Exploring the Impact of Rainfall on Vehicle Trajectory Patterns and Sideslip Risk: An Empirical Investigation

. Understanding the sideslip risks of various trajectory patterns, as well as the impact of rainfall on them, is critical for improving road safety. However, the lack of precise classifcation indicators hampers systematic analysis of the variations in vehicle trajectory patterns. To address this, this study proposes a parameterized classifcation method for trajectories on curved segments, employing the radius and ofset of the trajectory as the primary classifcation features and dividing the trajectories into nine patterns. Tese patterns represent variations from smaller to larger radii and inside to outside lane ofsets, refecting diferent driving behaviors and vehicle stability during vehicle cornering. Concurrently, the friction coefcient utilization rate is used to efectively compare vehicles’ sideslip risk under diferent weather conditions. Based on this, we construct a framework using computer vision technology for automatically identifying trajectory patterns and measuring sideslip risk. We conducted an empirical study on a highway-curved segment with high sideslip risk in China and collected two datasets under clear and rainy conditions for analysis. Te classifcation results show that the proposed method can efectively classify trajectories according to nine trajectory patterns. Comparative analysis reveals that vehicle trajectories in both the inside and outside lanes are notably more afected by rainfall compared to the middle lane. Meanwhile, trucks demonstrate a higher susceptibility to rainfall than cars. In addition, the analysis of the sideslip risk for diferent trajectory patterns discovers several high-risk patterns. Tis study provides an efective approach for monitoring and analyzing the sideslip risk on curved segments, thereby contributing to the enhancement of road design and trafc safety management.


Introduction
Rainfall poses a severe threat to highway driving safety, primarily due to its reduced skid resistance between tires and road pavement and the visibility of vehicles [1,2].Among these, the reduced skid resistance is often difcult to detect, causing drivers to overlook the crash risk.Driving on wet roads increases the likelihood of vehicle accidents and intensifes their severity.Fatal trafc crashes are 34% more likely to occur under rainy conditions than in clear weather conditions, and this likelihood increases by 27% even in light rain [3].About 11% of car crashes and nearly a quarter of fatal cases occur during periods of rainfall each year in Louisiana [4].Te reasons behind the frequent crashes under rainy conditions not only lie in the reduced skid resistance leading to longer braking distances for vehicles but also in the fact that vehicles easily sideslip on curved segments [5].Previous studies have shown that rainfall can increase the expected number of severe crashes on curved segments by 3.26 times [6,7].Terefore, comprehending the impact of rainfall on vehicle sideslip risk on curved segments is crucial for enhancing road design and trafc safety management.
Factors afecting the sideslip risk on curved segments under rainy conditions mainly include road surface conditions [8], road geometry [9,10], vehicle performance [11], and rainfall [12].Previous studies have achieved many results by extensively investigating these factors using various methods, such as crash report data [13,14], simulation experiments [15,16], and actual vehicle experiments [17].However, crash report data cannot reveal the impact of rainfall on vehicle sideslip risk at the microlevel.Meanwhile, there are limitations in the data quantity and quality of simulation experiments and actual vehicle experiments.Researchers are turning to mass-collected trajectory data under naturalistic driving conditions [18,19].Based on this, the impact of rainfall on vehicles has been analyzed using more microscopic indicators from trajectories, including speed, lateral acceleration, lateral ofset of vehicles, and intervehicle distance [20,21].Nevertheless, previous studies have overly focused on the trajectories of a small number of abnormal vehicles, neglecting the overall changes in trajectory features [22,23].Te classifcation of trajectory patterns is an essential basis for observing the overall changes in trajectory features.Previous studies have explored methods for classifying trajectories on curved segments, encompassing both single-indicator-based and mixed classifcation methods.Te single-indicator-based classifcations, such as trajectory radius [24], lane departure [25], and curve-cutting position [26], ofer specifc advantages such as efective sideslip risk description, simplicity and practicality, and utilization of turning characteristics, respectively.On the other hand, mixed classifcation [27], exemplifying patterns such as cutting, swinging, drifting, correcting, and normal and ideal behavior, provides a systematic categorization of vehicle trajectories on curved segments, being widely utilized despite its limitations in precise indicator defnition and potential overlap in trajectory categories.However, the classifcation methods proposed in previous studies still have limitations: (a) some methods are only applicable to specifc curved segments (e.g., hairpin curved roads) and cannot be universally applied to all cases; (b) many classifcations do not fully consider the characteristics of curved trajectories, namely, the radius of the trajectory and lateral ofset; and (c) many classifcation methods only provide illustrations without giving classifcation indicators.
With the advancement of computer vision technology, vehicle trajectories can be extracted more accurately from video [28,29].Tis overcomes the limitations of technologies like radar, such as difculty in coordinate conversion and low data accuracy of trajectories [30].Consequently, a more detailed and accurate driving behavior analysis based on vehicle trajectories is possible [31,32].Although the video collected by drones is more straightforward regarding coordinate conversion, it is difcult to collect video of vehicle movements under rainy conditions.Te closed-circuit television (CCTV) systems widely deployed along highways efectively mitigate data collection constraints under rainy conditions.Terefore, this study uses computer vision methods to process road surveillance videos to obtain vehicle trajectories under clear and rainy weather conditions as the databases for this study.
In essence, this study aims to explore the impact of rainfall on the safety of driving on curved segments using real driving trajectory data.To address the gaps in previous research, this study frst proposes a parameterized trajectory classifcation method.It categorizes vehicle trajectories on curved segments into nine patterns based on the trajectory's radius and ofset.Te proposed method is more explicit and suitable for automated data extraction and processing than previous methods.Meanwhile, the friction coefcient utilization rate is used to efectively compare vehicles' sideslip risk under diferent weather conditions.Based on this, we construct a framework for automatic trajectory pattern identifcation and sideslip risk measurement based on machine vision technology.We use it to collect two datasets of the same curve under clear and rainy conditions, containing the trajectories of 970 and 1021 vehicles, respectively.Based on these datasets, we investigated the proportion and safety of trajectory patterns under clear versus rainy weather conditions.Tis study has the following two-fold contributions: First, at the methodological level, we proposed a parametric classifcation method for the trajectories on curved segments.Tis method flls the void of quantitative indices in previous trajectory classifcations and enables a more accurate analysis of trajectory characteristics.Simultaneously, it facilitates the swift identifcation of vehicle trajectory patterns using computer vision technology.Secondly, at the application level, we extensively analyze the various vehicle trajectory patterns, revealing the strategies employed by drivers to safely navigate curved segments during rainfall.Tis innovative approach would facilitate enhancing the design of highway curved segments and optimizing trafc management strategies.
Te rest of the paper is structured as follows.Section 2 provides a literature review.Section 3 introduces the methodology.Section 4 gives the data preparation and implementation details.Te results and discussion are presented in Section 5. Finally, Section 6 provides the conclusions of our work.

Literature Review
2.1.Sideslip Risk Measurement.Vehicle sideslip, a dangerous situation of lateral vehicle instability, is most common when a vehicle passes around a curved segment [33].Te vehicle may sideslip if the friction between the tires and the road surface is insufcient to counteract the centrifugal force during vehicular curve driving.Figure 1 shows a typical example of a vehicle sideslip on a highway curve.Sideslip risk measurement indicators generally fall into two categories based on vehicle states [34,35] and based on friction coefcient [36,37].Among the indicators based on vehicle status, the sideslip angle provides the most direct refection of the sideslip condition, which is defned as the angle between the actual direction of vehicle motion and the wheel orientation.However, a signifcant drawback of the sideslip angle is its delayed response, as its value increases abruptly only when the sideslip occurs [37].To measure the sideslip risk of a vehicle before the sideslip occurs, Chen et al. [15] proposed the sideslip index, which can indicate the risk trend before the sideslip occurs based on tire loads.However, such indicators are usually only available for simulation models or sensor-equipped vehicles due to the difculty of obtaining vehicle state data.

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Journal of Advanced Transportation Compared to vehicle state-based measurement, friction coefcient-based measurement is better suited for practical road design and maintenance work [15].Te friction coefcient of the road surface is directly utilized to measure its skid resistance performance [38].Furthermore, the friction coefcient required for maintaining vehicle balance can be estimated based on the vehicle's speed and motion radius.Road geometry design extensively applies the required friction coefcient [39].However, it is less used in measuring real vehicle sideslip risk.Tis is because it necessitates realtime data on the vehicle's trajectory, radius, and speed.Nevertheless, the required friction coefcient of the vehicle provides a metric foundation for employing computer vision to measure vehicle sideslip risk [40,41].

Trajectories Classifcation.
Regarding geometric characteristics, curved segment trajectories display more distinct feature variations than straight segments.Currently, there are mainly two types of classifcation methods for curved segment vehicle trajectory patterns: one is based on single indicators, such as the trajectory radius [42] and lane departure [43]; the other is mixed classifcation method [44], which is the most commonly applied method, and categorizes the trajectories into six patterns, as shown in Figure 2. In addition, Table 1 presents a detailed analysis of four representative classifcation methods for curved segment trajectories, examining their advantages and disadvantages.
According to the analysis results in Table 1 and the literature review, we identify three aspects where current trajectory classifcation methods need improvement to observe the overall variation of trajectory characteristics more efectively.(a) Enhancing the universality of classifcation methods.For example, the method "based on the position of the curve-cutting point" is mainly applicable to specifc road scenarios like "hairpin curves."(b) Integrating the characteristics of curved trajectories.For instance, the "based on lane departure" and the mixed classifcation methods show poor correlation with the characteristics of curved trajectories.(c) Defning precise classifcation indicators.For example, the mixed classifcation method lacks clear classifcation indicators, and many trajectory categories overlap, making it difcult to distinguish efectively.

Methodology
In this study, a framework for vehicle trajectory pattern identifcation and sideslip risk measurement in curved segments is developed based on computer vision technology, as shown in Figure 3. Te framework consists of three parts: (1) trajectory extraction; (2) trajectory pattern classifcation; and (3) sideslip risk measurement.Subsequent subsections present each of these components in Sections 3.1-3.3 in their respective order.

Trajectory Extraction.
In the trajectory extraction part, we employ computer vision techniques to extract vehicle trajectories from surveillance videos.Specifcally, this process can be divided into three steps.Firstly, we conduct vehicle detection and tracking to automatically capture the vehicle trajectories from videos in the image coordinate system.Secondly, we conduct camera calibration and coordinate transformation to convert trajectory coordinates into real-world coordinates based on the camera parameters.

Vehicle Detection and Tracking.
In step 1, we employ the YOLO (You Only Look Once) and StrongSORT algorithms to achieve automated vehicle detection and tracking.YOLO, as a leading vehicle detection algorithm currently [47,48], utilizes a single neural network to predict the bounding box and class probabilities of each object in a single forward pass.Compared to region-based convolutional neural network (R-CNN), YOLO algorithm boasts advantages of faster detection speed and easier deployment [49].Moreover, to maintain continuous vehicle trajectory tracking, it is necessary to integrate vehicle detection algorithms with multiobject tracking (MOT) algorithms.StrongSORT, as an advanced MOT algorithm, integrates Gaussian-smoothed interpolation (GSI) with DeepSORT to reduce detection loss efectively [50].In this study, we trained Yolo v5 Version 6.0 and StrongSORT V4.0 models based on a dataset of more than 12,000 vehicles.Te snapshot of object detection and tracking uses YOLO and Deep Sort, as shown in Figures 3(a

Camera Calibration and Coordinate Transformation.
In step 2, the trajectory data obtained in the image coordinate system (u-v coordinates) must be further converted into Frenet coordinates (s-d coordinate system) for better visualization and classifcation of the trajectories.By utilizing the spatial position and focal length of the surveillance camera, the mapping relationship between the image coordinate system and the world coordinate system can be deduced, as shown in (1) [51].Tus, accurate external parameters of the surveillance camera, such as focal length, rotation angle, pitch angle, and height, are crucial for coordinate transformation.Various camera parameter calibration methods have been applied in previous studies, which can be divided into multivanishing point methods and single-vanishing point methods [52].Due to the lack of efective reference objects on road segments, this study combines the previous research results to apply the "VWL" (one vanishing point, known width and length) method for camera parameter calibration [53,54].Te "VWL" method determines a vanishing point based on the road markings and then obtains the camera parameters based on the known length of the road markings and a set of parallel markings with known spacing, as shown in equations ( 2)-( 6) and Figure 4(a).
where (u, v) represents the coordinate values in the image coordinate system, and (x, y, z) represents the coordinate values in the world coordinate system.f is the camera focal length, σ is the camera rotation angle, ϕ is the camera pitch angle, h is the camera height (m), and ξ is the scaling factor.(u 0 , v 0 ) is the vanishing point image coordinate, (u i , v i ) and (u j , v j ) are the start and end point image coordinates of the length reference object, ∆u is the diference in intercepts of the two known parallel lines on the road in the coordinate axis, W is the distance between the two known parallel lines on the road, L is the actual length (m) of the length reference object, and T is an intermediate variable.
Although some reliable methods have been proposed in previous studies to transform the world coordinate system into trajectories on curved segments.It is also the most widely used classifcation method Te method lacks precise indicators.At the same time, some trajectory categories overlap and are difcult to distinguish.Te characteristics of the trajectory cannot be efectively described the Frenet coordinate system [55], signifcant lens distortion and irregular scaling exist due to the camera perspective of surveillance video.To accurately measure the trajectory in realworld units (e.g., meters), this study adopts a grid remapping method that directly transforms trajectory data from the u-v coordinate system to the s-d coordinate system [56].Based on the camera calibration results, the grid mesh is established by equation ( 1).Every grid is of equal size in real-world units, shown in Figure 3(e).Further, according to the road geometry, we reconstruct the vehicle trajectory in the x-y coordinate system, as shown in Figure 3(f).

Classifcation Indicators.
To efectively analyze highway driving safety, it is essential to consider the characteristics of the trajectory on curved segments when conducting systematic trajectory classifcation.Te trajectory on curved segments has the following three major characteristics: (a) smooth and continuous; (b) approximated as circular arcs; and (c) possibly ofset to one side.Te trajectory of a vehicle within a curved segment can be approximated as a circular arc.Tis circular radius of approximation R T , referred to as the "trajectory's approximation radius," is a signifcant characteristic for distinguishing diferent types of trajectories.For example, the "cutting" and "ideal behavior" trajectory types proposed in previous studies can be diferentiated using R T [57].In addition, compared to straight segments, vehicles often tend to be ofset to one side when driving on curved segments.Tis is related to the turning position and the steering wheel angle.Understanding the ofset trend of trajectories has reference value for improving road geometric design and setting safety facilities [43].Based on this, we propose a parameterized trajectory classifcation method that employs the turning beneft ratio (TBR) and trajectory ofset (∆d) as classifcation indicators for trajectories.
(1) Turning beneft ratio: Te turning beneft ratio TBR as a classifcation indicator refers to the ratio of the trajectory's approximate radius to the ideal radius, as shown in equation (7).According to the coordinates of the starting point (x s , y s ), midpoint (x m , y m ), and endpoint (x e , y e ) of the trajectory in the x-y coordinate system, the approximation radius R T can be computed using Heron's formula [58], as shown in equation (8).According to TBR, we can classify trajectories into three patterns, as shown in Figure 5(a) When TBR ≈ 1, the vehicle trajectory radius is close to the ideal radius.In the Frenet coordinate system, the trajectory is then a straight line.(b) When TBR ≫ 1, the trajectory radius is much larger than the ideal radius.In the Frenet coordinate system, the trajectory is a curve with the center on the right (the outer side of the curve).(c) When TBR ≪ 1, the vehicle's trajectory radius is much smaller than the ideal radius.In the Frenet coordinate system, the trajectory is a curve with the center on the left (the inner side of the curve).Using the Frenet coordinate system provides an intuitive way to distinguish between three diferent trajectories classifed by TBR.
where R C is the radius of the road horizontal curve and d B is the lateral distance between the vehicle and the road boundary when entering a curved segment.
(2) Trajectory ofset distance: Te trajectory ofset ∆d refers to the diference in lateral distance between the vehicle and the road boundary when entering and exiting a curved segment, i.e., ∆d � d B − d E .Based on ∆d, we can classify the trajectory into three categories, as shown in Figure 6.When ∆d ≈ 0, the distance between the vehicle and the road boundary at the entry and exit of the curved segment is similar, indicating that the vehicle can maintain stable control.When ∆d ≫ 0, the vehicle deviates toward the outer side of the curve when exiting the curved segment.When ∆d ≪ 0, the vehicle deviates toward the inner side of the curve when exiting the curved segment.Although it is insufcient to determine insufcient or excessive steering solely based on trajectory ofset, typically, when a vehicle has insufcient steering, it tends to deviate toward the outside of the curve.Conversely, when a vehicle has excessive steering, it tends to deviate toward the inside of the curve.

Trajectory Pattern Descriptions.
To classify trajectories according to TBR and ∆d, we introduce the standard deviation of two indicators, α and β, as deviation thresholds, respectively.When TBR ∈ (1 − α, 1 + α), it can be considered that the approximation radius R T of the trajectory is close to the ideal radius.When ∆d ∈ (−β, β), it can be considered that the distance between the vehicle and the road boundary is the same when the vehicle enters and exits the curved segment.In this study, vehicle trajectories are divided into nine patterns according to TBR and ∆d, as shown in Figure 7. Te characteristics of the nine trajectory patterns are described as follows: (1) Inside ofset-smaller radius (I-S): Tis pattern has the characteristic that the trajectory's approximate radius R T is much smaller than the ideal radius and deviates to the inside of the curve.Its ∆d ∈ (−∞, −β] and TBR ∈ (0, 1 − α].When the vehicle is oversteering, its trajectory may show this pattern. (

Friction Coefcient Utilization
Rate.Sideslip is the most common situation in lateral instability [59].Figure 8 illustrates the forces acting on a vehicle on a curved segment during its travel.Te vehicle experiences three primary forces in the lateral direction: the component of the vehicle's centrifugal force along the road's cross slope F v cos θ, the component of the vehicle's gravitational force along the road's cross slope G sin θ, and the lateral resistance caused by the friction between the tires and the road surface F f [60].Te centrifugal force is related to the vehicle's mass m, speed v, and trajectory radius R and can be represented as F v � mv 2 /R.When the vehicle is stable in the lateral direction, F v cos θ � G sin θ + F f .We defne the required friction coefcient f R for the vehicle to maintain lateral stability as the ratio of the lateral friction resistance F f to the normal reaction force N. Accordingly, it can be known that f R is related to the vehicle's instantaneous speed v i , instantaneous turning radius R i , and road cross slope e. Te required friction coefcient f R can be calculated according to equation (9).When the required friction coefcient f R is less than the maximum friction coefcient f max that can be provided between the tire and the road surface, the vehicle maintains lateral force balance.
When the required friction coefcient f R for a vehicle is greater than the maximum friction coefcient f max that the interaction between the tires and the road surface can provide, the vehicle will sideslip [61].To enable a comprehensive comparison of the sideslip risk under clear and rainy weather conditions, we employ the friction coefcient utilization rate μ to measure the sideslip risk, as shown in equation (10).A higher value of μ translates to a greater risk of vehicle sideslip.However, it is difcult to obtain an accurate value for f max .Terefore, this study only selects the representative value of f max under clear and rainy weather conditions for measuring the sideslip risk.According to related research [62], the maximum lateral friction coefcient for asphalt surfaces is approximately 0.85 under clear weather conditions and approximately 0.30 under rainy conditions.

Data Extraction and Optimization. Speed v i 􏼈 􏼉 and trajectory radius R i
of the vehicle are critical basic data for measuring vehicle sideslip risk.Tey can be calculated based on the trajectory coordinates in the x-y coordinate system using equations ( 11) and (12), respectively.Here, the x and y coordinates are respectively ftted using fourth-degree polynomials to obtain the frst and second derivatives.
where k i   is the curvature at each trajectory point.Due to the random noise introduced during trajectory extraction processing steps, this study chose fltering techniques to optimize the data.Filtering is a data processing technique used to remove noise and enhance the accuracy of information extraction [63].Commonly used algorithms include median and mean flters, wavelet thresholding, a cubic spline flter, and a Savitzky-Golay flter [64].Compared to other algorithms, the Savitzky-Golay flter has the advantage of preserving data features and allowing for fexible parameter adjustment.Moreover, it has been found efective for optimizing vehicle trajectory data [65].Te Savitzky-Golay flter is given in the following equation: where Q j is the fltered data; q j+1 is the original data; λ i is the i-th coefcient of the flter, which can be calculated by a polynomial with order ω; N is the flter length; and m is the half-flter length, which is equal to 0.5(N − 1).

Data Preparation and Implementation Details
Te data for this study originate from the S4 highway in China, a major arterial route with three lanes in each direction and a design speed of 120 km/h.According to the crash report, we selected a curved segment with six rainy sideslip crashes in two years as the study segment.Tis segment has a curvature radius of 2,200 m and a superelevation of 3%.Te data collected comprise surveillance video and highway design drawings.Two surveillance videos were used as case studies, which were recorded in diferent weather conditions: clear and rainy weather (as shown in Figure 9).Te videos were captured using gun-type surveillance cameras ftted with rain shields to ensure quality recording in rainy conditions.Each video boasts a resolution of 4,064 pixels × 3,040 pixels and a frame rate of 24 frames/ second, spanning a duration of two hours.Te trafc volume in both videos is approximately equivalent, recording 970 vehicles under clear weather conditions and 1,021 vehicles during rainy conditions.Vehicle types are classifed into two categories: "Car" and "Truck.""Car" refers to vehicles with no more than two axles or four wheels (e.g., private cars and vans), while "Truck" refers to vehicles with more than two axles or four wheels (e.g., trucks and buses).Meanwhile, road lanes are divided into left, middle, and right lanes according to the direction of vehicle travel.Te left lane is closer to the outside of the curve, and the right lane is closer to the inside of the curve.

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Journal of Advanced Transportation Te vehicle's lane is identifed based on the lane it occupied when entering the segment, as shown in Figure 9(a).
In addition, we calibrated the parameters of the camera that captured these surveillance videos using the method described in Section 3.1.2.Based on the road markings, we located a vanishing point with coordinates (u 0 , v 0 ) � (676.99,−102.14).Further, utilizing the known information from the image, a lane width of 3.75 m and a road marking length of 6.00 m, we applied equations ( 2)-( 6) to calculate the camera's rotation angle σ � 0.027, the pitch angle ϕ � 0.062, the focal length f � 10296.18,and the camera height h � 8.99m.Based on this, we further defned the efective trajectory observation area and determined the grid mesh using equation ( 1), as shown in Figure 9(b).

Classifcation Indicators and Deviation Tresholds.
In order to classify trajectories according to the two indicators TBR and ∆d proposed in this study, it is necessary to determine the deviation thresholds α and β.We performed a statistical analysis on TBR and ∆d under clear and rainy weather conditions.Table 2 presents the statistical results, including the mean, standard deviation, skewness, and kurtosis of TBR and ∆d.
According to Table 2, we can conduct a preliminary analysis of the impact of rainfall on the trajectory.It can be observed that rainy weather increases the mean and skewness of both TBR and ∆d when compared to clear weather.Tis suggests that vehicles tend to deviate towards the outside of curves and adopt a larger trajectory radius on curved segments under rainy weather.Furthermore, rainy weather causes a decrease in the kurtosis of ∆d and an increase in its standard deviation when compared to clear weather conditions.It also suggests that the ofset dispersion is greater as the vehicle crosses the curved segment under rainy conditions.Notably, rainy conditions also reduce the kurtosis and the standard deviation of TBR compared to clear weather conditions.Tis suggests that, under rainy conditions, the trajectory approximation radius R T dispersion increases within a limited range while extreme deviations decrease.
Considering that the trajectories collected under clear weather are more representative of the normal situation, this study chooses the standard deviations of two indicators as deviation thresholds: α � 0.093 and β � 0.685.Te collected vehicle trajectories are classifed into nine patterns using these thresholds.Figure 10 presents the classifcation results of the trajectory under rainy weather conditions, clearly illustrating the signifcant characteristics diferences among the nine patterns.Tese characteristics of the nine trajectory patterns are consistent with the features described in Section 3.2.2.Tis result validates the feasibility and efectiveness of the trajectory pattern classifcation method combined with computer vision technology proposed in this study.

Exploratory Analysis of Trajectory Pattern.
Based on the classifcation of trajectory patterns, we investigate the infuence of rainfall on trajectory patterns for vehicles with diferent vehicle types and lanes.Table 3  proportions of diferent trajectory patterns under clear and rainy weather conditions across various lanes.Te following observations are noteworthy:

displays the
(a) Under rainy conditions, the proportion of the "S-I" trajectory pattern (the theoretically ideal trajectory pattern) decreases in all three lanes.Tis suggests that rainfall may heighten the challenge of controlling the vehicle along the geometric path of the road on curved segments.
(b) Compared to clear weather conditions, the proportion of vehicles in the left lane that deviates to the outside (left side) of the curve increases signifcantly under rainy conditions, while the proportion of vehicles in the right lane that deviates to the inside (right side) of the curve increases slightly.Among them, the "O-I" trajectory pattern increases by 25% in the left lane, while the "I-I" trajectory pattern increases by 7.4% in the right lane.Tis fnding implies that vehicles on the left and right lanes may tend to stay away from the middle lane under rainy conditions.(c) Vehicles adopting the "S-L" trajectory pattern must shift towards the middle lane when in the left lane, whereas vehicles in the right lane must deviate from the middle lane.Compared to clear weather conditions, the proportion of the "S-L" trajectory pattern decreases by 4.8% in the left lane, while it increases by 4.3% in the right lane.Tis fnding supports the speculation that vehicles on the left and right lanes may tend to stay away from the middle lane under rainy conditions.(d) In the middle lane, aside from a 7.8% reduction in the proportion of the "S-I" pattern, changes in other trajectory patterns remain below 3%.Tis indicates that the vehicle trajectory pattern in the middle lane is least afected by rainfall.
Table 4 illustrates the proportions of diferent trajectory patterns for various vehicle types under clear and rainy weather conditions.Te following trends can be observed: (a) Compared to clear weather conditions, the proportion of the "S-I" trajectory pattern for cars decreases by 14.5%, while for trucks, it only decreases by 6.9%.Tis suggests that car trajectory patterns may be more susceptible to rainfall compared to trucks.
(b) Compared to clear weather conditions, under rainy conditions, the proportion of the "O-I" trajectory pattern for cars and trucks shows a substantial increase of 12.3% and 6.2%, respectively.Meanwhile, the proportion of trucks with the "I-L" trajectory pattern decreases by 4.8% under rainy days, and the "O-L" trajectory pattern increases by 4.2%.Tis may indicate a tendency for vehicles to deviate towards the outside of curved segments.

Sideslip Risk
Analysis.Tis section focuses on two aspects: (a) analyzing the infuence of rainfall on speed and the minimum curvature radius of trajectories and measuring the sideslip risk in diferent weather conditions and (b) comparing and analyzing the sideslip risk associated with different trajectory patterns.

Analysis of the Impact of Rainfall.
A vehicle's speed and turning radius are key factors that infuence the risk of a sideslip.Figure 11 illustrates the relationship between speed and trajectory curvature radius under clear and rainy   or that rainfall may increase the difculty for drivers to control their vehicles, consequently making it challenging for vehicles to adopt a larger trajectory radius when negotiating curves.(c) Although there is no apparent linear relationship between vehicle speed and the minimum curvature radius (R min ), there is a decreasing trend in vehicle speed with decreasing R min under rainy weather conditions.In contrast, no such trend is observed under clear weather conditions.Tis trend may be related to the lateral stability of vehicles.In clear weather, the road surface ofers higher skid resistance, and the vehicle is more laterally stable.However, in rainy conditions, with lower skid resistance, drivers may distinctly feel the risk of vehicle sideslip, especially when taking curves with smaller radius.As the risk of sideslip increases with smaller trajectory radius, drivers need to reduce speed to ensure the safe passage of the vehicle.
Figure 12 presents the distribution of the friction coefcient utilization rate μ under clear and rainy weather conditions.Under clear weather, the mean value of μ is 0.066  14 Journal of Advanced Transportation with a standard deviation of 0.032, whereas, under rainy conditions, the mean value of μ is 0.128 with a standard deviation of 0.068.Overall, there is a signifcant increase in μ for vehicles on curved segments under rainy conditions compared to clear weather conditions.Furthermore, the maximum value of μ under clear weather is 0.299, corresponding to a maximum required friction coefcient f R of 0.255.In contrast, the maximum μ under rainy conditions reaches 0.493, corresponding to a maximum f R of 0.148.Previous studies suggest that the maximum achievable coefcient of friction between tires and road surfaces under rainy conditions may be lower than 0.1 [66], indicating an elevated sideslip risk on such road segments under rainy conditions.

Analysis of Diferent Trajectory
Patterns.Figure 13 illustrates the friction coefcient utilization rate μ of diferent trajectory patterns under clear and rainy weather conditions.Due to the limited sample size for some trajectory patterns, the median values may not be representative.Terefore, this study primarily uses the mean of μ to measure the sideslip risk of various trajectory patterns.Te investigation reveals the following fndings: (a) Rainfall impacts sideslip risk diferently for various trajectory patterns.Although the trajectory pattern with the highest risk of sideslip is "O-S" under both clear and rainy conditions, the trajectory pattern with the lowest risk of sideslip is diferent.Te "S-L" trajectory pattern has the lowest sideslip risk under clear conditions.However, the "S-I" trajectory pattern has the lowest risk of sideslip under rainy conditions.(b) Among the trajectory patterns of the same category based on TBR classifcation, there is no signifcant correlation between ∆d and the mean of μ under clear weather conditions.However, under rainy conditions, the symmetrical trajectory patterns ("S-S," "S-I," and "S-L") exhibit a lower mean of μ.Tis fnding indicates that maintaining a constant lateral distance from the lane under rainy conditions can reduce the vehicle's sideslip risk.(c) Among the trajectory patterns of the same category based on ∆d classifcation, the mean value of μ for varying patterns displays a certain order under clear weather conditions: smaller radius > ideal radius > larger radius.However, under rainy conditions, the mean value of μ for diferent trajectory patterns follows this order: smaller radius > larger radius > ideal radius.It can be seen that there is a certain correlation between the μ of the vehicle and the TBR of the trajectory.Te trajectory pattern with smaller TBR exhibits a higher risk of sideslip under clear weather conditions.However, this correlation is not observed under rainy conditions.Under rainy conditions, there is a tendency for the TBR of a trajectory to be closer to 1.00, which lowers its sideslip risk.Tis fnding reveals that adopting a larger TBR under rainy conditions does not necessarily reduce vehicle sideslip risk.Conversely, this may increase the sideslip risk when using the "S-L," "I-L," and "O-L" trajectory patterns.

Conclusion
To sum up, this study aimed to investigate the infuence of rainfall on trajectory patterns and sideslip risk on highway curves.Te main contributions of this study are as follows.Firstly, the turning beneft ratio TBR and trajectory ofset ∆d were introduced as parameterized classifcation indicators for trajectories, addressing the lack of precise indicators for systematic trajectory classifcation on curved segments.Furthermore, the friction coefcient utilization rate μ was adopted as a measure for sideslip risk.Secondly, we proposed a computer vision-based framework for automatically identifying trajectory patterns and measuring sideslip risk.Using this approach, we collected the trajectories and their sideslip risk data from a curved segment of highway in China under clear and rainy weather conditions.Based on the data, a case study was conducted, and the following important conclusions were drawn:   Journal of Advanced Transportation that can automatically adjust a vehicle's trajectory in rainy weather to reduce the sideslip risk.
However, there are some limitations to this study.It only compares the changes in trajectory patterns under diferent weather conditions on the same road segment, without considering the sensitivity of road trajectories to rainfall at diferent radii.In addition, the study lacks actual measurements of lateral friction coefcients on the road surface, comparing only the relative relationships of sideslip risks among diferent trajectories.Future work aims to collect more research data and further investigate the impact of factors such as road radius and longitudinal slope on trajectory patterns, providing a clearer understanding of the efects of rainfall on vehicle safety on curved segments.

Figure 1 :
Figure 1: Example of a vehicle sideslip.Te white car sideslipped and lost control, eventually crashing into the orange car.During this incident, the white car corrected its direction twice.(a) Normal state.(b) Sideslip.(c) First correction.(d) Lost control.(e) Second correction.

Figure 3 :
Figure 3: Framework of trajectory pattern identifcation and sideslip risk measurement based on computer vision.

Figure 4 :
Figure 4: Schematic diagram of camera calibration and grid mesh establishment.(a) "VWL" method for camera parameter calibration.(b) Grid mesh establishment based on camera calibration.

Figure 5 :( 8 )
Figure 5: Schematic diagram of the vehicle trajectory with diferent TBR.In the Frenet coordinate system, the s-axis is located along the left boundary of the road, and the d-axis is perpendicular to the s-axis.Te coordinate s represents the distance along the road, while the d-axis represents the vehicle's distance from the outer boundary.(a) Cartesian coordinates.(b) Frenet coordinates.

5. 1 .
Trajectory Pattern Statistical Analysis.Tis section primarily includes two aspects: (a) conducting statistical analysis on two trajectory classifcation indicators to determine the classifcation threshold and (b) exploring the infuence of vehicle lane and vehicle type on trajectory patterns in diferent weather conditions.

Figure 8 :
Figure 8: Front view of vehicle model on a curved segment.(a) Lateral force of the vehicle on the curved segment.(b) Vehicle lateral stability at diferent speed.

Figure 9 :Figure 10 :
Figure 9: Surveillance video screenshots of study segment in diferent weather conditions.(a) Video screenshots under clear weather condition with lane location indication.(b) Video screenshots under rainy weather condition with grid mesh indication.

Figure 11 :
Figure 11: Comparative scatter distributions of vehicle speed and minimum curvature radius under clear and rainy weather conditions.Te red dashed line in the fgures is the diagonal line of the coordinate axis, which is convenient for observing the changes.(a) Clear weather condition.(b) Rainy weather condition.
friction coefficient utilization rate µ During clear weather conditions During rainy weather conditions

Figure 12 :
Figure 12: Comparative analysis of the friction coefcient utilization rate under clear and rainy weather conditions.
(a) According to the proposed trajectory classifcation indicators, the turning beneft ratio TBR, and trajectory ofset ∆d, vehicle trajectories can be efectively classifed into nine patterns using computer vision techniques.Meanwhile, in the Frenet coordinate system, these nine trajectory patterns show signifcant characteristic diferences, which is helpful for better observation of trajectory changes.Tis classifcation method provides trafc management authorities with an efective tool for more accurate and parameterized monitoring of vehicle behaviors and sideslip risks on highways, especially during rainy weather conditions.(b) Te impact of rainfall on trajectory patterns of vehicles in diferent lanes, ranked in descending order, is as follows: left lane (outer side of the curve) > right lane (inner side of the curve) > middle lane.Meanwhile, cars are more susceptible to these impacts compared to trucks.In addition, compared to clear weather conditions, vehicles on the left and right lanes may tend to stay away from the middle lane under rainy conditions.Tis fnding directs trafc management authorities towards prioritizing certain lanes for management and aids highway designers in improving road design to accommodate these observed behaviors.(c) Rainfall signifcantly increased the sideslip risk on the curved segments.However, the infuence of rainfall on sideslip risk varied among diferent trajectory patterns.Under rainy conditions, both the trajectory's approximation radius and the ofset have a signifcant impact on the sideslip risk.Te "S-I" trajectory pattern has the lowest risk of sideslip under rainy conditions.Symmetrical trajectory patterns are safer than asymmetrical patterns, and trajectories closer to the ideal radius are safer than others.Tus, maintaining a steady lateral distance under rainy conditions can contribute to avoiding vehicle sideslip.Tis fnding could help develop or improve advanced driver assistance systems (ADAS)

Figure 13 :
Figure 13: Comparative analysis of the friction coefcient utilization rate in diferent trajectory patterns.

Table 2 :
Statistical analysis of trajectory classifcation indicators.

Table 3 :
Distribution of trajectory patterns across diferent lanes and weather conditions (%).Note.n is the number of vehicles.Other values are in percent.Te data in bold mean that the change value is greater than 4%.

Table 4 :
Distribution of trajectory patterns across vehicle types and weather conditions (%).Note.n is the number of vehicles.Other values are in percent.Te data in bold mean that the change value is greater than 4%.