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Cut Through Traffic Like a Snake: Cooperative Adaptive Cruise
Control with Successive Platoon Lane Change Capability
Haoran Wanga, Xin Lib, Xianhong Zhangc, Jia Hua*, Xuerun Yana, and
Yongwei Fenga
a Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji
University, Shanghai, China; b College of Transportation Engineering, Dalian
Maritime University, Dalian, China; c Autonomous Driving Business Unit, SAIC Motor
Corporation Limited, Shanghai, China
CONTACT Jia Hu, hujia@tongji.edu.cn, Key Laboratory of Road and Traffic
Engineering of the Ministry of Education, Tongji University, Shanghai, 201804, China
Cut Through Traffic Like a Snake: Cooperative Adaptive Cruise
Control with Successive Platoon Lane Change Capability
A Cooperative Adaptive Cruise Control (CACC) system may be impeded by slow-
moving traffic in the application. To improve the mobility of CACC, this research
proposes a CACC controller with successive platoon lane-change capability. The
goal is to help a platoon cut through traffic successively like a snake via smaller
windows. The proposed controller has the following features: i) with successive
platoon lane-change capability; ii) with string stability and lateral stability; iii) with
consideration of vehicle dynamics. The proposed controller is evaluated on a
simulation platform with the context of traffic and a joint simulation platform
consisting of PreScan and Matlab/Simulink. Results demonstrate that compared to
the conventional controller: i) platoon lane-change competence is enhanced by
71.36% on arterials and 120.49% on freeways; ii) platoon lane-change efficiency
is enhanced by 25.05% on arterials and 41.36% on freeways; iii) the proposed
controller is more robust against congestion. Moreover, the computation time of
the proposed controller is approximately 15 milliseconds when running on a laptop
equipped with an Intel i7-8750H CPU. This indicates that the proposed controller
is ready for real-time implementation.
Keywords: CACC, platoon lane change, motion planning, space-domain, stability
analysis.
Introduction
Improving mobility, sustainability, and safety are three major goals of transportation
operations. However, 81.3% of cities around the world are experiencing an increasing
level of traffic congestion according to TomTom Traffic Index 2016 (TOMTOM, 2016).
Traffic congestion not only deteriorates transportation mobility (Treiterer & Myers, 1974)
but also drives up fuel consumption (Greenwood & Bennett, 1996) and emissions (Barth
& Boriboonsomsin, 2008). Moreover, in 2017, 37,133 traffic fatalities occurred in the US.
It indicates that 1.16 fatalities occurred per 100 million vehicle miles traveled (National
Center for Statistics and Analysis, 2019, June).
The emerging Cooperative Adaptive Cruise Control (CACC) system is able to
mitigate the aforementioned problems. CACC forms Connected Automated Vehicles
(CAVs) into a platoon. The platoon time-headway can drop to as low as 0.5seconds
according to field tests, much smaller compared to the 1-2seconds smallest time headway
of Human-driven Vehicles (HVs) (Nowakowski et al., 2010; Shladover et al., 2010;
Shladover et al., 2012). This time headway reduction leads to an increase in traffic
capacity to over 4,200veh/hr/ln, which nearly doubles conventional road capacity
(Vander Werf et al., 2002). Cooperation among vehicles is also expected to reduce 12%
fuel consumption and 14% carbon dioxide emission (Park et al., 2011). In addition,
CACC is forecasted to improve traffic safety by reducing the diversity of heterogeneous
driving behavior, which is regarded as one of the major causes of crashes (Dey et al.,
2015).
When CACC is implemented in the field, the lateral lane-change capability is
critical since a platoon needs to change lane from time to time. Lack of lane-change
capability forces a CACC platoon to disengage when lane-change cannot be avoided
(Bengtsson et al., 2015; Liu et al., 2018; Toy et al., 2002). This platoon disengagement
not only calls for unnecessarily additional human take-over maneuvers but also breaks
the running continuity of CACC systems (Hu et al., 2020).
However, only a few past studies make efforts on developing Platoon Lane
Change (PLC) controllers for CACC. One of the conventional PLC methods is to
disengage the platoon and control vehicles to change lane freely (H.-H. Hsu & Liu, 2004;
Lam & Katupitiya, 2013). This method breaks the platoon form and control continuity.
Hsu and Liu proposed a PLC controller to keep the form of a platoon when changing lane
(H. C.-H. Hsu & Liu, 2008). Whereas, this PLC controller plans lateral motions by
computing lateral acceleration without considering vehicle dynamics. Therefore, the
planned path may not be feasible for a vehicle to fulfill. This problem could be handled
by an artificial potential field based controller that considers vehicle dynamics (Huang et
al., 2018). However, this controller may result in a local optimum that leads to lateral
oscillations and along with it -- risks. Moreover, most PLC controllers only allow CAVs
in a platoon to change lane simultaneously. The capability of the Simultaneous Platoon
Lane Change (SiPLC) method highly relies on the chance of finding a big-enough gap.
For a CACC system with five CAVs, one-second headway, and 65 mph cruising speed,
the platoon length is about 150 meters. In order to maneuver the platoon to change lane
simultaneously, a gap greater than 150 meters is needed. This gap is in correspondence
with the traffic density rated at the level of service A. Therefore, the SiPLC method
significantly reduces the chance to conduct lane change and further damages mobility.
However, in scenarios shown in Figure 1, lane-change windows are not enough for a
SiPLC maneuver, while PLC is still needed. Hence, a next-generation Successive Platoon
Lane Change (SuPLC) is proposed in this paper to increase mobility and ensure the
running continuity of CACC. The SuPLC controller has a different logic compared to the
conventional SiPLC controller, as shown in Figure 2. It enables a platoon to change lane
in turn by adopting smaller lane-change gaps. Therefore, it increases the lane-change
capability of CACC.
Figure 1. Application examples of SuPLC
Figure 2. Time-space resources allocation comparison between SiPLC and SuPLC
Platoon stability is necessary for safeguarding against disturbances in the field
application of CACC (Ploeg et al., 2011). A variety of stabilities shall be ensured, such
as local stability (Y. Zhang et al., 2020), string stability (Ploeg et al., 2013; Seiler et al.,
2004; Shaw & Hedrick, 2007; Swaroop & Hedrick, 1996), stability margin (Hao &
Barooah, 2012; Hao et al., 2011), and behavior coherence (Bamieh et al., 2012; Fardad et
al., 2011). Past studies have developed various CACC controllers with stability under its
regular cruise status. However, CACC controllers with stability under lane-change status
have not yet been developed. Moreover, the SuPLC maneuver is longitudinal and lateral
coupled. Followers track both longitudinal and lateral motions of the leader. Therefore,
both longitudinal and lateral stability should be proven.
In this paper, a next-generation CACC controller is proposed with enhanced lane-
change capability. The proposed controller is with the following features:
With successive platoon lane-change capability;
Ensuring string and lateral stability;
With consideration of vehicle dynamics.
The remainder of this paper is organized as follows. Section “Research scope”
presents the research scope of this paper. Section “Problem formulation” formulates the
SuPLC controller and presents a solution algorithm. Section “Performance analysis”
analyses the performance of the proposed controller and proves its stability. Section
Evaluations evaluates the proposed controller in simulation. Section “Conclusion and
future research” makes a conclusion and provides future research ideas.
Research scope
The goal of the proposed controller is to maneuver a CACC platoon to cut through traffic
successively like a snake. There are three highlights of this controller:
Smaller lane-change window requirement: The proposed controller can
control CAVs to change lane in turn. Therefore, a moving lane-change window that is
large enough for a single CAV’s lane-change maneuver is also theoretically sufficient for
platoon lane-change maneuver. Lower platoon lane change window threshold provides
platoons a better chance to cut through traffic. It further improves the platoon lane-change
capability. This highlight is demonstrated in the evaluation.
Reducing needs for platoon disengagement: In scenarios such as utilizing a
left turn pocket or exit ramp, the platoon has to be disengaged under the control of
conventional controllers. However, the proposed SuPLC controller can keep running
continuity in these scenarios.
String and lateral stable: The proposed controller is proven with string and
lateral stability.
System structure
The proposed SuPLC controller is adopted in the CACC system as presented in Figure 3.
The system consists of three modules. Module 1 makes platoon lane change decisions.
Module 2 plans trajectories and motions for the SuPLC maneuver. Module 3 is an
execution layer. The research scope of this paper is module 2 – the planning of the SuPLC
maneuver.
Details of the system are provided as follows:
Module 1: This module makes decisions on platoon lane change. First, a
choice shall be made between SiPLC and SuPLC based on real-time traffic statuses. The
SuPLC method is adopted when there is not enough SiPLC space. Then, a PLC window
is selected for the SuPLC maneuver. This paper proposes a simple rule-based platoon lane
change window selection algorithm for the evaluation with traffic, as shown in Table 1.
The algorithm ensures that a SuPLC decision is made only when enough space could be
found to settle down the platoon. Conservative parameters are adopted to avoid a SuPLC
maneuver being cut-off.
Table 1. Platoon lane-change window selection algorithm
Scenario notations:
Initialize:
;
;
Input:
; ; ; for each window: , , and
Output: PLC window selection
for all windows in from near to far
if
then
if
then
if
then
The window is selected for SuPLC.
where
is the safe lane-change window between a vehicle and its lag vehicle;
is the safe lane-change window between a vehicle and its front vehicle;
is the
human reaction time;
is the CAV reaction time; is the delay of brake;
is the speed of the lag vehicle of the window; is the speed of the front vehicle
of the platoon;
is the minimum car-following time in traffic;
is platoon desired
speed; is the length of platoon; is the set of alternative PLC windows.
Module 2: After receiving SuPLC command and status information passed on
from module 1, the platoon leader is controlled by an Automated Lane-Change (ALC)
controller (Nilsson et al., 2015) and an Adaptive Cruise Control (ACC) controller
(Kesting et al., 2008). The longitudinal control objective of the leader is to maintain the
desired speed. The lateral control objective of the leader is to change lane through the
selected window. All followers are controlled by the proposed SuPLC controller with the
objective of following the trajectory of the leader. Control commands are passed on to
module 3.
Module 3: This module actuates control commands provided by module 2.
Figure 3. System logic
Problem statement
To achieve a successive platoon lane-change maneuver, followers shall track longitudinal
and lateral motions of the leader. Therefore, the objective of the proposed SuPLC
controller is decomposed into a longitudinal objective and a lateral objective as follows:
Longitudinal objective: Longitudinal objective of the proposed controller is
to maintain a constant and stable time headway between adjacent CAVs in a platoon.
Lateral objective: Lateral objective of the proposed controller is to minimize
lateral tracking error. Lateral tracking error is defined as follows:
(1)
where is the longitudinal position in a relative frenet coordinate as shown in Figure 4.
In this relative frenet coordinate system, the origin is on the centerline of the road.
is the lateral position of the jth follower at a longitudinal position ; denotes the
leading CAV.
Problem formulation
The formulation of the optimal control based SuPLC controller is presented in this section.
It is designed to minimize longitudinal and lateral following error while fulfilling the
ultimate goal: capable of maneuvering through smaller windows.
Figure 4 presents a typical scenario where a platoon makes a successive lane
change. Indices and parameters utilized hereafter are defined in TABLE 2. Some of them
are depicted in Figure 4 for better illustration purposes.
Figure 4. Scenario notations
Table 2. Indices and Parameters
Parameters
Definition
Acceleration
The acceleration of platoon leader
Commanded acceleration
Minimum acceleration
Maximum acceleration
State coefficient matrix in dynamics model at step i
State coefficient matrix in the dynamics of a platoon with n followers
The unit block of coefficient matrix
Control coefficient matrix in dynamics model at step i
Control coefficient matrix in the dynamics of a platoon with n
followers
The unit block of coefficient matrix
A constant matrix in vehicle dynamics model at step i
A constant matrix of vehicle dynamics in the continuous domain
A constant matrix in the dynamics of a platoon with n followers
The longitudinal component of constant matrix
The lateral component of constant matrix
The longitudinal component of constant matrix
The lateral component of constant matrix
Window size (meters)
Distance between the platoon leader and the lag vehicle of the window
(meters)
Distance between the platoon leader and the front vehicle of the
platoon (meters)
The safe lane-change window between a vehicle and its lag vehicle
(meters)
The safe lane-change window between a vehicle and its front vehicle
(meters)
A concomitant matrix in the solution algorithm
Natural logarithm
The tracking error of the jth vehicle (meters)
Heading angle error (rad)
Heading angle error of the platoon leader (rad)
A parameter matrix of the control vector in the frequency domain
A parameter matrix of the platoon control vector with n followers in
the frequency domain
System dynamics function
A parameter matrix of vehicle control vector in the frequency domain
A parameter matrix of platoon control vector with n followers in the
frequency domain
An x-domain function
The format of function in the frequency domain
The safe lateral gap between vehicles (meters)
Time headway (seconds)
The safe lane-change headway of a vehicle (seconds)
Desired time headway in a platoon (seconds)
Safe car-following headway in a platoon (seconds)
Minimum platoon lane change headway (seconds)
Minimum simultaneous platoon lane change headway (seconds)
Minimum successive platoon lane change headway (seconds)
Minimum car-following headway in traffic (seconds)
Average traffic headway (seconds)
Hamiltonian function
Control step index
Iteration index in the solution algorithm
Identity matrix
Vehicle index in a platoon ( denotes the platoon leader)
Total cost
The minimal total cost
The index of surrounding vehicles
Distance between the front axle and rear axle (meters)
The width of the vehicle (meters)
The length of the vehicle (meters)
The length of the platoon (meters)
Running cost function
Laplace operator
A binary decision variable
A set of in all steps
An optimal
An optimal
An infinite value
A parameter matrix of vehicle control vector in the frequency domain
A parameter matrix of platoon control vector with n followers in the
frequency domain
The number of followers in a platoon
Total control steps
Positive natural numbers set
A parameter matrix in the lateral oscillation transfer function between
the jth follower and the platoon leader for a platoon with n followers
A parameter matrix in the string oscillation transfer function between
the jth follower and the platoon leader for a platoon with n followers
Platoon lane change capacity
Simultaneous platoon lane change capacity
Successive platoon lane change capacity
, , ,
Weighting factors in diagonal matrix
The weighting factor matrix of state error in a cost function
The weighting factor matrix in the centralized platoon control
problem
A concomitant matrix in the solution algorithm
,
Weighting factors in the diagonal matrix
The weighting factor matrix of control cost
Weighting factor matrix in the centralized platoon control problem
Positive set
The co-state vector in the frequency domain for a platoon with n
followers
A complex variable
Longitudinal position in relative frenet coordinate (meters)
The longitudinal position of the kth surrounding vehicle (meters)
The longitudinal position of CAV j in a platoon (meters)
Control horizon in the spatial domain (meters)
Global time (seconds)
The global time of the jth CAV at step i (seconds)
The delay of brake (seconds)
The CAV reaction time (seconds)
The human reaction time (seconds)
Vehicle control vector in the SuPLC controller
Vehicle control vector at step i in the SuPLC controller
Vehicle control solution of the iteration number
An optimal vehicle control vector
An optimal vehicle control vector
Control vector
The control vector of a platoon with n followers
The optimal platoon control vector
The control vector of a platoon with n followers in the frequency
domain
Vehicle speed (m/s)
Speed limit (mph)
Traffic speed (mph)
Free-flow speed (mph)
Platoon desired speed (mph)
The speed of the lag vehicle of the window (m/s)
The speed of the front vehicle of the platoon (m/s)
Traffic volume (veh/hr/ln)
Traffic Volume-to-Capacity rate
The set of alternative windows
domain variable
State vector
The state vector of a platoon with n followers
The optimal platoon state vector
The state vector of a platoon with n followers in the frequency domain
Vehicle lateral position (meters)
The lateral position of the platoon leader (meters)
The lateral position of the jth follower in the platoon (meters)
The lateral position of the kth surrounding vehicle (meters)
The maximum lateral position (meters)
The minimum lateral position (meters)
A parameter matrix in concomitant matrices at step i
A parameter matrix in concomitant matrices at step i
A parameter matrix in concomitant matrices at step i
A parameter matrix in concomitant matrices at step i
A parameter matrix in concomitant matrices at step i
The oscillation transfer function between two adjacent CAVs and
in a platoon with n followers
Wheel angle oscillation transfer function between follower and
follower in a platoon with n followers
Wheel angle oscillation transfer function between follower and the
leader in a platoon with n followers
Acceleration oscillation transfer function between follower and
follower in a platoon with n followers
Acceleration oscillation transfer function between follower and the
leader in a platoon with n followers
Front-wheel angle (rad)
The front-wheel angle of the platoon leader (rad)
The minimum front-wheel angle (rad)
The maximum front-wheel angle (rad)
Commanded steering angle (rad)
Tolerance error
Imaginary number
Road curvature
Co-state vector
The co-state vector for a platoon with n followers
The optimal co-state vector
The control vector of the jth follower in the frequency domain
Vehicle state vector in the SuPLC controller
Vehicle state vector at step i in the SuPLC controller
Vehicle state solution of the iteration number
The desired vehicle state vector
Desired vehicle state vector at step i
Reciprocal of speed (s/m)
Reciprocal of the leader’s speed (s/m)
Reciprocal of the leader’s initial speed (s/m)
Heading angle (rad)
The terminal cost function in the centralized platoon controller
Angular speed in the Laplace domain
Control step-size (meters)
The domain of binary variable
State definition
Since the lateral objective is to minimize lateral tracking error, a space-domain-based
controller can improve control accuracy compared to a time-domain-based controller.
Therefore, state and control variables are defined in space-domain, as shown in Definition
1.
Definition 1 (State and control vector): The system state vector and control
vector are defined as follows:
(2)
(3)
where is longitudinal position; is lateral position; is heading angle error; is
global time; is acceleration; is front-wheel angle; is defined as follows:
(4)
The desired state of ego vehicle is the state of the leader at the same longitudinal
position, as presented in Definition 2. The longitudinal position could be absolute or
relative in regard to the earth, depending on how the coordinate system is defined.
Definition 2 (Desired state): The ego CAV is the jth follower in the platoon. The
desired state is defined as follows:
(5)
(6)
where is the index of the leader; is the desired arrival time at a position ;
is the desired headway in a platoon.
System dynamics
The dynamics of a CAV are formulated based on a bicycle model. The conventional
bicycle model is formulated as follows (Campion et al., 1996; Dang et al., 2020; Lamiraux
& Lammond, 2001; H. Wang et al., 2019; C. Zhang et al., 2019):
(7)
(8)
(9)
(10)
where is front-wheel angle; is heading angle; is the distance between the front
axle and rear axle; is road curvature.
By substituting equations (4) and (7) into equations (8), (9), and (10), a space-
domain vehicle dynamics model is given as follows:
(11)
(12)
(13)
(14)
By applying the Euler Method and Definition 1, equations (11), (12), (13), and
(14) are discretized as follows:
(15)
(16)
(17)
(18)
Equations (15), (16), (17), and (18) describe the vehicle dynamics for the
proposed controller.
Cost function
The objective of the proposed SuPLC controller is to minimize total state error in the
control horizon. The state cost is formulated into a quadratic form
. The control cost
is used to restrict control efforts. The objective function
is formulated as follows:
(19)
where is the desired state at step ; is the terminal state. Weighting factors
and are defined as follows:
(20)
(21)
where , , , , , and are non-negative constants.
Constraints
The proposed controller is bounded by collision avoidance, desired speed range,
geometry boundaries, acceleration capability, and yaw angle constraints.
Collision avoidance constraints
Longitudinal collision avoidance between adjacent CAVs in a platoon is achieved by
regulating time headway as follows:
(22)
where
is the safe following headway in a platoon.
Lateral collision avoidance between a CAV and a surrounding HV is ensured by
equations (23) and (24).
(23)
(24)
where is the index of surrounding vehicles;
is the lateral safe gap; is the
width of a vehicle; is the length of a vehicle; is the speed of traffic flow;
is the safe following headway in traffic. is a decision variable. . is a
constant and significantly large value. The value of is directly associated with the
relative position of ego vehicle and lag vehicle, as demonstrated in Figure 5. The value
of is determined by finding the best relative position that minimizes the cost (equation
(19)). This concept follows the big M method (Ding et al., 2013). The optimization solver
adopted iterates through both potential values of and picks a value that produces the
minimal cost.
Figure 5. Value of illustration
Geometry boundaries constraint
Vehicles should travel within road geometry boundaries as follows:
(25)
Acceleration constraint
The vehicle’s acceleration should be bounded by vehicle capability and comfort.
(26)
Front-wheel angle constraint
The vehicle’s front-wheel angle should be within its steering range.
(27)
Solution algorithm
A solution algorithm previously developed by this research group is borrowed (Yu Zhang,
2020). It was originally developed for the longitudinal automation of CACC. Here it is
enhanced for longitudinal and lateral coupled automation functions. It can accelerate
computation speed greatly by predetermining the optimal terminal state. In each iteration
cycle, this algorithm involves a backward calculation of concomitant matrices and a
forward calculation of the control vector and state vector. It can iterate several times to
decrease the effects of inconstant coefficient matrices. Moreover, inspired by the
conventional branch and bound method (Lawler & Wood, 1966), this algorithm is further
enhanced here: i) restricting the objective function into a form of quadratic terms only; ii)
considering constraints; iii) considering parameter-varying dynamics coefficients; iv)
capable of solving mixed-integer programming problems. These designs are better
compatible with the problem of interest. This algorithm is described as follows:
Table 3. Solution algorithm
Input: initial state ,desired state , , , number of control steps
Output: optimal control and state for each step
Initialize:
while do
use to calculate , , and for
initialize , the domain of :
while is not empty do
pick and remove from
solve QP problem QP:
(28)
(29)
For
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
For
(38)
(39)
if QP is infeasible then
prune current node
else if
then
prune current node
else if
then
,
else
choose integer
from
if
is feasible and
then
,
branch node
if then
if then
if then
if then
Performance analysis
String stability is a must for a CACC system to resist speed perturbations. It has been
proposed and proven by past studies on longitudinal only CACC. However, the proposed
SuPLC controller is longitudinal and lateral coupled. Followers are with a goal of tracking
the lateral position trajectory of the leader. Therefore, lateral stability should also be
proven to ensure lateral oscillation would not be amplified when propagating along a
vehicle string. Moreover, the lane-change capacity of the proposed SuPLC controller is
quantified theoretically in this section.
Pontryagin’s Minimum Principle (PMP) is a method to solve optimal control
problems (Khalil, 1994; M. Wang et al., 2014a, 2014b). It is presented in Definition 3 for
further analysis.
Definition 3 (PMP method): Necessary conditions for the optimal solution are
as follows:
(40)
(41)
(42)
(43)
(44)
where is state vector; is control vector; is a co-state vector; is the running cost
function; is the system dynamics function; is control horizon; is the terminal cost
function.
To analyse platoon performance, Proposition 1 is first adopted.
Proposition 1 (Platoon controller in continuous form): A centralized platoon
controller is presented in the following. This platoon controller is equivalent to the SuPLC
controller.
(45)
s.t.
(46)
where
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
(57)
(58)
Proof:
The cost function of the proposed SuPLC controller in equation (19) is
transformed into a continuous domain as follows:
(59)
(60)
To adopt the PMP method, the state vector is set to . The vehicle
dynamics model in equation (15) is transformed into a continuous domain as follows:
(61)
where
(62)
(63)
(64)
Definition 3 is utilized to derive the control law. For a quadratic optimal control
problem, the optimal value is where the derivative is zero. Therefore, equation (41)
indicates:
(65)
By substituting equations (40), (60), and (61) into equations (65) and (43), the
control law of the SuPLC controller is as follows:
(66)
(67)
By substituting equations (40), (46), and (47) into equations (65) and (43), the
control law of the centralized platoon controller in Proposition 1 is as follows:
(68)
(69)
where is the co-state vector of the centralized platoon controller in Proposition 1.
is defined as follows:
(70)
Substituting equations (48), (49), (50), (51), (52), (53), and (70) into equations
(68) and (69):
(71)
(72)
Equations (71) and (72) are the same with equations (66) and (67) for any follower
j. Therefore, the control law of the SuPLC controller is the same as the control law of the
centralized platoon control problem in Proposition 1. This platoon controller is equivalent
to the SuPLC controller. This concludes the proof.
Stability criteria
String stability and lateral stability describe the condition that state oscillation would not
be amplified backward in a platoon. Since oscillation is better analyzed in the frequency
domain, the Laplace transform is adopted as presented in Definition 4. Stability criteria
are then proposed in Definition 5.
Definition 4: (Laplace transform): Function with can be transformed
into the frequency domain as follows:
(73)
(
(74)
where denotes the natural logarithm; is a complex variable; denotes imaginary
number; denotes angular speed in the Laplace domain.
By adopting existing string stability definitions (Ploeg et al., 2011; Ploeg et al.,
2013), string stability and lateral stability criteria are defined as follows:
Definition 5 (Stability criteria): For , the platoon is string stable and
lateral stable if and only if:
, ,
(75)
, ,
(76)
where
is the acceleration oscillation transfer function between follower and
follower ;
is the front wheel angle oscillation transfer function between
follower and follower ; denotes 2-norm.
String stability and lateral stability analysis
Lemma 1, Lemma 2, Lemma 3, and Lemma 4 are first proposed. String stability and
lateral stability are then proven in Theorem 1. The logic of analysis is presented as follows:
Lemma 1 presents the oscillation transfer function between the follower
and the leader.
Lemma 2 discusses the propagation rule (, transfer function between
follower and the leader) with regard to various platoon sizes, as illustrated in the yellow
line in Figure 6.
Lemma 3 discusses the propagation rule (, transfer function between the
first follower and the leader) with regard to various platoon sizes, as illustrated in the
green line in Figure 7.
Lemma 4 discusses the propagation rule (, transfer function between
two adjacent followers) with regard to various platoon sizes, as illustrated in the red line
in Figure 7.
Theorem 1 proves that the oscillation transfer function is always less than 1
(). It concludes the stability analysis.
Figure 6. Relation of the transfer function between a follower and the leader
Figure 7. Relation of the transfer function between adjacent CAVs
Lemma 1 (Follower-leader transfer function ( )): Oscillation transfer
functions between follower and the platoon leader are as follows:
(77)
(78)
where
(79)
(80)
(81)
(82)
(83)
Proof:
By applying Definition 4, state vector, control vector, and co-state vector are
transformed into the frequency domain as follows:
(84)
(85)
(86)
(87)
By applying Definition 4 and equations (84), (85), (86), platoon dynamics
function (equations (46)) and control law (equations (68), (69)) are transformed into the
frequency domain as follows:
(88)
(89)
(90)
Substituting equations (88) and (90) into equation (89):
(91)
is longitudinal and lateral coupled as defined in equation (56). Following
analysis proves no interaction exists between longitudinal and lateral perturbation.
Applying Proposition 1 and equation (87):
(92)
(93)
where
(94)
(95)
(96)
Substituting equations (20), (21), (54), and (55) into (94), (95), and (96):
(97)
(98)
(99)
Therefore:
(100)
(101)
Equations (100) and (101) indicate that longitudinal and lateral behaviors are not
coupled in the space domain. Therefore, by substituting equations (56), (79), (80), and
(91) into equations (75) and (76), equations (77) and (78) are derived. This concludes the
proof.
Lemma 2 (The propagation rule ()): With regard to various platoon size ,
the oscillation transfer function between follower and the leader follows the
following propagation rule:
(102)
(103)
Proof:
Given equations (50), (51), (52), (53), (56), (57), and (58):
(104)
(105)
(106)
(107)
(108)
(109)
Substituting equations (104), (105), (106), and (107) into equations (81), (82), and
(83):
(110)
(111)
(112)
Given equations (79) and (80):
(113)
(114)
Substituting equations (105), (108), (109), (110), (111), (112), (113), and (114)
into equations (77) and (78):
(115)
(116)
This concludes the proof.
Lemma 3 (The propagation rule ()): The oscillation transfer function
between the first follower and the leader is constant over various platoon size :
(117)
(118)
Proof:
Given equations (79) and (80):
(119)
(120)
Substituting equations (105), (108), (109), (110), (111), (112), (119), and (120)
into equations (77) and (78):
(121)
(122)
This concludes the proof.
Lemma 4 (The propagation rule ()): With different platoon size , the
propagation rule of the transfer function between two adjacent followers is as
follows:
Proof:
Given Lemma 2,
(125)
(126)
Therefore,
(127)
(128)
Given equations (127) and (128), equations (123) and (124) are derived. This
concludes the proof.
Theorem 1 (String stability and lateral stability): The CACC platoon under
control of the proposed SuPLC controller is with string stability and lateral stability.
Proof:
As illustrated in Figure 7, Lemma 3 and Lemma 4 gives:
(129)
(130)
Since ,, ,, , and are all non-negative, the following inequalities are
always satisfied by substituting equations (97), (98), (99), (100) and (101) into equations
(77) and (78).
(131)
(132)
(123)
(124)
Applying equations (131) and (132) into equations (129) and (130):
(133)
(134)
The stability criteria in Definition 5 are satisfied. Therefore, the CACC platoon
under control of the proposed SuPLC controller is string stable and lateral stable
regardless of platoon size. This concludes the proof.
Platoon lane change capacity
Compared with the conventional SiPLC controller, the proposed SuPLC controller is
capable of maneuvering a platoon through smaller windows. Therefore, a moving lane-
change window that is large enough for a single CAV’s lane-change maneuver is also
theoretically sufficient for platoon lane-change maneuver. The minimum lane-change
window reduction enhances the maximum platoon lane change capacity. In this section,
this enhancement on lane-change capacity is quantified.
Platoon lane change capacity is defined as the probability of traffic headway
larger than the minimum platoon lane change headway
:
(135)
A traffic headway distribution model is utilized to compute this probability as
follows (Schuhl, 1955):
(136)
where
is the minimum traffic headway; is average traffic time-headway:
(137)
where is traffic volume in the unit of veh/hr/ln. It is computed as follows (Margiotta &
Washburn, 2017):
(138)
where is volume-to-capacity ratio;
is the free-flow speed in the unit of mph.
Therefore, the platoon lane change capacity is finally quantified as follows:
(139)
The minimum successive platoon lane change headway
and the minimum
simultaneous platoon lane change headway
are computed as follows:
(140)
(141)
where
is the safe lane-change headway of a single vehicle;
is the desired
headway in a platoon.
Following settings are adopted:
is set to 1.1 sec (Milanés & Shladover, 2014);
is set to 0.5 sec (Milanés & Shladover, 2014);
is set to 2.68 sec (Yang et al., 2019).
The platoon lane change capacity of the proposed SuPLC controller and
the platoon lane change capacity of conventional SiPLC controller are computed
as follows:
(142)
(143)
A sensitivity analysis is conducted by numerical simulation. It considers two
scenarios: i) arterial with a free-flow speed of 72km/h (45mph); ii) freeway with a free-
flow speed of 105km/h (65mph).
Simulation results are illustrated in Figure 8. It demonstrates that the SuPLC
controller enhances platoon lane change capacity by 20.44% on arterials and 18.81% on
freeways, compared to the SiPLC controller.
Figure 8. Platoon lane change capacity analysis
Evaluations
The proposed SuPLC controller is evaluated from the following aspects: i) platoon lane-
change capability; ii) successive platoon lane change function validation; iii) platoon
string stability; iv) platoon lateral stability. Following settings are adopted for the
evaluation.
Following headway in a platoon
: 0.7s;
Acceleration range: ;
Steering wheel angle range: (Cho, 2009);
Lane width: 3m (9.8ft);
Human reaction time
: 1.5s;
CAV reaction time
: 0.5s;
Delay of brake :0.35s.
Function verification with vehicle dynamics
A simulation is conducted on a joint simulation platform consisting of PreScan and
Matlab/Simulink. PreScan follows a systematic physics-based simulation approach.
Efficient parametric math models reproduce system-level vehicle dynamics behavior.
Therefore, this experiment is a function verification with vehicle dynamics embedded
into the platform. The goal of this evaluation is as follows:
Function validation: This test is to verify the function of changing lane
successively from the aspect of vehicle control performance.
Stability evaluation: This test is to evaluate the string stability and lateral
stability of the proposed controller.
Test scenarios
The test is designed on two road types:
Arterial: with a speed limit of 45mph (72km/h);
Freeway: with a speed limit of 65mph (105km/h).
Two scenarios are designed for different evaluation purposes.
Scenario 1: successive platoon lane change function validation:
As illustrated in Figure 9, in this scenario, a platoon has to change lane to avoid
running into the off-ramp. The leading and left-side HVs cannot provide the platoon with
big enough cut-through windows. Therefore, the platoon has to change lane successively
under the control of the proposed SuPLC controller.
Figure 9. Scenario 1: successive platoon lane change function validation
Scenario 2: stability evaluation:
As illustrated in Figure 10, in this scenario, a platoon is free-cruising on a single-
lane road. speed perturbation and lateral perturbation are applied on the
platoon leader respectively, as illustrated in Figure 11.
Figure 10. Scenario 2: stability evaluation
Figure 11. Longitudinal and lateral perturbation on the leader
Results
Results confirm that the proposed controller is: i) capable of maneuvering a platoon
through traffic by utilizing a smaller lane-change window; ii) with string stability; iii)
with lateral stability; iv) with 15 milliseconds average computation time on a laptop
equipped with an Intel i7-8750H CPU.
Figure 12 and Figure 13 verify that the proposed controller functions as promised.
It is able to maneuver a platoon through traffic. Figure 12 illustrates the trajectory of a
platoon lane change process. Performance of the proposed controller on both arterial and
freeway are demonstrated. As shown in the figures, all CAVs are able to change lane
successfully while keeping a safe following gap and maintaining the form of a platoon.
Figure 13 illustrates gap trajectories between CAVs and background vehicles (front
vehicle and left lag vehicle). No collision occurs in the platoon lane change process.
Figure 12. Vehicle trajectories
Figure 13. Gaps between CAVs and background vehicles
The string stability of the proposed controller is confirmed. Figure 14 illustrates
the acceleration trajectory of the platoon. Figure 15 illustrates the speed trajectory of the
platoon. The two figures demonstrate that acceleration and speed trajectories of followers
have lower peaks compared to the leader. Therefore, the proposed SuPLC controller is
string stable.
Figure 14. Acceleration trajectory
Figure 15. Speed trajectory
The lateral stability of the proposed controller is also confirmed. Figure 16
illustrates the steering angle trajectory of the platoon. Figure 17 illustrates the lateral
position trajectory of the platoon. The two figures demonstrate that the steering angle and
position of followers have lower peaks compared to the leader. Therefore, the platoon
under control of the proposed SuPLC controller is laterally stable.
Figure 16. Steering angle trajectory
Figure 17. Lateral position trajectory
Function verification with traffic
In order to verify the proposed SuPLC controller in traffic, an evaluation has been
conducted using a simulation platform. The platform is able to simulate CACC platoon
in the context of traffic and was published in Transportation Research Part C: Emerging
Technologies (Cui et al., 2018; Lai et al., 2020). This evaluation compares the lane-
change capability between the proposed SuPLC controller and conventional SiPLC
controller (H. C.-H. Hsu & Liu, 2008). The goal of the evaluation is detailed as follows:
Function validation: This test is to verify the function of changing lane
successively for the proposed controller.
Platoon lane-change capability evaluation: This test is to quantify the
advantage of the proposed SuPLC controller over the conventional SiPLC controller.
Test scenario
The test scenario is illustrated in Figure 18. In this scenario, the platoon is cruising in
traffic. The platoon is with the objective of keeping its desired speed. Therefore, platoon
lane change decision is intrigued when the platoon is impeded by slow-moving HVs.
Figure 18. Test scenario for PLC evaluation in traffic
Eight cases are designed as shown in Table 4. Road types considered include
arterial and freeway. Sensitivity analysis is conducted on congestion level (v/c ratio). For
each case, 30 times simulations are conducted with different random seeds. The duration
of each simulation run is 12 minutes.
Table 4. Cases definition
v/c=0.25
v/c=0.50
v/c=0.75
v/c=1.0
Arterial
,
,
Case1
Case2
Case3
Case4
Freeway
,
,
Case5
Case6
Case7
Case8
Controller types
All surrounding HVs are controlled by driver models developed explicitly for human
drivers (Cui et al., 2018; Lai et al., 2020; VISSIM, 2017). The platoon lane-change
maneuver is controlled by two types of PLC controllers.
Baseline controller: This controller is the conventional SiPLC controller. It
maneuvers a platoon to change lane simultaneously.
The proposed controller: This controller is the proposed SuPLC controller. It
maneuvers a platoon to change lane successively. A conservative lane-change gap
selection algorithm is adopted as shown in Table 1.
Measurements of effectiveness
The following Measurements of Effectiveness (MOEs) are adopted.
PLC competence: This MOE is quantified by the average number of PLC per
minute.
PLC efficiency: This MOE is quantified by PLC delay. It is defined as the
delay from the time when a PLC decision is made to the time when the PLC maneuver is
finished. This delay includes two durations: LC window approaching and PLC
maneuvering. The LC window approaching refers to the process of a platoon approaching
a target window in order to make a PLC maneuver.
Results
Results confirm that the proposed controller is: i) with successive platoon lane change
function; ii) superior to the conventional SiPLC controller.
Figure 19 and Figure 20 illustrate trajectories from Case 5 under the control of
SuPLC and SiPLC respectively. Under the control of the proposed controller, the platoon
fulfills successive platoon lane change maneuvers as demonstrated in Figure 19. Under
the control of the conventional SiPLC controller, the platoon changes lane simultaneously
as demonstrated in Figure 20. Moreover, over 95% SuPLC maneuvers are not impeded
by traffic including cases with high congestion levels. It indicates that the proposed
controller could be adopted in field implementation. This result is reasonable since human
drivers may not feel stressed and impeded when a faster vehicle cuts in. The SuPLC
maneuver may not reduce the mobility of lag vehicles.
Figure 19. A sample trajectory of SuPLC
Figure 20. A sample trajectory of SiPLC
Figure 21 illustrates the average number of platoon lane changes per minute for
all cases. It confirms that the proposed SuPLC controller does enhance the PLC
competence no matter the road type under all v/c ratios. The proposed SuPLC controller
enhances PLC competence by 120.49% on freeways and 71.36% on arterials. It means
that the performance of the proposed SuPLC controller is positively correlated with speed.
It makes sense as the proposed SuPLC controller is better at taking advantage of the
marginal increase of distance headway due to speed change. The increase in traffic speed
leads to a decrease in density. The decrease of density may provide a larger distance
window for PLC maneuver. Since the proposed SuPLC controller needs a much smaller
PLC window, a slight increase in headway may be enough for the SuPLC maneuver,
while insufficient for the SiPLC maneuver.
Figure 21. Platoon lane-change competence
Figure 22 illustrates platoon lane change delay distributions for all cases. It
confirms that the proposed SuPLC controller does enhance PLC efficiency, compared to
the conventional SiPLC controller. The proposed SuPLC controller reduces PLC delay
by 41.36% on freeways and 25.05% on arterials. Therefore, the proposed SuPLC
controller has a higher PLC efficiency enhancement on freeways than arterials. In
addition, more outliers are found with the SuPLC controller. It makes sense as the
performance of the proposed SuPLC controller heavily relies on the driving behavior of
surrounding HVs. In the case that the surrounding vehicles form a constant barrier
preventing the platoon from cutting through, there is just nothing that could be done even
for the proposed SuPLC controller. Since the efficiency of the SuPLC controller is
generally superior, more outliers are observed. It indicates that the path planner of the
leading vehicle is critical to PLC efficiency. Future studies could consider enhancing the
path planner to avoid being trapped in a constant vehicle barrier.
Figure 22. Platoon lane-change efficiency
A sensitivity analysis of performance enhancement to v/c ratio is presented in
Figure 23. Performances have been demonstrated in regard to both PLC competence and
PLC efficiency. The results show that the performance of the proposed SuPLC controller
deteriorates with the increase of v/c ratio. However, the proposed SuPLC controller is
always superior to the conventional SiPLC controller under all v/c ratios. It means the
proposed SuPLC function could always be turned on in real implementation. In addition,
the performance enhancement of the proposed SuPLC controller peaks when v/c ratio is
between 0.5 and 0.75. This indicates that the proposed controller is more robust against
congestion.
Figure 23. Sensitivity analysis of performance enhancement regarding v/c ratio
Conclusion and future research
This research proposes a CACC controller with platoon lane-change capability. The
proposed SuPLC controller is designed for maneuvering a platoon to change lane
successively like a snake. By adopting smaller lane-change gaps, the proposed controller
improves the lane-change capability of CACC. The proposed controller has the following
features: i) with successive platoon lane-change capability; ii) with string stability and
lateral stability; iii) with consideration of vehicle dynamics. The proposed controller is
evaluated on a simulation platform with the context of traffic and a joint simulation
platform consisting of PreScan and Matlab/Simulink. Results demonstrate that:
The proposed SuPLC controller functions as expected and is able to maneuver
a platoon through traffic via small windows.
The proposed SuPLC controller is confirmed with string stability and lateral
stability.
The proposed SuPLC controller enhances PLC competence by 120.49% on
freeways and 71.36% on arterials.
The proposed SuPLC controller enhances PLC efficiency by 41.36% on
freeways and 25.05% on arterials.
The performance of the proposed SuPLC controller peaks when v/c ratio is
between 0.5 and 0.75. This indicates that the proposed controller is more robust against
congestion.
The proposed SuPLC controller is ready for real-time application.
In this research, the communication delay and actuation delay are not
accommodated. Future research could consider enhancing the proposed controller
accordingly.
Acknowledgement
This paper is partially supported by Shanghai Municipal Science and Technology Major
Project 2021SHZDZX0100, Shanghai Oriental Scholar (2018), Tongji Zhongte Chair
Professor Foundation (No. 000000375-2018082), and the Fundamental Research Funds
for the Central Universities, National Natural Science Foundation of China (Grant No.
61903058).
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