c(t) = JT he

e2

e1

τ i

+

+ndash

ndash

+ ndash



NN output


variable


τ(t)

r(t)

y(t)


Complexity 5


F 12

Y(t) minus Ym(t)( 11138572

12(u(t))

2 (19)



10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(a)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(b)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(c)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(d)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(e)


6 Complexity

HiddenlayerInput

layerOutput

layer


NN algorithm

y(t)r(t)

c(t)τ(t)




02

ndash023 3

ndash3 ndash32 2


015

ndash015

01

ndash01

005

ndash0050k p

(a)

004

ndash0043 3

ndash3 ndash32 2


003

ndash003

002

ndash002

001

ndash0010k i

(b)

3 3

ndash3 ndash32 2


2

ndash2

15

ndash15

1

ndash1

05

ndash050k d

(c)


Complexity 7


Δbj u(t)wjhj

X minus Cj

2

b3j


Δcji u(t)wj

xj minus cji

b2j


(20)



zY

zu 1113944

m

j1wjhj

cji minus Δub2j

(21)





B [15039 03736 26521 24361 02505 15362]T

C


minus10356 minus02987 10536 05626 23748 05032

minus21452 minus02098 18364 26869 10592 21782

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(22)





2 4 6 8 100Time (s)

0

2

4

6

8

10

12

14

Con

stant

forc

e tra

ckin

g (N

)


ndash03

ndash02

ndash01

0

01

02

Forc

e tra

ckin

g er

ror (

N)

4 6 8 102Time (s)


8 Complexity







0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)


ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)


ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)


Complexity 9


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11



τi d

dt

zL

z _θi

minuszL

zθi

i 1 2 (7)


z _θ1 m1p

21 + m2l

211113872 1113873 _θ1 + m2l1p2 2 _θ1 + _θ21113872 1113873c2

+ m2p22

_θ1 + _θ21113872 1113873

zL


τ1 ddt

zL

z _θ1minus

zL

zθ1 D11


_θ1 _θ2

+D122_θ22 + D1

(8)


D11 m1p21 + m2p

22 + m2l

21 + 2m2l1p2c2

D12 m2p22 + m2l1p2c2

D112 minus2m2l1p2s2

D122 minusm2l1p2s2

D1 m1p1 + m2l1( 1113857gs1 + m2p2gs12

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)


τ2 ddt

zL

z _θ2minus

zL

zθ2 D21


_θ1 _θ2 + D211_θ22 + D2

(10)where

D21 m1p22 + m2l1p2c2

D22 m2p22

D212 0

D211 m2l1p2s2

D2 m2p2gs12

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)


D11 D12

D21 D221113890 1113891

euroθ1euroθ2

⎡⎣ ⎤⎦ +D112

D2121113890 1113891

_θ1 _θ2_θ2 _θ1

⎡⎣ ⎤⎦ +D211

D1221113890 1113891

_θ22

_θ21

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

+D1

D21113890 1113891

τ1τ2

1113890 1113891

(12)




D12prime I2 + D12

D21prime I2 + D21

D22prime I2 + D22

(13)





l1c1 + l2c12 l2c121113890 1113891 (15)








4 Complexity

















2

2b2j

⎛⎜⎜⎝ ⎞⎟⎟⎠ X [r(t) c(t)]T (17)




c(t) = JT he

e2

e1

τ i

+

+ndash

ndash

+ ndash



NN output


variable


τ(t)

r(t)

y(t)


Complexity 5


F 12

Y(t) minus Ym(t)( 11138572

12(u(t))

2 (19)



10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(a)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(b)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(c)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(d)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(e)


6 Complexity

HiddenlayerInput

layerOutput

layer


NN algorithm

y(t)r(t)

c(t)τ(t)




02

ndash023 3

ndash3 ndash32 2


015

ndash015

01

ndash01

005

ndash0050k p

(a)

004

ndash0043 3

ndash3 ndash32 2


003

ndash003

002

ndash002

001

ndash0010k i

(b)

3 3

ndash3 ndash32 2


2

ndash2

15

ndash15

1

ndash1

05

ndash050k d

(c)


Complexity 7


Δbj u(t)wjhj

X minus Cj

2

b3j


Δcji u(t)wj

xj minus cji

b2j


(20)



zY

zu 1113944

m

j1wjhj

cji minus Δub2j

(21)





B [15039 03736 26521 24361 02505 15362]T

C


minus10356 minus02987 10536 05626 23748 05032

minus21452 minus02098 18364 26869 10592 21782

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(22)





2 4 6 8 100Time (s)

0

2

4

6

8

10

12

14

Con

stant

forc

e tra

ckin

g (N

)


ndash03

ndash02

ndash01

0

01

02

Forc

e tra

ckin

g er

ror (

N)

4 6 8 102Time (s)


8 Complexity







0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)


ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)


ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)


Complexity 9


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11

















2

2b2j

⎛⎜⎜⎝ ⎞⎟⎟⎠ X [r(t) c(t)]T (17)




c(t) = JT he

e2

e1

τ i

+

+ndash

ndash

+ ndash



NN output


variable


τ(t)

r(t)

y(t)


Complexity 5


F 12

Y(t) minus Ym(t)( 11138572

12(u(t))

2 (19)



10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(a)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(b)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(c)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(d)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(e)


6 Complexity

HiddenlayerInput

layerOutput

layer


NN algorithm

y(t)r(t)

c(t)τ(t)




02

ndash023 3

ndash3 ndash32 2


015

ndash015

01

ndash01

005

ndash0050k p

(a)

004

ndash0043 3

ndash3 ndash32 2


003

ndash003

002

ndash002

001

ndash0010k i

(b)

3 3

ndash3 ndash32 2


2

ndash2

15

ndash15

1

ndash1

05

ndash050k d

(c)


Complexity 7


Δbj u(t)wjhj

X minus Cj

2

b3j


Δcji u(t)wj

xj minus cji

b2j


(20)



zY

zu 1113944

m

j1wjhj

cji minus Δub2j

(21)





B [15039 03736 26521 24361 02505 15362]T

C


minus10356 minus02987 10536 05626 23748 05032

minus21452 minus02098 18364 26869 10592 21782

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(22)





2 4 6 8 100Time (s)

0

2

4

6

8

10

12

14

Con

stant

forc

e tra

ckin

g (N

)


ndash03

ndash02

ndash01

0

01

02

Forc

e tra

ckin

g er

ror (

N)

4 6 8 102Time (s)


8 Complexity







0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)


ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)


ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)


Complexity 9


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11


F 12

Y(t) minus Ym(t)( 11138572

12(u(t))

2 (19)



10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(a)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(b)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(c)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(d)


10

05

00

NMNB


NS ZO


PS PM PB

Plots points 181

(e)


6 Complexity

HiddenlayerInput

layerOutput

layer


NN algorithm

y(t)r(t)

c(t)τ(t)




02

ndash023 3

ndash3 ndash32 2


015

ndash015

01

ndash01

005

ndash0050k p

(a)

004

ndash0043 3

ndash3 ndash32 2


003

ndash003

002

ndash002

001

ndash0010k i

(b)

3 3

ndash3 ndash32 2


2

ndash2

15

ndash15

1

ndash1

05

ndash050k d

(c)


Complexity 7


Δbj u(t)wjhj

X minus Cj

2

b3j


Δcji u(t)wj

xj minus cji

b2j


(20)



zY

zu 1113944

m

j1wjhj

cji minus Δub2j

(21)





B [15039 03736 26521 24361 02505 15362]T

C


minus10356 minus02987 10536 05626 23748 05032

minus21452 minus02098 18364 26869 10592 21782

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(22)





2 4 6 8 100Time (s)

0

2

4

6

8

10

12

14

Con

stant

forc

e tra

ckin

g (N

)


ndash03

ndash02

ndash01

0

01

02

Forc

e tra

ckin

g er

ror (

N)

4 6 8 102Time (s)


8 Complexity







0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)


ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)


ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)


Complexity 9


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11

HiddenlayerInput

layerOutput

layer


NN algorithm

y(t)r(t)

c(t)τ(t)




02

ndash023 3

ndash3 ndash32 2


015

ndash015

01

ndash01

005

ndash0050k p

(a)

004

ndash0043 3

ndash3 ndash32 2


003

ndash003

002

ndash002

001

ndash0010k i

(b)

3 3

ndash3 ndash32 2


2

ndash2

15

ndash15

1

ndash1

05

ndash050k d

(c)


Complexity 7


Δbj u(t)wjhj

X minus Cj

2

b3j


Δcji u(t)wj

xj minus cji

b2j


(20)



zY

zu 1113944

m

j1wjhj

cji minus Δub2j

(21)





B [15039 03736 26521 24361 02505 15362]T

C


minus10356 minus02987 10536 05626 23748 05032

minus21452 minus02098 18364 26869 10592 21782

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(22)





2 4 6 8 100Time (s)

0

2

4

6

8

10

12

14

Con

stant

forc

e tra

ckin

g (N

)


ndash03

ndash02

ndash01

0

01

02

Forc

e tra

ckin

g er

ror (

N)

4 6 8 102Time (s)


8 Complexity







0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)


ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)


ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)


Complexity 9


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11


Δbj u(t)wjhj

X minus Cj

2

b3j


Δcji u(t)wj

xj minus cji

b2j


(20)



zY

zu 1113944

m

j1wjhj

cji minus Δub2j

(21)





B [15039 03736 26521 24361 02505 15362]T

C


minus10356 minus02987 10536 05626 23748 05032

minus21452 minus02098 18364 26869 10592 21782

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(22)





2 4 6 8 100Time (s)

0

2

4

6

8

10

12

14

Con

stant

forc

e tra

ckin

g (N

)


ndash03

ndash02

ndash01

0

01

02

Forc

e tra

ckin

g er

ror (

N)

4 6 8 102Time (s)


8 Complexity







0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)


ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)


ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)


Complexity 9


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11







0 4 6 8 102Time (s)

3 5 7 91ndash2

ndash15ndash1

ndash050

05

151

2D

ispla

cem

ent o

f X (m

m)

(a)


1 2 3 4 5 6 7 8 9 100Time (s)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

(b)


ndash003

ndash002

ndash001

0

001

002

003

004

Posit

ion

trac

king

erro

r (m

m)

1 2 3 4 5 60Time (s)


ndash2ndash15

ndash1ndash05

005

115

2

Disp

lace

men

t of X

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

ndash2

ndash1

0

1

2

Disp

lace

men

t of Y

(mm

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b)


Complexity 9


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11


5 Conclusions


Data Availability




Acknowledgments


References


















0

2

4

6

8

10

12

14

Trac

king

(KN

)

2 4 6 8 100Time (s)


10 Complexity

























Complexity 11

























Complexity 11

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