Optimization based coordination of autonomous vehicles in conﬁned areas Stefan Kojchev1 Robert Hult2and Jonas Fredriksson3

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Optimization based coordination of autonomous vehicles in conﬁned

areas

Stefan Kojchev1, Robert Hult2and Jonas Fredriksson3

Abstract— Conﬁned areas present an opportunity for early

deployment of autonomous vehicles (AV) due to the absence of

non-controlled trafﬁc participants. In this paper, we present an

approach for coordination of multiple AVs in conﬁned sites.

The method computes speed-proﬁles for the AVs such that

collisions are avoided in cross-intersection and merge crossings.

Speciﬁcally, this is done through the solution of an optimal

control problem where the motion of all vehicles is optimized

jointly. The order in which the vehicles pass the crossings is

determined through the solution of a Mixed Integer Quadratic

Program (MIQP). Through simulation results, we demonstrate

the capability of the algorithm in terms of performance and

satisfaction of collision avoidance constraints.

I. INTRODUCTION

It is believed that fully automated vehicles (AV) have the

potential to drastically change the transport industry, both

in terms of increased safety and efﬁciency [1]. The most

drastic improvements are expected when a substantial part

of the vehicles on public roads are fully automated, as in

e.g., ”robo-taxis” and hub-to-hub transports on highways.

Unfortunately, managing the unpredictable conditions on

public roads in a reliably safe manner has proven to be

harder than initially expected, and the current state-of-the-

art exhibits a lack of production-level maturity.

However, conﬁned areas, such as mines, ports, and logistic

centers, lack many of the difﬁcult aspects of public road

driving, and present use cases for near-future, large-scale

deployment of automated vehicles. Within this context, the

AVs form a component in transport solutions for commercial

operations, where for example material-ﬂow can be handled

without human involvement.

One of the challenges in such systems is the efﬁcient

coordination of multiple AVs use of mutually exclusive

resources (MUTEX), such as intersections, narrow roads,

work-stations (e.g. crushers, loading/unloading spots, etc.)

and, in the case of electriﬁed AVs, charging-stations. Poor

coordination can lead to substantial decreases in productivity

and energy-efﬁciency, reducing the beneﬁts of automation.

The problem of handling mutual exclusive resources has

been addressed for industrial robots [16], [17], where dif-

*This work is partially funded by Sweden’s innovation agency Vinnova,

project number: 2018-02708.

1Stefan Kojchev is with Volvo Autonomous Solutions and the Mecha-

tronics Group, Systems and Control, Chalmers University of Technology

stefan.kojchev@volvo.com;kojchev@chalmers.se

2Robert Hult is with Volvo Autonomous Solutions, 41873 G¨

oteborg,

Sweden robert.hult@volvo.com

3Jonas Fredriksson is with the Mechatronics Group, Systems and

Control, Chalmers University of Technology, 41296 G¨

oteborg, Sweden

jonas.fredriksson@chalmers.se

ferent scheduling algorithms have been explored. The coor-

dination of multiple AVs, however, adds a different aspect

to the challenge, where for example, the dynamics of the

vehicles and the road topography play a signiﬁcant part in

the optimization problem. The coordination of automated

vehicles at intersections has been widely discussed in the

literature recently, see [2] for a comprehensive survey. In

general, the problem is difﬁcult to solve and has been

formally shown to be NP-hard in [15]. Often relying on

simplifying assumptions and heuristics, a number of methods

has been presented that solve the problem using, e.g., hybrid

system theory [3], reinforcement learning [4], scheduling [5],

model predictive control (MPC) [6], [7] or direct optimal

control (DOC) [8], [9].

Coordination of AVs in conﬁned areas has some distinct

differences compared to the intersection scenarios often

found in the literature. For instance, the full site-layout of

conﬁned areas is typically known at the planning stage,

and it can often be expected that no non-controlled actors

will disturb the execution of a plan once it’s formed. The

motion of each vehicle can therefore be planned from the

start of a transport mission to its end. Planning for conﬁned

sites is thus beneﬁted by methods that can handle long

planning horizons. This is in contrast to the intersection

coordination context found in the literature, where a cutout

around the intersection proper is most often considered, with

the vehicles arriving at speed [6], [7].

In this paper, we formulate the MUTEX-coordination

problem as an an optimal control problem. We adapt the

two-stage heuristic procedure proposed in [9] to the conﬁned-

site context and employ it to solve the problem. Besides

cross-intersections, we consider the merge-split MUTEX-

zones, where the vehicles ﬁrst join in on a common patch

of road which after some distance separate. The approach

is capable to optimize the vehicle trajectories over their full

path and there are no limitations on the model that is used

for the vehicles. Although the approach focuses on conﬁned

sites, the method of handling the mutual exclusion zones can

be extendable to other scenarios as well (e.g., public road

applications).

From similar scenarios that have been considered, the

authors in [10] and [11] propose an optimization approach

for handling merge scenarios, which is a subset of the merge-

split collision zone, and [12] uses a game-theoretical strategy

for optimizing trafﬁc ﬂow through multiple intersection

collision zones. The handling of multiple intersections and

zones of different types were also identiﬁed in the survey [2]

as topics for further work in this ﬁeld.

arXiv:2210.14738v1 [eess.SY] 26 Oct 2022

The remainder of the paper is organized as follows:

Section II formulates the problem that is solved in this paper.

In Section III the method for solving the stated problem is

presented, followed by Section IV where simulation results

illustrate the coordination algorithm. Section V concludes the

work and provides some possible extensions.

II. PROBLEM FORMULATION

We consider Nafully automated vehicles on a road

network with cross-intersection, path merges and path splits.

The road network is assumed to be fully in a conﬁned area,

such that non-controlled trafﬁc participants (e.g. manually

operated vehicles, pedestrians, bicyclists etc.) are absent. We

further assume that the paths of all vehicles, i.e., their routes

through the road network are known, that no vehicle reverses,

and that overtakes are prohibited.

A. Vehicle modelling

The motion of the vehicles along their path is described

˙pi(t) = vi(t)(1)

˙xi(t) = fi(pi(t), xi(t), ui(t)) (2)

0≤hi(pi(t), xi(t), ui(t)).(3)

where pi(t)∈Ris the position, xi(t)∈Rnthe vehicle

state, ui(t)∈Rmthe control input, with i∈ {1, . . . , Na}.

The state is subdivided as xi(t)=(vi(t), zi(t)), with the

speed along the path vi(t)∈Rand zi(t)∈Rn−1collecting

possible other states. The functions fiand hi, both assumed

smooth, describes the dynamics and constraints that capture,

e.g., actuator and speed limits respectively.

B. Vehicle modelling in the spatial domain

For conﬁned site optimization, it is beneﬁcial to optimize

the trajectories of the vehicles over their full paths. However,

the time it takes a vehicle to traverse a path is dependent

on the solution, and not known a-priori. Consequently, it

is inappropriate to plan the vehicle’s motion with time

as the independent variable. Due to this, the problem is

reformulated in the spatial domain, using that dpi

dt=vi(t)

and dt= dpi/vi(t). The formulation of the vehicle dynamics

(1) in the spatial domain is

dti

dpi

vi(pi)(4)

dxi

dpi

vi(pi)fi(pi, xi(pi), ui(pi)) (5)

0≤h(pi, xi, ui).(6)

where the position piis the independent variable.

C. Conﬂict zone modelling

Aconﬂict zone (CZ) is described by the entry and exit

position [pin

i, pout

i]on the path of each vehicle. From the

known positions, the time of entry and exit of vehicle iis

tin

i=ti(pin

i),tout

i=ti(pout

i), respectively. In this paper,

two types of conﬂict zones are considered, as depicted in

Figure 1: the “intersection-like” and the “merge-split”. We let

I={I1, I2, ..., Ir0}denote the set of all intersections in the

conﬁned site, with r0being the total number of intersection

CZs, and let Qr={qr,1, qr,2, ..., qr,l}denote the set of

vehicles that cross an intersection Ir. In the intersection-

like CZ, it is desired to only have one vehicle inside the

CZ, i.e., not allowing the vehicle jto enter the CZ before

vehicle i6=jexits the CZ, or vice-versa. The order in

which the vehicles cross the intersection Iris denoted OI

sr,1, sr,2, ..., sr,|Qr|, where sr,1, sr,2, ... are vehicle indices

and we let OI=OI

1,...,OI

r. A sufﬁcient condition

for collision avoidance for the r-th intersection CZ can be

formulated as

tsr,i (pout

sr,i )≤tsr,i+1 (pin

sr,i+1 ), i ∈I[1,|Qr|−1],(7)

where tis determined from (4). In the merge-split CZ case,

let M={M1, M2, ..., Mw0}denote a set of all merge-

split zones, with w0being the total number of merge-split

CZs in the site and Zw={zw,1, zw,2, ..., zw,h}denote

the set of vehicles that cross the merge-split CZ Mw.

For efﬁciency, it is desirable to have several vehicles in

the zone at the same time, instead of blocking the whole

zone. This requires having rear-end collision constraints

once the vehicles have entered the CZ safely. In this case,

the order in which the vehicles enter the zone is denoted

as OM

w=sw,1, sw,2, ..., sw,|Zw|, and we let OM=

OM

1,...,OM

w. The collision avoidance requirement for

this w-th CZ is described with the following constraints:

tsw,i (pin

sw,i )+∆t≤tsw,i+1 (pin

sw,i+1 +c)(8a)

tsw,i ,ki+ ∆t≤tsw,i+1 (psw,i,ki−pin

sw,i +pin

sw,i+1 +c),

kin

sw,i ≤ki≤kout

sw,i (8b)

tsw,i (pout

sw,i )+∆t≤tsw,i+1 (pout

sw,i+1 +c),(8c)

i∈I[1,|Zw|−1].

That is, while in the CZ, the vehicles must be separated

by at least a time-period ∆tiand a distance cior by ∆tj

and cj, depending on if vehicle jis in front of vehicle i

or vice versa. This is equivalent to the standard offset and

time-headway formulation often used in automotive adaptive

cruise controllers.

D. Discretization

The independent variable is discretized as pi=

(pi,1, . . . , pi,Ni), where pi,Niindicates the end position for

vehicle i, and the input is approximated using zero order

hold such that u(p) = ui,k, p ∈[pi,k, pi,k+1[. The equations

(4), (5) are (numerically) integrated on this grid, giving the

“discretized” state transition relation

ti,k+1

xi,k+1=F(xi,k , ui,k, pi,k , pi,k+1)(9)

where Fdenotes the integration of (4), (5) from pi,k to

pi,k+1.

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OptimizationbasedcoordinationofautonomousvehiclesinconnedareasStefanKojchev1,RobertHult2andJonasFredriksson3AbstractConnedareaspresentanopportunityforearlydeploymentofautonomousvehicles(AV)duetotheabsenceofnon-controlledtrafcparticipants.Inthispaper,wepresentanapproachforcoordinationofmultipleAV...

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Optimization based coordination of autonomous vehicles in conﬁned areas Stefan Kojchev1 Robert Hult2and Jonas Fredriksson3

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