B3A. INTELLIGENT TRAFFIC INTERSECTIONS

Intelligent Traffic Intersection Management UsingMotion Planning for Autonomous Vehicles

Viktor Tuul and John Dahlberg

Abstract—With the increasing advances in the field of au-tonomous vehicles it is alluring to ask if a possible vehicularparadigm shift is in the near future. Maximizing road capacitywith Intelligent Traffic Intersections that communicate withautonomous vehicles could become a reality, where the needfor traffic lights and stop signs is excluded. In this paper, anAutonomous Intersection Management system is introduced thatutilizes trajectory-based prioritization and motion planning tech-niques to manage traffic in an orthogonal single lane four-wayintersection. The developed system reduces the need for vehiclesto slow down or even stop before intersections, contrariwise, itlets all vehicles enter the intersection at the highest allowed speed.The proposed solution is shown to increase the capacity of in-tersections compared with contemporary intersections managedwith traffic lights.

I. INTRODUCTION

ROAD based traffic is increasing all around the globe,especially in densely populated urban areas [1]. The

resulting congestion, especially at traffic intersections, impliesincreased fuel consumption, pollution and in addition wastedtime [2]. The advances made in research and developmentwithin the fields of autonomous vehicles and communicationtechnology will in a near future enable a great deal ofpossibilities within transportation systems, including trafficmanagement [3].

When introducing the concept of Autonomous IntersectionManagement, improvement of efficiency in traffic manage-ment, compared with contemporary methods, by using con-trol of autonomous vehicles has been shown be possible byDresner and Stone [4]. The Autonomous Intersection Man-ager (AIM) approach may in theory also be used with non-autonomous vehicles [5][6] which implies that eliminatingtraffic lights is not necessarily limited to self-driving cars.However, this work examine how communication and controltheory can be utilized for exclusively smart autonomous vehi-cles in traffic intersections in order to optimize road capacityand minimize congestion.

First, the vehicle- and intersection models are presented inSection II. Then in Section III the Autonomous IntersectionManager and its interactions with autonomous vehicles areintroduced. In Section IV a developed control algorithm isproposed that utilizes motion planning techniques which areused to direct three critical tasks; how the AIM 1) takesthe possible trajectories into consideration, 2) determines theindividual vehicle prioritization, and 3) calculates the optimalspeed request for each individual vehicle in order to ensurehighest allowed speed when entering the intersection. SectionVII presents the results from simulations performed in our

developed simulation environment programmed in C#, bothwith our developed control algorithm applied and also withtraffic lights. Additionally, comparisons regarding the vehic-ular throughput for both systems are presented. The resultsare then discussed in Section VIII, and the conclusions arepresented in Section IX.

II. MODELS

This section covers the vehicle- and intersection models thatare used throughout this paper. In Section II-A the geometricand abstract models of the intersection are introduced, andalso how trajectory distances inside the intersection depends onvehicle priority. In Section II-B the vehicle model is presented,where the dynamics and variables of the vehicles that areimplemented in the intersection model are covered.

A. Intersection Model

Each vehicles predetermined parameter of trajectory Tridescribing boundary positions, i.e. start and target, isdefined by the syntaxes

<trajectory> ::= <start><target>

<start> ::= <N>|<E>|<S>|<W>

<target> ::= <N>|<E>|<S>|<W>,

provided that start 6= target and where <N>, <E>, <S> and<W> are the positions north, east, south and west relative originin a, to the intersection, locally defined coordinate system. Thepossible trajectories are defined in Figure 1.

Fig. 1: Visual representation of the intersection trajectories.

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

The intersection model intends one lane for each incomingdirection with lane width L. Some of the trajectories have arisk of collision with other trajectories. The trajectory relationsare presented in an adjacency matrix in Table I which is anabstraction of the geometric representation of the intersectionmodel in Figure 1. The purpose of the abstract model isto allow the AIM to handle trajectory-based prioritizationsregarding vehicles, and to include them in the motion planningalgorithm presented in Section IV.

TABLE I: A matrix that explicitly presents the trajectory con-flict relations where 1 represents a conflict while 0 indicatesno conflict between respective trajectory.

NW NS NE EN EW ES SE SN SW WS WE WN

NW 1 1 1 0 1 0 0 0 1 0 0 0

NS 1 1 1 0 1 1 0 0 1 1 1 1

NE 1 1 1 0 1 1 1 1 1 0 1 1

EN 0 0 0 1 1 1 0 1 0 0 0 1

EW 1 1 1 1 1 1 0 1 1 0 0 1

ES 0 1 1 1 1 1 0 1 1 1 1 1

SE 0 0 1 0 0 0 1 1 1 0 1 0

SN 0 0 1 1 1 1 1 1 1 0 1 1

SW 1 1 1 0 1 1 1 1 1 0 1 1

WS 0 1 0 0 0 1 0 0 0 1 1 1

WE 0 1 1 0 0 1 1 1 1 1 1 1

WN 0 1 1 1 1 1 0 1 1 1 1 1

The key purpose of the matrix is to identify which of thetrajectories that have collision risk (the ones) and whichones that does not (the zeros).

A vehicle Vi with the trajectory Tri that has a collision riskwith vehicle Vj with trajectory Trj has a specified distancedTrjTri

indicating how far the vehicle Vi can travel inside theintersection to a hypothetical point of collision with Vj occurs.Table II presents the traversing distance dTriTrj

, given thatvehicle Vj , with corresponding trajectory on the left hand side,is the prioritized vehicle, i.e. the particular vehicle is expectedto arrive to the hypothetical point of collision before Vi. Thedistances are thus either the quarter circle arc lengths inside theintersection or the distance straight through the intersection,where L is the lane width, which are geometrically identifiedin Figure 2. The figure additionally shows possible positionsfor points of collision risks.

Given that vehicle Vj is the prioritized vehicle, the depri-oritized vehicle Vi must give way for Vj . This implies thatthe distance d

TrjTri

in the intersection for vehicle Vi is givenby the values in Table III. The most trivial case is when thetraversing distance dTriTrj

does not depend on which vehiclethat has priority. Figure 3 shows a case for two vehicles inconflict with the trajectories WN and SN , which impliesthe traversing distances dSNWN and dWN

SN . According to TableII and Table III the traversing distances for the individualvehicles mostly are the same regardless of priority, which isalso illustrated in Figure 3, i.e. the distances do not change.

TABLE II: A matrix that shows the traversing distance insidethe intersection for a prioritized vehicle with the trajectory onthe left hand side until a hypothetical collision occurs with adeprioritized vehicle.

dTriTrjNW NS NE EN EW ES SE SN SW WS WE WN

NW 0 0 0 0πL

40 0 0

πL

40 0 0

NS 0 0 0 0 L 0 0 0 0 0 0 0

NE 0 0 0 0 03πL

8

3πL

4

3πL

4

3πL

40

3πL

8

3πL

8

EN 0 0 0 0 0 0 0πL

40 0 0

πL

4

EW 2L 2L 2L 0 0 0 0 0 L 0 0 0

ES 0π3L

8

3πL

80 0 0 0 0

π3L

8

3πL

4

3πL

4

3πL

4

SE 0 0πL

40 0 0 0 0 0 0

πL

40

SN 0 0 0 2L 2L 2L 0 0 0 0 0πL

4

SW3πL

4

3πL

4

3πL

40

3πL

8

3πL

80 0 0 0 0

3πL

8

WS 0πL

40 0 0

πL

40 0 0 0 0 0

WE 0 0πL

40 0 0 2L 2L 2L 0 0 0

WN 0 03πL

8

3πL

4

3πL

4

3πL

40

3πL

8

3πL

80 0 0

TABLE III: A matrix that shows the traversing distance insidethe intersection for a deprioritized vehicle with the trajectoryon the left hand side until a hypothetical collision occurs witha prioritized vehicle. The bold zeros empathizes the numericaldifferences from Table II.

dTriTrjNW NS NE EN EW ES SE SN SW WS WE WN

NW 0 0 0 0πL

40 0 0

πL

40 0 0

NS 0 0 0 0 L 0 0 0 0 0 0 0

NE 0 0 0 0 03πL

8

3πL

4

3πL

40 0

3πL

8

3πL

8

EN 0 0 0 0 0 0 0πL

40 0 0

πL

4

EW 2L 2L 2L 0 0 0 0 0 L 0 0 0

ES 03πL

8

3πL

80 0 0 0 0

π3L

8

3πL

4

3πL

40

SE 0 0πL

40 0 0 0 0 0 0

πL

40

SN 0 0 0 2L 2L 2L 0 0 0 0 0πL

4

SW3πL

4

3πL

40 0

3πL

8

3πL

80 0 0 0 0

3πL

8

WS 0πL

40 0 0

πL

40 0 0 0 0 0

WE 0 0πL

40 0 0 2L 2L 2L 0 0 0

WN 0 03πL

8

3πL

4

3πL

40 0

3πL

8

3πL

80 0 0

However, when comparing Table II and Table III, dTriTrjand

dTrjTri

differ for two trajectory combinations, i.e. the crossingtrajectories NE/SW and ES/WN . Figure 4 shows how thecrossing traversing distances dWN

ES and dESWN are dependenton which vehicle that has priority. In case a) the prioritizedvehicle Vj have the trajectory WN (bolded). This particularvehicle must cover its full trajectory length before the deprior-itized vehicle Vi with trajectory ES can enter the intersectionin order to avoid collision. This implies that dESWN > dWN

ES as

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

specified in Table II and Table III. However as in case b), ifthe prioritized vehicle Vj has the trajectory ES (bolded), thenvehicle Vi with the trajectory WN must give way before itcan enter the intersection, i.e. dWN

ES > dESWN .

Fig. 2: An illustration of possible points of collision risks forall trajectories.

Fig. 3: An illustration that shows an example for two trajec-tories in conflict where the traversing distances do not dependon which vehicle that has priority, i.e. WN in case a) and SNin case b).

Fig. 4: An illustration that shows an example for two trajecto-ries in conflict where the traversing distances depend on whichvehicle that has priority, i.e. WN in case a) and ES in case b).

B. Vehicle Model

The vehicle characteristics that are taken into account aredefined and described by the variables listed in Table IV,where the index k refers to the variables for any arbitraryvehicle Vk.

TABLE IV: Vehicle variables.

Notation Description Definition Unit

Xk =

(xk

yk

)x- and y coordinates [m]

vk =

(vx,k

vy,k

)x- and y velocity compo-nents

∂Xk

∂t[m/s]

Fk Force output [N]Fmaxk Force output limit [N]

ak Absolute accelerationFk

mk[m/s2]

Trk Trajectory <start><target>mk Mass [kg]lk Length [m]Akk Projected front area [m2]

The x- and y positions are defined to be relative the origin,the center of the intersection, see Figure 1.

Each vehicle’s throttle- and break dynamics are modeledwith an individual PID-controller which emulates the use ofcruise control [7]. The force output Fk of the vehicle dependson the current vehicle speed |vk| and a given speed request,vk,req , and also the upper force output limit Fmaxk . The errorin speed,

e = vk,req − |vk|

is fed into the PID-controller which returns a force output Fkthat depends on three parts, one that is proportional to theerror (P), one that is proportional to the integral of the error(I) and finally one that is proportional to the derivative of theerror (D). The output is given by the sum of the three, that is,

Fk = Kpe(t) +KI

∫ t

0

e(τ)dτ +Kd∂e(t)

∂t

for |Fk| ≤ Fmaxk , and where Kp, KD and KI are weightsthat distributes the influence of the proportional, derivative andaccumulated error to the force output.

The drag force is modeled with

Fd,k =1

2ρ|vk|2CDAk,

where ρ is the air density, CD is the drag coefficient, Ak isthe projected vehicle front area and |vk| is the absolute speed.

The rolling friction force is modeled with

Fw,k = Ckmkg,

where Ck is the coefficient of rolling resistance, mk is themass of the vehicle and g is the gravitational constant. Thisresults in the net force

Fnet,k = Fk − Fd,k − Fw,k

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

and resulting acceleration

ak =Fk − Fd,k − Fw,k

mk.

The integral of the vehicle acceleration gives the speed of thevehicle,

vk = vk0 +

∫ t

0

akdt

where vk0 is the speed of the vehicle entering the system, thatis, when |Xk| = Dc, which is the communication boundarybetween the vehicle and intersection manager, and t is theelapsed time in the system.

III. AUTONOMOUS INTERSECTION MANAGER

The AIM has a set of predefined parameters that describethe geometry of the system, e.g. the lane width and distanceboundary for the communication range. Additionally, the AIMhas dynamic parameters that describe distance- and speedmargins for the vehicles, as well as the delay for the AIM.Table V presents the set of parameters.

TABLE V: Geometry and margin variables.

Notation Description UnitL Lane width [m]Dc Communication boundary [m]dMi,j Vehicle- to vehicle distance margin [m]DM Minimum front-to-back distance between two ve-

hicles[m]

vmax Speed limit [m/s]Td System delay [s]

A. Data exchange

The exchange of data between the vehicles and the AIM iscomprehensively shown in the flowchart in Figure 5.

Fig. 5: Vehicle and AIM communication.

The vehicle-to-AIM data flow consists of the parameterspresented in Table VI.

TABLE VI: Vehicle-to-AIM transmitted state variables

Notation Description UnitID Unique vehicle IDFmax,k Force output limit [N]dk Absolute distance to the intersection [m]|vk| Absolute velocity [m/s]Trk Trajectorylk Vehicle length [m]

B. Computation of state variables

Based on the speed |vk| of each vehicle Vk and its distanceto the intersection, the estimated time to arrival is

Tk =dk|vk|

(1)

where dk is the vehicle’s distance to the intersection edge. Iftwo vehicles Vi, Vj will arrive approximately at the same timeto the intersection, then the time for vehicle Vi from enteringthe intersection to reaching the point hypothetical point ofcollision is

TTrjTri

=dTrjTri

|vi|(2)

where dTrjTriis the traversing distance inside the intersection

before a hypothetical collision with another vehicle. Thisimplies that TTrjTri

depends on the intersection size and thetrajectories of both vehicles in question, see Table II and TableIII in Section II-A. Lets also denote

dtoti,j ≡ di + dTrjTri

(3)

to compactly express the total distance for a vehicle i to ahypothetical point of collision with another vehicle. Note thatdTrjTri

and dtoti,j only makes sense if there is a risk of collisionbetween vehicles i and j.

By taking into account the individual vehicle-to-intersectiondistance di and the vehicle-to-vehicle distance for two vehiclesin conflict,

∆ij =√

(xi − xj)2 + (yi − yj)2 (4)

and the vehicle’s trajectory distances inside the intersectionbefore a hypothetical collision, the AIM calculates and sendsa speed request vreq to the not prioritized vehicle. This is donein order for the vehicles to fulfill the distance margin criterion,

dMij = DM +1

2(li + lj) + ε

where the vehicles trajectories cross, that is, the vehicle-to-vehicle distance ∆ij converges to dMij at that given point intime. The vehicles lengths li and lj are taken into considerationas safety-margins, and the parameter ε is supposed to becovering common expected error sources, such as the GPSsignal error, GPS receiver quality and environmental signalblockage between the vehicles and the AIM. The frequencyof which the speed requests are sent to the vehicles insidethe communication boundary is 1/Td where Td is the systemdelay that considers the data rate and quality in the softwareand hardware of the AIM.

The algorithm is explained in depth in following section.

IV. CONTROL ALGORITHM

In this section the control algorithm is presented that takesthree critical concepts into account; 1) trajectory conflicts 2)vehicle distance prioritization and 3) motion planning basedon speed requests.

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

One significant reason to enter an intersection at the highestallowed and possible speed is to maximize the vehicle flow bypreventing vehicles from unnecessarily slowing down, or evencompletely stopping, before entering the intersection [3]. Thecore of the algorithm is to identify trajectory conflicts betweenall of the vehicles in the system and specify the priority foreach vehicle.

A. Priority

When defining the priority of respective vehicle withinthe communication zone, shown in Figure 6, an appropriateparameter of priority is needed. In this case the vehicles followthe First Come, First Served (or FCFS) principle. For thatreason, this algorithm is determined to operate based on themomentary distance to the intersection edge for each vehicleVk, that is, dk(t). For a set

Vc ={Vc1 , Vc2 , . . . , Vcn

}where each trajectory of the vehicles is at least at one pointcoinciding with each other according to Figure 1 and Table Iin Section II-A, for instance vehicles arriving from a mutualstart position, it must hold that if

dc1(t0) < dc2(t0) < . . . < dcn(t0)

at some arbitrary time t = t0, then also

dc1(t) < dc2(t) < . . . < dcn(t),∀t

while each vehicle Vk remains within the communication zone.On the contrary, for a set

Vnc ={Vnc1 , Vnc2 , . . . , Vncn

}only including vehicles with non-coinciding trajectories arenot restricted by this, i.e. if at a certain time

ds(t0) < dt(t0),

where s, t are index of two vehicles Vs, Vt ∈ Vnc, then

ds(t0 + ∆t) > dt(t0 + ∆t)

at some later time t = t0+∆t is a possible outcome. Handlingeither of these two sets is a straight forward process; assuminga system only with vehicles belonging to the first set Vc, theneach vehicle Vk is allowed to enter the intersection one byone while prioritizing the ones with the shortest distancesdk(t) according to a simple queue. In a system only withvehicles in the latter set Vnc on the other hand, all of thevehicles are approved to enter the intersection simultaneouslywith maximum allowed speed vmax. However, for a hybridset

Vh ⊆{Vc ∪ Vnc

∣∣Vh 6= Vc and Vh 6= Vnc

},

some of the vehicles are in risk of collision whereas others arenot, which implies a more complex process of prioritizationand a systematic solution is needed.

Fig. 6: Illustration of the communication boundary zone fromthe simulation environment.

B. Linear Projection

When a conflict for two individual vehicles is detected, thatis, their trajectories are at some point coinciding, the AIMcalculates and sends speed requests to each of them whichis based on their distances and estimated times for reachingthe hypothetical point of collision defined in Section III-B.This is done according to a concept which is to be denotedby Linear Projection and is illustrated in Figure 7. It is usedto derive the expression for the speed request intended for avehicle Vi. The concept Linear Projection is a projection ofthe states of the vehicle Vi and another arbitrary vehicle Vj(observe that dj < di!) evaluated at some point of time afterthe momentary time t = t0 when the first vehicle Vj arrivesto the hypothetical point of collision, that is, at

t = t0 + Tj + TTriTrj≡ Ttoti,j .

The estimation is a linearization of the total distances to thehypothetical point of collision according to equation (3), i.e.

dtoti,j (t) ≈ dtoti,j (t0)− |vi,j(t0)|(t− t0).

This approximation is updated each iteration and thereforeconverges to the actual distances. The estimated distancebetween the vehicle in question and another arbitrary vehicleVj at any point in time is given by equation (4).

In order for the vehicles to have a predetermined distancemargin DM when vehicle Vj arrives at the hypotheticalcollision position, ∆ij must converge to and reach dMij at thatgiven instance. This is achieved by sending a speed requestto vehicle Vi, which is expected to arrive after vehicle Vj .First, the estimated time at t = t0 until intersection arrivalfor vehicle Vj , i.e. Tj defined in equation (1), is calculated.By taking the traversing time TTriTrj

of vehicle Vj defined inequation (2) into consideration, the speed request

vi,req =dtoti,j −DM

Ttoti,j(5)

is calculated for vehicle Vi, see Figure 7. This assuresthat when vehicle Vj arrives at the position of hypothetical

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

collision, the distance ∆ij between the two vehicles indeed isapproximately dMij .

Fig. 7: Linear Projection. State variables when vehicle Vjarrives to the intersection are estimated by linearization tocalculate required speed to avoid collision.

C. The algorithm

Since the FCFS principle is followed in events of limitedspace, each of the m vehicles V1, V2, . . . , Vm that are rel-evant, i.e. vehicles within the communication zone so thatdk ≤ Dc−L, are sorted as representing objects in an abstractdata queue with respect to their distance to the origin, implyingthat d1 < d2 < . . . < dm. This queue and the following stepsare illustrated in Figure 8.

Now, the first step is sending a speed request with themaximum allowed speed to the first vehicle V1 represented inthe queue, i.e. v1,req = vmax. The second object in the queueis then denoted as i so that i = 2 and is to be compared withthe object j = i − 1 = 1 ahead (Step 2.1, 2.2 in Figure 8).In a general case, i representing vehicle Vk is to be iterativelycompared with all of the objects j ∈

{i − 1, i − 2, . . . , 2, 1

}(Step k.2, k.3, . . . , k.k) ahead with shorter distances. This isperformed upwards in the queue by decreasing index j so thatj = j − 1 after every iteration. Each comparison is to detectwhether there is a risk of collision between the two vehiclesVi and Vj according to the adjacency matrix in Figure I. If norisk of collision is detected after iterating until the final indexj = 1, then the speed limit vmax is sent as speed requestvreq to Vi. However, if a risk of collision is detected withsome object j, the estimated trajectories of the two vehiclesVi and Vj are to be investigated with Linear Projection. Thisis repeated for the objects j until a hypothetical collision isconfirmed with vehicle Vq1 . For each comparison confirminga hypothetical collision, a proposed restricting speed requestvqk,req,∀k = 1, 2, . . . according to equation (5) is calculatedand appended to a list vres =

{vq1,req, vq2,req, . . .

}. If any

of the restricting speed requests are less than the maximumallowed speed, then the least value is sent as a speed requestto Vi, i.e. vi,req = min

∀k

{vqk,req, vmax

}, as it has to slow

down sufficient to bypass collision with each of the vehiclesin conflict that are to arrive before. When the speed request is

sent to Vi, it means that the current j-loop is terminated andobject i is to be shifted to the next object i+ 1 behind (Step2.1). Again as in the previous process it is to be comparedwith objects ahead starting with the new object j = i − 1.This new j-loop is again iterating until the index j = 1, i.e.it has reached the first object, while simultaneously savingrestricting speed requests before sending one. This double i, j-loop is carried on until object i = m which is the last objectand therefore all vehicles have been compared in the particularinstance. The next step at this state is to check whether anew relevant object is detected and in that case add it to thequeue. Also, current objects in the queue that have left theintersection are not relevant anymore and are to be removed.Note that a new local index is again introduced, so that therelevant vehicle objects are denoted by 1, 2, . . .m where mas before is the latest added object. The control algorithm isalso described with a flowchart in Figure 9. The process ofthe control algorithm is repeated resulting in conservation ofthe traffic flow while ensuring that the vehicles are keepingdistance margins to prevent collisions.

Fig. 8: Priority queue of the control algorithm.

V. SYSTEM ROBUSTNESS

Little’s law [8], which is a result of importance in queuingtheory, explains that the average number of objects in a systemN is equal to the average arrival rate λ times the average timeW an object is spends in the system, i.e.

N = λW.

Little’s law shows that the largest flow of vehicles an inter-section and included roads can manage is equal to the upperlimit of N divided by the lower limit of W,

λmax =NmaxWmin

Therefore there are two ways to raise the maximum flowof vehicles; either by increasing the average number of ofvehicles in the system, or by decreasing the average time avehicle is present in the system. This implies that a vehicleshould maintain a high speed in the system. However, asalready stated, the number of vehicles in the system can notbe arbitrarily large as there is a limit for how many vehiclesthe roads can fit.

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

Initializeempty queue.

Input: If newrelevant object ≡ m

detected: appendto queue. Removeirrelevant objects.

Sort the m objectsin queue by shortest

distance so thatd1 < d2 < . . . < dm.

Send v1,req = vmaxto V1.

Denote second objectby i, i.e. i = 2.

Initializeempty

list vres.

Denote otherobject byj, so thatj = i − 1.

Is objectj = 1,

i.e. first?

Is objecti = m,

i.e. last?

Shift objecti to nextbehind:

i = i + 1.

Compare objecti with object j.

Risk ofcollision forVi, Vj

according toadjacencymatrix?

Shift object jto next ahead:j = j − 1.

PerformLinear

Projection.

Appendcalculatedvreq to vres.

Send vi,req =min∀k

{vres,k, vmax

}to Vi.

Yes

No

No

No

Yes

Yes

Fig. 9: Flowchart of the control algorithm.

The theoretical upper limit for the number of vehicles inthe system, bounded by the communication boundary Dc, isgiven by

Nmax = 4Dcn∑n

i=1(li +DM )(6)

where 4Dc is the total road length of the four lanes, li iseach vehicles length, DM is the distance margin between eachvehicle and n is the momentary number of vehicles in thesystem at an arbitrary point in time. This implies that equation(6) calculates the total road distance divided by the averagevehicle length of the vehicles in the system in addition to thedistance margin. This holds assuming that the lanes are fullysaturated, that is, all vehicles are uniformly distributed acrossall four lanes with the distance margin DM to each other. Theimplication of this is that if N > Nmax congestion is builtup. Since the length of the vehicles in practice is not constant,Nmax is dynamic. The implication of this is that Nmax isthe maximum number of vehicles the system can hold withthe assumption that the length distribution for all vehicles isaccording to the actual vehicles present in the system, for everypoint in time.

In order for this requirement to be fulfilled, the averagethroughput, β, must converge to the average rate of incomingvehicles, λ, as

limτ→∞

∫ t0+τ

t0

(λ− β)dt = 0 (7)

where t0 is a starting point in time and τ is a time of measure.This implies that the upper limit of the average throughput thatcan be achieved, βmax, must be greater or equal to the averageincoming vehicle rate λ, that is,

λ ≤ βmax.

There are therefore two requirements to be fulfilled to maintainan unsaturated system without congestion over a long periodof time; {

N ≤ Nmaxβ → λ ≤ βmax

where β is the actual average throughput, which mustconverge to λ according to equation (7).

VI. TRAFFIC LIGHTS

In this section a model for traffic lights is introduced torepresent traditional traffic management. The purpose is togenerate data that the developed AIM can be compared with.It is assumed that all geometrical assumptions are identicaldefined as previous i.e. one lane etc.

To begin with, one cycle length C is the period of time forall signal indicators to serve all four lanes once, that is,

C = 4(G+ Y ),

where G is the time for green indicator and Y for the yellowindicator. The traffic light simulation can be in two possible

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

states; either one of the four signals of the lanes is greenwhile the others are red implying only vehicles in that lane areallowed to pass, or the other state where the two signals thatare about to shift from red to green or vice versa are yellowwhile the others are red, meaning no vehicle is allowed topass at all. Figure 10 shows the principle of a traffic light testperformed in our developed simulation environment, whereonly one lane at the time has free way.

Fig. 10: Image of a saturated test with traffic lights from thedeveloped simulation environment.

VII. EXPERIMENTAL RESULTS

In this section experimental results from simulations arepresented in order to evaluate the efficiency, robustness andliability of the system. The tests are performed in our devel-oped simulation environment programmed in C# which allowfor real time visual monitoring of the vehicles and the accessof dynamic vehicle parameters and statistical data.

Three tests are performed, two with the developed systemand one test that considers traffic lights. The first test isstable and is covered in Section VII-A. Then in Section VII-Bthe second test is performed with full vehicular saturationand reveals the limitations of the proposed system. Finallyin Section VII-C a saturated simulation with traffic lights ispresented to allow comparisons, which are covered in SectionVII-D.

For each simulation, vehicles are spawned from a randomposition, <N>, <E>, <S> or <W>, in intervals of the periodTspawn and with an incoming speed of venter. Additionally,the intersection speed limit vmax, the distance margin DM andthe maximum force output Fmax for each vehicle are predeter-mined and taken into consideration. With the assumption thatall vehicles have the mass m = 2000kg and an accelerationand retardation limit of amax = 3m/s2, consequently themaximum force output Fmax = mamax = 6000N . The PID-controller parameters are empirically selected to Kp = 500,KI = 1 and Kd = 10, i.e. are specified in a way such that thedynamics of the vehicles correspond to reasonable limitationsfor vehicles today. The purpose of the settings mentioned isthus also to enable the behavior of the simulated vehicles tocoincide with human way of driving. The parameters that areconsidered in all simulations are shown in Table VII.

TABLE VII: Predetermined simulation parameters for allconsidered tests.

Notation Description Value UnitSimulation time 360 [s]

Dc Communication boundary 350 [m]venter Incoming vehicle speed 20 [m/s]vmax Intersection speed limit 20 [m/s]DM Distance margin 5 [m]Fmax Force output limit 6000 [N]m Vehicle mass 2000 [kg]l Vehicle length 5 [m]ε Distance margin of common ex-

pected errors0 [m]

Td System delay 0.5 [s]Kp Proportional constant 500KI Integral constant 1Kd Derivative constant 10

A. Simulation 1: Stable

The first simulation shows a results from a stable test, wherethe system does not get saturated. The incoming vehicle timeinterval Tspawn and corresponding rate λ = 1/Tspawn areshown in Table VIII.

TABLE VIII: Vehicle inflow rate for Simulation 1: Stable.

Notation Description Value UnitTspawn Incoming interval 1 [s]λ Incoming rate 1 [1/s]

Figure 11 shows that the average rate of incoming vehiclesand the average rate of passing vehicles are converging, whichis in alignment with equation (7), that implies that the systemwill not get over saturated.

Fig. 11: Vehicle inflow and outflow, Simulation 1: Stable. Theoutflow converges to the inflow.

As shown in Figure 12 below, the number of incomingvehicles in the system N has a upper limit of 18, which isless than the saturated limit Nmax = 140 for l = 5.

Table IX shows the calculated robustness parameters alongwith the statistical data from the simulation.

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

Fig. 12: Vehicles in the system, Simulation 1: Stable.

TABLE IX: Results for Simulation 1: Stable.

Parameter Value CommentNmax 140 For l = 5

N 18 N < Nmax, not saturatedλ 1β [0, 0.95] for t ∈ [0, 360]s β converges to λ

B. Simulation 2: Saturated

The second simulation shows an over saturated scenario,where the number of incoming vehicles in the system exceedsthe limit Nmax, due to the lowered interval of incomingvehicles Tspawn, i.e. increased λ, see table X.

TABLE X: Vehicle inflow rate for Simulation 2: Saturated.

Notation Description Value UnitTspawn Spawn interval 0.5 [s]λ Incoming rate 2 [1/s]

Figure 13 shows that the rate of incoming vehicles andpassing vehicles do not converge as λ = 2 and β convergesto approximately 1.5, which is not in alignment with therobustness criteria, see equation (7), as β < λ.

Fig. 13: Vehicle inflow and outflow, Simulation 2: Saturated.

The fact that the throughput β does not converge to theinflow λ implies that the number of vehicles in the systemwill increase with time without any upper limit. Figure 14shows how the number of incoming vehicles in the systemaccumulates with time.

Fig. 14: Vehicles in the system, Simulation 2: Saturated.Congestion is accumulated as the number of vehicles in thesystem increases beyond the maximum upper limit Nmax.

The lowered interval of incoming vehicles Tspawn increasesthe inflow λ to an extent which the throughput β can not reachwhich is shown in Table XI. The over saturation of vehiclesimplies that the throughput β does converge to the maximumthroughput βmax, see Figure 13 and Table XI. However, asalready presented, since λ > β = βmax the robustnesscriterion in equation (7) is not fulfilled, and congestion istherefore accumulated.

TABLE XI: Results for Simulation 2: Saturated.

Parameter Value CommentNmax 140 For l = 5

N 181 N > Nmax, over saturatedλ 2βmax ≈ 1.5

β ≈ 1.5 β = βmax < λ

C. Simulation 3: Traffic lights

In order to compare the performance of the developedsystem, with intersection management based on traffic lights,a traffic light test in our developed simulation environment isalso performed. The simulation proposes a maximum capacitytest filling all four lanes with vehicles, with each vehiclehaving a front-to-end distance DM = 5m, to correspond toa realistic scenario with fully saturated roads. For this testthe signal time parameters G = 25s and Y = 5s are usedwhich gives a cycle length of C = 120s which is typical forintersections with high approach speeds [9]. Table XII presentthe test parameters, which is in alignment with the saturatedtest in Section VII-B.

TABLE XII: Parameters, Simulation 3: Traffic lights.

Notation Description Value UnitSimulation time 360 [s]

G Period of green signal 25 [s]Y Period of yellow signal 5 [s]C Period of complete cycle 120 [s]

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

Fig. 15: Vehicle outflow, Simulation 3: Traffic lights.

Fig. 16: Passed vehicles, Simulation 3: Traffic lights.

The simulation shows that the rate of passing vehiclesconverges to approximately 0.68 vehicles per second accordingto Figure 15. The total amount of passed vehicles over time isshown in Figure 16, where the five second yellow light delaywith a 25 second interval is spotted by observing the postponein passed vehicles. The statistical data for the simulation, thatis, the number of passed vehicles and the average throughputβ are presented in Table XIII.

TABLE XIII: Results, Simulation 3: Traffic lights.

Parameter Value CommentP 245 Passed vehiclesβ ≈ 0.68 Average throughput

D. Comparison

The following comparisons that considers the performanceof intersections managed by 1) our developed AIM, and 2)traffic lights, intends saturated flow, that is, the particularsimulations done in Section VII-B and Section VII-C. Thesimulations are, as already presented, performed with equiva-lent conditions, i.e. the basis of the vehicles are the same inorder to represent as equal tests as possible.

Plotting the average throughput over time shows that the rateof passing vehicles for the autonomous intersection is greaterthan for the regular traffic light case, see Figure 17.

Figure 18 presents the number of passed vehicles withrespect to time for both cases, comparing the two shows that

Fig. 17: Comparison of the vehicle outflow.

Fig. 18: Comparison of the numbers of passed vehicles.

the capacity is higher for the intersection managed with theAIM.

The results show that the AIM can perform better comparedthe scenario of simulating traffic lights, in the sense ofvehicle throughput. Table XIV presents the number of passedvehicles PAIM , PTL and the average outflows βAIM , βTL forrespective simulation.

In these particular tests where the used environmental andvehicular parameters are according to Table VII and Table XII,the relative quotient of the outflow for the two tests is

βAIM − βTLβTL

≈ 1.2

This implies that the AIM is approximately 120% more effi-cient than the use of traffic lights considering the throughputfor the specific setting.

TABLE XIV: Results regarding the throughputs and passedvehicles, comparison between Simulation 2: Saturated andSimulation 3: Traffic lights.

System Parameter Value CommentAIM PAIM 539 Passed vehiclesAIM βAIM ≈ 1.5 Average throughputTraffic lights PTL 245 Passed vehiclesTraffic lights βTL ≈ 0.68 Average throughput

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

VIII. DISCUSSION

In this section we discuss the developed system with respectto the obtained simulation results, the performance and itslimitations.

In Section VIII-A we discuss the advantages of our devel-oped system in a traffic flow point of view, in comparisonwith traffic light managed intersections. Then in VIII-B wediscuss our solution in a sustainability point of view; whichenvironmental effects, both positive and negative, that couldcome with implementing it in the real world. In section VIII-Cthe robustness of the system is discussed with regards to delaysin the vehicle-to-AIM communication. Section VIII-D coversthe factors that limit the use and implementation of our system,and finally, in Section VIII-E we propose what future work thatcould be done to solve them.

A. Advantages

By allowing the speed of autonomous vehicles to be con-trolled by Autonomous Intersection Managers, not only theneed for vehicles to slow down or even stop is reduced, but italso opens up the possibility for vehicles to enter intersectionsat highest allowed speed. This also implies that intersectionthroughput can be drastically improved. The fact that thevehicles can enter the intersection with high speed, from anydirection eliminates one of the main disadvantages with singlelane intersections managed by traffic lights, which is that atleast three lanes are always still which reduces the averagetraversing speed and throughput significantly.

The performed simulations clearly show that the proposedAIM solution can manage a larger inflow of vehicles, thanintersections managed by traffic lights can. Similar resultsare obtained with other simulation settings as well, that is,when changing the vehicle’s incoming speed, speed limit,performance and the distance margin between the vehicles.This implies that a potential future with only autonomousvehicles and intelligent traffic intersections could be a partof a solution for congestion free roads.

B. Sustainability

Removing the need for vehicles to drastically speed up, slowdown, or even completely stop before entering intersectionsimplies that less energy is wasted as well as reduced conges-tions which both have positive effects on the environment,especially in densely populated urban areas. Not only canautonomous driving and efficient intersections have positiveenvironmental impacts, it may also increase the productivity,primarily by eliminating the wasted manual driving time, andsecondly by reducing the time of travel from point A to B.However, by increasing the efficiency of traffic and locallyreducing the negative environmental impacts, the effects on alarger, global scale should be considered and studied. Improve-ment in efficiency enables for a larger number of vehicles tobe present which in itself could have negative impacts on theenvironment.

C. Robustness

The simulations performed with our proposed system showthat the motion planning algorithm significantly reduces theneed for vehicles to slow down or even stop. For each iterationof the algorithm the AIM calculates the corresponding speedrequest for each vehicle in the system which ensures thatthe incoming vehicles behave in a way to eliminate risks ofcollisions, why also the system is not sensitive for delays. Thisfollows from that one iteration is sufficient for a safe motionplanning, while following iterations serves the purposes ofslightly adjustment of planned motions or initialize motionplanning for new vehicles entering the system.

D. Limitations

The proposed system depends on that every vehicle enteringthe intersection is fully autonomous and connected to theAIM. These requirements are far from fundamental and tobe expected today. On the contrary; the technology is still notfully tested and too expensive for the majority of car owners.This paradigm shift of autonomous vehicles is simply toolarge-scaled to be implemented in one step. It is thereforenecessary to gradually upgrade the system by introducingsemi-autonomous vehicles, in the sense of vehicles that arecompatible with equipment of extended technology allowingthem to in some sufficient extent be automated for driving andcommunication.

In case of failure of any vehicle causing it to slow down orstop, the traffic flow through the system would unequivocallydecrease, however it would not imply any direct risk ofcollision since the AIM keeps track of the momentary veloc-ities and positions of the vehicles and would adapt for suchincident. Some common expected errors of data ε in signals isalso included in the distance margin DM according to SectionII-B, meaning these kind of errors would not put the system injeopardy while not exceeding ε. However, the AIM would notbe able to handle some unexpected errors in signals, such asincorrect information about vehicles current speed or position.For that reason, it is necessary that multiple independent safetymargins are implemented on different levels of the systemwith authority to overrule other instructions in crucial contexts.An individual autonomous vehicle should include its owndefined actions of safety, for instance rely on its own sensorsand immediately stop if an obstacle is detected too closeahead even if it get an faulty speed request from the AIMto accelerate.

E. Further work

As discussed in the previous section VIII-D, there are manycomplications during the implementation time of the AIM,since a great deal of systems need to be adapted. For instance,in order to manufacture the introduced semi-autonomous ve-hicles that are compatible, further research needs to be doneto implement such systems. In addition, it is assumed there isworking technology enabling the communication between thevehicles and the AIM. This should be specified in terms of e.g.frequency range, equipment for transferring data or eventual

B3A. INTELLIGENT TRAFFIC INTERSECTIONS

user interface between the driver and a semi-autonomousvehicle. Another subject to work further on is in what extentand where to implement the system to society. Since onlysome consumers will initially have access to the technologyof autonomous vehicles that meets the requirements to becompatible with the AIM, exclusively some chosen intersec-tions would be upgraded with the system to begin with. Thiswould raise discussions of great extend in technical, political,economical and even ethical aspects that need research.

IX. CONCLUSIONS

Autonomous Intersection Managers in conjunction withautonomous vehicles can be used to manage intersections.Utilizing motion planning techniques to determine individualincoming vehicles’ speeds can replace traditional traffic lightsand therefore enable growth in vehicular throughput, providedthat all road based traffic in consideration is autonomous andhas the ability to communicate with an intersection manager.

The performed simulations show that a significant increaseof throughput in single lane intersections is possible withthe implementation of our developed system, compared withintersections managed by traffic lights. With fully saturatedtraffic, we show that the developed control algorithm mayincrease the vehicular throughput in single lane intersectionsby 120%, if replacing traffic lights.

The proposed solution can be considered to be a smallcontribution to a new working system of intelligent trafficintersections. In the future, such solution should be imple-mented and evaluated for real autonomous vehicles in realworld conditions.

X. ACKNOWLEDGMENT

We would like to thank our supervisor Sofie Andersson forher support and guidance throughout the project.

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