Stability and Safety of Cooperative Adaptive Cruise Control Vehicular Platoon under Diverse Information Flow Topologies

For the cooperative adaptive cruise control (CACC) vehicular platoon, apart from decentralized controllers, the dynamics of a platoon can be a ﬀ ected substantially by the information ﬂ ow among connected and automated vehicles (CAVs). Existing research studies mainly focus on the stability analysis of platoons where CAVs only adopt the predecessor-following (PF) communication scheme; however, when CAVs “ look ” further ahead or behind than one vehicle, the stability of platoons might change. To this end, this study seeks to explore the stability and investigate the rear-end collision risk of CACC vehicular platoon under diverse information ﬂ ow topologies. The research ﬁ rst comprehensively reviews typical information ﬂ ow topologies for CAV platoons and platoon stability criteria for analyzing local and string stability of platoons. Moreover, the CACC longitudinal dynamic model is derived using the exact feedback linearization technique, which accommodates the inertial delay of powertrain dynamics. Accordingly, su ﬃ cient conditions of stability are mathematically derived to guarantee distributed frequency-domain-based control parameters. Simulation experiments are conducted to verify the correctness of derived su ﬃ cient stability conditions. The results show that platoons could better maintain stability with more vehicle information taken into consideration. However, when assessing the safety, it is found that the bidirectional type information ﬂ ow topology would increase rear-end collision risk for CAV platoon. Further, the information ﬂ ow topology of two-predecessor-leader following is the most recommended to enhance fully CAV platoon stability.


Introduction
Connected and automated vehicle (CAV) technologies, which incorporate communication technologies into autonomous systems to enable cooperative sensing and control [1,2], are expected to improve traffic mobility, safety, and sustainability [3,4]. A widely adopted method to achieve these purposes is adaptive cruise control (ACC), which is a vehicle-following control system that controls the speed in pace with the immediate preceding vehicles [5][6][7]. Moreover, the emergence of advanced communication (i.e., vehicle-to-everything (V2X) communication technology) has provided particularly promising for CAVs to receive additional information from other connected vehicles and could better form a platoon where CAVs travel as a string/ chain to enhance traffic efficiency and stability. It has recently attracted extensive research interest (see [8][9][10] and the references therein).
From the viewpoint of control, CAV platoon is a controlled multiagent system where several vehicles are aimed at traveling at a common speed while keeping a safe distance within them. In this context, Cooperative Adaptive Cruise Control (CACC), which is an extension of ACC functionality, is recognized as an effective means to maintain stable and desired headways between adjacent vehicles [7,11]. CACC has been modeled in a recent literature in different ways [12][13][14][15]. Among them, constant time headway (CTH) spacing policy is suggested to combine with CACC by many studies, where the equilibrium spacing is defined as the speed multiplied by a predefined constant time headway plus a standstill spacing, since it is more robust to error propagation through traffic [16,17].
On the other hand, for a CAV platoon, apart from decentralized controllers, the information flow among vehicles can significantly affect the dynamics, as well. In a platoon, agents exchange information (e.g., speed, intervehicle distance, and control inputs) with each other depending on how that information flows and a different communication scheme arises [18,19]. The information flow topologies can describe the configuration of V2X communication links from one CAV to one or more CAVs in the platoon. Earlystage platoons primarily focus on predecessor following type which means that a vehicle can only obtain the information of its nearest front vehicles [20][21][22][23]. Under the V2X communication framework, more types of information flow topologies are proposed, such as the two-predecessors following type [24], the predecessor-leader following type [25], and the bidirectional type [26,27]. For instance, an H∞ controller synthesis approach is developed by [28] to guarantee platoon stability for platoons under one-and two-CAV lookahead information flow topologies. Zheng et al. [18] study the relationship between information flow topology and the internal stability and scalability of CAV platoons moving in a rigid formation. However, to date, few studies have investigated the influence of different information flow topologies on CACC vehicular platoon stability.
The aim of stability analysis is to study how the perturbation of a leading vehicle evolves over time and space by assuming that vehicles travel on a single lane without overtaking. For the respective of CAV platoon, there exist two types of stability that have been defined in existing research: local stability and string stability [29,30]. More specifically, if the distance gap and speed fluctuations of a single follower decrease with time, the CAVs are considered locally stable, while the platoon is string stable if local perturbations decay everywhere even in an arbitrarily long vehicle platoon. Hence, for a platoon to be stable, it is not enough just to satisfy the local stability but also to satisfy the string stability.
In conclusion, to the best of the authors' knowledge, the stability of a CAV platoon under diverse information flow topologies so far has not been considered before while most of the investigation is focused on the predecessorfollowing communication scheme. Moreover, few researches have evaluated the safety of CAV platoon under diverse information flow topologies. The main objective of this paper is to fill these gaps. To this aim, several main contributions are provided, each one corresponding to a different section.
This paper is organized as follows. Section 2 introduces different types of information flow topologies and the Cooperative Adaptive Cruise Control low for a CAV platoon. Section 3 provides the stability analysis of vehicular platoon under diverse information flow topologies. Numerical simulations are shown in Section 4, and the safety assessment is presented in Section 5. Concluding remarks and future research directions are described in the final section.

Diverse Information Flow Topologies for Cooperative Adaptive Cruise Control Vehicular Platoon
According to diverse information flow topologies, the control laws of the CAV platoon are diverse. In this section, the form of diverse information flow topologies and the corresponding CACC model would be elaborated on.
2.1. Diverse Information Flow Topologies. Due to V2X communication, CAVs can form platoons in the single-lane freeway sections where there is no cut-in/cutout behavior. The platoon consists of N + 1 CAVs, as indicated in Figure 1, with a leading vehicle and N following vehicles. The platoon travels on a flat route and uses various information flow topologies. Figure 1 depicts six types of typical topologies as follows: (1) Predecessor following (PF) topology (2) Predecessor-leader following (PLF) topology (3) Two-predecessor following (TPF) topology (4) Bidirectional (BD) topology (5) Bidirectional-leader (BDL) topology (6) Two-predecessor-leader following (TPLF) topology Specifically, for the leader following type (i.e., PLF, BDL, and TPLF), a leader with information broadcasting functions is indispensable; for the bidirectional type (i.e., BD and BDL), the controller can control utilizing information from both the preceding and following vehicles; for the two-predecessor type (i.e., TPF and TPLF), the controller can control utilizing information from both the two preceding vehicles. Furthermore, many more topologies are not included here for the sake of brevity, although they may all be examined using similar methods.

CAV Longitudinal Control with CACC.
This section describes the formulation of the proposed distributed CAV longitudinal control. Assume that all CAVs in the platoon are of the same type. According to Zhou et al.'s [31] study, the CAV longitudinal control usually includes upper level and lower level controllers that govern CAVs to stay close and stable while compensating for vehicle longitudinal dynamics. The lower level controller prescribes the acceleration rate that can be realized after considering the vehicle longitudinal dynamics since the demanded acceleration may not be fully executed due to air drag force, gear position, etc.; meanwhile, the upper level controller, in particular, regulates a CAV to follow predefined equilibrium spacing at the same speed as the preceding vehicle via vehicle acceleration.
For the lower level controller, as suggested by Yi and Kwon [32], the generalized vehicle dynamic (GLVD) equation is leveraged to incorporate the nonlinear vehicle dynamics. The vehicle dynamics are specifically modeled using the first-order approximation as 2 Wireless Communications and Mobile Computing _ a n t ð Þ = − 1 T L a n t ð Þ + where K L represents the ratio of demanded acceleration that can be realized (which ideally equals to 1) for CAVs, T L is the time-lag for CAVs to realize the acceleration, _ a n ðtÞ is the jerk, a n ðtÞ stands for the realized acceleration of CAV n at time t, and u n ðtÞ is the demanded acceleration of CAV n at time t by the upper level controller, which also represents the control low of CAV n.
For the upper level controller, CACC combined with constant time gap policy which is shown to be more robust against disturbance propagation [17] is adopted. On the basis of the Society of Automotive Engineer (SAE) standard, the constant time gap policy is widely leveraged in CAV platoon control. According to the constant time gap policy [33], the equilibrium spacing is defined as where v n ðtÞ is the velocity of CAV n at time t, d * n ðtÞ is the target equilibrium spacing of CAV n at time t, τ * n presents the predefined constant time gap of CAV n, and l n is the standstill spacing of CAV n. Moreover, the relationships between acceleration, velocity, and position of CAV n are defined as where p n ðtÞ is position of CAV n. Based on Equations (1)-(4), a 3rd-order state space model is derived for each CAV in the platoon with control input u n ðtÞ as below: Further, each controller can only use the information specified by ℚ n which depends on the type of information flow topology applied to the platoon. The linear control law of CAV n (except the leading CAV) in the platoon is

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where k # (# = 1, 2, 3, Ja, Jv) is the control gain of the CACC and they all exceed zero. Except k 1 , k 2 , k 3 , for the predecessor-leader following type, the control gains are written as k lv and k la ; for two-predecessor following type, the control gains are written as k tv and k ta ; for the bidirectional type, the control gains are written as k bv and k ba .
More specifically, based on Equation (7), the linear control law for CAV n under PF topology is The linear control law for CAV n under PLF topology is described as The linear control law for CAV n under TPF topology is described as The linear control law for CAV n under BD topology is written as Likewise, the linear control law under TPLF topology is

Stability Analysis of CACC Vehicular Platoon
As alluded, the control gains should be prudently selected to guarantee that the CACC vehicular platoon is stable. In this section, the theory of stability is concisely recalled from the control theory standpoint, and the stability of the CACC vehicular platoon under diverse information flow topologies is explored.
3.1. Preliminaries for Stability. In general, there are two typical categories of stability, which have been widely adopted for maintaining safety under perturbances while controlling vehicular platoon, i.e., (1) Local stability: a platoon following linear timeinvariant dynamics is said to be locally stable concerning equilibrium state x n,e if and only if the closed-loop system has eigenvalues with strictly negative real parts [34].
(2) String stability: a platoon is said to be string stable if and only if the magnitude of a perturbance is not amplified for each leader-follower pair when propagating along the vehicular string [24,25], as defined in where k:k 2 denotes the H 2 norm; E n ðsÞ ∈ fa n ðsÞ, v n ðsÞ, where w > 0 represents frequency and j is the imaginary unit. Local stability can guarantee each CAV can eliminate deviation from desired spacing and speed difference activated locally through a single vehicle by perturbances (deviation from equilibrium spacing, speed difference, or acceleration). Local stability is essential for any acceleration model since driver behavior is locally stable in reality. Furthermore, to satisfy the local stability, the sufficient condition by Hurwitz criterion [35] is given below.
Unlike local stability, string stability considers how a small perturbation in the gap and speed of the leading vehicle influence the gap and speed of all the following vehicles in the platoon. Thus, the cohesion of the CAVs can be maintained, and the states of CAVs bounded can be kept by string stability. However, string stability is not always observed in empirical data. According to [7,36,37], string stability can also be divided into two kinds: strict string stability and head-to-tail string stability. Additionally, the detailed definition is as follows: Definition 1 (strict string stability). A platoon system of a finite number of vehicles N + 1 is said to be strict string stable if satisfying for any sufficiently small perturbation input acting upon CAV 0, for any n ∈ ð0, N: where F n ðsÞ is the transfer function describing perturbance propagation between CAV n − 1 and CAV n in the frequency domain.

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Definition 2 (head-to-tail string stability). A platoon system of a finite number of vehicles N + 1 is said to be head-totail string stable if satisfying for any sufficiently small perturbation input acting upon CAV 0, for any n ∈ ð0, N: where G n ðsÞ is the transfer function describing perturbance propagation from CAV 0 and CAV n in the frequency domain. Note that when F n ðsÞ = 1 and G n ðsÞ = 1, the CAV platoon is marginally H 2 -norm string stable.
For perturbances in the form of velocity and acceleration, the transfer functions F n ðsÞ and G n ðsÞ are of the general form: Except for the velocity and acceleration oscillation, the spacing error can be obtained through Laplace Transformation: As there is no spacing error in the leading CAV, the transfer function is defined for other CAVs from the first following CAV to CAV n. Thus, the transfer functions of spacing error can be formulated as follows: To address the string stability, based on [7], the worst cases, F n−1 ðsÞ = F n ðsÞ = 1 and G n−1 ðsÞ = 1, are taken into consideration to reduce the order in Equations (18) and (19). Therefore, we get On the basis of Equations (20) and (21), aiming at analyzing the string stability of the CACC vehicular platoon under different information flow topologies, we can only concern the transfer functions F n ðsÞ and G n ðsÞ.

Local Stability.
Based on Routh Hurwitz stability criterion, local stability requires that the real-part of the Eigenvalue of matrix A cn = ðA n + B n KÞ is less than zero, where K = ½k 1 , k 2 + ∑ J∈ℚ n k Jv , k 3 + ∑ J∈ℚ n k Ja .
According to control law as Equation (7), we have Hence, by applying the Hurwitz criterion, the CACC vehicular platoon is locally stable if the following inequalities are satisfied.
3.2.2. String Stability. Under diverse information flow topologies, the string stability needs to be analyzed separately.
(1) PF Topology. Based on Equation (8), for PF topology, the transfer function F n ðsÞ is presented by Building on Equations (14) and (24), the following sufficient condition for the H 2 -norm string stability is shown: Hence, the CACC vehicular platoon under PF topology is strict string stable if the following inequalities are satisfied.

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Under PF topology, the head-to-tail string stability can be written as When kF n PF ðsÞk 2 ≤ 1, kG n PF ðsÞk 2 is not greater than 1. In conclusion, the CAV platoon under PF topology is string stable if the inequations in Equations (26a), (26b), and (26c) are satisfied.
(2) PLF Topology. Under PLF topology, based on Equation (9), the first following CAV could be affected by the leading CAV, and the transfer function G 1 PLF ðsÞ is written as Differently, the second following CAV could be affected by both its predecessor and the leader. Consequently, through substituting Equation (28), its transfer function is formulated as For the rest following CAV n, the transfer function can be obtained by induction: Nevertheless, the above Equation (30), is a high-order transfer function which is too complicated to be analyzed directly. In this way, we have the following formulation assuming the worst case, G n−1 PLF ðsÞ = 1: Hence, the CACC vehicular platoon under PLF topology is head-to-tail string stable if the following inequalities are satisfied: Comparing Equations (32a) and (32b) with Equations (26a) and (26b), it can be found that the value ranges of k 1 , k 2 , k 3 (i.e., the stable region) under PLF topology are greater than that under PF topology, since CAVs could obtain more information (i.e., the leading vehicle information) under PLF topology than under PF topology.
Similarly, the transfer function F n PLF is presented as Consider the worst case: In brief, the CACC vehicular platoon under PLF topology is string stable if the inequations in Equations (32a), (32b), and (32c) are satisfied.
(3) TPF Topology. Correspondingly, under TPF topology, based on Equation (10), expressions of the transfer function are Considering the worst case, the CACC vehicular platoon under TPF topology is string stable if the following inequations are satisfied. Compared with the stable region under PF topology, the stable region under TPF topology is larger. As a matter of fact, we set the CACC controllers to make kF n+1 BD ðsÞk 2 < 1. Thus, kF n BD ðsÞk 2 < kF n PF ðsÞk 2 , and kG n BD ðsÞk 2 < kG n PF ðsÞk 2 . The stable region for BD topology is larger than the stable region for PF topology.
(5) BDL Topology. Based on PLF topology and BD topology, under BDL topology, based on Equation (12), the transfer function of F n BDL ðsÞ and G n BDL ðsÞ can be written as In addition, when F n+1 BDL ðsÞ = 1, F n BDL ðsÞ = F n PLF ðsÞ; when F n+1 BDL ðsÞ ≠ 1, the CACC vehicular platoon under BDL topology is string stable if inequations in Equations (41a), (41b), and (41c) are met. Therefore, the stable region for BDL topology is larger than the stable region for PF topology and even larger than the stable region for PLF topology.
On the whole, under diverse information flow topologies, with more information a CAV obtains, the larger stable region the platoon system has and the CAV platoon is easier to control for maintaining stability. However, according to Zheng et al.'s study [18], under bidirectional topology, the scalability of CAV platoon will be affected. Moreover, for a pure CAV platoon, the perturbances are often applied to the leading vehicle instead of internal CAVs. Hence, the leader-type information flow topology can play an important role in enhancing stability. In this study, the information flow topology of TPLF provides the largest stable region for CAV platoon control.

Numerical Simulation
Numerical simulations are conducted to illustrate the main results. The parameter setting for the CACC vehicular platoon is given in Table 1, according to the existing studies [28,31]. Unless otherwise specified, these parameters would not change. Since CAVs under PF topology are simplest for control and receive the least information, the CAV performance under PF topology is set as the baseline.
Based on the parameter setting, Figure 2 illustrates the regions of feedback gains for different information flow topologies to satisfy the local and string stability. The plots suggest empirically that with more information given to the CAV, the stable region will be larger, which supports the stability analysis results in Section 3.
To further verify the theoretical results presented in Section 3, we conduct two simulation scenarios on the CACC vehicular platoon: with simulated perturbation and with the human-driven vehicle trajectory. In both scenarios, we consider a CAV platoon with 11 identical CAVs (1 leader and 10 followers) interconnected by the six information flow topologies shown in Figure 1. In addition, the CAV length is equal to 3 m and the control gains k 1 , k 2 , k 3 are set as 2, 2, and 1, respectively.
The initial state of the platoon is set as the desired state; i.e., the initial spacing errors and velocity errors are all equal to 0.

Scenario II: With Human-Driven Vehicle Trajectories from NGSIM.
To explore CAV platoon stability in a more realistic situation, simulation experiments embedded with field data are carried out field. Hence, two sets of trajectory data of the real-world vehicle that experienced stop-and-go perturbances are obtained from the NGSIM dataset [38,39]. The leading vehicles adopt the data of vehicle #1992 in Lane 2 from 4:00 p.m. to 4:15 p.m. on April 13, 2005, for Interstate 80 (i.e., I-80) in Emeryville, and the data of vehicle #1635 in Lane 1 from 7:50 a.m. to 8:35 a.m. on June 15, 2005, for US101 freeway (i.e., US101) in Los Angeles, whose dynamic statuses are illustrated in Figures 6 and 7. Note that, due to the noise in the acceleration data, the data is handled via a low-pass filter with the lower bound of 0.5 Hz. From the above simulations, we can find that when stable conditions are satisfied, with more information a CAV obtains, the CAV platoon is more stable under perturbances, since the larger stable region means the system has a significant ability to resist the perturbances. In addition, we also find that the leader type could better maintain stability than other types. The reason is that the perturbances are applied to the leader in the platoon in this study.  Figure 15 (T L = 0:3 s), it can be found that with the same K L (setting K L = 1), the lesser the actuation lag, the more stable a CACC vehicular platoon is. Besides, the changes in stable regions under diverse information flow topologies accord with the theoretical analysis results in Section 3.
Then numerical simulations of three K L values were carried out based on the stability analysis in Section 3.2. It should be emphasized that the value of K L cannot exceed one. With the same T L (setting T L = 0:45 s), comparing the Figure 2 (K L = 1) with Figure 16 (K L = 0:9) as well as Figure 17 (K L = 0:8), the larger the ratio of demanded acceleration that can be realized, the more stable a CACC vehicular platoon is.

Safety Assessment
To thoroughly assess the risk probability of CAVs, Time-tocollision (TTC), the modified Time Integrated Time-tocollision (TIT) and the Time Exposed Time-to-collision (TET) are employed in this study. TTC is defined as the time required for two vehicles to collide if they keep their current velocity on the same path [40]: Herein, a larger TTC indicates a safer condition. It is worth mentioning that minimum TTC could only demonstrate the most dangerous degree of vehicles while ignoring the overall duration. Hence, we use minimum TTC value to evaluate the danger degree during moving.
Both TET and TIT are aggregated indexes from TTC for safety evaluation. Concretely, the expression of TIT and TET can be presented as [41] TIT where TTC * is the threshold of TTC value and Δt stands for the sampling time. Furthermore, a larger TIT/TET indicates a more dangerous condition. TET could present the rear-end collision risk, and TIT shows the duration of having the risk of rear-end collision over the whole operation. Aiming at further evaluating the safety of different information flow topologies, we select three more realworld single vehicles from NGSIM I-80 dataset randomly: Vehicle #12, #391, and #2385, as the leading vehicle. The trajectories of these vehicles are shown in Figure 18, and the entire traveling time for each vehicle is 70.5 s, 37.1 s, and 34.5 s, respectively.
The results of minimum TTC, TIT, and TET values for the CACC vehicular platoons under diverse information flow topologies with real-world vehicle trajectories described in Section 4 and Figure 18 are demonstrated in Table 2, where Δt = 0:1s, τ * n = 0:5 s (then, the desired intervehicular distance can be calculated by Equation (2) at each sampling time), and all CAVs are in equilibrium at the initial time with the same velocity as the leading vehicle. Moreover, the threshold of TTC value (i.e., TTC * ) is also set as 0.5 s, which is equal to the desired time headway.
The results in Table 2 suggest the following: (1) from the aspect of minimum TTC, with more information, CAV platoon could better reduce the risk of rear-end collisions. However, in general, there is less impact of information flow topologies on minimum TTC; (2) from the aspect of TET, predecessor-leader following type (i.e., PLF, BDL, and TPLF) could better maintain safety for CAV platoons. Meanwhile, the two-predecessor type topologies perform better than the bidirectional type topologies since the TET values of CAV platoon under TPF topology are larger than those under BD topology and the TET values of CAV platoon under TPLF topology are larger than those under BDL topology; (3) from the aspect of TIT, the bidirectional type topologies could bring adverse impact on reducing the risk of rear-end collision and predecessor-leader following type could also enhance the CAV platoon safety. In all cases, the TIT values under BD topology are larger than that under PF topology, and the TIT values under BDL are larger than that under PLF topology. It is worth noting that the CAV under TPLF topology could reduce TIT by half compared with the CAV under PF topology. Therefore, among the six information flow topologies, TPLF is the most recommended to enhance both stability and safety.      16 Wireless Communications and Mobile Computing

Conclusions
This paper first studies the stability of CACC vehicular platoon under diverse information flow topologies. The CACC longitudinal dynamic model is derived using the exact feedback linearization technique, which accommodates the inertial delay of powertrain dynamics. Accordingly, sufficient conditions of stability are mathematically derived to guarantee distributed frequency-domain-based control parameters. The paper demonstrates that under diverse information flow topologies, with more information a CAV obtains, the larger stable region the platoon system has and the CAV platoon is easier to control for maintaining stability and safety. Further, the information flow topology of TPLF is the most recommended to enhance platoon stability. Then, this paper assesses the safety of fully CAV platoon under diverse information flow topologies with real-world vehicles as leaders. The safety assessment results demonstrate that the bidirectional type topologies could bring adverse impact on reducing the risk of rear-end collision comparing with other types. Moreover, the predecessorleader following type topologies contribute to reducing rear-end collision risk. Thus, the information flow topology of TPLF is also recommended to enhance platoon safety.
Unsolved topics for future research include the string stability for heterogeneous platoons under diverse information flow topologies. In addition, there is a need to address CACC vehicular platoon with nonidentical controllers that possess communication delays.

Data Availability
This study is theoretical and does not use personal data. Herein, the simulation involves the open source NGSIM data sets, which can be found in the following website https://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm.