# Optimal Control of Connected Automated Vehicles with Event-Triggered Control
Barrier Functions: a Test Bed for Safe Optimal Merging

Ehsan Sabouni∗,1, H.M. Sabbir Ahmad∗,1, Wei Xiao2, Christos G. Cassandras1 and
Wenchao Li1 ∗These authors contributed equally to this paper.1Division of
Systems Engineering and Department of Electrical & Computer Engineering,
Boston University, Boston, MA, USA, {esabouni, sabbir92, cgc,
<EMAIL_ADDRESS>Science & Artificial Intelligence Laboratory,
Massachusetts Institute of Technology, Cambridge, MA, USA<EMAIL_ADDRESS>This
work was supported in part by NSF under grants ECCS-1931600, DMS-1664644,
CNS-1645681, CNS-2149511, by AFOSR under grant FA9550-19-1-0158, by ARPA-E
under grant DE-AR0001282, by the MathWorks and by NPRP grant (12S-0228-190177)
from the Qatar National Research Fund, a member of the Qatar Foundation (the
statements made herein are solely the responsibility of the authors).

###### Abstract

We address the problem of controlling Connected and Automated Vehicles (CAVs)
in conflict areas of a traffic network subject to hard safety constraints. It
has been shown that such problems can be solved through a combination of
tractable optimal control problems and Control Barrier Functions (CBFs) that
guarantee the satisfaction of all constraints. These solutions can be reduced
to a sequence of Quadratic Programs (QPs) which are efficiently solved on line
over discrete time steps. However, guaranteeing the feasibility of the CBF-
based QP method within each discretized time interval requires the careful
selection of time steps which need to be sufficiently small. This creates
computational requirements and communication rates between agents which may
hinder the controller’s application to real CAVs. In this paper, we overcome
this limitation by adopting an event-triggered approach for CAVs in a conflict
area such that the next QP is triggered by properly defined events with a
safety guarantee. We present a laboratory-scale test bed we have developed to
emulate merging roadways using mobile robots as CAVs which can be used to
demonstrate how the event-triggered scheme is computationally efficient and
can handle measurement uncertainties and noise compared to time-driven control
while guaranteeing safety.

##  I Introduction

The effective traffic management of Connected and Automated Vehicles (CAVs)
through control and coordination has brought the promise of resolving long-
lasting problems in transportation networks such as accidents, congestion, and
unsustainable energy consumption [1],[2],[3]. To date, both centralized [4, 5]
and decentralized [6, 7, 8] methods have been proposed to tackle the control
and coordination problem of CAVs in conflict area such as intersections,
roundabouts, and merging roadways; an overview of such methods may be found in
[9]. In decentralized methods, as opposed to centralized ones, each CAV is
responsible for its own on-board computation with information from other
vehicles limited to a set of neighbors [10]. One approach is to formulate a
constrained optimal control problem jointly minimizing travel time through
conflict areas and energy consumption, e.g., for optimal merging [11] or
crossing intersections [12]. The complexity of obtaining the solution,
however, necessitates the use of online control methods like Control Barrier
Functions (CBFs) [13], [14] and Model Predictive Control (MPC) techniques [15]
for real-world applications.

An approach combining optimal control solutions with CBFs (termed OCBF) was
recently presented in [16]. In this combined approach, the solution of a
_tractable_ optimal control problem is used as a reference control. Then, the
resulting control reference trajectory is optimally tracked subject to a set
of CBF constraints that ensure the satisfaction of all constraints of the
original optimal control problem. Finally, this optimal tracking problem is
efficiently solved by discretizing time and solving a simple Quadratic Problem
(QP) at each discrete time step over which the control input is held constant
[13]. However, the control update interval in the time discretization process
must be sufficiently small in order to always guarantee that every QP is
feasible. In practice, such feasibility can be often seen to be violated.

One way to remedy this problem is to use an _event-triggered_ scheme instead.
A general event-triggered framework for CBFs has been proposed in [17]. It
remains to be shown how this framework can be used for real CAVs with proper
event-triggered communication, optimization, and control in the presence of
measurement uncertainties, noise, and multiple safety constraints. In this
paper, we address this question by adapting the framework in [17] to CAVs and
implementing it in a laboratory-scale test bed we have developed focusing on
the problem of automated merging roadways and emulating CAVs through mobile
robots. Several optimal control methods have been developed for CAVs and
implemented on actual mobile robots. CAV maneuvers have been demonstrated in a
scaled environment with 2-3 CAVs in [18]. Highway driving conditions have been
created to validate CAV maneuvers in [19]. In [20] and [21] the unconstrained
optimal control solution has been implemented on robots for the merging
roadway scenario and the corridor scenario, respectively. To our knowledge,
there is no general _safety-guaranteed_ control algorithm for CAVs implemented
on real robots for merging roadways.

Our contribution: The contributions of the paper are as follows. $(i)$ We
follow the approach presented in [17] and introduce an _event-triggered_
framework for coordination and control of CAVs in merging roadways. The
underlying concept behind the approach is to define _events_ associated with
states reaching a certain bound, at which point the next QP instance is
triggered. This approach provides manifold benefits, namely $(a)$ guaranteeing
safety constraints are satisfied in the presence of measurement noise and
model uncertainties which are inevitable in practical applications, as opposed
to the time-driven scheme, and $(b)$ significant reduction in the number of
QPs solved, thereby reducing the need for unnecessary communication among CAVs
and overall computational overload. $(ii)$ In addition to validating the
event-triggered framework and aforementioned benefits through simulation, we
have also implemented it in a laboratory-scale test bed using mobile robots as
shown in Fig 1. The test bed allows us to test a variety of cooperative
control algorithms for CAVs under multiple safety constraints (such as rear-
end safety and safe merging) in the presence of disturbances that cannot be
captured in a simulated setting. Although the experiment was set up for a
merging scenario as illustrated in Fig. 3, it can be readily extended to other
bottleneck points. In this test bed, all computation, including checking for
bound violations and re-solving a QP to obtain a new control input, are
performed on-board in a real-time manner.

Figure 1: The merging experiment setup where the control zone (CZ) starts from
point $O$ in the main road and point $O^{\prime}$ in the merging road and ends
at the merging point $MP$.

The paper is organized as follows. In Section II, we provide an overview of
the time-driven decentralized constrained optimal control for CAVs through the
OCBF approach in merging roadways as an example of a conflict area, providing
the motivation for the event-triggered approach. In section III, the event-
triggered scheme is presented. In section IV, implementation details,
including the communication and motion control of the robots, are discussed.
In section V, the implementation results are provided to demonstrate the
effectiveness of the proposed event-triggered method in experiments with
actual robots in terms of improved feasibility and guaranteed safety
constraints. Additionally, extensive simulation results are provided to
validate the proposed approach.

##  II Problem Formulation and Event-triggered Control

In this section, we review the setting for CAVs whose motion is cooperatively
controlled at conflict areas of a traffic network. This includes merging
roads, signal-free intersections, roundabouts, and highway segments where lane
change maneuvers take place. Although the framework can be used for all
conflict areas, the focus on this paper is on merging roadways. Following
[22], we define a _Control Zone_ (CZ) to be an area within which CAVs can
communicate with each other or with a coordinator which is responsible for
facilitating the exchange of information within this CZ. As an example, Fig. 2
shows a conflict area due to vehicles merging from two single-lane roads and
there is a single Merging Point (MP) which vehicles must cross from either
road [22]. In such a setting, assuming all traffic consists of CAVs, a finite
horizon constrained optimal control problem can be formulated aiming to
determine trajectories that jointly minimize travel time and energy
consumption through the CZ and guaranteeing safety constraints are always
satisfied.

Figure 2: Illustration of cooperative control of CAVs at merging roadways.

Let $S(t)$ be the set of First-In-First-Out (FIFO)-ordered indices of all CAVs
located in the CZ at time $t$ along with the CAV that has just left the CZ
(whose index is 0 as shown in Fig.2). $N(t)$ is defined as the cardinality of
$S(t)$. Thus, a CAV arriving at time $t$ is assigned $N(t)$ as its index. All
CAV indices in the CZ $S(t)$ decrease by one when a CAV crosses the MP and the
vehicle whose index is $-1$ is dropped.

The vehicle dynamics for each CAV $i\in S(t)$ take the following form

$\left[\begin{array}[c]{c}\dot{x}_{i}(t)\\\
\dot{v}_{i}(t)\end{array}\right]=\left[\begin{array}[c]{c}v_{i}(t)\\\
u_{i}(t)\end{array}\right],$ (1)

where $x_{i}(t)$ denotes the distance from the origin at which CAV $i$
arrives, $v_{i}(t)$, and $u_{i}(t)$ denotes the velocity and control input
(acceleration) of CAV $i$, respectively.

###### Remark 1

The dynamics of CAVs and mobile robots in real-world applications are more
complex than those in (1). In the test bed we have developed, the inadequacies
of simple dynamics are compensated for through a local controller.

Let $t_{i}^{0}$ and $t_{i}^{f}$ denote the time that CAV $i$ arrives at the
origin and leaves the CZ at its exit point, respectively. Constraints for any
conflict area can be listed as follows: Constraint 1 (Safety constraints): Let
$i_{p}$ denote the index of the CAV which physically immediately precedes $i$
in the CZ (if one is present). We require that the distance
$z_{i,i_{p}}(t):=x_{i_{p}}(t)-x_{i}(t)$ be constrained by:

$z_{i,i_{p}}(t)\geq\varphi v_{i}(t)+\delta,\text{ \ }\forall
t\in[t_{i}^{0},t_{i}^{f}]$ (2)

where $\varphi$ denotes the reaction time and $\delta$ is a given minimum safe
distance. If we define $z_{i,i_{p}}$ to be the distance from the center of CAV
$i$ to the center of CAV $i_{p}$, then $\delta$ depends on the length of these
two CAVs.
Constraint 2 (Safe merging): Whenever a CAV crosses a MP, a lateral collision
is possible and there must be adequate safe space for the CAV to avoid such
collision, i.e.,

$z_{i,i_{c}}(t_{i}^{m})\geq\varphi v_{i}(t_{i}^{m})+\delta,$ (3)

where $i_{c}$ is the index of the CAV that may collide with CAV $i$ at the
merging point $M$ as shown in Fig. 2. The determination of CAV $i_{c}$ depends
on the policy adopted for sequencing CAVs through the CZ. For the merging
roadway in Fig. 2 under FIFO, we have $i_{c}=i-1$ and $t_{i}^{m}=t_{i}^{f}$
since the MP defines the exit from the CZ.
Constraint 3 (Vehicle limitations): Finally, there are constraints on the
speed and acceleration for each $i\in S(t)$:

$\displaystyle v_{min}\leq v_{i}(t)\leq v_{max},\forall
t\in[t_{i}^{0},t_{i}^{f}]$ (4) $\displaystyle u_{{min}}\leq u_{i}(t)\leq
u_{{max}},\forall t\in[t_{i}^{0},t_{i}^{f}]$ (5)

where $v_{\max}>0$ and $v_{\min}\geq 0$ denote the maximum and minimum speed
allowed in the CZ, $u_{{\min}}<0$ and $u_{\max}>0$ denote the minimum and
maximum control, respectively.

Decentralized Optimal Control Problem formulation. Our goal is to determine a
control law jointly minimizing the travel time and energy consumption subject
to constraints 1-3 for each $i\in S(t)$ governed by the dynamics (1).
Expressing energy through $\frac{1}{2}u_{i}^{2}(t)$ and normalizing travel
time and energy, we use the weight $\alpha_{i}\in[0,1]$ to construct a convex
combination as follows:

$\begin{aligned}
\min_{u_{i}(t),t_{i}^{f}}J_{i}(u_{i}(t),t_{i}^{f})=\int_{t_{i}^{0}}^{t_{i}^{f}}\left(\alpha_{i}+\frac{(1-\alpha_{i})\frac{1}{2}u_{i}^{2}(t)}{\frac{1}{2}\max\\{u_{max}^{2},u_{min}^{2}\\}}\right)dt\end{aligned}.$
(6)

Letting
$\beta_{i}:=\frac{\alpha_{i}\max\\{u_{max}^{2},u_{min}^{2}\\}}{2(1-\alpha_{i})}$,
we obtain a simplified form:

$\min_{u_{i}(t),t_{i}^{f}}J_{i}(u_{i}(t),t_{i}^{f}):=\beta_{i}(t_{i}^{f}-t_{i}^{0})+\int_{t_{i}^{0}}^{t_{i}^{f}}\frac{1}{2}u_{i}^{2}(t)dt,$
(7)

where $\beta_{i}\geq 0$ is an adjustable weight to penalize travel time
relative to the energy cost of CAV $i$. Note that the solution is
_decentralized_ in the sense that CAV $i$ requires information only from CAVs
$i_{p}$ and $i_{c}$ required in (2) and (3).

The OCBF approach. Firstly, we derive the CBFs that ensure the constraints
(2), (3), and (4) are always satisfied, subject to the vehicle dynamics (1) by
defining $f(\boldsymbol{x}_{i}(t))=[v_{i}(t),0]^{T}$ and
$g(\boldsymbol{x}_{i}(t))=[0,1]^{T}$. Each of these constraints can be easily
written in the form of $b_{q}(\boldsymbol{x}(t))\geq 0$, $q\in\\{1,...,n\\}$
where $n$ stands for the number of constraints and
$\boldsymbol{x}(t)=[\boldsymbol{x}_{1}(t),\boldsymbol{x}_{2}(t),...,\boldsymbol{x}_{N(t)}(t)]$.
The CBF method (details provided in [16]) maps a constraint
$b_{q}(\boldsymbol{x}(t))\geq 0$ onto a new constraint which is _linear_ in
the control input and takes the general form

$L_{f}b_{q}(\boldsymbol{x}(t))+L_{g}b_{q}(\boldsymbol{x}(t))u_{i}(t)+\gamma(b_{q}(\boldsymbol{x}(t)))\geq
0.$ (8)

To obtain the CBF constraint for the safety constraint (2) we set
$b_{1}(\boldsymbol{x}_{i}(t),\boldsymbol{x}_{i_{p}}(t))=z_{i,i_{p}}(t)-\varphi
v_{i}(t)-\delta=x_{i_{p}}(t)-x_{i}(t)-\varphi v_{i}(t)-\delta$ and since
$b_{1}(\boldsymbol{x}_{i}(t),\boldsymbol{x}_{i_{p}}(t))$ is differentiable,
the CBF constraint for (2) is

$\small\underbrace{v_{i_{p}}(t)-v_{i}(t)}_{L_{f}b_{1}(\boldsymbol{x}_{i}(t),\boldsymbol{x}_{i_{p}}(t))}+\underbrace{-\varphi}_{L_{g}b_{1}(\boldsymbol{x}_{i}(t))}u_{i}(t)+\underbrace{k_{1}(z_{i,i_{p}}(t)-\varphi
v_{i}(t)-\delta)}_{\gamma_{1}(b_{1}(\boldsymbol{x}_{i}(t),\boldsymbol{x}_{i_{p}}(t)))}\geq
0,$ (9)

where the class-$\mathcal{K}$ function $\gamma_{1}(\boldsymbol{x})$ is chosen
here to be linear.

Deriving the CBF constraint for the safe merging constraint (3) poses a
technical challenge due to the fact that it only applies at a certain time
$t_{i}^{m}$, whereas a CBF is required to be in a continuously differentiable
form. To tackle this problem, by following the technique used in [16], we
replace $\phi$ in (3) by $\Phi(x_{i}(t))=\varphi\frac{x_{i}(t)}{L}$, where $L$
is the length of the road traveled by the CAV from the CZ entry to the MP, and
use
$b_{2}(\textbf{x}_{i}(t),\textbf{x}_{i_{c}}(t))=x_{i_{c}}(t)-x_{i}(t)-\Phi(x_{i}(t))v_{i}(t)-\delta$
so that the CBF constraint for safe merging (3) becomes

$\displaystyle\underbrace{v_{i_{c}}(t)-v_{i}(t)-\frac{\varphi}{L}v_{i}^{2}(t)}_{L_{f}b_{2}(\boldsymbol{x}_{i}(t),\boldsymbol{x}_{i_{c}}(t))}+\underbrace{-\varphi\frac{x_{i}(t)}{L}}_{L_{g}b_{2}(\boldsymbol{x}_{i}(t))}u_{i}(t)+$
$\displaystyle\underbrace{k_{2}(z_{i,i_{c}}(t)-\varphi\frac{x_{i}(t)}{L}v_{i}(t)}_{\gamma_{2}(b_{2}(\boldsymbol{x}_{i}(t),\boldsymbol{x}_{i_{c}}(t)))}\geq
0.$ (10)

The speed constraints in (4) are also easily transformed into CBF constraints
by defining $b_{3}(\boldsymbol{x}_{i}(t))=v_{max}-v_{i}(t)$ and
$b_{4}(\boldsymbol{x}_{i}(t))=v_{i}(t)-v_{min}$. This yields:

$\displaystyle\underbrace{-1}_{L_{g}b_{3}(\boldsymbol{x}_{i}(t))}u_{i}(t)+\underbrace{k_{3}(v_{max}-v_{i}(t))}_{\gamma_{3}(b_{3}(\boldsymbol{x}_{i}(t)))}\geq
0,$
$\displaystyle\underbrace{1}_{L_{g}b_{4}(\boldsymbol{x}_{i}(t))}u_{i}(t)+\underbrace{k_{4}(v_{i}(t)-v_{min})}_{\gamma_{4}(b_{4}(\boldsymbol{x}_{i}(t)))}\geq
0,$ (11)

for the maximum and minimum velocity constraints, respectively.

As a last step in the OCBF approach, we use a Control Lyapunov Function (CLF)
associated with tracking the CAV speed to a desired value $v_{i}^{ref}(t)$
setting $V(\boldsymbol{x}_{i}(t))=(v_{i}(t)-v_{i}^{ref}(t))^{2}$ and express
the CLF constraint as follows:

$L_{f}V(\boldsymbol{x}_{i}(t))+L_{g}V(\boldsymbol{x}_{i}(t))\boldsymbol{u}_{i}(t)+c_{3}V(\boldsymbol{x}_{i}(t))\leq
e_{i}(t),$ (12)

where $e_{i}(t)$ makes this a soft constraint.

Finally, we can formulate the OCBF problem as follows:

$\small\min_{u_{i}(t),e_{i}(t)}J_{i}(u_{i}(t),e_{i}(t)):=\int_{t_{i}^{0}}^{t_{i}^{f}}\big{[}\frac{1}{2}(u_{i}(t)-u_{i}^{ref}(t))^{2}+\lambda
e^{2}_{i}(t)\big{]}dt$ (13)

subject to vehicle dynamics (1), the CBF constraints (9), (II), (II), and CLF
constraint (12). In this approach, $(i)$ we solve the unconstrained optimal
control problem in (7) which yields $u_{i}^{ref}$, $(ii)$ the resulting
$u_{i}^{ref}$ is optimally tracked such that constraints including CBF
constraints (9), (II), (II) satisfied, and $(iii)$ this tracking optimal
control problem is efficiently solved by discretizing time and solving a
simple QP at each discrete time step.

As already mentioned, a common way to solve this dynamic optimization problem
is to discretize $[t_{i}^{0},t_{i}^{f}]$ into intervals
$[t_{i}^{0},t_{i}^{0}+\Delta],...,[t_{i}^{0}+k\Delta,t_{i}^{0}+(k+1)\Delta],...$
with equal length $\Delta$ and solving (13) over each time interval. The
decision variables $u_{i,k}=u_{i}(t_{i,k})$ and $e_{i,k}=e_{i}(t_{i,k})$ are
assumed to be constant on each interval and can be easily calculated at time
$t_{i,k}=t_{i}^{0}+k\Delta$ through solving a QP at each time step:

$\displaystyle\min_{u_{i,k},e_{i,k}}$
$\displaystyle[\frac{1}{2}(u_{i,k}-u_{i}^{ref}(t_{i,k}))^{2}+\lambda
e_{i,k}^{2}]$ (14)

subject to the constraints (9), (II), (II), (12), and (5) where all
constraints are linear in the decision variables. We refer to this as the
_time-driven_ approach. As pointed out earlier, the main problem with this
approach is that there is no guarantee for the feasibility of each CBF-based
QP, as it requires a small enough discretization time which is not always
possible to achieve. Also it is worth mentioning that requiring
synchronization among all CAVs can be difficult to impose in real-world
applications.

##  III Event-triggered control

In this paper, we adopt an _event-triggering_ scheme whereby a QP (with its
associated CBF constraints) is solved when one of three possible events (as
defined next) is detected. We will show that this provides a guarantee for the
satisfaction of the safety constraints which cannot be offered by the time-
driven approach described earlier. As introduced in [17], the key idea is to
ensure that the safety constraints are satisfied while the state remains
within some bounds and define events which coincide with the state reaching
these bounds, at which point the next instance of the QP in (14) is triggered.

Let $t_{i,k}$, $k=1,2,...$, be the time instants when CAV $i$ solves the QP in
(14). Our goal is to guarantee that the state trajectory does not violate any
safety constraints within any time interval $(t_{i,k},t_{i,k+1}]$ where
$t_{i,k+1}$ is the next time instant when the QP is solved. We define a subset
of the state space of CAV $i$ at time $t_{i,k}$ such that:

$\boldsymbol{x}_{i}(t_{i,k})-\boldsymbol{s}_{i}\leq\boldsymbol{x}_{i}(t)\leq\boldsymbol{x}_{i}(t_{i,k})+\boldsymbol{s}_{i},$
(15)

where $\boldsymbol{s_{i}}\in\mathbb{R}_{>0}^{2}$ is a parameter vector whose
choice will be discussed later. We denote the set of states of CAV $i$ that
satisfy (15) at time $t_{i,k}$ by

$S_{i}(t_{i,k})=\bigl{\\{}\boldsymbol{y}_{i}\in\textbf{X}:\boldsymbol{x}_{i}(t_{i,k})-\boldsymbol{s}_{i}\leq\boldsymbol{y}_{i}\leq\boldsymbol{x}_{i}(t_{i,k})+\boldsymbol{s}_{i}\bigl{\\}}.$
(16)

In addition, let $C_{i,1}$ be the feasible set of our original constraints
(2),(3) and (4) defined as

$C_{i,1}:=\Bigl{\\{}\mathbf{x}_{i}\in\mathbf{X}:~{}b_{q}(\mathbf{x})\geq 0,\
q\in\\{1,2,3,4\\}\Bigl{\\}},$ (17)

Next, we seek a bound and a control law that satisfies the safety constraints
within this bound. This can be accomplished by considering the minimum value
of each component of (8), where $q\in\\{1,2,3,4\\}$, as shown next.

Let us start with the first term in (8), $L_{f}b_{q}(\boldsymbol{x}(t))$,
rewritten as $L_{f}b_{q}(\boldsymbol{y}_{i}(t),\boldsymbol{y}_{r}(t))$ with
$\boldsymbol{y}_{i}(t)$ as in (16) and $r$ in $\boldsymbol{y}_{r}(t)$ stands
for “relevant” CAVs affecting the constraint of $i$ (i.e.,
$r\in\\{i_{p},i_{c}\\}$ in Fig. 2). Let $b^{min}_{q,f_{i}}(t_{i,k})$ be the
minimum possible value of the term $L_{f}b_{q}(\boldsymbol{x}(t))$ over the
time interval $(t_{i,k},t_{i,k+1}]$ for each $q\in\\{1,2,3,4\\}$ over the set
$\bar{S_{i}}({t_{i,k}})\cap\bar{S_{r}}({t_{i,k}})$:

$b^{min}_{q,f_{i}}(t_{i,k})=\displaystyle\min_{\boldsymbol{y}_{i}\in\bar{S}_{i}({t_{i,k}})\atop\boldsymbol{y}_{r}\in\bar{S}_{r}({t_{i,k}})}L_{f}b_{q}(\boldsymbol{y}_{i}(t),\boldsymbol{y}_{r}(t)),$
(18)

where $\bar{S}_{i}({t_{i,k}})$ is defined as follows:

$\bar{S}_{i}({t_{i,k}}):=\bigl{\\{}\boldsymbol{y}_{i}\in C_{i,1}\cap
S_{i}(t_{i,k})\bigl{\\}}$ (19)

Similarly, we can define the minimum value of the third term in (8):

$b^{min}_{\gamma_{q}}(t_{i,k})=\displaystyle\min_{\boldsymbol{y}_{i}\in\bar{S}_{i}({t_{i,k}})\atop\boldsymbol{y}_{r}\in\bar{S}_{r}({t_{i,k}})}\gamma_{q}(\boldsymbol{y}_{i}(t),\boldsymbol{y}_{r}(t)).$
(20)

For the second term in (8), note that $L_{g}b_{q}(\boldsymbol{x}_{i})$ is a
constant for $q\in\\{1,3,4\\}$, therefore there is no need for any
minimization. However, $L_{g}b_{2}(\boldsymbol{x}_{i})$ in (II) is state
dependent and needs to be considered for the minimization. Since $x_{i}(t)\geq
0$, note that $L_{g}b_{2}(\boldsymbol{x}_{i})=-\varphi\frac{x_{i}(t)}{L}$ is
always negative, therefore, we can determine the limit value
$b^{min}_{2,g_{i}}(t_{i,k})\in\mathbb{R},$ as follows:

$\displaystyle\small
b^{min}_{2,g_{i}}(t_{i,k})=\begin{cases}\displaystyle\min_{\boldsymbol{y}_{i}\in\bar{S}_{i}({t_{i,k}})\atop\boldsymbol{y}_{r}\in\bar{S}_{r}({t_{i,k}})}L_{g}b_{2}(\boldsymbol{x}_{i}(t)),\
\textnormal{if}\ u_{i,k}\geq 0\\\ \\\
\displaystyle\max_{\boldsymbol{y}_{i}\in\bar{S}_{i}({t_{i,k}})\atop\boldsymbol{y}_{r}\in\bar{S}_{r}({t_{i,k}})}L_{g}b_{2}(\boldsymbol{x}_{i}(t)),\
\ \ \textnormal{otherwise},\end{cases}$ (21)

where the sign of $u_{i,k},\ i\in S(t_{i,k})$ can be determined by simply
solving the CBF-based QP (13) at time $t_{i,k}$.

Thus, the condition that can guarantee the satisfaction of (9),(II) and (II)
in the time interval $\left(t_{i,k},t_{i,k+1}\right]$ is given by

$b^{min}_{q,f_{i}}(t_{i,k})+b^{min}_{q,g_{i}}(t_{i,k})u_{i,k}+b^{min}_{\gamma_{q}}(t_{i,k})\geq
0,$ (22)

for $q\in\\{1,2,3,4\\}$. In order to apply this condition to the QP (13), we
just replace (8) by (22) as follows:

$\displaystyle\min_{u_{i,k},e_{i,k}}$
$\displaystyle[\frac{1}{2}(u_{i,k}-u_{i}^{ref}(t_{i,k}))^{2}+\lambda
e_{i,k}^{2}]$ $\displaystyle\textnormal{s.t.}\ \
\eqref{CLF_constraint},\eqref{minCBF},\eqref{VehicleConstraints2}$ (23)

It is important to note that each instance of the QP (III) is now triggered by
one of the following three events where $k=1,2,\ldots$ is an event (rather
than time step) counter:
Event 1: the state measurement of CAV $i$ reaches the boundary of
$S_{i}(t_{i,k-1})$.
Event 2: the state measurement of CAV $i_{p}$ reaches the boundary of
$S_{i_{p}}(t_{i,k-1})$ (if $i_{p}$ exists for CAV $i$).
Event 3: the state measurement of CAV $i_{c}$ reaches the boundary of
$S_{i_{c}}(t_{i,k-1})$ where $i_{c}$ is the index of the CAV that may collide
with $i$ in (3), e.g., $i_{c}=i-1\neq i_{p}$ in the merging problem case, if
such a CAV exists.

The state measurements are obtained from external sensors which introduce
noise in the state values. Thus, we denote the state measurements of CAV $i$
by $\tilde{\mathbf{x}}_{i}(t)$ defined as follows:

$\tilde{\boldsymbol{x}}_{i}(t)=\boldsymbol{x}_{i}(t)+\boldsymbol{w}(t)$ (24)

where $\boldsymbol{w(t)}$ is a random but bounded noise term. As a result,
$t_{i,k},k=1,2,...$ is unknown in advance but can be determined by CAV $i$
through:

$\displaystyle
t_{i,k}=\min\Big{\\{}t>t_{i,k-1}:|\tilde{\boldsymbol{x}}_{i}(t)-\tilde{\boldsymbol{x}}_{i}(t_{i,k-1})|=\boldsymbol{s}_{i}$
(25) $\displaystyle\text{or}\ \
|\tilde{\boldsymbol{x}}_{i_{p}}(t)-\tilde{\boldsymbol{x}}_{i_{p}}(t_{i,k-1})|=\boldsymbol{s}_{i_{p}}$
$\displaystyle\text{or}\ \
|\tilde{\boldsymbol{x}}_{i_{c}}(t)-\tilde{\boldsymbol{x}}_{i_{c}}(t_{i,k-1})|=\boldsymbol{s}_{i_{c}}\Big{\\}},$

where $t_{i,1}=0$.

The following theorem formalizes our analysis by showing that if new
constraints of the general form (22) holds, then our original CBF constraints
(9),(II) and (II) also hold. The proof follows the same lines as that of a
more general theorem in [17] and, therefore, is omitted.

###### Theorem 1

Given a CBF $b_{q}(\mathbf{x(t)})$ with relative degree one, let $t_{i,k+1}$,
$k=1,2,\ldots$ be determined by (25) with $t_{i,1}=0$ and
$b^{min}_{q,f_{i}}(t_{i,k})$, $b^{min}_{\gamma_{q}}(t_{i,k})$,
$b^{min}_{q,g_{i}}(t_{i,k})$ for $q=\\{1,2,3,4\\}$ obtained through (18),
(20), and (21). Then, any control input $u_{i,k}$ that satisfies (22) for all
$q\in[1,2,3,4]$ within the time interval $[t_{i,k},t_{i,k+1})$ renders the set
$C_{i,1}$ forward invariant for the dynamic system defined in (1).

###### Remark 2

: Expressing (22) in terms of the minimum value of each component separately
may become overly conservative if each minimum value corresponds to different
points in the decision variable space. Therefore, an alternative approach is
to calculate the minimum value of the whole term.

The selection of the parameters $\mathbf{s}_{i}$ captures the trade-off
between _computational cost_ and _conservativeness_ : the larger the value of
each component of $\mathbf{s}_{i}$ is, the smaller the number of events that
trigger instances of the QPs becomes, thus reducing the total computational
cost. At the same time, the control law must satisfy the safety constraints
over a longer time interval as we take the minimum values in (18)-(21), hence
rendering the approach more conservative. The choice of parameter
$\mathbf{s}_{i}$ can also render our algorithm more robust to measurement
uncertainties and noise, hence making it useful for real-world implementation.
To account for measurement uncertainties, the value of each bound should be
greater than or equal to the measurement noise, i.e., the inequality
$\mathbf{s}_{i}\geq\sup\mathbf{w}(t)$ should hold componentwise.

##  IV Test bed for Coordinated Merging

The event-triggered coordinated merging process described in the last section
was implemented in a lab environment using mobile robots as CAVs, while the
coordinator was implemented on a laptop located in the vicinity of the robots.
The overall implementation can be divided into two main parts: $(i)$
communication and $(ii)$ control. A block-level illustration of the
architecture of the robots firmware is presented in Fig. 3 and provides the
basis for designing similar test beds for related research.

Communication. The OCBF approach only requires vehicle-to-infrastructure (V2I)
communication, whereby the coordinator is responsible for exchanging
information among robots. The communication was implemented using a 5G
wireless LAN where the messages were exchanged using ROS Topics. In an event-
triggered scheme, frequent communication is generally not needed, since it
occurs only when an event is triggered. The events are state-dependent, thus
each robot in the CZ requires its position information. Since the robots are
not equipped with GPS, the position and orientation (POSE) information was
obtained using the Optitrack localization system by the coordinator using ROS
topics over Ethernet as shown in Fig. 3. The coordinator uses the localization
information to index the robots upon arrival in the CZ for constructing the
queue table based on the FIFO scheme as illustrated in Fig. 2) and transmits
the indices to them. Additionally, the POSE information of robots is
transmitted to themselves periodically while in the CZ.

Each robot is responsible for checking its own state to detect any violation
in its state bounds in (16). When such an event occurs, the robot updates its
control input by re-solving the QP in (III) and informs the coordinator with
its newly obtained state (i.e., velocity and position) by publishing a ROS
topic. The structure of data packets sent by any robot with index $i$ is
described using the vector
$(i,\tilde{x}_{i,long},\tilde{x}_{i,lat},\tilde{v}_{i})^{T}$ where
$\tilde{x}_{i,long}$ and $\tilde{x}_{i,lat}$ denote the position along and
orthogonal to the direction of the main road, respectively and $\tilde{v}_{i}$
denotes the linear speed along the direction of the road of robot $i$. It then
becomes the responsibility of the coordinator to provide this information to
the relevant robots (i.e., those that might be affected). There can be at most
two robots that can be relevant to any robot $i$, the one which is immediately
following it in the same lane and the robot that will merge after it from the
other lane. Note that a triggered event due to an update in robot $i$’s state
can affect only robots $l>i$. The data are published as ROS topics by the
coordinator separately for each one (or two) of the robot(s) in the form
$(\tilde{x}_{{i_{p}},long},\tilde{x}_{{i_{p}},lat},\tilde{v}_{i_{p}},\tilde{x}_{i_{c},long},\tilde{x}_{i_{c},lat},\tilde{v}_{i_{c}})^{T}$
where $\tilde{x}_{{i_{p}},long}$ and $\tilde{x}_{{i_{p}},lat}$ denote the
lateral and longitudinal position and $\tilde{v}_{i_{p}}$ denotes the linear
speed of the preceding robot $i_{p}$, and $\tilde{x}_{{i_{c}},lat}$ and
$\tilde{x}_{{i_{c}},long}$ denote the lateral and longitudinal position and
$\tilde{v}_{i_{c}}$ denotes the linear speed of robot $i_{c}$ that will merge
before the robot. The robots subscribe to the topics published for them using
the index as the identifier. As can be seen from Fig. 3, the robot firmware
contains a dedicated thread to handle the ROS messages from the coordinator.
Finally, the notified robots decide whether they need to re-solve their QP or
maintain their control input until the next triggering event by checking for a
potential bound violation. As a result, they either need to re-solve their QP,
and send their updated state to the coordinator or they carry on with their
current control input until the next event.

Control. The event-triggered QP in (III) for any robot $i$ returns the
acceleration ${u}^{*}_{i,k}$, which has to be controlled along with the
trajectory of every robot in the CZ to prevent them from deviating from the
road. The two control objectives are achieved using feedback control. The
linear acceleration for the robots cannot be directly controlled, hence the
desired acceleration ${u}^{*}_{i,k}$ for any robot $i$ is mapped onto its
desired linear velocity $v^{*}_{i}(t)\ \forall t\in[t_{i,k},t_{i,k+1})\
\textnormal{and}\ \forall k$ using the mapping below.

$v^{*}_{i}(t+\Delta T)={u}^{*}_{i,k}\Delta T+v^{*}_{i}(t)\\\ \ \ \
t\in[t_{i,k},t_{i,k+1}),\ \forall k$ (26)

where $\Delta T$ is the period of the control loop execution. In order to
track the acceleration $u^{*}_{i,k}$ of robot $i$ the value of $\Delta T$
needs to be generally chosen to be five times smaller than the inter-event
time $(t_{i,k+1}-t_{i,k})$. However, since events occur asynchronously,
$\Delta T$ is chosen to be a fifth of the sensor sampling rate at which the
measurements are received, as the event-triggered approach solves a more
conservative version of the time-driven OCBF algorithm. The velocity of the
robot at any time $t$ is computed using the on-board wheel encoders and used
to control the velocity by manipulating the motor speed using a PID
controller.

It is desired that the robot follow the center of the lane while in the CZ,
which is programmed as a set of way-points. The deviation from the center of
the lane for any robot $i$ at time $t$ is denoted by
$\tilde{e}_{i}=\tilde{x}_{i,lat}(t)-L_{c}$, where $L_{c}$ is the center of the
road. The error can be negated by manipulating the steering angle using a PID
controller whose gain is set through trial and error. Due to the nature of
event-triggered control, two threads are dedicated in every robot to achieve
the control tasks where the first thread is responsible for controlling the
motion of the robot (executed every $\Delta T$ seconds) and the other for the
event-triggered control (executed asynchronously upon the occurrence of any
event as shown in Fig. 3).

Figure 3: Illustration of the testbed implementation.

##  V Numerical Results

All algorithms in this paper were implemented using Python. We used QP from
the cvxopt library for solving (III). Besides that, lingprog from the scipy
library has been used to solve the linear program in (18), (20) and (21),
fsolve was used for a nonlinear optimization problem arising when (18) and
(20) become nonlinear respectively, and solve_ivp was used to solve for the
vehicle dynamics (1). Note that for results obtained in the test bed, all
algorithms were executed on board each robot.

We have considered the merging problem shown in Fig. 1 where robots arrive to
the predefined CZ according to Poisson arrival processes with a given arrival
rate. The initial speed $v_{i}(t_{i}^{0})$ is also randomly generated with a
uniform distribution over $[0.1\textnormal{m/s},1\textnormal{m/s}]$ at the
origins $O$ and $O^{\prime}$, respectively. The parameters for (13) and (III)
are:
$L=3.04\textnormal{m},\varphi=0.18\textnormal{s},\delta=15\textnormal{cm},u_{max}=2\
\textnormal{m/s}^{2},u_{min}=-2\
\textnormal{m/s}^{2},v_{max}=1\textnormal{m/s},v_{min}=0\textnormal{m/s},k_{1}=k_{2}=k_{3}=k_{4}=1,\lambda=10$.
The sampling rate of the localization data is $30$Hz, sufficiently high to
avoid missing any triggering event (in general, a proper sampling rate can
always be calculated given the CAV specifications, i.e., bounds on velocity
and acceleration). The control update period for the time-driven control is
$\Delta=0.05$s and $\Delta T=0.25$s. For the event-triggered control, we let
the bounds $S=[s_{x},s_{v}]$ be the same for all robots in the network and set
to $[0.25,0.05]$.

###  V-A Simulation results

We first used a Matlab-based simulation environment using the python modules
developed for actual implementation in the test bed. It is worth noting that
some of the parameters for simulation and implementation are different as we
had used mobile robots and did the implementations in a lab environment.
Moreover during simulation, we included the computation of a more realistic
energy consumption model [23] to supplement the simple surrogate $L_{2}$-norm
($u^{2}$) model in our analysis:
$f_{v}(t)=f_{cruise}(t)+f_{accel}(t),f_{cruise}(t)=\omega_{0}+\omega_{1}v_{i}(t)+\omega_{2}v^{2}_{i}(t)+\omega_{3}v^{3}_{i}(t),f_{accel}(t)=(r_{0}+r_{1}v_{i}(t)+r_{2}v^{2}_{i}(t))u_{i}(t)$
where we used typical values for parameters
$\omega_{1},\omega_{2},\omega_{3},r_{0},r_{1}$ and, $r_{2}$ as reported in
[23]. Note that in the test bed implementation, the energy term used in (6)
does not signify the energy dissipation (as it is difficult to measure the
energy expended by the robots); rather, this term captures the control effort
(i.e. acceleration/deceleration) as it is related to the energy consumed.

Table I summarizes our results from 8 separate simulations corresponding to
both the event-triggered and time-driven methods under the same conditions
with different values for the relative weight of energy vs time as shown in
the table. We observe that by using the event-triggered approach we are able
to significantly reduce the number of infeasible QP cases (up to $86\%$)
compared to the time-driven approach. At the same time, the overall number of
instances when a QP needs to be solved has also decreased up to $50\%$ in the
event-triggered case. Note that the large majority of infeasibilities are due
to holding the control input (i.e. acceleration/develeration) constant over
the sampling period, which can invalidate the forward invariance property of
CBFs over the entire time interval. These infeasible cases were eliminated by
the event-triggering approach. However, another source of infeasibility is due
to conflicts that may arise between the CBF constraints and the control bounds
in a QP. This cannot be remedied through the event-triggered approach; it can,
however, be dealt with by the introduction of a sufficient condition that
guarantees no such conflict, as described in [24].

| Item | Event triggered | Time Triggered
---|---|---|---
$\alpha=0.1$ | Ave. Travel time | 15.53 | 15.01
Ave. $\frac{1}{2}u^{2}$ | 5.16 | 3.18
Ave. Fuel consumption | 31.04 | 31.61
Computation load (Num of QPs solved) | 34% (12168) | 100% (35443)
| Num of infeasible cases | 43 | 315
$\alpha=0.25$ | Ave. Travel time | 15.53 | 15.01
Ave. $\frac{1}{2}u^{2}$ | 14.25 | 13.34
Ave. Fuel consumption | 51.42 | 55.81
Computation load (Num of QPs solved) | 48% (13707) | 100% (28200)
| Num of infeasible cases | 28 | 341
$\alpha=0.4$ | Ave. Travel time | 15.53 | 15.01
Ave. $\frac{1}{2}u^{2}$ | 18.22 | 17.67
Ave. Fuel consumption | 52.42 | 56.5
Computation load (Num of QPs solved) | 49% (13573) | 100% (27412)
| Num of infeasible cases | 25 | 321
$\alpha=0.5$ | Ave. Travel time | 15.17 | 14.63
Ave. $\frac{1}{2}u^{2}$ | 24.93 | 25.08
Ave. Fuel consumption | 53.21 | 56.93
Computation load (Num of QPs solved) | 50% (13415) | 100% (26726)
| Num of infeasible cases | 20 | 341
TABLE I: CAV metrics under event-driven and time-driven control.

###  V-B Test Bed Implementation Results

Although simulation results indicate the efficacy of the event-triggered OCBF
approach, the results of the lab test bed implementation are significant in
two ways: firstly, they demonstrate that the control algorithm can be
implemented and executed in real-time, which is essential for safety-critical
systems. Secondly, they demonstrate the robustness of the algorithm to the
various noise sources present in real-world applications. To illustrate the
aforementioned points, two different experiments have been considered. The
first experiment involves 5 robots for comprehensive implementation and the
results are depicted in Fig. 4 illustrating the satisfaction of both rear-end
safety ($b_{1}(\textbf{x}_{i}(t),\textbf{x}_{i_{p}}(t))$) and merging
constraints ($b_{2}(\textbf{x}_{i}(t),\textbf{x}_{i_{c}}(t))$) during the
whole experiment duration. In Fig. 4, each line specified by the CAV number
represents a time plot constraint for that CAV in the CZ. As can be observed,
all lines are above zero, therefore, both rear end and merging constraints are
satisfied at all times in the CZ. Note that all the CAVs are not present in
the plot, as not all the CAVs are constrained. In the second experiment, the
initial conditions of the 2 robots (that have to merge together safely coming
from two different roads) are deliberately chosen to demonstrate that _the
time-driven approach cannot guarantee safety_ due to uncertainties and
imperfect measurements while the proposed event-triggered scheme with a proper
choice of bounds (as discussed in section III) can maintain a safe merging
distance ($b_{2}(\textbf{x}_{2}(t),\textbf{x}_{1}(t))$) between two robots as
depicted in Fig 5.

The following web link contains a video of our implementation of the event
triggered scheme for a merging roadway in a lab environment:
https://www.youtube.com/watch?v=qwhLjEskPS8. The video contains two scenarios
with different arrival sequences where we have used A FIFO passing sequence.
In the first scenario, three robots (with indices 1, 2 and 5) arrive from the
main road and two robots (with indices 3 and 4) arrive from the side road and
they safely merge at the merging point. For the second scenario, a different
arrival order was chosen, whereby three CAVs (with indices 1, 2 and 4) arrive
from the main road and two CAVs (with indices 3 and 5) arrive from the side
road. The proposed scheme is able to safely merge traffic from both roads as
depicted in the video. In this case, CAV 5 has both rear end and merging
constraint, to make room for CAV 4 to allow it to merge ahead of it while also
staying safe to CAV 3 that is physically preceding it. This represents the
most computationally extensive case where a CAV has both constraints thus
highlighting the real time feature of the proposed approach.

Figure 4: Rear end and safe merging constraints of 5 robots. Figure 5: Safe
merging constraints of 2 robots under event-triggered scheme with 2 different
event bounds and time-driven control schemes. Note the safe merging constraint
violation under time-driven control. On the other hand, for both the choice of
bounds, safe merging constraint was satisfied at all times by the event-
triggered scheme.

##  VI Conclusion

The problem of controlling CAVs in conflict areas of a traffic network subject
to hard safety constraints by the use of CBFs can be solved through a sequence
of QPs. However, the lack of feasibility guarantees for each QP, as well as
control update synchronization, motivate an event-triggering scheme. In this
paper, we have presented an event-triggered framework for the automated
control of merging roadways (which can be generalized to any conflict area)
and designed a lab test bed for the cooperative control of CAVs using mobile
robots. As part of ongoing work, we will Validate this algorithm in an
extensive test bed comprised of multiple conflict points using mobile robots.

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