2.1 Lane-changing decision-making models

Given the critical role of lane-changing decision-making in autonomous driving systems, a substantial body of research has been devoted to modeling and understanding lane-changing behavior. Existing approaches can be broadly categorized into rule-based, data-driven, and game-theoretic paradigms, each offering distinct advantages and limitations in terms of interpretability, adaptability, and interaction modeling.

Early studies primarily relied on rule-based formulations, where lane-changing decisions are governed by predefined logical conditions. Representative work by Gipps [9] established a foundational framework for modeling lane-selection priorities in multilane traffic environments, while subsequent studies further decomposed the lane-changing process into multiple stages, including motivation generation, target lane selection, gap acceptance, and execution [10]. With the increasing availability of real-world trajectory datasets, such as NGSIM, later works incorporated empirical validation into rule-based models [11] and enhanced decision-making by integrating safety and efficiency considerations through artificial potential field formulations [12]. Despite their computational efficiency and interpretability, these approaches generally treat surrounding vehicles as passive entities, limiting their ability to capture interactive and dynamic traffic behaviors.

In contrast, data-driven approaches, particularly those based on neural networks, have demonstrated strong capability in capturing complex and nonlinear lane-changing behaviors directly from data. Early studies applied neural networks to predict lane-changing decisions in simplified scenarios [13], while subsequent work improved prediction accuracy by incorporating asymmetric lane-changing characteristics and traffic features [14]. More recent advances employ deep learning architectures, such as recurrent neural networks combined with driver behavior models, to better capture temporal dependencies and contextual interactions [15]. However, these approaches are primarily designed for behavior prediction rather than decision-making and control, and their black-box nature limits interpretability in safety-critical applications.

Rule-based approaches are among the earliest methods for modeling lane-changing behavior. These methods typically rely on predefined rules or heuristics derived from traffic regulations or empirical driving data. For example, gap acceptance models and safety distance rules are commonly used to determine whether a lane change is feasible. Although rule-based methods are computationally efficient and easy to implement, they often lack flexibility and struggle to capture complex interactions among multiple vehicles in dynamic traffic environments [8,16].

Optimization-based approaches formulate lane-changing as an optimal control or trajectory planning problem, where an objective function—such as minimizing travel time, energy consumption, or tracking error—is optimized subject to vehicle dynamics and safety constraints. These methods can generate smooth and dynamically feasible trajectories and have been widely applied in autonomous driving systems. However, most optimization-based approaches assume passive surrounding vehicles and therefore have limited capability in explicitly modeling interactive behaviors among multiple agents [17].

To explicitly account for interactions among vehicles, game-theoretic approaches have gained increasing attention. By modeling vehicles as rational agents engaged in strategic interactions, these methods provide a principled framework for capturing competitive and cooperative behaviors in traffic systems. Building upon classical equilibrium concepts, such as Wardrop equilibrium [18] and its relation to Nash equilibrium, game-theoretic models have been developed to describe lane-changing behavior under interactive environments. In mandatory lane-changing scenarios, where conflicts between vehicles are inevitable, these models are particularly effective in representing negotiation processes between lane-changing and surrounding vehicles. For instance, non-cooperative game models have been proposed to capture driver perception, judgment, and behavioral characteristics [19], while subsequent studies incorporated connected vehicle information into payoff functions to enable dynamic decision-making [20]. In addition, recent research has explored integrated frameworks that jointly optimize lane-changing decisions and motion planning within a prediction horizon [21], as well as models that explicitly incorporate conflict risks and roadway factors into utility design. Furthermore, earlier studies have also investigated the characteristics of mandatory lane-changing behavior using game-theoretic formulations, emphasizing the role of conflict time in utility design [22].

Despite these advances, several challenges remain. Existing rule-based and data-driven approaches often lack explicit modeling of vehicle interactions, while many game-theoretic models primarily focus on decision-level analysis without tightly integrating trajectory planning and control. Moreover, the coupling between decision-making and control under dynamic and interactive environments remains insufficiently addressed. These limitations motivate the development of unified frameworks capable of simultaneously achieving interaction-aware decision-making and robust trajectory execution in complex mandatory lane-changing scenarios.

2.2 Trajectory tracking control

In the field of vehicle trajectory tracking control, sliding mode control (SMC) has been widely adopted due to its structural simplicity, inherent robustness to model uncertainties, and ease of implementation. By explicitly coupling lateral displacement and heading angle errors in Cartesian coordinates, SMC-based approaches are capable of achieving high-precision trajectory tracking performance. Recent studies have further enhanced the robustness of SMC frameworks through the integration of higher-order sliding mode techniques and disturbance observers, which effectively attenuate the influence of external disturbances and measurement noise under uncertain operating conditions [23,24].

Despite these advantages, conventional SMC schemes are prone to chattering phenomena, which may degrade control accuracy and adversely affect ride comfort. To address this limitation, a range of improved SMC strategies have been developed from different perspectives. These efforts include redesigning sliding surfaces and incorporating state estimation mechanisms to improve control smoothness and reduce chattering effects [25], as well as integrating SMC with advanced control frameworks to enhance multi-objective tracking capability and convergence performance [26]. In addition, terminal sliding mode formulations have been explored to achieve finite-time convergence while jointly considering lateral and heading tracking errors, demonstrating improved tracking accuracy and robustness [27].

Overall, existing studies have significantly advanced the performance of SMC-based trajectory tracking control in terms of robustness and accuracy. However, most approaches primarily focus on controller design and performance improvement, while the integration of SMC with higher-level decision-making and interaction-aware control strategies remains insufficiently explored, particularly in complex and dynamic traffic environments.

2.3 Motivation and contribution of this paper

Most existing studies on lane-changing decision-making model autonomous vehicles as isolated agents, treating surrounding vehicles as passive moving obstacles and thereby neglecting the intrinsic characteristics of traffic flow and the dynamic interactions among vehicles. However, lane-changing is inherently an interactive and adaptive process, where surrounding vehicles continuously respond to the maneuver of the subject vehicle. Such interactions not only influence the feasibility and safety of the planned trajectory, but also induce deviations between the desired and actual vehicle states, which in turn affect subsequent decision-making and control actions. The lack of an integrated framework that simultaneously captures interaction-aware decision-making and trajectory evolution limits the applicability of existing approaches in complex and dynamic traffic environments.

To address this limitation, this paper proposes a unified lane-changing decision and trajectory planning framework based on stage game theory. Specifically, payoff functions are constructed to characterize the strategic interactions between the autonomous vehicle and the following vehicle in the target lane. By discretizing the lane-changing process into multiple decision stages, the proposed framework enables the autonomous vehicle to dynamically update its driving strategy by jointly considering the surrounding traffic conditions and the interactive feedback from competing vehicles. As a result, both the lane-changing decision and the desired trajectory can be adaptively adjusted in real time, enhancing the interaction awareness and robustness of the planning process.

In terms of trajectory tracking control, existing methods predominantly focus on steady-state tracking accuracy, with insufficient attention to transient performance. However, during lane-changing maneuvers, the intensified interactions among vehicles significantly elevate collision risks, making large transient tracking errors unacceptable from a safety-critical perspective. To this end, a preset performance–based control framework is developed to explicitly regulate both transient and steady-state behaviors. A composite tracking error, integrating lateral displacement and heading angle deviations, is defined and constrained within a prescribed performance boundary. The constrained error dynamics are subsequently transformed into an equivalent unconstrained form, upon which a sliding mode controller is designed to ensure robust, smooth, and safety-critical trajectory tracking throughout the entire lane-changing process.

3.1 The vicinity of signalized intersections

As a critical node in urban road networks, the operational efficiency of signalized intersections has a significant influence on travelers’ experience and overall traffic performance. Owing to the high concentration of vehicle conflicts and the complexity of control and decision-making in the vicinity of signalized intersections, these areas are particularly prone to reduced traffic efficiency and elevated safety risks [28].

In practice, such issues are not solely determined by the geometric and control characteristics of signalized intersections, but are also closely related to heterogeneity in drivers’ driving styles and operational objectives [29,30]. When vehicles approach the guide lane, drivers must simultaneously account for signal phase transitions and the motion states of adjacent vehicles. Inadequate handling of these competing demands may lead to indecision between proceeding and stopping, a phenomenon commonly referred to as the dilemma zone problem. To address this challenge, recent studies have extended the research scope beyond the guide lane itself and defined the guide lane together with its adjacent upstream and downstream regions as the vicinity of signalized intersections (VSI) [31,32].

In the VSI scenario illustrated in Fig 1, an autonomous vehicle is required to execute a right-turn maneuver at the intersection. To complete this task, the vehicle must first perform a lane change from its initial lane to the target lane. However, this lane-changing maneuver is subject to spatial constraints imposed by the guide lane. Specifically, the subject vehicle (SV) must complete the lane change before reaching position L3 in order to satisfy the intended driving task; otherwise, the vehicle must either stop and wait or continue straight toward the next intersection. To accurately characterize lane-changing behavior in the VSI, the control region is partitioned into three subareas: a lane-change execution area, a lane-change prohibition area, and the guide lane. The lane-change execution area specifies the region within which the vehicle is required to initiate the lane-changing maneuver, whereas the lane-change prohibition area denotes regions where lane-changing conditions are no longer satisfied and the maneuver must be abandoned. It should be noted that, according to the dynamic trajectory planning framework introduced in Section III, the boundaries of these regions are not fixed but vary dynamically with the vehicle’s speed and position.

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Fig 1. Vehicle lane change in the vicinity of traffic signals.

https://doi.org/10.1371/journal.pone.0350209.g001

3.2 Lane-changing decision at cyber layer and trajectory tracking control model at physical layer

When the subject vehicle (SV) is required to execute a right-turn maneuver at the current intersection, it must first perform a lane change to reach the target lane. However, this maneuver inevitably encroaches on the right-of-way of the following vehicle in the target lane, thereby disturbing its normal driving behavior and inducing a reactive response. As a result, the lane-changing process gives rise to pronounced conflicts of interest between the two vehicles.

From a decision-making perspective, this interaction reflects a trade-off among safety, efficiency, and comfort. Specifically, safety is associated with maintaining sufficient inter-vehicle distance and avoiding collision risk, efficiency relates to completing the lane change within a limited time window to meet route requirements, and comfort is reflected in smooth acceleration and deceleration profiles. These competing objectives are inherently coupled, as aggressive maneuvers may improve efficiency but compromise safety and comfort, while overly conservative actions may ensure safety at the expense of efficiency.

Accordingly, this paper focuses on modeling the interactive game relationship between the SV and the following vehicle, where the payoff function is designed to explicitly capture the balance among safety, efficiency, and comfort under dynamic interaction conditions.

As illustrated in Fig 2, the subject vehicle SV performs a lane change from Lane 1 to Lane 2. At a given time instant, the lateral position of SV is depicted in Fig 2. Under this configuration, the strategy set of the subject vehicle SV is denoted by , while the strategy set of the following vehicle in the target lane, denoted as LV, is represented by , as defined below.

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Fig 2. Cubic polynomial trajectory.

https://doi.org/10.1371/journal.pone.0350209.g002

Then, the game payoff matrix of vehicles SV and LV is obtained as shown in Table 1, where denotes the payoff of vehicle SV when SV selects strategy S_j and vehicle LV selects strategy L_i, and U_LV(i, j) represents the payoff of vehicle LV under the same strategy combination.

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Table 1. Payoff matrix.

https://doi.org/10.1371/journal.pone.0350209.t001

Furthermore, in the vicinity of signalized intersections, the primary objective of the vehicle’s lane-changing maneuver is to ensure successful completion of the intended route. Failure to complete the lane change within the available control region incurs a substantially higher cost to the vehicle. Accordingly, these considerations are incorporated into the formulation of the payoff function for the lane-changing vehicle, which is defined as follows:

(1)

where denotes the weighting coefficient characterizing the driving style of vehicle SV, with larger values indicating more aggressive driving behavior. U_sa1 represents the safety benefit of vehicle SV, while U_sp1 denotes the space benefit of vehicle SV. The corresponding formulations of U_sa1 and U_sp1 are given as follows:

(2)

(3)

where d denotes the lateral distance between the SV and the LV, W represents the vehicle width, a is the predefined safe lateral distance between the two vehicles, and u₁ denotes the safety factor associated with vehicle SV, which is computed as follows:

(4)

where T_h denotes the headway between the SV and LV, T_e1 represents the expected headway of vehicle SV, which is defined as where T₁ denotes the initial headway between the FV and the SV. The corresponding formulations of T_h and T₁ are given as follows:

(5)

(6)

where P_SV and P_LV denote the longitudinal positions of the subject vehicle (SV) and the following vehicle (LV), respectively, and v_SV and v_LV represent their corresponding longitudinal velocities.

As the vehicle approaches the lane-change prohibition region, its urgency to complete the lane-changing maneuver increases. To successfully accomplish the lane-change task under increasingly restrictive spatial conditions, the driving style of the lane-changing vehicle is required to become more aggressive. Even under extreme conditions, as long as the safety constraints are satisfied, the lane-change maneuver should be executed to fulfill the task requirements. Accordingly, the driving style coefficient of the lane-changing vehicle is defined as follows:

(7)

where denotes the default driving style coefficient, is a positive constant, and d_x represents the longitudinal distance between the current position of vehicle SV and position L2. When the subject vehicle SV performs a lane-changing maneuver, it inevitably encroaches on the driving space of the following vehicle LV and may introduce potential collision risks. Consequently, driving space and safety are the primary concerns for vehicle LV. The payoff function of vehicle LV is therefore defined as [33]:

(8)

where denotes the weighting coefficient characterizing the driving style of the LV driver, with larger values corresponding to more aggressive driving behavior.U_sa2 denotes the safety benefit of the following vehicle (LV), and U_sp2 represents the space benefit of vehicle LV. The corresponding formulations of U_sa2 and U_sp2 are defined as follows:

(9)

(10)

where d denotes the lateral distance between the subject vehicle (SV) and the following vehicle (LV), w represents the vehicle width, a is the predefined safe lateral distance between the two vehicles, and T_h denotes the headway of vehicle SV relative to vehicle LV. The parameter u₂ represents the safety factor associated with vehicle LV, which is computed as follows:

(11)

where T_e2 denotes the expected headway of vehicle LV, defined as . In solving the game model, each participant selects a strategy to optimize its respective objective. LV in the target lane, the Stackelberg equilibrium is employed. The resulting optimal strategies are given by

(12)

where and denote the strategy sets of the subject vehicle (SV) and the following vehicle (LV) in the target lane, respectively. is the optimal strategy of vehicle SV (the leader), and is the optimal response strategy of vehicle LV (the follower) to a given . U_SV and U_LV represent the decision payoffs of vehicles SV and LV, respectively.

The best response strategy set of vehicle LV corresponding to a given strategy is defined as

(13)

Accordingly, the optimal strategies of vehicles SV and LV can be obtained based on the above formulation.

3.3 Optimal lane-changing trajectory

In this paper, a cubic polynomial curve is adopted to generate the lane-changing trajectory of the vehicle. Such trajectories exhibit second-order smoothness [34], ensuring continuity of both position and velocity throughout the lane-changing process. Compared with higher-order polynomial trajectories, the cubic polynomial requires fewer parameters, thereby reducing the amount of information needed for trajectory generation. The cubic polynomial trajectory is formulated as follows:

(12)

where x(t) and y(t) denote the longitudinal and lateral positions of the vehicle as functions of time t, respectively, and a_i and b_i (i = 0,1,2,3) are the coefficients of the cubic polynomial.

The cubic polynomial lane-changing trajectory of the autonomous vehicle and the corresponding trajectory curvature are illustrated in Fig 2.

To enable dynamic trajectory planning for vehicle lane-changing maneuvers, a moving coordinate system is adopted in this paper. Whenever trajectory planning is performed, a local coordinate frame is established with the current vehicle position taken as the origin. Furthermore, to simplify the trajectory generation process, it is assumed that the vehicle maintains a constant longitudinal speed throughout the lane-changing maneuver. Accordingly, the following boundary conditions are imposed:

(13)

where T denotes the time required for the vehicle to complete the lane-changing maneuver from the current position, and represents the initial heading angle of the vehicle at the j-th trajectory planning step.

(14)

where denotes the initial heading angle of the vehicle at the current time step, which can be obtained from the vehicle state, and Y represents the lateral distance from the local coordinate origin to the centerline of the target lane.

The lateral distance Y from the vehicle to the centerline of the target lane can be obtained from onboard sensors. The longitudinal distance X and the lane-changing duration T are treated as decision variables; therefore, the values of X and T jointly determine the final lane-changing trajectory.

In practical lane-changing scenarios, lane-changing efficiency and passenger comfort are two critical criteria for evaluating trajectory quality. However, these objectives are often conflicting and cannot be simultaneously optimized. As the vehicle approaches the lane-change prohibition region, the urgency to complete the maneuver increases, and priority should be given to minimizing the lane-changing duration rather than maximizing passenger comfort. Accordingly, the following cost function is defined:

(15)

where J denotes the cost function, is the maximum curvature of the lane-changing trajectory, T represents the duration of the lane change, and v is the vehicle speed. The weighting coefficient reflects the urgency of the lane change and is defined as a function of the distance to the lane-change boundary:

(16)

where d_x denotes the longitudinal distance between the current vehicle position and position L2, d₂ is a constant representing the boundary of the lane-change region, is a positive constant, and is the initial value of the weighting coefficient. As illustrated in Fig 3, the curvature of the lane-changing trajectory reaches its maximum value at the completion of the maneuver. Accordingly, the maximum curvature of the lane-changing trajectory is defined as

(17)

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Fig 3. Simulation diagram.

https://doi.org/10.1371/journal.pone.0350209.g003

The solution of the cost function defined in (15) can be formulated as a nonlinear optimization problem. In this paper, the artificial fish swarm optimization (AFSO) algorithm is employed to solve the resulting optimization problem. By applying the AFSO algorithm, the optimal value of the cost function J^* can be obtained, along with the corresponding optimal lane-changing distance X^* and lane-changing duration T^*.

3.4 Lane changing area calculation

Based on the previous analysis, the vehicle lane-changing region in the vicinity of the signalized intersection has been identified. To further characterize the vehicle’s operational state within this region, the lane-changing area is further subdivided according to the maximum lateral acceleration generated along the lane-changing trajectory.

By adopting a cubic polynomial lane-changing trajectory, the lateral acceleration of the vehicle can be expressed as

(18)

where v denotes the vehicle speed, and x(t) and y(t) represent the longitudinal and lateral positions of the vehicle at time t, respectively.

When the vehicle completes the lane-changing maneuver, the trajectory curvature reaches its maximum value; consequently, the maximum lateral acceleration is attained at this time instant and is given by

(19)

where T denotes the time required for the vehicle to complete the lane-changing maneuver. Considering the relationship between vehicle speed and displacement, the following expression can be further derived:

(20)

where X denotes the lane-changing distance of the vehicle.

To determine the length of d₂ in Fig 1, the maximum lateral acceleration that can be tolerated by passengers without inducing vehicle rollover is denoted by . By jointly considering (19) and (20), the minimum feasible lane-changing distance can be obtained as , which corresponds to the length of d₂. Accordingly, the minimum lane-changing duration is denoted as .

By jointly considering lane-changing efficiency and passenger comfort, the optimal lane-changing distance X^* is obtained using the artificial fish swarm optimization algorithm. As a result, the length of region d₁ can be computed as

(21)

The length of the guide lane d₃ is assumed to be known a priori.

3.5 Prescribed performance control

To facilitate controller design under performance constraints, the constrained tracking error is transformed into an equivalent unconstrained form through a predefined error transformation function. Intuitively, this transformation maps the original error, which is required to evolve within a prescribed performance boundary, into a new variable defined over an unconstrained domain. As a result, the difficulty of directly handling state constraints is avoided, and standard control design techniques can be applied more conveniently.

Moreover, this transformation ensures that as long as the transformed error remains bounded, the original tracking error will automatically satisfy the predefined performance constraints, including both transient and steady-state requirements. This provides a systematic way to enforce safety-related bounds on tracking errors while maintaining analytical tractability. Compared with directly constrained control methods, the proposed transformation simplifies the control design process and enhances robustness, making it particularly suitable for safety-critical and highly interactive lane-changing scenarios.

(22)

where m denotes the vehicle mass, is the front-wheel steering angle, v_x and v_y represent the longitudinal and lateral velocity components in the vehicle coordinate system, respectively, denotes the yaw rate, and I_z is the moment of inertia of the vehicle about the vertical axis. a and b denote the distances from the vehicle centroid to the front axle and rear axle, respectively, represents the vehicle sideslip angle at the centroid, k₁ and k₂ are the cornering stiffness coefficients of the front and rear wheels, respectively.

In the trajectory tracking process of autonomous vehicles, it is necessary to regulate both the lateral tracking error and the heading angle error. Accordingly, the lane-changing trajectory tracking error model is defined as follows:

(23)

where e_y denotes the lateral tracking error, is the heading angle error, represents the actual heading angle of the vehicle, and denotes the desired heading angle along the reference trajectory.

The core idea of prescribed performance control (PPC) is to confine the system tracking error within a predefined performance envelope, such that the tracking error converges to a prescribed residual set with bounded overshoot and guaranteed minimum convergence rate. To alleviate the complexity introduced by the performance constraints in controller design, an error transformation function is introduced to map the constrained nonlinear system into an unconstrained form:

(24)

where e(t) denotes the initial tracking error of the controlled system, is the transformed error variable, and is the overshoot suppression parameter.

By substituting the lateral tracking error and heading angle error into (24), the corresponding equivalent error dynamics can be obtained as follows:

(25)

Recognizing that regulating only the front-wheel steering angle is insufficient to simultaneously suppress both lateral deviation and heading angle error, a more comprehensive control strategy is adopted that jointly accounts for these two error components. Building upon this concept, this study further introduces a unified error metric that integrates the equivalent transformations of both heading angle error and lateral deviation, thereby enabling more effective and coordinated error regulation.

(26)

where e_m denotes the comprehensive tracking error, and x_n and x_m are positive proportional coefficients corresponding to the lateral deviation and the heading angle error, respectively.

Assuming that the heading angle deviation remains sufficiently small during vehicle motion, the small-angle approximation can be applied, yielding

(27)

Owing to its favorable characteristics, including fast dynamic response and strong robustness against external disturbances and model uncertainties, sliding mode control is adopted to design the trajectory tracking controller. The sliding surface is constructed as

(28)

where S denotes the sliding surface, and c₁, c₂, and c₃ are positive design coefficients of the sliding surface. e_m represents the comprehensive transformed tracking error of the system.

In this paper, a general reaching law is adopted, which is expressed as

(29)

where is a positive constant, f(S) > 0, and .

To mitigate the chattering phenomenon inherent in sliding mode control, the saturation function is employed to replace the discontinuous sign function:

(30)

where denotes the boundary layer thickness. By combining the above formulations, a prescribed-performance-based sliding mode controller is obtained as

(31)

(32)

(33)

(34)

(35)

(36)