path tracking control of automatic parking cloud model

Research ArticlePath Tracking Control of Automatic Parking Cloud Modelconsidering the Influence of Time Delay

Yiding Hua1 Haobin Jiang12 Yingfeng Cai2 Xupei Zhang1 Shidian Ma2 and Di Wu1

1School of Automobile and Traffic Engineering Jiangsu University Zhenjiang 212013 China2Automotive Engineering Research Institute Jiangsu University Zhenjiang 212013 China

Correspondence should be addressed to Haobin Jiang jianghbujseducn

Received 25 October 2016 Revised 6 December 2016 Accepted 19 December 2016 Published 14 February 2017

Academic Editor Francisco Valero

Copyright copy 2017 Yiding Hua 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

This paper establishes the kinematic model of the automatic parking system and analyzes the kinematic constraints of the vehicleFurthermore it solves the problem where the traditional automatic parking system model fails to take into account the time delayFirstly based on simulating calculation the influence of time delay on the dynamic trajectory of a vehicle in the automatic parkingsystem is analyzed under the transverse distance119863lateral between different target spaces Secondly on the basis of cloud model thispaper utilizes the tracking control of an intelligent path closer to human intelligent behavior to further study the Cloud Generator-based parking path tracking control method and construct a vehicle path tracking control model Moreover tracking and steeringcontrol effects of the model are verified through simulation analysis Finally the effectiveness and timeliness of automatic parkingcontroller in the aspect of path tracking are tested through a real vehicle experiment

1 Introduction

In recent years the ldquoparking difficultyrdquo problem in moderncities has become more and more notable Parking operationin a parking area near crowded and narrow urban roadsand community roads is not easy for many drivers [1 2]Therefore assisted parking driving technology has becomeone of the research hotspots in the car engineering field andinvolves knowledge about electromagnetism environment-aware sensors signal processing information fusion modelidentification automatic control and electric power steeringdirection and automotive electronics [3ndash5] Car makers paymuch attention to assisted parking system and successivelylaunched their own assisted parking driving system

Automatic parking and its control have been studiedwidely in the domestic and foreign automobile industrywhich has yielded fruitful achievements However some keytechnologies in the research of automatic parking have notbeen researched such as time delay of automatic parking pathtracking and control system

At present some research studies have been carried outon the parking path tracking control Some intelligent controlmethods are used more extensively such as fuzzy logic

controller neural network controller and genetic algorithms(GAs) Yasunobu and Murai [6] proposed a human experi-ence based on fuzzy logic control theory A fuzzy logic controlalgorithm was employed to design the parking controller [7]based on a model car test or simulation work Jenkins andYuhas [8] introduced a simplified neural network controllertrained on the basis of kinematics data Daxwanger andSchmidt [9] employed fuzzy and neural network schemesto develop a visually based model car backward-parkingcontroller with online autoparking steering angle commandTayebi and Rachid [10] designed a robust time-varying statefeedback parking controller by using the Lyapunov stabilityrule for a wheel mobile robot Nevertheless the fuzzy orneural network controller must be designed on the basisof expertise and trial-and-error work or on a complicatedvehicle dynamics model This is not convenient for practicalapplication At the same time neither one of the papersmentioned above addressed the issue about time delay ofautomatic parking path tracking

Time delay of automatic parking path tracking mainlyincludes the following aspects (1) transmission time delay ofmeasurement signal from the sensor to the control computer(2) time delay caused by the calculation of control law (3)

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 6590383 14 pageshttpsdoiorg10115520176590383

2 Mathematical Problems in Engineering

transmission time delay of control sign from computer toactuator (4) time delay of actuator (5) time required toestablish control [11] Time delay greatly affects the perfor-mance of the system [12] It might even cause instabilityof feedback control system and so the vehicle will fail totrack the ideal parking path This leads to the failure ofthe automatic parking system and even collision accidentswhich seriously affect the practicability and accuracy of theautomatic parking system

The influence of time delay on the automatic parkingsystem has been reflected in some existing research forexample Gutjahr and Werling [13] designed time optimaltrajectories for braking based on the prediction of the futurevehicle motion to compensate for actuator time delay Choiand Song [14] designed a fault detection andhandlingmethodfor automatic valet parking to resolve packet loss and timedelay of communication Song et al [15] designed a lateralcontroller to be robust enough to compensate for the noiseand time delay However the uncertainty of time delays wasnot considered in the papers mentioned above [16ndash18]

Cloud model is an uncertain transformation modelbetween the qualitative concept proposed by AcademicianLi Deyi and its quantitative value Characterized by thecoexistence of the uncertainty and certainty as well as thestability during the course of knowledge representationthe cloud model has reflected the fundamental principle ofthe biological evolution in nature [19ndash21] By virtue of itscharacteristics such as easy implementation of derivationprocess simple rules and strong robustness it has beenwidely utilized in the field of intelligent control and sub-jective evaluation [22] Normal cloud model is one of themost important among cloud models Due to its excellentmathematical characteristics it can describe a large numberof uncertain phenomena in different disciplines [23]

Thus this paper proposes a novel path tracking controllerbased on cloud model in order to solve the shortcomingsof the existing the fuzzy and neural network path trackingcontroller and to consider the uncertainties of time-delayproblems

The paper is organized as follows In the next section avehicle kinematics model is built and kinematic constraintsare analyzedThen the influence of time delay on the parkingtrajectory is simulated with MATLAB In order to meet thecomplex control requirements one-dimensional and two-dimensional single rules are organically combined to formthemultirule reasoningmodel Subsequently control strategyis verified with straight path and circular path trackingAt last a real vehicle experiment is presented to verifythe effectiveness and timeliness of automatic parking pathtracking controller based on cloud model The paper closeswith the conclusions and references

2 Analysis of the Vehicle Kinematics Modeland Kinematic Constraints

This paper establishes a kinematicmodel of vehicles based onwhich the path planningmethod of automatic parking systemis analyzed As shown in Figure 1 (119909119903 119910119903) and (119909119891 119910119891) are the

H

X

Y

h

W

L S

C

v

L2

120579

L1

120593

yr

yf

xf xr

Figure 1 Simplified car model for parking

midpoint coordinates of rear and front axles of the vehiclerespectively119882 is indicated as the wheel tread119867 is the widthof the road 1198711 and 1198712 are separately the width and lengthof the target parking space ℎ is the distance between themidpoint of the rear axle and the lateral barrier 119878 representsthe distance between the midpoint of the rear axle and thetail end of the barrier at the front of the target parking space120579 denotes the course angle of the vehicle 120593 is the Ackermanangle and clockwise direction is positive

The rear-wheel lateral velocity (vertical to wheel direc-tion) is zero and the vehicle motion equation in the verticaldirection can be obtained

119903 sdot sin 120579 minus 119903 sdot cos 120579 = 0 (1)

According to Ackerman steering geometry principleAckerman angle 120593 in the process of vehicle steering approxi-mates to the steering angle of midpoint of vehicle front axleThe central steering angle of the front axle of the vehicle islinearly proportional to steering wheel angle 120574 Hence

120574 = 119870 sdot 120593 (2)

where 119870 is proportional constantThe midpoint of the rear axle of the parked vehicle

is the origin of coordinates and the coordinate system isestablished as Figure 1 The positional relationship betweenthe midpoints of front and rear axles of the vehicle can beobtained Hence 119909119903 = 119909119891 + 119871 sdot cos 120579

119910119903 = 119910119891 + 119871 sdot sin 120579 (3)

Formula (3) is differentiated to obtain the velocity rela-tionship between the midpoints of front and rear axles of thevehicle Therefore

119903 = 119891 minus 119871 sdot sin 120579119903 = 119891 + 119871 sdot cos 120579 (4)

Mathematical Problems in Engineering 3

Substituting (4) into (1) we can obtain the vehiclekinematics relation

119891 sdot sin 120579 minus 119891 sdot cos 120579 minus 119871 sdot = 0 (5)

In addition at a certain moment of parking in a garagethe velocity of midpoint of the front axle along the directionof axis is as follows

119891 = V sdot cos (120579 + 120593) 119891 = V sdot sin (120579 + 120593) (6)

Substituting (6) into (5) we can obtain

= minusV sdot sin120593119871 (7)

Substituting (6) and (7) into (5) we can obtain the velocityof midpoint of front axle along the direction of axis

119903 = V sdot cos 120579 cos120593119903 = V sdot sin 120579 cos120593 (8)

Formulas (7) and (8) are denoted as the kinematicequation of vehicle Hence

119903 = V sdot cos 120579 cos120593119903 = V sdot sin 120579 cos120593 = minusV sdot sin120593119871

(9)

Formula (9) is utilized to integrate time 119905 and theobtained movement locus equations of midpoint of rear axleare

119909119903 (119905) = minus119871 sdot cot 120579 sdot sin(V sdot sin 120579119871 sdot 119905) 119910119903 (119905) = 119871 sdot cot 120579 sdot sin(V sdot sin 120579119871 sdot 119905) minus 119871 sdot cot 1205791199091199032 + (119910119903 + 119871 sdot cot 120579)2 = (119871 sdot cot 120579)2

(10)

According to the geometric relation between parametersof the vehicle and positions of each coordinate in Figure 1movement locus equations of four vehicle wheels and enve-lope points can be obtained Thereby the actual movementlocus of vehicle in the whole process from starting point toterminal point of parking can be calculated

Parking movement process should not only meet thevehicle kinematic and geometric characteristics but alsogive full consideration to external factors such as securityaccuracy and efficiency of parking In this section theinfluence of barrier in parking environment on the processof parking is analyzed

Based on the planned parking path namely the rearparking locus function the theoretical curvature 120588 of vehicle

at arbitrary point in the process of parking in a garage can begiven as

120588 = 11991010158401015840[1 + (1199101015840)2]32 (11)

According to the relation of Ackerman angle

tan120593 = 119871119877 (12)

where 119871 is wheel base 119877 is the radius of turning circle and119877 = 1120588According to formulas (11) and (12) Ackerman angle of

the vehicle at arbitrary point is

120593 = arctan( 119871 sdot 11991010158401015840[1 + (1199101015840)2]32) (13)

The movement locus of A B C and D can be obtainedbased on coordinates of midpoints of rear axle and theirmutual relationship Hence

119909119860 = 119909 + (119871119891 + 119871) sdot cos 120579 + 1198822 sdot sin 120579119910119860 = 119910 minus (119871119891 + 119871) sdot sin 120579 + 1198822 sdot cos 120579119909119861 = 119909 + (119871119891 + 119871) sdot cos 120579 minus 1198822 sdot sin 120579119910119861 = 119910 minus (119871119891 + 119871) sdot sin 120579 minus 1198822 sdot cos 120579119909119862 = 119909 minus 119871119903 sdot cos 120579 minus 1198822 sdot sin 120579119910119862 = 119910 + 119871119903 sdot sin 120579 minus 1198822 sdot cos 120579119909119863 = 119909 minus 119871119903 sdot cos 120579 + 1198822 sdot sin 120579119910119863 = 119910 + 119871119903 sdot sin 120579 + 1198822 sdot cos 120579

(14)

On the basis of the parking kinematic model establishedin the above section there are four positionswhere the dangerexists in the process of parallel parking (as shown in Figure 2)(a) collision between point C and the left boundary of thecarriageway or the collision between the boundary beyondthe current carriageway and vehicle running on the leftcarriageway in the process of parking (b) collision betweentail of vehicle which has not been parked and the barrieror vehicle at the front of right available parking space (c)collision between point B of the vehicle and the barrier afterwhich the vehicle is parked in the space (d) collision betweenpoint D as well as point A of the vehicle and the barrier afterwhich the vehicle is parked in the space

In order to ensure the safety in the parking processand reduce the risk of collision between the vehicle and


CB

A

D

a

C

B

A

D

b

CB

A

Dc

C

B

A

Dd

Figure 2 Parallel parking possible collisions

the barrier locus function needs to meet the followingconditions

When 119909119860 isin [0 119904119900 minus 1198712] 119910119860 lt ℎ0When 119909119860 isin [0 119904119900 minus 119871119903] 119910119860 lt ℎ0 + 1198711When 119909119861 isin [0 119904119900 minus 1198712] 119910119861 lt ℎ0When 119909119861 isin [0 119904119900 minus 119871119903] 119910119861 lt ℎ0 + 1198711When 119909 isin [0 119904119900 minus 119871119903] 119910119862 lt ℎ0 + 1198711 minus ℎWhen 119909 = 119904119900 minus 119871119903 119910119862 gt ℎ0When 119909 = 119904119900 minus 119871119903 119910119863 gt ℎ0

(15)

The analysis of the constraints of kinematics in theprocess of parking lays the foundation for vehicle trajectoryplanning and path tracking in the process of parking andensures the safety of the running vehicle during parking

3 Influence of Time Delay onthe Parking Trajectory

Due to the variation of arc length of vehicle wheels ineach process generated from the time delay this system cancalculate the passing arc length of wheels at the three stages ofparking on the basis of analysis of the difference in distance119863lateral between the vehicle and target parking space

In actual parking process there are time errors in eachstage especially at the points between any two stages Δ1198711Δ1198712 and Δ1198713 are assumed as the arc length deviations ateach stage under the average velocity speed of 2 kmh When119863lateral is 05m 1m and 15m respectively the influence oftime delay on the deviation of parking path is analyzed

By analyzing Figures 3ndash5 it can be known that evena short time delay can affect the parking trajectory Thered trajectory in the figure indicates the theoretically idealparking trajectory obtained from simulation and the bluetrajectory represents the actual parking trajectory affectedby time error The time delay most significantly influencesthe trajectory between the first and the second stages andthe influence between the second and the third stages is

less significant Furthermore we can realize that the greaterthe time delay the more obvious the trajectory error If thedeviation is excessive the tested vehicle cannot be accuratelyparked in the target parking space theoretically At presentonly the influence of the time delay between every two stageson the whole parking trajectory deviation is consideredHowever the effect of superposition of time delay on parkingtrajectory has not been analyzed

4 Control Strategy

This paper constructs the tracking control model and adjuststhe characteristic parameters of the cloud on the basis ofuncertain cloud reasoning model The controller can enablethe vehicle to rapidly and accurately track the expected path

41 Basic Principles of Cloud Model The cloud is a modelusing the linguistic value to represent the uncertainty con-version between a qualitative concept and its quantitativerepresentation Suppose119880 is a quantitative domain expressedin precise values and 119860 is a qualitative concept in 119880 If aquantitative value 119909 isin 119880 is a random realization of thequalitative concept 119860 and the membership of 119909 to 119860 120583(119909) isin[0 1] is a random number with a stable tendency 120583 119880 rarr[0 1] forall119909 isin 119880 119909 rarr 120583(119909) then the distribution of 119909 ondomain 119880 is called the cloud and each 119909 is called a clouddroplet [24] Particularly it is assumed that 1198771(1198641 1198642) can beexpressed as a random function obeying normal distributionwhere 1198641 is expected value and 1198642 is standard deviation If119909 (119909 isin 119880) and 120583(119909) satisfy the equations which can beexpressed as

119909 = 119877 (119864119909 119864119899) 119901 = 119877 (119864119899 119867119890) 120583 = 119890minus(12)((119909minus119864119909)119901)2

(16)

then the distribution of 119909 on domain 119880 is called the normalcloud [25] 120583 is the degree of membership cloud for inputvariable 119909 Cloud numerical features can be utilized to reflectthe overall characteristics of the qualitative concept expressed


0 2 4 6 8 10

minus2

minus1

0

1

2

3

4

5

x (m)

y(m

)

(a)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(b)

minus2

minus1

0 1 2 3 4 5 6 7 8 9 10

0

1

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x (m)

y(m

)

(c)

minus2

minus1

0 2 4 6 8 10

0

1

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3

4

5

x (m)

y(m

)

(d)

Figure 3 Effect of time delay on parking trajectory when 119863lateral = 05m (a) The time delay is 03 s between the first and the second stages(b) The time delay is 09 s between the first and the second stages (c) The time delay is 03 s between the second and the third stages (d) Thetime delay is 09 s between the second and the third stages

by the cloud model The cloud droplets have three numericalfeatures In (16) 119864119909 119864119899 and 119867119890 denote the expectationentropy and hyperentropy respectively which are used todescribe the numerical characteristics of cloud 119864119909 is theexpectation of cloud droplets in the distribution of thedomain and is the most typical point that represents thisqualitative concept 119864119899 is the uncertain measurement of thequalitative concept and reflects the relevance of fuzziness andrandomness119867119890 is the uncertainmeasurement of entropy andis determined by the fuzziness and randomness1198772 is assumed to be a two-dimensional random functionobeying normal distribution where 1198641 and 1198642 are expectedvalue and 1198643 and 1198644 are standard deviation Hence(119909 119910) = 1198772 (1198641 1198642 1198643 1198644)

(119901119909 119901119910) = 1198772 (1198643 1198644 119867119890119909 119867119890119910) 1205831015840 = 119890minus(12)[(119909minus1198641)21199012119909+((119910minus1198642)21199012119910)]

(17)

The cloud model constructed by data meeting for-mula (17) on drop(119909 119910 1205831015840) is a two-dimensional normalcloud model which is abbreviated as two-dimensional nor-mal cloud The data constructing this could model fordrop(119909 119910 1205831015840) is called two-dimensional cloud droplets [26]

Three normal clouds with different characteristics areshown in Figure 6 Compared with the three clouds it can befound that the smaller the value of 119864119899 is the more divergentthe cloud will be And it can also be found that the bigger thevalue of 119867119890 is the more divergent the cloud will be Normalcloud is composed of some cloud droplets which can reflectthe fuzziness Cloud model is not described through certainfunctions therefore to enhance the processing capacity foruncertainty

The process of reasoning about the uncertain rules ofthe cloud model is the target rules inferred and calculatedthrough the known conditions in a certain environment [27]Cloud Generator (CG) is the algorithm of cloud model The


0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

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minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

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minus2

minus1

y(m

)

x (m)

(d)


inputs of the generator are the three numerical character-istics The outputs are cloud droplets CG can realize themapping from qualitative data to quantitative data Thereare many CGs such as Forward Cloud Generator BackwardCloud Generator 119883 Condition Cloud Generator and 119884Condition Cloud Generator

Consider the one-dimensional cloudmodel of single-rulereasoning if 119909 then 119910 Its structure is shown in Figure 7(a)where 119909 condition cloud model is (119864119909 119864119899119909 119867119890119909) and 119910condition cloud model is (119864119910 119864119899119910 119867119890119910) When the rule isactivated more than once by the quantitative input value 119909CG119883 produces a set of 120583119894 values while 120583119894 generates a set of 119910119894values through CG119884 CG119883 can be presented by formula (16)CG119884 can be expressed as follows

119875 = 119877 (119864119899119910 119867119890119910)119910119894 = 119864119910 plusmn radicminus2 ln (120583119894) sdot 119875 (18)

Both 119883 and 119884 CGs are the basis of the construction ofcloud model uncertainty reasoning which form a single-rulegenerator after being connected The two-dimensional cloudmodel of single-rule reasoning is shown in Figure 7(b)

42 Path Tracking Control Model The basic idea of theparking path tracking control model is as follows the vehicleruns at a certain speed According to the known path andthe running parameters of the vehicle proper steering anglecontrol parameters are calculated The driving direction ofthe vehicle is changed and it runs along the preplannedpath In this process the steering control parameters areobtained through relevant inference based on the givenrunning parameters The mapping relation of individualone-dimensional or two-dimensional single-rule reasoningbetween input and output is simple and can hardly meetthe complex control requirements In this paper the above-mentioned one-dimensional and two-dimensional single


0 2 4 6 8 10

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(a)

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(b)

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minus1

y(m

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x (m)

(c)

0 2 4 6 8 10

0

1

2

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minus2

minus1

y(m

)

x (m)

(d)


rules are organically combined to form the multirule reason-ing model shown in Figure 8 This model can complete bothlinear andnonlinear reasoning requirements Amore difficultprediction at a higher level can be carried out by multirulededuction

PID controller has the advantages of simple structureconvenient adjustment and parameter tuning contact inengineering but the choice of traditional PID regulatorparametersmainly depends on repeated tests and experiencewhen the state of the object changes PID controller consistsof the proportion of unit P integral unit I and differential unitD In this paper in order to facilitate the comparison the PIDcontroller is designed PID controller of automatic parking isapplied to adjust actual steering wheel angle output by setting119896119901 119896119894 and 119896119889

The working process of the model is as follows thecontrolled quantity relevant parameters 1199091 1199092 and 1199093 areassumed as the input of the multirule generator according

to which the multirule derivation is carried out The resultsare applied to the adjustment of constant coefficients 119896PDand 119896I so as to adjust and control the output parameterrange Finally the weighted average output steering electricalmachine is controlled in real time

Input variables 1199091 1199092 and 1199093 activate different 119883 con-ditional cloud PCG119860119894119861119894 and CG119886119895 generating different 120583119894and 120583119895 respectively The cloud droplet is generated through119884 conditional cloud CG119888119894 and CG119887119895 Through the weightedaverage after adjustment of constant coefficient the inferen-tial results could be obtained and the generator process iscompleted

Inference rule of rule base of multidimensional cloudmodel is formulated

If 119860 119894 119861119895 and 119886119896 then 119877119894119895119896 where the value ranges of 119894 119895and 119896 are 1sim5 The corresponding rules of the front and rearpart are constructed in Figure 9The consequents of the ruleslibrary are listed in Table 1


Table 1 The consequents of the rules library

11987711119896 = (1198621 1198871) 11987712119896 = (1198621 119887119896) 11987713119896 = (1198621 119887119896) 11987714119896 = (1198622 119887119896) 11987715119896 = (1198622 119887119896)11987721119896 = (1198621 119887119896) 11987722119896 = (1198621 119887119896) 11987723119896 = (1198622 119887119896) 11987724119896 = (1198623 119887119896) 11987725119896 = (1198623 119887119896)11987731119896 = (1198622 119887119896) 11987732119896 = (1198622 119887119896) 11987733119896 = (1198623 119887119896) 11987734119896 = (1198624 119887119896) 11987735119896 = (1198624 119887119896)11987741119896 = (1198623 119887119896) 11987742119896 = (1198624 119887119896) 11987743119896 = (1198624 119887119896) 11987744119896 = (1198625 119887119896) 11987745119896 = (1198625 119887119896)11987751119896 = (1198624 119887119896) 11987752119896 = (1198625 119887119896) 11987753119896 = (1198625 119887119896) 11987754119896 = (1198625 119887119896) 11987755119896 = (1198625 119887119896)

5 10 15 20 25 30 350

010203040506070809

1

16 17 18 19 20 21 22 23 240

010203040506070809

1

5 10 15 20 25 30 350

010203040506070809

1

Ex = 20

En = 3

He = 01

Ex = 20

En = 1

He = 01

Ex = 20

En = 3

He = 03

Figure 6 Three examples of the normal cloud

119860 119894 means deviation of actual steering wheel angle andtheoretical steering wheel 119861119895 means deviation differential ofactual steering wheel angle and theoretical steering wheel 119886119896means deviation integral of actual steering wheel angle andtheoretical steering wheel 119862119894 means control outputs of PDtype two-dimensional cloud model mapper and 119887119895 meanscontrol outputs of I type two-dimensional cloud modelmapper

According to the contents discussed above it is knownthat if parking path is given Ackerman angle of the vehicleat arbitrary point on the path could be obtained Parkingkinematicmodel is taken as the controlled object of trajectorytracking control model In order to control the vehicle

tracking path through controlmodel the deviation120583betweenAckerman angle120593 of the vehicle and the theoretical value1205930 istaken as the input of the control model among which 1205930 canbe calculated through formula (13) By inputting the deviation120583 deviation integral 120583119894 and deviation differential 120583119888 into thecloud model generator the output controlled variable can berapidly adjusted based on variation of deviation so that thevehicle can be driven tracking the path

In actual operation in order to facilitate the measure-ment the variable input of the controlmodel is converted intoangle deviation value of steering wheel according to formula(13) To avoid the adverse effects of three aspects of time delaythat is (1) time delay in calculation of steering control law (2)transmission time delay of control sign from microprocessorto steering motor and (3) time required to establish controlalgorithm the range of deviation120583 is selected as (minus10 10)minus10minus5 0 5 and 10 are respectively taken as the five expectedvalues of the front five rules The numerical characteristicvalue of cloud is inputThe algorithm can reduce the negativeinfluence of time delay on the basis of guaranteeing highercontrol precision Hence

1198601 = 1198611 = 1198861 = (minus10 5 01) 1198602 = 1198612 = 1198862 = (minus5 3 01) 1198603 = 1198613 = 1198863 = (0 3 01) 1198604 = 1198614 = 1198864 = (5 3 01) 1198605 = 1198615 = 1198865 = (10 5 01)

(19)

Similarly the output of the control model is the adjust-ment amount of the steering angle and the numericalcharacteristic value is output Therefore

1198621 = 1198871 = (minus12 5 01) 1198622 = 1198872 = (minus6 3 01) 1198623 = 1198873 = (0 3 01) 1198624 = 1198874 = (6 3 01) 1198625 = 1198875 = (12 5 01)

(20)

Figure 10 is the structural diagram of path trackingcontrol model constructed in MATLABSimulink in thispaper Its running process is as follows the current stateparameters (119909119888 119910119888 120579119888 and 120593) of the midpoint of the rearaxle are obtained through a vehicle-mounted sensor Vectorquantity (119909119888 119910119888) is input into path tracking module throughwhich the ideal steering angle 120593119894 of the current reference


Weighted averagex y

(Ex EnxHex) (Ey EnyHey)

120583i drop(yi 120583i)CGx CGy

(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)

(Ex1 Enx1Hex1) (Ex2 Enx2Hex2)

PCGx1x2 CGy

(b)

Figure 7 Single positive cloud rule generator

Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1

(ExAn EnAnHeAn) (ExBn EnBnHeBn) (Eycn Encn Hecn)

(Exan Enan Hean) (Eybn Enbn Hebn)

PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1

(ExA1 EnA1HeA1) (ExB1 EnB2HeB2) (Eyc1 Enc1 Hec1)

(Ex1 En1He1) (Eyb1 Enb1 Heb1)

Figure 8 Multiple rule reasoning model

point could be calculated Furthermore Δ120593 is obtained bycomparing with the actual steering angle 120593 and taken asthe input of multirule generator to obtain the adjustmentamount of the steering angle in the parking process Withthe movement of the target point in the planning path Δ120593is repetitively calculated until the tracking is completed

Parking path tracking control model is established onMATLABSimulink simulation platform It is assumed thatthe vehicle is running at the speed of 2ms and the trackingsimulations on different paths are carried out respectively

The trajectory is assigned as 119910 = 119909 and 119909 isin (0 4) Thestarting point of the path is (0 0) and direction angle is 0∘The results of tracking simulation are shown in Figure 11where the full line is the prescribed path and the dotted lineis the tracking trajectory Ten sample points are selected tocalculate the standard deviation which is 015

The trajectory is specified as 1199092 + 1199102 = 1 The startingpoint of the path tracking is (minus2 0) and the direction angle is0∘The simulation results are shown in Figure 12 Twenty-five

sample points are selected to calculate the standard deviationwhich is about 01

According to Figures 11 and 12 as for the straight linetracking since the steering angle variation is 0 after therapid convergence at the starting point in tracking thefluctuation of the tracking points is less As for the circulartrajectory tracking the steering angle significantly changesThe tracking model can achieve rapid convergence and thetracking error is within the reasonable range

5 Experiment and Result Analysis

Real vehicle experiment of automatic parking consists of thefollowing components automatic parking control systemparking space scene detection equipmentWAVEBOOK512Hportable dynamic data acquisition system (including a note-book computer and DASlab80 data analysis software) Agi-lentU1620Aportable oscilloscope tape connectingwire andHaima Familia test vehicle


R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151

middot middot middot




middot middot middotmiddot middot middot middot middot middot

middot middot middotmiddot middot middotmiddot middot middot middot middot middot





















middot middot middot middot middot middot























Figure 9 Rule library

1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y

One-dimensional cloud model

Xa

XbY

Double-dimensional cloud model

Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus

Figure 10 Path tracking control model

Hardware circuit board of the automatic parking con-troller is developed according to the circuit principle dia-gram The parking controller is loaded on the vehicle tocarry out the test Experimental diagramof parking controllerhardware circuit board and parking controller loading test isshown in Figure 13

The ultrasonic sensor of the system is mainly arranged inthe front and rear as well as the right side of the vehicle Thefront and rear sensors are mainly short range radar and theright one is mainly long distance sensor which is shown inFigure 14

In the parking test field the test site is arranged accordingto the size of target parking space of the tested vehicledetermined in the second section The system is loaded on

the tested vehiclewith adaptability adjustmentThe automaticparking system is started for actual parking experimentwhich is shown in Figure 15

At the starting point of the vehicle parking the geometricmethod is utilized to plan the ideal parking trajectory Thesegmented curve is fit to the parking path Path trackingcontrol model is used to track the path The starting point is(71 23) and direction angle is 0∘ when 119863lateral = 05m

Three different lateral distances (119863lateral) of the modelcar are planned to execute the roadside parallel autoparkingexperiments Experimental results and pictures for a fixedvehicle speed of 5 kms are shown here for explanationExperimental results are listed in Table 2 for comparison withthe data from the PID controller and cloud model controller


Table 2 Path tracking control performance comparison

Control scheme

Experimental data for the following lateral distances (119863lateral)119863lateral 05m 119863lateral 10m 119863lateral 15mMinimum of 119884

error (m)Maximum of 119884

error (m)Minimum of 119884



error (m)Maximum of119884 error (m)

No control minus103 minus001 minus098 minus002 minus102 minus004PID control minus01 02 minus009 015 minus012 018Cloud modelcontrol minus001 006 minus002 008 minus002 01

0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)

Ideal valueSimulation value

Figure 11 Straight path tracking

minus2 minus1 0 1 2 3 4 5minus2

minus15minus1

minus050

051

152

x (m)

y (m

)


Figure 12 Circular path tracking

and without controller Four instantaneous motion picturesare shown in Figure 15

Experimental process without controller the wheel speedsignal outputted by the wheel speed sensor in the ABS(antilock brake system) system is collected The number ofeach square wave pulse signal outputted by the wheel is fixedin every circular that is the travelled distance (arc length)is certain So the number of pulses acquired can be usedto calculate the distance the wheel travels (arc length) Inthe absence of control there is a corresponding relationshipbetween the distance travelled (arc length) by the wheel and

the steering anglewhen119863lateral is fixedTherefore once the arclength of each stage is reached the microcontroller outputs afixed steering wheel angle to achieve the parking process Atthis time the trajectory can be calculated without controller

The trajectory of the vehiclersquosmotion the tracking error ofminimumof119884 error and the tracking error ofmaximumof119884error are shown in Figure 16 It can be observed that themodelvehicle follows the specified autoparking path backwards tothe target position with a small tracking errorTheminimumand maximum of 119884 error tracking of cloud model controllerare minus001 006 minus002 008 and minus002 01 respectively Itcan also be observed from Table 2 that the dynamic controlperformance of the cloud model controller is better than thatof PID controller The trajectory without controller reflectsthe adverse impact of time delay on automatic parkingSimultaneously it can be observed that the cloud modelcontroller can be better to effectively reduce the adverseimpact of time delay

Because the control parameters of the PID controller arefixed it is impossible to adjust quickly and rationally forthe uncertainty of the time delay The situation of lag andovershoot in the control process can be found In the case ofno controller the problem of delay will further be aggravatedin the process of automatic parking path trackingThe reasonis that all the time delays effect will be superimposed result-ing in increasing error and even leading to parking accidentfinallyHowever under the cloudmodel controller because ofthe establishment of cloudmodel reasoning rule base and thecombination of the one-dimensional and two-dimensionalcloud model mapper can solve the adverse effect of timedelay and improve the robustness of the system Furthermorecloud model controller can guarantee the stability of thesystem work effectively

6 Conclusions

In this paper the adverse impact of time delay on automaticparking is analyzed In order to solve the uncertainty of timedelays uncertain cloud reasoning model is proposed Theprecision of the position of the mapping point of midpoint atthe rear axle on the ground is taken as the controlled targetand the deviation between the ideal steering wheel angle andactual steering angle is taken as the input The single-rulereasoning of multidimensional cloud model is organicallyintegrated to design a multirule tracking control modelAccording to the known path the simulation experiment


Figure 13 Parking controller hardware circuit board and integrated controller loading test chart

(a) (b)

(c)

Figure 14 Ultrasonic sensor placement in test vehicle (a) Front of ultrasonic radar sensor layout (b) Rear ultrasonic radar sensor layout(c) Side ultrasonic radar sensor layout

21

43

Figure 15 Actual test of the automatic parking system


0 1 2 3 4 5 6 7

minus1minus05

005

115

225

Ideal trajectoryNo control

PID controlCloud model control

xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m

Figure 16 The test results of path tracking

of the parking tracking control model is carried out underconstant velocity The results show that the tracking controlmodel has a good tracking control performance and canquickly track the specified pathThe real vehicle experimentalresults show that cloud model controller can achieve rea-sonable tracking control accuracy for this automatic parkingsystemwith uncertainty of time delays Generally the experi-mental results are better than traditional PID controller Pathtracking control of automatic parking system with variablespeed will be the topic of future work

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work is financially supported by the National Natu-ral Science Fund (nos U1564201 and 51675235) and theCEEUSRO InnovativeCapital Project in Science-TechBureauof Jiangsu Province (no BY2012173)

References

[1] B Wang C Shao J Li D Zhao and M Meng ldquoInvestigatingthe interaction between the parking choice and holiday travelbehaviorrdquo Advances in Mechanical Engineering vol 7 no 6 pp1ndash11 2015

[2] M Roca-Riu E Fernandez and M Estrada ldquoParking slotassignment for urban distribution models and formulationsrdquoOmegamdashInternational Journal of Management Science vol 57pp 157ndash175 2015

[3] X Sun L Chen H Jiang Z Yang J Chen and W ZhangldquoHigh-performance control for a bearingless permanent-magnet synchronous motor using neural network inversescheme plus internal model controllersrdquo IEEE Transactions onIndustrial Electronics vol 63 no 6 pp 3479ndash3488 2016

[4] X Sun L Chen and Z Yang ldquoOverview of bearinglesspermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 60 no 12 pp 5528ndash5538 2013

[5] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[6] S Yasunobu and Y Murai ldquoParking control based on predictivefuzzy controlrdquo in Proceedings of the 3rd IEEE Conference onFuzzy Systems pp 1338ndash1341 June 1994


[7] M Sugeno T Murofushi T Mori T Tatematsu and J TanakaldquoFuzzy algorithmic control of a model car by oral instructionsrdquoFuzzy Sets and Systems vol 32 no 2 pp 207ndash219 1989

[8] R E Jenkins and B P Yuhas ldquoA simplified neural networksolution through problem decomposition the case of the truckbacker-upperrdquo IEEETransactions onNeuralNetworks vol 4 no4 pp 718ndash720 1993

[9] W A Daxwanger and G K Schmidt ldquoSkill-based visual park-ing control using neural and fuzzy networksrdquo in In Proceedingsof the IEEE International Conference on Systems Man AndCybernetics 1995 pp 1659ndash1664 ancouver Canada October1995

[10] A Tayebi and A Rachid ldquoTime-varying-based robust controlfor the parking problem of a wheeled mobile robotrdquo in Proceed-ings of the 13th IEEE International Conference on Robotics andAutomation pp 3099ndash3104 IEEE Minneapolis Minn USAApril 1996

[11] M-C Pai ldquoRBF-based discrete sliding mode control for robusttracking of uncertain time-delay systems with input nonlinear-ityrdquo Complexity vol 21 no 6 pp 194ndash201 2016

[12] H Wu ldquoDecentralised robust stabilisation of uncertain large-scale interconnected time-delay systems with unknown upperbounds of uncertaintiesrdquo International Journal of Systems Sci-ence vol 47 no 12 pp 2816ndash2826 2016

[13] B Gutjahr andMWerling ldquoAutomatic collision avoidance dur-ing parking and maneuveringmdashan optimal control approachrdquoin Proceedings of the 25th IEEE Intelligent Vehicles Symposium(IV rsquo14) pp 636ndash641 Dearborn Mich USA June 2014

[14] H Choi and B Song ldquoFault detection and handling forsensor and communication faults of an automatic valet parkingsystemrdquo in Proceedings of the 13th International Conference onControl Automation and Systems (ICCAS rsquo13) pp 1338ndash1341IEEE Gwangju Republic of Korea October 2013

[15] B Song D Kim and H Choi ldquoCooperative lateral control forautomatic valet parkingrdquo in Proceedings of the 11th InternationalConference on Control Automation and Systems (ICCAS rsquo11) pp567ndash570 Gyeonggi-do Korea October 2011

[16] X Miao and L Li ldquoAdaptive observer-based control for uncer-tain nonlinear stochastic systemswith time-delayrdquo Journal of theFranklin Institute vol 353 no 14 pp 3595ndash3609 2016

[17] Y Batmani and H Khaloozadeh ldquoOn the design of suboptimalsliding manifold for a class of nonlinear uncertain time-delaysystemsrdquo International Journal of Systems Science vol 47 no 11pp 2543ndash2552 2016

[18] H Lv Q Zhang and J Ren ldquoDelay-dependent H infin controlfor a class of uncertain time-delay singular Markovian jumpsystems via hybrid impulsive controlrdquo International Journal ofControl Automation and Systems vol 14 no 4 pp 939ndash9472016

[19] Q Zhang and P-C Li ldquoAdaptive grouping chaotic cloudmodel shuffled frog leaping algorithm for continuous spaceoptimization problemsrdquoKongzhi yu JueceControl and Decisionvol 30 no 5 pp 923ndash928 2015

[20] B Fu D G Li and M K Wang ldquoReview and prospect onresearch of cloud modelrdquo Application Research of Computersvol 28 no 2 pp 420ndash426 2011

[21] M-Z Liu X Zhang M-X Zhang M-G Ge and J HuldquoRescheduling decision method of manufacturing shop basedon profit-loss cloud modelrdquo Control and Decision vol 29 no 8pp 1458ndash1464 2014

[22] Y-J Zhang S-F Shao and N Julius ldquoCloud hypermutationparticle swarm optimization algorithm based on cloud modelrdquo

Pattern Recognition and Artificial Intelligence vol 24 no 1 pp90ndash96 2011

[23] G-W Zhang RHe Y LiuD-Y Li andG-S Chen ldquoEvolution-ary algorithm based on cloud modelrdquo Jisuanji XuebaoChineseJournal of Computers vol 31 no 7 pp 1082ndash1090 2008

[24] S Marston Z Li S Bandyopadhyay J Zhang and A GhalsasildquoCloud computingmdashThe business perspectiverdquo Decision Sup-port Systems vol 51 no 1 pp 176ndash189 2011

[25] C Guang B Xiaoying H Xiaofei et al ldquoCloud performancetrend prediction by moving averagesrdquo Journal of Frontiers ofComputer Science and Technology vol 6 no 6 pp 495ndash5032012

[26] Z Xiao W Song and Q Chen ldquoDynamic resource allocationusing virtual machines for cloud computing environmentrdquoIEEE Transactions on Parallel and Distributed Systems vol 24no 6 pp 1107ndash1117 2013

[27] D Xu S Yang and H Luo ldquoA fusion model for CPU loadprediction in cloud computingrdquo Journal of Networks vol 8 no11 pp 2506ndash2511 2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


transmission time delay of control sign from computer toactuator (4) time delay of actuator (5) time required toestablish control [11] Time delay greatly affects the perfor-mance of the system [12] It might even cause instabilityof feedback control system and so the vehicle will fail totrack the ideal parking path This leads to the failure ofthe automatic parking system and even collision accidentswhich seriously affect the practicability and accuracy of theautomatic parking system

The influence of time delay on the automatic parkingsystem has been reflected in some existing research forexample Gutjahr and Werling [13] designed time optimaltrajectories for braking based on the prediction of the futurevehicle motion to compensate for actuator time delay Choiand Song [14] designed a fault detection andhandlingmethodfor automatic valet parking to resolve packet loss and timedelay of communication Song et al [15] designed a lateralcontroller to be robust enough to compensate for the noiseand time delay However the uncertainty of time delays wasnot considered in the papers mentioned above [16ndash18]

Cloud model is an uncertain transformation modelbetween the qualitative concept proposed by AcademicianLi Deyi and its quantitative value Characterized by thecoexistence of the uncertainty and certainty as well as thestability during the course of knowledge representationthe cloud model has reflected the fundamental principle ofthe biological evolution in nature [19ndash21] By virtue of itscharacteristics such as easy implementation of derivationprocess simple rules and strong robustness it has beenwidely utilized in the field of intelligent control and sub-jective evaluation [22] Normal cloud model is one of themost important among cloud models Due to its excellentmathematical characteristics it can describe a large numberof uncertain phenomena in different disciplines [23]

Thus this paper proposes a novel path tracking controllerbased on cloud model in order to solve the shortcomingsof the existing the fuzzy and neural network path trackingcontroller and to consider the uncertainties of time-delayproblems

The paper is organized as follows In the next section avehicle kinematics model is built and kinematic constraintsare analyzedThen the influence of time delay on the parkingtrajectory is simulated with MATLAB In order to meet thecomplex control requirements one-dimensional and two-dimensional single rules are organically combined to formthemultirule reasoningmodel Subsequently control strategyis verified with straight path and circular path trackingAt last a real vehicle experiment is presented to verifythe effectiveness and timeliness of automatic parking pathtracking controller based on cloud model The paper closeswith the conclusions and references

2 Analysis of the Vehicle Kinematics Modeland Kinematic Constraints

This paper establishes a kinematicmodel of vehicles based onwhich the path planningmethod of automatic parking systemis analyzed As shown in Figure 1 (119909119903 119910119903) and (119909119891 119910119891) are the

H

X

Y

h

W

L S

C

v

L2

120579

L1

120593

yr

yf

xf xr

Figure 1 Simplified car model for parking

midpoint coordinates of rear and front axles of the vehiclerespectively119882 is indicated as the wheel tread119867 is the widthof the road 1198711 and 1198712 are separately the width and lengthof the target parking space ℎ is the distance between themidpoint of the rear axle and the lateral barrier 119878 representsthe distance between the midpoint of the rear axle and thetail end of the barrier at the front of the target parking space120579 denotes the course angle of the vehicle 120593 is the Ackermanangle and clockwise direction is positive

The rear-wheel lateral velocity (vertical to wheel direc-tion) is zero and the vehicle motion equation in the verticaldirection can be obtained

119903 sdot sin 120579 minus 119903 sdot cos 120579 = 0 (1)

According to Ackerman steering geometry principleAckerman angle 120593 in the process of vehicle steering approxi-mates to the steering angle of midpoint of vehicle front axleThe central steering angle of the front axle of the vehicle islinearly proportional to steering wheel angle 120574 Hence

120574 = 119870 sdot 120593 (2)

where 119870 is proportional constantThe midpoint of the rear axle of the parked vehicle

is the origin of coordinates and the coordinate system isestablished as Figure 1 The positional relationship betweenthe midpoints of front and rear axles of the vehicle can beobtained Hence 119909119903 = 119909119891 + 119871 sdot cos 120579

119910119903 = 119910119891 + 119871 sdot sin 120579 (3)

Formula (3) is differentiated to obtain the velocity rela-tionship between the midpoints of front and rear axles of thevehicle Therefore

119903 = 119891 minus 119871 sdot sin 120579119903 = 119891 + 119871 sdot cos 120579 (4)







= minusV sdot sin120593119871 (7)





(9)



(10)





120588 = 11991010158401015840[1 + (1199101015840)2]32 (11)


tan120593 = 119871119877 (12)



120593 = arctan( 119871 sdot 11991010158401015840[1 + (1199101015840)2]32) (13)



(14)




CB

A

D

a

C

B

A

D

b

CB

A

Dc

C

B

A

Dd




(15)







4 Control Strategy



119909 = 119877 (119864119909 119864119899) 119901 = 119877 (119864119899 119867119890) 120583 = 119890minus(12)((119909minus119864119909)119901)2

(16)



0 2 4 6 8 10

minus2

minus1

0

1

2

3

4

5

x (m)

y(m

)

(a)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(b)

minus2

minus1

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

x (m)

y(m

)

(c)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(d)




(17)





0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(d)








0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(d)











11987711119896 = (1198621 1198871) 11987712119896 = (1198621 119887119896) 11987713119896 = (1198621 119887119896) 11987714119896 = (1198622 119887119896) 11987715119896 = (1198622 119887119896)11987721119896 = (1198621 119887119896) 11987722119896 = (1198621 119887119896) 11987723119896 = (1198622 119887119896) 11987724119896 = (1198623 119887119896) 11987725119896 = (1198623 119887119896)11987731119896 = (1198622 119887119896) 11987732119896 = (1198622 119887119896) 11987733119896 = (1198623 119887119896) 11987734119896 = (1198624 119887119896) 11987735119896 = (1198624 119887119896)11987741119896 = (1198623 119887119896) 11987742119896 = (1198624 119887119896) 11987743119896 = (1198624 119887119896) 11987744119896 = (1198625 119887119896) 11987745119896 = (1198625 119887119896)11987751119896 = (1198624 119887119896) 11987752119896 = (1198625 119887119896) 11987753119896 = (1198625 119887119896) 11987754119896 = (1198625 119887119896) 11987755119896 = (1198625 119887119896)

5 10 15 20 25 30 350

010203040506070809

1

16 17 18 19 20 21 22 23 240

010203040506070809

1

5 10 15 20 25 30 350

010203040506070809

1

Ex = 20

En = 3

He = 01

Ex = 20

En = 1

He = 01

Ex = 20

En = 3

He = 03






1198601 = 1198611 = 1198861 = (minus10 5 01) 1198602 = 1198612 = 1198862 = (minus5 3 01) 1198603 = 1198613 = 1198863 = (0 3 01) 1198604 = 1198614 = 1198864 = (5 3 01) 1198605 = 1198615 = 1198865 = (10 5 01)

(19)


1198621 = 1198871 = (minus12 5 01) 1198622 = 1198872 = (minus6 3 01) 1198623 = 1198873 = (0 3 01) 1198624 = 1198874 = (6 3 01) 1198625 = 1198875 = (12 5 01)

(20)



Weighted averagex y



(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)


PCGx1x2 CGy

(b)


Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1



PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1













R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















= minusV sdot sin120593119871 (7)





(9)



(10)





120588 = 11991010158401015840[1 + (1199101015840)2]32 (11)


tan120593 = 119871119877 (12)



120593 = arctan( 119871 sdot 11991010158401015840[1 + (1199101015840)2]32) (13)



(14)




CB

A

D

a

C

B

A

D

b

CB

A

Dc

C

B

A

Dd




(15)







4 Control Strategy



119909 = 119877 (119864119909 119864119899) 119901 = 119877 (119864119899 119867119890) 120583 = 119890minus(12)((119909minus119864119909)119901)2

(16)



0 2 4 6 8 10

minus2

minus1

0

1

2

3

4

5

x (m)

y(m

)

(a)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(b)

minus2

minus1

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

x (m)

y(m

)

(c)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(d)




(17)





0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(d)








0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(d)











11987711119896 = (1198621 1198871) 11987712119896 = (1198621 119887119896) 11987713119896 = (1198621 119887119896) 11987714119896 = (1198622 119887119896) 11987715119896 = (1198622 119887119896)11987721119896 = (1198621 119887119896) 11987722119896 = (1198621 119887119896) 11987723119896 = (1198622 119887119896) 11987724119896 = (1198623 119887119896) 11987725119896 = (1198623 119887119896)11987731119896 = (1198622 119887119896) 11987732119896 = (1198622 119887119896) 11987733119896 = (1198623 119887119896) 11987734119896 = (1198624 119887119896) 11987735119896 = (1198624 119887119896)11987741119896 = (1198623 119887119896) 11987742119896 = (1198624 119887119896) 11987743119896 = (1198624 119887119896) 11987744119896 = (1198625 119887119896) 11987745119896 = (1198625 119887119896)11987751119896 = (1198624 119887119896) 11987752119896 = (1198625 119887119896) 11987753119896 = (1198625 119887119896) 11987754119896 = (1198625 119887119896) 11987755119896 = (1198625 119887119896)

5 10 15 20 25 30 350

010203040506070809

1

16 17 18 19 20 21 22 23 240

010203040506070809

1

5 10 15 20 25 30 350

010203040506070809

1

Ex = 20

En = 3

He = 01

Ex = 20

En = 1

He = 01

Ex = 20

En = 3

He = 03






1198601 = 1198611 = 1198861 = (minus10 5 01) 1198602 = 1198612 = 1198862 = (minus5 3 01) 1198603 = 1198613 = 1198863 = (0 3 01) 1198604 = 1198614 = 1198864 = (5 3 01) 1198605 = 1198615 = 1198865 = (10 5 01)

(19)


1198621 = 1198871 = (minus12 5 01) 1198622 = 1198872 = (minus6 3 01) 1198623 = 1198873 = (0 3 01) 1198624 = 1198874 = (6 3 01) 1198625 = 1198875 = (12 5 01)

(20)



Weighted averagex y



(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)


PCGx1x2 CGy

(b)


Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1



PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1













R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










CB

A

D

a

C

B

A

D

b

CB

A

Dc

C

B

A

Dd




(15)







4 Control Strategy



119909 = 119877 (119864119909 119864119899) 119901 = 119877 (119864119899 119867119890) 120583 = 119890minus(12)((119909minus119864119909)119901)2

(16)



0 2 4 6 8 10

minus2

minus1

0

1

2

3

4

5

x (m)

y(m

)

(a)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(b)

minus2

minus1

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

x (m)

y(m

)

(c)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(d)




(17)





0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(d)








0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(d)











11987711119896 = (1198621 1198871) 11987712119896 = (1198621 119887119896) 11987713119896 = (1198621 119887119896) 11987714119896 = (1198622 119887119896) 11987715119896 = (1198622 119887119896)11987721119896 = (1198621 119887119896) 11987722119896 = (1198621 119887119896) 11987723119896 = (1198622 119887119896) 11987724119896 = (1198623 119887119896) 11987725119896 = (1198623 119887119896)11987731119896 = (1198622 119887119896) 11987732119896 = (1198622 119887119896) 11987733119896 = (1198623 119887119896) 11987734119896 = (1198624 119887119896) 11987735119896 = (1198624 119887119896)11987741119896 = (1198623 119887119896) 11987742119896 = (1198624 119887119896) 11987743119896 = (1198624 119887119896) 11987744119896 = (1198625 119887119896) 11987745119896 = (1198625 119887119896)11987751119896 = (1198624 119887119896) 11987752119896 = (1198625 119887119896) 11987753119896 = (1198625 119887119896) 11987754119896 = (1198625 119887119896) 11987755119896 = (1198625 119887119896)

5 10 15 20 25 30 350

010203040506070809

1

16 17 18 19 20 21 22 23 240

010203040506070809

1

5 10 15 20 25 30 350

010203040506070809

1

Ex = 20

En = 3

He = 01

Ex = 20

En = 1

He = 01

Ex = 20

En = 3

He = 03






1198601 = 1198611 = 1198861 = (minus10 5 01) 1198602 = 1198612 = 1198862 = (minus5 3 01) 1198603 = 1198613 = 1198863 = (0 3 01) 1198604 = 1198614 = 1198864 = (5 3 01) 1198605 = 1198615 = 1198865 = (10 5 01)

(19)


1198621 = 1198871 = (minus12 5 01) 1198622 = 1198872 = (minus6 3 01) 1198623 = 1198873 = (0 3 01) 1198624 = 1198874 = (6 3 01) 1198625 = 1198875 = (12 5 01)

(20)



Weighted averagex y



(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)


PCGx1x2 CGy

(b)


Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1



PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1













R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10

minus2

minus1

0

1

2

3

4

5

x (m)

y(m

)

(a)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(b)

minus2

minus1

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

x (m)

y(m

)

(c)

minus2

minus1

0 2 4 6 8 10

0

1

2

3

4

5

x (m)

y(m

)

(d)




(17)





0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(d)








0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(d)











11987711119896 = (1198621 1198871) 11987712119896 = (1198621 119887119896) 11987713119896 = (1198621 119887119896) 11987714119896 = (1198622 119887119896) 11987715119896 = (1198622 119887119896)11987721119896 = (1198621 119887119896) 11987722119896 = (1198621 119887119896) 11987723119896 = (1198622 119887119896) 11987724119896 = (1198623 119887119896) 11987725119896 = (1198623 119887119896)11987731119896 = (1198622 119887119896) 11987732119896 = (1198622 119887119896) 11987733119896 = (1198623 119887119896) 11987734119896 = (1198624 119887119896) 11987735119896 = (1198624 119887119896)11987741119896 = (1198623 119887119896) 11987742119896 = (1198624 119887119896) 11987743119896 = (1198624 119887119896) 11987744119896 = (1198625 119887119896) 11987745119896 = (1198625 119887119896)11987751119896 = (1198624 119887119896) 11987752119896 = (1198625 119887119896) 11987753119896 = (1198625 119887119896) 11987754119896 = (1198625 119887119896) 11987755119896 = (1198625 119887119896)

5 10 15 20 25 30 350

010203040506070809

1

16 17 18 19 20 21 22 23 240

010203040506070809

1

5 10 15 20 25 30 350

010203040506070809

1

Ex = 20

En = 3

He = 01

Ex = 20

En = 1

He = 01

Ex = 20

En = 3

He = 03






1198601 = 1198611 = 1198861 = (minus10 5 01) 1198602 = 1198612 = 1198862 = (minus5 3 01) 1198603 = 1198613 = 1198863 = (0 3 01) 1198604 = 1198614 = 1198864 = (5 3 01) 1198605 = 1198615 = 1198865 = (10 5 01)

(19)


1198621 = 1198871 = (minus12 5 01) 1198622 = 1198872 = (minus6 3 01) 1198623 = 1198873 = (0 3 01) 1198624 = 1198874 = (6 3 01) 1198625 = 1198875 = (12 5 01)

(20)



Weighted averagex y



(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)


PCGx1x2 CGy

(b)


Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1



PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1













R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(d)








0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(d)











11987711119896 = (1198621 1198871) 11987712119896 = (1198621 119887119896) 11987713119896 = (1198621 119887119896) 11987714119896 = (1198622 119887119896) 11987715119896 = (1198622 119887119896)11987721119896 = (1198621 119887119896) 11987722119896 = (1198621 119887119896) 11987723119896 = (1198622 119887119896) 11987724119896 = (1198623 119887119896) 11987725119896 = (1198623 119887119896)11987731119896 = (1198622 119887119896) 11987732119896 = (1198622 119887119896) 11987733119896 = (1198623 119887119896) 11987734119896 = (1198624 119887119896) 11987735119896 = (1198624 119887119896)11987741119896 = (1198623 119887119896) 11987742119896 = (1198624 119887119896) 11987743119896 = (1198624 119887119896) 11987744119896 = (1198625 119887119896) 11987745119896 = (1198625 119887119896)11987751119896 = (1198624 119887119896) 11987752119896 = (1198625 119887119896) 11987753119896 = (1198625 119887119896) 11987754119896 = (1198625 119887119896) 11987755119896 = (1198625 119887119896)

5 10 15 20 25 30 350

010203040506070809

1

16 17 18 19 20 21 22 23 240

010203040506070809

1

5 10 15 20 25 30 350

010203040506070809

1

Ex = 20

En = 3

He = 01

Ex = 20

En = 1

He = 01

Ex = 20

En = 3

He = 03






1198601 = 1198611 = 1198861 = (minus10 5 01) 1198602 = 1198612 = 1198862 = (minus5 3 01) 1198603 = 1198613 = 1198863 = (0 3 01) 1198604 = 1198614 = 1198864 = (5 3 01) 1198605 = 1198615 = 1198865 = (10 5 01)

(19)


1198621 = 1198871 = (minus12 5 01) 1198622 = 1198872 = (minus6 3 01) 1198623 = 1198873 = (0 3 01) 1198624 = 1198874 = (6 3 01) 1198625 = 1198875 = (12 5 01)

(20)



Weighted averagex y



(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)


PCGx1x2 CGy

(b)


Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1



PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1













R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(a)

0 2 4 6 8 10

0

1

2

3

4

5

minus2

minus1

y(m

)

x (m)

(b)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(c)

0 2 4 6 8 10

0

1

2

3

4

5

6

minus2

minus1

y(m

)

x (m)

(d)











11987711119896 = (1198621 1198871) 11987712119896 = (1198621 119887119896) 11987713119896 = (1198621 119887119896) 11987714119896 = (1198622 119887119896) 11987715119896 = (1198622 119887119896)11987721119896 = (1198621 119887119896) 11987722119896 = (1198621 119887119896) 11987723119896 = (1198622 119887119896) 11987724119896 = (1198623 119887119896) 11987725119896 = (1198623 119887119896)11987731119896 = (1198622 119887119896) 11987732119896 = (1198622 119887119896) 11987733119896 = (1198623 119887119896) 11987734119896 = (1198624 119887119896) 11987735119896 = (1198624 119887119896)11987741119896 = (1198623 119887119896) 11987742119896 = (1198624 119887119896) 11987743119896 = (1198624 119887119896) 11987744119896 = (1198625 119887119896) 11987745119896 = (1198625 119887119896)11987751119896 = (1198624 119887119896) 11987752119896 = (1198625 119887119896) 11987753119896 = (1198625 119887119896) 11987754119896 = (1198625 119887119896) 11987755119896 = (1198625 119887119896)

5 10 15 20 25 30 350

010203040506070809

1

16 17 18 19 20 21 22 23 240

010203040506070809

1

5 10 15 20 25 30 350

010203040506070809

1

Ex = 20

En = 3

He = 01

Ex = 20

En = 1

He = 01

Ex = 20

En = 3

He = 03






1198601 = 1198611 = 1198861 = (minus10 5 01) 1198602 = 1198612 = 1198862 = (minus5 3 01) 1198603 = 1198613 = 1198863 = (0 3 01) 1198604 = 1198614 = 1198864 = (5 3 01) 1198605 = 1198615 = 1198865 = (10 5 01)

(19)


1198621 = 1198871 = (minus12 5 01) 1198622 = 1198872 = (minus6 3 01) 1198623 = 1198873 = (0 3 01) 1198624 = 1198874 = (6 3 01) 1198625 = 1198875 = (12 5 01)

(20)



Weighted averagex y



(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)


PCGx1x2 CGy

(b)


Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1



PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1













R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11987711119896 = (1198621 1198871) 11987712119896 = (1198621 119887119896) 11987713119896 = (1198621 119887119896) 11987714119896 = (1198622 119887119896) 11987715119896 = (1198622 119887119896)11987721119896 = (1198621 119887119896) 11987722119896 = (1198621 119887119896) 11987723119896 = (1198622 119887119896) 11987724119896 = (1198623 119887119896) 11987725119896 = (1198623 119887119896)11987731119896 = (1198622 119887119896) 11987732119896 = (1198622 119887119896) 11987733119896 = (1198623 119887119896) 11987734119896 = (1198624 119887119896) 11987735119896 = (1198624 119887119896)11987741119896 = (1198623 119887119896) 11987742119896 = (1198624 119887119896) 11987743119896 = (1198624 119887119896) 11987744119896 = (1198625 119887119896) 11987745119896 = (1198625 119887119896)11987751119896 = (1198624 119887119896) 11987752119896 = (1198625 119887119896) 11987753119896 = (1198625 119887119896) 11987754119896 = (1198625 119887119896) 11987755119896 = (1198625 119887119896)

5 10 15 20 25 30 350

010203040506070809

1

16 17 18 19 20 21 22 23 240

010203040506070809

1

5 10 15 20 25 30 350

010203040506070809

1

Ex = 20

En = 3

He = 01

Ex = 20

En = 1

He = 01

Ex = 20

En = 3

He = 03






1198601 = 1198611 = 1198861 = (minus10 5 01) 1198602 = 1198612 = 1198862 = (minus5 3 01) 1198603 = 1198613 = 1198863 = (0 3 01) 1198604 = 1198614 = 1198864 = (5 3 01) 1198605 = 1198615 = 1198865 = (10 5 01)

(19)


1198621 = 1198871 = (minus12 5 01) 1198622 = 1198872 = (minus6 3 01) 1198623 = 1198873 = (0 3 01) 1198624 = 1198874 = (6 3 01) 1198625 = 1198875 = (12 5 01)

(20)



Weighted averagex y



(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)


PCGx1x2 CGy

(b)


Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1



PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1













R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Weighted averagex y



(a)

Weighted averagey

(Ey EnyHey)

120583ix1

x2

drop(yi 120583i)


PCGx1x2 CGy

(b)


Wei

ghte

d av

erag

e

CGan

CGa1

CGbn kI

kI

x3

kPD

kPD

x2

x1

x3

x2

x1



PCGAnBn

PCGA1B1

CGcn

CGb1

CGc1













R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R555

R155R155

R551R551

R111

R511

A

R111

a

B

R151

R115

R151R151



















































1+

sx(t)

y(t)

120579(t)

fi(t)

f(xy)fi(t)

Path tracking

dudt

Derivative

Integrator

x y


Xa

XbY


Ki

Gain 1

Kpd

Gain 2

K

+Gain 3

+1

Out 1

2Out 2

3Out 3

4Out 4

Slowmodel

minus










Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Control scheme








0 05 1 15 2 25 3 35 40

051

152

253

354

x (m)

y (m

)




minus15minus1

minus050

051

152

x (m)

y (m

)








6 Conclusions




(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a) (b)

(c)


21

43



0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7

minus1minus05

005

115

225



xr (m)

yr

(m)

(a) 119863lateral = 05m



0 1 2 3 4 5 6 7

minus1

0

1

2

3 minus

xr (m)

yr

(m)

(b) 119863lateral = 10m

0 1 2 3 4 5 6 7minus1

minus050

051

152

253

35



xr (m)

yr

(m)

(c) 119863lateral = 15m



Competing Interests


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









path tracking control of automatic parking cloud model

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