Cooperative Adaptive Cruise Control Using Delay-Based Spacing Policy: A Robust Adaptive Non-Singular Terminal Sliding Mode Approach

This study proposes two speed controllers based on a robust adaptive non-singular terminal sliding mode control approach for the cooperative adaptive cruise control problem in a connected and automated vehicular platoon. The delay-based spacing policy is adopted to guarantee that all vehicles in the platoon track the same target velocity profile at the same position while maintaining a predefined time gap. Factors such as nonlinear vehicle longitudinal dynamics, engine dynamics with time delay, undulating road profiles, parameter uncertainties, and external disturbances are considered in the system modeling and controller design. Different control objectives are assigned to the leading and following vehicles. Then, controllers consisting of a sliding mode controller with parameter adaptive laws based on the ego vehicle’s state deviation and linear coupled state errors, and a Smith predictor for time delay compensation are designed. Both inner stability and strong string stability are guaranteed in the case of nonlinear sliding manifolds. Finally, the effectiveness of the proposed controllers and the benefits of 44.73% shorter stabilization time, 11.20% less speed overshoot, and virtually zero steady-state inner vehicle distance deviation are illustrated in a simulation study of a seven-vehicle platoon cooperative adaptive cruise control and comparison experiments with a coupled sliding mode control approach.


Introduction
Significant developments in connected and automated vehicle (CAV) technology have been realized during the last decade. Intelligent CAVs integrated by connectivity and automation can not only drive by themselves with on-board computers and sensing sensors, but also communicate with each other through vehicle-to-vehicle (V2V) communications. Cooperative adaptive cruise control (CACC) is one of the most meaningful technologies for CAVs and has complemented adaptive cruise control (ACC) with cooperative maneuvers. In CACC systems, CAVs share their own parameters and real-time state information (e.g., acceleration, speed, and position) with other CAVs in the network through V2V communications.
Using CACC systems, potential benefits can be achieved, such as increasing traffic capacity by reducing the gaps between vehicles, reducing energy consumption due to unnecessary speed changes, and reducing aerodynamic drag on the following vehicles, which can achieve up to 10% energy conservation [1] and improve driving safety [2][3] .
In order to ensure the safe CACC operation of such platoons with shorter vehicle-following gaps, cooperative automation of vehicle longitudinal dynamics is required. The two fundamental aspects of the resulting behavior are the spacing policy and cooperative control approach.
Among the above controller design approaches, SMC, a well-known nonlinear method with robustness against bounded matched uncertainties, has attracted considerable attention and has been adopted by different researchers to solve the CACC problem. Kwon and Chwa [11] proposed a coupled SMC (CSMC) to improve the performance and string stability of bidirectional platoon control. The coupled sliding surfaces, which are in the form of a linear combination of the ego vehicle's sliding surface and its adjacent vehicles, are designed to guarantee string stability. Wu et al. [12] proposed a distributed SMC method for homogeneous vehicular platoons based on nonlinear dynamics and generic topologies whose associated matrices are positive definite. The distributed SMC design is divided into two parts: topological sliding surface and topological reaching law designs. Considering uncertain and time-varying driving resistance, Yan et al. [13] proposed a distributed adaptive SMC for platoon control. Then, an adaptive control method was applied to estimate the uncertain resistance coefficients. A coupled sliding surface, which is adopted to link two standalone sliding surface of a single vehicle, ensures satisfactory convergence and string stability.
The above studies are based on bi-level sliding surfaces, which are a lower-level sliding surface constructed by a linear combination of ego-vehicle state errors and an upper level sliding surface constructed by the linear coupled lower sliding surfaces.
Based on the transformed error, two adaptive SMC schemes based on leader-predecessor and leaderbidirectional information flows were presented by Guo and Li [14] to ensure string stability and strong string stability, respectively. A new linear sliding surface with the proposed transformed error, which is forced into the allowable region, was adopted. Factors such as actuator saturation nonlinearity, which is approximated by a smooth hyperbolic tangent function, uncertain parameters, and unknown disturbances, are also considered. A terminal SMC such as a switching nonlinear sliding manifold was designed in Ref. [15]. The string stability is guaranteed by using the proposed novel spacing error of the vehicles when the sliding surface is reached in finite time.
Different approaches have been combined with SMC methods. Guo and Li [22] integrated Pontryagin's minimum principle-based OC method with SMC for CACC set-point optimization and tracking control. A twolayered control architecture consisting of a fuel-timeefficient speed-planning algorithm and distributed SMC speed tracking is presented. Fesharaki et al. [23] integrated MPC with SMC for constrained nonlinear systems. A tractable nonlinear MPC is designed based on the nominal system, and the SMC provides the robustness and recovery of the performance against modeling error and system uncertainties. Guo et al. [24] proposed a Chebyshev NN (CNN)-based distributive integral SMC with special consideration for input saturation. CNNs are used to approximate unknown nonlinear functions in the followers online.
However, the majority of the above-mentioned works use a time domain-based spacing policy such as CS or CTH. Only limited studies [6,10] are based on D-B policy. Motived by the fact that the slopes, curvatures, speed limits, and other infrastructure information are all position-based, rather than time domain, a D-B space policy that ensures the same velocity and acceleration in the same location for each vehicle is more desirable and reasonable to be chosen as the target profile. In what follows, two scenarios where the CS or CTH policies are adopted are chosen as an example, explaining the reason for selecting the D-B policy. For a platoon of heavy-duty vehicles traversing a hilly road segment, owing to the length of the platoon, the leading vehicle could accelerate in level or downhill front segments while the following vehicles pass through the steep uphill segments. Not all of the following vehicles can necessarily achieve speed acceleration owing to the road geometry and limitation of their engine power, although the two spacing policies command all vehicles to maneuver synchronously. This scenario was recognized in experiments of Ref. [1]. A lower speed limit can frequently exist in curvature segments, ensuring safety issues. However, the leading vehicle could accelerate after passing through the curvature segment, forcing the following vehicles to accelerate because of the two policies, even though they remain in the curvature segments.
Furthermore, vehicle nonlinear dynamics, parameter uncertainties, external disturbances, and pure delay processes caused by communication delays and engine actor lags are seldom considered simultaneously. To resolve the above issue in CACC with D-B policy, non-singular terminal SMC (NTSMC), which was first proposed by Feng et al. [25] with features that ensure states of the resulting closed-loop system converge to the equilibrium point in finite time while avoiding the singularity problem in conventional terminal SMC, is adopted in this study.
In this paper, two controllers, for the leading vehicle and following vehicles, are proposed for the different control objectives. Factors such as the response process with engine time delay, nonlinearities of the vehicle dynamics, parameter uncertainties, and road vertical profiles are all considered in the CACC controller design.
The remainder of this paper is organized as follows: Section 1 describes a space domain-based CACC problem and the longitudinal dynamics of vehicles in the platoon. Section 2 provides a detailed design of the CACC controllers, including system structure, control laws, parameter adaption laws, and proofs of the inner and string stability. Section 3 provides numerical simulations and Section 4 compares the proposed method with the CSMC approach. Section 5 concludes the paper.

Space Domain-Based CACC Problem and
Delay-Based Spacing Policy Consider a vehicular platoon consisting of n heterogeneous CAVs that traverse on a horizontal road on an undulate vertical segment profile with a predefined distance-based speed profile. The predecessorleader following the information flow topology with the optional last-to-ego vehicle communication link is adopted, as displayed in Fig. 1.
Categorized by the different roles in the vehicular platoon, different control objectives are adopted in the driving of a vehicle.
(1) For the leading vehicle, the objective is to track the predefined speed profile computed based on the ego position of the current time as the independent variable:  , and x r2,1 andẋ r2,1 are the target speed and acceleration, respectively. Without loss of generality, "→" denotes the tracking process.
(2) For the following vehicles, the objective is to track the D-B position and speed profile obtained from the preceding or leading vehicle, while considering the distance gap between the preceding vehicle, leading vehicle, and optional last vehicle. Thus, the target speed profile for the ith following vehicle x r2,i in the D-B spacing policy is given as where t g,i−1 > 0 is the time gap between the ith vehicle and its preceding vehicle.
(3) Stability performance: controllers for the leading and following vehicles must fulfill the inner stability; controllers for the following vehicles must fulfill the string stability.

Longitudinal Dynamics of a Vehicle
The longitudinal dynamics of the ith vehicle in a platoon can be obtained by Newton's second law as follows: where x 1,i and x 2,i are the position and velocity of the vehicle, m i is the static mass of the vehicle, u e,i is the actual driving force that the engine outputs, d i is the unknown external disturbances such as model errors, uncertainties of the vehicle, road profiles and unpredicted wind gusts, and f R,i is the total resistance of the vehicle which is attributed to the basic resistance f b,i and road vertical profile-based resistance f g,i .
The basic resistance f b,i is a consequence of the rolling and speed-related aerodynamic resistances. The road vertical profile-based resistance f g,i is a consequence of the road slope α(x 1,i ). In Eq. (4), c 0,i = r i g where r i is the rolling resistance coefficient, c 1,i = 0, and where h a,i is the air density, A i is the vehicle front area, C d,i is the air drag coefficient, and g is the acceleration due to gravity. Typically, the grade of the slope on the road is small, and approximations of α ≈ tan α ≈ sin α and cos α ≈ 1 are used above.
In this study, we assume that the unknown disturbance satisfies |d i | d i , whered i is the upper bound of the disturbance.
Owing to the system inertia, mechanical properties, and transmission delay, the engine dynamics are modeled as a first-order plus dead time (FOPDT) process: whereu e,i is the time derivative of u e,i , u c,i is the control command input such as the throttle/brake command, T i is the engine time constant, and τ i is the pure delay time (including the wireless communication network owing to the queuing, contention, transmission, and propagation) for the control command.

Controller Structure
The proposed controller is composed of two main parts: the adaptive NTSMC-based trajectory tracking controller and Smith predictor (SP), as indicated in Fig. 2. Based on the different roles of the vehicles, different data are chosen as the target input, such as target speed and acceleration based on the position of the leading vehicle, and D-B target and state profile of the leading, preceding, and last vehicles. In every control cycle, the control command computed by the adaptive NTSMC is sent to the simulation (or real) vehicle, as well as the two nominal models with and without the engine delay process. Then, a modified state,  (3) and (5) Vehicle state  3) and (5) with ; then, the total resistance of the vehicle can be rewritten as . With unknown disturbance terms d i neglected, the vehicle dynamic model for the adaptive NTSMC design is then described aṡ andu

Cruise Controller for Leading Vehicle 2.2.1 Control Law Design for Leading Vehicle
For the leading vehicle (i = 1), based on the three-order model in Eqs. (6) and (7) with a predefined distance-based reference speed profile x r2,i and its derivativeẋ r2,i given, the speed tracking error e 2,i and its derivative, acceleration tracking error, e 3,i of the leading vehicle are defined as Thus, a non-singular terminal sliding manifold for the leading vehicle is defined as where the sliding manifold parameters satisfy p i and q i are constants of positive odd integers, and k 0,i < 0 is a constant. Thus, the time derivative of the sliding manifold in Eq. (9) iṡ whereė 3,i =ẍ 2,i −ẍ r2,i is the vehicle jerk error and jerk of the target speed profiles, andẍ r2,i is the derivative of the target acceleration profile. Define the time derivative of the total resistance aṡ Then from Eq. (6), the jerk of the leading vehicle is and the actual driving force that the engine outputs is By substituting Eqs. (7) and (12) into Eq. (11), we obtainẍ Then, Eq. (10) can be rewritten aṡ Based on the nature of the SMC, the control law includes two terms: where u eq,i is the equivalent control law for the leading vehicle and u sw,i is the nonlinear switching term. By deriving fromṡ i = 0, the equivalent control law is designed as and a constant rate reaching law is adopted here for the nonlinear switching term: where K i > 0 is a parameter greater than the norm bound of the total matched disturbance, satisfying the condition K i d i .

Proof:
We choose the Lyapunov candidate function for the leading vehicle as Then, the time derivative of Eq. (18) iṡ Then, by substituting Eq. (14) into Eq. (19), the time derivative of the Lyapunov candidate function can be changed tȯ where e wi−1 3,i > 0 for ∀e 3,i = 0 owing to the odd integers p i and q i , and sgn(s i )s i = |s i | > 0 for ∀s i = 0. Thus, V i < 0, the system is stable and can reach the sliding manifold s i = 0 in a finite time.
End of Proof.

Parameter Adaptation for Leading Vehicle
Owing to model uncertainties, the parameter adaptive law is designed as follows. Based on the control law designed in Eq. (15), two auxiliary variables, μ θ,i = T i θ 1,i and μ m,i = T i m i are defined. The parameters for the adaptive updates are chosen as follows: wherem i ,θ i ,μ θ,i , andμ m,i are the estimated values, and Δm i , Δθ i , Δμ θ,i , and Δμ m,i are the errors between the estimations and their real values. Then, the designed control law can be rewritten based on the parameter estimation as Choose the new Lyapunov candidate function with parameter uncertainties: where λ m,i , λ θ,i = diag(λ θ1,i , λ θ2,i , λ θ3,i ), λ μθ,i = diag(λ μθ1,i , λ μθ2,i , λ μθ3,i ), and λ μm,i are the designed constants of the adaptive learning rate. Then, the time derivative of Eq. (24) iṡ Considering the time derivative of the sliding manifold in Eq. (14), we obtain Then, substituting Eqs. (23) and (26) into Eq. (25), we obtaiṅ Thus, the adaptive law is designed aṡ and the time derivative of the Lyapunov candidate function becomesV which guarantees the stability of the system.

Cooperative Controller for Following Vehicles
Because the sliding manifold designed for the leading vehicle in Eq. (9) with the inner stability derived from Eqs. (18)-(21) cannot guarantee strong string stability, we introduce another NTSMC and its associated parameter adaption laws for the following vehicles in the platoon to ensure strong string stability with a nonlinear sliding manifold based on the modified input error idea.

Control Law Design for Following Vehicles
For the following vehicles (i = 2, 3, · · · , n), based on the same three-order model, Eqs. (6) and (7), and with a D-B reference position profile and real-time position information from the known leading, preceding, and last vehicles, the modified position error from the ego (ith) vehicle to its preceding vehicle is defined as Similarly, we have the modified position error between the ego vehicle and leading vehicle as and the error between the ego vehicle and last vehicle of the platoon as Then, we define a linear combination of Eqs. (31)-(33) as the input error z 1,i for the closed-loop system for the following vehicles: where α > β 0 are the coefficients of the linear coupling. It is worth noting that the case of β = 0 indicates that the transformed input error regards only the ego vehicle itself and the leading vehicle, which also means that the optional communication link with the last vehicle mentioned in Subsection 1.1 is not activated. Based on the same idea as in controller design for the leading vehicle, a non-singular terminal sliding manifold is also adopted and designed with new input errors as where z 2,i =ż 1,i is the time derivative of the input error z 1,i . Then, we define an auxiliary variable π i and its derivativeπ i as Thus, we have Then, the time derivative of the sliding manifold for following vehicles iṡ Taking steps that are similar to those described in Subsection 2.2, we can obtain the time derivative of the sliding manifold: Therefore, the equivalent control law for the sliding manifold of the following vehicles, designed by assuming thatṡ i = 0 is The constant rate reaching law is also chosen as the nonlinear switch part of the control law, which is where K i > 0 is a parameter greater than the norm bound of the total matched disturbance, satisfying the condition K i d i . Thus, the control law is designed as the summation of the equivalent control law in Eq. (40) and nonlinear switch part Eq. (41) as u c,i = u eq,i + u sw,i . (42)

Proof:
Like the proof in Subsection 2.2.1, a Lyapunov candidate function is chosen as and the time derivative of Eq. (43) iṡ By substituting Eq. (39) into Eq. (44), the time derivative of the Lyapunov candidate function can be changed tȯ and by substituting the designed control law in Eq. (42) into Eq. (45), the time derivative of Eq. (43) can be obtained asV where z wi−1 2,i > 0 for ∀z 2,i = 0, owing to the odd integers p i and q i , and sgn(s i )s i = |s i | > 0 for ∀s i = 0. Thuṡ V i < 0, the system is stable and can reach the sliding manifold s i = 0 in a finite time.
End of Proof.

Parameter Adaptation for Following Vehicles
Similarly, parameter adaptation is designed for the following vehicles. Using the same definition of parameter uncertainties as in Eq. (22), the new Lyapunov candidate function is designed as We consider the time derivative of the sliding manifold with parameter uncertainties: Then, the designed control law with parameter uncertainties is Similarly, note that Δṁ i = −ṁ i , Δθ i = −θ i , Δμ θ,i = −μ θ,i , and Δμ m,i = −μ m,i , and substituting them into Eq. (51), we obtaiṅ Thus, the adaptive law is designed aṡ Then, the time derivative of the Lyapunov candidate function becomeṡ which guarantees the stability of the system.

Numerical Simulation
In this section, numerical simulations are conducted to illustrate the effectiveness of the developed control schemes. A vehicular platoon with seven CAVs was configured. The parameters of the simulated vehicles were obtained from Ref. [14], with m i = 1 607 kg, c i = 0.414, f i = 236.229, T i = 0.25 s, and τ i = 0.3 s. The acceleration and jerk of the vehicles were bounded within ±2 m/s 2 and ±4 m/s 3 , respectively. The undulate road profile, target velocity, and target acceleration trajectories for the vehicular platoon are indicated in Fig. 3. We selected the vehicle position and speed with random errors as the initial state for the simulation to test the proposed adaptive NTSMC controller in a more general condition. The initial state of the leading vehicle was set as (x 1,1 , x 2,1 ) = (300, 14). The states of the other six following vehicles were set as (x 1,2 , x 2,2 ) = (250, 13. The simulation and control cycles were both 0.1 s and the numerical integrations in the simulation were computed by the Runge-Kutta method. The parameters of the adaptive NTSMC were set to p i = 15, q i = 13, and k 0,i = −0.1 for the leading vehicle and k 0,i = −0.5 for the following vehicles. These are based on the requirements for p i and q i to be positive odd integers and the sliding manifold reaching time t r given from Ref. [25],     presented with respect to their position owing to the adopted D-B policy. Stabilized by the proposed adaptive NTSMC controllers, different initial velocity errors converge after 1 500 m, as indicated in Fig. 4(b). All vehicle trajectories remained the same in the space domain. Figure 4(c) indicates that the time gaps tended to be consistent with the predefined value, which is reflected by a set of parallel time profiles after the position at 1 000 m. Because different controllers were adopted in the leading and following vehicles, the tracking performances described by the state deviations and sliding manifolds are separately presented in Figs. 5 and 6 in the time domain, presenting the convergence properties of the proposed controllers. The initial speed error of the leading vehicle converged to a zero-state within 30 s, which is faster than the converging process for the following vehicle position errors because the modified position errors of the following vehicles are not only based on ego error, but also considering the error between the leading, preceding, and last vehicles. The sliding manifolds of the following vehicles converged to the zero-state after 100 s.
The control command inputs for all vehicles are displayed in Fig. 7. It can be observed that the command inputs are smooth and no clear collisions or chattering phenomena occurred. The control input of a vehicle is a time-delay similar to its preceding vehicle after approximately the first 30 s of the stabilizing process, which consists of the D-B policy adoption for the target profile generation. Thus, it can be perceived that the realized controllers can potentially be used in practice.

Comparison Experiment
Because studies based on SMC for D-B policy are rare, a scenario with a constant target speed and zero acceleration was adopted for the simulation comparison. In a constant target speed setting, the reference speed and desired inner vehicle distance for the following vehicles exhibit the same behavior under both the CS and D-B policies.
The CSMC method proposed in Ref. [11] was chosen as the comparative following vehicle controller. In this comparison scenario, the constant target speed was set to 16 m/s and the total simulation time was 500 s. Other than these two modifications, the initial state of each vehicle and other configurations remained unchanged, as described in Section 3. The desired inner vehicle distance was set to 80 m in the CS policy, which is equivalent to the 5 s time gap in the D-B policy.
In addition to ST, the maximum speed overshoot (MSO), the integrals of time multiplied by the absolute value of error (ITAEs) for the vehicle speed and acceleration which are defined as    Figs. 10 and 11, respectively, it can be seen that with the same initial states, the proposed controller required approximately 44.73% less time to stabilize the vehicle platoon. The speed overshoot was 11.20% less than that of the CSMC controller, and the maximum steady-state inner vehicle distance deviation was virtually zero for the proposed controller, whereas the CSMC controller had a maximum steady-state error of approximately 1.94% of the desired distance. The control command inputs using the proposed method were smooth and exhibited no clear chattering phenomena. The five MOE indices listed in Table 1 indicate the performance of the proposed controller.

Conclusion
The CACC control problem in a vehicular platoon was studied. Two new CACC controllers based on the robust adaptive NTSMC approach were proposed for the leading and following vehicles. The D-B policy was adopted to ensure the same target speed tracking in the space domain and target time headway for all vehicles in the platoon. Different control objectives, which were the given speed and acceleration profile tracking for the leading vehicle, and proper distance and headway maintenance for the following vehicles were designed. By considering the engine dynamics modeled as the FOPDT process, a three-order nonlinear vehicle longitudinal dynamics model was adopted. To compensate and eliminate the influence of pure command delay, an SP consisting of nominal vehicle models with and without time delay was connected in parallel in the control loop. A nonlinear non-singular terminal sliding manifold was designed for the leading vehicle based on its speed and acceleration errors, and the control and parameter adaptive laws were derived by Lyapunov functions that ensured the inner stability and vehicle velocity tracking performance. Using the predecessor-leader following and optional last-to-ego vehicle information topology, the string stability was guaranteed in a similar nonlinear sliding manifold for the following vehicles through a linear coupled position error. Finally, numerical simulation results of a seven-CAV platoon and comparison experiments with a CSMC controller confirmed the effectiveness and outperformance of the proposed approach.
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