Optimizing Vehicle Re-Ordering Events in Coordinated Autonomous Intersection Crossings Under CAVs' Location Uncertainty

Road intersections represent the primary bottleneck in transportation systems and connected autonomous vehicles (CAVs) have the potential to aleviate the problem through communication and coordination. As such, this work proposes a novel framework where, accounting for CAVs' location uncertainty, an intersection manager (IM) controls CAVs approaching a road crossing so as to maximize the number of admitted vehicles, while ensuring a guaranteed (tunable) level of safety. To fully exploit the communication links among the IM and the CAVs, several features are included in the proposed framework: (i) periodic re-optimizations of the CAVs' applied controls; (ii) periodic re-ordering of the intersection crossing sequence; and (iii) event-based control and ordering optimizations to achieve the best possible trade-off between complexity and performance. The proposed framework is able to improve both the number of admitted CAVs to the intersection and the CAVs' average speeds as compared to relevant state-of-the-art solutions. Importantly, when event-triggering is applied, most of the benefits introduced by periodic optimizations are retained, while at the same time the number of re-optimizations required are reduced by 47.6% (34.18%) during average (heavy) traffic conditions.


I. INTRODUCTION
W IRELESS communications have the potential to improve the performance of autonomous vehicles (AVs), especially at dangerous road segments, e.g., at intersections, when on-board sensing is not sufficient to fully represent the area-of-interest [1]. Connected autonomous vehicles (CAVs) are therefore the focus of current research efforts for a more efficient and safe intersection management.
Several techniques have been exploited for coordinating CAVs at intersections: (i) reservation 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 in roundabouts or intersections [3]; and (iii) optimizations based on model predictive control to minimize a given metric, e.g., the intersection travel time or the gas consumption [4]. Interestingly, and in an effort to reduce complexity, the CAV's location uncertainty both at present and at future times steps is generally not taken into account. Among the few studies that consider the CAV's location uncertainty, [5] and [6] present algorithms able to plan a path for a CAV respecting pre-determined movement constraints, while [7] extends this framework to include also uncertainties due to unreliable wireless communication. Several approaches also model the uncertainty of human-driven vehicles at intersections, e.g., [8], for planning the trajectory of autonomous vehicles. The only framework that accounts for uncertainty in CAVs trajectories when optimizing intersections' performance is our previous work in [9]. However, in that work CAVs' trajectories are optimized only once, at the entrance to the area around the intersection. Our work in [10], [11] extends [9], accounting for demand management, i.e., selecting the best moment for a CAV to enter the intersection, and proposing a preliminary study on how communication may help control re-optimizations. Building on our previous efforts, this work considers an adaptive scheme, where an intersection manager (IM) collects the CAVs' states and associated uncertainties and, with the obtained holistic view of the area, decides the CAVs' future controls for crossing an intersection. Assuming a linear-Gaussian motion model, the IM propagates to future time instants the location uncertainties of all CAVs for which the controls have already been decided and determines collision-free areas in the intersection that the CAV under analysis can safely use to move. Amongst all possible safe trajectories, an optimization is used to increase the capacity of the intersection by choosing the optimal time the CAV enters the intersection (i.e., demand management is performed). Contrary to our work in [10], [11], the communication among the IM and the CAV is fully exploited in this work, and the optimization of the CAVs' intersection crossing order is also considered. Indeed, several alternative implementations are evaluated: (i) in AVOID-PERIOD, periodic optimizations of the CAVs' applied controls are performed to exploit the updated CAVs' location information; (ii) in AVOID-ORDER, in order to improve system-wide performance indicators, rather than focusing on each CAV independently, the CAVs' crossing order at the center of the intersection is also modified and optimized; (iii) to improve scalability in case of congested intersections and minimize computational and communication complexity, in both AVOID-EVENT and AVOID-ORDER EVENT approaches, event-triggering is applied to update the CAVs' controls or intersection crossing order only when favorable conditions are present. Specifically, event-triggering is used to update the CAVs' controls and crossing orders when: (i) an update originating from a possibly colliding CAV reduces the uncertainty associated with its future system state predictions and the center of the intersection can be crossed safely earlier than expected; (ii) the distance traveled by a vehicle is limited only by the preceding CAV traveling in the same lane and an update allows to safely get closer to the leading vehicle.
To summarize, the contributions of this work are as follows: r building on our uncertainty-aware mathematical framework in [9] and exploiting information communicated from CAVs as in [11], a periodic control optimization for autonomous intersection management is proposed, i.e., the AVOID-PERIOD approach; r exploiting also the optimization of CAVs' intersection crossing order, a novel framework maximizing intersection capacity is also proposed, i.e., the AVOID-ORDER approach. A toy example is also used to showcase how, even if some CAVs may be penalized, re-ordering is critical for the system-wide performance; r the AVOID-EVENT and AVOID-ORDER EVENT approaches are also developed, in order to extend the corresponding periodic optimizations by exploiting eventtriggering. This is done in an effort to find the best possible trade-off between performance and computational/communication complexity; r an extensive simulation campaign is presented to validate the proposed approaches in different traffic density regimes, exploring in detail their advantages and disadvantages against multiple state-of-the-art approaches. In the rest of the paper, Section II reviews the state of the art on autonomous intersection management approaches, Section III presents the system model, Section IV showcases the optimization framework underlying the periodic re-optimizations AVOID-PERIOD and AVOID-ORDER, Section V shows how event-triggering can be applied to the presented frameworks, while Section VI demonstrates the performance of the proposed optimizations, including a comparison with AVOID-DM, presented in our work in [10], and with an optimization-based approach that does not consider uncertainty [12]. Finally, Section VII concludes the work.

II. RELATED WORK
In [13] the autonomous intersection management problem is addressed by modeling autonomous intersections as multi-agent problems. Therein, an arbiter handles reservations by vehicles that try to book specific areas within the intersection for specified time periods. In a similar vein, several other works [14], [15], [16] extend the reservation approach, aiming to lessen the occurrence of cases where some of the reservation requests are not accommodated by the arbiter.
In a different approach, works in [17], [18] present an optimization framework that aims to minimize the amount of time the vehicles travel in the system. Contrary to the approach presented herein, in those works only one vehicle at a time is allowed to enter the intersection. An extension is considered in [19], where a FIFO policy is used to serve the vehicles. Rather than the amount of time the vehicles travel within the system, the objective of this work is to minimize the vehicles' overlap time within the intersection. Another alternative approach is examined in [20], having as an objective the reduction of the vehicle's energy consumption (by minimizing the variations in vehicles' speed).
A more recent effort in [21] (and its decentralized version in [22]) presents an isolated signalized intersection with mixed traffic (with human-driven vehicles and CAVs coexisting) where vehicles' trajectories and traffic light signals are jointly optimized. Further, the work in [23] presents a centralized signalfree intersection control technique that can maximize the intersection throughput when CAVs are present at the intersection. The works in [12] and [24] extend [23] by examining a decentralized approach and considering as well driver comfort and gas consumption minimization. Finally, the work in [25] examines a scenario where a new traffic light phase exists, i.e., the white phase, that allows conventional vehicles to follow CAVs through the intersection.
As done in this work by optimizing the CAVs' intersection crossing order (i.e., in the AVOID-ORDER and AVOID-ORDER EVENT approaches), several approaches also try to optimize aggregate system objectives. In [26], vehicles decide sequentially their controls to minimize their travel time, subject to a constraint that imposes each of the conflicting areas, within the vehicle trajectory, to be free. As a byproduct, the aggregate gas emission decreases. Instead, the work in [27] aims to minimize a system-wide objective function through the use of sequential optimizations and permutations of the optimization order. Even though this approach is effective for small-case scenarios, it does not scale for scenarios with a large number of vehicles as the complexity of the approach is combinatorial. Contrary to [27], as described later, our proposed AVOID-ORDER and AVOID-ORDER EVENT approaches only select a subset of the possible CAVs' intersection crossing re-orderings, avoiding the unnecessary evaluation of all permutations that intuitively will not provide any additional gain.
It should be noted that the focus of all approaches presented above is on optimizing trajectory and motion planning irrespective of any uncertainties. Thus, in those works, the safety constraints used are fixed, and they are based on the size of the vehicles or the size of the conflicting areas. On the contrary, in this work, two fundamental uncertainties are taken into consideration when the vehicles' trajectories are created: (i) uncertainties related to vehicle localization, i.e., vehicles transmit state information estimates based on inaccurate sensor readings; and (ii) prediction errors, i.e., vehicles are only able to approximately follow the planned motion.
To date, only a few works in the literature account for uncertainties; the work in [7] examines the probability for a vehicle to reach a target location outside the intersection area, assuming Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. uncertainties in the model, as well as a wireless communications channel that is unreliable. Furthermore, the works in [5], [6] present algorithms for path planning that guarantee respecting pre-defined movement constraints with a target probability, in the presence of vehicle location uncertainty. The authors of [28], instead, compute safe distances among CAVs that allow to bound the collision probability in case of unexpected hard braking and unreliable communication channel. Finally, in a distributed approach, in the case of a mixed-traffic highway scenario, a Monte-Carlo Tree Search is used [29] to account for humandriven maneuvering uncertainty. Contrary to the aforementioned studies, only our work in [9], [10], [11] focuses on improving capacity and safety in autonomous intersection crossings, while considering location uncertainty due to sensor and prediction errors.
The extensions proposed in this work aim at fully exploiting the communication between the IM and the CAVs, with the objective of maximizing system-wide performance metrics. With this in mind, a new optimization dimension is explored, i.e., the optimization of the CAVs' crossing order from the center of the intersection. Thanks to the IM-CAVs' communication, a system-wide coordination among CAVs is utilized, so that CAVs are not crossing the intersection in a FIFO order, but rather in an order the maximizes the overall intersection capacity. Furthermore, similar to [11], instead of optimizing the controls of the CAVs only once (at the entrance to the area supervised by the IM), herein the updated state estimates are transmitted by CAVs and used for periodic system re-optimizations. Such update system state estimates bear less uncertainty and propagate a smaller error to future time instances compared to the ones computed at the entrance to the area managed by the IM, hence unlocking better performance without compromising the CAVs' safety. Finally, as shown in AVOID-ORDER and AVOID-ORDER EVENT, introducing smart event-triggering techniques heavily reduces the computational and communication complexity of periodic re-optimizations, even though a high percentage of the performance gains on our previous work [9], [10] can be retained.

III. SYSTEM MODEL
In this work, an IM coordinates CAVs approaching an intersection by selecting their controls. Contrary to most of the works in the state of the art, herein location uncertainty is accounted for when deciding vehicles' trajectories. In the following sections, the CAV's position state estimation and prediction, together with the associated errors, are modeled.

A. CAV's System State Characterization
CAV j traveling through the intersection follows discrete-time linear dynamics, with sampling interval δτ : where x j τ = [x j ,ẋ j ] τ ∈ R 4 , with x j τ = [p x , p y ] j τ ∈ R 2 denoting the position andẋ j τ = [ν x , ν y ] j τ ∈ R 2 the velocity of CAV j in 2D Cartesian coordinates at time τ . To control CAV j at time τ , an acceleration vector is applied, denoted as denotes the Gaussian disturbance (with zero mean and covariance matrix Σ j w ) acting on the CAVs. Additive Gaussian disturbances are a realistic assumption in robotics [30], [31], [32], since they model present and future system state estimation/prediction errors, such as: (i) CAVs' localization errors, which are due to estimations from noisy sensor readings, (ii) the uncertainty in the system model which arises due to modeling approximations and mismatches with the true underlying model, and finally (iii) unpredictable environmental disturbances (e.g., wind drag or friction with the road pavement) acting on the CAV during operation.
In this model, it is also assumed that CAVs do not switch lanes and do not turn while traversing the intersection, traveling either from north-to-south/south-to-north or from west-to-east/east-towest. Thus, Φ and Γ represent the following unidimensional linear relationship among x j τ and the pair x j τ −1 , u j τ −1 : where Φ = Φ H and Γ = Γ H are used in case CAV j travels east-to-west/west-to-east and Φ = Φ V and Γ = Γ V are used in case CAV j travels north-to-south/south-to-north. Based on the expression of (1), the CAV dynamics obey the Markov property (i.e., the CAV's state at the next time step depends only on its present state and the control input). Therefore, for a specified (known) initial state of the CAV, x 0 , and a control inputs' sequence u 0:T −1 over a planning horizon consisting of T time-steps, the CAV's state x t , t ∈ [1, .., T ] can be obtained by recursively applying (1) (with index j omitted for notational clarity): (4) Then, the trajectory of the CAV over T , , is a stochastic process, with each future state of the CAV, x t , distributed according to x t ∼ N (μ t , Ξ t ), where μ t = [µ,μ] t and Ξ t are specified as: and where Σ 0 denotes the uncertainty of CAV j's initial system state. Interestingly, if the motion model of the CAV follows (5), the covariance matrix Ξ t can be easily pre-computed, as it does not depend on the applied controls, u 0:T −1 . At any time τ , CAV j computes a valid estimation of its own state. As an example, a Kalman Filter (KF) may be used for this purpose. In such a case, CAV j may exploit a measurement of the applied acceleration (tightly similar but not identical to the control chosen in advance by the IM) and a GPS measurement to estimate its own state. With Gaussian measurement errors, the obtained state estimates are assumed to follow a multivariate Gaussian distribution and when transmitted to the IM (via Cooperative Awareness Messages (CAMs) through the wireless channel), they are used as the initial state for subsequent control decisions, i.e., as x 0 and Σ 0 in (5).
Specifically, if at time τ − δτ , x 0 and Σ 0 are the last CAV j's estimated state and corresponding uncertainty, and a 0 is the measured acceleration (with associated measurement covariance error equal to Q), then the CAV j's KF a priori predicted statex 0 and covarianceΣ 0 for time τ are: If R GP S and z 0 are the GPS measurement and covariance of CAV j's location, respectively, then the final a posteriori state x 0 and Σ 0 covariance estimations at time τ are: with: where K 0 and S 0 are, respectively, the optimal Kalman Gain and the innovation covariance matrix at time τ , and H is the observation model.

IV. THE OPTIMIZATION FRAMEWORK
The optimization framework AVOID-ORDER used in this work for maximizing a specific intersection's achievable capacity is presented in this section, with its workflow summarized in Fig. 1.
An IM managing an intersection is responsible for coordinating the CAVs traversing the area. At every time slot τ , CAVs update their system state x 0 and uncertainty Σ 0 and the IM evaluates new controls for the CAVs in the area. First, it generates a set of candidates CAVs' crossing orders O τ through the center of the intersection. For each crossing order o τ ∈ O τ , the IM computes the optimal controls to be applied by CAVs through the intersection with the AVOID-PERIOD optimization. Fixing the crossing order o τ , the AVOID-PERIOD optimization maximizes the average speed experienced by each CAV, consequently improving the intersection capacity, while minimizing the CAVs' gas consumption as soon as multiple optimal trajectories are available. Such maximization generates safe trajectories even in the presence of CAVs' system state uncertainty while optimizing the entry time of CAVs in the danger zone, i.e., solving the demand management problem. Among all crossing orders O τ , the IM chooses the one that maximizes the aggregate average speed of CAVs in the intersection and transmits the corresponding optimal controls to the CAVs.
In the following, the description of the environment where the IM acts is presented in Section IV-A. Then, the different components of AVOID-ORDER are presented: (i) Section IV-B describes how uncertainty is dealt with; (ii) the demand management problem is described in Section IV-C; and (iii) Section IV-D presents the constraints allowing safe trajectories and introduces the AVOID-PERIOD formulation. Last, Section IV-E presents the set of possible planning orders selected by AVOID-ORDER, and Section IV-F showcases how the different presented framework components are combined.

A. An Overview of the Optimization Scenario and the IM
In this work, an IM is used to coordinate the CAVs within an area around its assigned intersection, with the CAVs interacting with the IM as soon as they reach l p m from the center of the intersection (i.e., upon entry into the intersection's pre-danger zone). A second (smaller) area around the intersection is also defined, located l d m from the center of the intersection (with l d < l p ), denoted as the danger zone. The system scenario is depicted in Fig. 2.
The IM's objective is to maximize the intersection's achievable capacity, and this can be accomplished by maximizing the average number of CAVs that can be admitted safely in the intersection. Thus, the IM maximizes the distance traveled by each CAV j, after entering the pre-danger zone, over a specific planning horizon T , which is equivalent to maximizing the CAV's average speed. Specifically, the IM applies a receding horizon approach; upon CAV j's entry into the pre-danger zone, the IM computes the controls u j 0:T −1 over the entire planning horizon T , and communicates to CAV j the control u j 0 for the next time slot. Immediately after receiving the new CAM messages from the CAVs with the updated states, estimated after applying the previous controls, the IM computes the new optimal crossing order and the corresponding CAVs' control profiles. For CAV j, the IM computes a control profile over a planning horizon of length T − 1, i.e., u j 1:T −1 , and transmits u j 1 to CAV j. The IM repeats this operation T times, i.e., up to the end of CAV j's original planning horizon.

B. Uncertainty Characterization
To account for location uncertainty when deciding CAV j's controls, for any present and future time slot, the IM characterizes a 2D area in the intersection containing its barycenter with probability 1− , with arbitrarily small. This is straightforward if the covariance matrix Ξ j t is known. Indeed, assuming that Ξ j t is given, and exploiting well-known statistical results on multivariate-Gaussian distributions, the present and future 2D areas containing (with fixed probability) the CAV's barycenter can be depicted as ellipses [33]. Specifically, taking into account numerical integration, the ellipse's two semi-axes, that contain with probability 1 − CAV j's barycenter location at the t-th time slot of its planning horizon, are equal to: where λ + t,j and λ − t,j denote the largest and the smallest eigenvalues of the Ξ j t 's location sub-matrix, respectively, and K denotes the inverse of the cumulative density function (CDF) of the chi-squared distribution with two degrees of freedom that is computed at 1− .
Selecting controls for CAV j such that the ellipse containing its barycenter never intersects with any other CAV i's ellipse guarantees that a bound exists on the probability of collision between CAVs j and i. Specifically, it is ensured that a collision between the two CAVs will not occur when both of them are inside their ellipses. Consequently, the maximum collision probability, P c , between CAVs i and j is given by: Nevertheless, due to demand management (as detailed later), Ξ j t (and hence the size of CAV j's ellipse) is not always available prior to IM decisions. Hence, the objective of the IM is to jointly determine, given demand management, the uncertainty Ξ j t associated with the CAV's present and future locations, and to select controls for CAV j such that the ellipse containing its barycenter with fixed probability never intersects with any other CAV's ellipse.

C. Demand Management
In the proposed approach, there is a clear distinction between the pre-danger and the danger zone. The pre-danger zone is considered as any other section of the road preceding the intersection, since only rear/front collisions are possible. Given the fact that such collisions can be avoided only by exploiting on-board sensors, the IM considers that the uncertainty in this area around CAV j's present and future mean expected location μ j is distributed as a zero-mean multivariate Gaussian distribution having a constant covariance matrix. Specifically, the uncertainty covariance matrix is equal to is a worst-case approximation of the covariance matrix of the one-step Kalman Filter state error estimation. On the contrary, within the danger zone, on-board sensors are not always enough for avoiding collisions, as lateral collisions (i.e., sudden collisions) are also possible. To cope with this challenge, as soon as CAV j enters the danger zone, the IM propagates the error in future time slots as in (5). This difference in uncertainty characterization does not allow to know prior to planning the uncertainty Ξ j t associated with μ j , since CAV j's entry time into the danger zone depends on the selected controls. Nevertheless, as shown in the following derivations, it is possible to define a constraint that automatically embeds the computation of CAV j's entry time in the danger zone and, hence, of Ξ j t over the entire planning horizon. Without loss of generality, let us assume that time τ corresponds to the ω-th time slot in the receding planning horizon of CAV j and that an auxiliary binary variable b t for each of the future time slots t ∈ [ω, . . ., T ] is introduced. For all t, if b t = 0 when CAV j is located within the pre-danger zone, and b t = 1 otherwise, then Eq. 13 can be used to compute the correct ellipse's semi-major axes at time t: Considering that a CAV does not travel backwards, hence, a CAV does not leave the danger zone after entering it, the different terms in (13) cancel each other out, with the only term remaining being the semi-major axes corresponding to the number of time-slots that CAV j spends inside the danger zone at time t ∈ [ω, . . ., T ]. Importantly, (13) is valid even in the case where CAV j is already in the danger zone and all auxiliary binary variables are equal to 1. Furthermore, it is easy to ensure that the auxiliary binary variables take, when planning is performed, their expected values. Without loss of generality, if it assumed that CAV j traverses the intersection west-to-east (hence −l d represents the entrance to the danger zone), (13) can be associated with the following inequality: In (14), the auxiliary binary variable is multiplied by M (i.e., a large constant). If at time t CAV j is in the danger zone, b t must turn to 1 to satisfy the constraint. If it is not yet in the danger zone, (14) is always satisfied and the auxiliary binary variable b t can take any value. Nevertheless, to reduce the size of the ellipse associated with CAV j's location and intrinsically allow for a larger distance to be traveled, b t turns automatically to 0 in the pre-danger zone.
Compared to an optimization that only considers the danger zone around the intersection, including also the pre-danger zone optimization allows selecting, for each CAV, the optimal entry time into the danger zone, effectively managing the demand in order to maximize intersection capacity.

D. The AVOID-PERIOD Optimization
This section presents the AVOID-PERIOD optimization. Herein, AVOID-PERIOD is presented within the optimization framework that optimizes crossing orders, i.e., AVOID OR-DER. Nevertheless, AVOID-PERIOD could be also applied periodically in a standalone fashion, to update CAVs' controls to new system state estimations. At a specific time slot τ , AVOID-PERIOD is applied to a sorted list o τ of CAVs in the pre-danger zone. By indexing CAVs in o τ as the order used for planning decisions, the IM uses AVOID-PERIOD to choose CAVs' controls that maximize the capacity of the intersection while respecting safety constraints. Specifically, the IM decides the trajectory for CAV j by deciding its acceleration profile u j ω−1:T −1 (with time τ corresponding to the ω−th time slot in the planning horizon of CAV j) ensuring that CAV j's ellipse does not intersect with any other ellipse of CAVs {1, . . ., j − 1} at any time slot in its planning horizon.
In order to achieve such objective, it is sufficient that in o τ , the order of CAVs in the same lane is respected and that the information related to the expected trajectories of CAVs for which controls have been already decided is used to obtain safe trajectories for all the vehicles that follow in the planning order. Then, two collision categories are present: lateral, when CAVs cross the intersection in perpendicular directions, and frontal, when CAVs follow each other. For all the lateral collisions, a precise collision area, B i,j , for CAV j can be computed for each lane that crosses its path. For example, Fig. 2 illustrates the possible collision areas for CAV j, that travels horizontally through a four-way intersection. If the ellipses, depicting the expected barycenter locations of two CAVs possibly colliding laterally, are never simultaneously in the corresponding collision area, then they never cross each other, and thus the probability of collision bound of (12) is respected. Similarly, if a constraint exists that always ensures that the distance between the ellipses of CAV j and its preceding CAV v is at least d min = f v + f j + s (with f j denoting the distance between CAV j's barycenter and its front bumper, f v denoting the distance between CAV v's barycenter and its back bumper, and s denoting the safety distance), then also the probability of frontal collisions is bounded by (12).
In the following, i ∈ {1, . . ., j − 1}, t ∈ {ω, . . ., T }, and v is the CAV preceding j in the same lane. Then, if CAV j travels through the intersection from west-to-east, without loss of generality, AVOID-PERIOD is formulated as follows (with the x direction subscript omitted to simplify the notation): subject to: AVOID-PERIOD exploits a multi-objective function as shown in (15a). The most important component in the objective function regards safety. Indeed, even though the objective of the IM is to maximize intersection capacity, safety always comes first. Due to the adopted receding horizon approach, uncertainty may lead the system to a situation where vehicles following one another cannot choose controls respecting the target bound on collision probability. Nevertheless, in order to provide a solution, slack variables ξ t are introduced. The slack variables are included in the objective function and multiplied by a very large constant M . In case slack variables are necessary, the optimization finds a solution that minimizes the summation of the slack variables, i.e., the amount of violation of the safety constraints, practically ignoring any other component. Automatically, if slack variables are not necessary, ξ t = 0 ∀t ∈ {ω, . . ., T } and AVOID-PERIOD maximizes the distance traveled by vehicle j in the optimization window T . This is obtained in the objective function of (15a) as the maximization of CAV j's x-axis mean location at the end of the optimization window T . Most of the times, several acceleration profiles are able to achieve the same distance traveled by j. If vehicle j is constrained by the movement of another vehicle i crossing j's path, this means that vehicle j will not be able to pass the corresponding collision area before CAV i, even if it accelerates to its maximum possible speed. Thus, there exist multiple acceleration profiles that allow CAV j to cross the corresponding collision area immediately after CAV i. To make a choice amongst these multiple (optimal) acceleration profiles, two additional terms are also included in the objective function. The first term maximizes the distance traveled by the CAV at each time slot over T (i.e., CAVs are pushed as soon as possible closer to the danger zone, making room for newly arriving CAVs), while the second term minimizes the difference between successive CAV accelerations (thus allowing for smoother applied controls and lower gas consumption). It should be noted that, as the latter term sums over a series of absolute values, these are transformed into a series of additional linear constraints [34], without however negatively affecting the computational complexity of the proposed approach.
AVOID-PERIOD is subject to the following constraints: (i) the mean system state predictions respect (5); (ii) the CAV's acceleration and speed respect bounds as set in (15c); (iii) consecutive acceleration controls cannot differ more than Δa m/s 2 (15d) (i.e., ensuring user comfort); (iv) demand management constraints respect (15e)-(15g); and (v) the coordination among CAVs is ensured by the last three constraints, where the values of α i and α v are known at the moment of optimization. Specifically, (15h) constraints the distance between the barycenters of the ellipses of CAV j and its preceding CAV v (i.e., always larger than d min ). If this constraint cannot be satisfied, a slack variable ξ t ≥ 0 allows for a small violation, enabling the derivation of a viable solution. Further, (15j) ensures that CAV j traverses area B i,j before or after i to avoid collisions, i.e., imposes that, while CAV i traverses the intersection, CAV j's predicted location, plus (minus) the larger axes of its elliptical uncertainty description, is not within the B i,j area.

E. Optimizing the Planning Order
The order o τ used to plan the trajectories of CAVs affects the achieved performance. Indeed, when using a planning order, the order used by the CAVs to cross the center of the intersection is typically very similar. Therefore, when choosing a planning order, intrinsically the set of CAVs considered by CAV j as obstacles is also chosen.
In this work, a selected number of planning orders O τ are tested at each time slot τ . In order to avoid considering all possibilities, that are combinatorial in number, the crossing order of the CAVs through the center of the intersection at time τ , namely I τ , is computed and only a few permutations of this order are considered as possible alternative planning orders. The overall objective is to test only planning orders where CAVs expected to be in close proximity to each other at the center of the intersection, swap their crossing orders. Specifically, for each CAV j, a new planning order is considered such that it crosses the intersection before all CAVs that were planned to cross the center of the intersection at most T W seconds before CAV j.
When considering possible planning orderings, two critical characteristics of the solutions provided by AVOID-PERIOD are taken into account. First, the longer the time between CAV j entering the danger zone and CAV j transiting through the center of the intersection, the larger the uncertainty associated with CAV j's predicted system state. As a result, in order to favor the maximization of the distance traveled by CAVs, AVOID-PERIOD chooses a trajectory for CAV j such that CAV j travels at constant speed in the danger zone equal to v MAX . Thus, considering planning orders where CAVs in the danger zone interchange their positions, does not provide any benefit, since all CAVs already travel at maximum speed. Second, when considering a planning order that swaps CAV j with any CAV i with intersecting trajectory, CAV j should also be considered before any other CAV that follows the same trajectory as CAV i and traverses the center of the intersection between CAVs i and j. Indeed, any of those CAVs represents an obstacle for CAV j and, if not considered after CAV j, the considered planning order would be clearly sub-optimal.
Based on the above considerations, Algorithm 1 is presented for defining set O τ . As input, apart from the crossing order I τ , also the vectors C τ and D τ are considered. Vector C τ , represents if j in pre-danger zone then 6:  (5), while vector D τ , represents the crossing direction by CAVs, i.e., "vertical" or "horizontal". The values in C τ and D τ are sorted as in I τ . For example, in Fig. 2, CAV j crossing the intersection west-to-east/east-to-west would have D τ (j) = 1, while CAV i crossing the intersection north-to-south/south-to-north would have D τ (i) = 2. For each CAV in the pre-danger zone, a possible alternative planning order is considered for testing in O τ . To obtain such a planning order, the following four sets are computed: r I I includes CAVs that are expected to pass through the center of the intersection more than T W s before CAV j; r I 1 includes CAVs with intersecting trajectories that are expected to pass through the center of the intersection less than T W s before CAV j; r I 2 includes CAVs traveling in the same direction that are expected to pass through the center of the intersection less than T W s before CAV j; r I F includes CAVs that pass through the center of the intersection after CAV j. Then, if set I 1 is not empty, the additional planning order included in O τ is: {I I , I 2 , j, I 1 , I F }. As it is clear from Algorithm 1, the number of tentative planning orders included in O τ only grows linearly with the number of CAVs in the system. Hence, even in the case of very dense traffic scenarios, selecting the controls of the CAVs requires a finite amount of time.

F. Combining All Pieces Together: The AVOID-ORDER Optimization Framework
The different components presented above are used by the IM to distribute to all CAVs in the pre-danger and danger zones their controls for the next planning time slot. To summarize, the proposed optimization framework, named hereinafter AVOID-ORDER, considers the following steps: r each CAV estimates its own system state for the present and for each time slot in its planning horizon, based on on-board measurements and the control decisions taken by the IM at time τ − δτ . Such system state estimations are wirelessly transmitted to the IM; r the IM computes the crossing time of each CAV (vector C τ ) and the CAV's crossing order at the center of the intersection (vector I τ ). Given the CAVs' traveling direction D τ , the IM computes the set of candidate planning orders O τ using Algorithm 1; r following each planning order o τ ∈ O τ , the IM computes the controls of the CAV in the system, u j ω−1:T −1 , using AVOID-PERIOD. The AVOID-PERIOD optimization maximizes the experienced average speed for each CAV, embedding in the optimization the demand management problem and tight security constraints; r the IM selects the controls computed by the planning order maximizing the overall distance traveled by all CAVs in the system and wirelessly distributes them to the CAVs.

V. AN EVENT-TRIGGERING APPROACH FOR EFFECTIVE AUTONOMOUS INTERSECTION MANAGEMENT
The output of AVOID-ORDER allows determining, even though not over the entire number of possibilities, the best CAVs' crossing order through the center of the intersection, as well as the corresponding optimal control profiles. Nevertheless, re-optimizing the system at each time τ leads to a number of optimizations and to a communication overhead that rapidly grow with vehicle density. To cope with this challenge, event-triggering is proposed as a possible solution. The objective of the proposed simplified approach is to retain the benefits of AVOID-ORDER, while greatly reducing its communication and computational overhead. In the following, Section V-A presents the conditions that trigger CAVs' controls re-planning, while Section V-B showcases how such conditions are used to reduce the complexity of the AVOID-PERIOD and of the AVOID-ORDER optimizations.

A. Defining the Re-Planning Conditions
The idea behind exploiting event-triggering is to re-optimize the controls of a single CAV or the crossing order of the CAVs through the center of the intersection only when a favorable situation is present, and not unconditionally. To help in this process, three observations are described. First, CAV j may collide laterally at the center of the intersection with other CAVs having intersecting trajectories. In this case, if the CAVs colliding with j continue on their expected trajectories and with the same expected uncertainty, then re-optimizing controls is superfluous. Second, if CAV j is not obstacled by another CAV crossing on a perpendicular direction, then its movement may be affected by the expected future system states of the CAV preceding j, i.e., CAV v. From the perspective of CAV j, CAV v represents an obstacle during the entire duration of the planning horizon of v. So, if there is no change in CAV v's system state or uncertainty at the end of CAV v's planning window, then a new planning decision would not present any benefit to CAV j.
Third, if no new CAV enters the system and all CAVs follow their predicted trajectories with no modification on the associated uncertainty, trying to modify the crossing order of CAVs at the intersection is most probably ineffective.
Based on the above observations, the following conditions are determined for re-planning a vehicle's trajectory or for modifying the order of CAVs crossing the center of the intersection. First, for what concerns lateral collisions, a new metric is introduced, i.e., the intersection occupancy, B j τ . At each time τ , B j τ represents the last future time slot in which, before the crossing of CAV j, any of the possible collision areas B i,j is occupied by another CAV with a probability larger than P . Specifically, B j τ is computed as follows: where τ corresponds to the ω i -th time slot in the planning horizon of CAV i, ∀i ∈ {1, . . ., j − 1}, and where τ * corresponds to the t * ≥ ω i time slot in the planning horizon of CAV i. Hence, based on I τ , B j τ is computed exclusively on the set of CAVs crossing the intersection before j. When comparing B j τ with the intersection occupancy computed at the moment in which CAV j last updated its controls, namely B j , if B j τ < B j , then the center of the intersection is free earlier than originally planned. Because of this additional space, both re-planning the controls only of CAV j or changing the crossing order of CAVs at the center of the intersection may lead to an improvement of the system performance.
The second condition is relative to frontal/rear collisions. If CAV j is not blocked by another CAV crossing on a perpendicular direction, then the expected future states of the CAV preceding j, i.e., CAV v, represent the impediment to CAV j's trajectory. Let us denote with d v T (τ ) the point of the ellipse of v that is closer to CAV j at the end of CAV v's planning horizon, as computed at time τ , and d v T the same quantity, as computed the last time the controls of j were updated. Then, a beneficial re-planning of CAV j's controls is certain when d v T (τ ) > d v T ; i.e., re-planning is triggered if more space is available for CAV j. Nevertheless, given the fact that this condition typically affects only two CAVs, changing the planning order may be redundant.
It has to be noted that, these first two events occur in one of following two circumstances: (i) the CAVs possibly colliding with CAV j are able to accelerate, leaving more space at the center of the intersection; (ii) the uncertainty related to the future states of the CAVs possibly colliding with j reduces. Recalling that in the danger zone system state prediction errors are propagated to all future time slots, when a CAV updates its state from the danger zone, errors are propagated for less time slots, hence reducing the associated uncertainty.
Finally, the third condition relates to CAVs entering the pre-danger zone. The output of AVOID-ORDER alternates the different movement directions at the intersection based on the largest obtained agglomerate distance traveled by CAVs. Once a CAV enters the pre-danger zone, it is possible that the previously decided alternation of movement directions is not the best anymore. Hence, considering that the planning order mostly corresponds to the order with which CAVs cross the center of the intersection, a CAV's entry into the pre-danger zone represents a definite opportunity to possibly change the planning order and improve system performance.

B. Exploiting Event-Triggering: The AVOID-EVENT and the AVOID-ORDER EVENT Optimizations
Given the three favorable conditions described above, two approaches accounting for a different trade-off between computational complexity and performance are introduced hereafter. The first approach, namely AVOID-EVENT, aims at improving performance, while maintaining the minimum computational complexity. The aim of this approach is to re-compute single CAV controls only in favorable situations. Specifically, only in time slots where better conditions are detected for CAV j, i.e., AVOID-PERIOD is applied to obtain a re-optimization of the controls of j.
The second approach, namely AVOID-ORDER EVENT, aims at achieving similar performance compared to the AVOID-ORDER approach, while limiting the number of planning order optimizations only to the aforementioned favorable conditions; i.e., in AVOID-ORDER EVENT, the planning order optimization mechanism described in Algorithm 1 is applied only when a new CAV enters the pre-danger zone and when, for at least one CAV in the system, B j τ < B j . At any time τ , for both AVOID-EVENT and AVOID-ORDER EVENT, in case no control optimization is triggered, CAV j does not receive a new set of controls from the IM. Within the pre-danger zone, in order to cope with any deviation due to uncertainty, CAV j tracks the last expected trajectory and speed received from the IM to safely traverse the intersection. Specifically, CAV j minimizes the error between all future expected states and the target states with the following Car-Follow optimization: subject to: In order to minimize the error to the target trajectory, the absolute values in the objective function in (17a) are again modified into linear constraints, where the expected location and speed to track are represented by m j t andṁ j t , respectively. The CAV's future system state prediction is obtained by exploiting its motion dynamics, as in (5). Constraints imposed include limits on the controls' values and on the CAVs' speed ( (17c)). Furthermore, to avoid frontal collisions, CAV j must if B j τ < B j then 6: The IM computes and transmits u j ω−1: The IM computes and transmits u j ω−1:  ( (17d)). Again, a set of minimized slack variables ξ t , ∀t ∈ [ω, . . ., T ], is introduced to obtain a solution even when the uncertainty is such that (17d) cannot be respected.
A summary of the presented AVOID-EVENT and AVOID-ORDER EVENT approaches is given in Algorithms 2 and 3, respectively.

VI. PERFORMANCE EVALUATION
This section presents the evaluation of the performance of the AVOID-PERIOD and the AVOID-ORDER approaches, together with their version exploiting event-triggering, i.e., AVOID-EVENT and AVOID-ORDER EVENT. First, the reference scenario is illustrated (Section VI-A). Then, Section VI-B explains in detail, in a toy example, how vehicle re-ordering may improve the intersection utilization. Subsequently, the complexity and performance trade-off between the periodic and event-triggered implementations of the proposed framework is shown in Section VI-C. Section VI-C also showcases a performance comparison with our previous optimization framework AVOID-DM [10], which accounts for CAVs' location uncertainty and demand management, but not for CAVs' control re-optimizations. Finally, Section VI-D showcases how the proposed AVOID-ORDER compares against the performance of [12], i.e., an approach that exploits a completely different optimization framework.

A. Simulation Scenario
In order to evaluate the performance of the proposed approaches, a 4-way intersection as in Fig. 2 is used. CAVs enter the pre-danger zone when they are at l p = 300 m from the center of the intersection and enter the danger zone when they are at l d = 150 m from the center of the intersection. An optimization window of T = 56 s is chosen, so that a CAV traveling with an average speed of 8 m/s can traverse the entire danger zone. Furthermore, the sampling rate δτ = 0.5 s.
The time interval between CAVs entering the pre-danger zone is exponentially distributed, with average Λ s varying in the range 0.33 − 5 s, depending on the simulation at hand. The traffic rate in the selected range is comparable to the average arrival rate at a centrally-located intersection of a medium-size city during peak hour [35]. The selection of the CAVs' initial speed (between v MIN = 0 and |v MAX | = 14 m/s) and entry lane follows a uniform distribution. To obtain meaningful simulations, given the random initial system states, a CAV can enter the pre-danger zone if there exists at least one feasible acceleration profile that avoids a collision with the CAV that preceeds it. The same set of arrivals is used for all tested approaches.
The uncertainty of the initial state and of the acceleration/GPS measurements that are used by the CAVs in the KF and by the IM in the state prediction, follows standard sensor sensitivities [36]. Consequently: (i) the worst-case approximation of the KF's covariance is Σ 0 = [0.6 m 2 , 0.2(m/s) 2 ; 0.2 m 2 , 0.06(m/s) 2 ], and (ii) the dynamics error covariance is Σ w = [0.0125δτ 4 0.025δτ 3 ; 0.025δτ 3 0.5δτ 2 ]. For both cases, the covariance matrix holds for location and speed in the direction of the CAV's movement, and is zero otherwise. Finally, the assumption is that all CAVs have the same size and that their barycenters must be at least at a distance of d min = 8 m from each other in order to ensure that the vehicles have at least 4 m of safe distance. The acceleration's absolute value is also constrained to the range 0 − 3m/s 2 , while Δa = 1 m/s 2 . Finally, is fixed to = 10 −5 and the weights used in the multi-objective functions are {γ = 10 −6 , β = 10 −5 }.
To assess the performance of the introduced approaches four metrics are used: (i) as a measure of the selected objective function, the CDF of the average speed experienced by CAVs in the planning horizon T ; (ii) for estimating the achieved intersection capacity, the average time between subsequent admitted CAVs; (iii) for safety, the minimum distance between CAVs' barycenters that share, at any time, a potential collision area; and (iv) the computation and communication overhead imposed by the proposed approaches. To obtain such performance metrics, a MATLAB simulator has been developed, following the framework in Section III and modeling the predicted, estimated, and real position of CAVs in the intersection at each time slot. The GUROBI [37] solver is used to solve, when required, the aforementioned optimizations.

B. Why Re-Ordering is Important: A Toy Example
In order to understand how re-ordering may affect performance, a toy example is initially presented. In this toy example, Λ s = 1 s, simulating a situation where "heavy traffic" needs to be accommodated by our AVOID-ORDER optimization framework. For what concern the simulation set-up, all parameters are as the ones described in Section VI-A. Therefore, two types of collisions are possible: (i) between vehicles that follow each other in the same lane; (ii) between CAVs crossing each other's trajectory. In the former case, CAVs respect a dynamic pre-computed distance (see (15h) in Section IV-D) in order for the border of the corresponding ellipses to be, at any time, at least d min m apart. In the latter case, the trajectories of the vehicles are such that ellipses are never simultaneously in the same collision area, B i,j . Depending on the lanes where the two CAVs are traveling, the two corresponding ellipses must not intersect in an area between 285 and 315 m from the start of the pre-danger zone. Specifically, with Hence, if two ellipses are not in the corresponding collision area at the same time, they do not intersect, and collisions are avoided with a guaranteed probability. Fig. 3 presents a summary of the results obtained at a specific time instance, before and after the AVOID-ORDER optimization framework is applied. The blue-shaded area depicts the distance traveled by vehicles traversing the intersection horizontally as a function of time, while the red-shaded region depicts the distance traveled by the vehicles traveling vertically. Lines represent the actual distance traveled by the vehicles, while pink-shaded and light-blue-shaded regions are the expected distances traveled in the future, within a 99.999% confidence interval. Therefore, collisions occur if: (i) two shaded regions of the same color that belong to the same lane overlap at any time; or (ii) two shaded regions of different colors are present simultaneously in their corresponding collision area B i,j , as described above.
In the toy example examined, AVOID-ORDER considers the possibility of swapping CAV 1 with CAV 3 in Fig. 3(a) when setting the order chosen to update the controls to be applied by the CAVs. As a result, Fig. 3(b) shows that, after re-ordering, CAV 3 crosses the center of the intersection before CAV 1, which in turn awaits for CAV 3 to pass from the corresponding collision Fig. 3. The effect of re-ordering on system performance. Swapping the crossing order of CAVs 1 and 3 improves the overall aggregate distance traveled, since CAVs 5-9 can access earlier the center of the intersection, hence traveling longer. area before accessing the center of the intersection. Fig. 3(c) focuses on CAVs 2 and 4 before the change in the order of the controls update. The movements of CAV 2 are limited by the movements of CAV 1, which crosses the intersection in a perpendicular direction. Similarly, CAV 4 awaits for CAVs 2 and 3 to pass before accessing the center of the intersection. When the order of controls update is swapped between CAVs 1 and 3, the situation changes, as shown Fig. 3(d). Given the initial speed at the time of the re-ordering, CAV 2 is not able to pass safely the intersection before CAV 1. CAV 4, instead, exploits a larger initial speed to move in the large gap that the re-ordering opened up prior to CAV 3 traversing the center of the intersection. Further, Fig. 3(e) shows the distance traveled by all remaining CAVs in the system at the time of re-ordering. Ultimately, CAV 4 represents an obstacle for CAVs 5-9. CAV 5 crosses the intersection right after CAV 4, while CAVs 6 and 7 are in the same lane as CAV 5, so they select controls that ensure the absence of frontal collisions in the selected optimization window. Finally, CAVs 8 and 9 cross the intersection immediately after CAV 7. Fig. 3(f) illustrates how the re-ordering improves the performance from a system perspective. Indeed, compared to the no re-ordering case, CAV 4 crosses the intersection much earlier, practically improving the distance traveled by all subsequent CAVs 5-9.
In total, the CAVs changing their crossing order at the center of the intersection, i.e., CAVs 1-4, improve in aggregate their expected distance traveled by only 23.7 m. Nevertheless, by changing the order of control updates, the total aggregate distance traveled by all CAVs in the area under the control of the IM improves by 258.3 m. As a final consideration, it is interesting to note the clear effect of γ on the trajectory selected by the CAVs at the start of the pre-danger zone. All CAVs, among all solutions maximizing the distance traveled, always prefer the trajectory that allows them to leave as much space as possible to new incoming CAVs at the start of the pre-danger zone.

C. Applying Event-Triggering and Control Update
Re-Ordering: Performance Evaluation 1) Λ s = 2.5 s: Fig. 4 shows the obtained results when Λ s = 2.5 s. Fig. 4(a) showcases the CDF of the CAVs' average speed in the optimization window T . Given the fact that T is constant for all CAVs, this metric is directly proportional to the distance traveled by the CAVs. Exploiting the updated system state and uncertainty prediction communicated by the CAVs to the IM, the AVOID-PERIOD approach is able to consistently improve the distance traveled by CAVs as compared to the state-of-the-art approach, AVOID-DM. Specifically, by utilizing AVOID-PERIOD all CAVs admitted in the pre-danger zone experience a gain, equal to 6.38% on average, and 10.1% for the 10-th percentile of the CDF (i.e., at the tail of the distribution, where CAVs experience temporary congestion at the intersection). Similarly, AVOID-EVENT also improves the obtained performance, while also presenting significantly lower complexity. Specifically, in the case of AVOID-EVENT the gain is equal to 3.81% on average. Changing the order of the control updates at CAVs, i.e., when considering the AVOID-ORDER approach, further improves performance. Indeed, the CAVs' average speed improves by 9.13%, and by 14.81% for the 10-th percentile of the CDF. Interestingly, the performance of AVOID-ORDER EVENT is practically indistinguishable from the performance of AVOID-ORDER. As a confirmation, the t-test among the average speeds obtained by each pair of approaches was computed. The null-hypothesis was rejected in all cases, i.e., the performance obtained by each approach is statistically different, except for the AVOID-ORDER EVENT and AVOID-ORDER pair. Indeed, in that case, the p-value was equal to 0.15 and the hypothesis that the average speed experienced by CAVs was generated by the same population could not be rejected. This means, that by carefully modifying the order of the CAVs' controls update instead of performing this operation at every time slot, does not significantly alter the system's performance.
Interestingly, even though the analyzed scenario is quite far from congestion (the only discarded arrivals are for CAVs entering the pre-danger zone in the same lane and at the same time and the average time among vehicles admitted to the pre-danger zone is the same for all approaches and equal to 2.7 s), accounting for additional location information and control update re-ordering delivers consistent advantages. Finally, as intuitively mentioned in Section IV-E, it is important to note that in the danger zone all CAVs travel at maximum speed (leaving aside small modification due to the uncertainty associated to the CAVs' system state estimation). Hence, the contention among CAVs occurs in the pre-danger zone, where the times and the order of intersection crossings are decided.
Further, Fig. 4(b) presents the CDF of the distance between CAVs that share a collision area, in case of crossing CAVs, or CAVs in the same lane that are on the same side of the center of the intersection for at least a time slot. Interestingly, all CAVs, with any of the approaches considered, respect the minimum distance of 8 m between CAVs. In general, the minimum distance between CAVs following each other is smaller than the distance between CAVs crossing each other's trajectories. This is a consequence of the fact that the presented optimizations exploit a conservative approach for CAVs at the center of the intersection, imposing at most one ellipse in each of the collision areas between CAVs. Moreover, as expected, AVOID-ORDER and AVOID-ORDER EVENT reduce the distances between CAVs crossing each other's trajectories compared to the rest of the approaches. Indeed, because of the additional knowledge exploited at the moment of planning, and due to a careful selection of CAVs crossing order at the center of the intersection, CAVs can safely get closer to each other. Fig. 5 enables a better understanding of the trade-off between computation/communication complexity and performance reached by the AVOID-EVENT and AVOID-ORDER EVENT approaches. On average, for each CAV traversing the intersection, the average number of events triggering a new control update under the AVOID-EVENT approach is equal to 6.76 and the average number of optimizations performed for each CAV under the AVOID-ORDER EVENT approach (accounting also for all tentative not-optimal re-ordering attempts) is equal to 35.09. The AVOID-PERIOD optimization, instead, triggers a new control at each time slot that a CAV is in the pre-danger zone, i.e., 37.91 times for each CAV. As a result, even though the performance of AVOID-EVENT is similar, especially at the tail of the traveled distance CDF, the computational complexity of AVOID-EVENT is 82.1% less than the one of AVOID-PERIOD. The complexity of AVOID-ORDER reaches, instead, up to 66.94 optimizations for each traversing CAV. Hence, the AVOID-ORDER EVENT's complexity is not only inferior to the AVOID-PERIOD's complexity (even though the achieved speed CDF is nearly-optimal), but it also requires 47.6% less optimizations compared to its periodic alternative.
Finally, in terms of communication, AVOID-EVENT and AVOID-ORDER EVENT also result in reduced overhead; while CAVs are required to send an update on their system state prediction at every slot for all approaches, i.e.. the uplink traffic is always the same, the IM needs to send control updates only when a new CAV's control sequence is obtained.
2) Λ s = 1.25 s: This section shows how results change when the arrivals to the pre-danger zone double, i.e., when Λ s = 1.25 s. The selected density exceeds the traffic density that can be handled by any of the approaches. Indeed, the long-term time between admitted CAVs is equal to: (i) 1.63 s for both AVOID-DM and AVOID-EVENT; (ii) 1.61 s for AVOID-PERIOD; (iii) 1.58 s for AVOID-ORDER EVENT; and (iv) 1.57 s for AVOID-ORDER. As a result, under traffic densities close or above the capacity of the road infrastructure, re-ordering, in conjunction with the use of event-triggering, can increase the number of admitted CAVs. Specifically, when using the AVOID-ORDER optimization framework, the number of admitted CAVs increases by 4.77%, as compared to the benchmark technique used, i.e., AVOID-DM. Fig. 6 demonstrates the performance of all presented approaches with the selected traffic density, in terms of the CAVs' average speed as well as the minimum distance between possibly colliding CAVs. Even though the number of CAVs admitted to the pre-danger zone is larger compared to the benchmark approach, AVOID-DM, Fig. 6(a) shows that all proposed approaches ensure a larger average speed for the admitted CAVs. Given the fact that the average speed of the CAVs in the AVOID-DM approach is lower, the relative improvements ensured by the proposed approaches are similar to the ones previously presented with Λ s = 2.5 s. Specifically, with a higher number of admitted CAVs, the CAVs' average speed improves by 4% for AVOID-EVENT, by 7.5% for AVOID-PERIOD, by 8.58% for AVOID-ORDER EVENT, and, finally, by 12.19% for AVOID-ORDER. Again, running the t-test among the average speeds obtained by each pair of approaches, certifies that the performance obtained are statistically different. With Λ s = 1.25 s, all null-hypothesis were rejected. Nevertheless, the pair AVOID-ORDER EVENT and AVOID-ORDER registered a p-value of 0.01, proving that the two approaches obtain similar performance in heavy-traffic scenarios as well. Fig. 6(b) presents instead the CDF of the distance between CAVs that may collide at any point in time, as previously defined. Even though safety is ensured and distances between CAVs are always greater than 8 m, the CDFs have a very different profile for both the CAVs following each other and the CAVs having intersecting trajectories, compared to the one obtained with Λ s = 2.5 s. As shown, at least 70% of the CAVs following each other in the same lane have a minimum distance less than 25 m. This means that, at some point, CAVs are "queued," one behind the other, waiting for the right moment to cross the intersection. In addition, a much smaller portion of CAVs sharing at some point a possible collision area, B i,j , present minimum pair-wise distances below 150 m. This can be explained by the fact that CAVs take longer to reach the center of the intersection once entering the pre-danger zone, due to the increased traffic density. Hence, considering that the average speed of CAVs in the danger zone is again constant and close to v MAX , this means that the minimum distance between a CAV entering the pre-danger zone and all the CAVs within the danger zone at that moment increases with traffic density.
Finally, Fig. 7 demonstrates how traffic density affects computational effort. First, the number of events triggered for each CAV in AVOID-EVENT is almost constant (7.07). This is due to the fact that the behavior of CAVs in the danger zone is independent of the traffic density (CAVs travel almost at constant speed), hence CAVs find with similar frequency additional space at the center of the intersection, triggering a similar number of control updates. As previously mentioned, the number of optimizations per CAV in AVOID-PERIOD is equal to the average number of time slots the CAV spends within the pre-danger zone. Hence, this number grows with traffic density, i.e., it almost doubles, reaching 61.98 optimizations per CAV. The higher density also affects the computations of AVOID-ORDER. In this case, not only CAVs spend more time in the pre-danger zone, hence being involved for a longer time in re-ordering attempts, but also the number of possible orderings attempted at each time slot increases (due to a larger number of CAVs in the pre-danger zone). For what concerns the complexity of AVOID-ORDER EVENT, the difference in terms of the number of optimizations decreases, as compared to AVOID-ORDER. Indeed, even though the number of events triggered by each CAV is constant, the probability that at least one event is triggered at a time slot increases (due to a larger number of CAVs in the pre-danger zone). Specifically, the number of optimizations per CAV triggered by AVOID-ORDER EVENT is 229.86, i.e., 34.18% less compared to the AVOID-ORDER approach.
It is important to note that, even though re-ordering is considered, the number of optimizations performed by any of the proposed approaches does not grow combinatorially; this is also true when operating close to the capacity region of the road infrastructure.
3) Varying Λ s : In this set of simulations, the average time between CAVs, i.e., Λ s , changes every 500 s (in the range Λ s = 5 − 0.33 s), with an increase of 0.2 veh/s at each step. The objective is to evaluate the performance of the proposed approaches when the road infrastructure experiences changing traffic conditions, from light traffic conditions to traffic congestion. For this reason, both the average number of admitted CAVs in the pre-danger zone and the average speed of admitted CAVs over time (considering a moving average over 2500 s) are depicted in Fig. 8.
As Fig. 8(a) illustrates, before congestion, all approaches admit a very similar number of CAVs into the pre-danger zone. This is mainly due to the fact that there is space for all incoming CAVs. Nevertheless, the average speed of the admitted CAVs is proportional to the efficiency of the approaches presented, as shown in Fig. 8(b). By exploiting event-triggering, control updates re-ordering, and periodic re-optimizations (or a combination of the three) CAVs can travel faster through the intersection, as compared to the performance of the benchmark approach, i.e., AVOID-DM. The situation changes as soon as traffic conditions approach congestion. In this regime, the number of admitted CAVs consistently improves compared to AVOID-DM for all proposed approaches. Considering the best possible approach from a performance point of view, i.e., AVOID-ORDER, the time between admitted CAVs decreases from 1.27 s to 1.22 s, i.e., improving the number of admitted CAVs in the intersection by 4.1%. Further, even though in this regime the number of CAVs in the pre-danger zone is larger with all approaches compared to AVOID-DM, the average speed of admitted CAVs also improves (from 6.76 m/s to 7.4 m/s, i.e., by 9.4%).
Hence, the proposed approaches are able to improve both the capacity of the intersection and the average speed of the admitted CAVs in all traffic regimes (even at congestion), when compared to AVOID-DM, which already exploits an efficient demand management optimization. [12] In this section, the performance obtained by the proposed approach is compared with a state-of-the-art solution, i.e., the optimization proposed in [12], which exploits a different framework. In short, [12] proposes an optimization which allows to optimize intersection capacity while minimizing the CAVs' total applied controls. Contrary to our approach, the work in [12]: (i) exploits continuous time; (ii) imposes the same order between CAVs entering the pre-danger zone and CAVs entering the intersection; and (iii) does not consider system state uncertainty.

D. Assuming No Uncertainty: A Comparison with
Hence, for a fair comparison, the same assumption of no uncertainty is applied to both the solution in [12] and the solution in this work. In the following, given the absence of uncertainty, the AVOID-ORDER optimization does not trigger every time slot a planning re-ordering, but only when a new CAV enters the pre-danger zone. For what concerns the state-of-the-art solution, the optimization (8) in [12] is implemented, with the speed of the first CAV obtained so as to maximize intersection capacity and with v MAX being the speed imposed on the CAVs when entering the center of the intersection.
Similar to Section VI-C3, simulations are obtained varying every 1000 s the average time between CAVs, i.e., Λ s , in the range Λ s = 5 − 1s with an increase of 0.2 veh/s at each step. Fig. 9 summarizes the obtained results. Accounting for planning re-ordering optimization allows AVOID-ORDER to obtain substantial gains over the performance obtained with [12]. This is especially true close to system saturation. Indeed, with Λ s = 1s, AVOID-ORDER simultaneously increases the number of CAVs admitted to the pre-danger zone, i.e., the number of CAVs for which it exists a feasible solution at the entrance to the predanger zone, as well as their average speed. For what concerns this latter metric, AVOID-ORDER is able to improve the average speed of CAVs traversing the intersection by more than 30%, even though the average number of CAVs in the intersection is larger. Apart from the different objective function used by AVOID-ORDER (i.e., AVOID-ORDER forces CAVs entering the pre-danger zone to leave space to other incoming CAVs, increasing de-facto the feasibility region of the optimization), the main difference between the two approaches is represented by planning re-ordering. Hence, also in this set of simulations, planning re-ordering proves to be critical for improving the capacity of autonomous intersections.
The results of this section showcase the flexibility of the proposed framework, since, not only AVOID-ORDER is able to accommodate for CAVs' system state uncertainty, but it is also able to obtain competitive results when no uncertainty is accounted for.

VII. CONCLUSION
In this work, a new framework for autonomous intersection management is presented, that considers simultaneously the CAVs' location uncertainty and demand management, with the following features: (i) exploiting a receding horizon, it performs periodic optimizations for all CAVs' controls (AVOID-PERIOD); (ii) exploiting control update re-ordering, it performs a periodic optimization of the CAVs' crossing order at the center of the intersection (AVOID-ORDER); (iii) in order to reduce computational complexity and communication overhead, an event triggering technique is implemented that re-optimizes CAVs' controls and CAVs' crossing order at the center of the intersection (AVOID-EVENT and AVOID-ORDER EVENT, respectively) only when favorable conditions exist.
Even though the benchmark approach, AVOID-DM, already improves substantially the performance as compared to any other state-of-the-art approaches, the results in this work demonstrate that by also accounting for controls and crossing order re-optimizations the system's performance can improve even further. Indeed, the capacity of the intersection and the average speed of the admitted CAVs (even though on average a larger number of CAVs are admitted) increase when periodic re-optimizations are exploited. This is also true even when no uncertainty is considered, and comparisons are performed with optimization-based approaches exploiting completely different frameworks. This gain is retained even when only a limited, nevertheless carefully chosen, number of control re-computations are executed, as in the extensions AVOID-EVENT and AVOID-ORDER EVENT. The proposed event-triggering techniques not only consistently improve performance, but they also have the ability to easily scale when heavy-traffic conditions are considered.
Future research directions involve the study of solutions that include also CAVs turning, as well as mixed-traffic scenarios (i.e., also including human-driven vehicles). For what concerns the former, the analysis requires including control-dependent uncertainty, so that steering can be correctly modeled. A possible solution includes the use of polynomial chaos [38]. For what concerns the latter, thanks to precise short-term trajectory predictions, e.g., exploiting [39], human-driven vehicles can be considered as moving obstacles and accounted for during optimization.