Microscopic Traffic Dynamics and Platoon Control Based on Bond

Microscopic Traffic Dynamics and Platoon Control Based on BondGraph Modeling

P. Kumar, R. Merzouki, B. Ould Bouamama, and H. Haffaf

Abstract— Modeling of traffic dynamic is important for thegood traffic management which leads to sustainable transport.Traffic models are classified based on level of details theyprovide as microscopic models and macroscopic models. Amicroscopic model of traffic flow describes the behavior ofindividual vehicle in response to motion of the vehicle precedingit, while, a macroscopic model describes the behavior of thetraffic as a whole, but the behavior of individual vehicle is notdescribed.

In the present work, we develop a microscopic model ofcar-following behavior of the vehicles and introduce the sub-microscopic model of traffic, in which the dynamic model ofeach vehicle is developed, which is not considered in most ofthe existing microscopic models. Then, a model based controlstrategy is proposed for the local control of the platoon of theintelligent autonomous vehicles (IAVs). This model based con-trol strategy analytically provides the calculation of necessaryeffort for the follower IAV to maintain the safe inter-distancewith the leader IAV.

I. INTRODUCTIONModeling of traffic dynamic is important for the good traf-

fic management. A good traffic model helps in reducing thecongestion and the pollution. Also, it helps in infrastructureoptimization by providing cooperative driving (platooning).When a transport system is equipped with information andcommunication technology (ICT) tools, it becomes intelli-gent transport system (ITS). Nowadays, IAVs can be used forthe platooning of vehicles in the confined space like containerterminal to improve the performance of the system. In thispaper, we model the microscopic traffic dynamic based onthe communication between vehicles.

Microscopic model describes both the space-time behaviorof the vehicles as well as their interactions at a high levelof detail (individually). These models are based on supposedmechanisms describing the process of one vehicle followinganother. The follower vehicle’s motion (position, speed andacceleration) is determined according to the motion of theleader vehicle.

Microscopic models are also called ’Car-following’ mod-els, in which leader vehicle influences the driving behavior of

P. Kumar, R. Merzouki and B. Ould-Bouamamaare with the LAGIS, UMR CNRS 8219,Ecole Polytechnique Universitaire deLille 1, Avenue Paul Langevin, 59655Villeneuve d’Ascq, Lille, France(e-mail: [email protected];[email protected];[email protected])H. Haffaf is with the Faculty of Science,Computer Science Department Oran UniversityEl MNaouar, Oran, Algeria (e-mail:[email protected])

the follower vehicle. Various car-following models have beendeveloped since early 1950s. In 1953, Pipes [1] developedsafe-distance car-following models describe the dynamicsof a vehicle in relation to its leading vehicle. Other safe-distance models are also presented in the contributions:Forbes (1959) [2] presented an improved safe-distance modelwith taking into consideration the driver reaction time. Asimilar approach was proposed by Kometani (1959) [3]assuming that vehicle separation is proportional to both speedof the subject vehicle and the leading vehicle. Gipps (1981)[4] derived the model by setting the limits of performanceof the driver and vehicle.

In 1961, Gazis, Herman and Rothery [5] proposed ageneric stimulus-response model. The model is known asGHR (Gazis-Herman-Rothery) model. GHR model is themost well known model in traffic literature. The modelsdescribe acceleration and deceleration response of a followervehicle due to driving action of the leader vehicle. Anothercar-following model which is known as optimal velocitymodel (OVM) was proposed by Bando (1995) [6]. In thismodel, the legal velocity function is introduced, which is afunction of the distance headway between the leader and thefollower vehicles. Li (2008) [7] presented a modified OVMby adding an acceleration-adjustment term.

For platooning of vehicles some recent contributions in-clude; Yi (2005) [8] proposed an impedance control sys-tem utilizing a spring-damper system for vehicle platoon.Avanzini (2009) [9] proposed a global platoon controlstrategy, supported by inter-vehicle communications. Contet(2009) [10] proposed a local control approach to linearplatoons for the control of inter-vehicle distance and commontrajectory matching.

In the above literature, the dynamic model of the vehicleis not considered in most of the model. We propose a modelwhich describes the dynamics of car-following behaviorincluding the dynamics of individual vehicle and proposea model based local control of the platoon of vehicles. Forthis, we use a energy based graphical approach named bondgraph (Mukherjee 2006) [11]. The characteristics of the bondgraph modeling approach are as following:

• The Bond graph modeling is based on the power transferprinciple between the different elements of the studiedsystem. The system is modeled using inertia element I,compliance element C, and dissipative element R.

• The mathematical equations (differential equations) canbe deduced systematically from the Bond graph model.

• This modeling technique allows to strongly simplifythe analysis techniques and to calculate the formal

Proceedings of the 16th International IEEE Annual Conference onIntelligent Transportation Systems (ITSC 2013), The Hague, TheNetherlands, October 6-9, 2013

WeC6.5

978-1-4799-2914-613/$31.00 ©2013 IEEE 2349

expressions of the control laws, offering a physicalcomprehension of the obtained properties.

• The Bond graph is very powerful to model in a unifiedway the physical systems of various natures and inde-pendently of the considered field (mechanical, electrical,thermal etc.).

A. Paper Organization

In Section II, we develop the bond graph model ofthe traffic system, starting from the sub-microscopic modelto microscopic model. In Section III, the platoon controlstrategy is described and the simulation results are discussedin Section IV. The conclusion and future work is describedin Section V.

II. BOND GRAPH MODELING

A. Sub-microscopic Model

At sub-microscopic level, we develop two dimensionalmodel of an IAV named RobuCar as shown in Fig. 1,the RobuCar has four traction wheels actuated with fourindependent DC motors. There are two steering systems, firstfor the two front wheels and second for the two rear wheels.The vehicle is equipped with an inertial sensor to measureits longitudinal, lateral and yaw speeds. Also, sensors aremounted to measure the angular speed of the each wheeland the current drawn by the each motor.

Fig. 1: IAV RobuCar at LAGIS

In Fig. 2, we can see the schematic top view of the IAVin which x-y is body fixed frame and X-Y is inertial frame.The orientation of x axis with respect to X axis is given byθ. The dimensions of the vehicle are denoted by a, b and c,while G represents the centre of mass (CM) of the vehicle towhich x-y frame is fixed. Angle α denotes the steering angleof the wheel. Flj and Fcj (where j is wheel number =1, 2,3 or 4) denote the longitudinal force and the cornering forcetransmitted to the wheel respectively.

For the modeling of the IAV following modeling assump-tions are made: (i) the vehicle moves in a plane surface(ii) the road is uniform and the suspension, roll and pitchdynamics are not considered (iii) each wheel is independentlydriven by DC motor and (iv) all the four wheels are steerable.Finally, we consider the following dynamics: (i) tractionactuator, slip and steering dynamics of the wheel and (ii)longitudinal, lateral and yaw dynamics of the vehicle body(CM). The word bond graph for the considered dynamics ofthe vehicle is shown in Fig. 3.

Fig. 2: Schematic diagram of the IAV

In the wheel-j dynamics part, voltage source providesvoltage Umj and current ij to the electrical part of the motor,which gives output (voltage Uj) to the mechanical part ofthe motor. The output of the motor (torque τj and angularspeed ωj) and the effect of steering and slip dynamics (forceFsj and slip velocity vsj), generate wheel velocity vwj andforce Fj , which is transmitted to the vehicle body. Thedynamics of the four wheels generate longitudinal, lateraland yaw dynamics of the vehicle body. Pathak (2008) [12]and Loureiro (2012) [13], developed the dynamic model ofan autonomous vehicle using the bong graph technique.

Fig. 4: Considered scheme of jth motor and wheel system

The vehicle is composed of four independent quarters ofvehicles (wheel-1, wheel-2, wheel-3 and wheel-4). Let usstart with the dynamics of wheel-j (j=1, 2, 3 and 4). Fig.4 shows the considered scheme of the motor and wheelsystem. The corresponding bond graph model of the wheel-j dynamics is shown in Fig. 5. In electrical part of themotor, Umj , Imj , Rmj and kj represent voltage, inductance,resistance and torque constant of the motor respectively. Inmechanical part of the motor, Iaj and Raj represent polarmoment of inertia and friction of the wheel-axle respectively.Angle α is the steering angle and r is the radius of thewheel. Rxj and Ryj represent the slip contribution in x and ydirection respectively. The full headed arrows correspondingto current ij in motor and angular speed of the wheelaxle ωj represent the sensors to measure the value of thecorresponding parameter.

The dynamic equations of the system are systematicallydeduced from the bond graph model in Fig. 5, where enand fn are the effort and flow in the corresponding bond

978-1-4799-2914-613/$31.00 ©2013 IEEE 2350

Fig. 3: Word bond graph of the vehicle

Fig. 5: Bond graph model of the motor and wheel dynamics

number n (n=1, 2, 3...). The state-space equations of thesystem are derived by applying junction law correspondingto bonds 1,2,3,4 and bonds 5,6,7,8 as following:

p︷ ︸︸ ︷[pmj

paj

]=

A︷ ︸︸ ︷[−Rmj

Imj− kj

Iajkj

Imj−Raj

Iaj

] p︷ ︸︸ ︷[pmj

paj

]

+

B︷ ︸︸ ︷[1 00 −1

] u︷ ︸︸ ︷[Umj

rFlj

]q︷ ︸︸ ︷[ijωj

]=

C︷ ︸︸ ︷[1

Imj0

0 1Iaj

] p︷ ︸︸ ︷[pmj

paj

](1)

The longitudinal and lateral speeds of the wheel xwj andywj respectively in conjunction with wheel’s spinning speedgenerate the longitudinal and lateral slip speeds xsj andysj respectively, and the dynamic relations can be derivedfrom the junction law corresponding to bonds 9,10,11,12 andbonds 17,18,19 as given below:

xsj = rωj − xwj cosα− ywj sinα (2)

ysj = −xwj sinα+ ywj cosα (3)

The longitudinal force Flj and cornering force Fcj arefunctions of the longitudinal and lateral slip speeds respec-tively and the dynamic equations can be derived from the

junction law corresponding to bonds 10,13 and bonds 17,20as given below:

Flj = Rxj xsj (4)

Fcj = Ryj ysj (5)

For small value of slip and not considering wheel camber,the values of Rxj and Ryj can be given as (Drozdz 1991)[14]: Rxj =

Cx

x and Ryj =Cy

x . The coefficients Cx and Cy

are dependent on vertical wheel load and x is the velocity ofCM of the vehicle in x direction. Fxj and Fyj are the forcesgenerated by the wheel in x and y directions respectively,and are transmitted to the body of the vehicle in x and ydirections respectively, the equations can be derived fromthe junction law corresponding to bonds 14,15,16 and bonds21,22,23 as given below:

Fxj = Flj cosα− Fcj sinα (6)

Fyj = Flj sinα− Fcj cosα (7)

The complete bond graph model of the IAV considering allthe dynamics is shown in Fig. 6. Symbols m and J representmass and polar moment of inertia of the IAV respectively.The dimensions of the IAV are denoted by a, b and c inmodulous of transformer elements. The full headed arrowscorresponding to the speeds (x, y, X , Y and θ) of the

978-1-4799-2914-613/$31.00 ©2013 IEEE 2351

Fig. 6: Complete bond graph model of the IAV

vehicle’s CM, represent the sensors to measure the valueof the corresponding parameter.

In Fig. 6, the longitudinal and lateral motions of the fourwheels are transformed to longitudinal, lateral and angularmotion of the vehicle body. The dynamic relations can bededuced from this bond graph. By applying junction law atlongitudinal, lateral and yaw dynamics junctions, we get thefollowing equations:

mx = Fx1 + Fx2 + Fx3 + Fx4 +mθy (8)

my = Fy1 + Fy2 + Fy3 + Fy4 −mθx (9)

Jθ = (Fy1 + Fy2) a− (Fy3 + Fy4) b− (Fx1 − Fx2 − Fx3 + Fx4) c (10)

The dynamic relations for longitudinal, lateral and yawmotions of the IAV are given by equations (8), (9) and (10)respectively. The velocity of the CM in inertial frame X-Y is obtained by the transformation of x-y frame by angleθ. In this way, now we have the complete two dimensionaldynamic model of an IAV.

B. Microscopic Model

At microscopic level of traffic flow, the interaction be-tween the vehicles is observed. A microscopic model oftraffic is ’car-following’ model, in which leader vehicle

influences the driving behavior of the follower vehicle. Therelative motion of the follower vehicle depends on the motionof the leader vehicle and follower vehicle always tries tomaintain a minimum safe separation (inter-distance) withleader vehicle.

In our approach of modeling at microscopic level, wemodel car-following behavior between the leader and thefollower vehicles based on the ’stick-slip’ dynamic. Thisstick-slip phenomenon is found mainly on the relative motionissued from the contact between different rigid mechanisms.This stick-slip motion generates a jerky phenomenon by thepresence of certain flexibility in the contact and representsa succession of jumps and stop. In this stick-slip dynamics,when the inter-distance reaches to a very small value thenfollower vehicle applies brakes and, when the inter-distanceincreases then follower vehicle again accelerate.

In this work, we develop the model of this stick-slipmotion based on inter-distance as the physical connectionbetween the vehicles. Actually, this physical connection isvirtual, and we emulate this connection by the spring-dashpotsystem to represent the stick-slip motion as shown in Fig.7. This virtual interconnection system represents in realitythe state information of the inter-distance variables collectedfrom the sensor and establishes the communication betweenthe IAVs.

978-1-4799-2914-613/$31.00 ©2013 IEEE 2352

Fig. 7: Platoon of IAVs virtually connected by spring-dashpot system

In Fig. 7, we can see the platoon of i-vehicles (i=1,2,3...),which are virtually connected by the spring-dashpot. Thespring stiffness, damping coefficient and position of thevehicles are denoted by ki, bi and xi respectively. Let usconsider the generic system of leader (nth) and follower(n+ 1th) vehicles. We connect sub-microscopic bond graphmodels of two vehicles by the bond graph model of thespring-dashpot system. The microscopic bond graph modelis shown in Fig. 8.

Fig. 8: Microscopic bond graph model connects sub-microscopic bond graphs of leader and follower vehicles witha virtual bond graph model of spring-dashpot system

In Fig. 8, bonds 1 and 2 connect the spring-dashpot systemto the longitudinal dynamic 1− junctions of vehicle(n+1)and vehicle(n) respectively. The flow f2 from the leadervehicle(n) is transmitted to the virtual spring-dashpot systemwith a full headed arrow bond, which represents the flowactivated bond and transfer only flow to the system anddoes not receive effort from the system; this restores theanisotropic property (motion of the leader vehicle is notaffected by the motion of the follower vehicle) of thetraffic flow. The effort e1 from the spring-dashpot system istransmitted to the follower vehicle(n+1) to determine its flowf1. The parameters kn and bn denote the spring stiffness anddamping coefficient respectively, which are used to calculatethe inter-distance. The dynamic relations for the microscopictraffic model can be derived from the bond graph shown inFig. 8. By applying the junction law corresponding to bonds3,4,5 we get the governing equation of the microscopic trafficdynamic and is given by the equation (16), as following:

m(n+1)x(n+1) + bn(x(n+1) − x(n)) + kn(x(n+1) − x(n)t)= Fx1(n+1) + Fx2(n+1) + Fx3(n+1) + Fx4(n+1)

−m(n+1)θ(n+1)y(n+1) (11)

The speeds x(n+1), x(n), y(n+1) and θ are measured bythe sensors mounted on the IAVs. The state value of inter-distance d (here we consider inter-distance as the distance

between the CM of the two IAVs) between the leader andfollower vehicles with respect to inerial frame is given byd = Xn −Xn+1.

III. PLATOON CONTROL

In this section, we use the microscopic bond graph modeldeveloped in previous section for the context of platooncontrol. The virtual physical inter-connection between theIAVs is the base of the local platoon control algorithm.This virtual inter-connection system analytically describesthe necessary effort calculated by the follower vehicle inorder to maintain the safe inter-distance. From the bondgraph model shown in Fig 8, we can develop the controlstrategy as shown in Fig. 9.

Fig. 9: Platoon Control Strategy

In Fig. 9, the first summation junction represents the 0-junction corresponding to bonds 1,2,3 and second summationjunction represents 1-junction corresponding to bonds 3,4,5of bond graph model in Fig. 8. At first summation junctionthe difference of speeds of the leader and the follower IAVsis taken as the error signal f3. A proportinal-integral (PI)control strategy is applied to compensate this error and tocalculate the necessary effort e3 to be applied on the followervehicle. The control equations are given below:

error in speed:

f3 = f2 − f1 = xn − xn+1 (12)

control effort applied on follower vehicle:

e3 = e4 + e5 = bn(xn − xn+1) + kn

∫(xn − xn+1) (13)

IV. SIMULATION RESULTS

For the simulation purpose, we use Symbols Shakti soft-ware. SYMBOLS acronym stands for SYstem Modeling byBOndgraph Language and Simulation. It is a modeling,simulation and control systems software for a variety ofscientific and engineering applications. In Fig. 10, the screenshot of the software plateform is shown.

978-1-4799-2914-613/$31.00 ©2013 IEEE 2353

Fig. 10: Software Plateform Symbols

The simulation parameters for the RobuCar are as fol-lowing: m = 390Kg, J = 160Kgm2, a = 0.96m, b =0.88m, c = 0.65m, and the values for spring stiffness anddamping coefficient are taken as kn = 100N/m and bn =0.1Ns/m. The position and speed of follower vehicle (n+1)are shown in Fig. 11 and Fig. 12 respectively, in response ofuniform motion of the leader vehicle (n). We can notice thatthe follower vehicle (n+1) decreases its speed, when it comescloser (decrease of inter-distance) to the leader vehicle (n)and again increases its speed when the separation is more(increase of inter-distance). In this way, this model basedcontrol of platoon performs well.

Fig. 11: Space-time behavior of a platoon of two IAVs

Fig. 12: Speed-time behavior of a platoon of two IAVs

V. CONCLUSION AND FUTURE WORK

In the present work, we proposed a microscopic modelof traffic dynamic based on graphical approach bond graphmodeling. The model describes the dynamic of car-followingbehavior including the dynamics of individual vehicle. Then,we propose a model based local control of the platoon ofIAVs, which analytically provides the calculation of neces-sary effort for the follower IAV to maintain the safe inter-distance with the leader IAV. The simulation results showthat this model based controller performs well and regulatesthe speed of follower IAV to maintain safe inter-distancewith the leader IAV. In the future work, we are interested toextend the model for macroscopic traffic dynamic using thisenergy based graphical modeling technique bond graph.

ACKNOWLEDGMENT

This work is supported and funded by European projectWeastflows (196G), Interreg IVB north-west Europe, [15].

REFERENCES

[1] Pipes, L.A., An Operational Analysis of Traffic Dynamics, Journal ofApplied Physics, vol. 24, no. 3, pp. 274-281, 1953.

[2] Forbes, T.W., Zagorski, M.J., Holshouser, E.L., Deterline W.A., Mea-surement of Driver Reaction to Tunnel Conditions, Proceedings of theHighway Research Board, vol. 37, pp. 345-357, 1959.

[3] Kometani, E., Sasaki, T., Dynamic Behaviour of Traffic with aNonlinear Spacing-Speed Relationship, Proceedings of the Symposiumon Theory of Traffic Flow, Research Laboratories, General Motorscorp., pp. 105-119, 1959.

[4] Gipps, P.G., A Behavioural Car Following Model for ComputerSimulation, Transportation Research B, vol. 15, pp. 105-111, 1981.

[5] Gazis, D.C., Herman, R., Rothery, R.W., Nonlinear Follow-The-LeaderModels of Traffic Flow, Operations Research, vol. 9, No. 4, pp. 545-567, 1961.

[6] Bando, M., Hasebe, K., Nakayama, A., Shibata, A., Sugiyama, Y.,Dynamical Model of Traffic Congestion and Numerical Simulation,Physical Review E, vol. 51, no. 2, pp. 1035-1042, 1995.

[7] Li, L., Xu, L., Linear Stability Analysis of a Multi-vehicle Car-following Traffic Flow Model, 15th International Conference on Man-agement Science and Engineering, Long Beach, USA, pp. 1642-1647,2008.

[8] Yi, S., Chong, K., Impedance control for a vehicle platoon system,Mechatronics, vol. 15, Issue 5, pp. 627-638, ISSN 0957-4158, June2005.

[9] Avanzini, P., Thuilot, B., Dallej, T., Martinet, P., Derutin, J.-P., On-line reference trajectory generation for manually convoying a platoonof automatic urban vehicles, IEEE/RSJ International Conference onIntelligent Robots and Systems, pp.1867-1872, 2009.

[10] Contet, J., Gechter, F., Gruer, P., Koukam, A., Reactive multi-agentapproach to local platoon control: stability analysis and experimen-tations, Int. J. of Intelligent Systems Technologies and Applications,vol.10, No.3, pp.231 - 249, 2011.

[11] Mukherjee, A., Karmakar, R., Samantaray, A. K., Bond graph inModeling, Simulation and Fault Identification. CRC Press, FL, USA,2006.

[12] Pathak, P. M., Samantaray, A. K., Merzouki, R., Ould-Bouamama, B.,Reconfiguration of Directional Handling of an Autonomous Vehicle,IEEE Region 10 Colloquium and the Third ICIIS, Kharagpur, India,December 2008.

[13] Loureiro, R., Merzouki, R., Bouamama, B.O., Bond Graph ModelBased on Structural Diagnosability and Recoverability Analysis: Ap-plication to Intelligent Autonomous Vehicles, Vehicular Technology,IEEE Transactions on, vol.61, no.3, pp.986-997, March 2012.

[14] Drozdz, W., Pacejka, H.B., Development and validation of a bondgraph handling model of an automobile, Journal of the FranklinInstitute, vol. 328, Issues 5-6, pp. 941-957, 1991.

[15] www.weastflows.eu, official website of Weastflows project, 2012.

978-1-4799-2914-613/$31.00 ©2013 IEEE 2354