1 Introduction

Traffic congestion and safety issues have been exacerbated by human driver errors and limitations. The development of adaptive cruise control (ACC) since the 1960s has significantly contributed to traffic safety and efficiency by enabling automatic following of the preceding vehicle, reducing driver effort [1]. To further enhance these aspects, the stop-and-go ACC and the cooperative adaptive cruise control (CACC) have been proposed [2].

CACC represents an enhancement of ACC that incorporates vehicle-to-vehicle (V2V) communication to utilize the rich information of the preceding vehicle and form platoon, improving road safety, reducing fuel consumption, and increasing traffic flow [3, 4]. Amoozadeh et al. [5] transmitted the preceding vehicle’s acceleration and maximum deceleration to the host vehicle to design the longitudinal control of the platoon, meeting the requirements of safety and theoretical throughput. Rajamani et al. [6] introduced the lead and preceding vehicles’ speed and acceleration to the host vehicle to design the longitudinal control of the National Automated Highway System Consortium (NAHSC) demonstration, realizing accurate speed tracking and spacing control. In addition, researchers have been trying to consider the information from every vehicle in platooning. Madeleine et al. [7] introduced unit-center-referenced formation and defined the “virtual leader” by averaging the positions of all vehicles in a two-vehicle platoon. Zhang et al. [8] proposed the “virtual centroid” of the platoon to describe the platoon kinematics considering all vehicles in front of the host vehicle, and thereby improving traffic efficiency. The CACC with its platoon kinematics demonstrates significant benefits and potential for enabling the host vehicle to respond promptly and reasonably, thereby improving overall traffic efficiency.

The spacing policy also plays an important role in improving traffic efficiency. Shladover et al. [9] studied a constant spacing policy of less than one meter to improve traffic capacity for close-formation platoons, which relied on a well-designed controller. The constant time headway policy was also applied to vehicle platoons to improve traffic efficiency by minimizing the values of time headway [10, 11]. Yanakiev et al. [12] proposed two nonlinear spacing policies, namely variable time headway and variable separation error gain, for automated heavy-duty vehicles platoon, leading to increased traffic throughput. In previous research, the integrated spacing policy was proposed to improve traffic efficiency by utilizing the safety redundancy [13]. The “virtual centroid” was proposed to describe platoon kinematics, and the control strategy was designed by combining the integrated spacing policy and virtual centroid, further improving traffic efficiency [8]. Both CACC with its platoon kinematics and spacing policy significantly influence traffic efficiency.

The CACC evaluation has also been under study. Tongue et al. [14] proposed the worst-case performance algorithm to evaluate the robustness of platoon controllers. Van et al. [3] utilized traffic throughput and roadway capacity to evaluate CACC's impact on traffic flow. Yang et al. [15] focused on CACC’s fuel economy, while Li et al. [16] proposed time-exposed time-to-collision (TET) and time-integrated time-to-collision (TIT) to evaluate the safety of CACC. In 2019, the ISO 20035 defined the performance evaluation methods of the CACC system from the perspective of production and application, including detection range, lane and range discrimination, and cooperative operating modes/states switching, but it notes that the coordinated strategies to control groups of vehicles, such as platooning, are not within the scope of ISO 20035 [17]. The platooning and platoon kinematics are vital to CACC, whereas the evaluation methods for CACC with platoon kinematics are an important and opening problem to be solved.

To improve traffic efficiency and evaluate the CACC with platoon kinematics, this paper attempts to explore the correlation mechanisms between safety spacing policy and time headway policy, establish the efficient spacing policy and its hierarchical control, and propose the evaluation methodology for CACC with platoon kinematics. The main contributions are as follows:

• The correlation mechanism between the safety spacing policy and time headway policy is revealed and the generic spacing model is proposed to unify these two policies, and the logic operation based efficient spacing policy is proposed for vehicle following control, avoiding the real-time solve of the critical velocity and the complex switching law depending on the critical velocity.

• The evaluation methodology for CACC with platoon kinematics is proposed, in which eight typical scenarios are reconstructed via three-layer decompositions, and the CACC synchronic index is proposed to evaluate the CACC performance with platoon kinematics.

• The hierarchical control of CACC including platoon kinematics is presented, in which the upper controller with platoon kinematics, the lower controller with vehicle nonlinear and time-variable characteristics, are established to improve traffic efficiency.

The subsequent sections of this paper are organized as follows: Sect. 2 delves into the correlation mechanism between safety spacing policy and time headway policy and proposes the efficient spacing policy. Section 3 establishes the upper controller for CACC with platoon kinematics and the lower controller with vehicle nonlinear and time-variable characteristics. Section 4 proposes the evaluation methodology for CACC with platoon kinematics. Section 5 illustrates the simulation test, and Sect. 6 draws conclusions.

2 Efficient Spacing Policy

The safety spacing policy and constant time headway policy pivotal for vehicle following control, dictating the distance that the host vehicle needs to maintain from the preceding vehicle. In this section, the correlation mechanism of the safety spacing policy and time headway policy is studied, and the generic spacing model and the efficient spacing policy are proposed.

2.1 Generic Spacing Model

The braking process is assumed as an extreme case: the preceding vehicle with velocity \({v}_{i-1}\) is at its maximum deceleration (−abmax), while the host vehicle with velocity \({v}_{i}\) is at its maximum acceleration (amax). The minimum centroid distance, \({D}_{{\text{min}}i}\), ensuring the safety for the vehicle mass point model without geometric considerations, is expressed by Eq. (1). The detailed derivation is elucidated in Appendix A.

$$\begin{aligned} {D}_{{\text{min}}i}&={S}_{i}-{S}_{i-1}+{d}_{{\text{a}}i}\\ &=\frac{{v}_{i}^{2}-{v}_{i-1}^{2}}{2{a}_{{\text{bmax}}}}+[{\tau }_{11}+{\tau }_{12}+{\tau }_{21}+\frac{{\tau }_{22}}{2} \\ & \quad +\frac{{a}_{{\text{max}}}}{{a}_{{\text{bmax}}}}\left({\tau }_{11}+\frac{{\tau }_{12}}{2}\right)]{v}_{i}-\frac{1}{24}{a}_{{\text{bmax}}}{\tau }_{22}^{2} \\ & \quad +{a}_{{\text{max}}}[\frac{{\tau }_{11}^{2}}{2}+{\tau }_{11}{\tau }_{12}+\frac{1}{6}{\tau }_{12}^{2}+\left({\tau }_{11}+\frac{{\tau }_{12}}{2}\right)\left({\tau }_{21}+\frac{{\tau }_{22}}{2}\right)+\frac{{a}_{{\text{max}}}}{2{a}_{{\text{bmax}}}}{\left({\tau }_{11}+\frac{{\tau }_{12}}{2}\right)}^{2}]+{d}_{{\text{a}}i} \\ \end{aligned}$$
(1)

where \({\tau }_{11}\) is the time required to detect that the preceding vehicle starts to brake, \({\tau }_{12}\) is the time for the driver to release the accelerator pedal completely, \({\tau }_{21}\) is the time to move the foot to the braking pedal and eliminate the mechanical clearance of the braking system, \({\tau }_{22}\) is the time to increase braking force, \({\tau }_{3}\) is the continuous braking time, \({\tau }_{4}\) is the braking release time, and \({d}_{{\text{a}}i}\) is a safety margin. It is worth noting that \({\tau }_{12}\) and \({\tau }_{21}\) are eliminated for CACC. For brevity, the last three terms of Eq. (1) are simplified into the static distance \({d}_{{\text{min}}}\), representing the centroid distance between the host vehicle and preceding vehicle when their velocities are 0.

According to Eq. (1), the spacing policy for emergency braking and the stable following are obtained:

  1. (1)

    Emergency braking case

    For an emergency braking case, it is assumed that a static vehicle or an obstacle suddenly appears in front of the host vehicle. At this moment, the velocity of the preceding target, \({v}_{i-1}=0\), and the host vehicle travels with a constant velocity \({v}_{i}\), the safe distance of the emergency braking case \({d}_{{\text{eme}}i}\) is derived as

    $$\begin{array}{l}{d}_{{\text{eme}}i}=\frac{{v}_{i}^{2}}{2{a}_{{\text{bmax}}}}+{v}_{i}\cdot \left({\tau }_{11}+{\tau }_{21}+\frac{{\tau }_{22}}{2}\right)+{d}_{{\text{min}}}\end{array}$$
    (2)

    Equation (2) can avoid the collision in emergencies, and is defined as the safety spacing policy \({d}_{{\text{saf}}i}\),

    $$\begin{array}{l}{d}_{{\text{saf}}i}={v}_{i}\left(\frac{{v}_{i}}{2{a}_{{\text{bmax}}}}+\tau \right)+{d}_{{\text{min}}}\end{array}$$
    (3)

    where \(\tau ={\tau }_{11}+{\tau }_{21}+{\tau }_{22}/2\).

  2. (2)

    Steady following case

    For a steady following case, the host vehicle and preceding vehicle have the same initial velocity \({v}_{i}\), and the safe distance of the steady following case \({d}_{{\text{ste}}i}\) is derived as

    $$\begin{array}{l}{d}_{{\text{ste}}i}={v}_{i}\cdot \left({\tau }_{11}+{\tau }_{21}+\frac{{\tau }_{22}}{2}+\frac{{a}_{{\text{max}}}}{{a}_{{\text{bmax}}}}{\tau }_{11}\right)+{d}_{{\text{min}}}\end{array}$$
    (4)

    And Eq. (4) is rewritten as the constant time headway policy \({d}_{{\text{cth}}i}\),

    $$\begin{array}{l}{d}_{{\text{cth}}i}={v}_{i}\cdot th+{d}_{{\text{min}}}\end{array}$$
    (5)

    where \({{th}}={\tau }_{11}+{\tau }_{21}+\frac{{\tau }_{22}}{2}+\frac{{a}_{{\text{max}}}}{{a}_{{\text{bmax}}}}{\tau }_{11}>\tau\).

    Comparing Eqs. (3) and (5), it is shown that the safety spacing policy is a variable time headway policy and its time headway is \({v}_{i}/2{a}_{{\text{bmax}}}+\tau\), which varies linearly with velocity. And Eqs. (3) and (5) are unified to the generic spacing model \({d}_{{\text{gen}}i}\),

    $$\begin{aligned} d_{{{\text{gen}}i}} & = v_{i} \cdot {{th}}_{{{\text{gen}}i}} + d_{{{\text{min}}}} \\ {{th}}_{{{\text{gen}}i}} & = \left\{ {\begin{array}{*{20}c} {{{th}}(v_{i} ),{\text{time}}\;{\text{headway}}} \\ {v_{i} /2a_{{{\text{bmax}}}} + \tau ,{\text{safety}}\;{\text{spacing}}} \\ \end{array} } \right. \\ \end{aligned}$$
    (6)

    where \({{th}}_{{\text{gen}}i}\) is the time headway of the generic spacing model, \({{th}}({v}_{i})\) is the time headway at the current velocity \({v}_{i}\).

    The generic spacing model, depicted schematically in Fig. 1, serves as a temporal headway policy that unifies both safety spacing and time headway policies. When the time headway of the generic spacing model \({{th}}_{{\text{gen}}i}\) is a constant, the generic spacing model is the constant time headway policy shown as \({{th}}_{{\text{gen}}i}={{th}({v}_{i})}_{{\text{constant}}}\), \({{th}}_{{\text{gen}}i}=\tau\), and \({{th}}_{{\text{gen}}i}=0{\text{s}}\); when the \({{th}}_{{\text{gen}}i}\) is variable, the generic spacing model is the variable time headway policy shown as \({{th}({v}_{i})}_{{\text{variable}}}\) and \({t}_{{\text{safe}}}\). The \({t}_{{\text{safe}}}\) is the safety spacing policy illustrated by the red long-dash line, and its tangent line’s slope is just its time headway at each velocity. And the details are as follows.

    Fig. 1
    figure 1

    The schematic diagram of generic spacing model

    Full size image
    1. •

      When \({v}_{i}=0\), the generic spacing model’s curves have the same starting point, \(\left(0,{d}_{{\text{min}}}\right)\), representing that the distance between the host vehicle and preceding vehicle equals \({d}_{{\text{min}}}\) when their velocities are 0. The tangent to the safety spacing policy is just the constant time headway curve with \({th}\left({v}_{i}\right)=\tau\) indicated as the pink dot-dash line, which is a special case of the safety spacing policy. \({th}\left({v}_{i}\right)=0\) is the ideal boundary denoted by the blue dotted line, meaning that the distance between the preceding and host vehicle is \({d}_{{\text{min}}}\) at any speed, and the platoon travels in a train-like form, reaching maximum traffic density.

    2. •

      When \({v}_{i}>0\), the slope of the constant time headway curve (depicted by the black double-dotted-dash line) and the safety spacing curve changes monotonically with increasing vehicle velocity, and \({th}\left({v}_{i}\right)>\tau\), so they only have one intersection \(\left({v}_{{\text{c}}},{d}_{{\text{c}}}\right)\) shown in Eq. (7). As shown in Fig. 1, when the vehicle velocity \({v}_{i}\) surpasses the critical velocity \({v}_{{\text{c}}}\), the constant time headway policy exhibits reduced safety redundancy. Conversely, when \({v}_{i}\) is less than \({v}_{{\text{c}}}\), the safety spacing policy manifests diminished safety redundancy. When \({v}_{i}\) equals \({v}_{{\text{c}}}\), the two policies have the same safety redundancy, as detailed in a previous work on the integrated spacing policy [13]. It can be concluded that the safety spacing curve only has one intersection with the constant time headway curve when \({th}\left({v}_{i}\right)>\tau\) and \({v}_{i}>0\). And for the variable time headway curve, shown as the olive green solid line, its slope may not change monotonically with increasing vehicle velocity. Multiple intersections may exist for the time headway curve and the safety spacing curve, especially when the time headway is velocity-varying with higher-order functions or lacks an explicit function. The real-time determination of critical velocities, as required by the critical velocity-based switching law and spacing policy, faces challenges and limitations. Therefore, an efficient spacing policy for the generic spacing model demands further investigation.

      $$\begin{array}{l}\left\{\begin{array}{l}{v}_{{\text{c}}}=2{a}_{{\text{bmax}}}\left({th}({v}_{i})-\tau \right)\\ {d}_{{\text{c}}}=2{a}_{{\text{bmax}}}\left({th}({v}_{i})-\tau \right){th}({v}_{i})+{d}_{{\text{min}}}\end{array}\right.\end{array}$$
      (7)

      where \({v}_{{\text{c}}}\) is the critical velocity, and \({d}_{\text{c}}\) is the distance between the preceding vehicle and host vehicle at critical velocity.

2.2 Efficient Spacing Policy Via Logic Operation

In this section, the author propose the efficient spacing policy via logic operation to get the generic spacing model’s safety redundancy. The logic operation is adopted to compare the distances of the safety spacing and constant time headway policies, and the minimum one is set as the spacing policy. As for the generic spacing model, a shorter time headway \({{th}}_{{\text{gen}}i}\) has less safety redundancy, and the efficient spacing policy \({d}_{{\text{ef}}i}\) is derived as presented in Eq. (8), representing the distance from the \(i\)th vehicle to its preceding vehicle. This efficient spacing policy can make use of the safety redundancy of various spacing policies, avoiding the need for solving intersections and the complex switching law depending on the critical velocity, thereby extending its work range.

$$\begin{aligned} {d}_{{\text{ef}}i}&={v}_{i}\cdot {\tau }_{{\text{ef}}i}+{d}_{{\text{min}}} \\ {\tau }_{{\text{ef}}i}&=\min\left\{\frac{{v}_{i}}{2{a}_{{\text{bmax}}}}+\tau, {{th}}({v}_{i})\right\} \\ \end{aligned}$$
(8)

3 CACC with Platoon Kinematics

In this section, the hierarchical control of CACC including platoon kinematics is established. The CACC upper control strategy with platoon kinematics is proposed to improve traffic efficiency, and its sliding mode controller is designed to allocate the longitudinal acceleration. Additionally, the lower controller, specialized for various driving and braking conditions, is developed to accurately and rapidly track the desired acceleration.

3.1 Hierarchical Control Framework

The hierarchical control framework in this paper is illustrated in Fig. 2. The upper controller, with platoon kinematics, allocates the host vehicle’s desired acceleration; and the lower controller tracks the desired acceleration by regulating the throttle opening and brake pressure.

Fig. 2
figure 2

The hierarchical control framework of CACC including platoon kinematics

Full size image

3.2 Upper Control Strategy

The upper control strategy is designed to allocate the desired distance and acceleration for each vehicle. In this section, the vehicle length is included in the presented efficient spacing policy for CACC. And it is assumed that the vehicles are homogeneous, and the information of the platoon is acquired by each vehicle without time delay.

In order to describe the platoon kinematics, the virtual centroid coordinates [8] for an n vehicles’ platoon has been proposed as

$$\begin{aligned} d_{{\text{G}}} & = \frac{{\sum _{{j = 1}}^{{n - 1}} W_{j} d_{n} + \sum _{{j = 1}}^{{n - 2}} W_{j} d_{{n - 1}} + \ldots + W_{1} d_{2} }}{{\sum _{{j = 1}}^{n} W_{j} }} \\ v_{{\text{G}}} & = \frac{{\sum _{{j = 1}}^{n} W_{j} v_{j} }}{{\sum _{{j = 1}}^{n} W_{j} }} \\ a_{{\text{G}}} & = \frac{{\sum _{{j = 1}}^{n} W_{j} a_{j} }}{{\sum _{{j = 1}}^{n} W_{j} }} \\ \end{aligned}$$
(9)

where \({d}_{{\text{G}}}\), \({v}_{{\text{G}}}\), and \({a}_{{\text{G}}}\) are the distance, velocity, and acceleration of the virtual centroid, respectively, \({d}_{i}\) is the centroid distance between the \(i\)th vehicle and its preceding vehicle, and \({W}_{j}\), \({v}_{j}\), and \({a}_{j}\) are the weight, velocity, and acceleration of the \(j\)th vehicle in the platoon, respectively. The virtual centroid, representing the whole platoon motion state, is then introduced to the CACC to improve traffic efficiency.

First, the desired distance is obtained. To improve traffic efficiency, the switching law of the following target is proposed: when \({d}_{i}\) falls below the lower threshold \({d}_{{\text{l}}}\), the \(i\)th vehicle’s following target is the preceding vehicle; when \({d}_{i}\) exceeds the upper threshold \({d}_{{\text{u}}}\), the following target is the virtual centroid; and when \({d}_{{\text{l}}}\le {d}_{i}\le {d}_{{\text{u}}}\), the \(i\)th vehicle keeps the current following target and desired distance to minimize frequent target switches [8]. Combining with this switching law, the efficient spacing policy considering vehicle length and platoon kinematics, \({d}_{{\text{des}}i}\), is:

$$\begin{array}{l}{d}_{{\text{des}}i}=\left\{\begin{array}{l}\begin{array}{l}{v}_{i}\cdot {\tau }_{{\text{ef}}i}+{d}_{{\text{min}}}+L,\; {d}_{i}<{d}_{{\text{l}}}\\ {\text{keep}}\;{\text{the}}\;{\text{current}}\;{\text{state}},\; { d}_{{\text{l}}}\le {d}_{i}\le {d}_{{\text{u}}}\end{array}\\ \frac{{\sum }_{j=1}^{i-1}{W}_{j}\left(i-j\right)}{{\sum }_{j=1}^{i-1}{W}_{j}}\left({v}_{i}\cdot {\tau }_{{\text{ef}}i}+{d}_{{\text{min}}}+L\right)\!, {d}_{i}>{d}_{{\text{u}}}\end{array}\right.\end{array}$$
(10)

with

$$\begin{array}{l}{d}_{{\text{l}}}={k}_{{\text{l}}}\left({v}_{i}\cdot {\tau }_{{\text{ef}}i}+{d}_{{\text{min}}}+L\right)\end{array}$$
(11)

and

$$\begin{array}{l}{d}_{{\text{u}}}={k}_{{\text{u}}}\left({v}_{i}\cdot {\tau }_{{\text{ef}}i}+{d}_{{\text{min}}}+L\right)\end{array}$$
(12)

where \({d}_{{\text{des}}i}\) is the desired centroid distance between the \(i\)th vehicle and following target, \({k}_{{\text{l}}}\) and \({k}_{{\text{u}}}\) are the lower threshold coefficient and upper threshold coefficient, respectively, which are tuned by experience, and \(L\) is the vehicle length.

Next, the vehicle longitudinal control is described as a first-order inertial system, as shown in Eq. (13). The centroid distance tracking error of the \(i\)th vehicle, \({e}_{i}\), is shown in Eq. (14), and the control objective is to regulate \({e}_{i}\) and its derivative to zero. To achieve this objective, a sliding mode controller is designed to allocate the desired acceleration of the \(i\)th vehicle, the sliding surface is defined by Eq. (16), and the desired acceleration of the \(i\)th vehicle is solved through Eq. (17), referred to as the control input, as shown in Eq. (18).

$$\begin{array}{l}{\dot{a}}_{i}=-\frac{1}{{\widehat{\tau }}_{s}}{a}_{i}+\frac{1}{{\widehat{\tau }}_{s}}{a}_{{\text{des}}i}\end{array}$$
(13)
$$\begin{array}{l}{e}_{i}={d}_{{\text{rea}}i}-{d}_{{\text{des}}i}\end{array}$$
(14)
$$\begin{array}{l}{d}_{{\text{rea}}i}={x}_{{\text{t}}}-{x}_{i}\end{array}$$
(15)
$$\begin{array}{l}S={e}_{i}+c{\dot{e}}_{i},c>0\end{array}$$
(16)
$$\begin{array}{l}\dot{S}=-\lambda S,\lambda >0\end{array}$$
(17)
$$\begin{array}{l}{a}_{{\text{des}}i}=\left\{\begin{array}{l}\begin{array}{l}{h}_{1}{e}_{i}+{h}_{2}{v}_{{\text{t}}}+{h}_{3}{v}_{i}+{h}_{4}{a}_{i}+{h}_{5}{a}_{{\text{t}}},{d}_{i}<{d}_{{\text{l}}}\\ {\text{keep}}\;{\text{the}}\;{\text{current}}\;{\text{state}},{d}_{{\text{l}}}\le {d}_{i}\le {d}_{{\text{u}}}\end{array}\\ {h}_{1}^{\mathrm{^{\prime}}}{e}_{i}+{h}_{2}^{\mathrm{^{\prime}}}{v}_{{\text{t}}}+{h}_{3}^{\mathrm{^{\prime}}}{v}_{i}+{h}_{4}^{\mathrm{^{\prime}}}{a}_{i}+{h}_{5}^{\mathrm{^{\prime}}}{a}_{{\text{t}}},{ d}_{i}>{d}_{{\text{u}}}\end{array}\right.\end{array}$$
(18)

with

$${h}_{1}=\frac{\lambda {\widehat{\tau }}_{s}}{c{\tau }_{{\text{ef}}i}},{h}_{2}=\frac{\left(\lambda c+1\right){\widehat{\tau }}_{s}}{c{\tau }_{{\text{ef}}i}},$$
$${h}_{3}=\frac{-\left(\lambda c+1+{\dot{\tau }}_{{\text{ef}}i}+\lambda c{\dot{\tau }}_{{\text{ef}}i}+c{\ddot{\tau }}_{{\text{ef}}i}\right){\widehat{\tau }}_{s}}{c{\tau }_{{\text{ef}}i}},$$
$${h}_{4}=1-\frac{{\widehat{\tau }}_{s}\left[\left(\lambda c+1\right){\tau }_{{\text{ef}}i}+c+2c{\dot{\tau }}_{{\text{ef}}i}\right]}{c{\tau }_{{\text{ef}}i}},{h}_{5}=\frac{{\widehat{\tau }}_{s}}{{\tau }_{{\text{ef}}i}},$$
$$h_{1}^{\prime } =\frac{{\sum }_{j=1}^{i-1}{W}_{j}}{{\sum }_{j=1}^{i-1}{W}_{j}\left(i-j\right)}\frac{\lambda {\widehat{\tau }}_{s}}{c{\tau }_{{\text{ef}}i}},$$
$$h_{2}^{\prime } =\frac{{\sum }_{j=1}^{i-1}{W}_{j}}{{\sum }_{j=1}^{i-1}{W}_{j}\left(i-j\right)}\frac{\left(\lambda c+1\right){\widehat{\tau }}_{s}}{c{\tau }_{{\text{ef}}i}},$$
$$h_{3}^{\prime } =\frac{-{\sum }_{j=1}^{i-1}{W}_{j}}{{\sum }_{j=1}^{i-1}{W}_{j}\left(i-j\right)}\frac{\left(\lambda c+1+{\dot{\tau }}_{{\text{ef}}i}+\lambda c{\dot{\tau }}_{{\text{ef}}i}+c{\ddot{\tau }}_{{\text{ef}}i}\right){\widehat{\tau }}_{s}}{c{\tau }_{{\text{ef}}i}},$$
$$h_{4}^{\prime } =1-\frac{{\widehat{\tau }}_{s}\left(\lambda c+1\right)}{c}-\frac{{\sum }_{j=1}^{i-1}{W}_{j}}{{\sum }_{j=1}^{i-1}{W}_{j}\left(i-j\right)}\left[\frac{{\widehat{\tau }}_{s}\left(c+2c{\dot{\tau }}_{{\text{ef}}i}\right)}{c{\tau }_{{\text{ef}}i}}\right],$$
$$h_{5}^{\prime } =\frac{{\sum }_{j=1}^{i-1}{W}_{j}}{{\sum }_{j=1}^{i-1}{W}_{j}\left(i-j\right)}\frac{{\widehat{\tau }}_{s}}{{\tau }_{{\text{ef}}i}}$$

where \({x}_{{\text{t}}}\), \({v}_{{\text{t}}}\), and \({a}_{{\text{t}}}\) are the position, velocity, and acceleration of following target separately, \({x}_{i}\) is the real position of the \(i\)th vehicle, \({d}_{{\text{rea}}i}\) is the real centroid distance between the \(i\)th vehicle and its following target, \({\widehat{\tau }}_{s}\) is the estimation of the vehicle response time.

The effects of the parameters on the upper control strategy are studied, including the sliding surface convergence speed, \(\lambda\), sliding surface parameter, \(c\), and weight of \(i\)th vehicle in the platoon, \({W}_{i}\). And it is shown that: first, \(\lambda\) and \(c\) mainly influence the transient performance of the controller, where larger \(\lambda\) and smaller \(c\) make the real distance converge to the desired distance faster; second, \({W}_{i}\) mainly influences the prospective performance of the controller, with smaller weights for vehicles near the \(i\)th vehicle, resulting in diminished acceleration and velocity amplitudes for the \(i\)th vehicle. However, this comes at the expense of an increased tracking error with respect to its preceding vehicle. The choice of \({W}_{i}\) necessitates a delicate balance to optimize performance. And the tuned values for the sliding mode controller parameters, determined through a trial-and-error process, are presented in Table 1.

Table 1 Control parameters
Full size table

String stability is important for the design of CACC system. Xiao et al. [18] designed a sliding model control law for the platoon, analyzed the negative effect of time delays on the homogeneous string stability, and obtained the string stable conditions. Specifically, the string stability could be ensured if the time headway \({h}_{i}\) and communication time delay \({\Delta }_{i}\) met \({h}_{i}>2{\Delta }_{i}\) and the sliding surface convergence speed fell within the range \({\lambda }_{i}\in [0,({h}_{i}-2{\Delta }_{i})/2{h}_{i}{\Delta }_{i}]\). In this paper, \({h}_{i}={\tau }_{{\text{ef}}i}=1.5{\text{s}}\), \({\lambda }_{i}=1\), and \({\Delta }_{i}\) is assumed as 0.3 s, and the relationship \({h}_{i}=1.5 {\text{s}}>2\times 0.3 {\text{s}}=2{\Delta }_{i}\), \({\lambda }_{i}=1\in [\mathrm{0,1}]\) are satisfied, showing that the proposed CACC can satisfy the string stability.

3.3 Lower Controller

In this section, the lower controller is established to track the desired acceleration, which is divided into braking and driving conditions. Achieving precise and rapid tracking of acceleration is imperative for ensuring following safety and string stability.

3.3.1 Driving Conditions Control

Here, the feedforward-feedback control is designed to control throttle opening during driving conditions to achieve accurate and swift tracking of the desired acceleration. To address the nonlinear and time-variable nature of the vehicle's longitudinal dynamics, stemming from components like the hydraulic torque converter, transmission, and main decelerator, a fuzzy PID controller is employed as the feedback control. This fuzzy PID controller adapts its parameter settings to effectively mitigate acceleration tracking errors.

As shown in Fig. 3, the inverse model of longitudinal dynamics computes the feedforward throttle opening \({u}_{{\text{throttle}},{\text{f}}}\), while the fuzzy PID controller calculates the feedback throttle opening \({u}_{{\text{throttle}},{\text{b}}}\), and the \({u}_{{\text{throttle}},{\text{f}}}\) and \({u}_{{\text{throttle}},{\text{b}}}\) constitute the throttle opening \({u}_{{\text{throttle}}}\). The control law for driving conditions is expressed as

$${u}_{{\text{throttle}}}={f}_{\text{e}}^{-1}\left({T}_{{\text{e}},{\text{des}}},{\omega }_{{\text{e}},{\text{des}}}\right)+\left({K}_{{\text{p}},{\text{o}}}+\Delta {K}_{{\text{p}},{\text{r}}}\right){e}_{{\text{a}}}+\begin{array}{l}\left({K}_{{\text{i}},{\text{o}}}+\Delta {K}_{{\text{i}},{\text{r}}}\right)\cdot {\int }_{0}^{t}{e}_{a}{\text{d}}t+\left({K}_{{\text{d}},{\text{o}}}+\Delta {K}_{{\text{d}},{\text{r}}}\right)\frac{{\text{d}}{e}_{{\text{a}}}}{{\text{d}}t}\end{array}$$
(19)

where \({f}_{\text{e}}^{-1}\) is the inverse model of engine operating characteristics, \({T}_{{\text{e}},{\text{des}}},{ \omega }_{{\text{e}},{\text{des}}}\) are desired engine torque, desired engine speed, \({K}_{{\text{p}},{\text{o}}}, { K}_{{\text{i}},{\text{o}}},{ K}_{{\text{d}},{\text{o}}}\) are the proportional coefficient, integral coefficient, differential coefficient whose values are 0.03, 0.06 and 0 respectively, \(\Delta {K}_{{\text{p}},{\text{r}}},\Delta {K}_{{\text{i}},{\text{r}}},\Delta {K}_{{\text{d}},{\text{r}}}\) are the actual parameter correction values of proportional, integral, and differential coefficient, \({e}_{{\text{a}}}\) is the acceleration tracking error.

Fig. 3
figure 3

The lower control diagram for driving conditions

Full size image

3.3.2 Braking Conditions Control

In the braking conditions, a feedforward-feedback control mechanism is formulated. Depicted in Fig. 4, the inverse model of vehicle longitudinal dynamics computes the feedforward brake master cylinder pressure \({u}_{{\text{b}},{\text{f}}}\), the PID controller calculates the feedback brake master cylinder pressure \({u}_{{\text{b}},{\text{b}}}\) to eliminate acceleration tracking errors, and the \({u}_{{\text{b}},{\text{f}}}\) and \({u}_{{\text{b}},{\text{b}}}\) constitute the brake master cylinder pressure \({u}_{{\text{b}}}\). The control law for braking conditions is expressed as

$$\begin{array}{l}{u}_{{\text{b}}}=\frac{{T}_{{\text{t}},{\text{des}}}-{T}_{{\text{t}},{\text{e}}}}{{K}_{{\text{b}}}}+{K}_{{\text{p}},{\text{b}}}{e}_{{\text{a}}}+{K}_{{\text{i}},{\text{b}}}{\int }_{0}^{t}{e}_{{\text{a}}}{\text{d}}t+{K}_{{\text{d}},{\text{b}}}\frac{{\text{d}}{e}_{{\text{a}}}}{{\text{d}}t}\end{array}$$
(20)

where \({{T}_{{\text{t}},{\text{des}}},T}_{{\text{t}},{\text{e}}}\) are the desired engine torques and actual engine torques acting on the driving wheels, \({K}_{{\text{b}}}\) is the proportional coefficient of braking torque and brake master cylinder pressure,\({K}_{{\text{p}},{\text{b}}},{K}_{{\text{i}},{\text{b}}},{K}_{{\text{d}},{\text{b}}}\) are the proportional, integral, and differential coefficient whose values are 0.5, 0.2, and 0, respectively.

Fig. 4
figure 4

The lower control diagram for braking conditions

Full size image

3.3.3 Validation of Lower Control

The lower controller's effectiveness is assessed through simulation in an urban scenario, specifically focusing on the 130-390 s segment of the Worldwide Harmonized Light-duty Test Cycle (WLTC), marked by the red dash line in Fig. 5(a). As depicted in Fig. 5, the controller adeptly tracks the reference speed and acceleration demonstrating smooth and rapid responses. The root-mean-square of the speed tracking error is 0.16 m/s, and the root-mean-square of acceleration tracking error is 0.10 m/s2. These results affirm that the presented lower controller can accurately and swiftly track the reference speed and acceleration in urban conditions.

Fig. 5
figure 5

Simulation results of lower control under WLTC

Full size image

4 Evaluation on CACC with Platoon Kinematics

In this section, eight simulation scenarios are reconstructed, and the CACC synchronic index is defined to evaluate the CACC performance with platoon kinematics.

4.1 Scenarios Re-construction

To systematize the urban traffic scene, eight typical scenarios are reconstructed using a three-layer decomposition, outlined in Table 2. From a macro perspective, these urban traffic scenarios can be categorized into two types based on the sources of disturbance.

Table 2 Reconstructed scenarios
Full size table
  1. (1)

    Disturbance inside the platoon

    When the disturbance is inside the platoon, the scenarios are divided into the steady following and vehicle cut-out scenarios. (a) The steady following scenario evaluates precision of following the preceding vehicle under steady conditions; and (b) the vehicle cut-out scenario assesses the safety of the process when a vehicle cuts out and subsequently re-establishes steady following.

  2. (2)

    Disturbance outside the platoon

    When the disturbance is outside the platoon, the scenarios are divided into three kinds of scenarios based on the state of the disturbance.

    The first scenario is a moving vehicle appears in front of the platoon. These scenarios are divided into three cases according to the vehicle speed and its initial distance from the platoon: (a) Approaching a low-speed vehicle. (b) Keeping away from a low-speed vehicle. (c) Keeping away from a high-speed vehicle. The above three cases are used to evaluate the safety of processes that the high-speed platoon is reconstructed into the low-speed platoon when a low-speed vehicle appears with a safe distance; the high-speed platoon is reconstructed into the low-speed platoon when a low-speed vehicle appears less than the safety distance; the low-speed platoon is reconstructed into the high-speed platoon when a high-speed vehicle appears less than the safety distance.

    The second scenario is a static disturbance that appears in front of the platoon, and these scenarios are divided into two cases according to the initial distance: (a) Approaching a static disturbance and (b) Emergency braking. The above two cases evaluate the safety of the platoon when decelerating to a stop via normal or emergency braking, respectively.

    And the third scenario is vehicle cut-in, which is used to evaluate the process’s safety from a vehicle cutting in to re-establishing steady following.

4.2 Evaluation Index

To evaluate response speeds of vehicle platoon with different lengths uniformly, the CACC synchronic index is proposed and defined as

$$\begin{array}{l}{I}_{{\text{s}}}=\frac{{T}_{m}-{T}_{n}}{m-n}\end{array}$$
(21)

where \({T}_{n}\) is the moment when the \(n{\text{th}}\) vehicle starts to change state, \({T}_{m}\) is the moment when the \(m{\text{th}}\) vehicle starts its response, and \(m>n\). The synchronic index shows the CACC’s ability to make promoting control and agile response according to the platoon kinematics. A smaller \({I}_{{\text{s}}}\) indicates improved synchronization and foresight within the vehicle platoon, leading to reduced response delay accumulation.

5 Simulation and Discussion

In this section, an urban stop-and-go case simulation test is studied to validate and evaluate the proposed CACC including platoon kinematics. A virtual test platform is developed via co-simulation with Matlab/Simulink and CarSim, considering a platoon with four vehicles. The CACC upper control strategy with platoon kinematics for stop-and-go (C-S&G) is expressed as the Eq. (17). The traditional ACC upper control strategy for stop-and-go (S&G) is represented by Eq. (22) [19]. The details of the lower controller are discussed in Sect. 3, and four SUVs are chosen as the test platoon using CarSim.

$$\begin{array}{l}{a}_{i,{\text{des}}}=\frac{1}{th}{e}_{i}+\frac{1}{th}\left({v}_{i-1}-{v}_{i}\right)\end{array}$$
(22)

The steady following scenario is selected as the simulation condition. The initial speed of the four vehicles is zero, and the initial distance from the front of the \(i\)th vehicle to the rear of its preceding vehicle is 2 m. The first vehicle travels at the reference speed, as shown by the solid blue line in Fig. 6. It stops completely during 0–5 seconds, then accelerates from 0 to 15 m/s, in ten seconds, maintains a constant speed during 15–25 seconds, and eventually decelerates to 0 in 10 s, and stops completely to the end. This scenario mainly studies the motion state of the rear three vehicles and verifies the tracking accuracy during steady following of the preceding vehicle. Table 3 shows the controller parameters. The simulation results of the steady following scenario are depicted in Fig. 6, where the black dotted-dash line represents the presented CACC for S&G (C-S&G), and the red dash line represents the traditional ACC for S&G (S&G). The velocity histories of the second, third, and fourth vehicles are shown in Fig. 6(a)–(c), the distance histories of the second, third, and fourth vehicles to their preceding vehicles are shown in Fig. 6(d)–(f), and the CACC synchronic index is presented in Table 4. The simulation results demonstrate that:

Fig. 6
figure 6

Velocity and distance histories of the following vehicles in platoon

Full size image
Table 3 Control parameters
Full size table
Table 4 Simulation result of CACC synchronization
Full size table

  • As shown in Fig. 6(d)–(f), both C-S&G and S&G ensure safety, with C-S&G exhibiting a smaller following distance compared to S&G. The average distance of C-S&G is 8.1 m less than S&G, indicating an enhancement in traffic efficiency and capacity.

  • Figure 6 illustrates that C-S&G responds more accurately and rapidly to the state of the first vehicle compared to S&G, showcasing the foresightedness of the proposed C-S&G.

  • As per Table 4, both the first vehicle of C-S&G and S&G starts at 5.00 s, the second vehicle of C-S&G starts at 5.15 s, the third vehicle of C-S&G responds to the first two vehicles in 0.26 s, the fourth vehicle responds to the first three vehicles in 0.36 s, and the fourth vehicle starts even faster than the second vehicle of S&G, which starts at 5.66 s. The C-S&G’s synchronic index IS is 85% lower than S&G’s, showing that the platoon kinematics can reduce the response delay, and the presented C-S&G exhibits superior synchronization. The proposed CACC synchronic index effectively reveals the synchronic ability of the CACC with platoon kinematics.

In summary, the proposed CACC, including platoon kinematics, demonstrates superior foresight and synchronization compared to traditional ACC, thereby improving traffic efficiency.

6 Conclusion

An efficient cooperative adaptive cruise control including platoon kinematics is presented in this paper, in which the efficient spacing policy is proposed via the presented generic spacing model, the CACC strategy considering platoon kinematics and its evaluation methodology are proposed and studied.

The safety spacing policy is essentially a variable time headway policy whose time headway varies linearly with velocity. The generic spacing model unifies safety spacing and time headway policies, offering a flexible variable time headway policy. The efficient spacing policy is devised through logical operations, leveraging safety redundancy from various spacing policies and avoiding the critical velocity solving and complex switching laws.

The hierarchical control of CACC with platoon kinematics is presented, featuring an upper controller with platoon kinematics for improved traffic efficiency and a sliding mode controller for acceleration allocation. The lower controller manages driving and braking conditions, ensuring accurate and fast tracking of desired acceleration.

The CACC synchronic index and eight typical scenarios are proposed for comprehensive evaluation. A steady following scenario is simulated, demonstrating that the proposed CACC with platoon kinematics enhances traffic efficiency. The evaluation methodology proves effective in assessing the synchronic ability of the CACC with platoon kinematics.

Future work needs to extend the research to heterogeneous platoons, considering real-world vehicle tests and accounting for communication time delays.