Research ArticleA119867-Infinity Control for Path Tracking withFuzzy Hyperbolic Tangent Model

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhao and Jian Li

School of Mechanical Engineering University of Science amp Technology Beijing Beijing 100083 China

Correspondence should be addressed to Jue Yang yangjueustbeducn

Received 6 May 2016 Revised 3 September 2016 Accepted 5 October 2016

Academic Editor Mario Russo

Copyright copy 2016 Guangsi Shi 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

To achieve the goal of driver-less underground mining truck a fuzzy hyperbolic tangent model is established for path tracking onan underground articulated mining truck Firstly the sample data of parameters are collected by the driver controlling articulatedvehicle at a speed of 3ms including both the lateral position deviation and the variation of heading angle deviationThen accordingto the improved adaptive BP neural network model and deriving formula of mediation rate of error estimator by the method ofCauchy robust the weights are identified Finally119867-infinity control controller is designed to control steering angle The results ofhardware-in-the-loop simulation show that lateral position deviation heading angle deviation and steering angle of the vehiclecan be controlled respectively at 0024m 008 rad and 021 rad All the deviations are asymptotically stable and error control is inless than 2 The method is demonstrated to be effective and reliable in path tracking for the underground vehicles

1 Introduction

Articulated vehicle is widely used in underground miningResearches on driver-less articulated vehicle have been car-ried out for many years to prompt efficiency and safety inunderground mine A key principle of autonomous drivingin underground tunnel is to find a path for articulated vehicleto track at a reasonable high speed and avoiding crash intosidewall Many automatic control algorithms can be appliedin this filed

In the field of fuzzy model Zeng and Singh [1] andWang[2] have theorized a fuzzy relational model and its purpose isto build a fuzzy model to approximate the ideal control beha-vior This fuzzy relationship can be seen as a fuzzy mappingfrom input space to the output space Its main disadvantageis that many important dynamic information systems areignored during the modeling process and it is difficult toobtain good performances of controller Takagi and Sugeno[3 4] have proposed a fuzzy model named T-S aiming toconstruct a series of linear equations to express each subsys-tem then to make a global model through the membershipfunctions of these subsystems Its main drawback is that thecomplexity of the structure itself to establish amodel requires

lots of work offline Fuzzy hyperbolic tangent model is betterthan the above models [5 6]

Path tracking problem is a very important issue in thecourse of unmanned field The former studies are based onarticulated vehicle kinematics andmany scholars have done alot of researches Nayl et al [7] Lee and Yoo [8] Ridley andCorke [9] and Xuan et al [10] derived kinematic models ofarticulated vehicle and the reference model errors of pathand they used method of model predictive control for track-ing control with simulation The work efficiency and safetyare decided by the accuracy of path tracking Path trackingcontrol design based on articulated vehicle kinematic char-acteristics has no good adaptability in underground miningenvironment Dynamics characteristics are significant forpath tracking control design [11] Literature [12] considersthe influence of articulated vehicle dynamics on sideslip butthere would be a big deviation as they only consider oneaspect and the lack of system modeling

To solve these problems above and to deal with thismulti-variable strong coupling highly complex nonlinear dynam-ical systems of underground mining vehicles we use fuzzyhyperbolicmodel and design a nonlinear quadratic controllerthrough testing field experiments by hardware-in-the-loop

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 9072831 9 pageshttpdxdoiorg10115520169072831

2 Journal of Control Science and Engineering

R

P1

P2

P3

Cx

D yPlowast

120576120579120575120579

Figure 1 Trajectory curve and parametric model image of under-ground mining articulated vehicle

(HIL) simulation to ensure the quality of control aiming atachieving the goal of unmanned underground mining arti-culated vehiclesThis method is to obtain kinematic relationsof vehicle via the driver information and the fact that driverrepeatedly driving the process can make the relationshipdynamics of the vehicle which are included in those databecomes increasingly evident This method can also be con-sidered to be intrinsically linked with a data mining toreduce the complexity of the modeling process which greatlyreduces the system error caused by the dynamics modelingThe coefficient matrix of fuzzy hyperbolic model nonlinearsystems becomes constant matrix The parameters of theidentification via neural network supervised learning meth-ods make the model more close to the real model Also it isconvenient to the control algorithm design Young [13] hasproposed variable structure controller for the first timewhich was applied to robot control Jafarov et al [14] haveproposed a new type of sliding PID controller Althoughthese methods can solve the control problem they could notconsider the robustness of the system119867-infinity control is a good robustness design methodwith clear design ideas good control effect and advantagesespecially on model perturbation of multiple input multipleoutput (MIMO) systems Finally hardware verification con-trol which results in the final loop (HIL) simulation is donein order to ensure quality control

2 Kinematic Model of UndergroundArticulated Vehicle

21 Steering Deviation Model Figure 1 shows the errorsbetween the real and reference path The circle centered by119862 is the reference path Ideally the vehicle should pass 1198751 1198752and 1198753 The variables are defined as follows [5 15]

(1) lateral displacement error the lateral displacementerror between the vehicle reference point and the cor-responding point119875 (the nearest point on the referencepath)

(2) orientation error the orientation error between thevelocity orientation of 119901 and the tangential orienta-tion of 119875

(3) steering angle relative rotation angle of undergroundmining articulated vehicle body before and after thehorizontal plane

These two variables can basically reflect the position andposture of articulated underground mining vehicles in thetunnel and building the trajectory of articulated vehiclesmodel can be achieved by the fuzzy hyperbolic method

Figure 2(a) is the sketch of an artificial tunnel for testing ofarticulated vehicle driver-less control system The truck usedin this test is shown in Figure 2(b)

As can be seen in Figures 3ndash5 at speed of 3m per secondby driver controlling the truck in the process of turning thelateral deviation is stable at 15m Heading angle deviationremains stable at around 0 rad while body roll is obviousThe steering angle remains stable at 02 rad with obviousadjustments As shown in Figures 6 and 7 at speed of 3m persecond rate of change of lateral deviation and the headingangle also is 0 but there are more noise and severe jitter

22 Kinematic Models Based on Fuzzy Hyperbolic ModelNear the origin fuzzy hyperbolic model is a global model[6] with a global approximation performance And it is notonly an essentially nonlinear model but also easy to expressthe dynamic characteristics of nonlinear systems Thereforeaccording to this model the controller can be designed sothat the whole system can achieve the optimal performancesCompared with other fuzzy models fuzzy hyperbolic modelis more suitable for the control object known as multivariablenonlinear finite object

Given a plant 119899 state variables number 119909 = (1199091(119905) 119909119899(119905))T and 119901 input variables numbers u = (u1(119905) u119901(119905))T and the fuzzy rules are used to describe this systemby the following conditions so this group of fuzzy rules istermed as the hyperbolic tangent type of fuzzy rules

Fuzzy rules for the following forms

IF 1199091 is 1198651199091 and 1199092 is 1198651199092 and 119909119899 is 119865119909119899 and u1 is 119865u1and u2 is 119865u2 and u119899 is 119865u119901THEN 119897 = plusmn1198881199091 plusmn 1198881199092 sdot sdot sdot plusmn 119888119909119899 plusmn 119888u1 plusmn 119888u2 sdot sdot sdot plusmn 119888u119901 (119897 =1 119899)

where 119865119909119894 (119894 = 1 119899) and 119865u119895 (119895 = 1 119901) are fuzzy sub-set including (119875) and (119873) corresponding to 119909119894 and u119895 and119888119909119894 (119894 = 1 119899) and 119888u119895 (119895 = 1 119901) are constant of 119865119909119894and 119865u119894 (119895 = 1 119901)

Let 120575119889 = 1199091 120575120579 = 1199092 and u = 120574There are certain mathematical relationships between

manned lateral deviation heading angle deviation the steer-ing angle and rate of change of the lateral deviation andheading angle deviation from the data observed from relevantpapers [10] And they can be roughly described as the follow-ing information through the professional experiences so as to

Journal of Control Science and Engineering 3

8255

R2573

5159

650

6

(a) (b)

Figure 2 (a) The plan view of the tunnel (b) The real truck

minus2

minus15

minus1

minus05

0

05

Man

ned

later

al d

evia

tion

(m)

20 40 60 80 100 1200Time (s)

Figure 3 Manned lateral deviation

20 40 60 80 100 1200Time (s)

minus04

minus03

minus02

minus01

0

01

02

03

Man

ned

head

ing

angl

e dev

iatio

n (r

ad)

Figure 4 Manned heading angle deviation

construct the corresponding fuzzy hyperbolic model of fuzzyrules

R1 IF 1199092 is 1198751199092 THEN 1 = 01R2 IF 1199092 is 1198731199092 THEN 1 = minus01R3 IF 1199091 is 1198751199091 IF 1199092 is 1198751199092 IF u is 119875u THEN 2 = 001R4 IF 1199091 is 1198731199091 IF 1199092 is 1198751199092 IF u is 119875u THEN 2 = 0R5 IF 1199091 is 1198731199091 IF 1199092 is 1198731199092 IF u is 119875u THEN 2 = 008

20 40 60 80 100 1200Time (s)

minus02

minus01

0

01

02

03

04

05

06

Man

ned

steer

ing

angl

e (ra

d)

Figure 5 Manned steering angle

Man

ned

rate

of c

hang

e

20 40 60 80 100 1200Time (s)

minus006minus005minus004minus003minus002minus001

0001002003

of th

e lat

eral

pos

ition

dev

iatio

n (m

middotsminus1)

Figure 6 Manned rate of change of the lateral position deviation

R6 IF 1199091 is 1198731199091 IF 1199092 is 1198731199092 IF u is 119873u THEN 2 =minus001R7 IF 1199091 is 1198751199091 IF 1199092 is 1198731199092 IF u is 119873u THEN 2 = 0R8 IF 1199091 is 1198751199091 IF 1199092 is 1198751199092 IF u is 119873u THEN 2 = minus008R9 IF 1199091 is 1198751199091 IF 1199092 is 1198731199092 IF u is 119875u THEN 2 = 01R10 IF 1199091 is 1198731199091 IF 1199092 is 1198751199092 IF u is 119873u THEN 2 = minus01

4 Journal of Control Science and Engineering

Man

ned

rate

of c

hang

etimes10minus3

20 40 60 80 100 1200Time (s)

minus8minus6minus4minus2

02468

of th

e hea

ding

dev

iatio

n (m

middotsminus1)

Figure 7 Manned rate of change of the heading deviation

Given a set of fuzzy rules of hyperbolic tangent define119875119911 and 119873119911 (where 119911 are arbitrary state variables or inputvariables) as membership function (where k119911 gt 0)

119875119911 (119909) = 119890minus(12)(119909minusk119911)119873119911 (119909) = 119890minus(12)(119909+k119911) (1)

Let k1199091 = k1199092 = ku = 1Membership function

u1198751199091 (119909) = 119890minus(12)(119909minus1)2 u1198731199091 (119909) = 119890minus(12)(119909+1)2 u1198751199092 (119909) = 119890minus(12)(119909minus1)2 u1198731199092 (119909) = 119890minus(12)(119909+1)2 u119875u (119909) = 119890minus(12)(119909minus1)2 u119875u (119909) = 119890minus(12)(119909+1)2

(2)

Then

= A tanh (k119909119909) + B tanh (kuu) (3)

where

A = [ 0 01001 minus01]

B = [ 0008]

k = [[[1

11]]]

tanh (119909) = 119890119909 minus 119890minus119909119890119909 + 119890minus119909

k119909 = diag (k1199091 k119909119899) ku = diag (k1199091 k119909119901)

(4)

Bku are linearized to Bu

= A tanh (k119909119909) + Bu[12] = [ 0 01

001 minus01] [tanh (1199091)tanh (1199092)] + [ 0

008] 120574 (5)

When the absolute value of 119909 is smaller tanh(119909) asymp 119909fuzzy hyperbolic model can be written as = A tanh(k119909119909) +Bu that the system is a linear model when it is close to theequilibrium point

3 Parameter Identification Based onImproved Adaptive BP Neural Network

Recognition technology evaluation includes two indicatorsOne is the identification accuracy and the other is speedidentification Compared to other models in terms of fuzzytopology of the neural network can be used to optimize theparameters for FHM [16] The 119860 weights of mediation rate ofadaptive BP neural network are deduced by Cauchy robusterror to eliminate the effect of outliers in the data and to fitthe original data better

Neural network topology is shown in Figure 8 of FHMControlled object state variables are 119909 = (1199091(119905) 119909119899(119905))Tand input variables u = (u1(119905) u119898(119905))T as the input ofthe neural network topology and model output is the rateof change of the state variables k119894 (119894 = 1 119899) 119892119895 (119895 =1 119898) 119888119894119895 (119894 119895 = 1 119899) and 119889119894119895 (119894 = 1 119899 119895 =1 119898) as the connection weights which need to be trained

A and B are constant matrix whose elements are theweights comprised of 119888119894119895 and 119889119894119895 If the hidden layer function1198911(119909) = tanh(119909) and output layer activation function 1198912(119909) =119909 then the following models can be obtained

= A tanh (k119909119909) + B tanh (kuu) (6)

Obviously when making u linearize variables can beobtained

= A tanh (k119909119909) + Bu (7)

Thus FHM may be built by the neural network modelbecause it can train FHM model parameters [6] Since thereare many jitters in the data of rate of change of horizontal andheading angle the samples show a large number of outliersSo from the statistical view of robust the traditional MSEaggravate the ldquooutliersrdquo of samples Therefore Cauchy robusterror estimator can be used

Let error of outputs be

119864 = 12119902sum

k=1ln [1 + (1199101015840k minus 119910k)2] (8)

Journal of Control Science and Engineering 5

x1

xn

u1

um

k1

kn

g1

gm

f1(middot)c11

c1n

cn1

cnm

d11

d1m

dnm

dn1

f2(middot)x1

xn

Figure 8 Fuzzy hyperbolic tangentmodel neural network topology

where 1199101015840k is ideal output for the network and 119910k is actualoutput

Cauchy error estimator obviously does not depend on theinitial weights and thresholds transition and it can effectivelyeliminate the negative impact of ldquooutliersrdquo while retainingthe main characteristics of the value of the output error andspeeding up the convergence rate

By the steepest descent method it can obtain weightsiterative equations of each layer of neurons

119882(119905 + 1) = 119882 (119905) minus uΔ119864 (9)

where Δ119864 = 120597119864120597120596119882 = 120596119894119895So get a rate adjustment based on the following model

parameters above BP algorithm

119888119894119895 (119905 + 1) = 119888119894119895 (119905) + 120592 sdot 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2 sdot tanh (k119894119909119894)

k119895 (119905 + 1) = k119895 (119905) + 120592 sdot sumk119888119894119895 sdot ( 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2)

sdot 119909119894tanh (sum119894 120596119895k119909119894)

(10)

In this paper learning rate 120592 is considered as the adaptivescheme in which its main idea is that the initial value of 120592being set higher generally about 07ndash09 with the increaseor decrement of the number of learning may change in alaw When 120592 decreases to a certain extent if 120592 has not stillbecome convergence or error has still no improvement 120592 isset again at about 05ndash07 120592 enters the learning process againuntil the end of the run The learning rate is associated withthe error function when the error is reduced increase thelearning rate when the error increases reduce the learningrate

120592(k) =

112120592(kminus1) 119864(kminus1) gt 119864(k)08120592(kminus1) 119864(kminus1) lt 119864(k)120592(kminus1) 119864(kminus1) = 119864(k)

(11)

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

20 40 60 80 100 1200Time (s)

minus006minus005minus004minus003minus002minus001

0001002003

of th

e lat

eral

dev

iatio

n (m

s)

Figure 9 Fitting of change rate of the lateral deviation

Identify the model parameters after obtaining the data after6000 steps of learning

A = [00012 0099600095 minus00999]

B = [minus000700803]

k = [[[1000

100909998

]]]

(12)

Then

[12] = [00012 0099600095 minus00999] [tanh (1199091)

tanh (1199092)]

+ [minus000700803] 120574(13)

The fitting results of Figures 9 and 10 substantially elimi-nate the effect of outliers with relative error of control in lessthan 1 and less than 10The trend is essentially coincidentwith the sample data

The existing experimental data can be applied to theabove-described method for fuzzy hyperbolic model param-eter optimization in order to be close to the actual model Itprovides the basis to the next controller design

4 Controller Design

Based on this model we can design conventional linearcontrollers or other nonlinear controllers In this paper thecontroller is designed with hyperbolic tangent function ofthe state variables so we can use language to describe theinformation of the controller Thus this given controller is afuzzy one [16]

6 Journal of Control Science and Engineering

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

times10minus3

20 40 60 80 100 1200Time (s)

minus8

minus6

minus4

minus2

0

2

4

6

8of

the l

ater

al d

evia

tion

(rad

middotsminus1)

Figure 10 Fitting of change rate of the heading deviation

The system can be stabilized by nonquadratic perform-ance index function [16] System performance can get to aminimum with u through the givenQ and R

= A119891 (119909) + Bu (14)

Definition 1 (see [17]) Let 119878119888 satisfy the following conditionsall sets of 119891(sdot) include R rarr R

(1) 119891 is continuous(2) 119891(0) = 0 for other 119909 isin R 119891(119909)119909 gt 0(3) when |119909| rarr infin int119909

0119891(119910)119889119910 rarr infin

Theorem 2 (see [17]) For nonlinear system

= A119891 (119909) + Bu +D119908 (15)

where 119891(119909) = (1198911(119909) 119891119899(119909))T and 119891(119894) isin 119878119888 and thenk gt 0 tanh(k119909) isin 119878119888 (119894 = 1 2 3 119899) and a performanceindex function can be proposed as

119869 (1199090 1199050 u) = intinfin1199050

119891T (119909 (119905))Q119891 (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(16)

where Q R and S are given as the definite positive symmetricconstant matrix

Design Optimal Control Vector [18] Considerlowast

u(t) = lowast

u(t)(119905119909(120591)119905120591=0) and there is L(1199090) with boundary Then

sup 119869 ( lowast

u (t) 119908) le L (1199090) (17)

If there is a diagonal positive definite matrix Pmin satisfyingRiccati equation

PA + ATP minus P (BRminus1BTP minusDSminus1DT)P +Q = 0 (18)

then [119860 119861QR] is optimized diagonal matrix

Given nonlinear system and nonlinear quadratic perfor-mance index function [119860 119861QR] is an optimized diagonalmatrix The optimal control vector is

lowast

u (t) = minusRminus1BTPmin119891 (119909 (119905)) (19)

where Pmin = diag(1199011 119901119899) is optimized diagonal matrixLet L(1199090) = 2sum119899119894=1 119901119894 int119909119894(1199050)0 119891119894(120591)119889120591

Thus tanh(k119909) isin 119878119888 and for fuzzy hyperbolic tangent thequadratic performance index function is

119869 (1199090 1199050 u) = intinfin1199050

tanhT (119909 (119905))Q tanh (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(20)

Let Q = [ 10 00 15 ] R = 08 and S = 800 with Riccatiequation then

P = [10972 27592759 1678] (21)

According to the closed-loop system equation of state

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (22)

Based on Lyapunov theorem and Riccati equation for thisequation

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (23)

It is easy to prove

Pmin (A minus BRminus1BTPmin) + (A minus BRminus1BTPmin)Pmin

= minus (Q + Pmin (BRminus1BTP minusDSminus1DT)Pmin) (24)

And (A minusBRminus1BTP) is diagonally stable thus the closed-loop system is asymptotically stable

Then

K = [11627 6135] (25)

5 HIL Simulation

Thedifference between simulation and hardware-in-the-loopsimulation is that there is a real-time simulation of thecircuitrsquos physical hardware Hardware-in-the-loop simulationis intended to provide real-time incentives as the true signalto the controller so that the controller is connected to its realaccused equipment and test its performance Figure 11 is a pic-torial diagram of the hardware simulation platform in whichc-RIO controller is recognized as the path tracking anglecontroller program compiled by the Simulink PXI platformsimulation runs the Adams which can build undergroundmining articulated vehicle model as the object of emulationPC can run the LabVIEWgraphical user interface for PXI andc-RIO real-time and display the data of simulation compiledfrom the process Speed can be achieved by PID control at

Journal of Control Science and Engineering 7

PCc-RIO

PXI

LabVIEW

Figure 11 HIL simulation platform

Travel path comparison

Practical pathTarget path

100 20 30minus20 minus10minus30x (m)

minus30

minus20

minus10

0

10

20

30

y (m

)

Figure 12 Simulation of driving route comparedwith the ideal path

V = 3ms So as to test this robust of control there is onepit per 20m on the road The reference trajectory is set asa round whose center is (0 0) and R is 25m and an initialparameter setting is simulation time 200 s and the startingpoint coordinates (minus3 minus25)

Derived from the simulation results Figure 12 is thecomparison chart of the actual trajectory and the referencetrajectory As can be seen from the simulation curve thearticulated vehicle traveling trace and the reference circularpath are consistent and the trajectory is relatively smoothThe speed which is controlled by PID fromFigure 13 becomesfinally steady at 30 seconds The lateral displacement erroris gradually flat from the beginning of violent shaking inFigure 14 The overshoot amount is 428 and it is at mostabout 0024mat 18 secondswith respect to the tread based on2280m and the error is 17 and then close to the origin stepby stepThe orientation error is gradually flat from the begin-ning of violent shaking in Figure 15The overshoot amount is49 and it is at the top at about 008 rad at 18 seconds andthen close to the origin step by step The articulated angleis gradually flat from the beginning of violent shaking in

0

05

1

15

2

25

3

35

Spee

d (m

middotsminus1)

20 40 60 80 100 120 140 160 180 2000Time (s)

Figure 13 Longitudinal velocity of midpoint of front axle

20 40 60 80 100 120 140 160 180 2000Time (s)

minus003

minus0025

minus002

minus0015

minus001

minus0005

0

0005

Late

ral d

ispla

cem

ent e

orrr

(m)

Figure 14 Lateral displacement error

Figure 16 The overshoot amount is 35 and it is eventuallystabilized at 021 rad at 22 seconds Compared with the artic-ulated vehicle steering based on angle 45∘ the error is 12And the peaks are caused by pits but then the system imme-diately recovers to the right path It has better robustness thanliterature [12] in which there are a lot of jitters to adapt tothe path as a similar situation The results of literature [10]are as follows lateral deviation was 1m and heading angledeviation is 01 rad Literature [19] also analyzed themodels ofarticulated vehicle to control it with synovial control methodwhose results of the simulation are as follows the lateraldeviation is 01m and heading angle deviation was 017 radSo there exists a large gap when compared with the results ofthis paper

6 Conclusions

This article provides a method used in unmanned systemsbased on theway of fuzzy hyperbolic pole control act for artic-ulated vehicle trajectory tracking accurately Conclusions areas follows

(1) Fuzzy hyperbolic tangent model can well describethe quantitative relation among lateral displacementerror the orientation error and the articulated angle

(2) Based on the method of Cauchy robust error estima-tor the weights of BP neural network can effectively

8 Journal of Control Science and Engineering

20 40 60 80 100 120 140 160 180 2000Time (s)

minus002

0

002

004

006

008

01

Orie

ntat

ion

erro

r (ra

d)

Figure 15 Unmanned orientation error

20 40 60 80 100 120 140 160 180 2000Time (s)

0

005

01

015

02

025

Art

icul

ated

angl

e (ra

d)

Figure 16 Articulated angle

be reduced by the influence of singular error of neuralnetwork learning and fitting error and relative errorbelow expectations can be decreased so that it canachieve the system of identification

(3) By the 119867-infinity controller design controller has agood control performance and it makes the systemkeep a better performance about robustness in orderto achieve the purpose of the comprehensively opti-mal control error

(4) This method can be used in articulated vehicle pathtracking effectively based on the119867-infinity controllerSimulation in hardware-in-the-loop shows that over-shoot and the response time are less than expectationsand are eventually to stabilize The controller has metthe requirement of the real-time control performance

Competing Interests

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhaoand Jian Li declare that there is no conflict of interestsregarding the publication of this manuscript

Acknowledgments

This work was financially supported by the National HighTechnology Research and Development Program (863 Pro-gram) of China under Award 2011AA060404 IntelligentUnderground Mining Truck and Fundamental ResearchFunds for the Central Universities FRF-TP-16-004A1research on path-planning and path-following algorithmfor underground mining vehicles based on reinforcementlearning

References

[1] X-J Zeng and M G Singh ldquoApproximation theory of fuzzysystemsmdashSISO caserdquo IEEE Transactions on Fuzzy Systems vol2 no 2 pp 162ndash176 1994

[2] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceeding of the IEEE International Conference on Fuzzy Sys-tems pp 1163ndash1170 1992

[3] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[4] M Sugeno and T Yasukawa ldquoFuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] S Scheding G Dissanayake E Nebot and H Durrant-WhyteldquoSlip modelling and aided inertial navigation of an LHDrdquo inProceedings of the IEEE International Conference on Roboticsand Automation pp 1904ndash1909 Institute of Electrical andElectronics Engineers Albuquerque NM USA April 1997

[6] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[7] TNayl GNikolakopoulos andTGustfsson ldquoSwitchingmodelpredictive control for an articulated vehicle under varying slipanglerdquo in Proceedings of the 20th Mediterranean Conference onControl amp Automation (MED rsquo12) pp 890ndash895 IEEE ComputerSociety Barcelona Spain July 2012

[8] J H Lee andW S Yoo ldquoPredictive control of a vehicle trajectoryusing a coupled vector with vehicle velocity and sideslip anglerdquoInternational Journal of Automotive Technology vol 10 no 2 pp211ndash217 2009

[9] P Ridley and P Corke ldquoLoad haul dump vehicle kinematicsand controlrdquo Journal of Dynamic Systems Measurement andControl vol 125 no 1 pp 54ndash59 2003

[10] Z Xuan Y Jue and Z Wenming ldquoFeedback linearizationcontrol for path tracking of articulated dump truckrdquo TELKOM-NIKA Telecommunication Computing Electronics and Controlvol 9 no 13 pp 922ndash929 2015

[11] D Piyabongkarn R Rajamani J A Grogg and J Y LewldquoDevelopment and experimental evaluation of a slip angleestimator for vehicle stability controlrdquo IEEE Transactions onControl Systems Technology vol 17 no 1 pp 78ndash88 2009

[12] Y Qi Trajectory Control for the Underground Articulated DumpTruck Based on Model Predictive Control University of Scienceand Technology Beijing Beijing China 2013

[13] K-K D Young ldquoController design for a manipulator usingtheory of variable structure systemsrdquo IEEE Transactions on Sys-tems Man and Cybernetics vol 8 no 2 pp 101ndash109 1978

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

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2 Journal of Control Science and Engineering

R

P1

P2

P3

Cx

D yPlowast

120576120579120575120579

Figure 1 Trajectory curve and parametric model image of under-ground mining articulated vehicle

(HIL) simulation to ensure the quality of control aiming atachieving the goal of unmanned underground mining arti-culated vehiclesThis method is to obtain kinematic relationsof vehicle via the driver information and the fact that driverrepeatedly driving the process can make the relationshipdynamics of the vehicle which are included in those databecomes increasingly evident This method can also be con-sidered to be intrinsically linked with a data mining toreduce the complexity of the modeling process which greatlyreduces the system error caused by the dynamics modelingThe coefficient matrix of fuzzy hyperbolic model nonlinearsystems becomes constant matrix The parameters of theidentification via neural network supervised learning meth-ods make the model more close to the real model Also it isconvenient to the control algorithm design Young [13] hasproposed variable structure controller for the first timewhich was applied to robot control Jafarov et al [14] haveproposed a new type of sliding PID controller Althoughthese methods can solve the control problem they could notconsider the robustness of the system119867-infinity control is a good robustness design methodwith clear design ideas good control effect and advantagesespecially on model perturbation of multiple input multipleoutput (MIMO) systems Finally hardware verification con-trol which results in the final loop (HIL) simulation is donein order to ensure quality control

2 Kinematic Model of UndergroundArticulated Vehicle

21 Steering Deviation Model Figure 1 shows the errorsbetween the real and reference path The circle centered by119862 is the reference path Ideally the vehicle should pass 1198751 1198752and 1198753 The variables are defined as follows [5 15]

(1) lateral displacement error the lateral displacementerror between the vehicle reference point and the cor-responding point119875 (the nearest point on the referencepath)

(2) orientation error the orientation error between thevelocity orientation of 119901 and the tangential orienta-tion of 119875

(3) steering angle relative rotation angle of undergroundmining articulated vehicle body before and after thehorizontal plane

These two variables can basically reflect the position andposture of articulated underground mining vehicles in thetunnel and building the trajectory of articulated vehiclesmodel can be achieved by the fuzzy hyperbolic method

Figure 2(a) is the sketch of an artificial tunnel for testing ofarticulated vehicle driver-less control system The truck usedin this test is shown in Figure 2(b)

As can be seen in Figures 3ndash5 at speed of 3m per secondby driver controlling the truck in the process of turning thelateral deviation is stable at 15m Heading angle deviationremains stable at around 0 rad while body roll is obviousThe steering angle remains stable at 02 rad with obviousadjustments As shown in Figures 6 and 7 at speed of 3m persecond rate of change of lateral deviation and the headingangle also is 0 but there are more noise and severe jitter

22 Kinematic Models Based on Fuzzy Hyperbolic ModelNear the origin fuzzy hyperbolic model is a global model[6] with a global approximation performance And it is notonly an essentially nonlinear model but also easy to expressthe dynamic characteristics of nonlinear systems Thereforeaccording to this model the controller can be designed sothat the whole system can achieve the optimal performancesCompared with other fuzzy models fuzzy hyperbolic modelis more suitable for the control object known as multivariablenonlinear finite object

Given a plant 119899 state variables number 119909 = (1199091(119905) 119909119899(119905))T and 119901 input variables numbers u = (u1(119905) u119901(119905))T and the fuzzy rules are used to describe this systemby the following conditions so this group of fuzzy rules istermed as the hyperbolic tangent type of fuzzy rules

Fuzzy rules for the following forms

IF 1199091 is 1198651199091 and 1199092 is 1198651199092 and 119909119899 is 119865119909119899 and u1 is 119865u1and u2 is 119865u2 and u119899 is 119865u119901THEN 119897 = plusmn1198881199091 plusmn 1198881199092 sdot sdot sdot plusmn 119888119909119899 plusmn 119888u1 plusmn 119888u2 sdot sdot sdot plusmn 119888u119901 (119897 =1 119899)

where 119865119909119894 (119894 = 1 119899) and 119865u119895 (119895 = 1 119901) are fuzzy sub-set including (119875) and (119873) corresponding to 119909119894 and u119895 and119888119909119894 (119894 = 1 119899) and 119888u119895 (119895 = 1 119901) are constant of 119865119909119894and 119865u119894 (119895 = 1 119901)

Let 120575119889 = 1199091 120575120579 = 1199092 and u = 120574There are certain mathematical relationships between

manned lateral deviation heading angle deviation the steer-ing angle and rate of change of the lateral deviation andheading angle deviation from the data observed from relevantpapers [10] And they can be roughly described as the follow-ing information through the professional experiences so as to

Journal of Control Science and Engineering 3

8255

R2573

5159

650

6

(a) (b)

Figure 2 (a) The plan view of the tunnel (b) The real truck

minus2

minus15

minus1

minus05

0

05

Man

ned

later

al d

evia

tion

(m)

20 40 60 80 100 1200Time (s)

Figure 3 Manned lateral deviation

20 40 60 80 100 1200Time (s)

minus04

minus03

minus02

minus01

0

01

02

03

Man

ned

head

ing

angl

e dev

iatio

n (r

ad)

Figure 4 Manned heading angle deviation

construct the corresponding fuzzy hyperbolic model of fuzzyrules

R1 IF 1199092 is 1198751199092 THEN 1 = 01R2 IF 1199092 is 1198731199092 THEN 1 = minus01R3 IF 1199091 is 1198751199091 IF 1199092 is 1198751199092 IF u is 119875u THEN 2 = 001R4 IF 1199091 is 1198731199091 IF 1199092 is 1198751199092 IF u is 119875u THEN 2 = 0R5 IF 1199091 is 1198731199091 IF 1199092 is 1198731199092 IF u is 119875u THEN 2 = 008

20 40 60 80 100 1200Time (s)

minus02

minus01

0

01

02

03

04

05

06

Man

ned

steer

ing

angl

e (ra

d)

Figure 5 Manned steering angle

Man

ned

rate

of c

hang

e

20 40 60 80 100 1200Time (s)

minus006minus005minus004minus003minus002minus001

0001002003

of th

e lat

eral

pos

ition

dev

iatio

n (m

middotsminus1)

Figure 6 Manned rate of change of the lateral position deviation

R6 IF 1199091 is 1198731199091 IF 1199092 is 1198731199092 IF u is 119873u THEN 2 =minus001R7 IF 1199091 is 1198751199091 IF 1199092 is 1198731199092 IF u is 119873u THEN 2 = 0R8 IF 1199091 is 1198751199091 IF 1199092 is 1198751199092 IF u is 119873u THEN 2 = minus008R9 IF 1199091 is 1198751199091 IF 1199092 is 1198731199092 IF u is 119875u THEN 2 = 01R10 IF 1199091 is 1198731199091 IF 1199092 is 1198751199092 IF u is 119873u THEN 2 = minus01

4 Journal of Control Science and Engineering

Man

ned

rate

of c

hang

etimes10minus3

20 40 60 80 100 1200Time (s)

minus8minus6minus4minus2

02468

of th

e hea

ding

dev

iatio

n (m

middotsminus1)

Figure 7 Manned rate of change of the heading deviation

Given a set of fuzzy rules of hyperbolic tangent define119875119911 and 119873119911 (where 119911 are arbitrary state variables or inputvariables) as membership function (where k119911 gt 0)

119875119911 (119909) = 119890minus(12)(119909minusk119911)119873119911 (119909) = 119890minus(12)(119909+k119911) (1)

Let k1199091 = k1199092 = ku = 1Membership function

u1198751199091 (119909) = 119890minus(12)(119909minus1)2 u1198731199091 (119909) = 119890minus(12)(119909+1)2 u1198751199092 (119909) = 119890minus(12)(119909minus1)2 u1198731199092 (119909) = 119890minus(12)(119909+1)2 u119875u (119909) = 119890minus(12)(119909minus1)2 u119875u (119909) = 119890minus(12)(119909+1)2

(2)

Then

= A tanh (k119909119909) + B tanh (kuu) (3)

where

A = [ 0 01001 minus01]

B = [ 0008]

k = [[[1

11]]]

tanh (119909) = 119890119909 minus 119890minus119909119890119909 + 119890minus119909

k119909 = diag (k1199091 k119909119899) ku = diag (k1199091 k119909119901)

(4)

Bku are linearized to Bu

= A tanh (k119909119909) + Bu[12] = [ 0 01

001 minus01] [tanh (1199091)tanh (1199092)] + [ 0

008] 120574 (5)

When the absolute value of 119909 is smaller tanh(119909) asymp 119909fuzzy hyperbolic model can be written as = A tanh(k119909119909) +Bu that the system is a linear model when it is close to theequilibrium point

3 Parameter Identification Based onImproved Adaptive BP Neural Network

Recognition technology evaluation includes two indicatorsOne is the identification accuracy and the other is speedidentification Compared to other models in terms of fuzzytopology of the neural network can be used to optimize theparameters for FHM [16] The 119860 weights of mediation rate ofadaptive BP neural network are deduced by Cauchy robusterror to eliminate the effect of outliers in the data and to fitthe original data better

Neural network topology is shown in Figure 8 of FHMControlled object state variables are 119909 = (1199091(119905) 119909119899(119905))Tand input variables u = (u1(119905) u119898(119905))T as the input ofthe neural network topology and model output is the rateof change of the state variables k119894 (119894 = 1 119899) 119892119895 (119895 =1 119898) 119888119894119895 (119894 119895 = 1 119899) and 119889119894119895 (119894 = 1 119899 119895 =1 119898) as the connection weights which need to be trained

A and B are constant matrix whose elements are theweights comprised of 119888119894119895 and 119889119894119895 If the hidden layer function1198911(119909) = tanh(119909) and output layer activation function 1198912(119909) =119909 then the following models can be obtained

= A tanh (k119909119909) + B tanh (kuu) (6)

Obviously when making u linearize variables can beobtained

= A tanh (k119909119909) + Bu (7)

Thus FHM may be built by the neural network modelbecause it can train FHM model parameters [6] Since thereare many jitters in the data of rate of change of horizontal andheading angle the samples show a large number of outliersSo from the statistical view of robust the traditional MSEaggravate the ldquooutliersrdquo of samples Therefore Cauchy robusterror estimator can be used

Let error of outputs be

119864 = 12119902sum

k=1ln [1 + (1199101015840k minus 119910k)2] (8)

Journal of Control Science and Engineering 5

x1

xn

u1

um

k1

kn

g1

gm

f1(middot)c11

c1n

cn1

cnm

d11

d1m

dnm

dn1

f2(middot)x1

xn

Figure 8 Fuzzy hyperbolic tangentmodel neural network topology

where 1199101015840k is ideal output for the network and 119910k is actualoutput

Cauchy error estimator obviously does not depend on theinitial weights and thresholds transition and it can effectivelyeliminate the negative impact of ldquooutliersrdquo while retainingthe main characteristics of the value of the output error andspeeding up the convergence rate

By the steepest descent method it can obtain weightsiterative equations of each layer of neurons

119882(119905 + 1) = 119882 (119905) minus uΔ119864 (9)

where Δ119864 = 120597119864120597120596119882 = 120596119894119895So get a rate adjustment based on the following model

parameters above BP algorithm

119888119894119895 (119905 + 1) = 119888119894119895 (119905) + 120592 sdot 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2 sdot tanh (k119894119909119894)

k119895 (119905 + 1) = k119895 (119905) + 120592 sdot sumk119888119894119895 sdot ( 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2)

sdot 119909119894tanh (sum119894 120596119895k119909119894)

(10)

In this paper learning rate 120592 is considered as the adaptivescheme in which its main idea is that the initial value of 120592being set higher generally about 07ndash09 with the increaseor decrement of the number of learning may change in alaw When 120592 decreases to a certain extent if 120592 has not stillbecome convergence or error has still no improvement 120592 isset again at about 05ndash07 120592 enters the learning process againuntil the end of the run The learning rate is associated withthe error function when the error is reduced increase thelearning rate when the error increases reduce the learningrate

120592(k) =

112120592(kminus1) 119864(kminus1) gt 119864(k)08120592(kminus1) 119864(kminus1) lt 119864(k)120592(kminus1) 119864(kminus1) = 119864(k)

(11)

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

20 40 60 80 100 1200Time (s)

minus006minus005minus004minus003minus002minus001

0001002003

of th

e lat

eral

dev

iatio

n (m

s)

Figure 9 Fitting of change rate of the lateral deviation

Identify the model parameters after obtaining the data after6000 steps of learning

A = [00012 0099600095 minus00999]

B = [minus000700803]

k = [[[1000

100909998

]]]

(12)

Then

[12] = [00012 0099600095 minus00999] [tanh (1199091)

tanh (1199092)]

+ [minus000700803] 120574(13)

The fitting results of Figures 9 and 10 substantially elimi-nate the effect of outliers with relative error of control in lessthan 1 and less than 10The trend is essentially coincidentwith the sample data

The existing experimental data can be applied to theabove-described method for fuzzy hyperbolic model param-eter optimization in order to be close to the actual model Itprovides the basis to the next controller design

4 Controller Design

Based on this model we can design conventional linearcontrollers or other nonlinear controllers In this paper thecontroller is designed with hyperbolic tangent function ofthe state variables so we can use language to describe theinformation of the controller Thus this given controller is afuzzy one [16]

6 Journal of Control Science and Engineering

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

times10minus3

20 40 60 80 100 1200Time (s)

minus8

minus6

minus4

minus2

0

2

4

6

8of

the l

ater

al d

evia

tion

(rad

middotsminus1)

Figure 10 Fitting of change rate of the heading deviation

The system can be stabilized by nonquadratic perform-ance index function [16] System performance can get to aminimum with u through the givenQ and R

= A119891 (119909) + Bu (14)

Definition 1 (see [17]) Let 119878119888 satisfy the following conditionsall sets of 119891(sdot) include R rarr R

(1) 119891 is continuous(2) 119891(0) = 0 for other 119909 isin R 119891(119909)119909 gt 0(3) when |119909| rarr infin int119909

0119891(119910)119889119910 rarr infin

Theorem 2 (see [17]) For nonlinear system

= A119891 (119909) + Bu +D119908 (15)

where 119891(119909) = (1198911(119909) 119891119899(119909))T and 119891(119894) isin 119878119888 and thenk gt 0 tanh(k119909) isin 119878119888 (119894 = 1 2 3 119899) and a performanceindex function can be proposed as

119869 (1199090 1199050 u) = intinfin1199050

119891T (119909 (119905))Q119891 (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(16)

where Q R and S are given as the definite positive symmetricconstant matrix

Design Optimal Control Vector [18] Considerlowast

u(t) = lowast

u(t)(119905119909(120591)119905120591=0) and there is L(1199090) with boundary Then

sup 119869 ( lowast

u (t) 119908) le L (1199090) (17)

If there is a diagonal positive definite matrix Pmin satisfyingRiccati equation

PA + ATP minus P (BRminus1BTP minusDSminus1DT)P +Q = 0 (18)

then [119860 119861QR] is optimized diagonal matrix

Given nonlinear system and nonlinear quadratic perfor-mance index function [119860 119861QR] is an optimized diagonalmatrix The optimal control vector is

lowast

u (t) = minusRminus1BTPmin119891 (119909 (119905)) (19)

where Pmin = diag(1199011 119901119899) is optimized diagonal matrixLet L(1199090) = 2sum119899119894=1 119901119894 int119909119894(1199050)0 119891119894(120591)119889120591

Thus tanh(k119909) isin 119878119888 and for fuzzy hyperbolic tangent thequadratic performance index function is

119869 (1199090 1199050 u) = intinfin1199050

tanhT (119909 (119905))Q tanh (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(20)

Let Q = [ 10 00 15 ] R = 08 and S = 800 with Riccatiequation then

P = [10972 27592759 1678] (21)

According to the closed-loop system equation of state

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (22)

Based on Lyapunov theorem and Riccati equation for thisequation

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (23)

It is easy to prove

Pmin (A minus BRminus1BTPmin) + (A minus BRminus1BTPmin)Pmin

= minus (Q + Pmin (BRminus1BTP minusDSminus1DT)Pmin) (24)

And (A minusBRminus1BTP) is diagonally stable thus the closed-loop system is asymptotically stable

Then

K = [11627 6135] (25)

5 HIL Simulation

Thedifference between simulation and hardware-in-the-loopsimulation is that there is a real-time simulation of thecircuitrsquos physical hardware Hardware-in-the-loop simulationis intended to provide real-time incentives as the true signalto the controller so that the controller is connected to its realaccused equipment and test its performance Figure 11 is a pic-torial diagram of the hardware simulation platform in whichc-RIO controller is recognized as the path tracking anglecontroller program compiled by the Simulink PXI platformsimulation runs the Adams which can build undergroundmining articulated vehicle model as the object of emulationPC can run the LabVIEWgraphical user interface for PXI andc-RIO real-time and display the data of simulation compiledfrom the process Speed can be achieved by PID control at

Journal of Control Science and Engineering 7

PCc-RIO

PXI

LabVIEW

Figure 11 HIL simulation platform

Travel path comparison

Practical pathTarget path

100 20 30minus20 minus10minus30x (m)

minus30

minus20

minus10

0

10

20

30

y (m

)

Figure 12 Simulation of driving route comparedwith the ideal path

V = 3ms So as to test this robust of control there is onepit per 20m on the road The reference trajectory is set asa round whose center is (0 0) and R is 25m and an initialparameter setting is simulation time 200 s and the startingpoint coordinates (minus3 minus25)

Derived from the simulation results Figure 12 is thecomparison chart of the actual trajectory and the referencetrajectory As can be seen from the simulation curve thearticulated vehicle traveling trace and the reference circularpath are consistent and the trajectory is relatively smoothThe speed which is controlled by PID fromFigure 13 becomesfinally steady at 30 seconds The lateral displacement erroris gradually flat from the beginning of violent shaking inFigure 14 The overshoot amount is 428 and it is at mostabout 0024mat 18 secondswith respect to the tread based on2280m and the error is 17 and then close to the origin stepby stepThe orientation error is gradually flat from the begin-ning of violent shaking in Figure 15The overshoot amount is49 and it is at the top at about 008 rad at 18 seconds andthen close to the origin step by step The articulated angleis gradually flat from the beginning of violent shaking in

0

05

1

15

2

25

3

35

Spee

d (m

middotsminus1)

20 40 60 80 100 120 140 160 180 2000Time (s)

Figure 13 Longitudinal velocity of midpoint of front axle

20 40 60 80 100 120 140 160 180 2000Time (s)

minus003

minus0025

minus002

minus0015

minus001

minus0005

0

0005

Late

ral d

ispla

cem

ent e

orrr

(m)

Figure 14 Lateral displacement error

Figure 16 The overshoot amount is 35 and it is eventuallystabilized at 021 rad at 22 seconds Compared with the artic-ulated vehicle steering based on angle 45∘ the error is 12And the peaks are caused by pits but then the system imme-diately recovers to the right path It has better robustness thanliterature [12] in which there are a lot of jitters to adapt tothe path as a similar situation The results of literature [10]are as follows lateral deviation was 1m and heading angledeviation is 01 rad Literature [19] also analyzed themodels ofarticulated vehicle to control it with synovial control methodwhose results of the simulation are as follows the lateraldeviation is 01m and heading angle deviation was 017 radSo there exists a large gap when compared with the results ofthis paper

6 Conclusions

This article provides a method used in unmanned systemsbased on theway of fuzzy hyperbolic pole control act for artic-ulated vehicle trajectory tracking accurately Conclusions areas follows

(1) Fuzzy hyperbolic tangent model can well describethe quantitative relation among lateral displacementerror the orientation error and the articulated angle

(2) Based on the method of Cauchy robust error estima-tor the weights of BP neural network can effectively

8 Journal of Control Science and Engineering

20 40 60 80 100 120 140 160 180 2000Time (s)

minus002

0

002

004

006

008

01

Orie

ntat

ion

erro

r (ra

d)

Figure 15 Unmanned orientation error

20 40 60 80 100 120 140 160 180 2000Time (s)

0

005

01

015

02

025

Art

icul

ated

angl

e (ra

d)

Figure 16 Articulated angle

be reduced by the influence of singular error of neuralnetwork learning and fitting error and relative errorbelow expectations can be decreased so that it canachieve the system of identification

(3) By the 119867-infinity controller design controller has agood control performance and it makes the systemkeep a better performance about robustness in orderto achieve the purpose of the comprehensively opti-mal control error

(4) This method can be used in articulated vehicle pathtracking effectively based on the119867-infinity controllerSimulation in hardware-in-the-loop shows that over-shoot and the response time are less than expectationsand are eventually to stabilize The controller has metthe requirement of the real-time control performance

Competing Interests

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhaoand Jian Li declare that there is no conflict of interestsregarding the publication of this manuscript

Acknowledgments

This work was financially supported by the National HighTechnology Research and Development Program (863 Pro-gram) of China under Award 2011AA060404 IntelligentUnderground Mining Truck and Fundamental ResearchFunds for the Central Universities FRF-TP-16-004A1research on path-planning and path-following algorithmfor underground mining vehicles based on reinforcementlearning

References

[1] X-J Zeng and M G Singh ldquoApproximation theory of fuzzysystemsmdashSISO caserdquo IEEE Transactions on Fuzzy Systems vol2 no 2 pp 162ndash176 1994

[2] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceeding of the IEEE International Conference on Fuzzy Sys-tems pp 1163ndash1170 1992

[3] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[4] M Sugeno and T Yasukawa ldquoFuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] S Scheding G Dissanayake E Nebot and H Durrant-WhyteldquoSlip modelling and aided inertial navigation of an LHDrdquo inProceedings of the IEEE International Conference on Roboticsand Automation pp 1904ndash1909 Institute of Electrical andElectronics Engineers Albuquerque NM USA April 1997

[6] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[7] TNayl GNikolakopoulos andTGustfsson ldquoSwitchingmodelpredictive control for an articulated vehicle under varying slipanglerdquo in Proceedings of the 20th Mediterranean Conference onControl amp Automation (MED rsquo12) pp 890ndash895 IEEE ComputerSociety Barcelona Spain July 2012

[8] J H Lee andW S Yoo ldquoPredictive control of a vehicle trajectoryusing a coupled vector with vehicle velocity and sideslip anglerdquoInternational Journal of Automotive Technology vol 10 no 2 pp211ndash217 2009

[9] P Ridley and P Corke ldquoLoad haul dump vehicle kinematicsand controlrdquo Journal of Dynamic Systems Measurement andControl vol 125 no 1 pp 54ndash59 2003

[10] Z Xuan Y Jue and Z Wenming ldquoFeedback linearizationcontrol for path tracking of articulated dump truckrdquo TELKOM-NIKA Telecommunication Computing Electronics and Controlvol 9 no 13 pp 922ndash929 2015

[11] D Piyabongkarn R Rajamani J A Grogg and J Y LewldquoDevelopment and experimental evaluation of a slip angleestimator for vehicle stability controlrdquo IEEE Transactions onControl Systems Technology vol 17 no 1 pp 78ndash88 2009

[12] Y Qi Trajectory Control for the Underground Articulated DumpTruck Based on Model Predictive Control University of Scienceand Technology Beijing Beijing China 2013

[13] K-K D Young ldquoController design for a manipulator usingtheory of variable structure systemsrdquo IEEE Transactions on Sys-tems Man and Cybernetics vol 8 no 2 pp 101ndash109 1978

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 3

8255

R2573

5159

650

6

(a) (b)

Figure 2 (a) The plan view of the tunnel (b) The real truck

minus2

minus15

minus1

minus05

0

05

Man

ned

later

al d

evia

tion

(m)

20 40 60 80 100 1200Time (s)

Figure 3 Manned lateral deviation

20 40 60 80 100 1200Time (s)

minus04

minus03

minus02

minus01

0

01

02

03

Man

ned

head

ing

angl

e dev

iatio

n (r

ad)

Figure 4 Manned heading angle deviation

construct the corresponding fuzzy hyperbolic model of fuzzyrules

R1 IF 1199092 is 1198751199092 THEN 1 = 01R2 IF 1199092 is 1198731199092 THEN 1 = minus01R3 IF 1199091 is 1198751199091 IF 1199092 is 1198751199092 IF u is 119875u THEN 2 = 001R4 IF 1199091 is 1198731199091 IF 1199092 is 1198751199092 IF u is 119875u THEN 2 = 0R5 IF 1199091 is 1198731199091 IF 1199092 is 1198731199092 IF u is 119875u THEN 2 = 008

20 40 60 80 100 1200Time (s)

minus02

minus01

0

01

02

03

04

05

06

Man

ned

steer

ing

angl

e (ra

d)

Figure 5 Manned steering angle

Man

ned

rate

of c

hang

e

20 40 60 80 100 1200Time (s)

minus006minus005minus004minus003minus002minus001

0001002003

of th

e lat

eral

pos

ition

dev

iatio

n (m

middotsminus1)

Figure 6 Manned rate of change of the lateral position deviation

R6 IF 1199091 is 1198731199091 IF 1199092 is 1198731199092 IF u is 119873u THEN 2 =minus001R7 IF 1199091 is 1198751199091 IF 1199092 is 1198731199092 IF u is 119873u THEN 2 = 0R8 IF 1199091 is 1198751199091 IF 1199092 is 1198751199092 IF u is 119873u THEN 2 = minus008R9 IF 1199091 is 1198751199091 IF 1199092 is 1198731199092 IF u is 119875u THEN 2 = 01R10 IF 1199091 is 1198731199091 IF 1199092 is 1198751199092 IF u is 119873u THEN 2 = minus01

4 Journal of Control Science and Engineering

Man

ned

rate

of c

hang

etimes10minus3

20 40 60 80 100 1200Time (s)

minus8minus6minus4minus2

02468

of th

e hea

ding

dev

iatio

n (m

middotsminus1)

Figure 7 Manned rate of change of the heading deviation

Given a set of fuzzy rules of hyperbolic tangent define119875119911 and 119873119911 (where 119911 are arbitrary state variables or inputvariables) as membership function (where k119911 gt 0)

119875119911 (119909) = 119890minus(12)(119909minusk119911)119873119911 (119909) = 119890minus(12)(119909+k119911) (1)

Let k1199091 = k1199092 = ku = 1Membership function

u1198751199091 (119909) = 119890minus(12)(119909minus1)2 u1198731199091 (119909) = 119890minus(12)(119909+1)2 u1198751199092 (119909) = 119890minus(12)(119909minus1)2 u1198731199092 (119909) = 119890minus(12)(119909+1)2 u119875u (119909) = 119890minus(12)(119909minus1)2 u119875u (119909) = 119890minus(12)(119909+1)2

(2)

Then

= A tanh (k119909119909) + B tanh (kuu) (3)

where

A = [ 0 01001 minus01]

B = [ 0008]

k = [[[1

11]]]

tanh (119909) = 119890119909 minus 119890minus119909119890119909 + 119890minus119909

k119909 = diag (k1199091 k119909119899) ku = diag (k1199091 k119909119901)

(4)

Bku are linearized to Bu

= A tanh (k119909119909) + Bu[12] = [ 0 01

001 minus01] [tanh (1199091)tanh (1199092)] + [ 0

008] 120574 (5)

When the absolute value of 119909 is smaller tanh(119909) asymp 119909fuzzy hyperbolic model can be written as = A tanh(k119909119909) +Bu that the system is a linear model when it is close to theequilibrium point

3 Parameter Identification Based onImproved Adaptive BP Neural Network

Recognition technology evaluation includes two indicatorsOne is the identification accuracy and the other is speedidentification Compared to other models in terms of fuzzytopology of the neural network can be used to optimize theparameters for FHM [16] The 119860 weights of mediation rate ofadaptive BP neural network are deduced by Cauchy robusterror to eliminate the effect of outliers in the data and to fitthe original data better

Neural network topology is shown in Figure 8 of FHMControlled object state variables are 119909 = (1199091(119905) 119909119899(119905))Tand input variables u = (u1(119905) u119898(119905))T as the input ofthe neural network topology and model output is the rateof change of the state variables k119894 (119894 = 1 119899) 119892119895 (119895 =1 119898) 119888119894119895 (119894 119895 = 1 119899) and 119889119894119895 (119894 = 1 119899 119895 =1 119898) as the connection weights which need to be trained

A and B are constant matrix whose elements are theweights comprised of 119888119894119895 and 119889119894119895 If the hidden layer function1198911(119909) = tanh(119909) and output layer activation function 1198912(119909) =119909 then the following models can be obtained

= A tanh (k119909119909) + B tanh (kuu) (6)

Obviously when making u linearize variables can beobtained

= A tanh (k119909119909) + Bu (7)

Thus FHM may be built by the neural network modelbecause it can train FHM model parameters [6] Since thereare many jitters in the data of rate of change of horizontal andheading angle the samples show a large number of outliersSo from the statistical view of robust the traditional MSEaggravate the ldquooutliersrdquo of samples Therefore Cauchy robusterror estimator can be used

Let error of outputs be

119864 = 12119902sum

k=1ln [1 + (1199101015840k minus 119910k)2] (8)

Journal of Control Science and Engineering 5

x1

xn

u1

um

k1

kn

g1

gm

f1(middot)c11

c1n

cn1

cnm

d11

d1m

dnm

dn1

f2(middot)x1

xn

Figure 8 Fuzzy hyperbolic tangentmodel neural network topology

where 1199101015840k is ideal output for the network and 119910k is actualoutput

Cauchy error estimator obviously does not depend on theinitial weights and thresholds transition and it can effectivelyeliminate the negative impact of ldquooutliersrdquo while retainingthe main characteristics of the value of the output error andspeeding up the convergence rate

By the steepest descent method it can obtain weightsiterative equations of each layer of neurons

119882(119905 + 1) = 119882 (119905) minus uΔ119864 (9)

where Δ119864 = 120597119864120597120596119882 = 120596119894119895So get a rate adjustment based on the following model

parameters above BP algorithm

119888119894119895 (119905 + 1) = 119888119894119895 (119905) + 120592 sdot 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2 sdot tanh (k119894119909119894)

k119895 (119905 + 1) = k119895 (119905) + 120592 sdot sumk119888119894119895 sdot ( 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2)

sdot 119909119894tanh (sum119894 120596119895k119909119894)

(10)

In this paper learning rate 120592 is considered as the adaptivescheme in which its main idea is that the initial value of 120592being set higher generally about 07ndash09 with the increaseor decrement of the number of learning may change in alaw When 120592 decreases to a certain extent if 120592 has not stillbecome convergence or error has still no improvement 120592 isset again at about 05ndash07 120592 enters the learning process againuntil the end of the run The learning rate is associated withthe error function when the error is reduced increase thelearning rate when the error increases reduce the learningrate

120592(k) =

112120592(kminus1) 119864(kminus1) gt 119864(k)08120592(kminus1) 119864(kminus1) lt 119864(k)120592(kminus1) 119864(kminus1) = 119864(k)

(11)

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

20 40 60 80 100 1200Time (s)

minus006minus005minus004minus003minus002minus001

0001002003

of th

e lat

eral

dev

iatio

n (m

s)

Figure 9 Fitting of change rate of the lateral deviation

Identify the model parameters after obtaining the data after6000 steps of learning

A = [00012 0099600095 minus00999]

B = [minus000700803]

k = [[[1000

100909998

]]]

(12)

Then

[12] = [00012 0099600095 minus00999] [tanh (1199091)

tanh (1199092)]

+ [minus000700803] 120574(13)

The fitting results of Figures 9 and 10 substantially elimi-nate the effect of outliers with relative error of control in lessthan 1 and less than 10The trend is essentially coincidentwith the sample data

The existing experimental data can be applied to theabove-described method for fuzzy hyperbolic model param-eter optimization in order to be close to the actual model Itprovides the basis to the next controller design

4 Controller Design

Based on this model we can design conventional linearcontrollers or other nonlinear controllers In this paper thecontroller is designed with hyperbolic tangent function ofthe state variables so we can use language to describe theinformation of the controller Thus this given controller is afuzzy one [16]

6 Journal of Control Science and Engineering

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

times10minus3

20 40 60 80 100 1200Time (s)

minus8

minus6

minus4

minus2

0

2

4

6

8of

the l

ater

al d

evia

tion

(rad

middotsminus1)

Figure 10 Fitting of change rate of the heading deviation

The system can be stabilized by nonquadratic perform-ance index function [16] System performance can get to aminimum with u through the givenQ and R

= A119891 (119909) + Bu (14)

Definition 1 (see [17]) Let 119878119888 satisfy the following conditionsall sets of 119891(sdot) include R rarr R

(1) 119891 is continuous(2) 119891(0) = 0 for other 119909 isin R 119891(119909)119909 gt 0(3) when |119909| rarr infin int119909

0119891(119910)119889119910 rarr infin

Theorem 2 (see [17]) For nonlinear system

= A119891 (119909) + Bu +D119908 (15)

where 119891(119909) = (1198911(119909) 119891119899(119909))T and 119891(119894) isin 119878119888 and thenk gt 0 tanh(k119909) isin 119878119888 (119894 = 1 2 3 119899) and a performanceindex function can be proposed as

119869 (1199090 1199050 u) = intinfin1199050

119891T (119909 (119905))Q119891 (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(16)

where Q R and S are given as the definite positive symmetricconstant matrix

Design Optimal Control Vector [18] Considerlowast

u(t) = lowast

u(t)(119905119909(120591)119905120591=0) and there is L(1199090) with boundary Then

sup 119869 ( lowast

u (t) 119908) le L (1199090) (17)

If there is a diagonal positive definite matrix Pmin satisfyingRiccati equation

PA + ATP minus P (BRminus1BTP minusDSminus1DT)P +Q = 0 (18)

then [119860 119861QR] is optimized diagonal matrix

Given nonlinear system and nonlinear quadratic perfor-mance index function [119860 119861QR] is an optimized diagonalmatrix The optimal control vector is

lowast

u (t) = minusRminus1BTPmin119891 (119909 (119905)) (19)

where Pmin = diag(1199011 119901119899) is optimized diagonal matrixLet L(1199090) = 2sum119899119894=1 119901119894 int119909119894(1199050)0 119891119894(120591)119889120591

Thus tanh(k119909) isin 119878119888 and for fuzzy hyperbolic tangent thequadratic performance index function is

119869 (1199090 1199050 u) = intinfin1199050

tanhT (119909 (119905))Q tanh (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(20)

Let Q = [ 10 00 15 ] R = 08 and S = 800 with Riccatiequation then

P = [10972 27592759 1678] (21)

According to the closed-loop system equation of state

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (22)

Based on Lyapunov theorem and Riccati equation for thisequation

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (23)

It is easy to prove

Pmin (A minus BRminus1BTPmin) + (A minus BRminus1BTPmin)Pmin

= minus (Q + Pmin (BRminus1BTP minusDSminus1DT)Pmin) (24)

And (A minusBRminus1BTP) is diagonally stable thus the closed-loop system is asymptotically stable

Then

K = [11627 6135] (25)

5 HIL Simulation

Thedifference between simulation and hardware-in-the-loopsimulation is that there is a real-time simulation of thecircuitrsquos physical hardware Hardware-in-the-loop simulationis intended to provide real-time incentives as the true signalto the controller so that the controller is connected to its realaccused equipment and test its performance Figure 11 is a pic-torial diagram of the hardware simulation platform in whichc-RIO controller is recognized as the path tracking anglecontroller program compiled by the Simulink PXI platformsimulation runs the Adams which can build undergroundmining articulated vehicle model as the object of emulationPC can run the LabVIEWgraphical user interface for PXI andc-RIO real-time and display the data of simulation compiledfrom the process Speed can be achieved by PID control at

Journal of Control Science and Engineering 7

PCc-RIO

PXI

LabVIEW

Figure 11 HIL simulation platform

Travel path comparison

Practical pathTarget path

100 20 30minus20 minus10minus30x (m)

minus30

minus20

minus10

0

10

20

30

y (m

)

Figure 12 Simulation of driving route comparedwith the ideal path

V = 3ms So as to test this robust of control there is onepit per 20m on the road The reference trajectory is set asa round whose center is (0 0) and R is 25m and an initialparameter setting is simulation time 200 s and the startingpoint coordinates (minus3 minus25)

Derived from the simulation results Figure 12 is thecomparison chart of the actual trajectory and the referencetrajectory As can be seen from the simulation curve thearticulated vehicle traveling trace and the reference circularpath are consistent and the trajectory is relatively smoothThe speed which is controlled by PID fromFigure 13 becomesfinally steady at 30 seconds The lateral displacement erroris gradually flat from the beginning of violent shaking inFigure 14 The overshoot amount is 428 and it is at mostabout 0024mat 18 secondswith respect to the tread based on2280m and the error is 17 and then close to the origin stepby stepThe orientation error is gradually flat from the begin-ning of violent shaking in Figure 15The overshoot amount is49 and it is at the top at about 008 rad at 18 seconds andthen close to the origin step by step The articulated angleis gradually flat from the beginning of violent shaking in

0

05

1

15

2

25

3

35

Spee

d (m

middotsminus1)

20 40 60 80 100 120 140 160 180 2000Time (s)

Figure 13 Longitudinal velocity of midpoint of front axle

20 40 60 80 100 120 140 160 180 2000Time (s)

minus003

minus0025

minus002

minus0015

minus001

minus0005

0

0005

Late

ral d

ispla

cem

ent e

orrr

(m)

Figure 14 Lateral displacement error

Figure 16 The overshoot amount is 35 and it is eventuallystabilized at 021 rad at 22 seconds Compared with the artic-ulated vehicle steering based on angle 45∘ the error is 12And the peaks are caused by pits but then the system imme-diately recovers to the right path It has better robustness thanliterature [12] in which there are a lot of jitters to adapt tothe path as a similar situation The results of literature [10]are as follows lateral deviation was 1m and heading angledeviation is 01 rad Literature [19] also analyzed themodels ofarticulated vehicle to control it with synovial control methodwhose results of the simulation are as follows the lateraldeviation is 01m and heading angle deviation was 017 radSo there exists a large gap when compared with the results ofthis paper

6 Conclusions

This article provides a method used in unmanned systemsbased on theway of fuzzy hyperbolic pole control act for artic-ulated vehicle trajectory tracking accurately Conclusions areas follows

(1) Fuzzy hyperbolic tangent model can well describethe quantitative relation among lateral displacementerror the orientation error and the articulated angle

(2) Based on the method of Cauchy robust error estima-tor the weights of BP neural network can effectively

8 Journal of Control Science and Engineering

20 40 60 80 100 120 140 160 180 2000Time (s)

minus002

0

002

004

006

008

01

Orie

ntat

ion

erro

r (ra

d)

Figure 15 Unmanned orientation error

20 40 60 80 100 120 140 160 180 2000Time (s)

0

005

01

015

02

025

Art

icul

ated

angl

e (ra

d)

Figure 16 Articulated angle

be reduced by the influence of singular error of neuralnetwork learning and fitting error and relative errorbelow expectations can be decreased so that it canachieve the system of identification

(3) By the 119867-infinity controller design controller has agood control performance and it makes the systemkeep a better performance about robustness in orderto achieve the purpose of the comprehensively opti-mal control error

(4) This method can be used in articulated vehicle pathtracking effectively based on the119867-infinity controllerSimulation in hardware-in-the-loop shows that over-shoot and the response time are less than expectationsand are eventually to stabilize The controller has metthe requirement of the real-time control performance

Competing Interests

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhaoand Jian Li declare that there is no conflict of interestsregarding the publication of this manuscript

Acknowledgments

This work was financially supported by the National HighTechnology Research and Development Program (863 Pro-gram) of China under Award 2011AA060404 IntelligentUnderground Mining Truck and Fundamental ResearchFunds for the Central Universities FRF-TP-16-004A1research on path-planning and path-following algorithmfor underground mining vehicles based on reinforcementlearning

References

[1] X-J Zeng and M G Singh ldquoApproximation theory of fuzzysystemsmdashSISO caserdquo IEEE Transactions on Fuzzy Systems vol2 no 2 pp 162ndash176 1994

[2] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceeding of the IEEE International Conference on Fuzzy Sys-tems pp 1163ndash1170 1992

[3] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[4] M Sugeno and T Yasukawa ldquoFuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] S Scheding G Dissanayake E Nebot and H Durrant-WhyteldquoSlip modelling and aided inertial navigation of an LHDrdquo inProceedings of the IEEE International Conference on Roboticsand Automation pp 1904ndash1909 Institute of Electrical andElectronics Engineers Albuquerque NM USA April 1997

[6] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[7] TNayl GNikolakopoulos andTGustfsson ldquoSwitchingmodelpredictive control for an articulated vehicle under varying slipanglerdquo in Proceedings of the 20th Mediterranean Conference onControl amp Automation (MED rsquo12) pp 890ndash895 IEEE ComputerSociety Barcelona Spain July 2012

[8] J H Lee andW S Yoo ldquoPredictive control of a vehicle trajectoryusing a coupled vector with vehicle velocity and sideslip anglerdquoInternational Journal of Automotive Technology vol 10 no 2 pp211ndash217 2009

[9] P Ridley and P Corke ldquoLoad haul dump vehicle kinematicsand controlrdquo Journal of Dynamic Systems Measurement andControl vol 125 no 1 pp 54ndash59 2003

[10] Z Xuan Y Jue and Z Wenming ldquoFeedback linearizationcontrol for path tracking of articulated dump truckrdquo TELKOM-NIKA Telecommunication Computing Electronics and Controlvol 9 no 13 pp 922ndash929 2015

[11] D Piyabongkarn R Rajamani J A Grogg and J Y LewldquoDevelopment and experimental evaluation of a slip angleestimator for vehicle stability controlrdquo IEEE Transactions onControl Systems Technology vol 17 no 1 pp 78ndash88 2009

[12] Y Qi Trajectory Control for the Underground Articulated DumpTruck Based on Model Predictive Control University of Scienceand Technology Beijing Beijing China 2013

[13] K-K D Young ldquoController design for a manipulator usingtheory of variable structure systemsrdquo IEEE Transactions on Sys-tems Man and Cybernetics vol 8 no 2 pp 101ndash109 1978

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Journal of Control Science and Engineering

Man

ned

rate

of c

hang

etimes10minus3

20 40 60 80 100 1200Time (s)

minus8minus6minus4minus2

02468

of th

e hea

ding

dev

iatio

n (m

middotsminus1)

Figure 7 Manned rate of change of the heading deviation

Given a set of fuzzy rules of hyperbolic tangent define119875119911 and 119873119911 (where 119911 are arbitrary state variables or inputvariables) as membership function (where k119911 gt 0)

119875119911 (119909) = 119890minus(12)(119909minusk119911)119873119911 (119909) = 119890minus(12)(119909+k119911) (1)

Let k1199091 = k1199092 = ku = 1Membership function

u1198751199091 (119909) = 119890minus(12)(119909minus1)2 u1198731199091 (119909) = 119890minus(12)(119909+1)2 u1198751199092 (119909) = 119890minus(12)(119909minus1)2 u1198731199092 (119909) = 119890minus(12)(119909+1)2 u119875u (119909) = 119890minus(12)(119909minus1)2 u119875u (119909) = 119890minus(12)(119909+1)2

(2)

Then

= A tanh (k119909119909) + B tanh (kuu) (3)

where

A = [ 0 01001 minus01]

B = [ 0008]

k = [[[1

11]]]

tanh (119909) = 119890119909 minus 119890minus119909119890119909 + 119890minus119909

k119909 = diag (k1199091 k119909119899) ku = diag (k1199091 k119909119901)

(4)

Bku are linearized to Bu

= A tanh (k119909119909) + Bu[12] = [ 0 01

001 minus01] [tanh (1199091)tanh (1199092)] + [ 0

008] 120574 (5)

When the absolute value of 119909 is smaller tanh(119909) asymp 119909fuzzy hyperbolic model can be written as = A tanh(k119909119909) +Bu that the system is a linear model when it is close to theequilibrium point

3 Parameter Identification Based onImproved Adaptive BP Neural Network

Recognition technology evaluation includes two indicatorsOne is the identification accuracy and the other is speedidentification Compared to other models in terms of fuzzytopology of the neural network can be used to optimize theparameters for FHM [16] The 119860 weights of mediation rate ofadaptive BP neural network are deduced by Cauchy robusterror to eliminate the effect of outliers in the data and to fitthe original data better

Neural network topology is shown in Figure 8 of FHMControlled object state variables are 119909 = (1199091(119905) 119909119899(119905))Tand input variables u = (u1(119905) u119898(119905))T as the input ofthe neural network topology and model output is the rateof change of the state variables k119894 (119894 = 1 119899) 119892119895 (119895 =1 119898) 119888119894119895 (119894 119895 = 1 119899) and 119889119894119895 (119894 = 1 119899 119895 =1 119898) as the connection weights which need to be trained

A and B are constant matrix whose elements are theweights comprised of 119888119894119895 and 119889119894119895 If the hidden layer function1198911(119909) = tanh(119909) and output layer activation function 1198912(119909) =119909 then the following models can be obtained

= A tanh (k119909119909) + B tanh (kuu) (6)

Obviously when making u linearize variables can beobtained

= A tanh (k119909119909) + Bu (7)

Thus FHM may be built by the neural network modelbecause it can train FHM model parameters [6] Since thereare many jitters in the data of rate of change of horizontal andheading angle the samples show a large number of outliersSo from the statistical view of robust the traditional MSEaggravate the ldquooutliersrdquo of samples Therefore Cauchy robusterror estimator can be used

Let error of outputs be

119864 = 12119902sum

k=1ln [1 + (1199101015840k minus 119910k)2] (8)

Journal of Control Science and Engineering 5

x1

xn

u1

um

k1

kn

g1

gm

f1(middot)c11

c1n

cn1

cnm

d11

d1m

dnm

dn1

f2(middot)x1

xn

Figure 8 Fuzzy hyperbolic tangentmodel neural network topology

where 1199101015840k is ideal output for the network and 119910k is actualoutput

Cauchy error estimator obviously does not depend on theinitial weights and thresholds transition and it can effectivelyeliminate the negative impact of ldquooutliersrdquo while retainingthe main characteristics of the value of the output error andspeeding up the convergence rate

By the steepest descent method it can obtain weightsiterative equations of each layer of neurons

119882(119905 + 1) = 119882 (119905) minus uΔ119864 (9)

where Δ119864 = 120597119864120597120596119882 = 120596119894119895So get a rate adjustment based on the following model

parameters above BP algorithm

119888119894119895 (119905 + 1) = 119888119894119895 (119905) + 120592 sdot 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2 sdot tanh (k119894119909119894)

k119895 (119905 + 1) = k119895 (119905) + 120592 sdot sumk119888119894119895 sdot ( 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2)

sdot 119909119894tanh (sum119894 120596119895k119909119894)

(10)

In this paper learning rate 120592 is considered as the adaptivescheme in which its main idea is that the initial value of 120592being set higher generally about 07ndash09 with the increaseor decrement of the number of learning may change in alaw When 120592 decreases to a certain extent if 120592 has not stillbecome convergence or error has still no improvement 120592 isset again at about 05ndash07 120592 enters the learning process againuntil the end of the run The learning rate is associated withthe error function when the error is reduced increase thelearning rate when the error increases reduce the learningrate

120592(k) =

112120592(kminus1) 119864(kminus1) gt 119864(k)08120592(kminus1) 119864(kminus1) lt 119864(k)120592(kminus1) 119864(kminus1) = 119864(k)

(11)

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

20 40 60 80 100 1200Time (s)

minus006minus005minus004minus003minus002minus001

0001002003

of th

e lat

eral

dev

iatio

n (m

s)

Figure 9 Fitting of change rate of the lateral deviation

Identify the model parameters after obtaining the data after6000 steps of learning

A = [00012 0099600095 minus00999]

B = [minus000700803]

k = [[[1000

100909998

]]]

(12)

Then

[12] = [00012 0099600095 minus00999] [tanh (1199091)

tanh (1199092)]

+ [minus000700803] 120574(13)

The fitting results of Figures 9 and 10 substantially elimi-nate the effect of outliers with relative error of control in lessthan 1 and less than 10The trend is essentially coincidentwith the sample data

The existing experimental data can be applied to theabove-described method for fuzzy hyperbolic model param-eter optimization in order to be close to the actual model Itprovides the basis to the next controller design

4 Controller Design

Based on this model we can design conventional linearcontrollers or other nonlinear controllers In this paper thecontroller is designed with hyperbolic tangent function ofthe state variables so we can use language to describe theinformation of the controller Thus this given controller is afuzzy one [16]

6 Journal of Control Science and Engineering

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

times10minus3

20 40 60 80 100 1200Time (s)

minus8

minus6

minus4

minus2

0

2

4

6

8of

the l

ater

al d

evia

tion

(rad

middotsminus1)

Figure 10 Fitting of change rate of the heading deviation

The system can be stabilized by nonquadratic perform-ance index function [16] System performance can get to aminimum with u through the givenQ and R

= A119891 (119909) + Bu (14)

Definition 1 (see [17]) Let 119878119888 satisfy the following conditionsall sets of 119891(sdot) include R rarr R

(1) 119891 is continuous(2) 119891(0) = 0 for other 119909 isin R 119891(119909)119909 gt 0(3) when |119909| rarr infin int119909

0119891(119910)119889119910 rarr infin

Theorem 2 (see [17]) For nonlinear system

= A119891 (119909) + Bu +D119908 (15)

where 119891(119909) = (1198911(119909) 119891119899(119909))T and 119891(119894) isin 119878119888 and thenk gt 0 tanh(k119909) isin 119878119888 (119894 = 1 2 3 119899) and a performanceindex function can be proposed as

119869 (1199090 1199050 u) = intinfin1199050

119891T (119909 (119905))Q119891 (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(16)

where Q R and S are given as the definite positive symmetricconstant matrix

Design Optimal Control Vector [18] Considerlowast

u(t) = lowast

u(t)(119905119909(120591)119905120591=0) and there is L(1199090) with boundary Then

sup 119869 ( lowast

u (t) 119908) le L (1199090) (17)

If there is a diagonal positive definite matrix Pmin satisfyingRiccati equation

PA + ATP minus P (BRminus1BTP minusDSminus1DT)P +Q = 0 (18)

then [119860 119861QR] is optimized diagonal matrix

Given nonlinear system and nonlinear quadratic perfor-mance index function [119860 119861QR] is an optimized diagonalmatrix The optimal control vector is

lowast

u (t) = minusRminus1BTPmin119891 (119909 (119905)) (19)

where Pmin = diag(1199011 119901119899) is optimized diagonal matrixLet L(1199090) = 2sum119899119894=1 119901119894 int119909119894(1199050)0 119891119894(120591)119889120591

Thus tanh(k119909) isin 119878119888 and for fuzzy hyperbolic tangent thequadratic performance index function is

119869 (1199090 1199050 u) = intinfin1199050

tanhT (119909 (119905))Q tanh (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(20)

Let Q = [ 10 00 15 ] R = 08 and S = 800 with Riccatiequation then

P = [10972 27592759 1678] (21)

According to the closed-loop system equation of state

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (22)

Based on Lyapunov theorem and Riccati equation for thisequation

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (23)

It is easy to prove

Pmin (A minus BRminus1BTPmin) + (A minus BRminus1BTPmin)Pmin

= minus (Q + Pmin (BRminus1BTP minusDSminus1DT)Pmin) (24)

And (A minusBRminus1BTP) is diagonally stable thus the closed-loop system is asymptotically stable

Then

K = [11627 6135] (25)

5 HIL Simulation

Thedifference between simulation and hardware-in-the-loopsimulation is that there is a real-time simulation of thecircuitrsquos physical hardware Hardware-in-the-loop simulationis intended to provide real-time incentives as the true signalto the controller so that the controller is connected to its realaccused equipment and test its performance Figure 11 is a pic-torial diagram of the hardware simulation platform in whichc-RIO controller is recognized as the path tracking anglecontroller program compiled by the Simulink PXI platformsimulation runs the Adams which can build undergroundmining articulated vehicle model as the object of emulationPC can run the LabVIEWgraphical user interface for PXI andc-RIO real-time and display the data of simulation compiledfrom the process Speed can be achieved by PID control at

Journal of Control Science and Engineering 7

PCc-RIO

PXI

LabVIEW

Figure 11 HIL simulation platform

Travel path comparison

Practical pathTarget path

100 20 30minus20 minus10minus30x (m)

minus30

minus20

minus10

0

10

20

30

y (m

)

Figure 12 Simulation of driving route comparedwith the ideal path

V = 3ms So as to test this robust of control there is onepit per 20m on the road The reference trajectory is set asa round whose center is (0 0) and R is 25m and an initialparameter setting is simulation time 200 s and the startingpoint coordinates (minus3 minus25)

Derived from the simulation results Figure 12 is thecomparison chart of the actual trajectory and the referencetrajectory As can be seen from the simulation curve thearticulated vehicle traveling trace and the reference circularpath are consistent and the trajectory is relatively smoothThe speed which is controlled by PID fromFigure 13 becomesfinally steady at 30 seconds The lateral displacement erroris gradually flat from the beginning of violent shaking inFigure 14 The overshoot amount is 428 and it is at mostabout 0024mat 18 secondswith respect to the tread based on2280m and the error is 17 and then close to the origin stepby stepThe orientation error is gradually flat from the begin-ning of violent shaking in Figure 15The overshoot amount is49 and it is at the top at about 008 rad at 18 seconds andthen close to the origin step by step The articulated angleis gradually flat from the beginning of violent shaking in

0

05

1

15

2

25

3

35

Spee

d (m

middotsminus1)

20 40 60 80 100 120 140 160 180 2000Time (s)

Figure 13 Longitudinal velocity of midpoint of front axle

20 40 60 80 100 120 140 160 180 2000Time (s)

minus003

minus0025

minus002

minus0015

minus001

minus0005

0

0005

Late

ral d

ispla

cem

ent e

orrr

(m)

Figure 14 Lateral displacement error

Figure 16 The overshoot amount is 35 and it is eventuallystabilized at 021 rad at 22 seconds Compared with the artic-ulated vehicle steering based on angle 45∘ the error is 12And the peaks are caused by pits but then the system imme-diately recovers to the right path It has better robustness thanliterature [12] in which there are a lot of jitters to adapt tothe path as a similar situation The results of literature [10]are as follows lateral deviation was 1m and heading angledeviation is 01 rad Literature [19] also analyzed themodels ofarticulated vehicle to control it with synovial control methodwhose results of the simulation are as follows the lateraldeviation is 01m and heading angle deviation was 017 radSo there exists a large gap when compared with the results ofthis paper

6 Conclusions

This article provides a method used in unmanned systemsbased on theway of fuzzy hyperbolic pole control act for artic-ulated vehicle trajectory tracking accurately Conclusions areas follows

(1) Fuzzy hyperbolic tangent model can well describethe quantitative relation among lateral displacementerror the orientation error and the articulated angle

(2) Based on the method of Cauchy robust error estima-tor the weights of BP neural network can effectively

8 Journal of Control Science and Engineering

20 40 60 80 100 120 140 160 180 2000Time (s)

minus002

0

002

004

006

008

01

Orie

ntat

ion

erro

r (ra

d)

Figure 15 Unmanned orientation error

20 40 60 80 100 120 140 160 180 2000Time (s)

0

005

01

015

02

025

Art

icul

ated

angl

e (ra

d)

Figure 16 Articulated angle

be reduced by the influence of singular error of neuralnetwork learning and fitting error and relative errorbelow expectations can be decreased so that it canachieve the system of identification

(3) By the 119867-infinity controller design controller has agood control performance and it makes the systemkeep a better performance about robustness in orderto achieve the purpose of the comprehensively opti-mal control error

(4) This method can be used in articulated vehicle pathtracking effectively based on the119867-infinity controllerSimulation in hardware-in-the-loop shows that over-shoot and the response time are less than expectationsand are eventually to stabilize The controller has metthe requirement of the real-time control performance

Competing Interests

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhaoand Jian Li declare that there is no conflict of interestsregarding the publication of this manuscript

Acknowledgments

This work was financially supported by the National HighTechnology Research and Development Program (863 Pro-gram) of China under Award 2011AA060404 IntelligentUnderground Mining Truck and Fundamental ResearchFunds for the Central Universities FRF-TP-16-004A1research on path-planning and path-following algorithmfor underground mining vehicles based on reinforcementlearning

References

[1] X-J Zeng and M G Singh ldquoApproximation theory of fuzzysystemsmdashSISO caserdquo IEEE Transactions on Fuzzy Systems vol2 no 2 pp 162ndash176 1994

[2] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceeding of the IEEE International Conference on Fuzzy Sys-tems pp 1163ndash1170 1992

[3] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[4] M Sugeno and T Yasukawa ldquoFuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] S Scheding G Dissanayake E Nebot and H Durrant-WhyteldquoSlip modelling and aided inertial navigation of an LHDrdquo inProceedings of the IEEE International Conference on Roboticsand Automation pp 1904ndash1909 Institute of Electrical andElectronics Engineers Albuquerque NM USA April 1997

[6] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[7] TNayl GNikolakopoulos andTGustfsson ldquoSwitchingmodelpredictive control for an articulated vehicle under varying slipanglerdquo in Proceedings of the 20th Mediterranean Conference onControl amp Automation (MED rsquo12) pp 890ndash895 IEEE ComputerSociety Barcelona Spain July 2012

[8] J H Lee andW S Yoo ldquoPredictive control of a vehicle trajectoryusing a coupled vector with vehicle velocity and sideslip anglerdquoInternational Journal of Automotive Technology vol 10 no 2 pp211ndash217 2009

[9] P Ridley and P Corke ldquoLoad haul dump vehicle kinematicsand controlrdquo Journal of Dynamic Systems Measurement andControl vol 125 no 1 pp 54ndash59 2003

[10] Z Xuan Y Jue and Z Wenming ldquoFeedback linearizationcontrol for path tracking of articulated dump truckrdquo TELKOM-NIKA Telecommunication Computing Electronics and Controlvol 9 no 13 pp 922ndash929 2015

[11] D Piyabongkarn R Rajamani J A Grogg and J Y LewldquoDevelopment and experimental evaluation of a slip angleestimator for vehicle stability controlrdquo IEEE Transactions onControl Systems Technology vol 17 no 1 pp 78ndash88 2009

[12] Y Qi Trajectory Control for the Underground Articulated DumpTruck Based on Model Predictive Control University of Scienceand Technology Beijing Beijing China 2013

[13] K-K D Young ldquoController design for a manipulator usingtheory of variable structure systemsrdquo IEEE Transactions on Sys-tems Man and Cybernetics vol 8 no 2 pp 101ndash109 1978

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 5

x1

xn

u1

um

k1

kn

g1

gm

f1(middot)c11

c1n

cn1

cnm

d11

d1m

dnm

dn1

f2(middot)x1

xn

Figure 8 Fuzzy hyperbolic tangentmodel neural network topology

where 1199101015840k is ideal output for the network and 119910k is actualoutput

Cauchy error estimator obviously does not depend on theinitial weights and thresholds transition and it can effectivelyeliminate the negative impact of ldquooutliersrdquo while retainingthe main characteristics of the value of the output error andspeeding up the convergence rate

By the steepest descent method it can obtain weightsiterative equations of each layer of neurons

119882(119905 + 1) = 119882 (119905) minus uΔ119864 (9)

where Δ119864 = 120597119864120597120596119882 = 120596119894119895So get a rate adjustment based on the following model

parameters above BP algorithm

119888119894119895 (119905 + 1) = 119888119894119895 (119905) + 120592 sdot 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2 sdot tanh (k119894119909119894)

k119895 (119905 + 1) = k119895 (119905) + 120592 sdot sumk119888119894119895 sdot ( 1199101015840k minus 119910k1 + (1199101015840k minus 119910k)2)

sdot 119909119894tanh (sum119894 120596119895k119909119894)

(10)

In this paper learning rate 120592 is considered as the adaptivescheme in which its main idea is that the initial value of 120592being set higher generally about 07ndash09 with the increaseor decrement of the number of learning may change in alaw When 120592 decreases to a certain extent if 120592 has not stillbecome convergence or error has still no improvement 120592 isset again at about 05ndash07 120592 enters the learning process againuntil the end of the run The learning rate is associated withthe error function when the error is reduced increase thelearning rate when the error increases reduce the learningrate

120592(k) =

112120592(kminus1) 119864(kminus1) gt 119864(k)08120592(kminus1) 119864(kminus1) lt 119864(k)120592(kminus1) 119864(kminus1) = 119864(k)

(11)

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

20 40 60 80 100 1200Time (s)

minus006minus005minus004minus003minus002minus001

0001002003

of th

e lat

eral

dev

iatio

n (m

s)

Figure 9 Fitting of change rate of the lateral deviation

Identify the model parameters after obtaining the data after6000 steps of learning

A = [00012 0099600095 minus00999]

B = [minus000700803]

k = [[[1000

100909998

]]]

(12)

Then

[12] = [00012 0099600095 minus00999] [tanh (1199091)

tanh (1199092)]

+ [minus000700803] 120574(13)

The fitting results of Figures 9 and 10 substantially elimi-nate the effect of outliers with relative error of control in lessthan 1 and less than 10The trend is essentially coincidentwith the sample data

The existing experimental data can be applied to theabove-described method for fuzzy hyperbolic model param-eter optimization in order to be close to the actual model Itprovides the basis to the next controller design

4 Controller Design

Based on this model we can design conventional linearcontrollers or other nonlinear controllers In this paper thecontroller is designed with hyperbolic tangent function ofthe state variables so we can use language to describe theinformation of the controller Thus this given controller is afuzzy one [16]

6 Journal of Control Science and Engineering

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

times10minus3

20 40 60 80 100 1200Time (s)

minus8

minus6

minus4

minus2

0

2

4

6

8of

the l

ater

al d

evia

tion

(rad

middotsminus1)

Figure 10 Fitting of change rate of the heading deviation

The system can be stabilized by nonquadratic perform-ance index function [16] System performance can get to aminimum with u through the givenQ and R

= A119891 (119909) + Bu (14)

Definition 1 (see [17]) Let 119878119888 satisfy the following conditionsall sets of 119891(sdot) include R rarr R

(1) 119891 is continuous(2) 119891(0) = 0 for other 119909 isin R 119891(119909)119909 gt 0(3) when |119909| rarr infin int119909

0119891(119910)119889119910 rarr infin

Theorem 2 (see [17]) For nonlinear system

= A119891 (119909) + Bu +D119908 (15)

where 119891(119909) = (1198911(119909) 119891119899(119909))T and 119891(119894) isin 119878119888 and thenk gt 0 tanh(k119909) isin 119878119888 (119894 = 1 2 3 119899) and a performanceindex function can be proposed as

119869 (1199090 1199050 u) = intinfin1199050

119891T (119909 (119905))Q119891 (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(16)

where Q R and S are given as the definite positive symmetricconstant matrix

Design Optimal Control Vector [18] Considerlowast

u(t) = lowast

u(t)(119905119909(120591)119905120591=0) and there is L(1199090) with boundary Then

sup 119869 ( lowast

u (t) 119908) le L (1199090) (17)

If there is a diagonal positive definite matrix Pmin satisfyingRiccati equation

PA + ATP minus P (BRminus1BTP minusDSminus1DT)P +Q = 0 (18)

then [119860 119861QR] is optimized diagonal matrix

Given nonlinear system and nonlinear quadratic perfor-mance index function [119860 119861QR] is an optimized diagonalmatrix The optimal control vector is

lowast

u (t) = minusRminus1BTPmin119891 (119909 (119905)) (19)

where Pmin = diag(1199011 119901119899) is optimized diagonal matrixLet L(1199090) = 2sum119899119894=1 119901119894 int119909119894(1199050)0 119891119894(120591)119889120591

Thus tanh(k119909) isin 119878119888 and for fuzzy hyperbolic tangent thequadratic performance index function is

119869 (1199090 1199050 u) = intinfin1199050

tanhT (119909 (119905))Q tanh (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(20)

Let Q = [ 10 00 15 ] R = 08 and S = 800 with Riccatiequation then

P = [10972 27592759 1678] (21)

According to the closed-loop system equation of state

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (22)

Based on Lyapunov theorem and Riccati equation for thisequation

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (23)

It is easy to prove

Pmin (A minus BRminus1BTPmin) + (A minus BRminus1BTPmin)Pmin

= minus (Q + Pmin (BRminus1BTP minusDSminus1DT)Pmin) (24)

And (A minusBRminus1BTP) is diagonally stable thus the closed-loop system is asymptotically stable

Then

K = [11627 6135] (25)

5 HIL Simulation

Thedifference between simulation and hardware-in-the-loopsimulation is that there is a real-time simulation of thecircuitrsquos physical hardware Hardware-in-the-loop simulationis intended to provide real-time incentives as the true signalto the controller so that the controller is connected to its realaccused equipment and test its performance Figure 11 is a pic-torial diagram of the hardware simulation platform in whichc-RIO controller is recognized as the path tracking anglecontroller program compiled by the Simulink PXI platformsimulation runs the Adams which can build undergroundmining articulated vehicle model as the object of emulationPC can run the LabVIEWgraphical user interface for PXI andc-RIO real-time and display the data of simulation compiledfrom the process Speed can be achieved by PID control at

Journal of Control Science and Engineering 7

PCc-RIO

PXI

LabVIEW

Figure 11 HIL simulation platform

Travel path comparison

Practical pathTarget path

100 20 30minus20 minus10minus30x (m)

minus30

minus20

minus10

0

10

20

30

y (m

)

Figure 12 Simulation of driving route comparedwith the ideal path

V = 3ms So as to test this robust of control there is onepit per 20m on the road The reference trajectory is set asa round whose center is (0 0) and R is 25m and an initialparameter setting is simulation time 200 s and the startingpoint coordinates (minus3 minus25)

Derived from the simulation results Figure 12 is thecomparison chart of the actual trajectory and the referencetrajectory As can be seen from the simulation curve thearticulated vehicle traveling trace and the reference circularpath are consistent and the trajectory is relatively smoothThe speed which is controlled by PID fromFigure 13 becomesfinally steady at 30 seconds The lateral displacement erroris gradually flat from the beginning of violent shaking inFigure 14 The overshoot amount is 428 and it is at mostabout 0024mat 18 secondswith respect to the tread based on2280m and the error is 17 and then close to the origin stepby stepThe orientation error is gradually flat from the begin-ning of violent shaking in Figure 15The overshoot amount is49 and it is at the top at about 008 rad at 18 seconds andthen close to the origin step by step The articulated angleis gradually flat from the beginning of violent shaking in

0

05

1

15

2

25

3

35

Spee

d (m

middotsminus1)

20 40 60 80 100 120 140 160 180 2000Time (s)

Figure 13 Longitudinal velocity of midpoint of front axle

20 40 60 80 100 120 140 160 180 2000Time (s)

minus003

minus0025

minus002

minus0015

minus001

minus0005

0

0005

Late

ral d

ispla

cem

ent e

orrr

(m)

Figure 14 Lateral displacement error

Figure 16 The overshoot amount is 35 and it is eventuallystabilized at 021 rad at 22 seconds Compared with the artic-ulated vehicle steering based on angle 45∘ the error is 12And the peaks are caused by pits but then the system imme-diately recovers to the right path It has better robustness thanliterature [12] in which there are a lot of jitters to adapt tothe path as a similar situation The results of literature [10]are as follows lateral deviation was 1m and heading angledeviation is 01 rad Literature [19] also analyzed themodels ofarticulated vehicle to control it with synovial control methodwhose results of the simulation are as follows the lateraldeviation is 01m and heading angle deviation was 017 radSo there exists a large gap when compared with the results ofthis paper

6 Conclusions

This article provides a method used in unmanned systemsbased on theway of fuzzy hyperbolic pole control act for artic-ulated vehicle trajectory tracking accurately Conclusions areas follows

(1) Fuzzy hyperbolic tangent model can well describethe quantitative relation among lateral displacementerror the orientation error and the articulated angle

(2) Based on the method of Cauchy robust error estima-tor the weights of BP neural network can effectively

8 Journal of Control Science and Engineering

20 40 60 80 100 120 140 160 180 2000Time (s)

minus002

0

002

004

006

008

01

Orie

ntat

ion

erro

r (ra

d)

Figure 15 Unmanned orientation error

20 40 60 80 100 120 140 160 180 2000Time (s)

0

005

01

015

02

025

Art

icul

ated

angl

e (ra

d)

Figure 16 Articulated angle

be reduced by the influence of singular error of neuralnetwork learning and fitting error and relative errorbelow expectations can be decreased so that it canachieve the system of identification

(3) By the 119867-infinity controller design controller has agood control performance and it makes the systemkeep a better performance about robustness in orderto achieve the purpose of the comprehensively opti-mal control error

(4) This method can be used in articulated vehicle pathtracking effectively based on the119867-infinity controllerSimulation in hardware-in-the-loop shows that over-shoot and the response time are less than expectationsand are eventually to stabilize The controller has metthe requirement of the real-time control performance

Competing Interests

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhaoand Jian Li declare that there is no conflict of interestsregarding the publication of this manuscript

Acknowledgments

This work was financially supported by the National HighTechnology Research and Development Program (863 Pro-gram) of China under Award 2011AA060404 IntelligentUnderground Mining Truck and Fundamental ResearchFunds for the Central Universities FRF-TP-16-004A1research on path-planning and path-following algorithmfor underground mining vehicles based on reinforcementlearning

References

[1] X-J Zeng and M G Singh ldquoApproximation theory of fuzzysystemsmdashSISO caserdquo IEEE Transactions on Fuzzy Systems vol2 no 2 pp 162ndash176 1994

[2] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceeding of the IEEE International Conference on Fuzzy Sys-tems pp 1163ndash1170 1992

[3] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[4] M Sugeno and T Yasukawa ldquoFuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] S Scheding G Dissanayake E Nebot and H Durrant-WhyteldquoSlip modelling and aided inertial navigation of an LHDrdquo inProceedings of the IEEE International Conference on Roboticsand Automation pp 1904ndash1909 Institute of Electrical andElectronics Engineers Albuquerque NM USA April 1997

[6] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[7] TNayl GNikolakopoulos andTGustfsson ldquoSwitchingmodelpredictive control for an articulated vehicle under varying slipanglerdquo in Proceedings of the 20th Mediterranean Conference onControl amp Automation (MED rsquo12) pp 890ndash895 IEEE ComputerSociety Barcelona Spain July 2012

[8] J H Lee andW S Yoo ldquoPredictive control of a vehicle trajectoryusing a coupled vector with vehicle velocity and sideslip anglerdquoInternational Journal of Automotive Technology vol 10 no 2 pp211ndash217 2009

[9] P Ridley and P Corke ldquoLoad haul dump vehicle kinematicsand controlrdquo Journal of Dynamic Systems Measurement andControl vol 125 no 1 pp 54ndash59 2003

[10] Z Xuan Y Jue and Z Wenming ldquoFeedback linearizationcontrol for path tracking of articulated dump truckrdquo TELKOM-NIKA Telecommunication Computing Electronics and Controlvol 9 no 13 pp 922ndash929 2015

[11] D Piyabongkarn R Rajamani J A Grogg and J Y LewldquoDevelopment and experimental evaluation of a slip angleestimator for vehicle stability controlrdquo IEEE Transactions onControl Systems Technology vol 17 no 1 pp 78ndash88 2009

[12] Y Qi Trajectory Control for the Underground Articulated DumpTruck Based on Model Predictive Control University of Scienceand Technology Beijing Beijing China 2013

[13] K-K D Young ldquoController design for a manipulator usingtheory of variable structure systemsrdquo IEEE Transactions on Sys-tems Man and Cybernetics vol 8 no 2 pp 101ndash109 1978

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Control Science and Engineering

Fitti

ng m

anne

d ra

te o

f cha

nge

FittingIdeal

times10minus3

20 40 60 80 100 1200Time (s)

minus8

minus6

minus4

minus2

0

2

4

6

8of

the l

ater

al d

evia

tion

(rad

middotsminus1)

Figure 10 Fitting of change rate of the heading deviation

The system can be stabilized by nonquadratic perform-ance index function [16] System performance can get to aminimum with u through the givenQ and R

= A119891 (119909) + Bu (14)

Definition 1 (see [17]) Let 119878119888 satisfy the following conditionsall sets of 119891(sdot) include R rarr R

(1) 119891 is continuous(2) 119891(0) = 0 for other 119909 isin R 119891(119909)119909 gt 0(3) when |119909| rarr infin int119909

0119891(119910)119889119910 rarr infin

Theorem 2 (see [17]) For nonlinear system

= A119891 (119909) + Bu +D119908 (15)

where 119891(119909) = (1198911(119909) 119891119899(119909))T and 119891(119894) isin 119878119888 and thenk gt 0 tanh(k119909) isin 119878119888 (119894 = 1 2 3 119899) and a performanceindex function can be proposed as

119869 (1199090 1199050 u) = intinfin1199050

119891T (119909 (119905))Q119891 (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(16)

where Q R and S are given as the definite positive symmetricconstant matrix

Design Optimal Control Vector [18] Considerlowast

u(t) = lowast

u(t)(119905119909(120591)119905120591=0) and there is L(1199090) with boundary Then

sup 119869 ( lowast

u (t) 119908) le L (1199090) (17)

If there is a diagonal positive definite matrix Pmin satisfyingRiccati equation

PA + ATP minus P (BRminus1BTP minusDSminus1DT)P +Q = 0 (18)

then [119860 119861QR] is optimized diagonal matrix

Given nonlinear system and nonlinear quadratic perfor-mance index function [119860 119861QR] is an optimized diagonalmatrix The optimal control vector is

lowast

u (t) = minusRminus1BTPmin119891 (119909 (119905)) (19)

where Pmin = diag(1199011 119901119899) is optimized diagonal matrixLet L(1199090) = 2sum119899119894=1 119901119894 int119909119894(1199050)0 119891119894(120591)119889120591

Thus tanh(k119909) isin 119878119888 and for fuzzy hyperbolic tangent thequadratic performance index function is

119869 (1199090 1199050 u) = intinfin1199050

tanhT (119909 (119905))Q tanh (119909 (119905))+ uT (119909 (119905))Ru (119909 (119905)) minus 119908T (119905) S119908 (119905) 119889119905

(20)

Let Q = [ 10 00 15 ] R = 08 and S = 800 with Riccatiequation then

P = [10972 27592759 1678] (21)

According to the closed-loop system equation of state

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (22)

Based on Lyapunov theorem and Riccati equation for thisequation

= (A minus BRminus1BTPmin) 119891 (119909 (119905)) (23)

It is easy to prove

Pmin (A minus BRminus1BTPmin) + (A minus BRminus1BTPmin)Pmin

= minus (Q + Pmin (BRminus1BTP minusDSminus1DT)Pmin) (24)

And (A minusBRminus1BTP) is diagonally stable thus the closed-loop system is asymptotically stable

Then

K = [11627 6135] (25)

5 HIL Simulation

Thedifference between simulation and hardware-in-the-loopsimulation is that there is a real-time simulation of thecircuitrsquos physical hardware Hardware-in-the-loop simulationis intended to provide real-time incentives as the true signalto the controller so that the controller is connected to its realaccused equipment and test its performance Figure 11 is a pic-torial diagram of the hardware simulation platform in whichc-RIO controller is recognized as the path tracking anglecontroller program compiled by the Simulink PXI platformsimulation runs the Adams which can build undergroundmining articulated vehicle model as the object of emulationPC can run the LabVIEWgraphical user interface for PXI andc-RIO real-time and display the data of simulation compiledfrom the process Speed can be achieved by PID control at

Journal of Control Science and Engineering 7

PCc-RIO

PXI

LabVIEW

Figure 11 HIL simulation platform

Travel path comparison

Practical pathTarget path

100 20 30minus20 minus10minus30x (m)

minus30

minus20

minus10

0

10

20

30

y (m

)

Figure 12 Simulation of driving route comparedwith the ideal path

V = 3ms So as to test this robust of control there is onepit per 20m on the road The reference trajectory is set asa round whose center is (0 0) and R is 25m and an initialparameter setting is simulation time 200 s and the startingpoint coordinates (minus3 minus25)

Derived from the simulation results Figure 12 is thecomparison chart of the actual trajectory and the referencetrajectory As can be seen from the simulation curve thearticulated vehicle traveling trace and the reference circularpath are consistent and the trajectory is relatively smoothThe speed which is controlled by PID fromFigure 13 becomesfinally steady at 30 seconds The lateral displacement erroris gradually flat from the beginning of violent shaking inFigure 14 The overshoot amount is 428 and it is at mostabout 0024mat 18 secondswith respect to the tread based on2280m and the error is 17 and then close to the origin stepby stepThe orientation error is gradually flat from the begin-ning of violent shaking in Figure 15The overshoot amount is49 and it is at the top at about 008 rad at 18 seconds andthen close to the origin step by step The articulated angleis gradually flat from the beginning of violent shaking in

0

05

1

15

2

25

3

35

Spee

d (m

middotsminus1)

20 40 60 80 100 120 140 160 180 2000Time (s)

Figure 13 Longitudinal velocity of midpoint of front axle

20 40 60 80 100 120 140 160 180 2000Time (s)

minus003

minus0025

minus002

minus0015

minus001

minus0005

0

0005

Late

ral d

ispla

cem

ent e

orrr

(m)

Figure 14 Lateral displacement error

Figure 16 The overshoot amount is 35 and it is eventuallystabilized at 021 rad at 22 seconds Compared with the artic-ulated vehicle steering based on angle 45∘ the error is 12And the peaks are caused by pits but then the system imme-diately recovers to the right path It has better robustness thanliterature [12] in which there are a lot of jitters to adapt tothe path as a similar situation The results of literature [10]are as follows lateral deviation was 1m and heading angledeviation is 01 rad Literature [19] also analyzed themodels ofarticulated vehicle to control it with synovial control methodwhose results of the simulation are as follows the lateraldeviation is 01m and heading angle deviation was 017 radSo there exists a large gap when compared with the results ofthis paper

6 Conclusions

This article provides a method used in unmanned systemsbased on theway of fuzzy hyperbolic pole control act for artic-ulated vehicle trajectory tracking accurately Conclusions areas follows

(1) Fuzzy hyperbolic tangent model can well describethe quantitative relation among lateral displacementerror the orientation error and the articulated angle

(2) Based on the method of Cauchy robust error estima-tor the weights of BP neural network can effectively

8 Journal of Control Science and Engineering

20 40 60 80 100 120 140 160 180 2000Time (s)

minus002

0

002

004

006

008

01

Orie

ntat

ion

erro

r (ra

d)

Figure 15 Unmanned orientation error

20 40 60 80 100 120 140 160 180 2000Time (s)

0

005

01

015

02

025

Art

icul

ated

angl

e (ra

d)

Figure 16 Articulated angle

be reduced by the influence of singular error of neuralnetwork learning and fitting error and relative errorbelow expectations can be decreased so that it canachieve the system of identification

(3) By the 119867-infinity controller design controller has agood control performance and it makes the systemkeep a better performance about robustness in orderto achieve the purpose of the comprehensively opti-mal control error

(4) This method can be used in articulated vehicle pathtracking effectively based on the119867-infinity controllerSimulation in hardware-in-the-loop shows that over-shoot and the response time are less than expectationsand are eventually to stabilize The controller has metthe requirement of the real-time control performance

Competing Interests

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhaoand Jian Li declare that there is no conflict of interestsregarding the publication of this manuscript

Acknowledgments

This work was financially supported by the National HighTechnology Research and Development Program (863 Pro-gram) of China under Award 2011AA060404 IntelligentUnderground Mining Truck and Fundamental ResearchFunds for the Central Universities FRF-TP-16-004A1research on path-planning and path-following algorithmfor underground mining vehicles based on reinforcementlearning

References

[1] X-J Zeng and M G Singh ldquoApproximation theory of fuzzysystemsmdashSISO caserdquo IEEE Transactions on Fuzzy Systems vol2 no 2 pp 162ndash176 1994

[2] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceeding of the IEEE International Conference on Fuzzy Sys-tems pp 1163ndash1170 1992

[3] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[4] M Sugeno and T Yasukawa ldquoFuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] S Scheding G Dissanayake E Nebot and H Durrant-WhyteldquoSlip modelling and aided inertial navigation of an LHDrdquo inProceedings of the IEEE International Conference on Roboticsand Automation pp 1904ndash1909 Institute of Electrical andElectronics Engineers Albuquerque NM USA April 1997

[6] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[7] TNayl GNikolakopoulos andTGustfsson ldquoSwitchingmodelpredictive control for an articulated vehicle under varying slipanglerdquo in Proceedings of the 20th Mediterranean Conference onControl amp Automation (MED rsquo12) pp 890ndash895 IEEE ComputerSociety Barcelona Spain July 2012

[8] J H Lee andW S Yoo ldquoPredictive control of a vehicle trajectoryusing a coupled vector with vehicle velocity and sideslip anglerdquoInternational Journal of Automotive Technology vol 10 no 2 pp211ndash217 2009

[9] P Ridley and P Corke ldquoLoad haul dump vehicle kinematicsand controlrdquo Journal of Dynamic Systems Measurement andControl vol 125 no 1 pp 54ndash59 2003

[10] Z Xuan Y Jue and Z Wenming ldquoFeedback linearizationcontrol for path tracking of articulated dump truckrdquo TELKOM-NIKA Telecommunication Computing Electronics and Controlvol 9 no 13 pp 922ndash929 2015

[11] D Piyabongkarn R Rajamani J A Grogg and J Y LewldquoDevelopment and experimental evaluation of a slip angleestimator for vehicle stability controlrdquo IEEE Transactions onControl Systems Technology vol 17 no 1 pp 78ndash88 2009

[12] Y Qi Trajectory Control for the Underground Articulated DumpTruck Based on Model Predictive Control University of Scienceand Technology Beijing Beijing China 2013

[13] K-K D Young ldquoController design for a manipulator usingtheory of variable structure systemsrdquo IEEE Transactions on Sys-tems Man and Cybernetics vol 8 no 2 pp 101ndash109 1978

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 7

PCc-RIO

PXI

LabVIEW

Figure 11 HIL simulation platform

Travel path comparison

Practical pathTarget path

100 20 30minus20 minus10minus30x (m)

minus30

minus20

minus10

0

10

20

30

y (m

)

Figure 12 Simulation of driving route comparedwith the ideal path

V = 3ms So as to test this robust of control there is onepit per 20m on the road The reference trajectory is set asa round whose center is (0 0) and R is 25m and an initialparameter setting is simulation time 200 s and the startingpoint coordinates (minus3 minus25)

Derived from the simulation results Figure 12 is thecomparison chart of the actual trajectory and the referencetrajectory As can be seen from the simulation curve thearticulated vehicle traveling trace and the reference circularpath are consistent and the trajectory is relatively smoothThe speed which is controlled by PID fromFigure 13 becomesfinally steady at 30 seconds The lateral displacement erroris gradually flat from the beginning of violent shaking inFigure 14 The overshoot amount is 428 and it is at mostabout 0024mat 18 secondswith respect to the tread based on2280m and the error is 17 and then close to the origin stepby stepThe orientation error is gradually flat from the begin-ning of violent shaking in Figure 15The overshoot amount is49 and it is at the top at about 008 rad at 18 seconds andthen close to the origin step by step The articulated angleis gradually flat from the beginning of violent shaking in

0

05

1

15

2

25

3

35

Spee

d (m

middotsminus1)

20 40 60 80 100 120 140 160 180 2000Time (s)

Figure 13 Longitudinal velocity of midpoint of front axle

20 40 60 80 100 120 140 160 180 2000Time (s)

minus003

minus0025

minus002

minus0015

minus001

minus0005

0

0005

Late

ral d

ispla

cem

ent e

orrr

(m)

Figure 14 Lateral displacement error

Figure 16 The overshoot amount is 35 and it is eventuallystabilized at 021 rad at 22 seconds Compared with the artic-ulated vehicle steering based on angle 45∘ the error is 12And the peaks are caused by pits but then the system imme-diately recovers to the right path It has better robustness thanliterature [12] in which there are a lot of jitters to adapt tothe path as a similar situation The results of literature [10]are as follows lateral deviation was 1m and heading angledeviation is 01 rad Literature [19] also analyzed themodels ofarticulated vehicle to control it with synovial control methodwhose results of the simulation are as follows the lateraldeviation is 01m and heading angle deviation was 017 radSo there exists a large gap when compared with the results ofthis paper

6 Conclusions

This article provides a method used in unmanned systemsbased on theway of fuzzy hyperbolic pole control act for artic-ulated vehicle trajectory tracking accurately Conclusions areas follows

(1) Fuzzy hyperbolic tangent model can well describethe quantitative relation among lateral displacementerror the orientation error and the articulated angle

(2) Based on the method of Cauchy robust error estima-tor the weights of BP neural network can effectively

8 Journal of Control Science and Engineering

20 40 60 80 100 120 140 160 180 2000Time (s)

minus002

0

002

004

006

008

01

Orie

ntat

ion

erro

r (ra

d)

Figure 15 Unmanned orientation error

20 40 60 80 100 120 140 160 180 2000Time (s)

0

005

01

015

02

025

Art

icul

ated

angl

e (ra

d)

Figure 16 Articulated angle

be reduced by the influence of singular error of neuralnetwork learning and fitting error and relative errorbelow expectations can be decreased so that it canachieve the system of identification

(3) By the 119867-infinity controller design controller has agood control performance and it makes the systemkeep a better performance about robustness in orderto achieve the purpose of the comprehensively opti-mal control error

(4) This method can be used in articulated vehicle pathtracking effectively based on the119867-infinity controllerSimulation in hardware-in-the-loop shows that over-shoot and the response time are less than expectationsand are eventually to stabilize The controller has metthe requirement of the real-time control performance

Competing Interests

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhaoand Jian Li declare that there is no conflict of interestsregarding the publication of this manuscript

Acknowledgments

This work was financially supported by the National HighTechnology Research and Development Program (863 Pro-gram) of China under Award 2011AA060404 IntelligentUnderground Mining Truck and Fundamental ResearchFunds for the Central Universities FRF-TP-16-004A1research on path-planning and path-following algorithmfor underground mining vehicles based on reinforcementlearning

References

[1] X-J Zeng and M G Singh ldquoApproximation theory of fuzzysystemsmdashSISO caserdquo IEEE Transactions on Fuzzy Systems vol2 no 2 pp 162ndash176 1994

[2] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceeding of the IEEE International Conference on Fuzzy Sys-tems pp 1163ndash1170 1992

[3] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[4] M Sugeno and T Yasukawa ldquoFuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] S Scheding G Dissanayake E Nebot and H Durrant-WhyteldquoSlip modelling and aided inertial navigation of an LHDrdquo inProceedings of the IEEE International Conference on Roboticsand Automation pp 1904ndash1909 Institute of Electrical andElectronics Engineers Albuquerque NM USA April 1997

[6] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[7] TNayl GNikolakopoulos andTGustfsson ldquoSwitchingmodelpredictive control for an articulated vehicle under varying slipanglerdquo in Proceedings of the 20th Mediterranean Conference onControl amp Automation (MED rsquo12) pp 890ndash895 IEEE ComputerSociety Barcelona Spain July 2012

[8] J H Lee andW S Yoo ldquoPredictive control of a vehicle trajectoryusing a coupled vector with vehicle velocity and sideslip anglerdquoInternational Journal of Automotive Technology vol 10 no 2 pp211ndash217 2009

[9] P Ridley and P Corke ldquoLoad haul dump vehicle kinematicsand controlrdquo Journal of Dynamic Systems Measurement andControl vol 125 no 1 pp 54ndash59 2003

[10] Z Xuan Y Jue and Z Wenming ldquoFeedback linearizationcontrol for path tracking of articulated dump truckrdquo TELKOM-NIKA Telecommunication Computing Electronics and Controlvol 9 no 13 pp 922ndash929 2015

[11] D Piyabongkarn R Rajamani J A Grogg and J Y LewldquoDevelopment and experimental evaluation of a slip angleestimator for vehicle stability controlrdquo IEEE Transactions onControl Systems Technology vol 17 no 1 pp 78ndash88 2009

[12] Y Qi Trajectory Control for the Underground Articulated DumpTruck Based on Model Predictive Control University of Scienceand Technology Beijing Beijing China 2013

[13] K-K D Young ldquoController design for a manipulator usingtheory of variable structure systemsrdquo IEEE Transactions on Sys-tems Man and Cybernetics vol 8 no 2 pp 101ndash109 1978

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Control Science and Engineering

20 40 60 80 100 120 140 160 180 2000Time (s)

minus002

0

002

004

006

008

01

Orie

ntat

ion

erro

r (ra

d)

Figure 15 Unmanned orientation error

20 40 60 80 100 120 140 160 180 2000Time (s)

0

005

01

015

02

025

Art

icul

ated

angl

e (ra

d)

Figure 16 Articulated angle

be reduced by the influence of singular error of neuralnetwork learning and fitting error and relative errorbelow expectations can be decreased so that it canachieve the system of identification

(3) By the 119867-infinity controller design controller has agood control performance and it makes the systemkeep a better performance about robustness in orderto achieve the purpose of the comprehensively opti-mal control error

(4) This method can be used in articulated vehicle pathtracking effectively based on the119867-infinity controllerSimulation in hardware-in-the-loop shows that over-shoot and the response time are less than expectationsand are eventually to stabilize The controller has metthe requirement of the real-time control performance

Competing Interests

Guangsi Shi Jue Yang Xuan Zhao Yanfeng Li Yalun Zhaoand Jian Li declare that there is no conflict of interestsregarding the publication of this manuscript

Acknowledgments

This work was financially supported by the National HighTechnology Research and Development Program (863 Pro-gram) of China under Award 2011AA060404 IntelligentUnderground Mining Truck and Fundamental ResearchFunds for the Central Universities FRF-TP-16-004A1research on path-planning and path-following algorithmfor underground mining vehicles based on reinforcementlearning

References

[1] X-J Zeng and M G Singh ldquoApproximation theory of fuzzysystemsmdashSISO caserdquo IEEE Transactions on Fuzzy Systems vol2 no 2 pp 162ndash176 1994

[2] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceeding of the IEEE International Conference on Fuzzy Sys-tems pp 1163ndash1170 1992

[3] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[4] M Sugeno and T Yasukawa ldquoFuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] S Scheding G Dissanayake E Nebot and H Durrant-WhyteldquoSlip modelling and aided inertial navigation of an LHDrdquo inProceedings of the IEEE International Conference on Roboticsand Automation pp 1904ndash1909 Institute of Electrical andElectronics Engineers Albuquerque NM USA April 1997

[6] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[7] TNayl GNikolakopoulos andTGustfsson ldquoSwitchingmodelpredictive control for an articulated vehicle under varying slipanglerdquo in Proceedings of the 20th Mediterranean Conference onControl amp Automation (MED rsquo12) pp 890ndash895 IEEE ComputerSociety Barcelona Spain July 2012

[8] J H Lee andW S Yoo ldquoPredictive control of a vehicle trajectoryusing a coupled vector with vehicle velocity and sideslip anglerdquoInternational Journal of Automotive Technology vol 10 no 2 pp211ndash217 2009

[9] P Ridley and P Corke ldquoLoad haul dump vehicle kinematicsand controlrdquo Journal of Dynamic Systems Measurement andControl vol 125 no 1 pp 54ndash59 2003

[10] Z Xuan Y Jue and Z Wenming ldquoFeedback linearizationcontrol for path tracking of articulated dump truckrdquo TELKOM-NIKA Telecommunication Computing Electronics and Controlvol 9 no 13 pp 922ndash929 2015

[11] D Piyabongkarn R Rajamani J A Grogg and J Y LewldquoDevelopment and experimental evaluation of a slip angleestimator for vehicle stability controlrdquo IEEE Transactions onControl Systems Technology vol 17 no 1 pp 78ndash88 2009

[12] Y Qi Trajectory Control for the Underground Articulated DumpTruck Based on Model Predictive Control University of Scienceand Technology Beijing Beijing China 2013

[13] K-K D Young ldquoController design for a manipulator usingtheory of variable structure systemsrdquo IEEE Transactions on Sys-tems Man and Cybernetics vol 8 no 2 pp 101ndash109 1978

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 9

[14] E M Jafarov M N A Parlakci and Y Istefanopulos ldquoA newvariable structure PID-controller design for robot manipula-torsrdquo IEEE Transactions on Control Systems Technology vol 13no 1 pp 122ndash130 2005

[15] P Petrov and P Bigras ldquoA practical approach to feedback pathcontrol for an articulated mining vehiclerdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems pp 2258ndash2263 Institute of Electrical and ElectronicsEngineers Maui Hawaii USA November 2001

[16] G-T Hui H-G Zhang G Wang X-P Xie and Z-N WuldquoResearch on fuzzy hyperbolic tangent model a reviewrdquo ActaAutomatica Sinica vol 39 no 11 pp 1849ndash1857 2013

[17] E Kaszkurewicz and A Bhaya ldquoRobust stability and diagonalLiapunov functionsrdquo SIAM Journal on Matrix Analysis andApplications vol 14 no 2 pp 508ndash520 1993

[18] K Ogata Modern Control Engineering Electronic IndustryPress Beijing China 2011

[19] X Zhao J Yang W Zhang and J Zeng ldquoSliding mode controlalgorithm for path tracking of articulated dump truckrdquo Tran-sactions of the Chinese Society of Agricultural Engineering vol31 no 10 pp 198ndash203 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of