Sight Distance of Automated Vehicles Considering Highway Vertical Alignments and Its Implications for Speed Limits

Most existing road infrastructures were constructed before the emergence of automated vehicles (AVs) without considering their operational needs. Whether and how AVs could safely adapt to as-built highway geometry are questions that remain inconclusive, and a plausible concern is a challenge from vertical alignments. To fill this gap, this study uses a virtual simulation to investigate the available sight distance (ASD) of AVs on vertical alignments subject to the current highway geometric design specification and its implications for speed limits. According to the scenario generation framework, several scenarios featuring vertical geometric elements and lidar sensors were created and tested. Moreover, the maximum speed for adequate ASD is calculated to determine the AV speed limit, considering safe sight distance and speed consistency requirements. The results indicate that crest curves are not disadvantaged in ASD compared with either sag curves or tangent grades. Only equipped with multichannel lidar and advanced perception algorithms enabling a lower detection threshold would a level 4 AV be compatible with the as-built vertical alignment with a design speed (Vd) of 100 km/h. However, a level 3 AV can only adapt to the vertical profile with Vd = 60 km/h. The findings of this study should be of interest to the road-oriented operational design domain and support road administrators in regulating AV safe speeds.


T
he International Society of Automotive Engineers has classified driving automation into six levels [1], i.e., from no driving automation (level 0 [L0]) to full driving automation (L5).Over the past few years, many features of lower automation levels have been available in the automobile market (e.g., the L2 Tesla).In the meantime, many features of higher levels are being tested on public roads (e.g., the L4 Baidu).According to the China automobile market statistics report [2], general automation is developing from L2 to L3.In other words, it is maturing from advanced driver-assist systems to automated driving systems (ADS).
Recent studies have revealed significant differences in the performance and features between automated vehicles (AVs) (i.e., vehicles equipped with ADS) and traditional humandriven vehicles (HVs), particularly in the perception-related functions (e.g., [3] and [4]).However, most as-built road infrastructures, especially their road geometry, were designed only considering the characteristics of human drivers or HVs [5].In this regard, the compatibility of AVs with as-built roads is gaining increasing interest from academia and industry.
As stated in previous studies (e.g., [6]), specifying the operational design domain (ODD) of AVs is essential to safely deploy them on as-built roads.The ODD refers to an AV's needed operating conditions, including environmental, traffic, and roadway characteristics, among others [1].However, only the less detailed ODD requirements regarding roadway characteristics were specified, such as merely mentioning the allowable road type without the speed limit for the specific geometric conditions.This might lead to consumer overreliance, suspicion, or confusion [7], [8].In addition, it is very challenging to address a lack of ODD standardization or elaborate on numerous conditions for every vehicle [9].These limitations hinder road administrators' application of this vehicle-based ODD concept for road design, management, and maintenance.
Since many road infrastructures were constructed before the emergence of AVs, it could be cost-effective for road administrators to adapt AVs to as-built roads.Therefore, it is necessary to adjust or improve the ODD concept from the road perspective-that is, offering such a road-oriented ODD concept by stating the specific road condition (e.g., the road geometry) and its matching AV operation (e.g., the driving speed).As García et al. [9] proposed, the road-oriented ODD concept can be defined as the operational road section.Previous studies focused mostly on horizontal alignments.They (e.g., [10]) highlighted the perception-related limitations of AVs and justified the available sight distance (ASD) for exploring AV compatibility.The ASD refers to the longest path distance at which a stationary obstruction along the roadway geometry can be detected.Given the limited angular resolution, range, and object detection threshold of AV sensors, AVs might not have sufficient ASD to ensure safe driving at high speeds on horizontal curved roads.
Due to vertical curves, vertical alignments may be subject to the same concerns as horizontal alignments.Although Wang et al. [10] demonstrated that either a higher sensor mounting height or a larger upward vertical field of view (VFoV) would induce a milder requirement of vertical in Figure 1.Such obstructions reduce the ASD and the corresponding maximum safe speed [11], [12], [13].However, the results on vertical alignments were quite inconclusive.Only a limited number of scenarios including crest vertical curves and advanced driver-assist systems were investigated in field tests [11].In addition, the analytical studies (e.g., [10]) might overestimate the actual AV's perception ability since some critical sensor-related factors (e.g., the angular resolution) were not considered.Given those motivations and limitations, further extending road geometry studies to vertical alignments is vital.
This study investigates whether and how AVs could safely adapt to as-built vertical alignment.In this regard, two intriguing questions arise: how far can AVs see, and how fast should they drive?A series of virtual simulations featuring AVs and vertical alignments were conducted to estimate the ASD.Based on that, we derived the maximum speed for adequate ASD (V max ) according to the safe sight distance (SD) requirement.That is, the ASD must not be less than the required stopping SD (RSD) [14].In addition, the V max -based speed limit on vertical alignments was determined considering speed consistency.Note that the road type in this study refers to a highway, as only the ego vehicle and geometric features were considered.The present study offers thorough and feasible answers to these two questions, which are expected to further advance the roadoriented ODD and support road administrators in regulating Avs' safe speeds on as-built vertical alignments.
The study is organized as follows.The next section presents the literature review.The "Methods" section intro-duces the simulation design.The "Results and Discussion" presents the simulation and analysis results, and the final section presents concluding remarks, limitations, and future work.

Related Work
Over the past few years, many efforts in the literature have been expended on improving the ODD concept from the road perspective and determining how AVs could safely adapt to the as-built roadway geometry.Recent studies can be classified into four types: analytical studies, empirical studies, computer-aided studies, and virtual simulation studies.
The empirical studies better captured AVs' natural characteristics [11], [19], [20].Previous studies investigated the Since many road infrastructures were constructed before the emergence of AVs, it could be cost-effective for road administrators to adapt AVs to as-built roads.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.maximum operational speeds for the automation system on public roads.In such field tests, although they have real alignment conditions, it is difficult to completely exclude the effect of nongeometric factors (e.g., the traffic flow and weather).In addition, those costly tests would limit the sample sizes regarding various sensor configurations and geometric design elements.
The computer-aided studies used actual lidar point cloud data of highways [12], [21].They created a simulation environment and then proposed automatic ASD estimation algorithms.Compared with analytical and empirical studies, computer-aided studies increased the FoV from two dimensions to three, save testing time, and expanded the sensorrelated sample size (e.g., the sensor quantity).Nevertheless, they failed to consider other sensor-related factors (e.g., the angular resolution).Also, the sensitivity investigation of geometric conditions might be insufficient due to the limited number of road scenarios modeled by the point clouds.
The virtual simulation studies avoided the mentioned issues and limitations [13], [22].These studies adopted a high-fidelity simulation technology to effectively simulate AVs' perception processes and, thus, estimated the ASD along different alignments.Importantly, they ascertained that lidar's angular resolutions and laser-point thresholds for object detection affect the ASD substantially.Moreover, this method is well recognized in the AV-testing domain [23].It can customize numerous scenarios effectively, which include road geometry, sensors, AV features, etc.

AVs' Lidar-Based Perception System
The lidar-based multisensor fusion and perception system has been recognized as one of the optimal perception solutions for AVs because of lidar's advantages over cameras and radar sensors [24].Generally, a camera is more susceptible to adverse weather and lighting conditions [25]; radar has a narrower FoV, especially the VFoV; a shorter range; a coarser angular resolution; and inferior performance on stationary target detection [26].In addition, camera data are mainly used for lane-marking detection, traffic sign identification, and traffic light recognition [24].
As stated, the ASD of AVs is a critical item in previous studies that can effectively reflect the safety margin provided by the road alignment from the SD perspective and inform AV safe driving speeds [27].As demonstrated by Wang et al. [13], [22], the primary cause of many crashes involving an AV colliding with a stationary obstruction is ASD < RSD.Furthermore, in the case of ASD testing, sensing results from lidar would play a priority role over those from either a camera or radar mainly due to the features of "long distance" and "stationary obstruction," respectively.
Therefore, referring to [13] and [22], the AV, in this context, also denotes an AV equipped with a lidar-based perception system (LAV).Note that effective object detection is closely related to a sufficient number of laser points impinging on that object, as highlighted in previous studies (e.g., [22]).

Overview
The workflow of this study is shown in Figure 2. First, a cosimulation platform is used to conduct virtual simulations.Then, the experiments are designed according to the basic framework of AV driving scenario generation (i.e., functional-logicalconcrete scenarios), as defined in the PEGASUS project [28].Those experiments consider vertical alignments and lidar elements.The vertical alignment elements are design speed (V d ), length and curvature of the crest curve (L CV and R CV ), length and curvature of the sag curve (L SV and R SV ), and tangent grade length (L TG ).The lidar elements are the number of vertical In this regard, two intriguing questions arise: how far can AVs see, and how fast should they drive?channels (N C ), laser-point threshold for vehicle detection (N T ), and mounting height (h mL ).After testing each trial, the ASD of the LAV is output.Second, "How far can the LAV see on highway vertical alignments?" is answered by investigating the relationships between the ASD and those variables.With the output of the ASD, "How fast should the LAV drive on highway vertical alignments?" is further addressed by setting the ASD equal to the RSD of the LAV.The answer is V max for the LAV.Finally, given a specific vertical alignment, V max related to the V d of that alignment is adopted as the speed limit.

Simulation Platform
As shown in Figure 2, we used the PreScan software package (version 2021.1.0)and MATLAB/Simulink (version 2018b) to establish the cosimulation platform.This platform is good at physics-based calculations of perception sensor inputs/ outputs.It is highly effective on cosimulations of roads, vehicle control, and AV systems [29] and has been employed in previous studies to investigate the ASD of LAVs (e.g., [22]).
Specifically, PreScan can offer a colossal actor database of vehicles, user-defined vehicle trajectories, roads with varied geometric characteristics, and top-notch sensor models [29].MATLAB/Simulink enables real-time data access from PreScan through a cluster communication portbased interface.The data herein include lidar outputs (e.g., the number of received laser points) and the vehicle's path information (e.g., the relative path distance).Based on these data, the ASD calculation can be programmed in MATLAB.Note that Wang et al. [13], [22] have verified the effectiveness of this cosimulation platform in investigating the ASD of LAVs.They compared the virtual ASD with the actual data, measured by Abdo et al. [30], under the same scenario.

Experimental Design
Given the virtual simulations conducted in this study, there is a need for a scenario-based experimental de-sign that can reflect various variables and handle welldesigned scenarios [31].In view of the advantages of the scenario-based design approach, a much-cited scenario design framework [28] was used, as shown in Figure 3.This scenario-based design can explicitly and systematically present the information required for the generation of simulation scenarios [32].
Specifically, the functional scenario has a high degree of abstraction, written or depicted in natural language, to clarify the scenario content, objective, and necessary components [28].Based on the components defined in the functional scenario, the types and value ranges (if any) of their representative parameters or variables are further defined in the logical scenario [28].Finally, a concrete scenario is established by sampling the parameters and variables defined in the logical scenario [28].

Functional Scenario
Regarding this study, the functional scenario (see Figure 4) can be expressed as a situation where the LAV drives along -Vehicles -Vertical Alignments -Lidar Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
a particular road alignment.A still target vehicle (TV), as a fixed obstacle, is on the desired path ahead of the LAV.Many previous findings have shown that a rear-end collision with another vehicle is the most common type of AV-involved crash [33].Moreover, the vehicle is also the representative object (or obstacle) in the HV-related ASD analysis [5], where the object height (0.6 m) for estimating the ASD refers to the vehicle's taillight height.Therefore, a TV was used in the scenario.
As shown in Figure 4, the passenger vehicle for the LAV and TV was selected according to the primary design vehicle in the highway geometric design [5].Also, the on-road passenger vehicle is one of the most important manifestations of automated driving technology [1].The vertical alignments include crest and sag curves as well as the tangent grade.In addition, a one-lane road segment was selected to enable the LAV's driving path to overlap the road centerline completely so as to explore the direct effect of geometry.

Logical Scenario
To serve the functional scenario defined previously, we further defined the components, including the vehicles (the LAV and TV), road geometry, and lidar.

Vehicles
Specific information on vehicles includes the dimensions, motion states, and locations.The Audi A8 vehicle model provided by the simulation platform was adopted for both the LAV and TV.The vehicle is 5.21 m long, 2.04 m wide, and 1.44 m high.
When measuring the ASD of the HV, the ego vehicle is usually set to drive at a uniform speed, closely related to the road environment, especially the geometry [5].Concerning the LAV, without external disturbances, its products (e.g., Tesla) usually drive at a uniform speed set by the driver's desire [34].To the authors' best knowledge, personalized automated driving strategies [35] that can imitate an HV's speed characteristics remain theoretical.
To maximize the safety benefit, the LAV is primarily mandated to drive with a minimal deviation from the lane centerline [36].This path feature is also consistent with that required in the HV-related ASD measurement.Therefore, a constant speed and a fixed driving path (i.e., the lane centerline) were used for the LAV independent of geometric conditions.
Very few efforts were expended to measure the AV's actual driving or operating speeds corresponding to various as-built geometric conditions, which are designed mainly by a V d -derived deterministic approach.In addition, to achieve better mobility, a speed closer to the safe margin supplied by the as-built road geometry could be the desired speed for AVs.Therefore, V d corresponding to the geometry was used as the LAV's driving speed.
Furthermore, the minimum speed within the AV's ODD is usually larger than 40 km/h [37].In addition, Varotto et al. [38] found an average speed of 107.2 km/h from the naturalistic driving data of AVs, which were collected on high-type highways and under normal driving conditions.Also, the driving risk is rarely attributable to the limited ASD solely when V d is larger than 100 km/h [39].Therefore, we adopted the driving speed (i.e., V d herein) for the LAV ranging from 40 to 100 km/h with an interval of 20 km/h.As for the stationary TV, it can be positioned anywhere along the driving path of the LAV from the end of the road segment until it is detected.

Road Geometry
To investigate the effect of as-built highway geometry, all adopted values of vertical geometric elements comply with the Chinese design specification for highway alignments [40].Also, other safety failure modes regarding vehicle dynamics (e.g., rollover) can be excluded.Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
TV and lidar FoV.Therefore, the ASD under different i G values is supposed to be the same as that on the tangent.Since the specified maximum i G decreases with an increase in V d [40], an | i G | of 4% corresponding to the maximum i G at V d = 100 km/h was adopted.
In addition, as mentioned, the ASD for i G = 4% is the same as that on the tangent.Wang et al. [13] have simulated the ASD of an LAV on the tangent at N C = 64, h mL = 1.44 m (referring to the roof of Audi A8), V d = 40-100 km/h, and N T = 10 and 20.Their results show that the driving speed of the LAV hardly affects the ASD on the tangent since there is neither lateral nor vertical relative movement between the vehicles.Based on their results, the average ASD values over V d = 40-100 km/h at N T = 10 and 20 are 71 and 43 m, respectively, which are much shorter than the minimum L TG (120 m at V d = 40 km/h) specified by [40].Therefore, to reduce the unnecessary sample size, we adopted L TG ranging from 0 to 70 m and 0 to 40 m at N T = 10 and 20, respectively, with an interval of 10 m.
■ Vertical curves: Regarding the vertical curve models in the simulation platform, the length (L V ) and curvature (R V ) of the vertical curve are related by where ~ is the algebraic difference in grades [40].To simplify the simulation, the i G before and after the point of vertical intersection (i G0 and i G1 , respectively) is assumed to be zero (or ±4%) and ±4% (or zero), respectively, and, thus, an ; ; ~ of 4% was used.Furthermore, to satisfy the requirements of both L CV (or L SV ) and R CV (or R SV ) specified by [40], their value ranges were selected or calculated based on the ; ; ~ of 4%, as listed in Table 1.Note that the minimum values of their ranges should not be less than their limited minimum values (L lim_Vmin and R lim_Vmin ) specified by [40]; the maximum values are determined by reference to the common minimum values (L com_Vmin and R com_Vmin ) specified by [40].The intervals of L V and R V are 10 and 250 m, respectively.Specifically, the following hold: ■ The maximum L V and R V at V d = 40-80 km/h use those at V d = 100 km/h to capture as many ASD features along the vertical alignment as possible.Therefore, the minimum L V and R V selections at V d = 100 km/h also apply to its maximum L V and R V .
■ Tangent grade and vertical curve: For the alignment of a tangent grade followed by a vertical curve, the values of their respective geometric design elements (e.g., L V ) are consistent with those stated earlier.In addition, since the LAV always drives along the lane centerline, and the lidar is mounted on the centerline of the LAV's width, the lane width and cross slope are not expected to affect the ASD.Therefore, a lane width of 3.75 m and a cross slope of 2% were adopted as per [40].

Lidar
According to the comparisons of various commercially available lidar products [24], [25], [41], [42], [43], the most significant difference among them is N C or vertical angular resolution (= VFoV/(N C -1) if vertical channels are uniformly distributed).The frequently used N C values are 32, 64, and 128 for the multichannel lidar, which have gradually become the requisite of high-level AVs (e.g., Waymo).On the contrary, many lidar manufacturers have reached a consensus on the design values of other technical parameters of their lidar products.Also, to eliminate the possible blind spot, they pay more attention to obtaining as good a horizontal-related sensing performance as possible, e.g., a wider horizontal FoV (HFoV).
According to these comparisons, the adopted values of the lidar technical parameters are shown in Table 2.Note that these values were selected concerning their general levels of current multichannel lidar products instead of referring to a specific product.Also, to simplify the lidar model setting, the uniform and symmetrical distribution of vertical channels was selected, which enables the adopted N C to correspond to vertical angular resolutions in order.For example, an N C of 32 corresponds to a vertical angular resolution of 0.97°.In addition, since the investigated road segment only extends longitudinally, an HFoV of 120° is sufficient.
Only one high-type lidar model with technical parameters listed in Table 2 was employed because that is the typical lidar configuration for detecting long-distance targets [44].Furthermore, two widely used mounting locations were considered: the front-end centers of the LAV's roof and bumper (or headlights) [10].Therefore, the adopted h mL values are 1.44 and 0.6 m, corresponding to the heights of Audi A8's roof and headlights, respectively.
As stated in the "Introduction" section, receiving sufficient laser points reflected from the target is essential for lidar-based detection algorithms.Although different algorithms might require more or fewer laser points, a generallevel N T could be determined by referring to previous studies.Specifically, Teichman et al. [45] found that the accuracy of the proposed algorithm decreases to approximately 80% when fewer than 50 laser points are received.The accuracy of the algorithm proposed by Suganuma et al. [46] drops to 85% when the number of points reduces to 25.Furthermore, Abdo et al. [30] and Wang et al. [13], [22] have investigated the effective range of lidar at many N T values, e.g., 10, 20, etc.Therefore, to reduce the sample size, two typical values of N T with a two-fold relation, 10 and 20, were adopted.

Concrete Scenario
Using the pair sampling approach [32], the values of parameters or variables were selected by considering all possible combinations within the set range according to the user-defined categories.Due to the different geometry properties of vertical curves and tangent grades, two categories-1) vertical curves and 2) tangent grades and vertical curves-were considered for the concrete scenarios.
Figure 6 shows the parametersvariables pair sampling for concrete scenarios.As noted, to reduce the sample size, 1) in the category of vertical curves, h mL = 0.6 m is only paired with N C = 64, and 2) in the category of tangent grades and vertical curves, only N C = 64 and h mL = 1.44 m are selected in the ranges of N C and h mL , respectively.Furthermore, the parameters and variables of each component are paired (see "+" in Figure 6), and variables between each component are paired (see solid or dashed straight arrows in Figure 6).

Experimental Process
Figure 7 shows the entire experimental process for each trial.As noted, first, according to the scenario design, two categories of concrete scenarios are established in the simulation platform.After starting the simulation, the ASD in specific geometric and lidar conditions is estimated by collecting N i , comparing it with N T , and outputting L i when N i meets the requirement.Finally, the ASD profiles for all concrete scenarios are created.More details regarding the programming of ASD estimation were described in previous studies [13], [22].

Results and Discussion
How Far Can the LAV See?

Vertical Curves
According to the results in the ASD profiles, Figure 8 shows the ASD at N C = 64 and h mL = 1.44 m along vertical curves with different R V (= L V /4%).As noted, the ASD increases Regarding the road segment covered by the lidar FoV, the overall curvature of its longitudinal profile decreases as the proportion of a tangent grade increases.
linearly as R V increases and then fluctuates around a general level.Specifically, the fluctuation amplitudes are about 20 and 10 m at N T = 10 and 20, respectively.Compared with the ASD results simulated by Wang et al. [13], [22], these fluctuations along vertical curves with different R V values are much larger than those along the horizontal curves with different radii (about 5 and 2 m at N T = 10 and 20, respectively), but the former frequency is lower.Also, there are considerable overlaps between ASD curves at different V d , which means that V d has little effect on the ASD along vertical curves.
The ASD, limited by the ; ; ~ of 4%, can cover the entire L V (= 4%R V ) at a small R V, but its actual features appear as R V increases to a critical R V (R Vcri ); i.e., ASD = L V if R V ≤ R Vcri .The features mentioned are attributed to the following factors: ■ Prerequisite: Instead of the horizontal angular resolution, the vertical angular resolution plays a significant role in determining the ASD along the vertical curves because there is no lateral movement between the vehicles.on the vertical curves is smaller than the yaw angle (or lateral displacement) on horizontal curves.Moreover, as shown in Figure 8(a) and (b), the ASD at N T = 10 is longer than at N T = 20, and the ASD on the crest curve is approximately the same as on the sag curve.Specifically, ASD ≈ 55-75 m and 40-50 m at N T = 10 and 20, respec-tively.This means that the as-built crest-curve pavement profile does not have a disadvantage in ASD (i.e., blocking the laser beams) compared with the sag-curve pavement.
Because of the insignificant effect of V d on ASD, the ASD at V d = 40 km/h is shown in Figure 9 to further explore the effects of N C and h mL .As shown in Figure 9 ASD increases with N C .Notably, more N C or less N T can reduce the difficulty of TV detection, thus increasing the ASD, which is consistent with the previous findings [13], [22].Figure 9(c) and (d) show that the ASD curve at h mL = 0.6 m is mostly above that at h mL = 1.44 m, especially when R VC or R VS is large.Due to the limited ~ and large R V , the closer the h mL (0.6 m) is to the middle height of the TV (1.44/2 = 0.72 m), the more the laser points can be emitted to it by the lidar with uniformly and symmetrically distributed vertical channels, as illustrated in Figure 10.This also aligns with the finding on horizontal curves [22].However, this would contradict the opinion that a higher h mL does enable a lidar to receive more target information without occlusion.To justify that opinion, it needs to assume that the TV can be detected once its boundary enters the lidar's FoV or it is applied in urban streets with several vehicles between the LAV and TV.To trade off the pros and cons of a higher or lower h mL , some lidar products (e.g., Velodyne's Alpha Prime) mounted on the vehicle roof use a VFoV with a downward offset and a more concentrated resolution distribution toward the road pavement, as depicted in Figure 10.
To capture the R Vcri and compare the ASD on sag curves with that on crest curves from the average perspective, Table 3 shows the R Vcri and the average ASD ASD ^h when R V > R Vcri .Note that, due to the adopted R V interval of 250 m, the resulting R Vcri might differ from the actual value.
As shown in Table 3, the R Vcri for crest curves is larger than that for sag curves in all conditions except at N C = 64 and h mL = 0.6 m.Also, the ASD on sag curves is basically the same as that on crest curves except that the former is longer than the latter at N C = 64 and h mL = 1.44 m.These indicate that a lower h mL does cause a shorter ASD on crest curves when R VC < R Vcri ; a higher h mL enables the ASD to cover a wider range of L VC , but the ASD substantially decreases when R VC > R Vcri , which is explained by Figure 11.
As shown in Figure 11, given the same R V , the FoV above the pavement of the crest curve is larger than that of the sag curve according to the geometric relation.However, on the crest curve, the laser pulses from the lower lidar chan-nels would be easily blocked, and most upper beams point to the air.On the contrary, most upper laser beams shoot to the TV on the sag curve.In addition, both vertical curves approach the tangent section as R V increases.It should be noted that the current V d -derived design of the crest curve cannot block the FoV of the LAV completely due to the limited , ~ which is required by vehicle dynamics safety [40].In Figure 11, although the area of the FoV above the pavement reduces as h mL lowers from 1.44 m to 0.6 m, the FoV can still cover the TV in the same position ahead.Also, the actual ASD on vertical curves (see ASD in Table 3) is significantly shorter than the range of lidar (200 m).
Furthermore, given the same N C , N T , and h mL conditions, ASD on vertical curves (see Table 3) is basically the same as that on tangent sections [13], but it is shorter than that on the horizontal curves [13], [22], especially for N C = 128 and N T = 10 (shorter by about 10 m).That further demonstrates the small effect of vertical curve pavement on the ASD.
In addition, it is noteworthy that the LAV's FoV might be obstructed by an overhead structure (e.g., a flyover) on sag curves, as illustrated in Figure 12.Therefore, whether such a structure reduces the ASD on sag curves is further examined.Note: a R Vcri corresponds to the critical L V (= 4%R Vcri ); b There is no R Vcri under those N C , h mL , and N T conditions.
Table 3.The critical curvature of vertical curves and average ASD for RV > RV cri .
Since that as-built structure must be designed to provide sufficient vertical clearance (h VC ) for the HV's RSD, it is convenient to calculate the ASD and then compare it with the ASD of the LAV.Specifically, the ASD of the LAV would be limited to ASD under if ASD under ≤ ASD.Otherwise, the structure would not obstruct the LAV's FoV.The ASD is given by [40] as follows: . R where ASD under is the ASD at the undercrossing, and h E and h O are the heights of the human driver's eye and object, respectively.The empirical values of h VC , h E , and h O are 4.5, 1.5, and 0.75 m, respectively [40].
Given the same R VS listed in Table 1, ASD under calculated by ( 1) is significantly longer than the ASD of the LAV.Accordingly, the ASD would not be affected by as-built structures on sag curves.

Tangent Grade and Vertical Curve
Based on the stated relationship between the ASD and R V , only four typical R V values, i.e., both minimum and maximum values of R V ranges at V d = 40 km/h (see Table 1), were adopted to investigate the independent effect of L TG .Figure 13 As shown in Figure 13(a) and (b), the ASD still fluctuates with L TG .In general, with an increase in L TG , 1) when R V adopts the maximum value, the ASD increases and decreases at N T = 10 and 20, respectively, and 2) when R V adopts the minimum value, the ASD still increases linearly (ASD = L TG + L V ) and then rises gradually (ASD < L TG + L V ).
Regarding the road segment covered by the lidar FoV, the overall curvature of its longitudinal profile decreases as the proportion of a tangent grade increases.Accordingly, the relative path distance within the FoV decreases based on the geometric relation, as with the ASD at larger R V and N T .Given that a large R V causes the longitudinal road profile to approach horizontal alignment, this aligns with the previous finding that, the higher the N T , the more consistent the change of the ASD with the relative path length [13], [22].On the contrary, a lower N T allows a longer ASD, but that ASD would fluctuate instead of complying with the change of the relative path length due to sparser channels.Furthermore, the ASD results at the minimum R V are consistent with those in Figure 8(a) and (b) because the additional tangent grade length can be considered the increasing R V .
Table 4 further lists ASD when ASD < L TG + L V .As shown in Table 4, all ASD results are basically the same as those corresponding results at N C = 64 and h mL = 1.44 m on complete vertical curves (see Table 3), except that the ASD (= 68.1 and 68.7 m) at N T = 10 on the tangent grade and crest curve are much larger than 61.9 m.It is necessary to highlight such an ASD reduction from a tangent grade followed by a crest curve to a complete crest curve at those lidar-related features.
Finally, both the ASD (see Figures 9 and 13) and ASD (see Tables 3  and 4) results answer, "How far can LAV see?"

How Fast Should the LAV Drive?
According to [1], AV automation levels that are still constricted by the ODD include L3 and L4.As proposed where t P_L3 and t P_L4 are the perception-brake reaction times of L3 and L4, respectively; t T is the driver's takeover time       Our answers to these two questions appear to fall well short of what the general public expects for an AV's potential advantages-i.e., it can see farther and drive faster.
highways needs to be comparable to that of HVs or vehicles with lower automation levels.The V d of a specific highway section can be tentatively regarded as a general driving speed of most vehicles.Therefore, to ensure speed consistency with V d (40-100 km/h), a typical V max range of (V d -20 km/h) to V d [49], [50] corresponding to the specified R V range was adopted [see Figure 14 Such a fluctuating V max along the alignment specified by a certain V d could be feasible in an individual AV's dynamic control, which improves the vehicle-based ODD.However, V max might not be attractive to road administrators due to its lack of generality.To obtain a more general V max for the speed limit, the average maximum speed Vmax ^h within the specified R V range was further calculated, as shown in Table 5.Note that Vmax was separately calculated using the maximum speed on the upgrade V _ max u ^h and downgrade .V _ max d ^h As shown in Table 5, Vmax with an exact value (see yellow or light red/green cells) can be used as a speed limit for vertical curves.In addition, the Vmax on the tangent grade and vertical curve are basically the same as those on vertical curves due to the same ASD except the case at N C = 64, h mL = 1.44, and N T = 10 on the tangent grade and crest curve.In that case, with ASD ≈ 68 m, Vmax would increase by about 4 km/h.However, more attention to a smaller Vmax should be paid on subsequent crest curves.Finally, the results of both V max (Figure 14) and Vmax (Table 5) answer the question, "How fast should the LAV drive?"Our answers to these two questions appear to fall well short of what the general public expects for an AV's potential advantages-i.e., it can see farther and drive faster.This is mainly attributed to the specific experimental designs related to lidar adopting only a single multichannel lidar and N T of more than one laser point, which were justified earlier.Therefore, how the LAV safely adapts to the as-built vertical alignment could be solved by referring to the proposed speed limits.Otherwise, to help the LAV see farther and drive faster on as-built vertical alignments, it is suggested to equip it with more lidars, incorporate more channels, or develop algorithms with fewer N T .In addition, we conjecture that detection capability beyond the SD can be achieved by deploying roadside monitoring sensors [51], which is also beneficial to improving the LAV's V max .

Concluding Remarks
This study adopts a virtual simulation method and conducts a scenario-based experimental design to answer the question, "How far can the LAV see on highway vertical alignments?"Then, the SD and speed consistency requirements are considered to answer the question, "How fast should it drive?"To the authors' best knowledge, this study is the first attempt that considers the LAV's ASD issues on vertical alignments and proposes the corresponding speed limits.Based on the study, the following comments are offered: ■ The answer to "How far can the LAV see on highway vertical alignments?" depends on variables regarding vertical geometric elements and lidar.Specifically, the ASD increases as N T decreases, N C increases, or h mL decreases.Importantly, the ASD on crest curves is the same as on sag curves or tangent sections, which means that the as-built crest-curve pavement profile would not limit the ASD.Also, since lidar's vertical angular resolution is generally coarser than the horizontal, the ASD fluctuates around ASD as R V increases, which is more noticeable than horizontal curves.Consequently, these results provide new insight into effects attributable to the features of the vertical geometry and lidar on an LAV's ASD variations.They further highlight the limited perception capability of current AVs from the perspective of road safety.
■ The answer two "How fast should it drive?' further depends on the LAV's automation level besides those variables.Specifically, ranges of V max for L3 and L4 are 30-55 km/h and 55-85 km/h, respectively, at N T = 10 and 20 -40 km/h and 45-75 km/h, respectively, at N T = 20.An important practical implication of this study is related to the proposed speed limit within the road section, which regulates L3 and L4 safe speeds and improves road-oriented ODD specifications.Also, only at N C = 128 and N T = 10 would L4 be compatible with asbuilt vertical alignment with V d = 100 km/h.However, even under these conditions, L3 can only adapt to the vertical profile with V d = 60 km/h.
■ The main limitation of this study is a lack of consideration of weather effects.On the one hand, adverse To help the LAV see farther and drive faster on as-built vertical alignments, it is suggested to equip it with more lidars, incorporate more channels, or develop algorithms with fewer N T .
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
weather (e.g., rainy days) would impair lidar functions and, thus, shorten the ASD [30].On the other hand, the weather might also impact the RSD due to the wet pavement or longer takeover time.In addition, this study only considers one AV type of passenger vehicle.Besides the ASD, speed limits for heavy-duty AVs on vertical alignment must account for their braking capacity and truck drivers' reaction times [10].Therefore, extensive tests that include weather effects and automated trucks should be conducted in the future.Also, in the future, we are interested in extending the road geometry from 2D to 3D, i.e., combined alignment, which would be more consistent regarding the reality of geometric conditions.

FIG 1
FIG1 Possible sightline obstructions for AVs on crest and sag vertical curves.lidar: light detection and ranging; TV: target vehicle.

■FIG 5
FIG5 The relative position between the LAV, TV, and lidar VFoV under different i G values.

:FIG 6
FIG6 The parameters-variables pair sampling for the concrete scenarios.

FIG 8 FIG 9 R
FIG 8 The ASD at N C = 64 and h mL = 1.44 m: (a) crest curves and (b) sag curves.

FIG 10 FIG 11 A
FIG10 The TV coverages by the lidar's VFoV at different h mL values.Note that the vehicles are simplified as rectangles, and the VFoV is simplified as a triangle.
(a) and (b) depicts the ASD at N C = 64 and h mL = 1.44 m along the tangent grades and vertical curves with different L TG values.

N T = 10 RFIG 13
FIG13 The ASD on (a) tangent grades and crest curves and (b) tangent grades and sag curves.
FIG14 V max : (a) for crest curves, N T = 10; (b) for crest curves, N T = 20; (c) for sag curves, N T = 10; (d) and for sag curves, N T = 20.Note that upgrade refers to the crest or sag curves with i G0 = 4% or zero, respectively, and downgrade refers to the crest or sag curves with i G0 = zero or -4%, respectively.

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40km/h adopts the L lim_Vmin , and the minimum R CV is calculated by (L CV /4%), while the minimum R CV at V d = 60-100 km/h uses R lim_Vmin , and the minimum L CV is calculated as 4% of the R CV .SV values at V d = 40 and 60 km/h use R lim_Vmin , and the minimum L SV is calculated as 4% R SV , while the minimum L SV values at V d = 80 and 100 km/h use L lim_Vmin , and the minimum R SV values are calculated as L SV /4%.
■ The minimum R

Table 1 .
Adopted ranges for the length and curvature of vertical curves.

Table 2 .
Adopted values for the lidar technical parameters.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Table 4 .
The average available SD for ASD < L TG + L V .

Table 5 .
Average maximum speeds of LAV for upgrades and downgrades (V max_u .V max_d ) (a)-(d)].Furthermore, given specific geometry, lidar, and automation level conditions, if V max > V d and (V d -20 km/h) ≤ V max ≤ V d , such an LAV driving with V d and V max , respectively, could satisfy both the SD and speed consistency requirements.However, if V max < (V d -20 km/h), the speed consistency for the LAV driving with max will fail.As shown in Figure14(a)-(d), generally, V max for L3 and L4 at N T = 10 ranges from 30 to 55 km/h and 55 to 85 km/h, respectively, and both V max values for L3 and L4 reduce by 10-15 km/h at N T = 20.