pathplanningandtrajectorytrackingstrategyof autonomousvehicles · stream(ansaldobreda)...

Research ArticlePath Planning and Trajectory Tracking Strategy ofAutonomous Vehicles

Peng Han 1 and Bingyu Zhang 2

1School of Air Traffic Management Civil Aviation University of China Tianjin 300300 China2Railway Motive Power Department Tianjin Railway Technical and Vocational College Tianjin China

Correspondence should be addressed to Peng Han tpl_hp163com

Received 11 September 2020 Revised 8 January 2021 Accepted 21 January 2021 Published 31 January 2021

Academic Editor Haipeng Peng

Copyright copy 2021 Peng Han and Bingyu Zhang is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

With the development of global urbanization and the construction of regional urbanization residents around urban cities areincreasingly making demands on urban public transportation system A new kind of modern public transportation vehicle namedMulti-Articulated Guided Vehicle based on Virtual Track (MAAV-VT) with the advantages of beautiful smart energy con-servation and environmental protection is proposed in this paper which aims at optimizing the public transportation systembetween and within urban areas erefore concentrating on the general design and control strategy the main contents of thispaper are as follows At first the design concepts and key technologies of MAAV-VTare introduced It is the fusion of urban railtransit operation mode and advanced automotive technologies which have the characteristics of 100 low-floor medium to highvelocity medium to big capacity and low construction cost en as the core subsystem to guarantee the properties of self-guiding and trajectory tracking of the new vehicle this paper is focused on the control system based on the dynamics andkinematics model of the whole multi-articulated vehicle e multi-trace-points cooperative trajectory tracking control strategyon the basis of the circulation of feasible path generation method is proposed and the lateral controller is designed for trajectorytracking e process of feasible path generation is conducted once the tracking error exceeded A simulation platform is builtconsidering the mechanical properties of each vehicle element and the characteristic of articulated mechanism Finally thefunction of control system is validated e tracking error of each vehicle elements would be reduced to make sure the wholemulti-articulated vehicle moves along the preset virtual track

1 Introduction

As the rapid growth of private transportation the con-struction level of urban road system is hardly to satisfy thedemand of transportation which caused the phenomenonof traffic environment deterioration and the imbalancebetween supply and demand is is typical in big cities ofemerging developing countries Developing the urbanpublic transportation system energetically is an effectivecountermeasure to mitigate the transportation press eguided vehicle system could play an important role inpublic transportation system e guided vehicle is guidedby external medium and operated without the driverrsquoscontrol [1] e external medium which plays the role of

guidance could be both contactless form such as optics andmagnetics and contact form including guide wheel and rail[2] e ldquoCIVISrdquo [3] and ldquoPhileasrdquo [4] are two typicalguided vehicles which had been used for engineeringapplications As well the AutoTram Extra Grand is also along guided vehicle developed by the Fraunhofer IVIcombining the advantages of rail and road-bound trans-port systems A few similar vehicles and the lines in op-eration are listed in Table 1

e advanced automation control technologies in thefield of intelligent vehicles such as motor control tech-nology navigation technology identification technology ofoperation environment and trajectory tracking technologyare developed rapidly in recent years

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 8865737 11 pageshttpsdoiorg10115520218865737

First of all the guided system and environment per-ception technology provide the operation lines informationand environment information around the bus to the controlsystem e guided system is mainly constituted by one ormore of the navigation methods including GPS navigationinertial navigation laser navigation magnetic navigationand visual navigation e researches on environmentperception technology are concentrated on the access of theobstacle information and traffic signal information mea-sured by the camera radar and other related sensors Reid[7] proposed an automatic guidance method for tractor bydifferential dynamic positioning technology with GPS in realtime Will [8] Makela [9] and Bakambu [10] designed theguided systems for agricultural machinery and submarinesseparately Chan [11] focused on two types of magneticsystems to identify the characteristics of these two sensingsystems and to offer a comparison of their distinct featuresHopstock [12] developed a permanent magnetic pavementmarking tape Sections of varying magnetization wavelengthwere installed in a 230-m linear array Magnetic field profileswere determined at lateral displacements from the tape outto 09m Soslashgaard [13] developed a laser optic position de-termination system (PDS) by mounting it on an agriculturaltractor and a sowing machine Se [14] described a kind ofvision-based mobile robot localization and mapping algo-rithm using scale-invariant image features as landmarks inunmodified dynamic environments Giovanni [15] reviewedthree approaches to vision-based self-localization used in theRoboCup middle-size league competition and described theresults they achieved in the robot soccer environment forwhich they had been designed

e control system of guided vehicle should exportaccuracy control commands to make the vehicle move alongthe expected trajectory e control system integrates thesesubsystems including information perception task plan-ning behavior decision and execution organization Rea-sonable decision support architecture could increase thedecision-making ability of the whole system Several kinds ofdecision structures are used in intelligent vehicles Rose-nblatt [16] developed a distributed architecture for mobilenavigation system Brooks [17] described subsumption ar-chitecture for controlling mobile robots Layers of controlsystem were built to let the robot operate at increasing levelsof competence which are made up of asynchronous modulesthat communicate over low-bandwidth channels Higher-level layers subsumed the roles of lower levels by suppressingtheir outputs Other scholars also developed the architecturelike hierarchical intelligent control structure [18] multilevelstructure system [19 20] and so on [21 22]

e researches on trajectory tracking are very rich at thefield of intelligent vehicles e intelligent vehicles aretypically nonholonomic constraint system which has thecharacteristics of highly nonlinear and complexityus it isfairly difficult to propose the control strategy based onprecise modeling e commonly used tracking controlstrategy decouples the lateral and longitudinal motionusthe path following and velocity control of the system couldbe individually controlled e path following could bedivided into preview control and compensation control epreview control methods [23ndash25] calculate the targetquantities which control the motion state of the intelligentvehicles first based on the dynamics model and the kine-matics model e real time condition monitoring systemshould give a feedback of the actual lateral acceleration yawvelocity sideslip angle and so on e controller makes thedifference between the calculated target and actual quantitiesdecrease to approach the target path Xia [9] proposed anovel approach combining the sliding mode control andextended state observer (ESO) for attitude control of amissile model Saber [26] developed trajectory tracking andconfiguration stabilization for the vertical takeoff andlanding (VTOL) aircraft which addressed global configu-ration stabilization for the VTOL aircraft with a strong inputcoupling using a smooth static state feedback e com-pensation control methods monitor the target trajectory andactual location of the vehicle en the actuators whichcontrol the motion of the vehicle are controlled directly tomake the vehicle move along the trajectory e steeringmodel kinematics model and dynamics model are oftenused in the control systeme control method could also beclassified based on the control strategy such as PID methodoptimum control sliding-mode control model predictivecontrol fuzzy control and neural network control All thesemethods are applied in intelligent vehicle control system andobtained good results

In recent years there are some safer and more robustalgorithms in the field of path following control especiallyfor autonomous vehicle For example Zhang [27] investi-gated the path following control problem for four-wheel-independent-drive electric vehicles with consideration ofmodeling errors and complex driving scenarios whichemployed a suppress twisting second-order sliding mode(SOSM) control strategy to suppress the heavy chatteringissue existing in the traditional sliding mode control (SMC)In addition this team provided a new solution for pathfollowing control of autonomous ground vehicles whichformulated a standard model and represented in a Taka-giSugeno fuzzy form to deal with the time-varying nature of

Table 1 A few guided vehicles and the lines in operation

Vehicle type Guidance mode Lines in operationO-bahn (Mercedes Volvo) [5] Physical guidance (guide wheel in side) Adelaide Leeds EssenCIVIS (Iris bus) Optical guidance Las vegas Rouen Clermont-FerrandPhileas (APTS) Magnetic guidance EindhovenStream (Ansaldobreda) Magnetic guidance TriesteAeg (Cegelec) Induction cable guidance Channel tunnel ShuttleToer [6] Optical guidance Rouen

2 Mathematical Problems in Engineering

the vehicle speed [28] To enhance the vehicle safety thisteam further investigated the problem of steering actuatorfault diagnosis for automated vehicles based on the approachof model-based support vector machine (SVM) classification[29]

e development of these advanced technologies pro-vides proper environment for the development of guidedvehicle On this background a new vehicle concept based onvirtual track which combines the advantages of intelligentguided vehicle and rail transportation system is proposed Inthis work the job designing of multi-articulated guidedvehicle is proposed to solve the problem of urban publictransportation congestion e design concepts and generaltechnologies involved in the design process of the newvehicle are proposed To solve the problem of uncoordinatedmovement of front and rear carriages during the operationof the multi-articulated vehicle a collaborative trackingalgorithm based on dynamic and kinematic characteristicsare proposed e contributions and highlights of this workare summarized as follows

(1) ree layers framework of control system includingidentification and monitoring feasible trajectoryplanning and execution are introduced

(2) e dynamic and kinematic model between the jointconstraints and each carriage is built which ulti-mately formats the whole characteristic of multi-articulated vehicle

(3) A feasible path planning method and trajectorytracking strategy of the multi-articulated vehicle isproposed and verified by the constructed simulationplatform

2 DesignConcepts andGeneral Technologies ofthe MAAV-VT

e multi-articulated guided vehicle is a new publictransportation vehicle which positions as an importantcomponent of public transport system e constructionmode is able to take full advantages of the city space anddevelop a comprehensive transportation system with theconnection of other vehicles between and within urbanareas e MAAV-VT could undertake different responsi-bilities including urban agglomerations transportation andinner-city transportation It is an extension and supplementof the urban public transport system e MAAV-VT is thefusion of operation mode of rail transit and automobileemerging technologies It has the advantages of high velocityand big capacity like railway vehicles Furthermore it couldoperate without steel rail which helps decrease the con-struction cost and keep the road neat and beautiful edesign concepts of the MAAV-VT are described as thefollowing six parts which are shown in Figure 1

(1) Multi-articulated connection for big capacity thenew vehicle should have a big carrying capacity ofabout 15 thousand persons per hour with the op-eration speed of 40ndash50 kmh In this situation theguided vehicle should have at least three units for

carriage us the multi-articulated connectionmethod is used here is characteristic helps thevehicle possess the ability like traditional railwayvehicles

(2) Rubber tyre support to operate without rail the newvehicle should adopt rubber tyre support mode isis the guarantee to operate without steel rail ere isno need to destroy the existing road surface in theconstruction progress e integrity of the road isreserved which saves a lot cost and keeps the roadbeautiful

(3) Virtual track guidance for self-guiding the lack ofsteel rail also brings loss of physical constraints of thevehicle us the guidance function which is pro-vided by steel rail should be reformed Virtual trackguidance here is used for self-guiding Virtual track isdefined as a series of continuous or discrete signalband It could be set as electronic map magnetic nailor vision based band to guide the vehicle

(4) Specially designed structure and control strategy fortrajectory tracking the structure of the vehicle andthe tracking controller should also be speciallydesigned to realize self-guidinge vehicle elementsare connected by articulated mechanism e tra-jectory following ability of the whole vehicle shouldbe guaranteed by the specially designed structure anda suitable control strategy

(5) Hub motors for independent driving independentdriving is an effective method to design 100 low-floor vehicle Hob motors is very suitable here forincreasing the traction efficiency and simplifying thestructure of the machinery drive system Even moreimportant isolated control of each wheel is conve-nient for the trajectory following of each vehicleelement

(6) Mixed road rights with existing road vehicles toimprove efficiency the vehicle should operate withother road vehicles in unban cities e synergeticservice of all these vehicles leads to great efficiency ofurban transport system

According to the design concepts a few technologiesinclude the identification and guidance of virtual track self-guiding and trajectory tracking and the control strategy ofhubmotors are combined in order to realize the designationAs the core of the Multi-Articulated Guided Vehicle basedon Virtual Track the control system is the key to guaranteethe properties of self-guiding and trajectory tracking erole of control system and the relationship between it andother subsystems of the MAAV-VT are focused on thispaper

e framework of control system based on multilevelhierarchical theory is shown in Figure 2 ere are threelayers in the system including identification and moni-toring feasible trajectory planning and execution to as-sure the vehicle move along the given virtual track einput of the control system is the identified results ofvirtual tracks

Mathematical Problems in Engineering 3

e geometry information of the virtual track is per-ceived and identified first such as the location road slopegrade super elevation and obstruction e vehiclersquos real-time operation status including the attitude and locationshould also be monitored en feasible trajectory of thewhole multi-articulated vehicle is planned according to the

boundary conditions of the kinematics and dynamics modelFinally the execution layer is the key to make sure trajectorytracking and following through the accurately control ofeach motor e objective velocity and torque of each wheelfor tracking the planned feasible trajectory are calculatedseparately e control of hub motors is based on the vehicle

Yaw moment

Yaw velocity

Acceleration

Rotational speed

Operation conditions

Control objective

Vehicle operationenvironment monitoring

Economic and energy saving

Anti-skid

Location of virtual track

Actual body centerRoad slope gradeIdentification of the

virtual track Actual body attitudeRoad super elevation

Road obstruction

e preset trajectory correction

Actual trajectory of each vehicleReal-time turning center

Ideal trajectory of body centerOperation speed

Ideal body attitude e speed of each wheel

Actual velocityWheel dynamics

e dynamics of the vehicle

Rolling angle of the body center

Tire model

Vehicle unitsNormal load

Hinge jointsSlip rate

Submodel of the driven and control of the hub motors

Figure 2 e three layers of the control system

Low construction cost (without steel rail)

Beautiful smart energy conservation andenvironmental protection

Medium operation speed ( 40~50kmh)Design tasks

Big capacity (15000 persons per hour)

Self-guiding and trajectory tracking

Intelligent traction control

100 low-floor

Keytechnologies

Induction and identification of virtual trackSpecially designed structure and control

strategy for trajectory tracking

Hub motors for independent driving

Virtual track guidance for self-guidingDesign

concepts

Rubber tire support to operate without rail

Mixed road rights with existing roadvehicles to improve efficiency

Multiarticulated connection for big capacity

Figure 1 e design tasks and concepts


system dynamics the friction model between wheel andground and the control theory e three layers formed aclosed loop from the identification and monitoring to thetrajectory tracking control strategy

3 Kinematics Model of MAAV-VT System

31 Description of Virtual Track e multi-articulatedguided vehicle is operated by the guidance of preset virtualtrack on the road e virtual track in front and the actualposition and attitude of the guided vehicle should beidentified based on the optical identification system A seriesof coding graphs are used to describe the virtual track eQR codes have the advantages of uniqueness and veracityFurthermore the codes could store the information ofvirtual track including the location in front road slopegrade and super elevation e actual position and attitudeof each vehicle element relative to the virtual track could bemeasured and calculated by the visual system

32 Kinematics Model of Joint Constraints As is shown inFigure 3 the kinematics constraints are acted between twovehicle elementsemoving coordinate system are denotedas OiXiYiZi and OjXjYjZj Taking the revolute joint as anexample the revolute joint has limited the three degrees offreedom of parallel motion and the rotational motions of

horizontal and longitudinal Only the vertical rotationalmotion is retained e axles of revolution of the two vehicleelements are recorded as ωi

rarr and ωjrarr Two orthogonal vectors

are selected in the second element and recorded as ωj1rarr and

ωj2rarr separately e equations of kinematics constraints inthe hinge point are expressed as follows

C qi qj1113872 1113873

riJ minus rjJ

ωi middot ωj1

ωi middot ωj2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

0 (1)

where riJ and rjJ represent the position vector in the hingepoint of the first and second vehicle elements Substitutingthe transformation matrix of the ground fixed coordinateand car body following coordinate which named as Ai andAj into Equation (1) we can get

Ri + Ai riJrarr

minus Rj + Aj rjJrarr

1113872 1113873

Aiωirarr

middot Ajωj1rarr

Aiωirarr

middot Ajωj2rarr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

0 (2)

whereRi andRj represent the position vectors of the originalpoint of the two car bodies following coordinates e Jacobmatrix of the constraint equation is expressed as follows

_Cq qi qj1113872 1113873 zCzq

I minusAiriJGi minusI AjrjJGj

0 minusAiωiGi middot Ajωj1 0 minusAiωi middot Ajωj1Gj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

CqiCqj

1113960 1113961 (3)

4 Dynamics Model of MAAV-VT System

e dynamics model of the MAAV-VTsystem is the reflectionof its real service status e instructions of control objectivesadopted from the feasible path planning layer are transferred tothe dynamics model e control performances of hub motorsand the mechanical characteristics of articulated mechanismbetween each vehicle are considered in the dynamics modelus an integrated simulation platform is built to verify theaccuracy of the path-following method raised above estructure and parameters of each vehicle element are nearly thesame as the electromobiles with the characteristics of four-wheelactive steering and all-wheel-drive Each element is constitutedof four independent hub motors and two sets of steeringmechanisms e vehicle elements are articulated by a series ofhinges and formed a unified whole us the dynamics modelof each vehicle is studied first en taking into account of theconnection mechanism between each vehicle the modelingmethod based on loop variables is used

As is shown in Figure 4 the articulated mechanism isconstituted of kinematic pairs and force elements e ar-ticulated mechanism should satisfy the needs of connection

force and the freedom of motion between each vehicle el-ement Taking the spring-damper system as an example themechanical properties of the articulated mechanism areanalyzed

e function of spring-damper system here is used todecrease the impact of longitudinal impulse between eachvehicle e acting points of the force element system in thefront and rear vehicle element are recorded as Si and Sj As isshown in Equation (4) the distance vector rSiSj

between Si

and Sj could be calculated by the position vector

rSiSj Ri + AiriSi

minus Rj + AjrjSj1113874 1113875 (4)

_rSiSj Ri minus Ai

1113957rSiG _θi minus Rj minus Aj

1113957rSjG _θj1113874 1113875 (5)

where _rSiSjis the velocity vector and _θj is the first-order

derivative of the attitude in the generalized coordinateusthe acting force FS could be expressed as follows

FS K rSiSjminus r0SiSj

1113874 1113875 + C _rSiSj1113874 1113875 (6)


where K and C are the stiffness matrix and the dampingmatrix separately r0SiSj

is the initial distance vectore formof generalized force vector could be described as follows

Qi minusQj

FSeq

minusGTATi

1113957rTSiFSeq

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (7)

e multi-articulated vehicle should consider theinfluence of the connection between each conjoint ele-ment As is shown in Figure 5 each vehicle element isregarded as a basic unit which is the same with the dy-namic model above in the modeling en the connec-tion forces are taken into account e modeling methodof long-large train based on the loop variable developedby CHI [30] is used here e basic idea of this method isto regard each vehicle element as central integral unite kinematic equation could be expressed as

M euroX + C _X + KX P (8)

where M C and K represent the mass matrix dampingmatrix and stiffness matrix of the vehicle element euroX _X andX are the generalized acceleration vector velocity vectorand displacement vector P is the generalized load vector

Equation (8) changes to (9) while considering the actingforces F between each vehicle element

M euroX + C _X + KX P + F (9)

Equation (9) is expanded for each basic integral unitwhich is expressed as follows

MieuroXi + Ci

_Xi + KiXi Pi + Fi (10)

In this way the integral of the whole multi-articulated ve-hicle is divided into small integral units While giving the initial

x

y z

O

SiSj

rsi

risirjsj

rsj

zi

xjyj

zj

Oj

RjRi

Oi xiyi

Figure 4 Spring-damper system

Firstcar

Carriage2

Carriagei

Lastcar

Connectionforce

helliphellip helliphellip

Figure 5 e calculation method of long-large train based on the loop variable

Jij

rjJriJ

rjJ

wj wi

rijrj1

rj2

Oi

zixjyj

zj

Oj

Rj Ri

xy

z

O

xiyi

Figure 3 Kinematics constraints of the articulated mechanism


value and acting force of each vehicle element the dynamicmodels of the whole multi-articulated vehicle could be built

5 Feasible Path Planning

eaims of feasible path planning are to generate a feasible pathof the multi-articulated vehicle which meeting the requirementof kinematics constraints boundary conditions and charac-teristic of actuators e problem of path planning could berepresented as the planning of a series of movement attitudesand gestures between the original state and the terminal statee multi-articulated vehicle could operate according to theplanning gestures to achieve the goal of path following

51 Comparisons of the Path Planning Methods ree kindsof path planning methods are compared including spline curvefitting Bessel curve fitting and polynomial fitting As shown inFigure 6 the results of cubic spline interpolation quarticB-spline interpolation and polynomial interpolation are rela-tively similar with a high degree of coincidence e maindifference of each interpolation curve is reflected in the be-ginning end It can be seen from the local enlarged view of thecurve that the cubic spline interpolation curve has thesmoothest transition followed by B-spline In order tomake thepolynomial interpolate each interpolation point the degree ofpolynomial is higher to seven degreee transition of the curvein this section is less gentle than that of cubic spline andB-spline In addition to the initial node and the target node thefitting curve obtained by using the Bezier function does not passthrough other control points which are only used to control theshape of the fitting curve erefore the curve fitted by theBezier function is the smoothest but the disadvantage is that thefunction value of each control point and the tangent directioncannot be strictly controlled

e curvature radius slope curvature value and changerate of curvature value obtained by each method are shownas follows As shown in Figure 6(a) the slope change of eachcurve is relatively gentle and the results of cubic spline curveinterpolation B-spline interpolation and seventh polyno-mial interpolation are relatively similar and the slopechange of Bessel fitting curve is the least e curvatureradius of each curve has little difference and each curvegeneration method can better meet the requirement of theminimum curvature radius e curvature radius of eachcurve generated from the above initial position to the targetposition is all greater than 10m As can be seen fromFigure 6(c) the curvature of each curve changes continu-ously so the curves generated by each method can all realizethe constraint conditions on the rate of curvature changeproposed by the aforementioned actuator characteristics Asshown in Figure 6(d)) the curve curvature change rate ofCubic Spline interpolation and B-Spline interpolation didnot show large peaks and troughs but showed a stable changetrend which was more conducive to meeting the require-ment of curvature radius change rate e interpolationmethod of seventh degree polynomial had a larger curvaturechange rate at the boundary

us considering from the feasible path constraintcubic spline curve interpolation and B-spline interpolationcan basically meet the requirements of planning and feasiblepath e planned path meets the requirements of trackingpoint function values with continuous second derivative andcontrols the minimum curvature radius and the maximumradius of curvature change rate better However the firstderivative value namely the trace point velocity directioncannot be controlled e higher order polynomial cansatisfy the requirement of function value and derivativevalue at the control point but the curvature change rate ofthe generated curve is difficult to guarantee ereforecombining the advantages of piecewise function and poly-nomial interpolation a piecewise quartic polynomial in-terpolation method is selected to generate feasible paths forself-guided tram

52 Piecewise Quartic Polynomial Interpolation Methode physical quantities which describe the motion of multi-articulated vehicle are labelled as set C e set contains thelocation coordinates of each trace point (xObm

yObm) the in-

stantaneous turn center (xOi yOi

) and the articulated mech-anism(xJi

yJi) e attitudes yaw velocity of each vehicle

element steering angle and velocity of each wheel whichrecorded separately as φi ωi δijk and vijk are all included emapping function which is labelled as Γ from the planned pathto these physical quantities which control themotion features ofthe multi-articulated vehicle could be written as follows

xOi yOi

1113872 1113873φiωi δijk vrarr

Ji xJi

yJi1113872 11138731113966 1113967 Γ g(x) v

rarrObm

xObm yObm

1113872 11138731113966 1113967

(11)

where g(x) is the planned path and vrarr

Obmis the velocity of

each trace point Parameter m is the number of trace pointsParameter i is the number of vehicle elements Parameter j

and k represent the location of each wheel which areassigned as f or r on behalf of front and rear and l or r onbehalf of left and right Figure 7

Cubic spline and B-Spline interpolation are appropri-ately considered the kinematic characteristics of the multi-articulated vehicle e two curves could fit the functionvalues in trace points And they are all second differentiatedto control the radius of curvatureese advantages are goodto control the continuity of the path However the first-order derivative is uncontrollable which means the directionof the curve is uncertaine higher-degree polynomial is allright to control both the value and the first-order derivativein trace points but the curvature of planned curve is hard toguarantee us a method of segmental interpolation basedon quartic polynomial is proposed here

e polynomial equation in each segment is expressed asfollows

Ph(x) 11139444

j0ahjx

j (12)

us the first- and second-order derivatives areexpressed as the following equations


Phprime(x) 1113944

4

j1j middot ahjx

jminus 1 (13)

PhPrime(x) 1113944

4

j2j middot (j minus 1) middot ahjx

jminus 2 (14)

e characteristics of the segmental interpolation based onquartic polynomial should guarantee the continuity of functionvalues and the first and second derivatives in the boundarypoints

6 Verification of the TrajectoryTracking Strategy

e coordinate trajectory tracking strategy of the multi-articulated vehicle based on the circulation of feasible pathplanning is verified by the constructed simulation

platform e simulation platform is built based on thedynamics model and trajectory calculation model evalues of control variables are calculated according to thereplanned path and then transferred to the dynamicsmodel e actual motion trajectory and the kinetic pa-rameters are calculated e operation velocity is con-trolled by the first trace point which is preset at a mediumconstant value

e trajectory of lemniscate is preset as the virtual trackis kind of track is always used in vehicle handling andstability testing It is suitable to verify the trajectory trackingability of the multi-articulated vehicle considering theexecutability and stability e equation of the preset track isshown in the following equation

l 60lowast

cos(2ψ)

1113969

(15)

Cubic splinePolynomial interpolationBndashsplineBessel curve

5 10 15 20 250Longitudinal track position (m)

ndash06

ndash03

00

03

06

Slop

e of e

ach

curv

e 1 (m

)

(a)


ndash100

ndash50

0

50

100

Curv

atur

e rad

ius (

m)


(b)


ndash02

ndash01

00

01

Curv

atur

e val

ue 1

(m)


(c)


ndash006

ndash003

000

003

006Ch

ange

rate

of c

urva

ture


(d)

Figure 6 Comparison of results of different curve generation algorithms (a) slope of each curve (b) curvature radius (c) curvature value(d) change rate of curvature


eminimum radius of the lemniscate is 20meters whileψ 0∘ ere is a transition curve before the lemniscatetrack e transverse span and longitudinal span of the trackare about 140 meters and 40 meters separately

e black thick dash line represents the preset lemniscatetrack and the others are the actual trajectory each trace pointmoved As is shown in Figure 8 the trajectory of each vehicleis highly consistent with the preset track e controller hasdetected sixteen times of the condition that the position orattitude error exceeded the preset threshold value Under thecircumstances the controller would conduct the command offeasible path replanning to adjust the position and attitude to

follow the guidance in front of the virtual track e speed ofthe first trace point is constant at 6ms e actual speed ofeach wheel controlled by hub motors is exported as Figure 8e fluctuation of the velocity curves agrees with the time ofpath replanning which means that the hub motors execute ascontrol instructions e multi-articulated vehicle moved atthe right side of the lemniscate first e processes of pathreplanning for trajectory tracking are increased obviously Asa result the velocity curves and the actual trajectory of themulti-articulated vehicle are unsmooth than that of the leftside e whole variant trend of the velocity curves of eachvehicle is nearly the same However there is a difference of

Preset trajectorye front trace pointe second trace pointe third trace pointe fourth trace point

ndash60 ndash30 0 30 60 90ndash90Vertical coordinate (m)

ndash40

ndash20

0

20

Hor

izon

tal c

oord

inat

es (m

)

(a)

Le front wheelRight front wheelLe rear wheelRight rear wheel

ird vehicle

Second vehicle

First vehicle

18

21

2418

21

24

18

21

24

Velo

city

(km

h)

15 30 45 60 750Time (s)

(b)

Figure 8 e comparison of preset and motion trajectory and the velocity of each wheel

O1

O2

O3

J1

J2

vJ1

VJ2

Of1 Or1

Or2

Or3 J1

J2

Replanned feasible path

Trajectory of previous cycle

Preset virtual track

Figure 7 Feasible path planning


time phase to guarantee each vehicle element to path throughthe planned feasible curve in turns

e trajectory tracking error including the position error ofeach trace points and the attitude error of each vehicle isdisplayed in Figure 9 e variant trend of each trace points indifferent vehicle elements is nearly the samewhich proved greatfollowing features of the whole multi-articulated vehicle eposition error and the attitude error are fluctuated within 02meter and 002 rade tracking error could also show that thetracking performances of the first half of the preset lemniscatetrack are better than that of the second half

7 Conclusions

A new kind of modern public transportation vehicle namedMulti-Articulated Guided Vehicle based on Virtual Track(MAAV-VT) is described in this article It is a fusion of theoperation model of urban rail transit and advance automotivetechnology e following works are conducted in this articlecentered on the vehicle system

1 e design concepts and general technologies ofthe MAAV-VT are generalized which concludesusing rubber tire support to simplify the con-struction virtual track guide to realize self-guidepermanent magnet in-wheel motor drive to makeeach wheel independent and mixed road rights toincrease efficiency

(2) As the core technology of the multi-articulatedguided vehicle the feasible path planning methodbased on the kinematics model of MAAV-VT isanalyzed e expected position determinationmethod of MAAV-VT is proposed first to locate thevehicle en the boundary constraint conditionsare analyzed and the curve generation method isproposed to generate feasible path of the whole

vehicle Finally the trajectory tracking based on thecirculation of feasible path planning is proposedecirculation condition and terminal boundary of thecirculation are analyzed

(3) e dynamics model of the MAAV-VT system isbuilt to reflect its real service status and verify thetrajectory tracking strategy e results show that thecoordinate traction control strategy of the multi-articulated vehicle based on the circulation of fea-sible path planning has fairly good effects in thepreset lemniscate track

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work began from the original idea of tutor Prof ZhangWeihua who also provided a lot of technical support of thiswork is work was supported by the Fundamental Re-search Funds for the Central Universities (3122018C035)

References

[1] M Burke ldquoProblems and prospects for public transportplanning in Australian citiesrdquo Built Environment vol 42no 1 pp 37ndash54 2016

[2] A Kersys ldquoSustainable urban transport system developmentreducing traffic congestions costsrdquo Engineering Economicsvol 22 no 1 pp 5ndash13 2015

[3] B Furman S Ellis L Fabian et al ldquoAutomated transitnetworks (ATN) a review of the state of the industry andprospects for the futurerdquo MTI Report pp 12ndash31 2015

e second trace point

e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)

e fourth trace point

e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)

Figure 9 e position and attitude error of tracking trajectory


[4] O Michler R Weber and G Forster ldquoModel-based andempirical performance analyses for passenger positioningalgorithms in a specific bus cabin environmentrdquo in Pro-ceedings of the 2015 International Models and Technologies forIntelligent Transportation Systems (MT-ITS) pp 200ndash208IEEE Budapest Hungary January 2015

[5] E D Dickmanns ldquoDetailed visual recognition of road scenesfor guiding autonomous vehiclesrdquo in Advances in Real-TimeSystems pp 225ndash244 Springer Berlin Germany 2012

[6] T Deng and J D Nelson ldquoRecent developments in bus rapidtransit a review of the literaturerdquo Transport Reviews vol 31no 1 pp 69ndash96 2011

[7] J F Reid Q Zhang N Noguchi and M Dickson ldquoAgri-cultural automatic guidance research in North AmericardquoComputers and Electronics in Agriculture vol 25 no 1-2pp 155ndash167 2000

[8] J D Will F A C Pinto N Noguchi et al ldquoSensor fusionframework for heading determination using gps and inertialmeasurementrdquo in Proceedings of the 2000 ASAE Annual In-ternational Meeting Milwaukee WI USA July 2000

[9] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding modecontrol for missile systems based on an extended state ob-serverrdquo IET Control 7eory amp Applications vol 5 no 1pp 93ndash102 2011

[10] J N Bakambu V Polotski and P Cohen ldquoHeading-aidedodometry and range-data integration for positioning of au-tonomous mining vehiclesrdquo in Proceedings of the 2000 IEEEInternational Conference pp 279ndash284 Anchorage AK USASeptember 2000

[11] C Y Chan B Bougler D Nelson P Kretz H S Tan andW B Zhang ldquoCharacterization of magnetic tape and mag-netic marker as a position sensing system for vehicle guidanceand controlrdquo in Proceedings of the American ControlConference Chicago IL USA June 2000

[12] D M Hopstock and L D Wald ldquoWald verification of fieldmodel for magnetic pavement marking taperdquo IEEE Trans-actions on Magnetics vol 32 no 5 pp 5088ndash5090 1996

[13] H T Soslashgaard ldquoEvaluation of the accuracy of a laser opticposition determination systemrdquo Journal of Agricultural En-gineering Research vol 74 no 3 pp 275ndash280 1999

[14] S Se D G Lowe and J J Little ldquoVision-based global lo-calization and mapping for mobile robotsrdquo IEEE Transactionson Robotics vol 21 no 3 pp 364ndash375 2005

[15] G Adorni S Cagnoni S Enderle et al ldquoVision-based lo-calization for mobile robotsrdquo Robotics and AutonomousSystems vol 36 no 2-3 pp 103ndash119 2001

[16] J K Rosenblatt ldquoDAMN a distributed architecture formobile navigation-thesis summaryrdquo Journal of Experimentaland 7eoretical Aitificial Intelligence AAAI Press vol 9no 2-3 pp 339ndash360 1997

[17] R A Brooks ldquoA robust layered control system for a mobilerobotrdquo IEEE Journal on Robotics and Automation IEEEJournal of Robotics and Automation vol 2 no 1 pp 14ndash231986

[18] M Piaggio ldquoNon-hierarchical Hybrid Architecture for In-telligent robotsrdquo in Proceedings of ATAL Workshop on Agent7eories Architectures and Languages Paris France July 1998

[19] G N Saridis ldquoToward the realization of intelligent controlsrdquoProceedings of the IEEE vol 67 no 4 pp 1115ndash1133 2003

[20] J S Albus H G McCain and R LumiaNASANBS StandardReference Model for Telerobot Control System Architecture(NASREM) National Institute of Standards and TechnologyGaithersburg MD USA 1989

[21] T Le-Anh and M B M De Koster ldquoA review of design andcontrol of automated guided vehicle systemsrdquo EuropeanJournal of Operational Research vol 171 no 1 pp 1ndash23 2006

[22] J H Xin S M Li Q B Liao et al ldquoe application of fuzzylogic in exploration vehiclerdquo in Proceedings of the 4th In-ternational Conference on Fuzzy Systems and KnowledgeDiscovery vol 4 pp 199ndash203 Haikou China August 2007

[23] J Wang J Steiber and B Surampudi ldquoAutonomous groundvehicle control system for high-speed and safe operationrdquo inProceedings of the 2008 American Control Conferencepp 218ndash223 Seattle WA USA June 2008

[24] J Wit C D Crane and D Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21no 8 pp 439ndash449 2004

[25] M H Hebert Corpe and A Stentz Intelligent UnmannedGround Vehiclesautonomous Navigation Research at CarnegieMellon Springer Science amp Business Media Berlin Germany2012

[26] R Olfati-Saber ldquoGlobal configuration stabilization for theVTOL aircraft with strong input couplingrdquo IEEE Transactionson Automatic Control vol 47 no 11 pp 1949ndash1952 2002

[27] J Chen Z Shuai H Zhang and W Zhao ldquoPath followingcontrol of autonomous four-wheel-independent-drive electricvehicles via second-order sliding mode and nonlinear dis-turbance observer techniquesrdquo IEEE Transactions on Indus-trial Electronics vol 68 no 3 pp 2460ndash2469 2021

[28] A-T Nguyen C Sentouh H Zhang and J-C PopieulldquoFuzzy static output feedback control for path following ofautonomous vehicles with transient performance improve-mentsrdquo IEEE Transactions on Intelligent Transportation Sys-tems vol 21 no 7 pp 3069ndash3079 2020

[29] Q Shi and H Zhang ldquoFault diagnosis of an autonomousvehicle with an improved SVM algorithm subject to unbal-anced datasetsrdquo IEEE Transactions on Industrial Electronicsp 1 2020

[30] C H I Mao-Ru J Yi-Ping Z Wei-Hua et al ldquoSystem dy-namics of long and heavy haul trainrdquo Journal of Traffic andTransportation Engineering vol 11 no 3 pp 35ndash40 2011


First of all the guided system and environment per-ception technology provide the operation lines informationand environment information around the bus to the controlsystem e guided system is mainly constituted by one ormore of the navigation methods including GPS navigationinertial navigation laser navigation magnetic navigationand visual navigation e researches on environmentperception technology are concentrated on the access of theobstacle information and traffic signal information mea-sured by the camera radar and other related sensors Reid[7] proposed an automatic guidance method for tractor bydifferential dynamic positioning technology with GPS in realtime Will [8] Makela [9] and Bakambu [10] designed theguided systems for agricultural machinery and submarinesseparately Chan [11] focused on two types of magneticsystems to identify the characteristics of these two sensingsystems and to offer a comparison of their distinct featuresHopstock [12] developed a permanent magnetic pavementmarking tape Sections of varying magnetization wavelengthwere installed in a 230-m linear array Magnetic field profileswere determined at lateral displacements from the tape outto 09m Soslashgaard [13] developed a laser optic position de-termination system (PDS) by mounting it on an agriculturaltractor and a sowing machine Se [14] described a kind ofvision-based mobile robot localization and mapping algo-rithm using scale-invariant image features as landmarks inunmodified dynamic environments Giovanni [15] reviewedthree approaches to vision-based self-localization used in theRoboCup middle-size league competition and described theresults they achieved in the robot soccer environment forwhich they had been designed

e control system of guided vehicle should exportaccuracy control commands to make the vehicle move alongthe expected trajectory e control system integrates thesesubsystems including information perception task plan-ning behavior decision and execution organization Rea-sonable decision support architecture could increase thedecision-making ability of the whole system Several kinds ofdecision structures are used in intelligent vehicles Rose-nblatt [16] developed a distributed architecture for mobilenavigation system Brooks [17] described subsumption ar-chitecture for controlling mobile robots Layers of controlsystem were built to let the robot operate at increasing levelsof competence which are made up of asynchronous modulesthat communicate over low-bandwidth channels Higher-level layers subsumed the roles of lower levels by suppressingtheir outputs Other scholars also developed the architecturelike hierarchical intelligent control structure [18] multilevelstructure system [19 20] and so on [21 22]

e researches on trajectory tracking are very rich at thefield of intelligent vehicles e intelligent vehicles aretypically nonholonomic constraint system which has thecharacteristics of highly nonlinear and complexityus it isfairly difficult to propose the control strategy based onprecise modeling e commonly used tracking controlstrategy decouples the lateral and longitudinal motionusthe path following and velocity control of the system couldbe individually controlled e path following could bedivided into preview control and compensation control epreview control methods [23ndash25] calculate the targetquantities which control the motion state of the intelligentvehicles first based on the dynamics model and the kine-matics model e real time condition monitoring systemshould give a feedback of the actual lateral acceleration yawvelocity sideslip angle and so on e controller makes thedifference between the calculated target and actual quantitiesdecrease to approach the target path Xia [9] proposed anovel approach combining the sliding mode control andextended state observer (ESO) for attitude control of amissile model Saber [26] developed trajectory tracking andconfiguration stabilization for the vertical takeoff andlanding (VTOL) aircraft which addressed global configu-ration stabilization for the VTOL aircraft with a strong inputcoupling using a smooth static state feedback e com-pensation control methods monitor the target trajectory andactual location of the vehicle en the actuators whichcontrol the motion of the vehicle are controlled directly tomake the vehicle move along the trajectory e steeringmodel kinematics model and dynamics model are oftenused in the control systeme control method could also beclassified based on the control strategy such as PID methodoptimum control sliding-mode control model predictivecontrol fuzzy control and neural network control All thesemethods are applied in intelligent vehicle control system andobtained good results

In recent years there are some safer and more robustalgorithms in the field of path following control especiallyfor autonomous vehicle For example Zhang [27] investi-gated the path following control problem for four-wheel-independent-drive electric vehicles with consideration ofmodeling errors and complex driving scenarios whichemployed a suppress twisting second-order sliding mode(SOSM) control strategy to suppress the heavy chatteringissue existing in the traditional sliding mode control (SMC)In addition this team provided a new solution for pathfollowing control of autonomous ground vehicles whichformulated a standard model and represented in a Taka-giSugeno fuzzy form to deal with the time-varying nature of

Table 1 A few guided vehicles and the lines in operation

Vehicle type Guidance mode Lines in operationO-bahn (Mercedes Volvo) [5] Physical guidance (guide wheel in side) Adelaide Leeds EssenCIVIS (Iris bus) Optical guidance Las vegas Rouen Clermont-FerrandPhileas (APTS) Magnetic guidance EindhovenStream (Ansaldobreda) Magnetic guidance TriesteAeg (Cegelec) Induction cable guidance Channel tunnel ShuttleToer [6] Optical guidance Rouen





















Yaw moment

Yaw velocity

Acceleration

Rotational speed


Control objective



Anti-skid




Road obstruction








Tire model











100 low-floor

Keytechnologies





concepts














C qi qj1113872 1113873

riJ minus rjJ

ωi middot ωj1

ωi middot ωj2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

0 (1)


Ri + Ai riJrarr


1113872 1113873

Aiωirarr

middot Ajωj1rarr

Aiωirarr

middot Ajωj2rarr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

0 (2)


_Cq qi qj1113872 1113873 zCzq




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

CqiCqj

1113960 1113961 (3)






between Si


rSiSj Ri + AiriSi

minus Rj + AjrjSj1113874 1113875 (4)

_rSiSj Ri minus Ai


1113957rSjG _θj1113874 1113875 (5)




1113874 1113875 + C _rSiSj1113874 1113875 (6)




Qi minusQj

FSeq

minusGTATi

1113957rTSiFSeq

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (7)







MieuroXi + Ci



x

y z

O

SiSj

rsi

risirjsj

rsj

zi

xjyj

zj

Oj

RjRi

Oi xiyi


Firstcar

Carriage2

Carriagei

Lastcar

Connectionforce



Jij

rjJriJ

rjJ

wj wi

rijrj1

rj2

Oi

zixjyj

zj

Oj

Rj Ri

xy

z

O

xiyi










yObm) the in-





xOi yOi


Ji xJi

yJi1113872 11138731113966 1113967 Γ g(x) v

rarrObm

xObm yObm

1113872 11138731113966 1113967

(11)







Ph(x) 11139444

j0ahjx

j (12)



Phprime(x) 1113944

4

j1j middot ahjx

jminus 1 (13)

PhPrime(x) 1113944

4


jminus 2 (14)






l 60lowast

cos(2ψ)

1113969

(15)



ndash06

ndash03

00

03

06

Slop

e of e

ach

curv

e 1 (m

)

(a)


ndash100

ndash50

0

50

100

Curv

atur

e rad

ius (

m)


(b)


ndash02

ndash01

00

01

Curv

atur

e val

ue 1

(m)


(c)


ndash006

ndash003

000

003

006Ch

ange

rate

of c

urva

ture


(d)








ndash40

ndash20

0

20

Hor

izon

tal c

oord

inat

es (m

)

(a)


ird vehicle

Second vehicle

First vehicle

18

21

2418

21

24

18

21

24

Velo

city

(km

h)

15 30 45 60 750Time (s)

(b)


O1

O2

O3

J1

J2

vJ1

VJ2

Of1 Or1

Or2

Or3 J1

J2








7 Conclusions






Data Availability




Acknowledgments


References





e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)


e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)


















































Yaw moment

Yaw velocity

Acceleration

Rotational speed


Control objective



Anti-skid




Road obstruction








Tire model











100 low-floor

Keytechnologies





concepts














C qi qj1113872 1113873

riJ minus rjJ

ωi middot ωj1

ωi middot ωj2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

0 (1)


Ri + Ai riJrarr


1113872 1113873

Aiωirarr

middot Ajωj1rarr

Aiωirarr

middot Ajωj2rarr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

0 (2)


_Cq qi qj1113872 1113873 zCzq




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

CqiCqj

1113960 1113961 (3)






between Si


rSiSj Ri + AiriSi

minus Rj + AjrjSj1113874 1113875 (4)

_rSiSj Ri minus Ai


1113957rSjG _θj1113874 1113875 (5)




1113874 1113875 + C _rSiSj1113874 1113875 (6)




Qi minusQj

FSeq

minusGTATi

1113957rTSiFSeq

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (7)







MieuroXi + Ci



x

y z

O

SiSj

rsi

risirjsj

rsj

zi

xjyj

zj

Oj

RjRi

Oi xiyi


Firstcar

Carriage2

Carriagei

Lastcar

Connectionforce



Jij

rjJriJ

rjJ

wj wi

rijrj1

rj2

Oi

zixjyj

zj

Oj

Rj Ri

xy

z

O

xiyi










yObm) the in-





xOi yOi


Ji xJi

yJi1113872 11138731113966 1113967 Γ g(x) v

rarrObm

xObm yObm

1113872 11138731113966 1113967

(11)







Ph(x) 11139444

j0ahjx

j (12)



Phprime(x) 1113944

4

j1j middot ahjx

jminus 1 (13)

PhPrime(x) 1113944

4


jminus 2 (14)






l 60lowast

cos(2ψ)

1113969

(15)



ndash06

ndash03

00

03

06

Slop

e of e

ach

curv

e 1 (m

)

(a)


ndash100

ndash50

0

50

100

Curv

atur

e rad

ius (

m)


(b)


ndash02

ndash01

00

01

Curv

atur

e val

ue 1

(m)


(c)


ndash006

ndash003

000

003

006Ch

ange

rate

of c

urva

ture


(d)








ndash40

ndash20

0

20

Hor

izon

tal c

oord

inat

es (m

)

(a)


ird vehicle

Second vehicle

First vehicle

18

21

2418

21

24

18

21

24

Velo

city

(km

h)

15 30 45 60 750Time (s)

(b)


O1

O2

O3

J1

J2

vJ1

VJ2

Of1 Or1

Or2

Or3 J1

J2








7 Conclusions






Data Availability




Acknowledgments


References





e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)


e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)







































C qi qj1113872 1113873

riJ minus rjJ

ωi middot ωj1

ωi middot ωj2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

0 (1)


Ri + Ai riJrarr


1113872 1113873

Aiωirarr

middot Ajωj1rarr

Aiωirarr

middot Ajωj2rarr

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

0 (2)


_Cq qi qj1113872 1113873 zCzq




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

CqiCqj

1113960 1113961 (3)






between Si


rSiSj Ri + AiriSi

minus Rj + AjrjSj1113874 1113875 (4)

_rSiSj Ri minus Ai


1113957rSjG _θj1113874 1113875 (5)




1113874 1113875 + C _rSiSj1113874 1113875 (6)




Qi minusQj

FSeq

minusGTATi

1113957rTSiFSeq

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (7)







MieuroXi + Ci



x

y z

O

SiSj

rsi

risirjsj

rsj

zi

xjyj

zj

Oj

RjRi

Oi xiyi


Firstcar

Carriage2

Carriagei

Lastcar

Connectionforce



Jij

rjJriJ

rjJ

wj wi

rijrj1

rj2

Oi

zixjyj

zj

Oj

Rj Ri

xy

z

O

xiyi










yObm) the in-





xOi yOi


Ji xJi

yJi1113872 11138731113966 1113967 Γ g(x) v

rarrObm

xObm yObm

1113872 11138731113966 1113967

(11)







Ph(x) 11139444

j0ahjx

j (12)



Phprime(x) 1113944

4

j1j middot ahjx

jminus 1 (13)

PhPrime(x) 1113944

4


jminus 2 (14)






l 60lowast

cos(2ψ)

1113969

(15)



ndash06

ndash03

00

03

06

Slop

e of e

ach

curv

e 1 (m

)

(a)


ndash100

ndash50

0

50

100

Curv

atur

e rad

ius (

m)


(b)


ndash02

ndash01

00

01

Curv

atur

e val

ue 1

(m)


(c)


ndash006

ndash003

000

003

006Ch

ange

rate

of c

urva

ture


(d)








ndash40

ndash20

0

20

Hor

izon

tal c

oord

inat

es (m

)

(a)


ird vehicle

Second vehicle

First vehicle

18

21

2418

21

24

18

21

24

Velo

city

(km

h)

15 30 45 60 750Time (s)

(b)


O1

O2

O3

J1

J2

vJ1

VJ2

Of1 Or1

Or2

Or3 J1

J2








7 Conclusions






Data Availability




Acknowledgments


References





e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)


e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)

































Qi minusQj

FSeq

minusGTATi

1113957rTSiFSeq

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ (7)







MieuroXi + Ci



x

y z

O

SiSj

rsi

risirjsj

rsj

zi

xjyj

zj

Oj

RjRi

Oi xiyi


Firstcar

Carriage2

Carriagei

Lastcar

Connectionforce



Jij

rjJriJ

rjJ

wj wi

rijrj1

rj2

Oi

zixjyj

zj

Oj

Rj Ri

xy

z

O

xiyi










yObm) the in-





xOi yOi


Ji xJi

yJi1113872 11138731113966 1113967 Γ g(x) v

rarrObm

xObm yObm

1113872 11138731113966 1113967

(11)







Ph(x) 11139444

j0ahjx

j (12)



Phprime(x) 1113944

4

j1j middot ahjx

jminus 1 (13)

PhPrime(x) 1113944

4


jminus 2 (14)






l 60lowast

cos(2ψ)

1113969

(15)



ndash06

ndash03

00

03

06

Slop

e of e

ach

curv

e 1 (m

)

(a)


ndash100

ndash50

0

50

100

Curv

atur

e rad

ius (

m)


(b)


ndash02

ndash01

00

01

Curv

atur

e val

ue 1

(m)


(c)


ndash006

ndash003

000

003

006Ch

ange

rate

of c

urva

ture


(d)








ndash40

ndash20

0

20

Hor

izon

tal c

oord

inat

es (m

)

(a)


ird vehicle

Second vehicle

First vehicle

18

21

2418

21

24

18

21

24

Velo

city

(km

h)

15 30 45 60 750Time (s)

(b)


O1

O2

O3

J1

J2

vJ1

VJ2

Of1 Or1

Or2

Or3 J1

J2








7 Conclusions






Data Availability




Acknowledgments


References





e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)


e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)






































yObm) the in-





xOi yOi


Ji xJi

yJi1113872 11138731113966 1113967 Γ g(x) v

rarrObm

xObm yObm

1113872 11138731113966 1113967

(11)







Ph(x) 11139444

j0ahjx

j (12)



Phprime(x) 1113944

4

j1j middot ahjx

jminus 1 (13)

PhPrime(x) 1113944

4


jminus 2 (14)






l 60lowast

cos(2ψ)

1113969

(15)



ndash06

ndash03

00

03

06

Slop

e of e

ach

curv

e 1 (m

)

(a)


ndash100

ndash50

0

50

100

Curv

atur

e rad

ius (

m)


(b)


ndash02

ndash01

00

01

Curv

atur

e val

ue 1

(m)


(c)


ndash006

ndash003

000

003

006Ch

ange

rate

of c

urva

ture


(d)








ndash40

ndash20

0

20

Hor

izon

tal c

oord

inat

es (m

)

(a)


ird vehicle

Second vehicle

First vehicle

18

21

2418

21

24

18

21

24

Velo

city

(km

h)

15 30 45 60 750Time (s)

(b)


O1

O2

O3

J1

J2

vJ1

VJ2

Of1 Or1

Or2

Or3 J1

J2








7 Conclusions






Data Availability




Acknowledgments


References





e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)


e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)































Phprime(x) 1113944

4

j1j middot ahjx

jminus 1 (13)

PhPrime(x) 1113944

4


jminus 2 (14)






l 60lowast

cos(2ψ)

1113969

(15)



ndash06

ndash03

00

03

06

Slop

e of e

ach

curv

e 1 (m

)

(a)


ndash100

ndash50

0

50

100

Curv

atur

e rad

ius (

m)


(b)


ndash02

ndash01

00

01

Curv

atur

e val

ue 1

(m)


(c)


ndash006

ndash003

000

003

006Ch

ange

rate

of c

urva

ture


(d)








ndash40

ndash20

0

20

Hor

izon

tal c

oord

inat

es (m

)

(a)


ird vehicle

Second vehicle

First vehicle

18

21

2418

21

24

18

21

24

Velo

city

(km

h)

15 30 45 60 750Time (s)

(b)


O1

O2

O3

J1

J2

vJ1

VJ2

Of1 Or1

Or2

Or3 J1

J2








7 Conclusions






Data Availability




Acknowledgments


References





e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)


e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)




































ndash40

ndash20

0

20

Hor

izon

tal c

oord

inat

es (m

)

(a)


ird vehicle

Second vehicle

First vehicle

18

21

2418

21

24

18

21

24

Velo

city

(km

h)

15 30 45 60 750Time (s)

(b)


O1

O2

O3

J1

J2

vJ1

VJ2

Of1 Or1

Or2

Or3 J1

J2








7 Conclusions






Data Availability




Acknowledgments


References





e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)


e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)

































7 Conclusions






Data Availability




Acknowledgments


References





e front trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(a)


e third trace point

00

01

02

00

01

02

Posit

ion

erro

r of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(b)

ird vehicle

Second vehicle

First vehicle

ndash006

ndash003

000

003

ndash002

000

002

ndash002

000

002

Attit

ude e

rror

of t

rack

ing

traj

ecto

ry (m

)

20 40 60 800Time (s)

(c)































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