Introduction

Research by the World Health Organization (WHO) reveals that road traffic accidents kill approximately 1.3 million people globally each year and result in $277 billion in economic losses1. Data from the U.S. National Highway Transportation Safety Administration (NHTSA) further indicates that 94% (± 2.2%) of these accidents are caused by driver error2. Autonomous driving technology can significantly reduce crashes attributed to human error by minimizing the need for driver maneuvers3,4. Its widespread adoption is expected to enhance vehicle safety, thereby reducing economic losses and alleviating traffic congestion5,6,7. The core objective of autonomous driving technology is to ensure vehicle safety, which must be embedded throughout the entire system. This includes key components such as environment sensing8, trajectory prediction9, decision-making, trajectory planning10, and control11. Current trajectory planning methods primarily rely on predefined safety rules as fixed constraints for different scenarios, but there is limited research on adaptively adapting these rules to dynamic traffic conditions. While excessive safety rules are necessary for extreme operating conditions, they can restrict the adaptability of autonomous vehicles under normal driving conditions, making it difficult to optimize decisions for specific situations. Thus, the ability to adapt safety rules to diverse traffic scenarios is critical for balancing safety and efficiency in autonomous driving.

Existing trajectory planning methods can be classified into two main categories: learning-based and model-based methods12. Reinforcement learning-based autonomous driving trajectory planning methods, such as deep reinforcement learning13, DRMCTS14, and continuous dominance learning, have demonstrated high efficiency and adaptability in dynamic obstacle avoidance and complex scenarios, but their core limitations include low interpretability and high data dependency. In recent years, online trajectory planning methods based on deep learning have been proposed to tackle sudden moving obstacles: these methods model the temporal features of vehicle motion sequences through LSTM and combine the attention mechanism to dynamically focus on the position and motion trend of sudden obstacles, enabling real-time environment sensing and trajectory adjustment15. Its advantages include the ability to process sensor data end-to-end, reducing the reliance on predefined rules or offline training, and the attention weights that can partially reveal the decision basis. Furthermore, a study proposed an adaptive strategy integrating deep reinforcement learning and model predictive control16, which generates a guided speed profile through three stages of coarse planning, smoothing, and optimization. This realizes coordinated control of lane-changing and car-following behaviors in signalized intersection scenarios to improve energy efficiency and traffic efficiency. However, these learning-based methods are still limited by the risk of attention allocation bias in complex scenarios. Although their interpretability surpasses that of pure black-box models, it remains sensitive to data noise and relies on high-quality labeled time-series data for training.

Model-based trajectory planning methods ensure that AVs make operational decisions complying with legal and safety standards in diverse traffic environments by defining safety rules and driving strategies in advance. Therefore, rule-based trajectory planning has been extensively studied in recent years. For example, Morsali17 proposed a model predictive control-based approach to search for the optimal feasible path. The method considers rollover constraints and enables the planning of optimal driving trajectories, focusing on vehicle rollover risk constraints. To address stability risks in complex driving scenarios, Nie18 proposed a novel lane-changing motion planning strategy. The strategy incorporates vehicle dynamics constraints and introduces a stability manipulation envelope and a safety index for the safe lane-changing region. A fifth-degree polynomial serves as a trajectory cluster generator to generate ideal driving trajectories, yet it overlooks longitudinal travel safety. Furthermore, a related study developed a hierarchical flexible eco-cruising framework19, which employs the Dijkstra algorithm to plan efficient lane sequences from a global perspective and optimizes local speed using trigonometric speed profiles, aiming to reduce driving costs under varying traffic flows. However, the risk assessment of this framework only focuses on interactions with preceding vehicles, failing to cover multi-dimensional risks at the vehicle dynamics level, such as rollover and skidding. Additionally, for overtaking scenarios at signalized intersections, the optimal driving lane is planned using the Markov decision process, and the speed trajectory is optimized in combination with Pontryagin’s minimum principle to mitigate the restriction of preceding vehicles on energy efficiency20. However, it tends to result in an insufficient safety margin in scenarios with multiple concurrent risks. Therefore, to address longitudinal collision safety, Zheng21 developed a trajectory planning algorithm based on quadratic Bessel curves and hazardous potential fields. The algorithm evaluates the collision risk of candidate paths by introducing a potential field function, incorporates this as a constraint into the sequential quadratic programming algorithm, balances driving comfort and safety for candidate paths, and ultimately plans ideal driving trajectories. To comprehensively incorporate potential driving risks, Yu22 proposed a safe trajectory planning and tracking algorithm for AVs. The algorithm accounts for obstacle avoidance constraints, curb constraints, and non-convex input constraints defined by wheel sideslip angle and input boundedness. By employing a control barrier function, it effectively handles state and input constraints to generate safe driving trajectories. Although the above studies have considered safety risks such as collision, rollover, and slip in trajectory planning, not all driving scenarios require strict compliance with these safety rules. Imposing excessive safety constraints on a vehicle traveling on normal roads may limit its maneuverability and reduce adaptability. Therefore, adaptive adjustment of safety rules for different driving environments is essential.

To adapt to dangerous and complex traffic scenarios, some researchers incorporate potential safety risks into a risk indicator, which are then defined as safety constraints. However, challenges remain regarding the accuracy of risk indicators and their adaptability to scenarios. For example, Wang23 proposed an automatic lane-changing trajectory planning method based on the elastic soft constraints of the safety domain. By designing elastic soft constraint indicators in the safety domain, this method reduces the sensitivity of the dynamic trajectory planning system while ensuring safety. To enable continuous and accurate assessment of potential surrounding threats, Zhaojie Wang24 developed a motion planning method based on the risk field. This approach uses a static risk matrix as a safety constraint to design a path cost function and evaluate candidate paths generated through longitudinal and lateral sampling, thereby identifying feasible target paths. However, it faces challenges in computational complexity and limited generalization. To address these issues, Zhang25 proposed a motion planning and tracking method based on an improved artificial potential field. The method first constructs an improved artificial potential field hazard assessment module to characterize driving risks in the Frenet-Serret coordinate system. Treating the influence of other traffic participants as additional safety constraints, it generates lane-changing paths using gradient descent based on the potential energy distributions of target obstacles and road boundaries. However, it remains insufficient in responsiveness to dynamic obstacles. To improve the accuracy of safety boundary determination, Wu26 proposed a lane-change trajectory planning method with time-varying safety margins. This method employs a simplified gray prediction model to estimate the driving states of surrounding vehicles and introduces time-varying safety margins during trajectory planning to ensure safe vehicle maneuvers.

The trajectory planning strategies described in previous studies define potential safety risks in scenarios as constraints and subsequently determine geometric paths from multiple candidate trajectories to generate optimal trajectory profiles. These methods advantageously consider potential safety risks in any driving scenario, thereby ensuring vehicle operational safety even in complex, unstructured environments. However, limitations exist regarding their generalizability, particularly in safe, risk-free scenarios, where predefined safety constraints may prove redundant, thereby reducing driving efficiency. To address rule redundancy, achieve safe and efficient trajectory planning, and enhance the adaptability of trajectory planning to diverse scenarios, we propose a dynamic risk-information-driven rule-adaptive trajectory planning method, as illustrated in Fig. 1.

Fig. 1
figure 1

The framework of lane-change trajectory planning with rule-adaptive.

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In contrast to previous studies, this study makes the following key contributions:

  1. (1)

    To address the difficulty in balancing safety and efficiency during lane-changing of autonomous vehicles, a novel rule-adaptive trajectory planning framework is proposed in this study. Safety constraints are dynamically adjusted based on real-time risk assessment results to break through the limitations of fixed rules, thereby adapting to diverse driving scenarios.

  2. (2)

    During the planning stage, collision risk, rollover risk, and slip risk are comprehensively considered, and rule constraints are adaptively selected to generate candidate trajectories.

  3. (3)

    The optimal trajectory is selected through comprehensive decision-making based on the cost function and total time. After candidate trajectories are preliminarily screened by the total cost function, total time is used to make the final decision on the trajectory.

The remainder of this paper is structured as follows: Sect. Vehicle dynamics model and risk indices introduces the safety risk indicators of the developed vehicle dynamics model; Sect. Vehicle dynamics model presents the proposed trajectory planning strategy; Sect. Collision risk assessment discusses the simulation results; and Sect. Rollover risk assessment concludes the paper.

Vehicle dynamics model and risk indices

Evaluating collision and instability risks based on vehicle dynamics models is a critical step in the trajectory-planning process of AVs. This process involves an accurate calculation of the probability of collision between the vehicle and other traffic participants in the roadway, as well as assessing the vehicle’s inherent stability risks during travel.

Vehicle dynamics model

The vehicle’s center of mass is assumed fixed, neglecting the influence of suspension nonlinear characteristics on vehicle dynamics. A three-way-coupled dynamic model as shown in Fig. 2 is used to describe the lateral, yaw, and roll dynamics of AVs. The system dynamics are given by Eq. (1):

$$\left\{ {\begin{array}{*{20}c} {\sum {F_{y} = ma_{y} + (l_{f} m_{f} - l_{r} m_{r} )\dot{r} + m_{s} h\ddot{\phi }} } \\ {\sum {M_{z} = I_{zz} \dot{r} + (l_{f} m_{f} - l_{r} m_{r} )a_{y} \, } } \\ {\sum {M_{x} = I_{xx} \ddot{\phi } + M_{k} + M_{c} - m_{s} gh\phi \, } } \\ \end{array} } \right.$$
(1)
Fig. 2
figure 2

Vehicle dynamic model.

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with

$$\left\{ {\begin{array}{*{20}c} {\sum {F_{y} = F_{yf} + F_{yr} \, } } \\ {m = m_{s} + m_{f} + m_{r} } \\ {a_{y} = \dot{v}_{y} + v_{x} r \, } \\ \end{array} } \right.\;\left\{ {\begin{array}{*{20}c} {F_{sjl} = \frac{1}{2}m_{sj} g - k_{j} \Delta x,j = f,r \, } \\ {F_{sjr} = \frac{1}{2}m_{sj} g + k_{j} \Delta x,j = f,r} \\ \end{array} } \right.$$
$$\left\{ {\begin{array}{*{20}c} {\sum {M_{z} } = l_{f} F_{yf} - l_{r} F_{yr} \, } \\ {\sum {M_{x} } = ma_{y} h \, } \\ {M_{i} = M_{if} + M_{ir} {, }i = k,c \, } \\ {M_{kj} = (F_{sjr} - F_{sjl} )\frac{L}{2},j = f,r \, } \\ {M_{cj} = \frac{L}{2}(c_{jc} + c_{jr} )\dot{x},j = f,r \, } \\ \end{array} } \right.\;\left\{ {\begin{array}{*{20}c} {I_{xx} = I_{R} + m_{s} h^{2} \, } \\ {I_{zz} = I_{zzR} + I_{zzf} + I_{zzr} + l_{f}^{2} m_{f} + l_{r}^{2} m_{r} } \\ {\Delta x = \frac{L}{2}\sin \phi \approx \frac{\phi L}{2} \, } \\ {\dot{x} = \frac{{L\dot{\phi }}}{2}\cos \phi \approx \frac{{L\dot{\phi }}}{2} \, } \\ \end{array} } \right.$$

where m is the mass of the vehicle, ms is the sprung mass. msj represents the sprung mass of the front and rear axles, mf and mr are the unsprung mass of the front and rear axles , respectively. lf and lr are the distances from the vehicle’s center of gravity to the front and rear axles, respectively. L is the distance between the left and right suspensions, and h is the vertical distance from the center of gravity to the lateral inclination center. cjc and cjr are the damping coefficients for the compression and rebound of the front and rear axle suspensions, respectively. kj is the stiffness coefficient of the front/rear axle suspensions. \(\Delta x\) is the vertical deformation of the spring, and \(\dot{x}\) is the rate of change of damping displacement. vx and vy are the longitudinal and lateral vehicle velocities, respectively. ay is the lateral acceleration, and g is the gravitational acceleration. Fyf and Fyr denote the lateral forces on the front and rear axle tires, respectively. Fsfl and Fsfr represent the spring forces of the front axle’s left and right suspensions, while Fsrl and Fsrr denote those of the rear axle’s left and right suspensions. Mk, Mkf, Mkr, Mc, Mcf, and Mcr correspond to the resistant moments of the springs, front axle springs, rear axle springs, suspension damping, front axle damping, and rear axle damping, respectively. \(\phi\) is the lateral tilt angle, and r is the yaw angular velocity. Ixx and Izz denote the vehicle’s lateral tilt and yaw moments of inertia, respectively. IzzR, Izzf, and Izzr are the yaw moments of inertia for ms, mf, and mr, respectively. IR represents the lateral tilt moment of inertia of ms about the yaw axis.

From the “Tire: Lateral Force: 215/55 R17” module in CarSim, the relationship between tire lateral force and lateral deflection angle is determined, thereby enabling the derivation of the linearized lateral deflection stiffness of the used tires, as shown in Fig. 3.

Fig. 3
figure 3

Nonlinear tire force.

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As shown in Fig. 3, the relationship between tire slip angle and lateral force approximates linear when the slip angle is less than 4 degrees. Thus, we assume they are linearly related under this condition. Within this range, the tire’s lateral force can be expressed as a linear function of the lateral slip angle, defined by the product of the tire lateral stiffness and the lateral slip angle. By restricting the lateral slip angle within this range, linearization is achieved, and the linearization is described in Eq. (2):

$$\left\{ {\begin{array}{*{20}c} {F_{yf} = C_{\alpha f} \alpha_{f} } \\ {F_{yr} = C_{\alpha r} \alpha_{r} \, } \\ \end{array} } \right.$$
(2)

The linearized model described above is applicable within the linear region depicted in Fig. 3. When the tire’s lateral slip angle falls outside this linear range, it indicates a slip risk during the slip risk assessment.

The lateral slip angles for the vehicle’s front and rear axles are expressed in Eq. (3):

$$\left\{ {\begin{array}{*{20}c} {\alpha_{f} = \arctan (\frac{{v_{y} + l_{f} r}}{{v_{x} }}) - \delta_{f} \approx \frac{{v_{y} + l_{f} r}}{{v_{x} }} - \delta_{f} } \\ {\alpha_{r} = \arctan (\frac{{v_{y} - l_{r} r}}{{v_{x} }}) \approx \frac{{v_{y} - l_{r} r}}{{v_{x} }} \, } \\ \end{array} } \right.$$
(3)

where \(\delta_{f}\) denotes the front wheel steering angle.

By combining Eqs. (1–3) and incorporating the state variables \(\xi = \left[ {\begin{array}{*{20}c} {v_{y} } & r & {\dot{\phi }} & \phi \\ \end{array} } \right]^{T}\), the vehicle equivalent dynamic model is derived as:

$$\begin{aligned} \left[ {\begin{array}{*{20}c} m & {l_{f} m_{f} - l_{r} m_{r} } & {m_{s} h} & 0 \\ {l_{f} m_{f} - l_{r} m_{r} } & {I_{{zz}} } & 0 & 0 \\ { - mh} & 0 & {I_{{xx}} } & {\frac{{L^{2} c}}{4}} \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\dot{v}_{y} } \\ {\dot{r}} \\ {\ddot{\phi }} \\ {\dot{\phi }} \\ \end{array} } \right] = \\ & \left[ {\begin{array}{*{20}c} {\frac{{C_{{\alpha f}} + C_{{\alpha r}} }}{{v_{x} }}} & {\frac{{C_{{\alpha f}} l_{f} + C_{{\alpha r}} l_{r} - mv_{x} ^{2} }}{{v_{x} }}} & 0 & 0 \\ {\frac{{C_{{\alpha f}} l_{f} - C_{{\alpha r}} l_{r} }}{{v_{x} }}} & {\frac{{(l_{r} m_{r} - l_{f} m_{f} )v_{x} ^{2} + C_{{\alpha f}} l_{f} ^{2} + C_{{\alpha r}} l_{r} ^{2} }}{{v_{x} }}} & 0 & 0 \\ 0 & {mhv_{x} } & 0 & {m_{s} gh - \frac{{L^{2} k}}{2}} \\ 0 & 0 & 1 & 0 \\ \end{array} } \right] \\ & \left[ {\begin{array}{*{20}c} {v_{y} } \\ r \\ {\dot{\phi }} \\ \phi \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} { - C_{{\alpha f}} } \\ { - C_{{\alpha f}} l_{f} } \\ 0 \\ 0 \\ \end{array} } \right]\delta _{f} \\ \end{aligned}$$
(4)

Collision risk assessment

In this section, a collision risk assessment function for lane-changing scenarios is developed using time to forward (TTF) to accurately assess collision risk.

The ego-vehicle’s position and velocity components are expressed as:

$$\left\{ {\begin{array}{*{20}c} {v_{ex}^{t} = v_{e}^{t} \cos \psi_{e}^{t} } \\ {v_{ey}^{t} = v_{e}^{t} \sin \psi_{e}^{t} } \\ \end{array} } \right.\;\left\{ {\begin{array}{*{20}c} {x_{e}^{t} = x_{s}^{t} + \Delta x_{m}^{t} } \\ {y_{e}^{t} = y_{s}^{t} + \Delta y_{m}^{t} } \\ \end{array} } \right.$$
(5)

where \(x_{e}^{t}\) and \(y_{e}^{t}\) denote the x- and y-coordinates of the ego-vehicle in the road coordinate system at time t. \(x_{s}^{t}\) and \(y_{s}^{t}\) represent the x- and y-coordinates of the obstacle vehicle in the same road coordinate system at time t. \(\Delta x_{m}^{t}\) and \(\Delta y_{m}^{t}\) define the relative position of the ego-vehicle to the obstacle vehicle in the target lane. \(\psi_{e}^{t}\) and \(\Delta \psi^{t}\) refer to the heading angle and relative heading angle of the ego-vehicle, respectively. \(v_{e}^{t}\) and \(v_{s}^{t}\) indicate the velocities of the ego-vehicle and the obstacle vehicle in the target lane.

The speed component of the obstacle vehicle in the target lane is expressed as:

$$\left\{ {\begin{array}{*{20}c} {v_{sx}^{t} = v_{s}^{t} \cos \psi_{s}^{t} } \\ {v_{sy}^{t} = v_{s}^{t} \sin \psi_{s}^{t} } \\ \end{array} } \right.$$
(6)

Then the TTF is expressed as:

$$TTF(t) = t{\prime} - t = \frac{{\Delta y_{q}^{t} + (w_{s} /2)}}{{v_{ey}^{t} }} = \frac{{\Delta y_{q}^{t} + (w_{s} /2)}}{{v_{e}^{t} \sin \psi_{e}^{t} }}$$
(7)

where \(\Delta x_{q}^{t}\) and \(\Delta y_{q}^{t}\) denote the relative longitudinal and lateral positions of the ego-vehicle and the front obstacle vehicle.

Thus, the collision risk assessment function is defined as:

$$R_{cx} = \frac{{d_{safe} }}{{d_{es} }}$$
(8)

The collision risk indicator quantifies risk using Rcx. When Rcx < 1, the current distance meets safety requirements; otherwise, there is a collision risk. This indicator intuitively reflects the degree of collision risk through the ratio of safe distance to actual distance, facilitating its inclusion in trajectory optimization constraints.

\(d_{safe}\) and \(d_{es}\) can be calculated from Eq. (9):

$$\left\{ {\begin{array}{*{20}c} {d_{es} = \Delta x_{q}^{t} + b_{s} + \left( {v_{ex}^{t} - v_{s}^{t} } \right)TTF} \\ {d_{safe} = v_{e}^{t} t + \frac{{(v_{e}^{t^{\prime}} )^{2} }}{{2a_{e}^{t} }} - \frac{{(v_{sx}^{t^{\prime}} )^{2} }}{{2a_{s}^{t} }} \, } \\ \end{array} } \right. \, \left\{ {\begin{array}{*{20}c} {v_{s}^{t^{\prime}} = v_{s}^{t} } \\ {v_{ex}^{t^{\prime}} = v_{ex}^{t} } \\ \end{array} } \right.$$
(9)

The calculation of des integrates the relative longitudinal position \(\Delta x_{q}^{t}\), obstacle vehicle length bs, and the product of the relative speed between the two vehicles and TTF, to reflect the distance where an actual collision may occur.The calculation of dsafe includes the ego-vehicle speed term and maximum braking distance, ensuring the vehicle has sufficient safe distance to avoid collisions.

Rollover risk assessment

The rollover risk is determined by whether the tires leave the ground; thus, the LTR serves as an indicator for rollover risk. By defining the condition where one wheel leaves the ground as the critical rollover point, LTR is defined as:

$$LTR = \frac{{F_{zo} - F_{zi} }}{{F_{zo} + F_{zi} }}$$
(10)

where \(F_{zo}\) and \(F_{zi}\) denote the vertical forces of the inner and outer wheels, respectively.

When the vehicle tends to roll over, the axle load shifts from the inner to the outer wheel. By taking moments about the roll center, the moment balance equation is established, from which the difference in vertical forces between the inner and outer wheels can be derived as:

$$(F_{zo} - F_{zi} )\frac{W}{2} = M_{k} + M_{c} + \left( {\sum {F_{y} } } \right)h_{R}$$
(11)

where W is the track width, and \(h_{R}\) is the height of the roll center.

Based on Eqs. (10–11), the following can be derived:

$$LTR = \frac{{2\left[ {M_{k} + M_{c} + \left( {\sum {F_{y} } } \right)h_{R} } \right]}}{mgW}$$
(12)

From Eq. (12), it follows that the vehicle is free from rollover risk when the LTR is within the range of [− 1, 1]. Therefore, by restricting the LTR value to the range of [− 1, 1], the rollover stability risk assessment index is defined as:

$$- 1 < \frac{{2\left[ {M_{k} + M_{c} + \left( {\sum {F_{y} } } \right)h_{R} } \right]}}{mgW} < 1$$
(13)

As shown in Eq. (1), the values of \(M_{k}\), \(M_{c}\) and \(\sum {F_{y} }\) are given by Eq. (14):

$$\left\{ {\begin{array}{*{20}c} {\sum {F_{y} = ma_{y} + (l_{f} m_{f} - l_{r} m_{r} )\dot{r} + m_{s} h\ddot{\phi } \, } } \\ {M_{k} = M_{kf} + M_{kr} = (k_{f} + k_{r} )\Delta xL = kL^{2} \phi /2 \, } \\ {M_{c} = M_{cf} + M_{cr} = \frac{L}{2}(c_{fc} + c_{fr} + c_{rc} + c_{rr} )\dot{x} = cL^{2} \dot{\phi }/4} \\ \end{array} } \right.$$
(14)

Substituting \(M_{k}\), \(M_{c}\) and \(\sum {F_{y} }\) into Eq. (13) yields the further rearranged index.

$${\rm M}_{1} \xi - {\rm M}_{2} < H_{1} \dot{\xi } < {\rm M}_{1} \xi + {\rm M}_{2}$$
(15)

where

\({\varvec{H}}_{1} = \left[ {\begin{array}{*{20}c} 1 & {\frac{{l_{f} m_{f} - l_{r} m_{r} }}{m}} & {\frac{{m_{s} }}{m}h} & {\frac{{L^{2} c}}{{4mh_{R} }}} \\ \end{array} } \right]\), \(\user2{\rm M}_{1} = \left[ {\begin{array}{*{20}c} 0 & { - v_{x} } & 0 & { - \frac{{L^{2} k}}{{2mh_{R} }}} \\ \end{array} } \right]\), \({\rm M}_{2} = \frac{gW}{{2h_{R} }}\).

Slip risk assessment

The risk of vehicle slip mainly results from the lateral slip angle generated by tire deformation. By constraining the tire lateral slip angle within the linear region \(\left[ {\begin{array}{*{20}c} { - \alpha_{t} } & {\alpha_{t} } \\ \end{array} } \right]\) during dynamic modeling, the tire meets the linearization condition. It is assumed that the rear wheel lateral slip angle operating within this linear region is necessary to meet the slip stability condition, as shown in Eq. (16):

$$- \alpha_{t} \le \alpha_{r} = \frac{{v_{y} - l_{r} r}}{{v_{x} }} \le \alpha_{t}$$
(16)

Under steady-state equilibrium conditions, the yaw moment must be balanced, i.e., the moment generated by the front-wheel lateral force equals that by the rear-wheel lateral force, as shown in the following equation.

$$F_{yf} l_{f} = F_{yr} l_{r}$$
(17)

Based on Eqs. (2) and (17), the following can be derived:

$$C_{\alpha f} \alpha_{f} l_{f} = C_{\alpha r} \alpha_{r} l_{r}$$
(18)

In dynamic stability constraints, the front-wheel cornering moment is generally required not to exceed the rear-wheel one, as shown in Eq. (19).

$$- C_{\alpha r} \alpha_{r} l_{r} /l_{f} \le C_{\alpha f} \alpha_{f} \le C_{\alpha r} \alpha_{r} l_{r} /l_{f}$$
(19)

Combining Eqs. (1) and (19), we can obtained as:

$$\sum {F_{y} = F_{yf} + F_{yr} \, } = C_{\alpha f} \alpha_{f} + C_{\alpha r} \alpha_{r} = ma_{y} + (l_{f} m_{f} - l_{r} m_{r} )\dot{r} + m_{s} h\ddot{\phi }$$
(20)

The slip stability region of the vehicle is obtained as:

$$- C_{\alpha r} \alpha_{r} (1 + l_{r} /l_{f} )/m \le \dot{v}_{y} + \frac{{l_{f} m_{f} - l_{r} m_{r} }}{m}\dot{r} + \frac{{m_{s} }}{m}h\ddot{\phi } + v_{x} r \le C_{\alpha r} \alpha_{r} (1 + l_{r} /l_{f} )/m$$
(21)

Equations (16) and (21) define four slip stability boundaries. If vy and r lie within these boundaries, the vehicle is considered slip-stable. From a vehicle handling perspective, when vy and r are within this region, the vehicle exhibits neither severe understeer nor oversteer.

Incorporating the state variables \({\varvec{\xi}}\), Eq. (21) is expressed in matrix form, and the slip risk assessment metric is formulated as:

$${\varvec{M}}_{3} {\varvec{\xi}} - M_{4} \le {\varvec{H}}_{2} \dot{\user2{\xi }} \le {\varvec{M}}_{3} {\varvec{\xi}} + M_{4}$$
(22)

where

\({\varvec{H}}_{2} = \left[ {\begin{array}{*{20}c} {\frac{1}{{v_{x} }}} & {\frac{{l_{f} m_{f} - l_{r} m_{r} }}{{mv_{x} }}} & {\frac{{m_{s} h}}{{mv_{x} }}} & 0 \\ \end{array} } \right]\), \({\varvec{M}}_{3} = \left[ {\begin{array}{*{20}c} 0 & { - 1} & 0 & 0 \\ \end{array} } \right]\), \(M_{4} = \frac{{C_{\alpha r} \alpha_{t} (1 + l_{r} /l_{f} )}}{{mv_{x} }}\).

When the vehicle operates within the slip-stable region, there is no risk of slipping. However, this does not imply that the vehicle will slip outside this region: when the vehicle briefly exceeds the slip-stable region, the tire lateral slip angle no longer satisfies the linearization conditions, causing it to operate in the nonlinear region. Under such conditions, the vehicle can still remain controllable within a short period.

Rules-adaptive trajectory planning

When no safety risks are detected, the predefined safety constraints are not activated. Once a specific safety risk is identified along the planned trajectory, the rule-adaptive trajectory planning module is triggered to re-plan the driving path, incorporating the corresponding risk function as new constraints into the safety framework. This adaptive constraint-setting approach, based on real-time safety risk assessment, ensures that safety constraints adapt to diverse complex driving scenarios and avoids rule redundancy. The rule-adaptive trajectory planning method comprises three components: lane-change fifth-degree polynomial generation, trapezoidal speed planning, and trajectory optimization selection, detailed as follows:

Lane-change trajectory planning

To ensure vehicle travel safety, a fifth-degree polynomial is used to model the single lane-change path:

$$y = D^{T} X$$
(23)
$$D = \left[ {\begin{array}{*{20}c} {d_{0} } & {d_{1} } & {d_{2} } & {d_{3} } & {d_{4} } & {d_{5} } \\ \end{array} } \right]^{T} \;X = \left[ {\begin{array}{*{20}c} 1 & x & {x^{2} } & {x^{3} } & {x^{4} } & {x^{5} } \\ \end{array} } \right]^{T}$$
(24)

where x and y denote the longitudinal and lateral coordinates of the lane-change path, and di(i = 0–5) represent the coefficients of the fifth-degree polynomial.

Boundary constraints for the fifth-degree polynomial are defined as follows:

$$\left\{ {\begin{array}{*{20}c} {y(x_{0} ) = 0} \\ {\dot{y}(x_{0} ) = 0} \\ {\kappa (x_{0} ) = 0} \\ \end{array} } \right. \, \left\{ {\begin{array}{*{20}c} {y(x_{n} ) = y_{n} } \\ {\dot{y}(x_{n} ) = 0 \, } \\ {\kappa (x_{n} ) = 0 \, } \\ \end{array} } \right. \, \kappa = \frac{{d^{2} y/dx^{2} }}{{[1 + (dy/dx)^{2} ]^{3/2} }}$$
(25)

where x0 and y0 are the longitudinal and lateral coordinates of the vehicle’s center of mass at the start time. xn and yn represent the longitudinal and lateral endpoint coordinates of the collision-free trajectory, and \(\kappa\) denotes the path curvature.

Substituting the boundary conditions into Eq. (23) yields:

$$\left[ {\begin{array}{*{20}c} 0 \\ {y_{n} } \\ 0 \\ 0 \\ 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & {x_{0} } & {x_{0}^{2} } & {x_{0}^{3} } & {x_{0}^{4} } & {x_{0}^{5} } \\ 1 & {x_{n} } & {x_{n}^{2} } & {x_{n}^{3} } & {x_{n}^{4} } & {x_{n}^{5} } \\ 0 & 1 & {2x_{0} } & {3x_{0}^{2} } & {4x_{0}^{3} } & {5x_{0}^{4} } \\ 0 & 1 & {2x_{n} } & {3x_{n}^{2} } & {4x_{n}^{3} } & {5x_{n}^{4} } \\ 0 & 0 & 2 & {6x_{0} } & {12x_{0}^{2} } & {20x_{0}^{3} } \\ 0 & 0 & 2 & {6x_{n} } & {12x_{n}^{2} } & {20x_{n}^{3} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ {10y_{n} /x_{n}^{3} } \\ { - 15y_{n} /x_{n}^{4} } \\ {6y_{n} /x_{n}^{5} } \\ \end{array} } \right]$$
(26)

This in turn yields the collision-avoidance trajectory expression:

$$y(x) = 10y_{n} \left( {\frac{x}{{x_{n} }}} \right)^{3} - 15y_{n} \left( {\frac{x}{{x_{n} }}} \right)^{4} + 6y_{n} \left( {\frac{x}{{x_{n} }}} \right)^{5}$$
(27)

Trapezoidal velocity planning

When a specific safety risk is identified in the planned trajectory, corresponding safety constraints are activated to determine the lane-changing trajectory. Trapezoidal velocity planning is employed, where the optimal constant velocity Vm,j for lane-changing is defined as the maximum velocity enabling the vehicle to safely complete the maneuver along path j. Vm,j is determined for each candidate path by solving the optimization problem in Eq. (28).

$$\begin{gathered} \min J_{j} = - V_{m,j} \hfill \\ {\text{s}}{\text{.t}}{. }\dot{\user2{\xi }} = f\left( {{\varvec{\xi}},\delta_{f} } \right), \hfill \\ or \, \left\{ {\begin{array}{*{20}c} {V_{y,\min ,j} \le V_{y,j} \le V_{y,\max ,j} } \\ {r_{\min ,j} \le r_{j} \le r_{\max ,j} } \\ \end{array} } \right., \hfill \\ or \, |LTR|_{\max ,j} < 1, \hfill \\ or \, R_{x,j} < 1. \hfill \\ \end{gathered}$$
(28)

where Vy,min,j and Vy,max,j are the minimum and maximum lateral velocities of the candidate trajectory, rmin,j and rmax,j are the minimum and maximum yaw angular velocities, Vy,j and rj boundaries are derived from slip-stable boundaries, \(|LTR|_{\max ,j}\) denotes the maximum allowable LTR value. Furthermore, selecting min Jj = − Vm,j as the optimization objective aims to maximize the constant lane-changing speed Vm,j under safety constraints to improve driving efficiency. The negative sign converts speed maximization into a minimization problem, facilitating solution via gradient descent. Next is system dynamics; the three safety constraints are handling stability boundaries, rollover constraints, and collision constraints. Notably, when rollover, collision, and slip risks coexist, the optimized minimum Vm,j is adopted as the final result. Using the minimum Vm,j as the priority criterion enables collaborative optimization of risk constraints under safety thresholds, ensuring overall safety and efficiency balance in multi-risk coupling scenarios.

Upon detecting a specific safety risk, corresponding constraints are incorporated into the optimization problem of Eq. (28) to determine the optimal constant velocity Vm,j. For multiple detected risks, multiple safety constraints are applied to enable multi-scenario rule-adaptive trajectory planning.

Given the trapezoidal speed profile, constant deceleration adec and acceleration aacc are applied for braking and acceleration phases, respectively. The time and distance for these phases are calculated as:

$$t_{dec,j} = \frac{{V_{m,j} - V_{0} }}{{a_{dec} }},\quad t_{acc,j} = \frac{{V_{0} - V_{m,j} }}{{a_{acc} }}; \, d_{dec,j} = \frac{{V_{m,j}^{2} - V_{0}^{2} }}{{2a_{dec} }}, \, d_{acc,j} = \frac{{V_{0}^{2} - V_{m,j}^{2} }}{{2a_{acc} }}$$
(29)

Excluding initial and final uniform motion segments, the vehicle’s motion is divided into three phases: deceleration, lane-change, and acceleration. The total time can thus be expressed as:

$$t_{total} = t_{dec,j} + t_{lc,j} + t_{acc,j}$$
(30)

where

$$t_{lc,j} = \frac{{x_{f,j} - x_{0} }}{{V_{m,j} }},d_{lc,j} = x_{f,j} - x_{0}$$
(31)

Trajectory selection and optimization strategies

With adaptively preset safety constraints, the vehicle ensures driving safety and scenario adaptability. To plan the final driving trajectory, we must also consider comfort, efficiency, and trajectory deviation indices for further trajectory selection and optimization. This comprehensive consideration ensures that the selected trajectory provides a comfortable ride and efficient driving path while maintaining safety.

Using jerk to formulate the comfort cost for lateral and longitudinal trajectories separately, the comfort cost function is defined as:

$$\cos J_{c} = k_{y} \int_{{t_{0} }}^{{t_{1} }} {\dddot y^{2} (t)} dt + k_{x} \int_{{t_{0} }}^{{t_{1} }} {\dddot x^{2} (t)} dt$$
(32)

where ky and kx are the weight coefficients for the lateral and longitudinal comfort cost functions, respectively.

The efficiency of the planned trajectory is reflected in the speed deviation from the desired speed, and the efficiency function is:

$$\cos J_{e} = k_{e} \int_{{t_{0} }}^{{t_{1} }} {(v_{r} - \sqrt {\dot{y}(t)^{2} + \dot{x}(t)^{2} } )} dt$$
(33)

where ke is the weight coefficient of the efficiency cost function and vr is the desired speed.

The deviation index quantifies the degree of deviation of the planned trajectory from the reference trajectory, with the deviation cost function formulated as:

$$\cos J_{d} = k_{l} \int_{{t_{0} }}^{{t_{1} }} {(y - y_{r} )^{2} } dt + k_{\varphi } \int_{{t_{0} }}^{{t_{1} }} {(\varphi - \varphi_{r} )} dt$$
(34)

where \(k_{l}\) and \(k_{\varphi }\) are the weight coefficients for the lateral position error cost and heading angle error cost, respectively. \(y_{r}\) and \(\varphi_{r}\) represent the desired lateral offset and heading angle of the trajectory, respectively.

Thus, the total cost function, incorporating comfort, efficiency, and deviation indices, is:

$$\cos J = \sum\limits_{n = 1}^{T/dt} {\left[ {\cos J_{c} (n) + \cos J_{e} (n) + \cos J_{d} (n)} \right]}$$
(35)

After constraining trajectories with adaptive rules, the optimal trajectory is obtained by evaluating all feasible trajectories to identify the minimum-cost one. Given that all trajectories meet safety requirements, the most efficient trajectory is selected as the optimal driving path. However, in high-safety-requirement scenarios, the optimal trajectory screened from the trajectory set may not be ideal due to the limited number of feasible options. Thus, the selected trajectories require further optimization to meet higher safety standards. The trajectory optimization model is:

$$\begin{gathered} \min \, \cos J_{j} \hfill \\ {\text{ s}}{\text{.t}}{. }\dot{\user2{\xi }} = f\left( {{\varvec{\xi}},\delta_{f} } \right) \hfill \\ \, V_{y,\min ,j} \le V_{y,j} \le V_{y,\max ,j} \hfill \\ \, r_{\min ,j} \le r_{j} \le r_{\max ,j} \hfill \\ \, |LTR|_{\max ,j} < 1 \hfill \\ \, R_{cx,j} < 1 \hfill \\ \end{gathered}$$
(36)

The flow of the trajectory planning algorithm is shown in Algorithm 1.

Algorithm 1
figure a

Rule adaptive trajectory planning

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Additionally, attention should be paid to the sensitive factors of the proposed trajectory planning method. Risk assessment thresholds affect constraint activation via the dynamic risk assessment module. Safety constraints are triggered when exceeding the thresholds: smaller thresholds result in more conservative decisions, while larger ones lead to more aggressive ones, requiring dynamic adaptation to scenarios. The setting of risk duration thresholds balances decision robustness and response timeliness. Excessively short durations tend to cause false triggers due to noise, while excessively long ones may delay risk handling. In practice, they are set based on vehicle dynamic response characteristics to ensure timely adjustment when risks continuously affect trajectory safety.

Simulation verification and analysis

The proposed trajectory planning method is validated in an obstacle avoidance scenario using the developed Matlab/Simulink-CarSim co-simulation platform. A C-Class Hatchback from the CarSim database is selected as the test vehicle, and the test scenario is configured with an obstacle in the center of the lane, as shown in Fig. 4. Vehicles are more susceptible to safety risks in high-speed scenarios, with the vehicle driven at an initial speed of 120 km/h. Vehicle parameters required for the algorithm are listed in Table 1.

Fig. 4
figure 4

Results of trajectory planning for different end states.

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Table 1 Vehicle parameter description.
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In terms of parameter sources, the parameters listed in Table 1 include vehicle total mass m, wheelbase lf/lr, suspension stiffness kj, and tire cornering stiffness Cαr, all inherent physical properties of the vehicle. These parameters are derived from the technical manuals of the target vehicle manufacturer, accurately reflecting its structural characteristics and dynamic properties. The slip threshold αt = 4° is determined based on the tire linearization model assumption, ensuring model input accuracy.

For parameter adjustment, adaptation is based on vehicle models or scenarios. When switching to other models, model-related parameters must be replaced to match new model characteristics. For different scenarios such as slippery roads, large obstacles, and high-speed driving, relevant parameters require adjustment through theoretical derivation. For example, tire cornering stiffness is reduced on slippery roads, safety distance parameters are adjusted to fit obstacle sizes, and the slip threshold is tightened in high-speed scenarios. Such adjustments ensure risk assessment and trajectory planning align with actual scenarios, and the adjustment logic is grounded in dynamic theory and scenario characteristics.

Simulation settings

To evaluate the effectiveness of the rule-adaptive trajectory planning algorithm, three simulation scenarios are designed to assess how safety constraint adaptation enhances driving efficiency while ensuring safe operation.

Scenario 1: No safety constraints are applied to the trajectory planning, and neither collision nor instability constraints are incorporated into the optimal control problem in Eq. (28). This scenario aims to identify potential safety risks in the unconstrained trajectory and establish a baseline for adaptive safety constraint design.

Scenario 2: Instability constraints are adaptively applied to trajectories exhibiting instability risks in Scenario 1, i.e., instability constraints are dynamically integrated into the optimization problem in Eq. (28). By comparing with Scenario 1, this scenario verifies whether the vehicle’s instability risk is mitigated and if driving efficiency improves compared to fixed instability constraints.

Scenario 3: Collision constraints are adaptively applied to trajectories with collision risks identified in scenario 1. Comparisons with scenario 1 evaluate collision risk reduction under adaptive constraints and assess efficiency improvements relative to fixed collision constraints.

Trajectories planned without constraints are shown in Fig. 5, with corresponding safety risk assessment results in Fig. 6.

Fig. 5
figure 5

Results of trajectory planning for scenarios without preset safety rules.

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Fig. 6
figure 6

Risk assessment results for scenarios without preset safety rules.

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Without preset safe rules

Figure 5 shows the generated candidate trajectories, plotted as curves in the X–Y plane. The color mapping represents vehicle speed along the paths: dark red for higher speeds and dark blue for lower speeds.

For different candidate paths, Vm,j varies from 35 km/h to 120 km/h due to differing deceleration/acceleration profiles, resulting in distinct displacement and time requirements for each trajectory. As shown in Fig. 5, trajectories 1 and 2 require no braking/acceleration for lane change, while trajectories 3, 4, and 5 necessitate deceleration due to shorter lane-change distances, followed by acceleration to normal speed. Details of the candidate trajectories without preset safety constraints are listed in Table 2. Trajectory 3 has the lowest total cost 2.41 × 105, making it the optimal choice based on minimum cost. However, minimizing cost does not guarantee safety, so risk assessment is also necessary for each trajectory.

Table 2 Trajectory cost function values.
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Figure 6 presents instability and collision risk assessments for the five trajectories. The columns represent candidate trajectories, and the rows depict vehicle states (Vy,j, and rj), the rollover indicator LTRj, and the collision risk indicator Rx,j. Blue solid lines show dynamic responses, with red dashed lines denoting safety boundaries. As shown in Fig. 6a and b, for candidate trajectories 1–4, the lateral velocity and the yaw rate are within the corresponding stability boundaries, and trajectory 5 reaches the slip stability boundary. This result indicates that if trajectory 5 is selected as the optimal traveling trajectory for the obstacle-avoidance scenario, the vehicle may experience a slip risk. As shown in Fig. 6c, the rollover indicator LTR reaches the boundaries in trajectories 3, 4, and 5, respectively, indicating that a potential rollover risk may occur if trajectories 3, 4, and 5 are selected as the optimal driving trajectories. As shown in Fig. 6d, trajectories 1 and 2 reach the boundaries in the lane-changing collision indicator Rx, and the collision indicators of trajectories 3, 4, and 5 are limited to the boundaries, which indicates that choosing trajectory 1 or 2 as the optimal one may lead to collision risk.

Therefore, instability constraints are adaptively applied to trajectories 3–5 in scenario 2, and collision constraints are applied to trajectories 1–2 in scenario 3, to assess whether the constraint-adaptive method enhances driving efficiency while ensuring safety.

Preset instability rules scenario

In scenario 1, trajectories 3–5 were found to exhibit instability risks. Therefore, in scenario 2, instability constraints are adaptively assigned to these trajectories to validate the constraint-adaptive approach’s ability to ensure safety.

As shown in Fig. 7, Eq. (28) is used to determine Vm,j for each candidate trajectory, followed by lane-changing trajectory planning with instability constraints applied. Vm,j varies from 70 km/h to 120 km/h, and velocity planning mitigates instability risks, thereby increasing the optimal constant speed Vm,j by 35 km/h compared to the unpreset instability constraint scenario. Trajectories 1–3 require no braking/acceleration for lane change, while Trajectories 4–5 necessitate deceleration for safety, followed by acceleration to normal speed. Details of candidate trajectories with preset instability constraints are listed in Table 2. Trajectory 4 has the lowest total cost (1.77 × 105), making it the optimal choice. Notably, the optimal trajectory under fixed safety constraints may not maximize travel efficiency even if safety is ensured.

Fig. 7
figure 7

Results of trajectory planning for scenarios with preset instability rules.

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Similar to Fig. 6, Fig. 8 shows vehicle states, rollover, and collision metrics for trajectories with adaptive instability constraints. Figure 8a–b: For all trajectories, lateral velocity and yaw rate remain within stability boundaries. Instability risk metrics (Vy, r, and LTR) are reduced from 1.323, 0.303, and 1 in the unpreset scenario to 0.098, 0.127, and 0.679 in the preset scenario, with maximum values further decreasing to 1.225, 0.176, and 0.321.

Fig. 8
figure 8

Risk assessment results of preset instability rule scenarios.

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This confirms that adaptive instability constraints significantly enhance safety for trajectories 3–5, eliminating instability risks. Among trajectories 3–5 (all safety-compliant), trajectory 3 is selected for its highest travel efficiency (2.1 s), compared to trajectory 4 (4.76 s) under fixed constraints, achieving a 55.9% efficiency improvement. Therefore, the constraint-adaptive trajectory planning method optimizes driving efficiency while ensuring safety.

Preset collision rule scenario

In scenario 1, trajectories 1 and 2 were found to pose collision risks. Therefore, in scenario 3, collision constraints are adaptively assigned to these trajectories to validate the constraint-adaptive approach’s safety assurance capability. Simultaneously, trajectory optimization is performed to assess the approach’s efficiency improvement potential.

As shown in Fig. 9, collision constraints are adaptively applied to trajectories 1–2, requiring the vehicle to decelerate during lane change and then accelerate to normal speed. Trajectories 3–5, being collision-risk-free, exhibit consistent Rx values with those in Fig. 8d under collision constraints. Candidate trajectories are evaluated using the designed cost function J, as listed in Table 2 under preset collision constraints. Trajectory 1 has the lowest total cost (2.32 × 105), making it the optimal choice. Notably, the optimal trajectory under fixed collision constraints may not optimize travel efficiency even when safety is ensured.

Fig. 9
figure 9

Results of trajectory planning for scenarios with preset collision rules.

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Figure 10 shows vehicle states, rollover, and collision metrics for trajectories with adaptive preset collision constraints to assess safety risks. As shown in Fig. 10d, collision risk metrics Rx for all trajectories remain within their respective boundaries, indicating no collision risk. Compared with the unpreset collision constraint scenario, Rx values are reduced: the maximum risk value Rx,max for trajectory 1 decreases from 1 in the unpreset scenario to 0.894 in the preset scenario. Under collision constraints, instability risk indicators (Vy, r, and LTR) remain consistent with those in Fig. 8, all within stability boundaries. For trajectories 1–2 (both safety-compliant), trajectory 2 is selected for its higher travel efficiency (3.73 s vs. 5.17 s for trajectory 1), achieving a 27.86% efficiency improvement over fixed constraints.

Fig. 10
figure 10

Risk assessment results of preset collision rule scenarios.

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Conclusion

In this study, a dynamic risk information-driven adaptive lane-changing trajectory planning method for AVs rules is proposed. The method dynamically presets and adjusts safety rules by integrating safety risk assessment information to achieve more efficient and adaptive trajectory planning for driving. The designed collision and instability risk assessment functions are utilized to accurately quantify the safety risks during driving and accordingly dynamically preset reasonable safety rule constraints for trajectory planning. This study shows potential in reducing the need for presetting unnecessary safety rules in non-hazardous scenarios. It may help address the issue of rule redundancy that arises from attributing all potential risks solely to fixed rules. These improvements could enhance the adaptability of safety rules to diverse driving scenarios and contribute to overall driving efficiency.

The proposed trajectory planning method has been validated by a joint simulation platform. The experimental results show that the method of dynamically presetting safety rules not only significantly improves the safety of the vehicle during driving, but also enhances its ability to adapt to changing driving environments. However, the effectiveness of the risk assessment module depends on the predictability of obstacle movement patterns in the environment. In unstructured roads or when obstacles perform sudden maneuvers such as abrupt direction changes and hard braking, the existing risk assessment tends to generate deviations due to the failed prediction of movement patterns, thereby affecting the timeliness and accuracy of safety constraint adjustments. To address this limitation, future research will integrate a trajectory prediction module. By conducting prospective trajectory predictions for traffic participants such as pedestrians and vehicles, the accuracy of risk assessment in unstructured environments and sudden scenarios will be improved, thus enhancing the algorithm’s adaptability to complex dynamic environments.