Road geometrical design out of standards: a preliminary study in a simulated context

When a road design solution is quite out of standards for the presence of insurmountable constraints, there is the need for an objective procedure aimed at achieving a certain level of safety for drivers. To avoid issues on judicial responsibility, designers aim to fully satisfy the existing standards, possibly without any exception. Traditional methodologies based on previous experience or road administrators’ guidelines generally caused problems due to the high subjectivity involved in the analysis. In this paper, to overpass these issues, a rational procedure based on vehicles telemetry data in a simulated environment is proposed. This process, through synthetic indices, allows the analysts to compare two road geometries, similar but different, because one includes curves with shorter residual circular arcs than threshold values imposed by Italian standards. The main results, derived from a two-way ANOVA with subsequent contrast analysis, suggest that a certain deviation respect to the standards did not determine any decay in the driver’s performance. Compared to the existing literature, in this study, a full objective procedure was proposed, based on a totally new indicator, which can be easily adapted to any context, involving driver, road and vehicle at the same time.


Introduction
The existence of appropriate road standards assures that, already in the design phase, it is possible to confer to an infrastructure an ensemble of proper physical and geometrical features to avoid potentially dangerous contexts and situations. But compliance with standards is not a trivial operation and, in any case, it does not protect against further unforeseen critical issues, detectable only through subsequent deepening (Rizaldi et al. 2017). On the other hand, there could be deviations from these requirements with a sufficiently safe driving behaviour (Weekley et al. 2016).
When satisfying all the standard prescriptions is too expansive, the problem of determining any criticisms for drivers' safety arises (Pellegrino 2009(Pellegrino , 2011Montella et al. 2012). In these cases, some standards provide specific indications to the designer while others propose only generic advice. In Italy, for instance, a certain design flexibility beyond imposed thresholds is permitted, as this choice must be supported by opportune safety analyses that have never been specified (MIT 2001).
Other International Road Standards do not specify analytical and objective procedures to justify the deviation from the prescribed limits. In these cases, designers analyse an aspect or a manoeuvre having similarities with the unsatisfied check that can justify the acceptance of out-of-range results.
The idea of taking in remarkable consideration important environmental, economic, or constructive constraints led to rethink mandatory and strictness of the regulations for building new roads or for maintaining existing ones. USA, around the 2000s, reported the studies on design flexibility in a wide research sector called Context Sensitive Solutions, in which there are all those situations that, for their complexity, abound in particularly anthropized territories (FHWA 2004;NCHRP 2004;Bosurgi et al. 2011). Their principle is that deviations from standard thresholds, admitted only when needed, do not necessarily induce a decay in users' safety but, instead, may represent the occasion for a better management of the overall process through the optimization of all the involved variables (NCHRP 2003;WSDOT 2005). Other countries, such as Great Britain, already for several years have proposed, beyond traditional "desirable minimums", less strict design scenarios, named Relaxations or Departures, for which a specific application field is defined (TD 9/93 2002). Some procedures are characterised by a remarkable empiricism, as in the case of Road Safety Audits (Huvarinen et al. 2017), where it is assigned a certain safety level to the infrastructure, according to a team of experts and some supporting manuals. Whether this level is difficult to reach, they have also to propose appropriate maintenance operations or activities for mitigating dangers or limiting the traffic.
Compared to this undeniable subjectivity, even though duly qualified, there is the need of applying analytical procedures for quantifying the users' performance, in real conditions representing the analysed scenario, often not predictable by road standards. In the recent past, with the progress of vehicle on board sensors, the estimation of the driving behaviour performance was entrusted to some indices able to synthetize the performance on a homogeneous element of the road that, preferably, was the horizontal curve (Chen et al. 2022;He and Donmez 2022). At this regard, one of the most used variables in the scientific research was the Lateral Position (LP), as the trajectory in curve directly influences the values of the lateral acceleration and steering and, in turn, the manoeuvre safety. Then the need to obtain a single value representative of the driving along a homogeneous geometric element (for instance, a curve) imposed the adoption of synthetic indicators such as the standard deviation or the average of LP, named SDLP and μLP, respectively (O'Hanlon 1984;Ramaekers 2003;Coutton-Jean et al. 2009;Verster and Roth 2012;Brookhuis 2014;Hu et al. 2017;Kazemzadehazad et al. 2019).
In recent years, other indicators related to driving behaviour have been proposed, such as the so-called Time to-Line Crossing (TLC), determining the minimum time for the vehicle to overpass a marginal line, without any corrective action (Godthelp et al. 1984;Van Winsum et al. 2000). However, this is a very complex function, and its average or standard deviation does not permit to interpret the actual driving behaviour. This index does not fit to statistical analyses, but it can help to simplify a descriptive interpretation of the observed phenomenon.
Among the literature models derived from LP, it is interesting that one proposed by Cerni and Bassani (2017), concerning an index for "dimensionless average curvature difference", that is the difference between the curve radius and the trajectory followed by the vehicle.
It is clear that the horizontal curve represents one of the most critical geometric elements in terms of safety (Bìl et al. 2019;Elvik 2019). Calvi (2015) analysed in a simulated environment the relationship between the driving behaviour and the curve features, like the radius, presence of clothoids, visibility or transversal section. In particular, he measured the driving performance by means of some indicators, such as mean and standard deviation of speed, with other recently proposed (Calvi 2010;D'Amico 2006, 2013) as the Pathologic Discomfort (PD) and the Dispersion of Trajectory (DT). As known, the lateral acceleration, when the user follows the trajectory represented by the lane axis, is only a function of radius (R) of the lane and speed (V) values, i.e. a L = V 2 /R. If its real measure exceeds this value, then the trajectory is different respect to the axis and, probably, there has been an incorrect interpretation of the curve by the driver. The analytical expression for PD consists in a subtraction between the two functions: the theoretical a L and the actual values measured by the telemetry. DT index has the same meaning, as it measures the deviation of the real trajectory from the theoretical one (i.e. the axis of the lane) along the curve.
Subsequently, there have been many studies relating the analysis of trajectories with the driving behaviour or the environment. For example, Bassani et al. (2019) related some representative variables of driving behaviour, such as trajectories, speed and sight distance when travelling on road curves. Among the results, it was ascertained that the improvement of visibility conditions resulted in an increase in speeds but in a lower Dispersion of the Trajectories. Khakzar et al. (2021) analysed the driving behaviour of a sample of users by examining the vehicle trajectories, concluding that these are more influenced by external conditions-such as traffic flows or activity in secondary tasksrather than by the individual characteristics of the drivers.
The most modern studies (Gouribhatla and Pulugurtha 2022;Miller et al. 2022) investigated the role of some driver assistance systems but the variables on which they have tested their effectiveness are always those related to driving activities (lane departure, speeding, stopping distance, turning, car following). The trajectory is always perceived as a variable of great interest not only because it is indicative of the driver's ability to interpret but, as in the case of the study by Yang et al. (2021), it can be assumed as an element of prediction for future accidents. In this regard, the authors have prepared a Deep Learning model capable of predicting any driving error with only vehicle trajectory data. The data set was derived from a simulation programme (PC-Crash) and made it possible to classify these events in six different classes with an accuracy of over 95%.
From an examination of the most pertinent literature, it is clear that the relationship between the trajectories can be highly representative of the driving behaviour, but it is opportune to relate them to the geometry of the road, to identify eventual critical configurations for safety.
Before the introduction of the driving simulators, speed prediction was performed through regressive model often based on survey campaigns performed only on few points of the road. Some basic hypotheses, such as constant speed on homogeneous elements, were highly limiting and unrealistic. For this reason, Montella et al. (2015), relying on experiments in a simulated environment, deduced the trends of speed and longitudinal acceleration, proposing a piecewise linear regression model, able to predict the operative speed trend on the entire alignment, identifying start and end points of constant speed sections. The results, in macroscopic terms, permit to deduce the operative speed variation law along curve as a function of the radius.
Finally, it is relevant to underline how a large part of the recent scientific production focuses on specific conditions of the road environment, to solve strongly localised problems, and thus, the results are impractical to be used in other contexts (Boruah et al. 2020).
In a future that is not imminent yet, road vehicles will be fully autonomous. However, we are experiencing a long period of transition that started with the first driving information devices and will end in an undefined time. The challenge for vehicle manufacturers and road infrastructure managers will be to design driving aid equipment that can improve user safety, without increasing conflicts between different component of the road system, workload or distractions in an unsustainable way (Vanderhaegen 2021). Then the identification of variables suitable for explaining driving behaviour can allow to properly design and calibrate these instruments. The road manager, for his part, can acquire the information coming from the ADAS of the vehicles and improve the active and passive safety of his infrastructure.
The proposed literature review evidences some limitations that should be overpassed: • Several road administrators rely on empirical procedures for controlling infrastructure safety, strongly based on experts' judgment and suggested by government offices. • When analytical procedures are applied, the research has focussed to a single component of the road system (man, vehicle or road), neglecting the others and, mainly, their mutual interactions. Performing tests on roads is the unique way for verifying road safety in general terms, considering real or simulated environment in which all the elements interact. • Synthetic indicators, required for statistical analyses, are only partially representative of the actual phenomenon to interpret.
• The proposed procedures are very often tested in very specific environments and may not be generalised.
This research responds to some of these critical issues, through the proposal of an analytical procedure allowing the analysts to verify the impact on safety when some legislation requirements are not satisfied. In detail, the driving behaviour in terms of trajectory along some horizontal transition curves was investigated. Some of the curves were perfectly in compliance with Italian standards, while other-almost similar-presented the residual circular arc shorter than imposed minimum values. The experimentation was performed in a simulated environment on a sample of 21 users. The authors analysed the results, even in terms of proper synthetic indicators, partly innovative respect to existing literature. These indicators went through appropriate statistical procedures for evidencing any critical criticisms in the driving behaviour. At this purpose, the authors believe that a slight deviation from standard limits of a transition curve geometry can be accepted only if the trajectory and the steering speed do not undergo a statistically significant worsening compared to the standard condition.
However, even if the methodology is easily generalizable in any type of context, the results of this preliminary study are not yet applicable on a large scale since there is still a need to expand the experiments appropriately.

Methods
It should be noted that a sharp distinction between man, vehicle and road components is always complex and often leads to incorrect conclusions. The experimental tests on real roads or in a simulated environment do not allow easily to separate their specific contribution and even if, in this research, the vehicle trajectory and the steering speed were assumed as the main variables, there is no doubt that this was also influenced by the type of the vehicle and the characteristics of the users' sample.

Synthetic indicators representative of the trajectory
As anticipated in the last part of the Introduction section, some researches on vehicle trajectories applied indicators as SDLP and μLP, neglecting some evident critical issues (Bobermin et al. 2021). Probably, these indicators contribute to a certain knowledge of the phenomenon, but do not permit an objective and valid quantification of the observed behaviour in all the situations.
To evidence these issues, in Fig. 1, three hypothetical trajectories (in truth they are sinusoids) in a road lane and 1 3 shoulder for overall 4.5 m wide, from which the following considerations are deduced: -Trajectories 1 and 2, despite the same μLP, are deeply different, as the first shows higher amplitude and, thus, a higher potential danger for lateral elements (obstacles, barriers or opposite vehicles). -Trajectories 1 and 3, instead, have the same SDLP. It is evident that trajectory 3 is moved towards the opposite lane and, thus, it is more dangerous in terms of a potential frontal collision. -The deviation of trajectory 3 (or 1), whether pointing towards the right side of the curve, would not involve high risks, owing to the absence of obstacles (beyond the barrier), but μLP nor SDLP provide any useful indications in this perspective.
At this regard, for right curves and with sufficiently wide lanes, the driver often voluntary moves towards the centre of the curve, for driving on a shorter trajectory. Then the ideal indicator should consider the left deviation only, as it causes potential negative effects. For this, it would be sufficient to delete right deviation data with respect to the lane axes, as represented in Fig. 2.
Consequently, the previous indicators may be redefined as SDLP L and μLP L , to indicate that only left deviations with respect to the lane axes will be considered.
Finally, it is fundamental to assess for how long time or space the driver moves on the left of the axes. This information may be easily derived, referring to the integral of the left deviation function, representing the area between the left part of the trajectory and the lane axes (Fig. 3). The novel index is identified with the symbol INT L . The authors think it is appropriate then to consider the three indicators (INT L , SDLP L and μLP L ) together, for deducing any deficit in driving along a road, as hardly one of them alone is representative of all the observed phenomena.

Synthetic indicator representative of driving behaviour
The study of the trajectory has been integrated with the analysis of an indicator called "Max speed steering" since the steering represents the component, together with the pedals, through which the driver transmits his intentions to the vehicle according to the road environment. A partial acquisition of necessary information can cause errors and indecisions that users directly transfer to the steering. This does not automatically mean that an accident occurs, also because compensatory mechanisms are involved, which in most cases allow corrective manoeuvres in total safety. The study of the trajectory alone could hide these situations, while the activity on the steering and, in particular, the maximum steering speed could bring out these critical driving behaviours. It is known that, from a theoretical point of view, the steering speed has a constant trend on the transition curve and a value equal to zero on the arc of circumference but only if the user perfectly follows the lane axis. In real cases, there are oscillations around the ideal values, but the average calculation along the curve would hide them. To solve these problems, it was, therefore, decided to refer to the maximum steering speed values.

Opportunities of the simulated environment
Road safety, as anticipated in the Introduction section, should consider all the components of the road system, i.e. the driver, the vehicle and the road context. Performing tests on real roads or accurate simulated environments represent the only ways for involving them simultaneously. For this research, the second option is selected, according to generally discussed reasons evidenced in literature (Maxwell et al. 2021): -Users' safety during tests.
-Repeatability and homogeneity of the investigated scenarios in terms of weather condition, light and traffic, hard to obtain in real contexts. -Complete control of vehicle telemetry, pavement condition and main driver's psycho-physiological factors (Graichen et al. 2022). -Possibility of testing particular geometric elements not yet realised. -Accurate design of the road geometry.
-Proper choice of the variable that may influence the road environment; at this regard, the external disturbance was limited (absence of elements that may distract the driver or represent an obstacle to visibility, absence of traffic, Obviously, the use of a driving simulator also has some disadvantages such as the absence of kinaesthetic feedback, eventual simulator sickness, need for results validity with respect to real context, driver's motivation and perceived risk level (Kuiper et al. 2020;Chinazzo et al. 2021).
Regarding a total fidelity with respect to real conditions, it should be noted that the proposed study has been based on a commercial and very performing software (SCANeR ® ), widely adopted in recent years by many automotive manufacturers. Furthermore, the deducted conclusions concern the comparison between scenarios present only in the simulated environment, avoiding comparing mixed conditions.

The driving simulator at University of Messina
The experimentation has been performed using the driving simulator named SimEASY ® , produced by AVSimulation, available in the Digital Laboratory for Road Safety (DiLaRS) of the University of Messina (Fig. 4). This simulator has the following features: -Three 29-inch full HD screens (1920 × 1080 pixels each) with a horizontal field of view of 130° and a frequency higher than 50 Hz. -A steering wheel characterised by a force feedback sensor to simulate the rolling motion of wheels and shocks. -Sound effects reproduced through several speakers and subwoofers. -The SCANeR ® studio software, used to design tracks, generate the environmental context and run trials.
-Data collected with a frequency of 10 Hz.
-A family car powered by a 130 hp gas engine, with six manual gears and automatic clutch.
In this preliminary phase of the research, the vehicle has not been the subject of investigations and has been selected according to the indications provided by the Italian road standard.

Features of the alignment
The experimental road is represented by about 5 km long alignment, characterised by a succession of 18 transition curves, all including in and out transition curves and a residual circular arc.
The road belongs to the type called F (local rural) in the Italian standard and consists of two lanes in each direction of travel 3.50 m wide each and two shoulders of 1.00 m each.
The first nine curves can be divided in four curves, with a 60 m radius (called R60), followed by other five curves with radius equal to 100 m (R100). They are in compliance with Italian standard and, in particular, with the minimum imposed length of the residual circular arc, defined as the space driven by the driver moving at the design speed for 2.5 s. This means that for R60, as the design speed is equal to 45 km/h (12.5 m/s), the minimum length is equal to 32 m. Analogously, for R100, the design speed is 56 km/h (15.5 m/s) and, thus, the minimum length of the circular arc is 39 m.
Without discontinuities, the alignment continues with other nine curves, in this case not in compliance with standards, in terms of residual circular arc length. The first four of these curves again have radius equal to 60 m (called R60out) and the following five equal to 100 m  Table 1 and Fig. 5) are not in compliance with standards, being excessively lower than previously indicated thresholds.
Therefore, in total, there are then four types of curves, named R60, R100, R60out and R100out, the features of which are listed in Table 1 and Fig. 5.
In this experiment, the radii of the curves equal to 60 and 100 m were used as they are the most frequent in this type of road (local rural) and, at the same time, they are sufficiently different from each other to highlight particular drivers' behaviours dependent on the radius of the curve.
The R60out and R100out curves are characterised by arc lengths close to zero (0.3 and 6 m, respectively), caused by a reduction in the angle of deviation from 80° to 40° (Fig. 5).
It should be emphasised that these curves are arranged along a circuit and the starting point of each of the drivers is not fixed but random so as not to create a dependence of the results on the succession of curves. Finally, it is specified that the experimentation was carried out in the absence of traffic, given that the Italian legislation refers to an isolated vehicle.
The road markings are made up of continuous lines both at the edge and in the middle of the cross-section and the surrounding terrain is flat in such a way to not constitute a barrier for sighting.

The drivers' sample
The driving tests involved 21 users, between 22 and 26 years old, selected in such a way as to constitute a homogeneous sample with respect to age, number of years of driving license, presence of light visual impairments as myopia (below two dioptres), number of accidents experienced, eventual car sickness recorded after the activity driving to the simulator. This research complied with the American Psychological Association Code of Ethics and an informed consent was obtained from each participant.
In Table 2, the main results have been reported and in the bottom row, the standard deviation shows a good consistence of the sample.
The experimental phase was characterised by the following steps: a) Complete a pre-and post-drive questionnaire. b) Drive on a first pre-selected track (the duration of this step was subjective since the driver keeps driving until felt comfortable with the driving commands). c) Drive on the main track (for about 10 min).
The calculation of the sample size should derive from considerations related to the variance and the magnitude of the confidence level which, generally, is assumed to be 95%.
Given the good homogeneity of the selected drivers (see Table 2), the sample size was calculated as a function of the desired precision p and the expected frequency F using the formula below.
In the present study, the expected frequency F was set at 5% and the absolute precision p at 10%, obtaining a minimum number of 18, lower than the actual sample size, equal to 21.  5 Geometrical scheme of the four types of curves used in the experimentation. The first two were built in accordance with the legislation while the last two do not respect it in relation to the length of the circular arc. In every configuration, the shape parameter A of the transition curve and the radius R of the curve have been indicated. The overall length of the alignment consisting of a tangent, a first transition curve, a circular arc and a second transition curve is shown on the extreme right of each row. All lengths are in metres As mentioned, this research aims to propose a procedure that has general validity, but the results deduced only for a particular sample (young people between 22 and 26 years), a single type of vehicle and good conditions of the pavement cannot be valid when the scenario is different. The future developments will also include cases different from the one being tested.

Two-way ANOVA
Since each user is measured more than once under all levels, the authors performed an ANOVA factorial design within subjects with the following factors: -Type of curve (four levels: R60, R100, R60out, R100out, according to Table 1 and Fig. 5). -Direction (two levels: left, right). The response variable (or dependent variable DV) is represented by the performance indexes already introduced: INT L , μLP L , SDLP L , Max Speed Steering.
The reliability of the results depends on the satisfaction of the assumptions based on ANOVA analysis. In this case, the assumptions regard the following: -The dependent variable must be measured at the continuous level. In our case, the measures of the indexes are expressed in squared metre for INT L , and in metre for μLP L and SDLP L , i.e. in a continuous way. -The two within-subjects' factors (i.e. two independent variables) should consist of at least two related groups that indicates that the same subjects are present in both groups. They have been divided in four levels (the Type variable: R60, R100, R60out, R100out) and two levels (the Direction variable: right, left), respectively. -The observations are independent, without relationship between the observations in each group or between the groups themselves. -Absence of significant outliers. -Tests for normality by means of residuals.
-Check the sphericity, i.e. the variances of the differences between all combinations of related groups, were equal. When these conditions are violated, the Mauchly tests for sphericity was performed, adjusting the analysis by a correction criterion as the Greenhouse-Geisser method.
Since the authors have to perform a two-way ANOVA and there is the effect of two independent variables and the effect of the independent variables on each other, there are three pairs of null or alternative hypotheses, as following: However, if an ANOVA test shows significant results, it cannot say where those differences lie. In these cases, the post hoc Tukey's HSD (Honestly Significant Difference) test was run to find out which specific groups' means (compared with each other) are different.

Limitations
In conclusion of this section, it should be emphasised that the experimental conditions applied to this research will lead to some limitations of the results: The "Accidents" column includes both accidents suffered and caused. The "License" column relates to the number of years of possession of the driving license. The "Myopia" column includes only values of 1 (presence of myopia less than 2 dioptres) or 0 (no pathology). In the same way, the "Car sickness" column includes only values of 1 (the driver got a little nauseous while driving) or 0 (no problem)

Driver
Age Accidents  License  Myopia  Car sickness   1  23  0  4  0  0  2  24  0  5  0  0  3  22  1  3  0  0  4  25  0  6  1  1  5  -The simulated environment, the advantages of which have already been illustrated, could lead to a less realistic driver's response, due to a lower perception of risk. -Although the trajectory is the result of an action by the driver on the vehicle based on the perception of the surrounding environment, it would be appropriate in a subsequent step of the research to investigate some human factors related to vision, reaction times and workload. -The vehicle on which the experimentation was carried out is that indicated by the Italian road standard, but, in future studies, it would be interesting to test other types of vehicles and, above all, the interaction between vehicles. -The surface characteristics of the tested road pavement are indicated by the Italian road standard but, also in this case, critical conditions could be tested as regards roughness and friction especially. -The road geometry influences the sight distance and, therefore, the trajectories which could be profoundly different in other configurations. Figure 6 represents LP of user 1 along curve 4 (left R60). It is possible to notice that the zero of the ordinate axes coincides with the lane axis, while the unit of the abscissa is the time. Since the curvature is measured in the time domain, its trend shows irregularities as the speed is not constant along the curve. The synthetic indicators, representing the ANOVA's dependent variables, were calculated for each curve and each user. In total, a dataset with 2 independent variables (type and direction of the curve) and 3 DVs (INT L , μLP L , SDLP L ) with 378 records (21 users × 18 curves) was defined (Table 3 is referred to only a driver).

Results
Three different two-way ANOVA were performed, in which the DV in turn was represented, respectively, by INT L , MLP L , SDLP L . All the assumptions listed in the Methods section were verified. For example, in Fig. 7, the quantiles of the residuals are plotted to verify the normal distribution. The normal probability plot of the residuals should approximately follow a straight line and in the case of INT L , this hypothesis is quite satisfacted. The other DVs (μLP L and SDLP L ) present similar trends and for the sake of brevity have not been inserted. In the following, the results of the three ANOVA are reported.

Two-way ANOVA with dependent variable INTL
There is a significant difference in the average, both for  Table 4, the Tukey's post hoc test did not reveal significant differences in driving along the four types of right curves (p value in couple comparisons between 0.4088 and 0.9991), while in left ones, R60 induces a different driving behaviour than other curves (p value < 0.0001). Figure 8, concerning the estimation of the marginal averages of the various levels of the independent variables, shows a substantially homogeneous driving behaviour in right curves, regardless of radius or length of residual arc. Among left curves, R60 presents an average INT L value equal to 2.77 m 2 , higher than other curves that, instead, exhibit not significant differences.

Two-way ANOVA with dependent variable μLP L
Also in this case, there is a significant difference in the average both for variable Type In Table 5, the Tukey's post hoc test was reported: it did not reveal significant differences in driving along four right curve types (p value in couple comparisons between 0.7121 and 1.0000), while in left ones, there is a different behaviour between couples R60-R100, R60-R60out and R60-R100out (p value < 0.0001). Furthermore, in this case too, there is a relevant difference between right and left curves. Figure 9 evidences a substantially homogeneous behaviour for driving along all right curves, while for R60 and R100 left curves, different behaviours than other two types were measured.  (Table 6) did not show significant differences for the four types of right curves (p value in couple comparisons between 0.3864 and 0.9991), while in the left ones, there is a different behaviour between couples R60-R100 and R100-R60out (p value between 0.0004 and 0.0007). Moreover, in this case too, right and left curves determine different behaviours. Figure 10 shows a substantially homogeneous trend in driving along right curves, regardless of radius or compliance with standards for length of residual arc. Considering left curves, R60 curves present very similar behaviours, while R100 presents minor deviations.

Two-way ANOVA with dependent variable max speed steering
Also in this case there is a significant difference in the average both for variable Type [F(3,210) = 28.96, p = 0.000], for Direction [F(1,210) = 10.13, p = 0.004].
Referring to the contents of Table 7, for the curves to the right (R), the Tukey's post hoc test highlights that the curves with R = 100 m exhibit a very similar steering behaviour. While a much higher steering speed is noted for the 60 m radius in the configuration that does not comply with the standards.
In the case of left-hand curves (L), complying and noncomplying curves cause similar driving behaviour, regardless of the radii used (Fig. 11).

Discussion
The Italian Road Standard (but also the international ones when they introduce the "relaxations" from the prescribed limits) admits an overcoming from the thresholds as long as the designer explains his own decisions which, however, are often based on personal considerations and experience, but with reduced scientific basis. The proposed procedure, on the other hand, intends to introduce an analysis methodology based on scientific and objective data that takes into account variables related to the actual travel of that road section by a representative sample of users. About this, the authors believe that a slight deviation respect to the road standard of some curves can be accepted if the trajectory or the steering speed does not undergo a statistically significant worsening compared to the standard condition. This result is valid, of course, only for the experimentation of this study, without any general meaning. However, the authors believe that the proposed procedure can be always applied by a road safety inspector or a road designer since their burden consists only in the collection of traffic data, in the identification of appropriate indicators and in the subsequent statistical processing. Even some existing roads, some elements of which do not comply with the standard requirements, could be verified through this methodology to ascertain the safety of users' manoeuvres.
It should be underlined that the results previously reported are not directly generalizable, even because they reflect some specific characteristics of this experimentation, hardly of common interest. However, the presented method may be applied to almost every scenario in which quantifications on infrastructure safety level are required, both in absolute terms and in comparative analyses.
In this study, the attention was focussed on the road component on which the road designer can effectively act, confining the other two elements-the vehicle and the human factor-within very limited variability. A broad generalisation of the results must include greater attention to these components as well and will be developed in the next stages of the research.
The advantages obtained through this procedure may be listed in the following: -Reproducibility and repeatability of tests, guaranteed by the homogeneity of the driving scenario for all users. -Final considerations derived from the real driving of drivers and not from purely theoretical hypotheses of the standards, that are far from reality (for example, the coincidence between lane axis and trajectory). -Objective procedure based on telemetry data, focussing on variables specifically indicative of the observed phenomenon. In this case, the attention is on trajectory, as it would be assumed that a too short circular arc would cause a more complex manoeuvre for the driver.

3
The statistical analysis referred to synthetic indicators is able to quantitatively represent the driver's performance in terms of his trajectory on a specific curve. They relied on the following features of the vehicle motion: -Capacity of discerning right and left deviation from the axes. In the first case, the crossing of the axis line of the lane can be fully voluntary to reduce the travel distance.
On the contrary, a left deviation could cause a danger against vehicles driving in the opposite direction. -"Meandering" from the lane axes represents a danger only if its amplitude is relevant. In any case, its average value may lead to an underestimation of the danger ( Figs. 1 and 2) -The time/space in which the vehicle deviates from the lane axes. A short deviation, followed by a quick fix, is almost physiological and does not cause any critical issue for safety (Fig. 3).
The considerations reported at the end of the Introduction section evidence that there is not a perfect synthetic index, and the best solution is considering more than one. For these reasons, novel indicators, as INT L , μLP L e SDLP L , Max Speed Steering were proposed, representing the DVs for the three different two-way ANOVA. The exam of the results confirms that focussing on left deviations has led to similar results with the three indicators, evidencing only the most critical phenomena for safety. In all the analyses H1 hypothesis was confirmed, i.e. at least one of the averages of the groups is different from the others. However, this result is not interesting from an engineering point of view, since it does not deepen the relationship between the dependent variables (INT L , MLP L , SDLP L or Max Speed Steering) with the single levels of the two independent variables (Type and Direction). In this case, a multiple comparison test among the averages of the involved groups permitted to complete the information required to interpret the phenomenon, as noticed in Tables 4, 5, 6 and 7 and Figs. 8, 9, 10 and 11.
Generally, there is not a remarkable difference in driving along right curves, regardless of radius or residual circular arc length. For INT L , not only the averages are very similar (Table 4, where right curve couple comparisons have values between 0.4088 and 0.9991), but also the averages of the dependent variable are very low (between 0.07 and 0.34 m 2 , Fig. 8) and this indicates a good vehicle performance during curve driving.
In the second elaboration, averages of μLP L for right curves are very similar (p value between 0.7121 and 1.0000 in couple comparisons- Table 5) and the averages of DV are very low (between 0.02 and 0.04 m- Fig. 9), indicating a good performance of the vehicle while driving along curve. SDLP L presents the same trend for right curves (p value between 0.3864 and 0.9991 in couple comparisons- Table 6) and, even in this case, the averages of the dependent variable are very low (between 0.03 and 0.06 m- Fig. 9), indicating again a good performance of the vehicle along curve.
The Max Speed Steering indicator has a slightly different meaning compared to the previous ones. In particular, the curves of R = 100 m are covered with a very similar behaviour, without there being a difference between the configuration that respects the standard and the non-conforming one. For the smaller radius (60 m), obviously, there is a greater driving difficulty that the user feels in the non-standard configuration and manifests it with greater frenzy on the steering (Fig. 11). This result is interesting, as this indicator highlighted the driver's difficulty in steering with compensatory manoeuvres that made it possible to mitigate any negative effects on the trajectory.
In left curves, instead, the driving behaviour is completely different.
First of all, larger deviations are measured than right direction. Further, there are also quite different values between different curves, despite not always statistically significant. For example, for INT L there is a different trajectory (p value < 0.0001) between R60 and the other three (R60out, R100 e R100out), while these show a very similar behaviour (p value between 0.0158 and 0.9377) or, however, not statistically significant differences. It should be underlined that the averages present higher values for right curves (between 1.37 and 2.78 m 2 - Fig. 8) as left curves are harder for drivers. This phenomenon may be easily explained: while driving along curves, the driver gazes the internal marginal point, with higher curvature. For right curves, it coincides with the barrier or other non-overpassing elements, still easily to be recognised, determining great confidence in curve interpretation. In left curves on two-ways roads, the internal margin is only represented by the markings-whether clearly visible. However, this element is not seen by the users as a fixed and non-overpassing obstacle. This prerogative, with the possible influence of opposite vehicles, produces more spread trajectories for left curves.
The μLP L and SDLP L indicators, for left curves, present slightly different values than INT L , as more criticisms emerge in driving along R60 and R60out curve with respect to R100 and R100out (Figs. 9 and 10). Statistically, for μLP L there is a significant difference between couples R60-R100, R60-R100out and R100-R100out (p value < 0.0001) with average values of deviations between 0.18 and 0.37 m (Fig. 9). A different outcome appeared for SDLP L , where significant different behaviours are noticed only for couples R60-R100 and R100-R60out (p value between 0.0004 and  Fig. 10), proof of a higher driving difficulty. The Max Speed Steering indicator, with regard to the curves on the left, highlights homogeneous behaviours between the compliant and non-compliant configuration. Obviously, the smaller radius (60 m) has higher values than the larger one, as there is greater difficulty in travelling (Fig. 11).
Although couple comparisons are extremely interesting, the aim of this research is to evidence any critical issues in curves not in compliance with guidelines in terms of length of residual circular arc, in identical radius or direction conditions. For this reason, the most interesting comparisons are between couples R60-R60out and R100-R100out. As already said, no difference raised for right curves. This means that the violation of the minimum value imposed by the standards did not induce any negative effect on driving behaviour, at least with the geometry of this road. It is recalled that the Italian standards fix a minimum length equal to 31.46 and 38.95 m, respectively, for R60 and R100. Instead, R60out residual circular arc was only 0.3 m long, while that of R100out was 6.0 m long, remarkably below than admitted values. For left curves, INT L does not evidence any difference between R100-R100out curves but does evidence one between R60-R60out (p value < 0.0001). Overall, the most unfavourable situation in terms of deviation from axes was measured for R60. The other two indicators, instead, did not evidence any significant difference between compliance and violation scenarios. Considering these results, for this experimentation, it is possible to assess that deviation from standards, regarding the selected limit, does not produce any negative effect in terms of trajectory.
However, it is necessary to remember that the scenario in which the experimentation took place does not consider all the values that the involved variables could assume. At this purpose, the trajectory could be influenced by the surface characteristics of the pavement (roughness and friction), the driving skills of drivers, the mechanical characteristics of the vehicles. An experiment that fully takes these factors into account would certainly be interesting and, presumably, will be carried out in the continuation of this research. Furthermore, at this stage, reference was made only to a sample of young users, as it is believed that this drivers' class is more at risk for driving experience and risk attitude. Regarding the vehicle type and the pavement characteristics, the scenario indicated by the Italian road standard was applied. This one specifies that the pavement must be considered perfectly regular, with values of the friction coefficient provided as a function of speed and type of road; furthermore, it speculates that there is no interaction between vehicles and all the characteristics of interest of the isolated vehicle are provided-such as mass, aerodynamic coefficient, etc. (MIT 2001). The results of the trials indicate that, in this research, there were no major difficulties in driving along curves not complying with the standards, at least for the right curves. There was only a greater activity in the use of the steering in the 60 m curves which, however, by means of appropriate compensatory manoeuvres did not cause dangerous trajectories, at least for the right curves. These results can be used to appropriately place warning messages to drivers or arrange physical separation of the lanes or protection elements, such as barriers. However, some warnings can be sent on board units in the vehicle, effectively avoiding a potentially dangerous manoeuvre.
The results of this research may be useful to the developers of ADAS. It is known that the three types of sensors present on board of modern vehicles (Lidar, video cameras and radar) allow the collection of a huge quantity of data, making it possible to apply particular Artificial Intelligence techniques, such as Deep Learning. The idea is to "predict" the difficulty of driving or human errors in advance, through the analysis of critical events along similar curves, to induce the driver to behave more virtuously. In any case, the raw data cannot be directly used, but it must be processed and synthesised through indexes to preserve the representativeness of the observed phenomenon. In this regard, it is believed that the indices proposed in this research are already fully usable for these purposes.

Conclusion
The examination of the literature has shown that accidents are caused by a bad interaction between human, technical and organisational factors which, in the case of road design, refer to drivers, construction aspects of roads and vehicles and traffic management.
Driving is a very difficult activity, within a work environment that has become increasingly complicated due to the introduction of new technologies that often distract the driver or, at least, increase his workload. Moreover, in the road environment, the consequence of driving errors can have effects not only on the driver himself, but also on the other vehicles interacting with him on the same infrastructure.
In this paper, the interaction within the road system between its aforementioned components has been studied by analysing the trajectories and the activity on the steering of drivers, to highlight eventual errors in the manoeuvre attributable to a slightly different geometry with respect to the road standards prescriptions.
In this case, the problem was to verify a design solution not in compliance with Italian standards in terms of minimum length of the residual circular arc in transition curves. The results proved that, in the examined case, the deviations from the minimum thresholds did not produce statistically significant effects on homogeneous drivers' sample, at least in the right curves.
Some of the issues of the existing literature and reported in the Introduction section have been solved: -This methodology does not apply empirical procedures such in Context Sensitive Solutions or Road Safety Audits (NCHRP 2004;Bosurgi et al. 2011) but purely analytical ones, based on drivers' behaviour and vehicle telemetry data. -All the components of the road system (driver, context, vehicle) have been simultaneously involved, even if only the trajectory and the steering speed were evaluated as synthetic variables representative of the driving behaviour influenced by the geometry of the road and the external context (Bassani et al. 2019;Yang et al. 2021;Khakzar et al. 2022). -The results obtained from this experimentation cannot be generalised. This is not a limitation of the present research but, rather, it is caused by the extreme complexity of road context, commonly difficult to be represented through a simplified mathematical model. Rather, the product of this study regards a reliable procedure which, contrary to what has been said for the results, can be easily adapted to any type of road, drivers' class, vehicle. -The choice of the trajectory among the representative variables of the driving behaviour is not sufficient to guarantee the quality of the results. The risk is that using some synthetic indices well known in the literature such as the mean or the standard deviation of the trajectory (O'Hanlon 1984;Ramaekers 2003;Coutton-Jean et al. 2009;Verster and Roth 2012;Brookhuis 2014;Hu et al. 2017;Kazemzadehazad et al. 2019), a certain part of the information could be lost. For this reason, beyond the traditional indices, the authors have proposed an original one allowing the analyst to judge the trajectory with greater completeness. This may evidence some issues on left deviations from trajectories, more and more dangerous than right ones because of possible collisions against opposite vehicles. In particular, the new index is named INT L and represents the time/space in which the user overpasses the lane axis on the left. -For the same reasons, the maximum steering speed (Max Speed Steering) was used as it helps to highlight any sudden manoeuvre on the steering. Its average value along the curve would have risked hiding such behaviour.
In conclusion, this approach could be useful for practitioners, as it provides an analytical procedure to solve design problems in presence of territorial constraints limiting the available solutions only outside the range imposed by the road standards. However, in the next steps of the research, the role of the drivers will be emphasised, considering aspects related to the visual perception of the road geometry. This research may also help from a theoretical perspective, as it guides the choice towards specific curve geometry (or of other elements) considering the real driving behaviour, renouncing to the wrong assumption of perfect overlap of actual trajectories with the lane axis.
In future investigations, some variables typically related to human factors (workload, driving errors, vision) will be also included directly in the statistical analyses to check their concordance with the vehicle trajectories.

Author contributions
The authors confirm contribution to the paper as follows: study conception and design: GB, SM, OP, GS; data collection: GB, SM, OP, GS; analysis and interpretation of results: GB, SM, OP, GS; draft manuscript preparation: GB, SM, OP, GS. All authors reviewed the results and approved the final version of the manuscript.
Data availability Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest
The authors declare no competing interest.