Abstract

In this paper, we present a decentralized control framework for the coordination of connected vehicles and traffic lights in urban intersections. The framework uses gradient-based methods to dynamically coordinate, at every time-step, the planned intersection arrival times of vehicles and the planned switching times of traffic lights. Assuming no constraints, vehicles then use analytical optimal control methods to determine a nominal acceleration profile intended to place them at the intersection at the desired arrival time. Finally, using control barrier functions, the agents' accelerations and arrival times are modified to ensure safety and feasibility. The work in this paper builds on existing research on safe-set nonlinear control methods, multi-agent gradient-based methods, and optimal control methods used to tackle the intelligent intersection management problem. Here, we integrate the use of these different approaches to present a comprehensive control architecture that can flexibly coordinate the timing of both vehicles and traffic lights, while maintaining safety and feasibility. Simulations show that the method can save between 4% and 15% in fuel consumption and between 55% and 70% in delay for different traffic conditions.

1 Introduction

As our vehicle fleet and traffic infrastructure become more autonomous and connected, opportunities to improve the efficiency of traffic networks through the use of planning, coordination, and control emerge. In urban networks specifically, the use of these technologies can change the way vehicles navigate traffic intersections, which are perhaps the most common areas where congestion occurs. Indeed, by assuming connectivity and automation, researchers propose multiple ways in which one can control vehicles, traffic lights, or both in order to reduce time and energy costs. This paper proposes an approach that combines ideas from gradient-based multi-agent control, optimal control, and nonlinear control to coordinate vehicles and traffic lights in a decentralized, computationally efficient manner.

A rich body of work explores the control of connected and autonomous vehicles approaching unsignalized intersections. In Refs. [1] and [2], comprehensive reviews of such approaches are presented. A common structure for the solution of this problem found across the literature consists of separating the problem into a timing coordination layer, and a tracking trajectory planning layer. Approaches of this type differ in how arrival times are negotiated and how speed trajectories are computed. Reservation-based approaches rely on a centralized controller that handles requests from incoming vehicles and decides on their arrival times through heuristic rules [3], optimization [4], or both [5]. In Ref. [6], the intersection coordinators are connected across multiple junctions using multi-agent consensus to improve the performance of the strategy at a network-wide level. Decentralized approaches negotiate arrival times using different methods. For example, in Ref. [7], vehicles decide their arrival times through heuristic rules like maintaining first-in-first-out (FIFO) queue orders. In Refs. [8] and [9], arrival times are determined through a recursive decentralized minimization of intervehicles time gaps, also under a FIFO constraint. In Ref. [10], Kuramoto synchronization arranges arrival times so that vehicles cross the intersections in interweaving patterns.

A common issue encountered by the above work lies in the computational complexity and connectivity requirements of optimally tracking a given arrival time while considering other vehicles. Indeed, the work presented in Ref. [11] explores in detail the analytical solution to the arrival time tracking trajectory planning problem, and shows the difficulty of finding closed-form solutions when considering all safety constraints. An alternative approach of particular relevance to the work presented here consists of generating a controller based primarily on safety considerations. For example, in Ref. [12], control barrier functions (CBFs) are used to synthesize a safety-critical decentralized control strategy for the vehicle merging problem, which is similar to the intelligent intersection management problem. The work is further extended in Ref. [13] to include a reference velocity that is explicitly related to coordinated desired arrival times.

While the work on unsignalized intersection control of connected and autonomous vehicles is promising, there is benefit in formulating the problem considering actuated traffic lights that coordinate their timing along with the vehicles. This configuration is tackled in the literature with similar approaches to the unsignalized case, where timing optimization is performed by a central agent, and trajectory optimization is performed by the vehicles [1419]. While the approaches differ in their chosen traffic light and vehicle models, they typically involve centralized optimization, especially for the timing coordination layer.

In this work, we consider the decentralized control of both vehicles and traffic lights. Moreover, we are interested in the predictive control of these agents through the use of a timing state that governs when vehicles plan to arrive at the intersection and when the lights plan to switch. Indeed, such a control strategy continuously modifies not only the acceleration of the vehicles but also the planned state trajectories into the future of both the vehicles and the lights. As we discussed in the previous paragraphs, considering safety and feasibility constraints over a planning time horizon becomes computationally prohibitive. To address this issue, in this paper, we make use of safe sets and CBFs to generate a safety regulator that modifies a nominal controller that negotiates arrival times and plans acceleration trajectories (without explicitly considering safety and feasibility constraints). The main differences between this and previous contributions that use CBFs (i.e., Refs. [12] and [13]) are the inclusion of traffic lights and the consideration of an explicit state variable that governs the arrival times of the vehicles.

In Ref. [20], we explore the coordinating and tracking control strategies that we use here as nominal controllers. There, coordination of arrival times is performed using gradient-based multi-agent methods, and arrival tracking is performed using analytical unconstrained optimal trajectory planning. In this previous work, car following safety constraints are implemented based on the Gipps model overriding the nominal acceleration commands, and actuation constraints are implemented by saturating the range of allowable arrival times of the vehicles. The interactions between different safety and feasibility constraints, and between traffic lights and vehicles were not formally analyzed. Thus, one of the main contributions of this works is the derivation of a safety regulating controller in conjunction with previously explored coordinating strategies.

The rest of the paper is organized as follows: In Sec. 2, we describe the system and its dynamics. In Sec. 3, we give an overview of the proposed strategy. In Sec. 4, we define our nominal controllers. In Sec. 5, we derive the safety regulator, which modifies the nominal control signals. Finally, in Sec. 6, we evaluate our controller in simulation.

2 System Model

The system consists of a multi-agent fleet of vehicles inside a control zone around a traffic intersection, plus the traffic light servicing the different flows at the intersection. The control zone is given by a maximum distance L from the intersection stop lines, along the lengths of the roads approaching it. In this section, we define the states, control inputs, and dynamics of these agents, as well as assumptions about their connectivity, autonomy, and allowed behavior.

We assume that once the vehicles enter the control zone, they can no longer change lanes. With regard to connectivity, we assume that all agents exchange their current state information with each other according to the following network topology. Vehicles in the same road can communicate with each other and with the upcoming traffic light. The traffic light, in turn, can exchange information with all vehicles approaching from all roads. As such, we can represent the network of interconnected agents as nodes in a bidirectional graph, and their connections as edges. We call Ni the set of neighbors of agent i, where a neighbor refers to another agent that agent i exchanges information with. To illustrate this network topology, we show in Fig. 1 an example set of agents and their interconnections. We note that this connectivity topology avoids communication between agents in different roads in the interest of reducing communication burden and finding a simple solution to the coordination control problem.

Fig. 1
Schematic of network connectivity topology
Fig. 1
Schematic of network connectivity topology
Close modal

2.1 Vehicle Dynamics.

We consider the set V(t) of all vehicle agents currently approaching the intersection. We assume the ith vehicle's state is given by its distance to the intersection si, its speed vi, and its current planned desired time till arrival (DTTA) τi. The vehicle can control its acceleration ai, and can change its DTTA by an added rate ui. The dynamics of the vehicle are then
si˙(t)=vi(t)vi˙(t)=ai(t)τi˙(t)=1+ui(t)
(1)

The DTTA of a vehicle is a virtual state variable that encodes when the vehicle is planning to reach the intersection, and it evolves in time according to the coordinating input ui. The underlying assumption of using the above longitudinal vehicle dynamics model is that lane keeping and powertrain controllers can keep the vehicle in its lane and meet the acceleration command used in Eq. (1). We show the state of three vehicles approaching a three-way intersection in Fig. 2. The set V here contains agents 1, 2, and 3. In the spatial representation, we can see the current distance to the intersection for each car s1, s2, and s3. In the timing domain, we can see their planned arrival times τ1, τ2, and τ3.

Fig. 2
Vehicles and traffic light represented in the spatial and timing domain
Fig. 2
Vehicles and traffic light represented in the spatial and timing domain
Close modal

2.2 Traffic Light Dynamics.

The traffic light is modeled by its desired times till switching (DTTS). For the purposes of the multi-agent control strategy described in Sec. 4.1, each switch is considered its own independent agent. This means that a single traffic controller is in charge of multiple virtual agents, each associated with a switch from one phase to the next of that traffic light. We therefore consider the set L(t) of all upcoming planned DTTS, governed by the same dynamics as the DTTA of the vehicles
τi˙(t)=1+ui(t)
(2)

We assume here that the actuated light follows a predetermined sequence of phases. If the intersection is organized around compatible movements grouped into P phases, we can index each phase from 1 to P, and assume that transitions happen in ascending order. Furthermore, we define a minimum fixed yellow time Tyellow that separates the transition between phases. When the leading DTTS reaches Tyellow, the light switches from green to yellow for the current phase. When it reaches 0, the light switches from yellow to red for the current phase, and from red to green for the next phase in the sequence. We assume that the traffic light plans some time horizon TH into the future, and that new planned traffic light switching events only spawn at TH.

We illustrate the state of a traffic light controlling three flows in Fig. 2. In this example, agents 4, 5, and 6 belong to L(t). In the spatial domain, we can only see the current state of the light, which is servicing phase 1 at this time. In the timing domain, we can see the planned switches of the light τ4, τ5, and τ6 over the coming TH seconds.

3 Proposed Approach Overview

Given the above system of interconnected vehicle and traffic light agents, our objective is to synthesize a decentralized control strategy that generates the inputs ai and ui for all agents. That is, in the decentralized strategy, each agent should decide its own control action based only on the information received by its neighbors. Compared to human-driven and fixed-time traffic light control strategies, the controller should reduce the energy and delay incurred by having to stop at the light, while maintaining safety and feasibility constraints.

The proposed approach consists of having the agents continuously negotiate their timing by changing their DTTA/S. Negotiation is done through decentralized multi-agent methods; here, we use a gradient-based strategy. Then, a decentralized safety regulator for each agent considers constraints on intervehicle spacing, vehicles arriving on green periods, and consistent traffic light timings.

For each vehicle agent, the proposed control strategy is illustrated in Fig. 3. A trajectory planning controller uses the current position, speed, and DTTA of the vehicle to compute the nominal acceleration ai* that would place the vehicle at the intersection at the current DTTA, assuming the underlying safety constraints are satisfied. Simultaneously, a coordinating controller uses the communicated DTTA/S information from neighboring agents to determine the nominal change in timing ui*, assuming feasibility of the current DTTA is satisfied. Finally, a quadratic program (QP) controller minimizes the deviation from the nominal control input under safety and feasibility barrier certificate constraints.

Fig. 3
Control architecture for a vehicle agent
Fig. 3
Control architecture for a vehicle agent
Close modal

For the traffic light, the proposed control strategy is illustrated in Fig. 4. The approach is similar to that of the vehicles with a few key differences. The traffic light controller does not need a tracking algorithm, since there are no physical constraints on the rate at which it can change its DTTS. Furthermore, while each vehicle only controls its own DTTA, the traffic light controls all of its planned switching times together.

Fig. 4
Control architecture for the traffic light
Fig. 4
Control architecture for the traffic light
Close modal

Remark 1. As shown in Figs. 3 and 4, and detailed in Secs. 4 and 5, the proposed strategy has attractive properties in terms of computational complexity. The nominal acceleration control is an algebraic function of the state of the vehicle. The nominal coordinating input is an algebraic function of the timing of the agent's neighbors. Finally, the safety regulator is a convex quadratic program with linear constraints.

4 Nominal Controller

The nominal controller aims to coordinate the vehicles and lights by modifying their timings, and tracking the desired arrival times through trajectory planning optimal control, assuming that the underlying safety regulator will handle the safety and feasibility constraints. We denote the nominal control signals as ui* and ai*.

4.1 Coordinating Controller.

For the nominal timing coordination, we choose a gradient-based multi-agent control strategy. This approach is commonly used in multi-agent navigation and formation control problems [21]. As such, the coordinating input ui* is given by the negative gradient with respect to τi of some suitably defined positive semidefinite potential field Vi, which is a function of the timing of all the agents that neighbor i. We note that the function Vi encodes the coordination objective of the network, and not necessarily a specific physical quantity that we might aim to explicitly optimize for. The agent's potential Vi is defined as the sum of edge tension functions Vij associated with each edge between agent i and its neighbors jNi
ui*(τi,τjNi)=τiVi=jNiτiVij(τi,τj)
(3)
Since the main goal of this paper is the derivation of the safety regulator, here we use the same coordinating potential function presented in Ref. [20]. Namely, the edge tension functions (ETFs) Vij between pairs of connect agents i and j are defined using the logistic function
Vij=Kij1+eκij(τiτj)
(4)

As a function of τi, the above expression looks like a smooth step function centered at τj. The sign of the growth rate parameter κij determines whether the function goes from Kij to 0, or from 0 to Kij. The magnitude of κij determines how steep the transition is. The direction of the step defines whether agent j hastens or delays agent i; the steepness of the step defines how close the agents need to be for the interaction to be non-negligible, because for large values of |τiτj| the gradient of the function approaches 0. By defining the ETFs using this logistic function, we can think of agent j as pushing agent i to the left or to the right in the timing domain based on the sign of κij. The function, and sign of its parameters, is chosen by taking into consideration the desired interaction we can heuristically stipulate the agents should have. These are:

  • The DTTA of vehicles should be hastened by traffic light switches to red (to catch an earlier green window, or stay in the current one) and delayed by switches to green (to catch a later green window, or stay in the current one). This drives a vehicle's arrival time toward the closest green interval.

  • The DTTA of vehicles is delayed by preceding vehicles and hastened by following vehicles. This encourages vehicles to maintain appropriate headways with each other.

  • Traffic light switching times should repel each other. This encourages the switches to preserve their ordering.

In Table 1, we summarize the sign κij based on the relative timing, position, and type of agents i and j. The set Gi used in Table 1 is defined for all vehicles iV as the set of traffic light DTTS jNiL such that jGi if τj represents a switch to green for vehicle i.

Table 1

Sign of ETF growth rate κij based on the type of agents i and j and their relative timing or position

Agent j
jVjL
Agent iiVsj < siκij=κ1jGiκij=κ3
sj > siκij=κ2jGiκij=κ4
iLiGjκij=κ5τi<τjκij=κ7
iGjκij=κ6τi>τjκij=κ8
Agent j
jVjL
Agent iiVsj < siκij=κ1jGiκij=κ3
sj > siκij=κ2jGiκij=κ4
iLiGjκij=κ5τi<τjκij=κ7
iGjκij=κ6τi>τjκij=κ8

To illustrate the workings of the timing coordination gradient-based controller, in Fig. 5, we show an example with three agents. We see in the timing domain (top plot) the DTTS of a light (black dot), the DTTA of two vehicles arriving in a given road (blue dots), and the DTTA of a third vehicle approaching from a different road. In the bottom, we show the total potential Vi of the traffic light's DTTS as a function of its planned timing. This total potential is the sum of the three ETFs Vij of the light with each of the three plotted agents. We can see that the ETFs we have defined using the logistic functions will delay the switching of the light through gradient descent so that all vehicles arrive in a green window.

Fig. 5
(Top) DTTS (middle black dot) of a traffic light, and DTTA (bottom blue dots and top red dot) of three vehicles approaching from two different roads. (Bottom) Total potential function Vi of the light's DTTS (bold black line) and its component ETFs with each vehicle agent (middle blue lines and bottom red line).
Fig. 5
(Top) DTTS (middle black dot) of a traffic light, and DTTA (bottom blue dots and top red dot) of three vehicles approaching from two different roads. (Bottom) Total potential function Vi of the light's DTTS (bold black line) and its component ETFs with each vehicle agent (middle blue lines and bottom red line).
Close modal

4.2 Optimal Tracking Controller.

At any time t, the nominal acceleration is the outcome of the following optimal control problem:
ai*(t)=argminai(ξ=t)tt+τi(t)12ai(ξ)2dξ
(5)
Subject to
s˙i(ξ)=vi(ξ)v˙i(ξ)=ai(ξ)si(ξ=τi(t))=0si(ξ=t)=si(t)vi(ξ=t)=vi(t)
(6)

The cost functional of Eq. (5) is the square of acceleration. While minimizing acceleration is not the same as minimizing fuel consumption (or minimizing energy usage for electric and hybrid vehicles), it can serve as simpler proxy for these quantities. The constraints in Eq. (6) consider the vehicle dynamics and ensure that the vehicle reaches the intersection at the current DTTA, starting from the current position and speed. We do not consider other safety and feasibility constraints here so the optimization problem remains tractable; these constraints are considered in Sec. 5. Indeed, following the work presented in Ref. [11], the above optimization problem can be solved analytically by defining costates λis and λiv, constructing the Hamiltonian, and imposing necessary conditions for optimality.

The Hamiltonian is
H=12ai(ξ)2λis(ξ)vi(ξ)+λiv(ξ)ai(ξ)ξ[0,τi]
(7)
From the input stationarity condition, we have that the optimal acceleration ai* follows:
aiH=ai*(ξ)+λiv(ξ)=0ai*(ξ)=λiv(ξ)
(8)
From the costate equations, we can solve for the costates in terms of integration constants c1 and c2
λis˙(ξ)=siH=0λiv˙(ξ)=viH=λisλis(ξ)=c1λiv(ξ)=c1ξc2ξ[0,τi]
(9)
It follows from Eqs. (8) and (9), as well as from further integration of acceleration and position, that the optimal trajectories are
ai*(ξ)=c1ξ+c2vi*(ξ)=12c1ξ2+c2ξ+c3si*(ξ)=16c1ξ312c2ξ2c3ξ+c4
(10)
We can solve for the integration constants by enforcing the boundary conditions: si*(0)=si(t),vi*(0)=si(t),si*(τi(t))=0 and λiv(τi(t))=0
c1=3c3τi(t)c4τi(t)3c2=c1τi(t)c3=vi(t)c4=si(t)
(11)
Finally, we can get the nominal acceleration input as the value of ai* at ξ = 0, which from Eqs. (10) and (11) yields
ai*(t)=3si(t)vi(t)τi(t)τi(t)2
(12)

5 Control Barrier Function Safety Regulator

Given the nominal acceleration and coordinating input, the safety regulator will modify this input to maintain safety and feasibility constraints. Specifically, the vehicle's safety regulator simultaneously regulates acceleration and change in DTTA so that: (1) the safe intervehicle distance is maintained, (2) vehicles stop at red lights, (3) speed and acceleration bounds are satisfied, (4) the DDTA is such that the nominal trajectory solution respects speed and acceleration bounds. The light's safety regulator ensures the switching of phases maintains the correct order and that switches to red do not occur for vehicles that can longer stop the light. To do this, we make use of safety-critical control methods. Specifically, we seek the forward invariance of a safe set in the state space by the defining control barrier functions and enforcing their induced barrier certificates.

5.1 Background: Control Barrier Functions.

The control barrier function method has been recently developed for safety-critical control applications, and a comprehensive introduction to the method can be found in Ref. [22]. Here, we recall the main definitions and theorems from Ref. [22] required in the formulation of our safety regulator.

We start by defining a control affine system, with states xDn, input uUm, and locally Lipschitz dynamics f: nn and g: nn×m
x˙=f(x)+g(x)u
(13)

Definition 1. A setCis forward invariant if for every initial conditionx(0)=x0C,x(t)Cfor all time t.

Definition 2. SetCis the superlevel set of a continuously differentiable function h:Dnif
C={xDn:h(x)0}
(14)

Definition 3. An extended classKfunction is a strictly increasing function κ:withκ(0)=0.

Definition 4. The continuously differentiable function h:Dnis a control barrier function (CBF) if there exists an extended classKfunction κ such that for the system (13)
supuU[Lfh(x)+Lgh(x)Tu+κ(h(x))]0
(15)

for allxD.

The above definition uses the Lie derivative formalism to express the time derivative of the function h. Indeed, for locally Lipschitz dynamics and CBFs: h˙(x,u)=Lfh(x)+Lgh(x)Tu

Intuitively, the condition says that, if there exists some input u and some extended class K function κ such that h˙(x,u)κ(h(x)), we call h a control barrier function. This is useful because applying any input u that makes h˙(x,u)κ(h(x)) will render the super-level set of h forward invariant. In fact, this constitutes the main theorem involving control barrier functions.

THEOREM 1 (from Ref. [22]). LetCbe the superlevel set of a continuously differentiable function h:Dn. If h is a control barrier function andxh(x)0for allxC, then any Lipschitz continuous controlleru(x)Kcbf(x)such that
Kcbf(x)={uU:Lfh(x)+Lgh(x)Tu+κ(h(x))0}
(16)

renders the setCforward invariant and asymptotically stable in D.

The constraints on the input that define the set of allowable controllers are affine in the input. Given this property, we can synthesize a safe and computationally efficient controller using a linear or quadratic program, as shown in Ref. [23]. Given a nominal controller u*(x) that is not guaranteed to be safe, but that aims to achieve some other control objectives, a QP problem can be designed to minimize the deviation from the nominal controller subject to the CBF constraint on the input
u(x)=argminuU12||u(x)u*(x)||22Subjectto:Lfh(x)+Lgh(x)Tu+κ(h(x))0
(17)

The solution to this problem is shown to be Lipschitz continuous in Ref. [23]. When we have multiple constraints, or other constraints on the input, we can define candidate CBFs and formulate the above QP problem with one input constraint for each candidate CBF. In this case, the candidate CBFs would be valid CBFs if the QP problem has a solution for all xD.

5.2 Vehicle Safety and Feasibility Constraints.

In this section, we make use of the above theoretical framework to synthesize a safety regulator for the vehicles. To do this, we introduce constraints on the states of the vehicles so that they remain in a safe configuration. The constraints are then translated into candidate control barrier functions, which are in turn transformed into barrier certificate constraints on the inputs.

Before defining the candidate barrier functions, we first establish the equivalence between our system (1) and the control affine system (13). For the vehicle agents, we use the following definitions for the state and input vectors and their dynamics:
xi=[si,vi,τi]Tui=[ai,ui]Tf=[vi,0,1]Tg=[001001]
(18)
In the following, each new candidate CBF we consider is denoted by hq(x), for q=1,2,..,8 (i.e., we define up to eight different constraints on the states). Each hq is associated with its own safe-set Ci,q, and its induced allowable input space Ui,q(x) as defined in Eq. (16). Since the barrier certificates are linear constraints on the input, we can organize them together into a single matrix inequality A(i)uib(i). The rows of the inequality define Ui,q(x)
Ui,q(xi)={ui2:Aq,1(i)ai+Aq,2(i)uibq(i)}
(19)
Once all constraints are considered, any input uiUi(xi) will maintain the vehicle i inside of the safe set, where
Ui(xi)=q=0QUi,q(xi)
(20)

where Q is the total number of considered constraints.

Constraint 0 (acceleration bounds). The first feasibility assumption we make for the vehicles is that their acceleration is constrained
amin,iaiamax,i
(21)

In practice, the values of amin,i and amax,i can be derived from the limitations of the vehicle's powertrain and the road.

Note that the acceleration constraints are not constraints on the states, but on the input, as such no CBF is associated with them. They define an initial admissible input set Ui,02
Ui,0={[ai,ui]T2:amin,iaiamax,i,ui}
(22)
Constraint 1 (speed limitations). Vehicles are constrained by the speed limits of the road, so that
vminvivmax
(23)
These constraints can be directly turned into candidate control barrier functions
h1(xi)=vivmin
(24)
h2(xi)=vmaxvi
(25)
To compute the corresponding barrier certificates, we choose a linear extended class K function κ(hp)=γi,php, where a different value of γ>0 can be used for each CBF
ai+γi,1(vivmin)0
(26)
ai+γi,2(vmaxvi)0
(27)
Using the notation of Eq. (19), we define the coefficients of the matrix inequality
A1,1(i)=1A1,2(i)=0b1(i)=γi,1(vivmin)
(28)
A2,1(i)=1A2,2(i)=0b2(i)=γi,2(vmaxvi)
(29)

Constraint 3 (arrival time feasibility). The next constraints we consider limit the range of possible desired times till arrival (DTTAs) τi to those that are consistent with the vehicle's speed and acceleration limitations. In fact, we will limit the DTTAs to those whose nominal acceleration ai* and speed vi* trajectories (Eq. (10)) do not violate acceleration and speed constraints (Eqs. (21) and (23)).

From Eq. (10), we can see that the nominal trajectories for acceleration and speed are affine and quadratic functions of time, respectively. What is more, for t<ξ<t+τi, acceleration does not change signs; it starts at its extremum at ξ=t, then either increases or decreases until ai*(ξ=t+τi)=0. It follows that maximum or minimum speed occurs at ξ=τi. From this analysis, we can find corresponding constraints on the state space
amin,iai*(ξ=t)amax,i
(30)
vminvi*(ξ=t+τi)vmax
(31)
Using Eqs. (10) and (11), we can rewrite these constraints as candidate CBFs
h3(xi)=amax,i3si(t)vi(t)τi(t)τi(t)2
(32)
h4(xi)=3si(t)vi(t)τi(t)τi(t)2amin,i
(33)
h5(xi)=vi(t)+3si(t)vi(t)τi(t)2τi(t)vmin
(34)
h6(xi)=vmaxvi(t)3si(t)vi(t)τi(t)2τi(t)
(35)
Taking time derivatives of hq, we can compute the corresponding coefficients of Eq. (19)
A3,1(i)=3τaiA3,2(i)=3viτi26(viτisi)τi3b3(i)=6(viτisi)τi3+γi,3h3(xi)
(36)
A4,1(i)=3τaiA4,2(i)=3viτi2+6(viτisi)τi3b4(i)=6(viτisi)τi3+γi,4h4(xi)
(37)
A5,1(i)=12A5,2(i)=3vi2τi3(viτisi)2τi2b5(i)=3(viτisi)2τi2+γi,5h5(xi)
(38)
A6,1(i)=12A6,2(i)=3vi2τi+3(viτisi)2τi2b6(i)=3(viτisi)2τi2+γi,6h6(xi)
(39)
Constraint 7 (safe car following). The next constraint we introduce considers the spacing between the vehicle and its leader. For notation, let l(i):=j if j is the directly ahead of vehicle i in its lane. We can then write a spacing constraint that requires vehicles to maintain a minimum distance Dmin between them
sisl(i)Dmin
(40)
We need a control barrier function that allows vehicles to follow this constraint while satisfying speed and acceleration bounds previously established and considering the possible behavior of the lead vehicle. In the Appendix, we present a detailed derivation of the barrier function that assumes a worst case scenario where the lead vehicle performs a maximum deceleration maneuver. The resulting expression can be stated in terms of the minimum braking distances Bi and Bl(i) of both vehicles
h7(xi,xl(i))=sisl(i)Dmin+Bi(vi)Bl(i)(vl(i))
(41)
where the minimum braking distance of a vehicle is given by
Bi(vi)={vi22amin,i+amin,i2γ12ifviamin,iγ1viγ1ifvi<amin,iγ1
(42)
Taking the time derivative of Eq. (41), and assuming the lead vehicle decelerates at a maximum rate, we get the corresponding coefficients of the constraints in the input
A7,1(i)={viamin,iifviamin,iγ11γi,1ifvi<amin,iγ1A7,2(i)=0b7(i)=vi+γ7h7(xi,xl(i))
(43)

Remark 2. This barrier function only needs to be considered when the speed of the vehicle is larger than its leader's. This is because in the domain where vi < vl constraint (40) will always be satisfied, and no certificate is needed. This is further explained in the Appendix.

Constraint 8 (safe crossing constraint). Finally, we introduce a constraint that guarantees that the vehicle only enters the intersection when it is planning to arrive on green.

The idea consists of reproducing the previous barrier function h7 but assuming the leader's distance to the intersection and speed are both zero, as if there was a vehicle stopped at the intersection stop line
h8(xi)={siDmin+vi22amin,i+amin,i2γ12vivi,TsiDmin+viγ1vi<vi,T
(44)
The induced admissible set of inputs is given by
A8,1(i)={viamin,iifvivi,T1γi,1ifvi<vi,TA8,2(i)=0b8(i)=vi+γi,8h8(xi)
(45)

The above constraint is only used when the vehicle is planning to arrive during a red period of time. Notice then that h8 can only become negative when the vehicle is planning to arrive on green. When h8 is negative, the vehicle will no longer delay its DTTA (i.e., ui0), and the traffic light will ensure that the light's planning does change in such a way that the vehicle is then forced to arrive on red. This is explained in Secs. 5.3 and 5.4.

Definition 5. Let G(i) be an indicator function that is 1 when vehicle i is planning to arrive on green, and 0 when it is planning to arrive on red.

5.3 Vehicle Quadratic Program Safety and Feasibility Regulator.

We can now define the QP safety regulator for each vehicle i
ui(xi,xj)=argminai,ui12(aiai*(xi))2+12(uiui*(xi,xj))2Subjectto:amin,iaiamax,iAq,1(i)ai+Aq,2(i)uibq(i)forq=1,,6A7,1(i)ai+A7,2(i)uib7(i)ifvi>vl(i)A8,1(i)ai+A8,2(i)uib8(i)ifG(i)=0ui0ifh8(xi)0
(46)

5.4 Traffic Light Safety Constraints.

For the traffic light, we also establish constraints on the allowable rate with which the planned switching times can be changed based on candidate CBFs. We then formulate a safety regulating QP controller that modifies the nominal change in timing generated by the coordinating gradient-based control strategy.

While we consider each DTTS of the light as separate agent for the purposes of gradient-based coordination, in this section, we derive a single QP safety regulator for all the DTTS of a single light. This is different from the regulator derived for the vehicles, which assumes only the acceleration and rate of change of τi are control variables. Here, ui for all iL(t) are optimization variables of the QP regulator. We therefore define the vector state of the light xL and the vector input uL as a concatenation of all τi and ui for all iL(t).

Constraint 9 Given that we are assuming the light follows a predetermined sequence of phases, and that transitions occur whenever the leading DTTS reaches Tyellow, we can guarantee the safe operation of the traffic light by ensuring that the DTTS do not overtake each other and always stay Tyellow seconds apart in the planned timing domain
h9(τi,τj)=τjτiTyellowi,j{i,jL(t):τi<τj}
(47)
The barrier certificate input constraints are then
ujuiγi,9h9(τi,τj)i,j{i,jL(t):τi<τj}
(48)
Constraint 10. The second set of constraints on the DTTS ensures that once a vehicle can no longer stop at the upcoming intersection, the light does not change its plan in such a way that the vehicle arrives in red. This constraint is used once a vehicle's h8 CBF is negative (which can only happen if the vehicle is planning to arrive on green). In other words, once the vehicle crosses a safe stopping threshold, the light cannot hasten or delay its switches and make the vehicle arrive on red
h10(τi,τj)={τiτjifτiτjτjτiifτi<τjiL(t)andj{jV(t):h8(xj)0}
(49)
Assuming that vehicle j uses uj = 0, the barrier certificate input constraints are then
{uiγi,10h10(τi,τj)ifτiτjuiγi,10h10(τi,τj)ifτi<τjiL(t)andj{jV(t):h8(xj)0}
(50)

5.5 Traffic Light Quadratic Program Safety Regulator.

We can now define the QP safety regulator controller for each traffic light
uL(xL,xj)=argminuL12||uLuL*(xL,xj)||2Subjectto:Eqs.(48)Eqs.(50)
(51)

Remark 3. The above QP controller for the traffic light simply attempts to preserve the order of the DTTS. When some vehicles can no longer stop at the intersection, it also attempts to preserve the order of the DTTA of these vehicles relative to the DTTS of the light. We can see that there exists at least one feasible solution by noting that ui=0iL satisfies all of the constraints. Indeed, if the agents start in a certain order, and they all set their inputs to 0, they should preserve that order, given their homogeneous affine dynamics.

5.6 Constraint Relaxations.

As we have discussed previously, the QP-CBF formulation has been derived using candidate CBFs, which are only valid CBFs in the presence of each other if the QP problem has a feasible solution (UiØ) for all points inside the safe set (xiCi). In this section, we consider the possibility of the set Ui being empty for the vehicles, and we introduce appropriate constraint relaxations to ensure that the QP always has a solution. Remark 3 in the previous subsection discusses how this is not necessary for the traffic light.

In Fig. 6, we show an example of the vehicle's constrained input space Ui(xi,xl(i)), for a given value of xi and xl(i). Each line corresponds to one of the barrier certificates, and the adjacent shaded blue region corresponds to the inputs that the certificate does not allow. The remaining white space delineates then the inputs that satisfy all barrier certificates.

Fig. 6
Example schematic of the allowable input set Ui(xi,xl(1)) for a given value of xi and xl(i)
Fig. 6
Example schematic of the allowable input set Ui(xi,xl(1)) for a given value of xi and xl(i)
Close modal

In Sec. 5.2, we derive constraints 7 and 8 to be explicitly compatible with constraints 0 and 1 using Theorem 2, and we can show their compatibility with constraint 2 as this constraint cannot become active when the vehicle is braking as long as vmax>vmin. This implies that in Fig. 6, the vertical region between Ui,1 and Ui,7 (and Ui,8) will always be nonempty. In other words, Ui,7 (and Ui,8) will always be to the right of Ui,1.

On the other hand, the derivation of constraints 3–6 does not consider any constraint interactions. As such, the polytope formed by these four lines might become empty, or it might not intersect with the vertical region formed by Ui,1 and Ui,7 (or Ui,8). Given this possibility, we introduce a constraint relaxation input δi,q in Ui,q, for q=3,4,5,6, as follows:
Ui,q(xi)={ui2:Aq,1(i)ai+Aq,2(i)ui+δi,qbq(i)}q{3,4,5,6}
(52)
We can relax the vehicle's QP safety regulator defined in Eq. (46) by adding the δi,q as optimization variables, replacing the appropriate constraints by the ones defined above (Eq. (52)), and replacing the optimization objective by the following:
12(aiai*(xi))2+12(uiui*(xi,xj))2+Rq=36δi,q2
(53)

The relaxation weight R is chosen to be a very large number to heavily discourage the controller form violating the constraint.

Remark 4. In practice, there are other reasons to relax the constraints and heavily penalize their violation. First, in the field and in simulation, this strategy will be implemented as a discrete-time controller where, near the boundary of the safe set, the forward invariance of the set may not be guaranteed if the integration time step size is not small enough. Second, in real roads with human drivers, some of the assumptions about the behavior of the leading traffic could be violated, unwillingly placing the vehicle outside its safe set.

6 Simulation Results

In this section, we evaluate our control strategy in simulation, using our own matlab [24]-based tool opentrafficlab [25]. We consider a three-way intersection controlled by a traffic light, the same configuration as shown in Fig. 2.

We assume that vehicles arrive in each lane according to a Poisson arrival process, given by an input flow rate in vehicles per hour. Arrivals are delayed to the extent that they would have the vehicle enter outside of the safe set. For the “arrival” of traffic light switching events, if the light plans a time horizon TH into the future, a new DTTS is added at τi=TH when the last DTTS is some minimum time Tmin into the planned timing domain.

Assuming a 2 s headway, the saturation flow rate (i.e., the maximum number of vehicles that can traverse the network at the nominal desired speed) of each lane is 1800 veh/hr. We simulate our proposed strategy and baseline at different ratios of the saturation flow rate using the parameters summarized in Table 2. For each saturation ratio, we generate 10 realizations of the arrival process, and we use the same generated arrival times for both the baseline and the proposed strategy.

Table 2

Parameter values used in simulation

ParameterValueUnits
L100m
amin,i−5m/s2
amax,i3m/s2
vmin0m/s
vmax15m/s
vdes10m/s
Dmin7m
κ1,κ81
Kij10
γi,1,γi,251/s
γi,3,γi,101
R106
TH5s
Tmin5s
Tyellow2s
ParameterValueUnits
L100m
amin,i−5m/s2
amax,i3m/s2
vmin0m/s
vmax15m/s
vdes10m/s
Dmin7m
κ1,κ81
Kij10
γi,1,γi,251/s
γi,3,γi,101
R106
TH5s
Tmin5s
Tyellow2s

6.1 Baseline.

The baseline strategy consists of human drivers and fixed time traffic lights at the intersection. The human drivers are modeled using the Gipps car following model [26], with a desired speed vdes. To account for traffic lights, the baseline car following model acts as if a virtual vehicle is stopped at the intersection when the light is red. We note that other baselines models can be used to simulate the behavior of traditional traffic, like the intelligent driver model or the Wiedemann model [27]. The timing of the traffic light is determined using Webster's method [28], based on the demand (i.e., arrival rate) at each road.

6.2 Results.

We can look at the savings in fuel and delay incurred if all the vehicles used our proposed strategy over the baseline. In Fig. 7, we show the average savings per vehicle in both delay and fuel, respectively. We can see that the coordination strategy reliably saves energy and time at all saturation ratios, and that in general, at higher volumes the savings are more substantial.

Fig. 7
Average fuel (top) and delay (bottom) savings per vehicle, for different saturation ratios
Fig. 7
Average fuel (top) and delay (bottom) savings per vehicle, for different saturation ratios
Close modal

Remark 5. Fuel consumption calculations are based on the use of longitudinal vehicle dynamics to map both vehicle speed and acceleration to a net propulsion and/or braking torque/power demand. From engine power, we use a brake-specific fuel consumption (bsfc) map to compute vehicle fuel consumption with good accuracy. Delay is calculated as the difference between the ideal travel time, defined as the time it would have taken the vehicle to traverse the network at a desired speed vdes, and the actual travel time. In this paper, we assume that the ideal travel time is always the free flow travel time at speed vdes, independent of the current traffic demand level.

To understand these results, we can first look at the time–space diagrams for one lane in one of the simulations. In Fig. 8, we plot the trajectories of all vehicles that traverse one of the lanes for both the baseline and proposed strategy. This type of diagram is used in traffic engineering to quickly look at congestion patterns like queues and shock waves [29]. Free flowing traffic would show up in the diagram as straight lines, while queues and shock waves can be seen as disturbances from this nominal behavior. We can look particularly at the right leaning triangular patterns of stopped traffic that correspond to the formation queues and the back propagation of the shock wave. Comparing the coordinated traffic (top plot) with the baseline (bottom plot), we can see that coordination reduces queue length and dissipates the shock waves. This reduction in congestion explains the improvements in delay and fuel consumption.

Fig. 8
Space-time diagram of the trajectories of vehicles in a road segment for both the coordinated strategy and baseline
Fig. 8
Space-time diagram of the trajectories of vehicles in a road segment for both the coordinated strategy and baseline
Close modal

From the savings results, we can also see the strategy saves significantly more in delay than it does in fuel consumption. We believe this happens because the control strategy sometimes accelerates vehicles in order to catch a green light, a behavior that is not usually modeled by the Gipps car following model. We can see that this is the case in Fig. 9. Here, we plot the distribution of speeds for all vehicles at all time steps for the 0.7 saturation rate simulation. We can see the coordinating strategy not only has the vehicles stopped less often, but it also sometimes sees the vehicles traveling at speeds higher than the desired speed.

Fig. 9
Proposed strategy and baseline distributions of speeds, for one simulation scenario at 0.7 saturation rate
Fig. 9
Proposed strategy and baseline distributions of speeds, for one simulation scenario at 0.7 saturation rate
Close modal

Finally, we can further characterize the behavior of our algorithm by looking at when and how often the regulator actually modifies the nominal input. In Fig. 10, we show 19 consecutive vehicle trajectories for one of the lanes in one of our simulations, and we have highlighted in bold (red) the portions of the trajectories where the regulator is active. There are different scenarios where regulation kicks in. Most notably, we can see the regulator is responsible for vehicles safely stopping for a red light and forming a queue (see seconds 420–460). Also, the regulator is active when vehicles speed up (see, for example, the fifth vehicle trajectory, or the first three vehicle trajectories near the intersection). This is probably an interplay between the coordinator attempting to accelerate the vehicles to reach an earlier green light, and the regulator maintaining speed and acceleration upper bounds.

Fig. 10
Example of vehicle trajectories with regions where the safety regulator is active highlighted
Fig. 10
Example of vehicle trajectories with regions where the safety regulator is active highlighted
Close modal

7 Conclusion

In this paper, we present a decentralized coordination strategy for vehicles and traffic lights at urban intersections. We describe the vehicles by their distance to the intersection, their speed, and their desired time till arrival. Similarly, the lights are described by their planned time till switching. Using this common framework, the agents, whether they are lights or vehicles, negotiate their timing through gradient-based multi-agent control methods. The main objective of the work is to create a safety regulating controller that takes this negotiating strategy, and modifies it in order to ensure safe operation. The regulator is synthesized using control barrier certificates implemented through a quadratic program that aims to minimize deviations from the nominal coordinating control. The control strategy is tested in simulation and compared to traditional traffic light and human-driven behavior to showcase the improvement in fuel consumption and delay. In future work, more extensive simulation studies can be performed to: (1) validate our strategy against different baseline models like the Intelligent Driver model, or the Wiedeman driver model and (2) study the impact of other parameters like different types of powertrains and vehicle sizes. Additionally, the algorithm can be extended to include interactions between agents according to different connectivity topologies (i.e., considering communication between vehicles in different roads and across multiple intersections).

Funding Data

  • Advanced Research Projects Agency (Grant No. DE-AR-0000801; Funder ID: 10.13039/100009224).

Appendix

In this Appendix, we present the derivation of the safe car-following control barrier function candidate. The constraint we wish to turn into a barrier function is
sisl(i)Dmin
(A1)
Normally, the corresponding candidate CBF would simply be
ρ7(xi,xl(i))=sisl(i)Dmin
(A2)
We denote the function using ρ7 instead of h7 because, unlike other barrier functions derived in this paper, this constraint presents a more apparent issue when transforming it directly into a candidate barrier function. Namely, the constraint is on position, and its time derivative will be input independent (i.e., Lgρ(xi)=0). This is not necessarily a problem as long as the definition of the control barrier function (15) is still satisfied in some domain Di,7, with Ci,7Di,7. In this case, the certificate is
ρ7˙(xi,xl(i))=vi+vl(i)γ7ρ7(xi,xl(i))
(A3)

However, at the boundary of Ci,7,ρ7(x)=0, the above inequality reduces to vl(i)>vi. This means that the largest possible domain where ρ7 is a valid CBF will be Di,7={xi,xl(i)3:vl(i)vi}. Intuitively, this makes sense: if the vehicle's leader is traveling faster than itself and it is more than Dmin distance away, the vehicle cannot possibly reduce its spacing without first overtaking its leader in terms of speed and entering a set where the function is no longer a valid CBF. Therefore, we need a different candidate CBF.

To synthesize this candidate CBF, we use another theorem that is specifically helpful in generating multiple CBFs that are valid in the presence of other input constraints, or on higher relative degree dynamics (both of which are the case here). The idea consists of synthesizing CBFs from an initial candidate that may not be a valid CBF in the desired domain, plus a nominal controller that satisfies other input constraints. In Ref. [22], the candidate function is called a performance function and is denoted by ρ. The nominal controller is denoted by β:DU. The state of the system for any time t > 0 is xβ(t,x), when it is initialized at x and input β is applied.

Theorem 2 (from Ref. [22]). Letρ(x):Dbe a continuously differentiable performance function,β(x):DUbe a nominal controller such thatx˙(x,β)is continuously differentiable, andxβ(t,x)be the state of the system at time t when it is initialized atxand input β is applied. If
h(x)=inft[0,)ρ(xβ(t,x))
(A4)

then

h is a CBF

C={xD:h(x)0}{xD:ρ(x)0}β(x)Kcbf(x)

Given the previous discussion, when vl(i)vi, no barrier certificate is needed to ensure safe car following. For vl(i)vi, we derive a barrier function by assuming a nominal controller ai=βi(xi) that represents the maximum braking maneuvers for both vehicles. The maximum braking maneuver is given by the constraints on acceleration and the barrier certificate on minimum velocity, in Eq. (26). Assuming vmin=0, this is given by the largest of two values: βi(vi)=max(amin,i,γi,1vi). We can then integrate the system forward in time under this control action, and compute the value of ρ7 into the future. We then define the new candidate barrier function as the infima in time of ρ7, as explained in Theorem 2.

We can characterize the trajectory of ρ7 as it is integrated into the future by noticing that, since 0vl(i)vi, ρ7 monotonically decreases until both vehicles are completely stopped. We can therefore find its minimum value by adding the braking distance of each vehicle to the initial value of ρ7. The braking distance of vehicle i under βi is based on whether or not it starts at a speed larger than vi,T=amin,iγi,1, which is the speed at which the braking controller transitions from being amin,i to being γ1vi. After computing the appropriate braking distance for each vehicle, based on their current speeds relative to vi,T and vl(i),T, we can define the new candidate CBF
h7(xi,xl(i))={forvivi,Tvl(i)vl(i),T:ρ7(xi,xl(i))+vi22amin,ivl(i)22amin,l(i)+amin,i2γ12amin,l(i)2γ1,l(i)2forvivi,Tvl(i)<vl(i),T:ρ7(xi,xl(i))+vi22amin,i+amin,i2γ12+vl(i)γ1,l(i)forvi<vi,Tvl(i)<vl(i),T:ρ7(xi,xl(i))+viγ1+vl(i)γ1,l(i)
(A5)

where vi,T corresponds to the speed where the maximum deceleration switches from being amin,i to being γ1vi. That is, vi,T=amin,iγ1.

Plainly, this CBF candidate is the original performance function ρ7, which is the current intervehicle spacing, plus each vehicle's maximum braking distance when they use their maximum deceleration profile. This maximum deceleration profile is itself the product of following the constraints on acceleration given by Eqs. (21) and (26). The maximum braking distance of each vehicle depends on its current velocity. It should be noted that the above CBF candidate is continuously differentiable with respect to the state; this can be verified by evaluating h7 and its gradient from both sides at each potential discontinuity point.

Differentiating in time, and assuming the leader uses its allowed maximum deceleration we get the barrier certificate constraint on the input
A7,1(i)={viamin,iifvivi,T1γi,1ifvi<vi,TA7,2(i)=0b7(i)=vi+γ7h7(xi,xl(i))
(A6)

In deriving the above barrier certificate, we make the conservative assumption that the lead vehicle uses its maximum allowable deceleration. While looser assumptions can be made, this approach guarantees safety even in the worst-case scenarios.

Remark 6. To get this constraint, we need to assume that the deceleration parameters of the leader amin,l(i) and γ1,l(i) are known.

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