The car-like model is the basis for studying the platoon in a real environment [15]. The car-like model in the two-dimensional space quantifies the deviation of the vehicle from the road in a given two-dimensional space. In longitudinal control, the car-like model is used as a placement platform for the on-board laser and radar sensors to establish a mathematical model of the inter-vehicle distance. The placed sensor is used to detect the inter-vehicle distance and then transmit the message to the platoon controller to maintain the constant time headway or constant distance strategy of the platoon, thereby ensuring the longitudinal safety of the platoon. For example, Godbole and Lygeros propose the longitudinal control laws to maintain safe spacing, track an optimal velocity, and perform various maneuvers (forming, breaking up platoons, and changing lanes) [16]. In the lateral control, the car-like model is usually used to establish the relationship between the steering angle and the road offset, and to calculate the optimal steering angle when the vehicle is turning. Then, the vehicle maintains the distance between the nearest point and the curved road to ensure the lateral safety of the platoon. When driving on a curved road, the platoon includes both longitudinal control and lateral control. For instance, Wei et al. realize the longitudinal and lateral vehicle following only by radar and V2V, independent of high-accuracy positioning system and road marking [17]. Bayuwindra et al. overcome cut corners by the look-ahead extended to a point perpendicular to the direction of the preceding vehicle and then present a novel look-ahead concept for combined longitudinal and lateral vehicle-following control for a car-like platoon [18]. Finally, establishing the lateral deviation under the car-like model is a necessary condition for establishing the longitudinal inter-vehicle distance error.
2.1 Platoon under two-dimensional space
The curved road with varying curvature is established under the two-dimensional space for vehicle platooning. This method is close to the actual road design and easy to implement in the program. The movement of the platoon in the two-dimensional space causes a change in the inter-vehicle distance. At this time, the platoon stability varies with the error in the inter-vehicle distance for platooning control. Typically, inter-vehicle distance errors in the platoon are related to changes in road curvature and control law.
The curved road having different curvatures are designed, and curves of different curvatures are connected into curved roads by adjusting curvature control points. Then, we build a car-like model in the two-dimensional space. Then, the inter-vehicle distance error is established under the curvature space, and a constant control law K is added to design a stable platoon.
2.1.1 Car-like model
Car-like model is composed of a motorized wheeled axle at the rear of the chassis and a pair of orientable front steering wheels. For an easier analytical representation of the Car-like model, we put this model into a two-dimensional space, as represented in Fig. 1. X and Y represent the horizontal and vertical axes of the Cartesian coordinate system, respectively. The simple kinematic model is described as follows:
$$\begin{aligned} \begin{aligned}{\dot{x}}_{i}=v_t^i \cdot \cos \gamma ^{i}_{t}, \end{aligned} \end{aligned}$$
(1a)
$$\begin{aligned} \begin{aligned}{\dot{y}}_{i}=v_t^i \cdot \sin \gamma ^{i}_{t},\end{aligned} \end{aligned}$$
(1b)
$$\begin{aligned} \begin{aligned}{\dot{\gamma }}^{i}_{t}=\frac{v_t^i}{L} \cdot \tan (\gamma ^{i}_{t}-\beta ^{i}_{t}),\end{aligned} \end{aligned}$$
(1c)
where \(v_t^i\) is the velocity of vehicle i at time t. L is the wheelbase of the vehicle. \(\gamma ^{i}_{t}\) is the angle between orientation of vehicle i at time t and horizontal line with respect to world frame. \(\beta ^{i}_{t}\) is the steering angle of vehicle i at time t with respect to world frame. \(\alpha ^{i}_{t}\) is the angle of between orientation of the tangent to the trajectory at the same point in relation and horizontal line to the absolute reference. \(S^{i}_{\Delta t}\) is the curved distance of vehicles i from time t to \(t+\Delta t\). Given that Cartesian distance is not monotonous for very curved trajectories while quantifying the spacing between two consecutive vehicles, it is necessary to construct the model in curvilinear space, that is, with respect to the curvilinear abscissa of the nearest point on the trajectory measured from the vehicle, as shown follows.
\(\overrightarrow{GW}\) represents the projection of the Cartesian velocity of the vehicle into curvilinear space. The vehicle and the road are offset from the angle of \(\theta ^{i}_{t}=\gamma ^{i}_{t}-\alpha ^{i}_{t}\). \(c_{i}\) stands for the respective local curvature of vehicle i, which corresponds to the reciprocal of the radius of curvature. \({\dot{S}}^{i}_{\Delta t}\) is the further projection by considering the lateral deviation \(D^{i}_{t}\).
$$\begin{aligned} \begin{aligned}{\dot{S}}^{i}_{\Delta t}=\frac{v_t^i\cos \theta ^{i}_{t}}{1+D^i_tc_i},\end{aligned} \end{aligned}$$
(2a)
$$\begin{aligned} \begin{aligned}{\dot{D}}^{i}_{t}=v_t^i \cdot \sin \theta ^{i}_{t}, \end{aligned} \end{aligned}$$
(2b)
$$\begin{aligned} \begin{aligned}{\dot{\theta }}^{i}_{t}=v_t^i\left( \frac{\tan (\gamma ^{i}_{t}-\beta ^{i}_{t})}{L}-\frac{c_i\cdot \cos \theta ^{i}_{t}}{1+c_iD^i_t}\right) . \end{aligned} \end{aligned}$$
(2c)
Proof See “Appendix A.”\(\square\)
2.1.2 Modeling of inter-vehicle distance error
A longitudinal platooning system can be considered as a combination of four important components: vehicle longitudinal dynamics, information exchange flow, decentralized or centralized controllers and inter-vehicle distance policies [19,20,21,22]. In order to analyze longitudinal inter-vehicle distance error, the assumptions of the other components are as follows:
-
(1)
Vehicle longitudinal dynamics include the engine, drive line, brake system, aerodynamics drag, etc. In this paper, we consider that every vehicle has the homogeneous double-integrator model [23].
-
(2)
Decentralized or centralized controller is adopted in this paper. Decentralized control is that each vehicle controls its own vehicle status based on other vehicles’ information. Centralized control is that all platoon members are controlled by leader.
Considering N homogeneous vehicles driving along the curved road, platoon member (vehicle i) is to follow its preceding vehicle driving at a desired inter-vehicle distance S. Two platoon members are schematically depicted in Fig. 2 with S being the distance between vehicle i and its preceding vehicle \(i-1\). The main objective of each vehicle is to follow its preceding vehicle at a desired distance S. But, inter-vehicle distance will be changed due to sensor delay, communication delay, weather and curved road, etc. causing a distance error \(\delta ^i_{\Delta t}\). Here \(S^i_{\Delta t}\) and \(S^{i-1}_{\Delta t}\) are the curved distance of vehicles i and \(i-1\) in time \(\Delta t\), respectively. Let us define the curved distance error for the ith vehicle as:
$$\begin{aligned} \begin{aligned}\delta ^{i}_{\Delta t}=S^{i-1}_{\Delta t}-S^{i}_{\Delta t}, \end{aligned} \end{aligned}$$
(3a)
$$\begin{aligned} \begin{aligned}{\dot{\delta }}^{i}_{\Delta t}={\dot{S}}^{i-1}_{\Delta t}-{\dot{S}}^{i}_{\Delta t}. \end{aligned} \end{aligned}$$
(3b)
When \(\Delta t \rightarrow 0\), combining Eq. (3b) with Eq. (2c), we can get the following equation.
$$\begin{aligned} \begin{aligned} {\dot{\delta }}^{i}_{\Delta t}=\frac{v^{i-1}_{t}\cos \theta ^{i-1}_{t}}{1+D^{i-1}_{t}c_{i-1}}-\frac{v_t^i \cos \theta ^{i}_{t}}{1+D^i_{t} c_i}, \end{aligned} \end{aligned}$$
(4)
let \({\dot{\delta }}^{i}_{\Delta t}=K \cdot \delta ^{i}_{\Delta t}\) and \(K>0\). The expression of vehicle speed is as follows:
$$\begin{aligned} \begin{aligned} v_t^i=\frac{1+D^i_{t}c_i}{\cos \theta ^{i}_{t}}\left( \frac{v^{i-1}_t \cos \theta ^{i-1}_{t}}{1+D^{i-1}_{t}c_{i-1}}-K\delta ^{i}_{\Delta t}\right) , \end{aligned} \end{aligned}$$
(5)
where the vehicle speed \(v^{i-1}_t\) and cured distance error \(\delta ^{i}_{\Delta t}\) are used as the control input of vehicle i.
In this section, the inter-vehicle distance error is converged by constant control law K to keep the desired inter-vehicle distance. In the next section, the cooperation between multiple vehicles based on different information flow topologies is discussed.
2.2 Platooning control schemes
For multi-vehicle platooning, different controller placement positions will lead to different information flow topologies. In other words, different information flow topologies have different suitable platoon controller placement positions. For example, the PF (predecessor-following) topology is a topology that does not obtain information about the dynamics of the leader. It is suitable for placing the controller on each vehicle of the platoon members, and the vehicle itself will calculate the desired inter-vehicle distance at each moment. Ghasemi et al. propose a hierarchical platoon controller design framework, where the second layer is composed of a decentralized bidirectional control controller [24]. Due to the lack of dynamics about the leader, the platoon is likely to lose stability as the number of vehicles increases [25]. In the PLF (predecessor–leader-following) topology, each vehicle of the platoon can receive broadcast information from the leader, so the controller can be placed in each vehicle or only in the leader. When the controller is only placed in the leader, the leader controls platoon members through the network, which is called a centralized control platoon. When the controller is placed in each vehicle, the leader only broadcasts its dynamics information to platoon members, which is a decentralized control platoon. In paper [26], a centralized and distributed control policy is proposed in which each vehicle’s control decision depends solely on its relative kinematics with respect to the leader. In paper [19], for each topology, Chehardoli et al. propose a new neighbor-based adaptive control law to deal with adaptive control and identification of 1-D platoon of non-identical vehicles. Therefore, platoon control schemes under multi-vehicle mainly focus on the control and communication topology.
Typical platooning information flow topologies include predecessor-following (PF) topology, predecessor–leader-following (PLF) topology, bidirectional (BD) topology, bidirectional–leader (BDL) topology, two-predecessor-following (TPF) topology, and two-predecessor–leader-following (TPLF) topology [27, 28]. These information flow topologies are classified from the mode of transmitting the information. The simplex mode includes PF, PLF, TPF, and TPLF, while the duplex mode includes BD and BDL. In simplex mode, TPF and TPLF extend from PF and PLF, respectively. Therefore, the most dominant topologies are PF and PLF. The information flow topology only shows the transmission path of the platooning information and must be combined with platooning control strategy to stabilize the platoon.
Platooning control strategies can be centralized or decentralized according to the position of the controller [29]. In the centralized control, vehicles get their control commands from central units. They are therefore not autonomous, and communication is fundamental: any loss or delay in communication is critical. While in decentralized control, each vehicle receives data from other vehicles and calculates its own control in a stand-alone manner, so that communication remains very important, its loss is not as critical as the centralized case [30]. In general, the topology should match the control strategies.
Combining platooning information flow topologies and platooning control strategies, three typical platooning control schemes are proposed in Fig. 3. The first vehicle of each platoon is called the leader, and the others are called platoon members. Platooning control scheme (a) is a combination of PF topology and decentralized control that platoon members themselves adjust the vehicular dynamics based on the received dynamic information of the preceding vehicle to maintain the desired distance between the vehicle and preceding vehicle. Platooning control scheme (b) is a combination of PLF topology and decentralized control that platoon members themselves adjust the vehicular dynamics based on the received dynamic information of the preceding vehicle and leader to maintain the desired distance between the vehicle and preceding vehicle. Scheme (c) is centralized control approach by the leader that all platoon members adjust the speed according to the control commands message of the leader to maintain platooning stability (Fig. 4).
There are two major spacing policies for the desired inter-vehicle distance S: the constant time headway policy and constant distance policy [31]. For the constant time headway policy, the desired inter-vehicle distance varies with vehicle velocity. In the constant distance policy, the desired distance between two consecutive vehicles is independent of vehicle velocity. Here, we consider the constant time headway policy used for scheme (a), because the vehicles can only get the information of the predecessor. In order to ensure the safety of the vehicles, inter-vehicle distance needs to change with the speed of the predecessor. For scheme (b), we consider a constant distance, which means that the vehicles are controlled to move in a rigid platoon while following a leader because the vehicles can get the information of the predecessor and leader. Scheme (c) is similar to scheme (b) but lacks the preceding vehicle information.
When \(S=T_s v_t^i\), the platooning control scheme (a) is structured as follows. This is constructed according to the previous literature by Segata et al. [32].
$$\begin{aligned} \begin{aligned} {\ddot{S}}_{i}^{des}(t+\Delta t)&=-\frac{1}{T_s}({\dot{\xi }}_i+\lambda \delta ^{i}_{\Delta t}),\\ \end{aligned} \end{aligned}$$
(6)
where \({\ddot{S}}_{i}^{des}(t+\Delta t)\) is desired acceleration of vehicle i at time \(t+\Delta t\), \(\lambda\) is a design parameter strictly greater than 0 (default set to 0.1). \({\dot{\xi }}_i=v_t^i-v_t^{i-1}\) is relative speed between vehicle i and vehicle \(i-1\). \(T_s\) is time headway. In order to ensure the safety of passengers, [31] shows that it must satisfies \(T_s\ge 2L\) , where L is actuation lag [33].
By combining Formula (5) and (6), then we get the following formula.
$$\begin{aligned} \begin{aligned} {\ddot{S}}_{i}^{des}(t+\Delta t)&=-\frac{v^{i-1}_t}{T_s}\left( \frac{\cos \theta ^{i-1}_{t}(1+D^{i}_{t}c_{i})}{\cos \theta ^{i}_{t}(1+D^{i-1}_{t}c_{i-1})}-1\right) \\&\quad -\frac{\delta ^{i}_{\Delta t}}{T_s}\left( \lambda -\frac{k(1+D^i_{t}c_i)}{\cos \theta ^{i}_{t}}\right) . \end{aligned} \end{aligned}$$
(7)
When S is the constant distance, the platooning control scheme (b) is structured as follows. This is derived based on a previous literature by Segata et al. [32],
$$\begin{aligned} \begin{aligned} {\ddot{S}}_{i}^{des}(t+\Delta t)&=a{\ddot{S}}_{i-1}(t)+b{\ddot{S}}_0(t)\\&\quad +c{\dot{\xi }}_i+d(v^i_t-v^0_t))\\&\quad +e\delta ^{i}_{\Delta t}. \end{aligned} \end{aligned}$$
(8)
By combining formula (5) and (8),
$$\begin{aligned} \begin{aligned} {\ddot{S}}_{i}^{des}(t+\Delta t)&=a{\ddot{S}}_{i-1}(t)+b{\ddot{S}}_0(t)\\&\quad +v^{i-1}_t \left( \frac{\cos \theta ^{i-1}_{t}(1+D^{i}_{t}c_{i})}{\cos \theta ^{i}_{t}(1+D^{i-1}_{t}c_{i-1})}(c+d)-c\right) -dv^0_t\\&\quad +\delta ^{i}_{\Delta t}\left( e-(c+d)\frac{k(1+D^i_{t}c_i)}{\cos \theta ^{i}_{t}}\right) , \end{aligned} \end{aligned}$$
(9)
where \(v^0_t\) and \({\ddot{S}}_0(t)\) are the speed and acceleration of leader, respectively. a, b, c, d, e are parameters depicted as follows:
$$\begin{aligned} \begin{aligned} a&=1-W,\\ b&=W,\\ c&=-(2\zeta -W(\zeta +\sqrt{\zeta ^2-1}))b_w,\\ d&=-W(\zeta +\sqrt{\zeta ^2-1})b_w,\\ e&=-b_w^2, \end{aligned} \end{aligned}$$
(10)
where W is weighting factor between the acceleration of leader and preceding vehicle. \(\zeta\) is drag coefficient which is set as 1. \(b_w\) is receiving signal frequency. Since the distance between the leader and vehicle i is different from that between the vehicle \(i-1\) and vehicle i , the delay times of vehicle \(i-1\) and leader transmitting information to the vehicle i are different.
For scheme (c), it does not receive the information of the preceding vehicle, and formula (8) is transformed into the following formula.
$$\begin{aligned} \begin{aligned} {\ddot{S}}_{i}^{des}(t+\Delta t)=b{\ddot{S}}_{0}(t)+(c+d)v^i_t-dv^0_t+e\delta ^{i}_{\Delta t}. \end{aligned} \end{aligned}$$
(11)
The desired acceleration \({\ddot{S}}_{i}^{des}(t+\Delta t)\) of the vehicle i can be used as the power input of the engine at time \(t+\Delta t\) in the simulation program. The flow chart of the program is as follows:
The flowchart is based on the travel time of the vehicle. This paper uses the speed and acceleration of vehicle \(i-1\) and leader at time t and the curved distance error \(\delta ^{i}_{\Delta t}\) within \(\Delta t\) to calculate the desired acceleration \({\ddot{S}}_{i}^{des}(t+\Delta t)\) of the vehicle i at time \(t+\Delta t\).
The reason for curved distance error is the information flow delay and the dynamic system delay. The main reason for information flow delay is the difference in information flow topology. For example, PF topology can easily cause cascading delay. In the PLF topology, inter-channel interference and signal transmission failures cause repeated information transmission delays. The second cause of curved distance error is the dynamic system delay. That is caused by the vehicle’s own mechanical transmission.
The main purpose of platooning control is to adjust the curved distance error \(\delta ^{i} _{\Delta t}\) according to the control law k to make it tend to 0. When the curve distance error \(\delta ^{i}_{\Delta t}\) is close to 0, the control law k has no effect on the speed, and a stable platoon is formed. In other words, the vehicle travels at a desired inter-vehicle distance. Then, when the platoon is driving on the curved road, the vehicle adjusts the transmission power of the OBU (On board Unit) according to the difference between the communication distance and the inter-vehicle curved distance. A curvature-driven communication mechanism is shown in the next section.
2.3 Curvature-driven communication mechanism
Among the AES of electric vehicles, communication system is a continuous energy consumption part. The communication power control is a method to reduce the energy consumption of AES [34]. Communication energy includes both the power consumption for receive and send packets. Whether the power for the receiver or sender is reduced, the energy consumption of the AES communication system will be reduced, and the cruise range of the platoon will be increased. In the past, transmit power control (TPC) is usually used to improve the reliability and accuracy of communication. In [35], Zander investigates the control of co-channel and adjacent channels interference and achieves acceptable carrier-to-interference ratios by TPC in all active communication links in the system. Paper [36] describes a centralized power control scheme that computes transmitter powers so as to have a common carrier-to-interference ratio for all the receivers. In [37], Rosberg et al. use the transmitter power control techniques in protecting other users from excessive interference as well as making receivers more tolerant to this interference. In [38, 39], Liu et al. propose a 5G-based IoT scheme to joint optimization problem of allocation factors and node powers to maximize the 5G transmission rate while the IoT transmission rate and the total power are constrained. In general, when the receiver’s power is constant, the dynamic sensing power range requirement of the receiver is reduced and the adjacent channels are protected. At the same time, the transmitter’s power control can reduce inter-communication interference and inter-channel interference. Therefore, besides improving the communication reliability and accuracy, communication power control can also be used to reduce AES energy consumption.
Communication modes among vehicles in the platoon include unicast, multicast, and broadcast. At this time, the signal transmission power conforms to the path loss model. The specific inter-vehicle communication distance model is as follows.
In the curve motion of the vehicle, the chord length d corresponding to the arc length is \(S+\delta ^i_{\Delta t}\) as described in Eq. (12a). The proof of (12a) is given in “Appendix B.” Based on the basic V2R path loss model given in [40, 41], we can get the following path loss Eq. (12).
$$\begin{aligned} d= & {} \frac{2}{c_i}\cdot \sin \left( \frac{(S+\delta ^i_{\Delta t}) \cdot c_i}{2}\right) , \end{aligned}$$
(12a)
$$\begin{aligned} W_i= & {} W_{i-1} + 16.7\log 10(d) + 18.2\log 10(fc), \end{aligned}$$
(12b)
where fc is in GHz. \(W_i\) is the receive power from vehicle \(i-1\) to vehicle i. When a stable platooning passes through the curved road with varying curvature, its inter-vehicle communication distance is different from that of the inter-vehicle curved distance. The difference between inter-vehicle curved distance and inter-vehicle communication distance is the starting point of Algorithm 1. According to this difference, the transmission power of the OBU is controlled to save the energy consumption of the communication system. Finally, a communication energy allocation algorithm for the adaptive radius of curvature is proposed.
The general idea of Algorithm 1 is as follows, the preceding vehicle computes the inter-vehicle communication distance at every moment according to the change of road curvature. Then, the preceding vehicle adjusts the transmission power under the condition of meeting the minimum receiving power of the rear vehicle through the power control based on the radius of curvature. Before this algorithm, the curved distance error is computed according to the car-like model, and it is considered into the platooning control scheme. The platooning consensus is ensured by adjusting the control law K.
In Algorithm 1, the desired inter-vehicle distance is input into the algorithm as the initial communication distance. The local curvature is set according to the changes of the road. In the whole process of Algorithm 1, the platooning consensus must be ensured. In the first and second step of Algorithm 1, the minimum transmission power of vehicle \(i-1\) relative to vehicle i is set. The first step is to compute the inter-vehicle communication distance under the inter-vehicle curved distance, and the second step is to compute the minimum transmission power of vehicle \(i-1\) according to the path loss model. The third step is to set the minimum receiving power of vehicle i as 0 dBm. The fourth step is to adjust the transmission power of vehicle \(i-1\) according to the change of inter-vehicle communication distance. In the fifth step, the ratio of vehicle energy consumption with adaptive curvature radius algorithm and without adaptive curvature scheme is output. Since there are N vehicles, the number of computing inter-vehicle distance is \(N-1\), and the number of vehicle communication energy consumption is N, the complexity of the algorithm 1 is N.
In summary, when the platooning is driving on the curved road, the communication distance and control distance of the inter-vehicle are different. For platooning, It is necessary to control the vehicles to maintain a desired inter-vehicle distance and achieve platooning consensus. Under platooning consensus, we adjust the transmission power of communication in real-time according to the inter-vehicle communication distance. In the case of ensuring the reliability of the communication link and platooning consensus, the energy consumption optimization of the communication system is the main contribution of this paper.
