## Abstract

This study focuses on the lateral safety of autonomous vehicles. The aim is to provide greater safety margins for the motion controller under hazardous conditions by improving the planning methods. The proposed lateral planning method employs a hierarchical structure. First, a graph search method that considers road curvature is presented. This method benefits from the design of jump nodes and can generate a safe corridor in real-time by backtracking. Next, a concise and effective optimization problem is constructed in the Frenet frame. It is designed to optimize the path by incorporating the peak, integral, and differential of curvature into the objective function. By solving this problem using CasADi, the path within the safe corridor is obtained. Finally, the proposed planning method is combined with a model predictive motion controller and tested through simulation and experimentation on a real vehicle platform. This planning method has good scalability, adaptability, and real-time performance. Simulation and experimental results show that this method can help avoid exceeding lateral limits of road surface friction, improve path-following accuracy, and enhance the safety of autonomous vehicles in dangerous conditions.

## 1 Introduction

With the rapid development of autonomous driving, planning plays an increasingly important role in ensuring the safety of autonomous vehicles [1]. The planning task can be divided into lateral planning and speed planning. Lateral planning mainly affects the lateral motion of autonomous vehicles and has a significant impact on the safety of maneuvering. Traditional lateral planning methods mainly focus on generating a smooth and feasible trajectory without considering the avoidance of exceeding the acceleration limit of the vehicle on a complex road environment [2]. However, during driving, the friction provided by the vehicle tires is limited [3]. It is possible to encounter situations where the lateral acceleration of the vehicle becomes too high, causing the vehicle to enter an extreme state and endangering safety during diving in situations such as driving on icy or snowy roads, avoiding obstacles continuously in curves, and U-turning. Therefore, lateral planning methods that consider lateral limits of road surface friction are desired in such scenarios.

To address these issues, one kind of approach is to build the vehicle dynamics model, add constraints imposed by obstacles, and solve for the motion of the chassis in real-time, such as Refs. [4] and [5]. These approaches simultaneously accomplish both planning and control tasks. However, for parallel development and system interchangeability, most autonomous vehicle systems adopt a modular design approach, where the planning module and control module are typically separated [6]. Planners are primarily concerned with higher-level tasks such as obstacle avoidance and maintaining within the boundaries of the road. The planning module generates a continuous, smooth, and trackable trajectory according to the environment information and vehicle state. And, controllers' main objective is to diligently track a continuously evolving target trajectory. The control module uses the trajectory output by planning the planning module as a target to calculate actuators' commands for the chassis. This pipeline provides a clear division of responsibilities, making it suitable for engineering applications and preventing error propagation. It has been adopted in autonomous driving systems, such as Apollo [7] and Autoware [8]. This work is also conducted within the scope of this pipeline.

Nevertheless, there are several factors contributing to the computational complexity of the planning problem: (1) The planning task is an NP-hard problem [9]; (2) In order to avoid obstacles, the planning module usually has a longer prediction horizon [10]; (3) The planning module and control module usually run asynchronously, so the trajectory output by the planning module needs to be accurate throughout the entire process not only around the current state. Hence, directly employing vehicle dynamics models typically cannot achieve real-time planning online. Planning approaches that directly utilized the vehicle dynamics model were typically reserved for offline optimal path computation [11]. In most online planning methods, simplified system models were adopted to construct the optimization problem [10,12,13]. While they can generate continuous and smooth collision free trajectories, they did not take into account the need to simultaneously avoid approaching the lateral limits imposed by road friction. Dixit et al. [14] employ a potential field approach to avoid hazards arising from obstacles and the ego state, but it is limited to specific maneuvering tasks. Curvature is an important trajectory feature related to vehicle lateral dynamics, which affects the lateral acceleration and yaw rate of vehicles [15]. Its integral was minimized at the planning level to improve the performance for autonomous racing [16], but it did not extend to avoid danger on a structured road.

This work aims to develop a lateral planning method that can guide autonomous vehicles to avoid collision with obstacles and exceeding friction limits. It should effectively keep lateral tracking errors small even under extreme conditions to guarantee safety, and able to be scalable to different scenarios on structured roads. To achieve these objectives, this work presents the following contributions:

Based on the Frenet frame, a hierarchical structure for lateral planning was proposed, which consisted of a novel safety corridor searching process and an optimization process both considering the influence of the lateral trend of the road. The planner focuses on ensuring the path is trackable and its force requirements stay within the motion performance boundaries. The controller achieves coordinating dimensions of vehicle dynamics and generates specific motions of actuators.

Different from previous works, the peak, integral, and differential of the curvature related to the lateral acceleration and yaw rate of the vehicle were introduced as the optimization objective at the planning level for the first time. It enhanced the safety of autonomous vehicles by helping them avoid exceeding friction limits and did not rely on specific vehicle dynamics parameters.

The rest of the paper is organized as follows: Section 2 provides an overview of related work in lateral planning for autonomous vehicles. Section 3 introduces the proposed lateral planning method in detail, including the safe corridor search method, and the optimization problem. Section 4 presents simulation and experimental results to demonstrate the effectiveness of the proposed method. Finally, Sec. 5 concludes the paper with a summary of the contributions and future work.

The definitions of parameters and variables utilized in this work are shown in nomenclature section.

## 2 Related Works

Lateral planning methods can be classified into three categories: search-based methods, optimization-based methods, and learning-based methods. Each of these methods has its unique strengths and limitations. The selection of the most suitable method depends on the specific application and requirements.

### 2.1 Search-Based Lateral Planning.

Search-based methods typically use graph search algorithms to find a feasible path in a search space. Classic search-based methods include the A* and rapidly-exploring random tree algorithm families. They provided a foundation for the further development of search-based planning methods in autonomous driving. To accommodate scenarios with high speeds and near maneuvering limits, Stahl et al. [17] developed a planning method based on a graph search method with a high update rate and a far planning horizon, which allowed complex overtaking maneuvers. Zhang et al. [18] adopted a hierarchical framework, first dividing the search space to generate the local destination and then performing a heuristic search on the local space to obtain the path like the human driver's behavior. Lim et al. [19] combined the advantages of sampling-based methods in terms of flexible cost functions and simplified search spaces while the vehicle dynamics were considered in the lower planner.

Search-based algorithms are versatile in adapting to environments with complex obstacle constraints. However, within certain computational limits, the optimality of graph search and sampling methods is constrained by the resolution of the grid map and the randomness of sampling. Therefore, it is often necessary to integrate optimization methods with search-based algorithms to solve planning problems in autonomous driving effectively.

### 2.2 Optimization-Based Lateral Planning.

Optimization-based methods formulate the lateral planning problem as an optimization problem and use mathematical optimization techniques to find the optimal path. Some of them used parametric curves to model the path. The fifth-order Bezier curve was utilized in Ref. [20] to generate the path for lane changing of autonomous vehicles. Trajectories based on polynomial function were optimized with the elastic soft constraint to improve safety and comfort [21]. Others optimized discrete sequences of waypoints or vehicle states. In Ref. [22], considering the longitudinal load transfer, the trajectory optimization problem under the limit conditions was modeled as a quadratic programming problem. Autonomous racing also needs to deal with driving near the performance limit [23]. This strategy is worth migrating to driving in dangerous situations where the lateral acceleration is approaching the limit.

Optimization-based planning methods can incorporate various constraints and generate optimal trajectories subject to vehicle dynamics. However, the lateral planning problem in autonomous driving is nonconvex and nonlinear, making it difficult to construct the optimization problem and find a solution.

### 2.3 Learning-Based Lateral Planning.

Learning-based methods use machine learning techniques to learn a policy for generating paths based on input data. Specific methods involve deep learning (DL), imitation learning (IL), and reinforcement learning (RL). Huang et al. [24] used DL to predict the trajectories of other vehicles in the traffic environment and used the predicted results of the ego vehicle as the initial planning. Chen et al. [25] proposed an IL framework that efficiently learns the driving policy in urban scenarios using offline connected driving data, incorporating a safety controller to ensure safety during testing. Huang et al. [26] utilized IL to navigate the autonomous vehicle in unstructured environments. For the decision-making and planning modules, the most applied method is RL. It continues to improve in terms of efficiency and performance [27]. Based on a sampling planning framework, Wang et al. [28] used an RL agent to update the reward function for sampled actions and trained in the simulation environment to accomplish behavior and motion planning. In Ref. [29], an RL agent was developed to generate collision-free paths with the assistance of potential fields.

Despite the great potential of learning-based planning methods, ensuring the safety of autonomous vehicles in corner cases remains a significant challenge for researchers.

## 3 Lateral Planning With Minimum Curvature Peak

### 3.1 Hierarchical Framework.

In the perception-planning-control architecture of autonomous driving systems, the planning module typically focuses on the high-level task and acts as the upstream module of control. It generates collision-free and smooth trajectories based on the surrounding environment. The control module, which is downstream from the planning module, outputs actuator commands according to the current vehicle's motion state to track the trajectories generated by the planning module. In this work, to enable autonomous driving through extreme conditions, the planning module is responsible for minimizing the motion performance requirements. It makes the trajectory easier to follow and provides the control module with a greater margin of safety. The control module coordinates various vehicle dynamics dimensions to track the trajectory and determines the final motions of the chassis actuators.

Lateral acceleration and yaw rate are crucial elements of vehicle dynamics that significantly influence motion performance limits [30]. They are inherently linked to a fundamental trajectory characteristic: curvature [16]. Therefore, curvature plays a critical role in ensuring the safety and comfort of the autonomous driving system.

The expression of curvature is complicated in the Cartesian coordinate system. Additionally, it is challenging to mathematically describe safety corridors in the Cartesian coordinate system. These factors hinder the modeling and solving of optimization problems. Therefore, this work adopted the Frenet frame, which reduced the complexity of modeling and the nonlinearity of the optimization problem. Frenet frame can effectively improve the adaptability of motion planning to curves on structured roads [31,32].

Different types of planning methods have various advantages. Sampling-based methods and searching-based methods are good at finding suboptimal feasible solutions in nonconvex spaces. Optimization-based methods can output smooth and local optimal solutions considering vehicle dynamics. This work adopted a hierarchical structure for lateral planning to assemble the advantages of the above two methods. The first layer of the hierarchy was responsible for searching the safe corridor while the second stage refined the path and added more detail. By dividing lateral planning into these smaller subtasks, each layer could focus on a specific aspect of the problem, allowing for more efficient use of computational resources and better handling of nonconvex. More detail of the hierarchical structure can be found in Fig. 1.

The central idea of the proposed hierarchical planning framework is to avoid exceeding the friction limits by minimizing the curvature of the path. It was referred to as minimum curvature planning (MinCP) for the sake of brevity.

### 3.2 Safe Corridor Search in Frenet Frame.

The safety corridor provides a general model to describe geometric constraints for the planning module to guarantee safety in different scenarios. Similar concepts were utilized in Refs. [7,33]. In this work, the safety corridor is established using a graph search method as Fig. 2.

*s*,

*l*) were as

Additionally, if a decision module was available, the OGM could be masked to enable the autonomous vehicle to execute the behavior of “turn left,” “turn right,” or “keep lane.”

*k*of $\kappa refk$ represented the

*k*-th waypoint on the reference line. Its

*s*should be near $isk\u2009rs$. If the gaps of the reference line were uniform, It can also be calculated by

*is*. The heuristic function of the grid with OGM indexes of (

*is, il*) was as follows:

The nonuniform cost functions described above can assist autonomous vehicles in selecting a better safe corridor when driving on curves. When the search reached the goal, the last searched node was returned as the termination node. Otherwise, the node with the maximum *is* in the closed list was chosen as the termination node.

Finally, the safe corridor is constructed by backtracking from the termination node. The safe corridor was described by the differential element method. In each differential element corresponding to one waypoint on the reference line, *bl* and *br* are boundaries of the extended safe area where vehicles can drive safely. To simplify the description, the clipping of grid nodes outside the OGM is omitted in the pseudo-code. The algorithm of the backtracking process is as follows.

Input: Termination Node n, Reference Line$R$_{t} |

Output: Safe Corridor $C$ |

1 Initialize $C$ |

2 p ← the last point on $R$_{last} |

3 backtrack (n$C$)_{t}, |

4 Reverse $C$ |

5 The process of backtrack (n, $C$) is as follows: |

6 ifn is not Nonethen |

7 n← parentNode(n)_{p} |

8 ifn_{p} is not Nonethen |

9 s ← ($isn$_{rear}p + 1) × r_{s} |

10 else |

11 s ← 0_{rear} |

12 end |

13 p ← previous point of _{ref}p on $R$_{last} |

14 whiles of p ≤_{ref}s_{rear}do |

15 ifn_{p} is not Nonethen |

16 $br\u2190[min(iln,ilnp)\u22120.5]\xd7rl+Lmin$ |

17 $bl\u2190[min(iln,ilnp)\u22120.5]\xd7rl+Lmin$ |

18 else |

19 $\u2003\u2009br\u2190(iln\u22120.5)\xd7rl+Lmin$ |

20 $\u2003bl\u2190(iln+0.5)\xd7rl+Lmin$ |

21 end |

22 $C$ ←$C$ ∪ < b >_{r}, b_{l} |

23 p ← _{last}p_{ref} |

24 p ← previous point of _{ref}p on $R$_{last} |

25 end |

26 backtrack (n$C$)_{p}, |

27 end |

Input: Termination Node n, Reference Line$R$_{t} |

Output: Safe Corridor $C$ |

1 Initialize $C$ |

2 p ← the last point on $R$_{last} |

3 backtrack (n$C$)_{t}, |

4 Reverse $C$ |

5 The process of backtrack (n, $C$) is as follows: |

6 ifn is not Nonethen |

7 n← parentNode(n)_{p} |

8 ifn_{p} is not Nonethen |

9 s ← ($isn$_{rear}p + 1) × r_{s} |

10 else |

11 s ← 0_{rear} |

12 end |

13 p ← previous point of _{ref}p on $R$_{last} |

14 whiles of p ≤_{ref}s_{rear}do |

15 ifn_{p} is not Nonethen |

16 $br\u2190[min(iln,ilnp)\u22120.5]\xd7rl+Lmin$ |

17 $bl\u2190[min(iln,ilnp)\u22120.5]\xd7rl+Lmin$ |

18 else |

19 $\u2003\u2009br\u2190(iln\u22120.5)\xd7rl+Lmin$ |

20 $\u2003bl\u2190(iln+0.5)\xd7rl+Lmin$ |

21 end |

22 $C$ ←$C$ ∪ < b >_{r}, b_{l} |

23 p ← _{last}p_{ref} |

24 p ← previous point of _{ref}p on $R$_{last} |

25 end |

26 backtrack (n$C$)_{p}, |

27 end |

### 3.3 Path Optimization.

*l*was used to represent the lateral position of the waypoint.

*s*was used as a variable for integration. Additionally, to ensure that the path conforms to the kinematic and dynamic characteristics of the vehicle, the relative heading angle and curvature of the waypoint were added to the state variables. The relative heading angle indicated the direction of the path extension and the attitude of the vehicle rotating along the

*z*-axis. Curvature is also essential as it determines the trend of the changing of heading angle and affects the lateral force of the vehicle. The state variables of each waypoint are expressed as

The path optimization of MinCP was constructed in the form of multishooting. Two components underlying the construction of the optimization problem were, respectively, the objectives and the constraints.

The vehicle's motion state changes due to external forces, mainly from the tires. These tire forces are generated by the friction between the tires and the road surface. In limited working conditions with a constant road adhesion coefficient, the forces acting on the vehicle must be within a certain range. If the required lateral force of the vehicle exceeds the limiting lateral force, the vehicle will become unstable and cannot follow the planned trajectory effectively. Therefore, the planning module needs to minimize the lateral acceleration of the vehicle while ensuring no collision with obstacles so that the vehicle can pass through the maneuvering section with better driving stability and following accuracy.

*Jp*, as a slack variable met the following constraints:

Minimizing the slack variable *J _{p}* can reduce the curvature peak during the maneuvers.

Using curvature differential as the objective provides a flexible approach for enhancing maneuvering smoothness. In contrast, the hard constraint approach involves directly constraining the rate of curvature change within a specific range. The soft constraint approach has several advantages over the hard constraint approach, such as not requiring the accurate range of curvature derivative, providing tunable control over the degree of maneuvering smoothness, and avoiding solver failure in producing a feasible solution.

The constraints of the MinCP contained path model constraints, safety corridor constraints, and start and end constraints.

At each planning frame, the OGM was updated to search for safe corridors, and the parameters were updated according to the safe corridor and ego vehicle state. The updated optimization parameters were then passed into IPOPT, a solver that uses the interior-point method, with CasADi [34] as the interface for mathematical solving. The solving process produced a sequence of pose information in the Frenet frame for each waypoint on the path.

where $xp$ and $yp$ are the coordinates of the waypoint in the Cartesian coordinate system; $xr$ and $yr$ are the coordinates of the waypoint's projection on the reference line in the Cartesian coordinate system.

The output of the single planning frame in the pylon course slalom was shown in Fig. 4. It can be seen that the maximum curvature of MinCPpeak was 29.36% and 31.03% smaller than that of the two control groups, respectively.

### 3.4 Model Predictive Control Tracking Controller.

*Fyf*and

*F*are the lateral tire forces of the front axle and rear axle, respectively. They can be expressed as

_{yr}*Q*is the weight of the state, and

*R*is the weight of control; state variables consist of $[vy,r,ey,e\psi ]T$; input variable is the steering angle of front axle $\delta f$

*d*is the disturbance as Eq. (28). By linearizing the vehicle dynamics model, discrete state equations can be obtained

where $\omega $ and $v$ represent the white noise of state and observation.

Since the control module is not the primary focus of this work, for a more detailed description and related analysis, please refer to Ref. [35].

## 4 Simulation and Experiment

### 4.1 Simulation of Pylon Course Slalom With Low Adhesion.

Simulations can avoid the limitations of the experimental site and actuator performance to test the vehicle under extreme conditions. Natural environments such as rain, snow, and ice often cause the road surface adhesion coefficient to become low. Such conditions are dangerous for autonomous vehicles. The simulation of pylon course slalom with low adhesion was carried out to show the performance difference of planning close to the lateral limits of road surface friction. The spacing between the cones of the pylon course slalom was 25 meters. The autonomous vehicle maneuvered repeatedly to pass through the test section of the pylon course slalom under the condition that the adhesion coefficient is 0.2. For the simulation, the algorithm's parameters and the CarSim dynamics model were based on the Hongqi E-HS9 vehicle.

Two sets of MinCP objective weights were selected to compare their effects on vehicle performance. MinCPpeak used the peak, integral, and differential of curvature as optimization objectives, while MinCPnorm only optimized the integral and differential of curvature. The polynomial curve can smoothly connect different road segments, while the sinusoidal curve is a standard curve used in vehicle testing. In this simulation, a polynomial planner with sinusoidal curve assistance was used as a control group. The values of the algorithm parameters are listed in Table 1. The control module used the MPC controller described in Sec. 3.4.

Classification | Parmeter | Value | Parmeter | Value |
---|---|---|---|---|

Redundant parameters | ds | 0.25 m | $\Psi s$ | 1 deg |

Vehicle measurements | W | 3.25 m | $Lfront/Lrear$ | 2.25/2.00 m |

Optimize weights | $\omega 1/\omega 2/\omega 3$ | 0/1/20 for MinCPnorm 1/1/20 for MinCPpeak |

Classification | Parmeter | Value | Parmeter | Value |
---|---|---|---|---|

Redundant parameters | ds | 0.25 m | $\Psi s$ | 1 deg |

Vehicle measurements | W | 3.25 m | $Lfront/Lrear$ | 2.25/2.00 m |

Optimize weights | $\omega 1/\omega 2/\omega 3$ | 0/1/20 for MinCPnorm 1/1/20 for MinCPpeak |

The simulation results, including the actual paths and the lateral acceleration profiles, are shown in Fig. 5. At 36 km/h, the test vehicle with the sinusoidal planner deviated significantly from the following target near the position with the *x* coordinate of 100 m. An obvious peak platform appeared on the lateral acceleration profile of the 36 km/h test at the marked area. In comparison, in the 35 km/h tests, the shape of the lateral acceleration profile was sharper at the same position. With the gradual velocity increase, the deviation gradually intensified, and the platform became more and more prominent. Although the test adopting MinCPnorm showed a tendency to deviate at 38 km/h, the offset of MinCPnorm tests was kept minor compared with the offset of the sinusoidal planner tests in the subsequent experiments of higher longitudinal velocities. The platform was also formed near the peak of the lateral acceleration profile of the MinCPnorm test when the traveling paths deviated significantly. The test vehicle using MinCPpeak as the planning module had kept effectively tracking the planned path at all the velocities of this group of tests. The lateral acceleration profile of the MinCPpeak experiment at 40 km/h has not formed a platform.

From Fig. 6(a), it can be observed that tire forces were approaching the friction circle boundary. From Fig. 6(b), it can be seen that the trajectories of lateral acceleration and yaw angular acceleration were near the diamond-shaped boundary as described in Ref. [36]. All of these phenomena indicated that the vehicle's performance is approaching its limits under this condition. In such conditions, MinCPpeak has significantly reduced the frequence at which tire utilization exceeds 80%, as shown in Fig. 6(c). Overall, the proposed planning method has improved the safety of autonomous driving vehicles under extreme conditions.

### 4.2 Simulation of Continuous Obstacle Avoidance on Curve.

This simulation aimed to test the effect of the proposed planning method on continuous multiple maneuvering. The situations of autonomous vehicles encountering emergencies on the highway or city ring road were simulated. There were three lanes on the road, and each lane was 3.5 m wide. The adhesion coefficient was 0.85. The longitudinal velocity was 85 km/h. The planning methods included MinCPpeak, MinCPnorm, and polynomial planner. MinCPpeak and MinCPnorm were the same in the previous simulation. The polynomial planner is a commonly used planning method that allows for smooth and continuous trajectories by representing the path as a piecewise fifth-degree polynomial function.

The results of the simulation are shown in Fig. 7. Based on the results, it was observed that the MinCPs had significantly lower lateral errors providing greater safety. Specifically, the lateral error of the MinCP_{norm} test remained below 0.3, with minor depressions at 20–30 m and 110–120 m along the reference line. The lateral error of the MinCP_{peak} test was consistently smaller than that of the other methods throughout the maneuver. Moreover, the vehicle sideslip angle of the autonomous vehicle using the polynomial planner was notably higher than others in the 40–80 m section. It suggested that the proposed planning method improved the maneuvering stability of autonomous vehicles. The proposed method is effective in this scenario which is close to practical application.

### 4.3 Experiment Deployment and Results.

An autonomous electric sport utility vehicle (Hongqi E-HS9) was used as the experimental platform. As shown in Fig. 8, the experimental autonomous vehicle was composed of the steering, driving, and braking system controlled by vehicle control unit (VCU), which could communicate with the controller area network bus and enable autonomous driving. The localization, control, and planning modules were deployed in the robot operating system working on an industrial computer with an Intel Core i7 8700 6-core CPU, up to 4.6 GHz, TDP 65 W. An inertial measurement unit and global navigation satellite system with real-time kinematic positioning support were equipped as the localization devices of the experimental vehicle. The localization module received the serial port signal from the localization device and sent the analyzed localization information to the topic of robot operating system. The planning method proposed in this work was implemented in the planning module. According to input from the localization module during the maneuver, the planning module transmitted the trajectory ahead of the vehicle's current position to the control module. The control module adopted MPC for lateral control and feed-forward PID for longitudinal control. It outputted control commands, including steering angle and expected acceleration to the vehicle's chassis through the controller area network bus. The running times profile of the experiment at 50 km/h is shown in Fig. 9. It indicates that the proposed planning module can meet the online computational requirements of 10 Hz.

Double lane change (DLC) is atypical vehicle handling and stability test. It was used to assess the safety performance of autonomous vehicles. The setting of DLC is as follows: the width of each lane was 3.5 m; the longitudinal distance of each lane change was 30 m; the longitudinal distance between vehicles on the left side after the lane change was set to 20 m. The test velocities were from 30 to 50 km/h. The control group with a polynomial planner used the polynomial curves to connect with the centerlines of the straight section. It is also the default path in the DLC of CarSim.

According to Figs. 10(a) and 10(c), when the test vehicle used the polynomial planner, it deviated from 30 km/h, and the degree of deviation gradually increased with the increase of the test velocity. When the test velocity reached 50 km/h, the lateral error of the test using the polynomial planner reached 1.61 m. According to Figs. 10(b) and 10(d), when MinCP was adopted for the test vehicle, it kept the stable following from 30 km/h to 50 km/h, and the lateral error oscillated only in a small range with a slightly increasing trend.

The yaw rate profiles can characterize the stability of the maneuver. Excessive heading errors aggravate the expansion trend of lateral error and endanger the safety of autonomous vehicles. To better reveal the change law of yaw rate and heading error under different vehicle velocities, they are respectively shown in Figs. 11(a) and 11(b) according to the test velocity. It can be found that the yaw rates and heading angle errors of the vehicle using MinCP are much smaller than those of the polynomial planner during the DLC.

With the increase of the test vehicle velocity, the upper and lower boundaries of the yaw rate continued to expand until the absolute values approached around 20 rad/s. When the yaw rate of the tests using the polynomial planner reached 20 rad/s, it stopped rising but hovered around the peak. Under the same test velocity, the yaw rate of the tests using the MinCP planner did not reach the limit value. The heading error of the control group increases significantly from the test at 45 km/h, which corresponds to the velocity when the yaw rate profile forms a prominent peak platform. The heading angle error was kept relatively near the zero position by MinCP until 50 km/h.

### 4.4 Analysis and Discussion.

In terms of kinematics, the curvature of the path is related to the expected steering angle and yaw rate. When the yaw rate demand exceeds the mechanical limit, the heading angle error will increase. It will further lead to lateral error increasing. In Fig. 11(a), peak platforms appeared on the yaw rate curve, indicating that the yaw rate of the vehicle cannot increase with the increase of its demand, which causes a chain reaction of heading error increasing as Fig. 11(b) and lateral error increasing as Fig. 10(c) step by step.

From the perspective of dynamics, the curvature directly affects the lateral acceleration of the vehicle. When the centrifugal force from the lateral acceleration reaches the friction limit, the vehicle will be hard to control due to the lateral sliding of the tires. Furthermore, the autonomous vehicle will deviate from the planned path. As shown in Fig. 7, the formation of the peak platforms of the lateral acceleration profiles indicates that the vehicle has reached the lateral limits of road surface friction, accompanied by the lateral error increasing rapidly.

Synthetically, the change in curvature reflects the change in lateral driving performance. Reducing the curvature peak of the path under the safety corridor constraint helps the autonomous vehicle keep safe from the perspectives of both dynamics and kinematics in dangerous conditions. The proposed planner reduces the curvature peak while maintaining smooth and collision-free. It effectively helped the autonomous vehicle avoid exceeding lateral friction limits as possible and improve the critical velocity that the vehicle can achieve through a dangerous section.

## 5 Conclusions

This work investigated the lateral planning for autonomous vehicles under extreme conditions approaching lateral friction limits. The proposed planning method obtained the safe corridor by searching and optimized the path from the perspectives of the curvature's peak, integral, and differential. The optimization focused on the properties of the path and did not rely on vehicle dynamics parameters. In addition, this method has good scalability, which can be flexibly integrated with decision making and speed planning and can work in multiple scenarios. The simulation and experimental results demonstrate that utilizing MinCP as the lateral planning method in the autonomous driving system provides better smaller lateral following errors and yaw rates. The proposed planning method can enable autonomous vehicles to more safely pass through dangerous sections approaching lateral limits of road surface friction.

In future research, the coupling relationship between the lateral and longitudinal motion will be considered to plan the trajectory carrying both the lateral and longitudinal information, which will further release the maneuvering performance of autonomous vehicles and improve intelligent safety.

## Funding Data

National Natural Science Foundation of China (Grant Nos. 52172386 and 52372354; Funder ID: 10.13039/501100001809).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*a*_{y}=lateral acceleration of the vehicle

*b*_{l}*/b*_{r}=distance from the left/right of the safety corridor to the reference line

- $C$ =
safe corridor

*C*_{f}/*C*_{r}=tire lateral stiffness of the front/rear axle

*c*=cost of an edge between two grids

*d*_{s}=redundant distance for safety

*F*_{x}*/F*_{y}*/F*_{z}=longitudinal/lateral/ vertical tire force

*F*_{xy}=tire force in the longitudinal and lateral plane

*F*_{yf}*/F*_{yr}=front/rear axle lateral tire force

*h*=heuristic of a grid

*I*_{z}=rotational inertia of the vehicle

*i*_{s}*/i*=_{l}OGM longitudinal/lateral index

*J*=overall optimization objective

*J*_{p}/*J*_{i}/*J*_{d}=objective of path curvature peak/integral/differential

*k*=*k*-th waypoint on the path- $Lmin$ =
minimal coordinate

*l*of the OGM*L*_{front}*/L*_{rear}=length from the front/end of the vehicle body to the center of gravity

*l*=signed distance from the waypoint to the reference line

*l*/_{f}*l*_{r}=front/rear wheelbase of the vehicle

*N*=number of waypoints on the path

*n*=node based on a grid in the graph searching

*n*=_{t}termination node

*p*=state variables of each waypoint

- $R$ =
reference line

*r*=yaw rate of the vehicle

*r*/_{s}*r*=_{l}*s*/*l*resolution of the OGM*s*=distance of the waypoint along the reference line

- $sfront/srear$ =
the front/rear Frenets of the safe corridor section

*v*_{x}/*v*_{y}=longitudinal/lateral velocity of the vehicle

*W*=width of the vehicle body

- $\delta f$ =
steering angle of the front wheels

- $\Delta il/\Delta is$ =
index difference of

*s*/*l*between two grids- $\Delta \psi $ =
relative heading angle

- $\kappa /\kappa ref$ =
curvature at the waypoint/projection on the reference line

- $\psi /\psi ref$ =
heading angle of the waypoint/projection on the reference line

- $\psi s$ =
redundant heading angle for safety

*ω*_{1}/*ω*_{2}/*ω*_{3}=weight of each optimization objective