Periodic and Event-Triggering for Joint Capacity Maximization and Safe Intersection Crossing

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Periodic and Event-Triggering for Joint Capacity Maximization and
Safe Intersection Crossing
Christian Vitale, Panayiotis Kolios, and Georgios Ellinas
Abstract— Intersection crossing represents a bottleneck for
transportation systems and Connected Autonomous Vehicles
(CAVs) may be the groundbreaking solution to the problem.
This work proposes a novel framework, i.e, AVOID-PERIOD,
where an Intersection Manager (IM) controls CAVs approach-
ing an intersection in order to maximize intersection capacity
while minimizing the CAVs’ gas consumption. Contrary to most
of the works in the literature, the CAVs’ location uncertainty is
accounted for and CAVs controls are re-optimized periodically,
allowing for the optimization of system performance while
creating safe trajectories. To improve scalability for high-traffic
intersections, an event-triggering approach is also developed
(AVOID-EVENT), which reaches a better trade-off among per-
formance and computational and communication complexity.
In a realistic simulation scenario, AVOID-EVENT reduces the
number of re-optimizations required by 92.2%, while retaining
most of the benefits introduced by AVOID-PERIOD.
I. INTRODUCTION
Wireless communications have the potential to improve
the performance of CAVs, especially in dangerous road
sectors such as intersections, when on-board sensing is not
sufficient to fully ascertain the status of an area-of-interest
[1]. Thus, CAVs have been the focus of research works
aiming towards a more efficient and safe management of
intersection crossings.
A number of approaches have been investigated for CAV
coordination while crossing an intersection: (i) reservation-
based schemes, with intersections modeled as multi-agent
problems and vehicles booking a specific area for intersection
crossing [2]; (ii) fuzzy controllers that navigate the CAVs
at intersections [3]; and (iii) optimization-based schemes
based on Model Predictive Control, that aim at minimizing
specific metrics, such as the time to cross the intersection
[4]. Nevertheless, the CAVs’ present and future locations are
generally considered error-free.
As in our previous approach in [5], in this work, such sim-
plification is not considered. An IM collects CAVs’ system
status and associated uncertainties (e.g., due to imprecise
measurements) and decides the CAVs’ future controls for
crossing an intersection. In order to obtain safe trajectories,
the IM accounts for possible future location prediction errors
(e.g., due to unexpected forces acting on the vehicle or
The authors are with the Department of Electrical and Computer
Engineering and the KIOS Research and Innovation Center of Excel-
lence, University of Cyprus. {vitale.christian, pkolios,
gellinas}@ucy.ac.cy.
This work was supported in part by the European Union’s Horizon
2020 Research and Innovation Programme under Grant 739551 (KIOS
CoE - TEAMING) and Grant 101003439 (C-AVOID), and in part by the
Government of the Republic of Cyprus through the Deputy Ministry of
Research, Innovation and Digital Policy.
due to underlying simplified motion models). Using a pre-
determined control decision order, the IM propagates in
future time instants the location uncertainties of all vehicles
for which controls have been already decided. Under this
framework, and assuming a linear-Gaussian motion model,
the IM is able to characterize, for each time instant, collision-
free areas in the intersection that the CAV under optimization
can safely use to move. Among all possible feasible safe
trajectories, CAVs’ controls are selected to maximize the in-
tersection capacity as the primary objective and to minimize
gas consumption as a secondary objective.
Contrary to [5], where the communication/computation
complexity is minimized at the expense of system per-
formance, i.e., CAVs’ controls are selected only once at
the entrance of the area supervised by the IM, in this
work, a better trade-off between performance and commu-
nication/computation cost is explored. First, a novel opti-
mization framework (AVOID-PERIOD) is used to choose
the optimal controls for the CAVs traversing the inter-
section. AVOID-PERIOD focuses on maximizing the sys-
tem performance, at the expenses of a greater communi-
cation/computation complexity, by utilizing updated system
state estimations transmitted by CAVs to periodically recom-
pute their controls. When compared with the corresponding
forecasted system states at the entrance of the area under the
IM’s control, the updates transmitted by the CAVs exhibit
less uncertainty and propagate a smaller error to future
time instances, improving intersection capacity without com-
promising safety. Further, an event-triggering optimization
framework, i.e., AVOID-EVENT, that intelligently selects
when to trigger CAVs’ controls updates, is showcased.
AVOID-EVENT identifies two main cases where triggering
new controls for a CAV could be beneficial: (i) if an
update from a possibly colliding CAV reduces the uncertainty
associated with its future system state predictions, such that
the center of the intersection can be crossed safely earlier
than expected; (ii) if the distance traveled by a CAV is
limited only by a preceding CAV on the same lane and
an update allows to safely get closer to it. As a result,
on average, AVOID-EVENT triggers a small percentage of
the control updates triggered by AVOID-PERIOD, while
retaining most of the performance gains when compared to
[5]. Summarizing, this work:
•introduces a new uncertainty-aware optimization frame-
work to maximize road capacity in the use-case of
autonomous intersections;
•extends the mathematical framework proposed in [5]
(where safe trajectories were obtained minimizing

computation/communication complexity), introducing
AVOID-PERIOD optimization that maximizes perfor-
mance, periodically selecting CAVs’ controls based on
an updated and less error-prone view of the intersection;
•develops an event-triggering-based optimization
(AVOID-EVENT), that trades performance for
computational/communication complexity;
•presents an extensive simulation campaign to validate
the proposed optimizations and showcase in detail their
advantages and disadvantages.
II. RELATED WORK
Many approaches already exploit CAVs to enforce coor-
dination in an intersection crossing. Nevertheless, contrary
to what is proposed in this work, most of such proposals
rely on a strong assumption: the CAVs’ present and future
locations are considered error-free. Among the studies that
consider CAVs’ location uncertainty, [6] and [7] present
path planning algorithms able that respect pre-determined
movement constraints, while [8] also considers uncertainties
due to unreliable wireless communication. Other approaches
model the uncertainty of human-driven vehicles at inter-
sections during trajectory planning [9]. While all previous
works provided useful insights, when location uncertainty is
accounted for, only our work in [5] considers a framework
that optimizes the choices of CAVs aiming to maximize
intersection capacity. Nevertheless, as explained in Section
I, the approach in [5] focuses on reducing the computa-
tion/communication complexity, overlooking the possibility
of system re-optimizations, exploiting more precise informa-
tion transmitted by CAVs while traversing the intersection.
On a different use case, the trade-off between communi-
cation and performance was recently explored in [10], where
CAVs having crossing trajectories leave each other a gap so
that, in a pre-determined safety region, only one vehicle is
present at any time. A safety function is analytically defined
representing, at any point in the trajectory, the maximum
reaction time for a CAV to avoid being in the same safety
region with another CAV, even in case of emergency braking.
As a result, the gap left among CAVs is inversely propor-
tional to the transmission frequency of CAVs’ system state
updates, since the higher the number of transmissions, the
higher the probability that an update concerning emergency
braking is received on time. Nevertheless, contrary to our
work, CAVs’ present and future location are assumed to be
perfectly known, discarding the realistic possibility that two
CAVs may end up in the same safety region even without
the occurrence of an unexpected event.
III. SYSTEM MODEL
The primary objective of the IM is to maximize the
intersection’s achievable capacity, i.e., the average number
of CAVs safely admitted into the intersection. To achieve
this objective, the distance that is traveled by each CAV j
after entering the area managed by the IM is maximized,
over a planning horizon T(i.e., the intersection traversal
time by a CAV is minimized). Furthermore, when multiple
optimal solutions are available, the IM prefers trajectories
that minimize the CAVs’ gas consumption. The IM computes
the controls applied by CAVs to follow such optimal trajec-
tories accounting for location uncertainties. In the following,
Section III-A presents the CAVs’ motion model adopted in
this work, while Section III-B derives the CAVs’ location
uncertainty originated by the selected motion model.
A. CAV’s System State
CAV jfollows discrete-time linear dynamics, with sam-
pling interval δτ , as shown below:
xj
τ= Φxj
τ−1+ Γuj
τ−1+wj
τ−1(1)
where xj
τ= [xj,˙
xj]⊤
τ∈R4consists of position xj
τ=
[px, py]j
τ∈R2and velocity ˙
xj
τ= [νx, νy]j
τ∈R2components
in 2D Cartesian coordinates at time τ. To control each CAV,
acceleration controls are applied, with uj
τ= [aj
x,aj
y]⊤
τ∈
R2denoting the applied acceleration vector at time τand
wj
τ= [wj,˙
wj]⊤
τ∈R4∼ N (0,Σj
w)denoting the Gaussian
disturbance on the system because of uncontrolled forces
on the CAV, that has zero mean and covariance matrix Σj
w.
Being managed by the IM, with the chosen controls, it
is assumed that CAVs only travel on north-to-south/south-
to-north or west-to-east/east-to-west directions and do not
change lanes while traversing the intersection. Thus, Γand
Φrepresent the unidimensional linear relationship between
xj
τand pair xj
τ−1-uj
τ−1[5].
The dynamics of each CAV observe the Markov property
(Eq. (1)), i.e., a CAV’s state at the next time step only
depends on its current state and control input. Thus, to
compute a vehicle’s state xt, t ∈[1, .., T ]given a known
initial state x0and a sequence of control inputs u0:T−1over
the planning horizon T, Eq. (1) must be recursively applied:
xt= Φtx0+
t−1
X
k=0
Φk[Γut−k−1+wt−k−1],∀t(2)
where CAV index jis not included in order to simplify the
notation. Then, the trajectory of the CAV, XT={xt}, t ∈
[1, ..T ]over T, is a stochastic process, where each future
state xtfollows the distribution xt∼ N (µt,Ξt)having µt=
[µ,˙
µ]⊤
tand where Ξtis given as follows:
µt= Φtx0+
t−1
X
k=0
ΦkΓut−k−1,Ξt= Σ0+
t−1
X
k=0
ΦkΣw(Φ⊤)k.
(3)
In Eq. 3, Σ0denotes the uncertainty associated with the
initial system state of vehicle j. It should also be noted that
can be easily pre-computed, since Ξtdoes not depend on the
applied controls u0:T−1.
Finally, as in our work in [5], with Gaussian measure-
ment errors associated with the on-board sensors and GPS
measurements through Kalman Filter (KF), CAV jobtains
at any time τan estimation of its own system state which
follows a multivariate Gaussian distribution with mean µ′
τ
and covariance matrix Ξ′
τ. Such estimation is communicated
to the IM and can be used as x0and Σ0in Eq. (3) (i.e., as
the starting point of the control decisions).

B. Uncertainty Characterization
To account for potential collisions during path planning,
the 2D area that contains a CAV’s barycenter at any time
τis modeled as an ellipse based on the CAV’s location
distribution, with probability 1−ϵ(ϵarbitrarily small). Based
on the IM’s selected controls, if the ellipses containing the
two CAVs’ barycenters never intersect, then their collision
probability is bounded (i.e., there is a possibility of a colli-
sion only in the case where at least one of the two vehicles
lies outside the ellipse). Thus, the maximum probability of
collision, Pc, between vehicles iand jis given by:
Pc≤1−(1 −ϵ)2≤2ϵ−ϵ2≤2ϵ(4)
Since Ξtcan be pre-computed ∀t∈[1, ..T ], the size of
the associated ellipse can be obtained by utilizing established
statistical results, that consider the numerical integration of
the distribution of the CAV’s location [11]. Specifically, with
probability 1−ϵat time t, with t∈[1, ..T ], the two semi-
axes of the ellipse that contains the location of the CAV’s
barycenter are equal to:
α+
t,j =qKϵλ+
t,j ;α−
t,j =qKϵλ−
t,j ;(5)
with λ+
t,j and λ−
t,j denoting the largest and smallest eigen-
values of the Ξt’s location sub-matrix and Kϵdenoting the
inverse of the cumulative density function of the chi-squared
distribution having two degrees of freedom computed at 1−ϵ.
IV. OPTIMIZATION FRAMEWORK
A. IM’s Optimization Scenario Overview
In order to coordinate CAVs traversing an intersection, an
IM is utilized. The area around the intersection is divided
into the pre-danger zone at a distance of lpmeters from the
intersection’s center and the danger zone, at a distance of ld
m from the center of the intersection, with ld<lp. Interaction
with the IM is initiated when the vehicles enter the pre-
danger zone, as illustrated in Fig. 1.
Fig. 1: Overview of the IM’s optimization scenario.
With AVOID-PERIOD, to maximize the intersection ca-
pacity, the IM applies a receding horizon approach. When
CAV jenters the pre-danger zone, the IM computes the
controls uj
0:T−1over Tthat maximize the distance travelled
by jover the planning horizon, and communicates to jthe
control uj
0for the next time slot. Subsequently, once the IM
receives the CAV’s updated system state estimation, obtained
after applying the previous control decision, it computes the
new control profile, i.e., uj
1:T−1, over a planning horizon of
length T−1, and transmits uj
1to CAV j. This operation is
repeated for the original planning horizon (i.e., Ttimes).
B. AVOID-PERIOD Optimization
The AVOID-PERIOD technique jointly tackles both the
danger zone demand management and the intersection ca-
pacity maximization at each time slot, in a receding horizon
fashion, thus, extending the capabilities of a state-of-the-art
intersection management algorithm, i.e., AVOID-DM, that we
previously presented in [5] where joint danger zone demand
management and intersection capacity maximization takes
place only when the CAVs enter the pre-danger zone.
In AVOID-PERIOD, CAVs’ trajectories are computed
sequentially, and the expected trajectories’ information can
be exploited to obtain safe trajectories for all the ones that
follow in the planning order. Since the solutions chosen
by the IM impose that CAVs do not change lanes while
passing through the intersection, two collisions categories are
studied: lateral collisions when vehicles cross the intersection
following directions perpendicular to each other, and frontal
collisions, when vehicles follow one another. In the case
of lateral collisions for CAV j, a collision area Bi,j can
be defined for each lane that crosses its path (see example
in Fig. 1). Subsequently, the probability of lateral collision
bound in Eq.(4) is respected in the case where the ellipses
that depict the expected barycenter locations of two CAVs
with intersecting trajectories are never in the corresponding
collision area at the same time. Similarly, the probability
of frontal collision bound in Eq.(4) is respected in the
case where the distance between CAV j’s ellipse and the
preceding CAV v’s ellipse is at least dmin =fv+fj+s
(where fjdenotes the distance between the barycenter of
vehicle jand its front bumper, fvdenotes the distance
between the barycenter of vehicle vand its back bumper,
and sis the pre-defined safety distance).
To plan safe trajectories, the order used by the IM for
deciding the CAVs’ controls, at any time τ, is critical. In
this work, the planning order chosen at each time τrespects
the order in which CAVs cross the center of the intersection.
Thus, among two CAVs that may collide laterally, for the one
passing first from the collision area, the IM does not need to
know the trajectory of the other, while, for the one passing
second, the IM needs to know the interval of time the colli-
sion area is occupied, in order to avoid ellipses intersecting
one another. If the planning order respects the crossing order
of CAVs from the center of the intersection, and considering
that CAVs do not change lanes, CAVs following one another
can also choose a safe trajectory. Indeed, at the moment of
planning, the IM knows the expected locations, for any future
time slot, of the preceding CAV’s ellipse, and it can easily
impose a constraint so to avoid frontal collisions.
Summarizing, at each time slot τ, the IM determines
the crossing order Oτof CAVs through the center of the
intersection, based on the decision taken at τ−1. For this
purpose, the mean expected location of the CAVs’ barycenter
is used (µin Eq. (3)). Then, the IM plans a safe trajectory

for each CAV in the planning order, with the objective of
maximizing intersection capacity. Indexing CAVs in Oτas
the order used for planning decisions, the IM decides the
trajectory for jthrough its acceleration profile uj
ω−1:T−1,
with time τcorresponding to the ω−th time slot in the
planning horizon of j, and ensures that the ellipse of jdoes
not intersect with any ellipse of CAVs {1, ..., j −1}in any
time slot in T.
The size of the ellipses (Eqs. 3 and 5) characterizes the
uncertainty around future CAVs’ system states. It should be
noted that while the CAV is within the pre-danger zone,
it exploits on-board sensors to avoid the only possible
collision, i.e., with the preceding CAV. Thus, the pre-danger
area is treated as any other road section that precedes the
intersection and the IM considers that the uncertainty around
the mean expected location µfollows a multivariate Gaussian
distribution with zero-mean and constant covariance matrix
equal to Σ0+ Σj
w, with Σ0as the worst-case covariance
matrix of the one-step Kalman Filter system state error
estimation. On the contrary, within the danger zone, sen-
sors on-board the vehicles cannot always be used to avoid
possible sudden lateral collisions. Thus, upon the CAV’s
entry into the danger zone, the IM propagates the error
in future time slots as in Eq. (3). The reader should note
that, even though the moments when CAVs {1, ..., j −1}
enter the danger zone are known, this is not the case for
the CAV junder optimization prior to planning. Therefore,
in AVOID-PERIOD the computation of the optimal time
for vehicle jto enter the danger zone is embedded in
the optimization, effectively managing the demand at the
entrance to the danger zone. To accomplish this, a binary
variable btis defined for each time t∈[ω, ..., T ]in the
planning horizon of CAV j, that takes the value 0if vehicle
jis in the pre-danger zone, and the value 1otherwise.
Subsequently, for computing the correct ellipse’s semi-major
axes at any time tthe following equation suffices:
αj(t) = α+
0,j −
t
X
k=ω
bk(α+
t−k,j −α+
t−k+1,j ).(6)
Since a CAV traverses the danger zone upon entry, without
returning to the pre-danger zone, the different terms in Eq.
(6) cancel each other out, with the only remaining term
corresponding to the number of time-slots during which the
CAV is in the danger zone at time t∈[ω, ..., T ]. Importantly,
Eq. (6) holds even in the case when vehicle jis already in
the danger zone and all auxiliary binary variables equal 1.
Based on the discussion above, AVOID-PERIOD is pre-
sented, that considers the ω−th time slot in the planning
horizon of CAV j, which, w.l.o.g. traverses the intersection
west-to-east. If not otherwise specified, i∈ {1, ..., j −1},
t∈ {ω, ..., T }, and CAV vprecedes jin the same lane.
Problem AVOID-PERIOD :
max
aj
ω−1:T−1
µj
T−
T
X
t=ω
ξtM+γ
T
X
t=ω
µj
t−β
T
X
t=ω+1
|aj
t−aj
t−1|(7a)
subject to:
µi
t, µj
tas in (3) (7b)
aj
t∈[aMI N , aMAX ],˙µj
t∈[vMI N , vMAX ](7c)
|aj
t−aj
t−1| ≤ ∆a(7d)
αj(t)as in (6) (7e)
µj
t<−lD+btM(7f)
bt∈ {0,1}(7g)
µv
t−µj
t≥αj(t) + αv(t) + dmin −ξt(7h)
ξt≥0(7i)
µj
t−αj(t)≥Bi,j || µj
t+αj(t)≤Bi,j
∀t|µi
t±(αi(t)+fi+s)∈Bi,j (7j)
where the xdirection subscript is not included for nota-
tional simplicity. To cope with the adopted receding horizon
approach, slack variables ξtare introduced to always find
a solution and respect the presented safety constraints. As
safety is paramount, the slack variables are multiplied by
a large constant Min the objective function (Eq. 7a).
If slack variables are necessary, a solution is found that
minimizes the slack variables’ summation, i.e., the amount
of violation of the safety constraints, practically ignoring any
other component. Otherwise, ξt= 0 ∀t∈ {ω , ..., T }.
A multi-objective function is utilized when all cases where
slack variables are not needed. The maximization of the
distance traveled by vehicle jis realized by maximizing
CAV j’s x-axis mean location at the end of the optimization
window T(Eq. 7a). Further, the following fact is considered:
if vehicle ithat crosses the intersection constraints the
movement of vehicle j, then vehicle jwill be unable to
pass the corresponding collision area before vehicle i, even
if vehicle jaccelerates to the maximum possible speed. For
this reason, multiple optimal acceleration profiles allow j
to cross the corresponding collision area right after i. To
chose amongst all these profiles, two additional terms are
included in the optimization objective function. The first
term is utilized to push vehicle jas close as possible to
the danger zone, starting from the moment it enters the pre-
danger zone, in order to clear the way for new CAVs. This
is done by summing-up the distance traveled by vehicle jat
each time slot, over the planning horizon T(travelling earlier
on larger distance increases such summation). The second
term improves the smoothness of the controls by minimizing
the difference between consecutive accelerations applied by
vehicle j. It should be noted that, in our implementation, the
absolute values that are summed in this term are transformed
into a series of additional linear constraints [12]; this is done
without impairing the proposed approach’s complexity.
In terms of AVOID-PERIOD’s constraints, system state
predictions respect Eq. (3), the acceleration and speed of j
respect valid bounds (Eq. (7c)) and successive acceleration
controls do not vary more than ∆am/s2(Eq. (7d)) in order
to increase user comfort. Further, Eq. (7e) determines the
ellipse’s size that contains vehicle j’s barycenter with fixed
probability, due to the determination of j’s entry time in the
danger zone, while in Eq. (7f), if vehicle jis present in the
danger zone at t, binary variable bt(that multiplies large

constant M) becomes 1, in order to satisfy the constraint.
On the other hand, if vehicle jis outside the danger zone,
the constraint is always satisfied and binary variable bt
can take any value. Nevertheless, to reduce the size of the
ellipse associated to jand allow a larger traveled distance,
btturns automatically to 0in the pre-danger zone. Finally,
to ensure the coordination amongst CAVs the last three
constraints are utilized, with the αiand αvvalues known
when the optimization initiates. Specifically, Eq. (7h) ensures
that the distance between the barycenters of the ellipses of
vehicle jand its preceding vehicle vis larger than dMI N .
If this constraint cannot be satisfied, in order to obtain a
viable solution a slack variable ξt≥0is used, allowing a
small violation. Finally, by using Eq. (7j) vehicle jtraverses
collision area Bi,j , before or after i; i.e., when vehicle i
traverses Bi,j, vehicle j’s predicted location (plus (minus)
the larger axes of its ellipse) is outside Bi,j.
V. ANEV ENT TRIGGERING AP PROAC H
An event-triggering technique (AVOID-EVENT) is further
developed, aiming to retain the benefits of AVOID-PERIOD,
while greatly reducing communication and computational
overhead. Specifically, AVOID-EVENT triggers a new CAV
optimization only in the pre-danger zone and only if it
is strictly required. No optimization is triggered in the
danger zone, since errors are propagated and a safe, albeit
suboptimal, trajectory is always possible. In the pre-danger
zone, in case the CAVs colliding with jcontinue on their
expected trajectories and with the same expected uncertainty,
re-optimizing controls may be superfluous. Hence, j’s con-
trols are re-optimized only if there exist changes between
the intersection’s expected future status at time τand its
expected status at the moment j’s trajectory was selected.
For lateral collisions, the intersection occupancy metric,
Ij
τ, is introduced, representing at each time τ, the last future
time slot in which, before the crossing of j, any of the
possible collision areas Bi,j are occupied by another CAV
with a probability larger than 1−ϵ. Specifically,
Ij
τ= max
i∈{1,...,j−1}
hmax
τ∗|t∗∈[ωi−1:T−1] ZBi,j
N(µi
t∗,Ξi
t∗)dxdy > 1−ϵi
(8)
where τ∗corresponds to the t∗-th time slot in the planning
horizon of CAV i,∀i∈ {1, ..., j −1}and where Ij
τ
is computed exclusively on the set of CAVs crossing the
intersection before j. Hence, for lateral collisions, a new
optimization is triggered if Ij
τ< Ij, with Ijbeing the
intersection occupancy at the moment the controls for CAV j
were last updated. In practice, a new optimization is triggered
if the center of the intersection, from the perspective of
CAV j, is free earlier compared to when its controls were
last updated. If this is the case, CAV jcan: (i) anticipate
its entrance in the center of the intersection, (ii) reduce its
intersection traversal time, and (iii) ultimately increase the
capacity of the intersection.
If jis not impeded laterally, AVOID-EVENT may trigger
a new optimization if there exist changes on the expected
future system states of CAV vthat precedes j(i.e., the
location of vis such that the distance that can be traveled by
jincreases). This occurs when dv
T(τ)> dv
T, where dv
T(τ)
is the point of v’s ellipse that is closer to the entrance to
the pre-danger zone at the end of its planning horizon T, as
computed at time τand dv
Tis the point of v’s ellipse that is
closer to the entrance of the pre-danger zone at the end of its
planning horizon T, as computed the last time the controls
of jwere updated.
In general, the events triggered by AVOID-EVENT may
occur in two main cases: (i) the CAVs possibly colliding
with jare able to accelerate, leaving more space to j; (ii)
the uncertainty related to the future system states of the
CAVs possibly colliding with jreduces. Recalling that in the
danger zone system state prediction errors are propagated to
all future time slots, when a CAV updates its system state
from the danger zone, errors are propagated for fewer time
slots, hence reducing the associated uncertainty. In practice,
with the events belonging to the aforementioned second
case, any time that additional knowledge on colliding CAVs
is available, AVOID-EVENT tries to exploit the possibility
to increase the intersection capacity and triggers a new
optimization for j.
If no new optimization is triggered, to cope with any devi-
ation due to uncertainty within the pre-danger zone, jtracks
the last information (i.e., expected trajectory and speed)
received from the IM to traverse safely the intersection area.
Specifically, through the exploitation of a receding horizon
approach and an updated system state estimation utilizing
a KF, vehicle jcan minimize the error between the target
system states that are computed through AVOID-PERIOD
and all of vehicle j’s future expected system states following
the technique described below:
Problem Car-Follow :
min
aj
w−1:T−1
T
X
t=ω
|µj
t−mj
t|+δτ
T
X
t=ω
|˙µj
t−˙mj
t|+
T
X
t=ω
ξtM(9a)
subject to:
µv
t, µj
tas in (3) (9b)
aj
t∈[aMI N , aMAX ],˙µj
t∈[vMI N , vMAX ](9c)
µv
t−µj
t≥α+
0,j αv(t) + dmin −ξt(9d)
ξt≥0(9e)
For minimizing the error to the target trajectory the
objective function’s (Eq. (9a)) absolute values are modified
into linear constraints, where the expected location and
speed that have to be tracked are represented by mj
tand
˙mj
t, respectively. To obtain the CAV’s future system state
predictions, the motion dynamics of the CAV are exploited
(Eq. (3)). Constraint (9c) limits the values assumed by the
controls and by the speed of the CAV, while Constraint
(9d) is used to avoid frontal collisions, by forcing CAV j
to respect a minimum distance with its preceding CAV v.
Again, a set of minimized slack variables ξt,∀t∈[ω, ..., T ],
is introduced to obtain a solution even when the uncertainty
is such that Eq. (9d) cannot be respected. A summary of
AVOID-EVENT is given in Alg. 1.

Algorithm 1 The AVOID-EVENT approach.
1: Ij,dv
Tinitialized to +∞,−∞, resp., when jenters pre-danger zone
2: for ∀time τdo
3: for ∀j∈Oτdo
4: if jin pre-danger zone then IM computes Ij
τand dv
T(τ)
5: if Ij
τ< Ijthen
6: IM computes uj
ω−1:T−1w/ AVOID-PERIOD, sends to
j
7: Ij=Ij
τ
8: else if Ij= +∞&dv
T(τ)> dv
Tthen
9: IM computes uj
ω−1:T−1w/ AVOID-PERIOD, sends to
j
10: dv
T=dv
T(τ)
11: else jcomputes uj
ω−1:T−1w/ CAR-Follow
12: end if
13: end if
14: if jin danger zone then japplies last received uj
ω−1:T−1
15: end if
16: end for
17: end for
VI. PERFORMANCE EVALUATION
A. Simulation Scenario
Figure 1 illustrates the 4-way intersection used for
the evaluation of AVOID-PERIOD,AVOID-EVENT, and
AVOID-DM [5], with the pre-danger and danger zones start-
ing at 300m and at 150m from the center of the intersection,
respectively. The optimization window chosen, T= 56s,
allows for a CAV to traverse the entire danger zone when
traveling with an average speed of 8m/s. Further, the sam-
pling rate δτ = 0.5s. The inter-arrival times for CAVs
entering the danger zone follow the exponential distribution,
with a mean equal to 2s, that corresponds to the average
arrival rate of cars at a centrally-located intersection during
peak hour within a medium-size city [13]. The CAVs’ initial
speed (0−14m/s) and their lane are selected using a uniform
distribution. Finally, CAVs can enter the pre-danger zone
only if at least one acceleration profile is feasible so that
a collision with the preceding CAV can be avoided.
Moreover, typical sensor sensitivity data [14] are uti-
lized to model the uncertainty of the initial state, and
of acceleration/GPS measurements used in the KF by
the CAVs and in the system state prediction by the IM.
Thus, (i) the worst-case approximation of the KF’s co-
variance is Σ0= [0.6m2,0.2(m/s)2; 0.2m2,0.06(m/s)2],
while (ii) the dynamics error’s covariance is Σw=
[0.0125δτ 40.025δτ 3; 0.025δτ 30.5δτ 2]. For both cases,
the covariance matrix holds for location and speed in the
direction of the CAV’s movement, and it is zero otherwise.
Same-size CAVs are assumed, having a distance between
their barycenters of dmin = 8m to ensure a safe distance of
at least 4m. Also, the acceleration takes (absolute) values in
the range 0−3m/s2, while ∆a= 1m/s2. Finally, ϵ= 10−5,
γ= 10−6,β= 10−5. Each simulation involves 104CAVs.
The reader should note that, AVOID-DM, that selects the
controls of the CAVs (under location uncertainty) only upon
arrival to the pre-danger zone, without further exploiting the
updated system state estimations communicated by the CAVs
to the IM, is used in these simulations as a benchmark.
200 300 400 500 600 700 800 900
Distance travelled [m]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CDF
Fig. 2: CAVs’ traveled distance in T.
B. Performance of AVOID-PERIOD and AVOID-EVENT
The performance of the presented approaches was evalu-
ated utilizing a MATLAB simulator that was developed to
model at each time slot the predicted, estimated, and real
position of CAVs in the intersection, while all optimizations
were solved using GUROBI.
Figure 2 shows the cumulative distribution function (CDF)
of the total distance traveled by the CAVs in their plan-
ning horizons. Exploiting the updated system state and
uncertainty prediction communicated by CAVs to the IM,
AVOID-PERIOD performs better than the state-of-the-art,
AVOID-DM, reaching a gain of 12.26% in the tail of
the distribution, i.e., where it counts the most. Similarly,
AVOID-EVENT also improves the distance traveled by CAVs
with gains up to 11.15% in the first percentiles of the
CDF. Such improvement is critical, especially in high-density
scenarios, with both AVOID-EVENT and AVOID-PERIOD
exploiting additional information during planning by the IM.
Indeed, accounting for a more accurate system state estima-
tion and prediction when compared with the one available
at the entrance to the pre-danger zone, allows originating
trajectories that result in CAVs traveling longer distances.
Also, in AVOID-EVENT, on average, for each CAV, the
number of events triggering a new control update equals
8.786, contrary to AVOID-PERIOD that triggers a new
control each time slot, i.e., 112 times for each CAV. Thus,
while both techniques exhibit similar performance, especially
at the tail of the traveled distance CDF, the computational
complexity of AVOID-EVENT is reduced by 92.2%. Addi-
tionally, as the IM sends control updates only when a new
event is triggered, the CAVs’ downlink traffic is also reduced
by 92.2% (no effect on uplink traffic as the CAVs send an
update on their system state prediction at every slot).
8 25 50 75 100 125 150 175 200 225 250 275 300
Distance between vehicles [m]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CDF
Fig. 3: Minimum distance between CAVs that share a possible
collision area.
All CAVs, for all approaches considered, respect the
minimum distance of 8m between CAVs that allows avoiding

collisions (Fig. 3). Further, this distance is smaller for CAVs
that follow each other, as compared to CAVs that cross each
other’s trajectory. This is the case, as the presented optimiza-
tions exploit a conservative approach for CAVs in the center
of the intersection, allowing at most one ellipse in each of the
collision areas. Further, as expected, AVOID-PERIOD and
AVOID-EVENT perform better than AVOID-DM, since the
additional knowledge exploited at the moment of planning,
allows CAVs to safely get closer to each other.
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
Smoothness pre-danger [m/s2]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PDF
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Smoothness danger [m/s2]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PDF
Fig. 4: CAVs’ consecutive acceleration controls in intersection.
Finally, Fig. 4 shows the acceleration applied by CAVs
for AVOID-PERIOD and AVOID-EVENT. In the pre-danger
zone, AVOID-PERIOD, that recomputes the CAVs’ controls
at each slot, obtains less variable applied controls than
AVOID-EVENT (Fig. 4 (top)). This is due to the fact that
AVOID-EVENT, when no event is triggered, tries to tightly
track the safe trajectory decided by the IM, applying more
diverse accelerations while coping for the deviations intro-
duced by uncertainty. In the danger zone, AVOID-EVENT
presents almost no control changes (Fig. 4 (bottom)), as the
CAVs, after choosing the appropriate time to enter the danger
zone, proceed at constant speed (approximately vMAX =
14m/s). Instead, even though the intended trajectory is sim-
ilar, AVOID-PERIOD ensures that the CAVs’ speeds are
exactly vMAX by re-planning at each time slot, triggering
continuous adjustments due to uncertainty.
Clearly, smoother applied acceleration results in reduced
gas consumption, when this is associated with a smaller
average acceleration used (in absolute value) [15]. Over-
all, accounting for both pre-danger and danger zones,
AVOID-EVENT shows a 10.4% reduction in gas consump-
tion compared to AVOID-PERIOD, as the overall average
absolute value of the difference among consecutive acceler-
ations is equal to 0.227 m/s2and 0.254 m/s2, respectively.
VII. CONCLUSIONS
In this work, an optimization framework aiming at maxi-
mizing intersection capacity while accounting for uncertainty
in the location of CAVs, is presented, extending our work
in [5] by now fully exploiting the communication between
the CAVs and the IM. First, AVOID-PERIOD focuses on
maximizing the system performance at the expenses of a
greater communication/computation complexity. This is a
centralized approach that exploits periodic communication of
update system state estimations and predictions between the
CAVs and the IM to periodically re-optimize CAVs’ trajec-
tories. Second, AVOID-EVENT provides an optimal trade-
off among complexity and performance. In AVOID-EVENT,
the performance gain introduced by AVOID-PERIOD is
mostly retained even if only a limited, intelligently chosen,
number of trajectory re-computations are executed. The
event-triggering approach, not only improves consistently the
intersection capacity, but it can easily scale for high-traffic
scenarios.
Future work includes considering CAVs turning, i.e., sys-
tems where the uncertainty may also depend on the controls
applied.
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