# A Constrained Optimal Control Framework for Vehicle Platoons with Delayed
Communication

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###### Abstract.

Vehicle platooning using connected and automated vehicles (CAVs) has attracted
considerable attention. In this paper, we address the problem of optimal
coordination of CAV platoons at a highway on-ramp merging. We present a
single-level constrained optimal control framework that optimizes fuel economy
and travel time of the platoons while satisfying the state, control, and
safety constraints. We also explore the effect of delayed communication among
the CAV platoons and propose a robust coordination framework to enforce
lateral and rear-end collision avoidance constraints in the presence of
bounded delays. We provide a closed-form analytical solution of the optimal
control problem with safety guarantees that can be implemented in real time.
Finally, we validate the effectiveness of the proposed control framework using
a high-fidelity commercial simulation environment.

###### Key words and phrases:

Optimal control, Platoon coordination, Intelligent transportation systems.

###### 1991 Mathematics Subject Classification:

Primary: 58F15, 58F17; Secondary: 53C35.

The authors are supported by ARPAE grant DE-AR0000796

∗ Corresponding author: A M Ishtiaque Mahbub

A M Ishtiaque Mahbub∗, Behdad Chalaki,

and Andreas A. Malikopoulos

University of Delaware

130 Academy Street, Newark, DE-19716, USA

##  1\. Introduction

###  1.1. Motivation

Traffic congestion has increased significantly over the last decade [62].
Bottlenecks such as urban intersections, merging roadways, highway on-ramps,
roundabouts, and speed reduction zones along with the driver responses [66,
33] to various disturbances in the transportation network are the primary
sources of traffic congestion [52]. Emerging mobility systems, e.g., connected
and automated vehicles (CAVs), lay the foundation to improve safety and
transportation efficiency at these bottlenecks by providing the users the
opportunity to better monitor the transportation network conditions and make
optimal decisions [67, 21, 74]. Having enhanced computational capabilities,
CAVs can establish real-time communication with other vehicles and
infrastructure to increase the capacity of critical traffic corridors,
decrease travel time, and improve fuel efficiency and safety [7, 13, 44, 40,
82]. However, the cyber-physical nature of emerging mobility systems, e.g.,
data and shared information through vehicle-to-vehicle (V2V) and vehicle-to-
infrastructure (V2I) communication, is associated with significant technical
challenges and gives rise to a new level of complexity [48] in modeling and
control [24].

There have been two major approaches to utilizing the connectivity and
automation of vehicles, namely, coordination and platooning. The concept of
coordination through different traffic bottlenecks is enabled by the vehicle-
to-everything communication protocol among the CAVs and the surrounding
infrastructure. On the other hand, real-time computation and automation of
CAVs enables safe and comfortable trajectories with extremely short headway in
the form of CAV platoon, which consists of a string [10] of consecutive CAVs
traveling together at a constant headway and speed.

In this paper, we employ the concepts of CAV coordination and platooning to
address the problem of minimizing traffic congestion at the traffic
bottlenecks in an energy-efficient manner. In particular, we aim at optimally
coordinating platoons of CAVs at highway on-ramp merging [56] in the presence
of bounded delays among platoon leaders while guaranteeing state, control, and
safety constraints.

###  1.2. Literature Review

CAV coordination is an approach that has been explored to mitigate the speed
variation of individual CAVs throughout the transportation network [55]. Early
efforts [37, 5] considered a single string of vehicles that was coordinated
through a merging roadway by employing a linear optimal regulator. In $1993$,
Varaiya [72] outlined the key features of an automated intelligent
vehicle/highway system, and proposed a basic control system architecture. In
$2004$, Dresner and Stone [19] proposed the use of the reservation scheme to
control a signal-free intersection of two roads. Since then, several research
efforts [20, 18, 27, 6, 3] have extended this approach for coordination of
CAVs at urban intersections. More recently, a decentralized optimal control
framework was presented in [49, 45] for coordinating online CAVs at different
traffic scenarios such as on-ramp merging roadways, roundabouts, speed
reduction zones and signal-free intersections. The framework uses a
hierarchical structure consisting of an upper-level vehicle coordination
problem to minimize travel time and a low-level energy minimization problem.
The state and control constraints in the coordination problem has been
addressed in [49, 50, 41, 43, 16, 15] by incorporating the constraints in the
low-level optimization problem, and in [51, 46] by incorporating the
constraints in the upper-level optimization problem. Detailed discussions of
the research reported in the literature to date on coordination of CAVs can be
found in [60] and [25].

The aforementioned coordination strategies are vehicle-centric approaches
focusing on control of individual CAVs within the network, whereas platooning
can leverage the full potential of CAVs to enhance current optimal
coordination of CAVs. The concept of platoon formation gained momentum in the
1980s and 1990s as a system-level approach to address traffic congestion [65,
73, 59] and has been shown to have significant benefits [36, 2]. Shladover et
al. [65] presented the concept of operating automated vehicles in closely
spaced platoons as part of an automated highway system, and pioneered the
California Partners for Advanced Transportation Technology (PATH) program to
conduct heavy-duty truck platooning from $2001$ to $2003$. Rajamani et al.
[59] discussed the lateral and longitudinal control of CAVs for the automated
platoon formation. From a system point of view, platooning of vehicles yields
additional mobility benefits. It has been shown that capacity at a traffic
bottleneck, such as an intersection, can be doubled or even tripled by
platooning of vehicles [38]. Moreover, platooning improves the fuel efficiency
of the vehicles due to the reduction of the aerodynamic drag within the
platoon, especially at high cruising speeds[1, 64, 69, 8]. Various research
efforts in the literature have addressed vehicle platooning at highways to
increase fuel efficiency, traffic flow, driver comfort, and safety. To date,
there has been a rich body of research focusing on exploring several methods
of forming and/or utilizing platoons to improve transportation efficiency [71,
75, 29, 31, 80, 78, 58, 4, 11, 42]. A detailed discussion on different
approaches in vehicle platooning systems at highways can be found in [9, 81,
32].

###  1.3. Objectives and Contributions of the Paper

In this paper, we address the problem of coordinating CAV platoons at a
highway on-ramp merging. The main objective is to leverage the key concepts of
CAV coordination and platooning, and establish a control framework for platoon
coordination aimed at improving network performance while guaranteeing safety.

The key contributions of this paper are (i) the development of a
mathematically rigorous optimal control framework for platoon coordination
that completely eliminates stop-and-go driving behavior, and improves fuel
economy and traffic throughput of the network, (ii) the derivation and
implementation of the optimal control input in real time that satisfies the
state, control, and safety constraints subject to bounded delayed
communication, and (iii) the validation of the proposed control framework
using a commercial traffic simulator by evaluating its performance compared to
a baseline scenario.

###  1.4. Comparison With Related Work

To the best of our knowledge, this paper is the first attempt to establish a
rigorous constrained optimal control framework for coordination of vehicular
platoons at a highway on-ramp merging in the presence of bounded inter-platoon
delays. This paper advances the state of the art as follows. First, in
contrast to other efforts that neglected state/control constraints [35, 34,
68], our framework guarantees satisfaction of all of the state, control, and
safety constraints in the system. Second, our framework unlike the several
efforts in the literature at highway on-ramp merging scenario [61, 57, 77, 79]
does not impose a strict first-in-first-out queuing policy to ensure lateral
safety. Third, in this paper, we consider the bounded delay in the inter-
platoon communication, which most of the studies in the coordination of
vehicular platoons neglect [35, 26, 17]. Finally, our framework yields a
closed-form analytical solution while satisfying all of the system
constraints, and thus it is appropriate for real-time implementation on-board
the CAVs [39].

###  1.5. Organization of the paper

The remainder of the paper is organized as follows. In Section 2, we present
the modeling framework and formulate the problem. In Section 3, we provide a
detailed exposition of the optimal control framework and the algorithm to
implement the closed-form analytical solution to the constrained optimal
control problem. In Section 4, we evaluate the effectiveness of the proposed
approach in a simulation environment. Finally, we draw conclusions and discuss
the next steps in Section 5.

##  2\. Modeling Framework

Figure 1. On-ramp merging with a single merging point for platoons of CAVs.
The control zone is highlighted in light blue color, the entry time and exit
time to the control zone are depicted with circles, and example sets of
platoon leaders and followers are shown.

We consider the problem of coordinating platoons of CAVs in a scenario of
highway on-ramp merging (Fig. 1). Although our analysis can be applied to any
traffic scenario, e.g., signal-free intersections, roundabouts, and speed
reduction zones, we use a highway on-ramp as a reference to present the
fundamental ideas and results of this paper.

The on-ramp merging includes a control zone, inside of which platoons of CAVs
communicate with the coordinator. The coordinator does not make any decision
for the CAVs, and only acts as a database for the CAVs. The paths of the main
road and the ramp road intersect at a point called conflict point, indexed by
$n\in\mathbb{N}$, at which lateral collision may occur. We consider that CAVs
have formed platoon upstream of the control zone in a region called platooning
zone. We refer interested readers to [42, 28, 70] for further details on
platoon formation.

###  2.1. Network Topology and Communication

Figure 2. Network topology for information flow: (i) bidirectional inter-
platoon communication (dashed double-headed arrow) between the platoon leaders
via the coordinator, and (ii) unidirectional intra-platoon communication
(solid single-headed arrow) from platoon leader to the platoon followers.

In our modeling framework, we impose the following communication topology
based on the standard V2V and V2I communication protocol as shown in Fig. 2.

  1. (1)

Bidirectional inter-platoon communication: The leaders of each platoon can
exchange information with each other via the coordinator through a V2I
communication protocol. The flow of information is bidirectional.

  2. (2)

Unidirectional intra-platoon communication: The following CAVs of each platoon
can subscribe to the platoon leader’s state and control information. The flow
of information is unidirectional from the platoon leader to the following CAVs
within that platoon.

When a platoon leader enters the control zone, it subscribes to the
bidirectional inter-platoon communication protocol to connect with the
coordinator and access the information of platoons which are already in the
control zone. After obtaining this information, the leader derives its optimal
control input (acceleration/deceleration) to cross the control zone without
any lateral or rear-end collision with the other CAVs, and without violating
any of the state and control constraints. The leader then communicates its
derived control input and trajectory information to its followers using the
unidirectional intra-platoon communication protocol so that the following CAVs
can compute their control input. Finally, the platoon leader transmits its
information to the coordinator so that the subsequent platoon leaders can plan
their trajectories accordingly. In this paper, we enhance our framework to
consider delayed transmission during the inter-platoon communication protocol
due to the physical distance among the platoons. On the other hand, since the
CAVs within each platoon are closely spaced, we consider that there is an
instantaneous flow of information within the intra-platoon communication
protocol. In our modeling framework, we make the following assumption
regarding the nature of delay during the inter-platoon communication protocol.

######  Assumption 1.

The communication delay during the bidirectional inter-platoon communication
between each platoon leader and the coordinator is bounded and known a priori.

Assumption 1 enables the determination of upper bounds on the state
uncertainties as a result of sensing or communication errors and delays, and
incorporate these into more conservative safety constraints, the exposition of
which we provide in Section 3.3.

###  2.2. Dynamics and Constraints

Next, we provide some definitions that are necessary in our exposition.

######  Definition 2.1.

The queue that designates the order in which each platoon leader entered the
control zone is given by $\mathcal{L}(t)=\\{1,\ldots,L(t)\\}$, where
${L}(t)\in\mathbb{N}$ is the total number of platoons that are inside the
control zone at time $t\in\mathbb{R}_{\geq 0}$. When a platoon exits the
control zone, its index is removed from $\mathcal{L}(t)$.

######  Definition 2.2.

CAVs within platoon $i\in\mathcal{L}(t)$ are indexed with set
$\mathcal{N}_{i}=\\{0,1,\dots,m_{i}\\}$, where $0$ and $m_{i}\in\mathbb{N}$
denote the leader and last CAV of the platoon $i$, respectively. The size of
each platoon $i\in\mathcal{L}(t)$ is thus the cardinality of set
$\mathcal{N}_{i}$, and denoted by $M_{i}:=m_{i}+1$.

In our analysis, we consider that the dynamics of each CAV
$j\in\mathcal{N}_{i}$ in platoon $i\in\mathcal{L}(t)$ is governed by a double
integrator,

$\displaystyle\dot{p}_{i,j}(t)$ $\displaystyle=v_{i,j}(t),$
$\displaystyle\dot{v}_{i,j}(t)$ $\displaystyle=u_{i,j}(t),$ (1)

where $p_{i,j}(t)\in\mathcal{P}$, $v_{i,j}(t)\in\mathcal{V}$, and
$u_{i,j}(t)\in\mathcal{U}$ denote position, speed, and control input at
$t\in\mathbb{R}_{\geq}0$, respectively. The sets $\mathcal{P},\mathcal{V},$
and $\mathcal{U}$, are compact subsets of $\mathbb{R}$.

######  Remark 1.

In what follows, to simplify notation, we use subscript $i$ instead of ${i,0}$
to denote the leader of platoon $i\in\mathcal{L}(t)$.

Let $t_{i,0}^{0}=t_{i}^{0}\in\mathbb{R}_{\geq 0}$ be the time that leader of
platoon $i\in\mathcal{N}(t)$ enters the control zone, and
$t_{i,0}^{f}=t_{i}^{f}>t_{i}^{0}\in\mathbb{R}_{\geq 0}$ be the time that
leader of platoon $i$ exits the control zone. Since each CAV
$j\in\mathcal{N}_{i},~{}i\in\mathcal{L}(t)$, has already formed a platoon in
the platooning zone, when the leader enters the control zone at time
$t_{i}^{0}$, we have $v_{i,j-1}(t_{i}^{0})-v_{i,j}(t_{i}^{0})=0$ and
$p_{i,j-1}(t_{i}^{0})-p_{i,j}(t_{i}^{0})-l_{c}=\Delta_{i}$, where $l_{c}$
denote the length of each CAV $j$, and $\Delta_{i}$ is the safe bumper-to-
bumper inter-vehicle gap between CAVs $j,j-1\in\mathcal{N}_{i}$ within each
platoon $i\in\mathcal{L}(t)$. This bumper-to-bumper inter-vehicle gap is
imposed by the platoon forming control in platooning zone upstream of the
control zone. After exiting the control zone at $t_{i}^{f}$, the leader of
platoon $i$ cruises with constant speed $v_{i}(t_{i}^{f})$ until the last
follower in the platoon exits the control zone. Afterwards, each platoon
member $j\in\mathcal{N}_{i},i\in\mathcal{L}(t)$ is controlled by a suitable
car-following model [76] which ensures satisfying rear-end safety constraint.

For each CAV $j\in\mathcal{N}_{i}$ in platoon $i\in\mathcal{L}(t)$ the control
input and speed are bounded by

$\displaystyle u_{\min}$ $\displaystyle\leq u_{i,j}(t)\leq u_{\max},$ (2)
$\displaystyle 0<v_{\min}$ $\displaystyle\leq v_{i,j}(t)\leq v_{\max},$ (3)

where $u_{\min},u_{\max}$ are the minimum and maximum control inputs and
$v_{\min},v_{\max}$ are the minimum and maximum speed limit, respectively.

To ensure rear-end safety between platoon $i\in\mathcal{L}(t)$ and preceding
platoon $k\in\mathcal{L}(t)$, we have

$\displaystyle p_{k,m_{k}}(t)-p_{i}(t)\geq\delta_{i}(t)=\gamma+\varphi\cdot
v_{i}(t),$ (4)

where $m_{k}$ is the last follower in the platoon $k$ physically located in
front of platoon $i$ and $\delta_{i}(t)$ is the safe speed-dependent distance,
while $\gamma$ and $\varphi\in\mathbb{R}_{>0}$ are the standstill distance and
reaction time, respectively.

Similarly, to guarantee rear-end safety within CAVs inside each platoon
$i\in\mathcal{L}(t)$, we enforce

$\displaystyle p_{i,j-1}(t)-p_{i,j}(t)\geq\Delta_{i}+l_{c},\quad\forall
j\in\\{1,\dots,m_{i}\\}.$ (5)

Finally, let $k\in\mathcal{L}(t)$ correspond to another platoon that has
already entered the control zone and may have a lateral collision with platoon
$i\in\mathcal{L}(t)$ at conflict point $n$. For the first case in which
platoon $i$ reaches the conflict point after platoon $k$, we have

$t_{i}^{f}-t_{k,m_{k}}^{f}\geq t_{h},$ (6)

where $t_{h}\in\mathbb{R}_{>0}$ is the minimum time headway between any two
CAVs entering node $n$ that guarantees safety, $t_{i}^{f}$ is the time that
leader of platoon $i$ exits the control zone (recall that the conflict point
$n$ is at the exit of control zone), and $t_{k,m_{k}}^{f}$ is time that the
last CAV in the platoon $k$ exits the control zone. Likewise, for the second
case in which platoon $i$ reaches the conflict point $n$ before platoon $k$,
we have

$t_{k}^{f}-t_{i,m_{i}}^{f}\geq t_{h}.$ (7)

######  Remark 2.

Given the time $t_{i}^{f}$ that the platoon leader of platoon
$i\in\mathcal{L}(t)$ exits the control zone, we compute the time
$t_{i,m_{i}}^{f}$ that the last platoon member $m_{i}\in\mathcal{N}_{i}$ exits
the control zone as

$t_{i,m_{i}}^{f}=t_{i}^{f}+\frac{(M_{i}-1)(\Delta_{i}+l_{c})}{v_{i}(t_{i}^{f})}.$
(8)

To guarantee lateral safety between platoon $i$ and platoon $k$ at a conflict
point $n$, either (6) or (7) must be satisfied. Therefore, we impose the
following lateral safety constraint on platoon $i$,

$\displaystyle\min\Bigg{\\{}$ $\displaystyle
t_{h}-(t_{i}^{f}-t_{k,m_{k}}^{f}),~{}t_{h}-(t_{k}^{f}-t_{i,m_{i}}^{f})\Bigg{\\}}\leq
0.$ (9)

With the state, control and safety constraints defined above, we now impose
the following assumption:

######  Assumption 2.

Upon entering the control zone, the initial state of each CAV
$j\in\mathcal{N}_{i}(t),~{}i\in\mathcal{L}(t)$, is feasible, that is, none of
the speed or safety constraints are violated.

This is a reasonable assumption since CAVs are automated; therefore, there is
no compelling reason for them to violate any of the constraint by the time
they enter the control zone.

###  2.3. Information Structure

In this section, we formalize the information structure that is communicated
between the CAV leaders and the coordinator inside the control zone.

######  Definition 2.3.

Let $\boldsymbol{\phi}_{i}$ be the vector containing the parameters of the
optimal control policy (formally defined in Section 3.1) of the leader of
platoon $i\in\mathcal{L}(t_{i}^{0})$. Then, the _platoon information set_
$\mathcal{I}_{i}$ that the leader of platoon $i$ can obtain from the
coordinator after entering the control zone at time $t=t_{i}^{0}$ is

$\mathcal{I}_{i}=\\{{\boldsymbol{\phi}}_{1:L(t_{i}^{0})-1},~{}M_{1:L(t_{i}^{0})},~{}t_{1:L(t_{i}^{0})}^{0},~{}t_{1:L(t_{i}^{0})-1}^{f}\\},$
(10)

where
${\boldsymbol{\phi}}_{1:L(t_{i}^{0})}:=[{\boldsymbol{\phi}}_{1},\ldots,{\boldsymbol{\phi}}_{L(t_{i}^{0})-1}]^{T}$,
$M_{1:L(t_{i}^{0})}:=[{M}_{1},\ldots,{M}_{L(t_{i}^{0})}]^{T}$,

$t_{1:L(t_{i}^{0})}^{f}:=[{t}_{1}^{0},\ldots,{t}_{L(t_{i}^{0})}^{0}]^{T}$ and
$t_{1:L(t_{i}^{0})-1}^{f}:=[{t}_{1}^{f},\ldots,{t}_{L(t_{i}^{0})-1}^{f}]^{T}$.

######  Remark 3.

The information structure $\mathcal{I}_{i}$ for each platoon
$i\in\mathcal{L}(t_{i}^{0})$ indicates that the control policy, entry time to
the control zone $t_{j}^{0}$, exit time of the control zone $t_{j}^{f}$, and
the platoon size $M_{j}$ of each platoon
$j\in\mathcal{L}(t_{i}^{0})\setminus\\{i\\}$ already existing within the
control zone is available to the leader of platoon $i$ through the
coordinator. Note that, although the leader of platoon $i$ knows the
endogenous information $t_{i}^{0}$ and $M_{i}$, it needs to compute the vector
of its own optimal control input parameters $\boldsymbol{\phi}_{i}$ and the
merging time $t_{i}^{f}$, which we discuss in section 3.

######  Definition 2.4.

The _member information set_ $\mathcal{I}_{i,j}(t)$ that each platoon member
$j\in\mathcal{N}_{i}\setminus\\{0\\}$ belonging to each platoon
$i\in\mathcal{L}(t)$ at time $t\in[t_{i}^{0},t_{i}^{f}]$ can obtain is

$\displaystyle\mathcal{I}_{i,j}=\\{p_{i,0}(t),v_{i,0}(t),u_{i,0}(t)\\}.$ (11)

######  Remark 4.

The unidirectional intra-platoon communication protocol allows each platoon
member $j\in\mathcal{N}_{i}\setminus\\{0\\}$ belonging to platoon
$i\in\mathcal{L}(t)$ to access the state and control input information of its
platoon leader in the form of $\mathcal{I}_{i,j}$ at each time
$t\in[t_{i}^{0},t_{i}^{f}]$. The set $\mathcal{I}_{i,j}$ is subsequently used
to derive the optimal control input $u_{i,j}^{*}(t)$ of each platoon member
$j$, which we discuss in detail in Section 3.2.

##  3\. Optimal Coordination Framework

In what follows, we introduce our coordination framework which consists of two
optimal control problems. The first problem is to develop an energy-optimal
control strategy for the platoon leaders to minimize their travel time while
guaranteeing that none of their state, control, and safety constraints becomes
active. The second problem is concerned with the optimal control of followers
within each platoon in order to maintain the platoon formation while ensuring
safety and string stability.

###  3.1. Optimal Control of Platoon Leaders

In this section, we extend the single-level optimization framework we
developed earlier for coordination of CAVs in [46] to establish a framework
for coordinating platoons of CAVs. Upon entrance to the control zone, the
leader of platoon $i\in\mathcal{L}(t)$ must determine the exit time
$t_{i}^{f}$ (recall that based on Remark 1, this is the time that the leader
of platoon $i$ exits the control zone). The exit time $t_{i}^{f}$ corresponds
to the unconstrained energy optimal trajectory for the platoon leader ensuring
that the resulting trajectory does not activate any of (2.2) - (4) and (9).
The _unconstrained solution_ of the leader of platoon $i$ is given by [46]

$\displaystyle u_{i}(t)$ $\displaystyle=6a_{i}t+2b_{i},$ $\displaystyle
v_{i}(t)$ $\displaystyle=3a_{i}t^{2}+2b_{i}t+c_{i},$ (12) $\displaystyle
p_{i}(t)$ $\displaystyle=a_{i}t^{3}+b_{i}t^{2}+c_{i}t+d_{i},$

where $a_{i},b_{i},c_{i},d_{i}$ are constants of integration. The leader of
platoon $i$ must also satisfy the boundary conditions

$\displaystyle p_{i}(t_{i}^{0})$ $\displaystyle=p_{i}^{0},\quad$
$\displaystyle v_{i}(t_{i}^{0})$ $\displaystyle=v_{i}^{0},$ (13)
$\displaystyle p_{i}(t_{i}^{f})$ $\displaystyle=p_{i}^{f},\quad$
$\displaystyle u_{i}(t_{i}^{f})$ $\displaystyle=0,$ (14)

where $p_{i}$ is known at $t_{i}^{0}$ and $t_{i}^{f}$ by the geometry of the
road, and $v_{i}^{0}$ is the speed at which the leaders of platoon $i$ enters
the control zone. The final boundary condition, $u_{i}(t_{i}^{f})=0$, results
from $v_{i}(t_{i}^{f})$ being left unspecified [12]. There are five unknown
variables that determine the optimal trajectory of the leader of the platoon
$i$, four constants of integration from (3.1), and the unknown exit time
$t_{i}^{f}$. The value of $t_{i}^{f}$ guarantees that the unconstrained
trajectories in (3.1) satisfy all the state, control, and safety constraints
in (2), (3) and (4), respectively, and the boundary conditions in (14). In
practice, for the leader of each platoon $i\in\mathcal{L}(t)$, the coordinator
stores the optimal exit time $t_{i}^{f}$ and the corresponding coefficients
$a_{i},b_{i},c_{i},d_{i}$. We denote the coefficients of the optimal control
policy for leader of platoon $i\in\mathcal{L}(t)$ by vector
$\boldsymbol{\phi}_{i}=[a_{i},b_{i},c_{i},d_{i}]^{T}$, which is an element of
platoon information set for the leader of platoon $i\in\mathcal{L}(t)$
(Definition 2.3). We formally define our single-level optimization framework
for platoon leaders as follows.

######  Problem 1.

Upon entering the control zone, each leader of platoon $i\in\mathcal{L}(t)$
accesses the information set $\mathcal{I}_{i}$ and solves the following
optimization problem at $t_{i}^{0}$

$\displaystyle\min_{t_{i}^{f}\in\mathcal{T}_{i}(t_{i}^{0})}t_{i}^{f}$ (15)
subject to:
$\displaystyle\eqref{eq:rearendInterPlatoon},\eqref{eq:lateralMinSafety},\eqref{eq:optimalTrajectory},$

where the compact set
$\mathcal{T}_{i}(t_{i}^{0})=[\underline{t}_{i}^{f},\overline{t}_{i}^{f}]$ is
the set of feasible solution of leader of platoon $i\in\mathcal{N}(t)$ for the
exit time that satisfy the boundary conditions without activating the
constraints, while $\underline{t}_{i}^{f}$ and $\overline{t}_{i}^{f}$ denote
the minimum and maximum feasible exit time computed at $t_{i}^{0}$.

######  Remark 5.

We can derive the optimal control input of the platoon leaders using the
solution of Problem 1, $t_{i}^{f}$, the boundary conditions (13)-(14) and
(3.1).

In what follows, we continue our exposition by briefly reviewing the process
to compute the compact set $\mathcal{T}_{i}(t_{i}^{0})$ at time $t_{i}^{0}$
using the speed and control input constraints (2)-(3), initial condition (13),
and final condition (14). Details regarding the derivation of the compact set
$\mathcal{T}_{i}(t_{i}^{0})$ can be found in [14].

The lower-bound $\underline{t}_{i}^{f}$ of $\mathcal{T}_{i}(t_{i}^{0})$ can be
computed by considering the state and control constraints and boundary
conditions as

$\underline{t}_{i}^{f}=\min\left\\{t_{i,u_{\max}}^{f},t_{i,v_{\max}}^{f}\right\\},$
(16)

where,

$\displaystyle t_{i,v_{\max}}^{f}$
$\displaystyle=\frac{3(p_{i}(t_{i}^{f})-p_{i}(t_{i}^{0}))}{v_{i}(t_{i}^{0})+2v_{\max}},$
$\displaystyle t_{i,u_{\max}}^{f}$
$\displaystyle=\frac{\sqrt{9{v_{i}(t_{i}^{0})}^{2}+12(p_{i}(t_{i}^{f})-p_{i}(t_{i}^{0}))u_{\max}}-3{v_{i}(t_{i}^{0})}}{2u_{\max}}.$

Here, $t_{i,v_{\max}}^{f}$ and $t_{i,u_{\max}}^{f}$ are the times which leader
of platoon $i\in\mathcal{L}(t)$ achieves its maximum speed at the end of
control zone and its maximum control input at the entry of the control zone,
respectively. Similarly, we derive the upper-bound $\overline{t}_{i}^{f}$ as

$\overline{t}_{i}^{f}=\begin{cases}t^{f}_{i,v_{\min}},&\text{ if }\
9{v_{i}(t_{i}^{0})}^{2}+12(p_{i}(t_{i}^{f})-p_{i}(t_{i}^{0}))u_{\min}<0,\\\
\max\\{t_{i,u_{\min}}^{f},t_{i,v_{\min}}^{f}\\},&\text{otherwise,}\\\
\end{cases}$ (17)

where

$\displaystyle t^{f}_{i,v_{\min}}$
$\displaystyle=\frac{3(p_{i}(t_{i}^{f})-p_{i}(t_{i}^{0}))}{v_{i}(t_{i}^{0})+2v_{\min}},$
$\displaystyle t^{f}_{i,u_{\min}}$
$\displaystyle=\frac{\sqrt{9{v_{i}(t_{i}^{0})}^{2}+12(p_{i}(t_{i}^{f})-p_{i}(t_{i}^{0}))u_{\min}}-3v_{i}(t_{i}^{0})}{2u_{\min}}.$

Similar to the previous case, $t_{i,v_{min}}^{f}$ and $t_{i,u_{\min}}^{f}$ are
the times at which leader of platoon $i\in\mathcal{L}(t)$ achieves its minimum
speed at the end of control zone and its minimum control input at the entry of
the control zone, respectively.

Note that, the solution to the optimal control problem 1 yields the optimal
control input $u_{i}^{*}(t)$ for each platoon leader $i\in\mathcal{L}(t)$ for
$t\in[t_{i}^{0},t_{i}^{f}]$. However, the solution to this problem does not
consider the stability criteria of the platoon [54], which is essential to
guarantee safety within the platooning CAVs. In the following section, we
first introduce the notion of stability during platoon coordination, and then
propose a control structure $u_{i,j}(t)$ for each platoon member
$j\in\mathcal{N}_{i}\setminus\\{0\\}$, $i\in\mathcal{L}(t)$ that is optimal
subject to constraints, and satisfies the stability properties.

###  3.2. Optimal Control of Followers Within Each Platoon

Stability properties of the platoon system are well discussed in the
literature [23, 53, 54]. In general, there are two types of stability: (a)
local stability, which describes the ability of each platoon member converging
to a given trajectory, and (b) string stability, where any bounded disturbance
introduced into the platoon are not amplified while propagating downstream
along the vehicle string. In this paper, we adopt the following definition of
platoon stability that encompasses the above stability notions [23].

######  Definition 3.1.

A platoon $i\in\mathcal{L}(t)$ is stable if, for any bounded initial
disturbances to all the CAVs $j\in\mathcal{N}_{i}$, the position fluctuations
of all the CAVs remain bounded (string stability) and approach zero as time
goes to infinity (local stability).

With the stability properties Definition 3.1, we introduce the control problem
of each platoon member $j\in\mathcal{N}_{i}\setminus\\{i\\}$,
$i\in\mathcal{L}(t)$.

######  Problem 2.

Each platoon member $j\in\mathcal{N}_{i}\setminus\\{0\\}$,
$i\in\mathcal{L}(t)$ needs to derive its control input $u_{i,j}(t)$ for all
$t\in[t_{i}^{0},t_{i}^{f}]$ that

  1. (1)

is energy and time-optimal subject to the state and control constraints in
(2)-(3), and rear-end collision avoidance constraint in (5), and

  2. (2)

satisfying the stability properties according to Definition 3.1.

We provide the following proposition that addresses the Problem 2.

######  Proposition 1.

For each platoon member $j\in\mathcal{N}_{i}\setminus\\{0\\}$ in the platoon
$i\in\mathcal{L}(t)$, the optimal control input $u_{i,j}(t)=u^{*}_{i,0}(t)$,
where $u^{*}_{i,0}(t)$ is the solution to Problem 1, is an optimal solution to
Problem 2.

Next, we provide the proof of Proposition 1 using the following Lemmas.

######  Lemma 3.2.

For each platoon member $j\in\mathcal{N}_{i}\setminus\\{0\\}$ in each platoon
$i\in\mathcal{L}(t)$, the control input $u_{i,j}(t)=u^{*}_{i,0}(t)$ for all
$t\in[t_{i}^{0},t_{i}^{f}]$ is energy- and time-optimal subject to the control
(2), state (3) and safety constraint (5).

###### Proof.

(a) Optimality: We derive the control input $u^{*}_{i,0}(t)$ of the leading
CAV $i\in\mathcal{L}(t)$ by solving Problem 1. The optimal trajectory of the
leader of platoon $i\in\mathcal{L}(t)$ is given by (3.1) for all
$t\in[t_{i}^{0},t_{i}^{f}]$. Thus, for each platoon member
$j\in\mathcal{N}_{i}\setminus\\{0\\}$, the control input $u_{i,j}(t)$ such
that $u_{i,j}(t)=u^{*}_{i,0}(t)$ also generates optimal linear control,
quadratic speed and cubic position trajectories as in (3.1).

(b) Constraint satisfaction: Since $u_{i,j}(t)=u^{*}_{i,0}(t)$, for each
platoon member $j\in\mathcal{N}_{i}\setminus\\{0\\}$, $i\in\mathcal{L}(t)$, we
have $v_{i,j}(t)=v_{i,0}^{*}(t)$ for all $t\in[t_{i}^{0},t_{i}^{f}]$. The
trajectories $v_{i,0}^{*}(t)$ and $u_{i,0}^{*}(t)$ do not violate any
constraints in (2)-(3) since they are derived by solving Problem 1. Therefore,
the trajectories $v_{i,j}(t)$ and $u_{i,j}(t)$ of each platoon member $j$ are
ensured to satisfy constraints in (2)-(3). Additionally, if
$u_{i,j}(t)=u_{i,0}^{*}(t)$, the inter-vehicle gap
$p_{i,j-1}(t)-p_{i,j}(t)-l_{c}$ between two consecutive platoon members
$j,j-1\in\mathcal{N}_{i},~{}i\in\mathcal{L}(t)$ is time invariant, and equal
to $\Delta_{i}$. Thus, the rear-end safety within CAVs within the platoon $i$
in (5) is guaranteed to be satisfied. ∎

######  Lemma 3.3.

Each platoon member $j\in\mathcal{N}_{i}\setminus\\{0\\}$ for each platoon
$i\in\mathcal{L}(t)$ with the control input $u_{i,j}(t)=u^{*}_{i,0}(t)$ for
all $t\in[t_{i}^{0},t_{i}^{f}]$ is locally stable, and the resulting platoon
$i$ is string stable.

###### Proof.

(a) Local stability: Since $u_{i,j}(t)=u_{i,0}^{*}(t)$, for each platoon
member $j\in\mathcal{N}_{i}\setminus\\{0\\}$, $i\in\mathcal{L}(t)$, we have
$v_{i,j}(t)=v_{i,0}^{*}(t)$ for all $t\in[t_{i}^{0},t_{i}^{f}]$. Since there
is no communication delay within the unidirectional intra-platoon
communication protocol, the speed of each platoon member $j$ converges
instantaneously to the speed of the platoon leader $v_{i,0}^{*}(t)$, which
implies local stability.

(b) String stability: A sufficient condition for the string stability of a
platoon $i\in\mathcal{L}(t)$ containing CAVs $j\in\mathcal{N}_{i}$ is
$\|\frac{u_{i,j}(s)}{u_{i,j-1}(s)}\|_{\infty}\leq 1$ [53], where $u_{i,j}(s)$
is the Laplace transform of the control input $u_{i,j}(t)$. Since
$u_{i,j}(t)=u_{i,0}^{*}(t)$ for all $t\in[t_{i}^{0},t_{i}^{f}]$, we have
$u_{i,j}(s)=u_{i,0}^{*}(s)$ for all $j\in\mathcal{N}_{i}\setminus\\{0\\}$,
which yields $\|\frac{u_{i,j}(s)}{u_{i,j-1}(s)}\|_{\infty}=1$. Thus, each
platoon $i\in\mathcal{L}(t)$ is string stable. ∎

###  3.3. Delay in Platoon Communication

In this section, we enhance our framework to include delay in the bi-
directional inter-platoon communication. From Assumption 1, we know that delay
is bounded and this bound is known a priori. In particular, suppose the delay
in bi-direction communication of platoon leaders takes values in
$[\tau_{\min},\tau_{\max}]$, where $\tau_{\min}\in\mathbb{R}_{\geq 0}$ and
$\tau_{\max}\in\mathbb{R}_{\geq 0}$ correspond to the minimum and maximum
communication delay, respectively. To account for the effects of communication
delays in our framework, we consider the worst-case scenario. Namely, we
consider that it takes $0.5~{}{\tau_{\max}}$ until the coordinator receives
the request from the platoon leader, and it takes an extra
$0.5~{}{\tau_{\max}}$ for the leader of platoon $i\in\mathcal{L}(t)$ to
receive the platoon information $\mathcal{I}_{i}$. Thus, the leader needs to
cruise with the constant speed that it entered the control zone for
$\tau_{\max}$ until it receives the platoon information $\mathcal{I}_{i}$ to
plan its optimal trajectory. After receiving this information, the platoon
leader computes the compact set of the feasible solution $\mathcal{T}_{i}$ at
time $t_{i}^{0}+\tau_{\max}$ with initial condition
$v_{i}(t_{i}^{0}+\tau_{\max})$ and $p_{i}(t_{i}^{0}+\tau_{\max})$. Using the
compact set $\mathcal{T}_{i}(t_{i}^{0}+\tau_{\max})$ of the feasible solution,
the leader derives its optimal control policy by solving Problem 1. Then, it
sends the computed trajectory at time $t_{i}^{0}+\tau_{\max}$ to the
coordinator. In the worst-case scenario, the coordinator receives this
information after $0.5{\tau_{\max}}$ at $t_{i}^{0}+1.5\tau_{\max}$. To ensure
that new arriving platoons have access to this information, we need to have
the following constraint on the initial conditions of platoons upon entrance
the control zone.

######  Proposition 2.

Let platoons $i$ and $j$, $i,j\in\mathcal{L}(t)$, enter the control zone at
time $t_{i}^{0}$ and $t_{j}^{0}>t_{i}^{0},$ respectively. In the presence of a
bi-directional inter-platoon communication delay, which takes value in
$[\tau_{\min},\tau_{\max}]$, the optimal trajectory of platoon $i$ is
accessible to platoon $j$, if $t_{j}^{0}-t_{i}^{0}\geq\tau_{\max}.$

###### Proof.

Platoon $i$ computes its optimal trajectory at $t_{i}^{0}+\tau_{\max}$, but in
the worst-case scenario, due to delay in communication, this information
becomes available to the coordinator at $t_{i}^{0}+1.5~{}\tau_{\max}$. On the
other hand, upon entrance the control zone, platoon $j$ sends a request to the
coordinator to receive platoon information $\mathcal{I}_{j}$. However, the
coordinator receives this request at $t_{j}^{0}+0.5~{}\tau_{\max}$. In order
to have the optimal trajectory of platoon $i$ accessible to platoon $j$ we
need to have $t_{j}^{0}+0.5~{}\tau_{\max}\geq t_{i}^{0}+1.5~{}\tau_{\max}$,
and the result follows. ∎

######  Remark 6.

We can ensure that the condition in Proposition 2 holds by using an
appropriate controller in the platooning zone upstream of the control zone.

###  3.4. Implementation of the Optimal Coordination Framework

In Sections 3.1, 3.2 and 3.3, we provided the exposition of the intricacies of
our proposed control framework for optimal platoon coordination. In this
section, we introduce the approach that can be applied to implement this
framework in real time.

While entering the control zone at time $t_{i}^{0}$, platoon leader
$i\in\mathcal{L}(t)$ obtains the platoon information $\mathcal{I}_{i}$ from
the coordinator and solves the optimization problem (1) by constructing the
feasible set $\mathcal{T}_{i}(t)$ and iteratively checking the safety
constraint. The resulting optimal exit time $t_{i}^{f}$ is then used along
with the initial (13) and boundary (14) conditions to derive the vector of
control input coefficients $\boldsymbol{\phi}_{i}$ using (3.1). Subsequently,
each CAV $j\in\mathcal{N}_{i}$ in platoon $i\in\mathcal{L}(t)$ computes its
optimal control input $u_{i,j}(t)$ at each time instance
$t\in[t_{i}^{0},t_{i}^{f}]$ using $\boldsymbol{\phi}_{i}$. In what follows, we
provide an algorithm that delineates the step-by-step implementation of the
proposed optimal platoon coordination framework.

Algorithm 1 Vehicular Platoons Coordination Algorithm

1:for $i\in\mathcal{L}(t)$ do

2: for $j\in\mathcal{N}_{i}$ do

3: if j=0 then$\triangleright$ Platoon leader

4: $u_{i,j}=0\quad\forall t\in[t_{i}^{0},t_{i}^{0}+\tau_{\max})$
$\triangleright$ Cruise with constant speed

5: Compute $\mathcal{T}_{i}(t_{i}^{0}+\tau_{\max})$ $\triangleright$ Based on
(16)-(17)

6: $t_{i}^{f},\boldsymbol{\phi}_{i}\leftarrow$ Platoon Leader Control()
$\triangleright$ Algorithm 2

7: $[a_{i},b_{i},c_{i},d_{i}]\leftarrow\boldsymbol{\phi}_{i}$

8: $u_{i,j}(t)\leftarrow 6a_{i}t+2b_{i}$ $\triangleright$ $\forall
t\in[t_{i}^{0}+\tau_{\max},t_{i}^{f}]$

9: else$\triangleright$ Platoon followers

10: $u_{i,j}(t)=u_{i,0}(t)$

11: end if

12: end for

13:end for

Algorithm 2 Platoon Leader Control

Input: Platoon Information set $\mathcal{I}_{i}$, Compact feasible set
$\mathcal{T}_{i}(t_{i}^{0}+\tau_{i})=[\underline{t}_{i}^{f},\overline{t}_{i}^{f}]$

Output: Exit time $t_{i}^{f}$, Coefficients of the optimal control policy
$\boldsymbol{\phi}_{i}$

1:$t_{i}^{f}\leftarrow\underline{t}_{i}^{f}$

2:$k\leftarrow$ platoon physically located in front of platoon $i$

3:$p_{k,m_{k}}(t)\leftarrow
p_{k}(t)-(M_{k}-1)(\Delta_{k}+l_{c})$$\triangleright$ Position of the last
follower

4:while $p_{k,m_{k}}(t)-p_{i}(t)<\delta_{i}(t)$ do $\triangleright$ Rear-end
safety

5: $t_{i}^{f}\leftarrow t_{i}^{f}+dt$

6:end while

7:lateral $\leftarrow$ list of all platoons $j<i$ from the other road

8:for $j\in~{}$ lateral do

9: Compute $t_{j,m_{j}}^{f}$ and $t_{i,m_{i}}^{f}$ from (8)

10: while $t_{i}^{f}-t_{j,m_{j}}^{f}<t_{h}$ AND
$t_{j}^{f}-t_{i,m_{i}}^{f}<t_{h}$ do$\triangleright$ Lateral safety

11: $t_{i}^{f}\leftarrow t_{i}^{f}+dt$

12: end while

13:end for

14:Compute $\boldsymbol{\phi}_{i}$ $\triangleright$ From (3.1)-(14)

##  4\. Simulation Example

###  4.1. Simulation Setup

To evaluate and validate the performance of our proposed optimal platoon
coordination framework, we employ the microscopic traffic simulation software
VISSIM v$11.0$ [22]. We create a simulation environment with a highway on-ramp
merging, which has a control zone of length $560$ m. In our simulation
framework, we use VISSIM’s component object model (COM) interface with Python
$2.7$ to generate platoons of CAVs on the main road and on the on-ramp at
different time intervals. The time interval between two consecutive platoon
generations is randomized with a uniform probability distribution, and the
bounds can be controlled to increase or decrease the traffic volume in each
roadway. The length of each platoon is also randomly selected from a set of
$2$ to $4$ vehicles with equal probability. The speed limit of each roadway is
set to be $16.67$ m/s, and the maximum and minimum acceleration limit is $3$
m/s2 and -$3$ m/s2, respectively. Vehicles enter the main road and the on ramp
with a traffic volume of $700$ and $650$ vehicle per hour per lane with random
initial speed uniformly chosen from a set of $13.89$ to $16.67$ m/s. Videos of
the experiment can be found at the supplemental site,
https://sites.google.com/view/ud-ids-lab/CAVPLT.

Figure 3. The position trajectories of the optimally coordinated CAV platoons
at the (a) main road and (b) ramp road are shown. Figure 4. Speed profiles of
$400$ vehicles traveling through the on-ramp merging scenario for three cases:
(a) baseline without platooning, (b) baseline with platooning and (c) optimal
platoon coordination. Figure 5. Comparison of performance metrics: (a)
cumulative fuel consumption of optimal coordination (black) vs. baseline with
platooning (red), and (b) total travel time distribution of optimal
coordination (blue) vs. baseline with coordination (maroon).

To evaluate the performance of the proposed optimal control framework, we
simulate the following control cases.

(a) Baseline 1: All vehicles in the network are human-driven vehicles. In this
scenario, the Wiedemann car-following model built in VISSIM [22] is applied.
The conflict point of the on-ramp merging scenario has a priority mechanism,
where the vehicles on the ramp road are required to yield to the vehicle on
the main road within a certain look-ahead distance. Vehicles enter the network
individually without forming any platoons.

(b) Baseline 2: Similar to the above case, all the vehicles are human driven
vehicles integrated with the Wiedemann car-following model, and follow the
priority mechanism set at the conflict point of the on-ramp merging scenario.
The difference is that, when vehicles enter the network, they have already
formed platoons. Note, we consider this case to simulate the same initial
condition of the optimal coordination case, which we discuss next.

(c) Optimal Coordination: All the vehicles present in the network are
connected and automated. They enter the network forming platoons of different
sizes and optimize their trajectories based on the optimal coordination
framework presented in Section 3.

We use COM application programming interface to interact with the VISSIM
simulator externally, and implement the proposed optimal coordination
framework. At each simulation time step, we use the VISSIM-COM interface to
collect the required vehicle attributes from the simulation environment and
pass them to the external python script. The external python script implements
the proposed single-level optimal control algorithm (Section 3.4) to compute
the optimal control input of each CAV within the control zone. Finally, the
speed of each platooning CAV is updated in VISSIM traffic simulator in real-
time using the COM interface.

###  4.2. Results and Discussion

In Fig. 4, the position trajectories of the optimal coordinated CAV platoons
traveling through the main road and the ramp road of the considered on-ramp
merging scenario are shown. The spatial gaps between the trajectory paths
indicate that our framework satisfies the rear-end collision avoidance
constraint without any violation.

To visualize the performance of the proposed coordination framework in
comparison with the baseline cases, we focus on Figs. 4 and 5. In Fig. 4, the
speed trajectories of all the vehicles in the network are shown. In Figs. 4
(a)-(b), both baseline cases show stop-and-go driving behavior close to the
conflict point of the on-ramp merging scenario. In contrast, with the optimal
coordination framework, we are able to completely eliminate stop-and-go
driving behavior, as shown in Fig. 4(c). The elimination of the stop-and-go
driving behavior has associated benefits, namely, the minimization of
transient engine operation and travel time, as shown in Fig. 5. In Fig. 5 (a),
the baseline case with vehicle platoons (red) show sudden increase in fuel
consumption near the conflict point due to the transient engine operation
induced by the stop-and-go driving behavior. In contrast, the cumulative fuel
consumption trajectories of the optimally coordinated CAVs (black) remain
steady throughout the their path. Note that, we use the polynomial metamodel
proposed in [30] to compute the fuel consumption of each vehicle. In Fig. 5
(b), we illustrate the distribution of total travel time of the vehicles for
the baseline (maroon) and the optimal coordination (blue) framework. The high
variance of the travel time for the baseline case compared to the optimal
coordination approach indicates increased traffic throughput of the network.

Finally, we provide the summary of the performance metrics in Table 1. Based
on the simulation, the optimal coordination framework shows significant
improvement over the baseline cases in terms of average travel time and fuel
consumption.

Table 1. Summary of performance metrics Performance Metrics |  Avg. travel time [s] |  Avg. fuel consumption [gallon]
---|---|---
Baseline 1 |  57.33 |  0.042
Baseline 2 |  52.79 |  0.05
Optimal Coordination |  46.1 |  0.022
Improvement (baseline 1) [%] |  19.6 |  46.9
Improvement (baseline 2) [%] |  12.7 |  38.2

##  5\. Concluding Remarks

In this paper, we leveraged the key concepts of CAV coordination and
platooning, and established a rigorous optimal platoon coordination framework
for CAVs that improves fuel efficiency and traffic throughput of the network.
We presented a single-level optimal control framework that simultaneously
optimizes both fuel economy and travel time of the platoons while satisfying
the state, control, and safety constraints. We developed a robust coordination
framework considering the effect of delayed inter-platoon communication and
derived a closed-form analytical solution of the optimal control problem using
standard Hamiltonian analysis that can be implemented in real time using
leader-follower unidirectional communication topology. Finally, we validated
the proposed control framework using a commercial simulation environment by
evaluating its performance. Our proposed optimal coordination framework shows
significant benefit in terms of fuel consumption and travel time compared to
the baseline cases.

Ongoing work addresses the delay in intra-platoon communication and its
implications on platoon stability. A potential direction for future research
includes relaxing the assumption of $100$% CAV penetration considering the
inclusion of human-driven vehicles. Investigating the impact of connected
vehicle technologies on the size of the components of the vehicle’s powertrain
[47] and the implications on traveler’s decisions [63] is another potential
direction for future research.

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Received xxxx 20xx; revised xxxx 20xx.