Determining Passing Sight Distance on Upgraded Road Sections over Single and Platooned Heavy Military Vehicles

Abstract

Although truck platooning enhances transportation efficiency, reduces fuel consumption, and lowers freight transport costs, it can also create limited overtaking opportunities, potentially leading to risky overtaking maneuvers. The present study examines the impact of platooned heavy military vehicles on the quantification of Passing Sight Distance (PSD). Two distinct cases are examined: single and platooned military vehicles passing, the latter formed by five trucks. The authors, by realistically modeling the passing task, examined the interaction between vehicle dynamic parameters and roadway grade utilizing an existing vehicle dynamics model. The analysis of various speed values revealed significant PSD variations depending on the examined impeding (overtaken) vehicle’s platooning configuration and utilized grade. The present assessment accurately quantifies the grade impact on the required PSDs for such special vehicle arrangements and can be applied to any vehicle platooning configuration. Moreover, a preliminary tool is introduced to assist road designers in accurately assessing the impact of roadway grade on the passing process. This tool, when combined with a more in-depth analysis of additional factors, can help justify the need for an extra lane in road sections where platooning regularly occurs.

1. Introduction

In Europe, two-lane rural roads account for the highest proportion of road fatalities [1]. Regarding accident severity, the most prevalent crashes are linked to failures during the passing process, including head-on collisions or collisions between the overtaking vehicle and the impeding (overtaken) vehicle traveling in the same direction, e.g., [2,3].

Passing opportunities occur in designated areas known as passing zones, which require a minimum sight distance, referred to as passing sight distance (PSD). PSD is defined as the minimum distance drivers must be able to see ahead to safely and efficiently initiate and complete passing maneuvers of slower vehicles on two-lane rural roads while temporarily using the lane designated for opposing traffic [4].

A vast body of research on the passing process exists in the literature, with numerous models developed to quantify the relationships between key parameters. Many studies have collected passing maneuver data using either an instrumented vehicle representing the overtaking vehicle, e.g., [5,6], or video recordings, e.g., [7,8], primarily to evaluate critical distance, time, and speed parameters. Additionally, some researchers have employed driving simulators to support their findings, e.g., [9,10]. Beyond providing demographic insights into drivers, simulator-based studies offer the advantage of highly precise data collection related to the passing task [10].

When modeling passing zones and Passing Sight Distances (PSDs), numerous studies have yielded valuable insights. However, due to the variability of key influencing parameters, the proposed passing distance values are often derived from critical assumptions and should be interpreted with caution. For instance, many studies assume a constant speed for the overtaking vehicle or a fixed speed differential between the overtaking and impeding vehicles, while others incorporate acceleration data—an inherently challenging parameter to measure—by either treating them as constant or estimating them indirectly through passing time measurements. Similarly, current road design practices also rely on several simplifying assumptions, which may not always fully capture real-world driving conditions.

In the 2012 German RAL design guidelines [11], passing sight distance (PSD) is determined based on the homogeneity of the proposed road design classes rather than vehicle speed, resulting in a standardized PSD requirement of 600 m. This value was derived from the PSD needs of a passenger car traveling at 100 km/h while executing a passing maneuver on a two-lane rural road, alongside a truck moving at 70 km/h. Simultaneously, an opposing passenger car is assumed to be traveling at 100 km/h. The required PSD accounts for the total distance covered by both passenger cars during the maneuver, plus a safety margin of 100 m.

In contrast, the AASHTO [12] design guidelines establish PSD requirements based on field observations, defining PSD as the sum of four distinct distance components:

• d₁—the distance traveled during the driver’s perception–reaction time and the initial acceleration phase.
• d₂—the distance traveled while the overtaking vehicle is fully occupying the left lane.
• d₃, safety margin—the distance between the overtaking vehicle and the opposing vehicle at the end of the maneuver.
• d₄—the distance traveled by the opposing vehicle during the passing maneuver, typically set as two-thirds of d₂.

The PSD values adopted in AASHTO are derived from specific speed combinations between the overtaking and impeding vehicles, based on an assumed speed differential of 19 km/h (12 mph). This standardized approach simplifies design calculations but may not fully capture the variability of real-world passing maneuvers, where speed differentials can fluctuate depending on traffic conditions, driver behavior, and roadway characteristics.

Safety during the passing process can be compromised in various ways. Road sections with limited passing opportunities not only pose safety concerns but also lead to operational inefficiencies. In such cases, certain drivers may be tempted to execute risky overtaking maneuvers—either late within a designated passing zone or on road segments not intended for passing—increasing the likelihood of critical situations [9].

The impact of short passing zones, though not extensively validated through crash data, appears to be significant in terms of smooth versus abrupt reintegration of the overtaking vehicle into the through lane. A recent study [8] found that in short passing zones, 92% of passing maneuvers extended beyond the designated zone, compared to just 21% in longer passing zones (over 300 m). Similarly, research has shown that the proportion of forced and abrupt returns increased from 10% in 270 m passing zones to 45% in 200 m passing zones [13].

Another critical issue is excessive passing speeds. Several studies, e.g., [9,14], have reported that overtaking vehicles often exceed the posted speed limit, further exacerbating safety risks.

The most effective approach to improving both safety and operational efficiency during the passing process is the implementation of additional passing lanes or, at a minimum, protected passing zones through the use of a continuous three-lane cross-section. These configurations provide designated passing opportunities, reducing the likelihood of risky maneuvers and improving overall traffic flow.

On steep grades, auxiliary lane configurations—commonly referred to as climbing lanes—play a crucial role in facilitating the safe overtaking of slow-moving heavy vehicles. Beyond enhancing passing opportunities, these lanes help mitigate congestion, reduce delays, and improve roadway capacity, ultimately contributing to a smoother and safer driving environment.

As far as vehicle passing on level grades is concerned, the well-known 2+1 road sections are utilized.

However, implementing such road layouts is not always feasible in every roadway environment due to economic, topographical, or environmental protection constraints [15]. In certain cases, the addition of a third lane in a three-lane cross-section arrangement requires further investigation to determine its actual impact on passing maneuver safety. While additional lanes generally enhance overtaking opportunities and reduce risky maneuvers, factors such as traffic volume, driver behavior, and geometric design may influence whether safety improvements are realized.

This issue becomes even more significant in the context of truck platooning on two-lane rural highways. Truck platooning intends to enhance traffic efficiency by reducing aerodynamic drag, leading to lower fuel consumption, reducing emissions, and contributing to overall cost savings for freight transport. Additionally, it improves traffic flow by maintaining consistent speeds, minimizing abrupt lane changes, and leveraging vehicle-to-vehicle communication for coordinated braking and acceleration [16,17].

However, truck platooning with limited passing opportunities can lead to traffic delays, increased driver frustration, and risky overtaking attempts. The presence of closely spaced heavy vehicles restricts sight distance for following drivers and may necessitate longer passing maneuvers, raising safety concerns. Consequently, a more comprehensive evaluation is needed to assess the effectiveness of alternative road configurations, such as strategically placed passing lanes or dynamic lane management, to accommodate both truck platooning and mixed-traffic conditions while minimizing adverse safety and operational impacts.

To this end, it is important to recognize that PSD standards are primarily designed to accommodate a single vehicle overtaking a slower-moving impeding vehicle. However, research on passing maneuvers involving multiple vehicles or heavy vehicles remains scarce in the literature, despite the growing prevalence of truck platooning and its implications for passing sight distance requirements.

In one of those research studies [18], the authors developed a simulation model to analyze passenger car overtaking maneuvers on a two-lane highway with a 90 km/h speed limit, considering different truck lengths (20 m, 30 m, and 37 m) and various traffic conditions. Their study incorporated factors such as vehicle acceleration, driver aggression, and opposing lane traffic volumes. Results indicated that under moderate opposing traffic, with average gaps of 17.28 s, the likelihood of failing to pass a 37 m truck was two to six times higher than that of a 20 m truck. This suggests that drivers would either need to wait for a larger gap, forgo overtaking, or risk an aborted pass or potential crash.

Llorca et al. [6] studied the overtaking of two passenger cars on two-lane highways using a data-equipped vehicle and roadside cameras. The research recorded vehicle positions, headways, and speed differentials to develop a mathematical PSD model. The results showed that 20% of passing maneuvers involved overtaking multiple vehicles, requiring 43% longer passing sight distances and 36% more time in the opposing lane compared to passing a single vehicle.

Haq et al. [19] applied a kinematics-based PSD model to determine the required PSD for overtaking up to four platooned trucks on highways with a 110 km/h speed limit, finding distances ranging from 422 m to 585 m depending on platoon size and spacing. Results were validated using VISSIM simulations, showing less than 3% discrepancy. To mitigate the impact of truck platoons, this study recommended the implementation of passing lanes.

The aforementioned research studies evaluate PSD requirements based on operational considerations or by assuming acceleration rates that are either constant or derived from design guidelines. To the best of the authors’ knowledge, there is a notable gap in the literature regarding PSD analysis for multiple vehicles, specifically in relation to the interaction between vehicle dynamic parameters and roadway geometry, particularly in terms of grade. Consequently, a comprehensive assessment of the examined vehicles’ ability to perform under these conditions appears to be lacking.

Within this context, the present study examines the impact of platooned heavy vehicles on the quantification of PSD. This research was particularly motivated by the unique traffic conditions in Greece, where numerous military camps are situated along high-speed, two-lane rural roads, leading to frequent military vehicle platooning for training purposes. Although the platooning vehicles travel relatively short distances on the main arterial—ranging from approximately 5 km to 15 km—the resulting passing conditions are often not only impractical but also pose significant safety risks due to the limited opportunities for safe overtaking.

Military vehicle platooning presents unique operational characteristics, including rigid and lengthy convoy structures, variable vehicle sizes, and specific speed regulations, which differentiate it from other types of vehicle platooning. These factors influence passing sight distance (PSD) requirements and overtaking opportunities, necessitating a separate investigation to accurately assess the associated safety risks. Moreover, understanding these distinct characteristics can provide valuable insights applicable to other forms of vehicle platooning, such as industrial convoys, operating under similar conditions on high-speed rural roads.

By analyzing the interaction between vehicle dynamic parameters and roadway grade during the passing process of heavy vehicles, the authors seek to identify critical parameter combinations that influence safety and efficiency. This analysis aims to provide deeper insights into the factors affecting passing maneuvers, ultimately contributing to the development of safer and more effective road design strategies for managing heavy vehicle interactions.

More specifically, by analyzing the motion of the most common military single-unit heavy vehicle as well as its platooning, the present research accurately quantifies the grade impact on the required PSDs for such special vehicle arrangements.

The most significant contribution of this research is the development of a preliminary tool for road designers to accurately quantify the impact of roadway grade on the passing process. This tool also provides a data-driven foundation for justifying the implementation of an additional lane in road sections where platooning occurs regularly (even if only over short distances), enhancing both traffic flow and safety.

2. Methodology

The proposed passing sight distance (PSD) investigation is grounded in a safe and realistic representation of the passing process on tangent road sections. This study specifically examines the speed variation of both the overtaking and impeding vehicles throughout the maneuver, ensuring a more accurate and dynamic assessment of passing behavior under real-world conditions.

In their previous research, the authors developed a vehicle dynamics model to assess vehicle safety by examining the interaction between road geometry, tire–pavement friction, and vehicle parameters [20,21,22]. This model aimed to understand how these factors influence vehicle performance, particularly on grades, and highlighted the importance of considering tire–pavement friction, which varies with vehicle speed and slip ratio during maneuvers such as acceleration, braking, or cornering.

2.1. Vehicle Dynamics Approach

In the vehicle dynamics model developed by the authors, all forces and moments applied to the vehicle were analyzed within a moving three-dimensional coordinate system, coinciding at the vehicle’s center of gravity and formed by the vehicle’s longitudinal (X), lateral (Y), and vertical (Z) axes. This approach allows for a comprehensive assessment of how various vehicle technical characteristics, road geometry, and tire friction influence vehicle performance. The key parameters considered in the model include the following: vehicle speed, wheel drive, sprung and unsprung mass, its position of gravity center, aerodynamic drag, vertical lift, track width, wheelbase, roll center, suspension roll stiffness, cornering stiffness, grade, superelevation rate, rolling resistance, tire–road adhesion values, and horsepower supply.

Moreover, the model incorporates variables related to vehicle steering and tire sideslip angles, which are crucial for understanding the vehicle’s response to steering inputs and lateral forces. The actual wheel load, influenced by lateral load transfer during maneuvers, is also considered, as it affects the lateral force on each wheel. This comprehensive analysis leads to a four-wheel vehicle dynamics model that accurately represents the interactions between the vehicle’s components and the road surface [23,24,25].

The model’s outputs were validated against data from several distinct cases, demonstrating the model’s robustness and accuracy. The validation cases included the following:

• The final climbing speed of a truck on a grade [20].
• The output data from CARSIM Simulation Software [26].
• Field measurements of passenger cars [22].

All cases revealed a satisfying match.

The available tractive effort of the vehicle—defined as the driving force minus tire rolling resistance—acts on either the front or rear axle, depending on the vehicle’s drivetrain configuration. This effort is influenced by both the vehicle’s speed and the net power available at the driving wheels. However, due to tire skidding, a vehicle cannot always operate at its full horsepower capacity. To account for this limitation, the horsepower utilization factor (n) was introduced, leading to the following equation:

$$\mathrm{F}\mathrm{x}=745.60{\displaystyle \frac{\mathrm{P}}{\mathrm{V}}}{\displaystyle \frac{\mathrm{n}}{100}}$$

(1)

where

• Fx: driving force (Nt);
• P: engine horsepower available at the driven axle (around 94% of the nominal value (hp));
• V: vehicle speed (m/s);
• n: horsepower utilization factor (%).

By applying the laws of mechanics, the vehicle’s instantaneous acceleration can be expressed as a fourth-degree polynomial equation, formulated as a function of both the vehicle’s instantaneous speed and driven distance. This results in the following differential equation, which is solved using the numerical Runge–Kutta method [27]:

$$\mathrm{a}\left(\mathrm{v}\right)={\displaystyle \frac{\mathrm{d}\mathrm{v}}{\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}}\mathrm{V}$$

(2)

where

• a(v): acceleration (m/s²);
• V: speed (m/s);
• dist: distance (m).

Solving Equation (2) with respect to the initial speed yields the vehicle’s speed variation as a function of the driven distance. This process is carried out under impending skid conditions by matching the computed demanded longitudinal friction (f_Tdem) to the maximum available roadway friction (f_max). Additionally, the horsepower utilization factor (n) from Equation (1) is dynamically adjusted throughout the computation.

It is evident that during the vehicle’s tractive mode, its speed remains variable regardless of whether it operates at full net engine horsepower (n = 100%) or a reduced level (n < 100%).

As long as the vehicle utilizes part of the total net engine horsepower (n < 100%), it operates under impending skid conditions, as any further increase in “n” would result in vehicle skidding.

However, when n = 100%, the vehicle has reached its maximum horsepower utilization and cannot draw additional power. Despite this, its speed continues to fluctuate due to the presence of acceleration and deceleration, albeit at a reduced rate, as the total available horsepower is consistently applied. Beyond this point, the vehicle is no longer operating under impending skid conditions [22].

During the motion of loaded trucks on tangent sections, which is the case of the present research, previous research [20,21] has found that

• n = 100%, even for poor friction values of fmax = 0.35, which means that the pavement friction does not affect the examined heavy vehicle’s motion;
• The parameters that mainly determine vehicle speed variation, along with the driven distance, are vehicle gross weight-to-horsepower rate [W/hp (kgr/hp)], road grade [s (%)], vehicle drag force [Ad (Nt)], and tire rolling resistance force [Fkr (Nt)].

From the above, it is worth mentioning that the vehicle dynamics model is capable of calculating, along with the driven distance, the examined vehicle’s spot speed until the final constant speed (known as crawl speed for heavy vehicles) is reached. Therefore, within the present analysis, the acceleration is not considered constant.

2.2. Examined Vehicles

The assessment was conducted by analyzing the motion of two distinct vehicles, as illustrated in Figure 1: a C-class passenger car (Toyota C-HR with an automatic transmission) and a widely used single-unit military heavy vehicle (Steyr 680M). The specific parameters used for each vehicle are detailed in

Table 1. Vehicle parameters inserted into the model [28].

	Military Truck	C-Class Passenger Car
	Steyr 680M	Toyota CH-R
LO (m)	6.750	4.360	Length
L (m)	2.800	2.640	Wheelbase
tf (m)	1.752	1.550	Front track width
tr (m)	1.628	1.570	Rear track width
m (kgr)	10,000	1500	Vehicle mass
lf (m)	1.756	1.161	Position of GC from front axle
h (m)	1.800	0.620	Position of GC from surface
Kφf (Nm/rad)	453,711	27,502	Suspension roll stiffness (front)
Kφr (Nm/rad)	453,711	14,324	Suspension roll stiffness (rear)
Caf (kp/rad)	14,099.7	2295.7	Cornering coef. (front)
Car (kp/rad)	11,857.8	2120.7	Cornering coef. (rear)
muf (kgr)	293	92	Unsprung mass (front)
mur (kgr)	196	120	Unsprung mass (rear)
hRf (m)	0.530	0.020	Roll center height (front)
hRr (m)	0.530	0.410	Roll center height (rear)
rdyn (m)	0.450	0.290	Dynamic radius
Af (m2)	5.702	1.850	Frontal area
cd	0.90	0.32	Aerodynamic drag
P (hp)	150 × 90%	120 × 94%	Horsepower (net)

.

Regarding the vehicle parameters presented in Table 1, an effort was made to source them directly from the vehicle industry (bolded values) or from the relevant literature [24]. In all cases, further details on the parameter values, along with their roles in the equations governing the vehicle dynamics model, can be found in the authors’ previous research [20,22].

For the utilized horsepower rates, different assumptions were applied. Since the examined military heavy vehicle operates continuously for approximately 40 years, the utilized horsepower rate available at the driven axle was set to 90% of the relevant net value. This assumption is not far from a similar study [29], where, based on energy deficits, the value of the utilized horsepower rate available at the driven axle of a passenger car was found to be around 94% of the respective nominal value. As a result, regarding the military vehicle, the weight-to-horsepower rate (W/hp) was set to 74.1 kgr/hp. It must be stressed that this value is less conservative compared to the relevant value of 90.8 kgr/hp (200 lb/hp) adopted by AASHTO [12].

The passing process for both vehicles was based on the outputs of the vehicle dynamics model, and more specifically on their speed–distance performance for upgrades [Figure 2a,b].

The military vehicle was examined for grade values of 4%, 6%, and 8%, assuming an initial speed equivalent to the military heavy vehicle’s posted speed of 70 km/h [Figure 2a]. For grade values equal to or less than 3%, it was found that the examined heavy vehicle could maintain or even exceed the 70 km/h speed.

The speed–distance outputs for the examined passenger car, initialized at 50 km/h and tested on road grades of 0%, 4%, 6%, and 8%, are presented in Figure 2b. The results indicate that, in all scenarios, the vehicle operates under acceleration and rather rapidly reaches the posted speed limit of 110 km/h for two-lane rural roads. It must be stressed that 110 km/h is considered an excessive posted speed for two-lane rural roads in Greece. The respective typical value is 90 km/h.

2.3. Passing Assessment and Assumptions

The passing assessment, besides utilizing a single vehicle, was also performed for platooning military vehicles, and more specifically for the most common configuration formed by 5 vehicles. This military vehicle configuration, applied for daily training reasons, is typical near military camps. The headway between each military vehicle in the platoon, provided by [30], follows the 2 s driver’s perception–reaction assumption. However, an empirical rule is more applicable, according to which the headway is equal to half of the speed (km/h). The formula for calculating the overall platoon length is as follows:

$${\mathrm{L}}_{\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{n}}=\mathrm{N}{\mathrm{L}}_{\mathrm{O}}+(\mathrm{N}-1){\displaystyle \frac{3.6\mathrm{V}}{2}}$$

(3)

where

• L_platoon: overall platoon length (m);
• L_O: vehicle length (m);
• V: speed (m/s);
• N: number of vehicles in the platoon.

The overall length of the passing military vehicle(s) with and without platooning for various speed values is shown in

Table 2. Overall length of passing vehicle [28].

V (km/h)	Single Vehicle Length [N = 1] (m)	Platoon Length [N = 5] (m)
50	6.75	133.75
60	6.75	153.75
70	6.75	173.25

.

The analysis aims to deliver a tool for rationalizing the design of additional lane configurations in order to facilitate passing maneuvers and thus quantify more accurately the required PSDs, especially near military premises.

Therefore, the assessment of the passing process was conducted exclusively through the interaction between vehicle dynamics and road geometry, incorporating decision passing distance as a key factor. The analysis, assuming free-flow conditions, considers the involvement of three vehicles: the overtaking vehicle, the impeding (overtaken) vehicle(s), and the opposing vehicle.

All three vehicles exhibit distinct motion characteristics, with the following criteria and assumptions applied:

• The speed of all three vehicles never exceeds the posted speed of the roadway.
• Depending on the initial speed and grade value, the motion of the military vehicle (overtaken vehicle) is under deceleration mode or steady-state conditions (V = 70 km/h for s ≤ 3%).
• The motion of the opposing vehicle is modeled under steady-state conditions, traveling at the posted speed limit of the roadway.
• The overtaking vehicle’s motion during the passing maneuver begins in an acceleration phase, with its initial speed matching that of the overtaken vehicle. The speed then increases continuously until it reaches the roadway’s posted speed, at which point steady-state conditions are applied.
• The headway (dist₁) between the front of the overtaking vehicle and the front of the overtaken vehicle at the beginning of the passing maneuver is assumed to be 16.5 m. This includes 9.5 m (Llorca et al., 2013) plus 7.0 m, approximately the length of the utilized overtaken military vehicle, or 9.5 m + L_platoon.
• The headway (dist₂) between the front of the overtaking vehicle and the front of the overtaken vehicle at the end of the passing maneuver is assumed to be 28.5 m. This consists of 24 m, as referenced in (Llorca et al., 2013), plus approximately 4.5 m, the length of the utilized overtaking vehicle.
• The safety margin was set to a constant value of 100 m [15], which can be interpreted as a safety margin of approximately 3.5 s at a speed of 100 km/h.

The distance criteria for the passing process involving a single overtaking vehicle, as outlined above, are illustrated in Figure 3.

3. Analysis

As already stated in the previous section, the posted speed must not be exceeded during the passing process. Therefore, having in mind the 70 km/h speed limit of the examined military vehicles and aiming to assess realistic but at the same time unfavorable conditions, the PSD analysis was performed for two-lane rural roads where the posted speed value was set to 90 km/h, which represents a rather common value for most two-lane rural roads.

Utilizing a posted speed of 80 km/h will result in excessive PSDs. On the other hand, a posted speed of 100 km/h is not an unfavorable case since, based on the recently established German rural road design guidelines RAL [11], both two-lane rural roads with posted speed values of 90 km/h and 100 km/h (referring to EKL2 and EKL3 design classes, respectively) adopt the same PSD of 600 m.

In order to assess the grade impact of heavy vehicles’ motion during the passing process, four different grade values were investigated (s = 0%, s = 4%, s = 6%, and s = 8%). For the level roadway (s = 0%), the speed of the heavy vehicle(s) was assumed to be constant and equivalent to the 70 km/h speed limit.

As already mentioned, pavement friction does not affect the motion of the examined heavy vehicles. However, regarding passenger cars, the authors’ previous research [22] has shown that pavement friction has a mild impact on their acceleration performance.

More specifically, in [22], the authors examined the interaction between vehicle dynamics parameters and road geometry during the passing process of single passenger cars. The analysis focused on certain cases involving four independent variables: vehicle horsepower rates [P (hp)], difference between the overtaken vehicle’s speed (also the initial speed of the overtaking vehicle) and the roadway’s posted–design speed [ΔV1 (km/h)], peak friction supply coefficients (fmax), and grade values [s (%)]. Each independent variable was associated with three different values, where, in total, 81 different cases per roadway design class were examined. In terms of friction, three values of peak friction supply coefficients fmax were used [0.35, 0.50, and 0.65, assessing pavements with poor friction performance under both wet (0.35) and dry (0.65) pavement conditions]. For the same ΔV1 and P and grade values, the PSD results differed less than 5%, indicating that wet vs. dry pavements have a moderate effect on PSD.

The authors, in their same previous research [22], aiming to define the effect of the assessed parameters, developed two different models: one for 100 km/h and one for 90 km/h (EKL2 and EKL3 design classes, respectively, based on the German rural road design guidelines). A histogram of the response variable led to the identification of a clearly skewed density function, suggesting a lognormal distribution. Both models utilized the following parameters combinations: ΔV1, P × fmax, and s × ΔV1. A collinearity test was conducted to ensure that the independent variables were not correlated with each other. The parameter estimates of the main effects suggest that an increase in one interaction (i.e., s × ΔV1) increases PSD, while an increase in others (i.e., ΔV1, P × fmax) decrease PSD. The likelihood ratio test leads to accepting the model compared to the null model, and an adjusted R-squared, equal to 0.94 for both cases, revealed a satisfactory outcome.

The present research primarily focuses on assessing the dynamic characteristics of the accelerating (overtaking) and decelerating (overtaken) vehicles from the perspective of roadway grade. To this end, the pavement friction supply coefficient, representing the acceleration performance of the passenger car (overtaking vehicle), was set to a constant value of fmax = 0.80 for all examined cases.

The passing process was assessed in line with the speed variation of both vehicles, based on the speed–distance performance in accordance with grade, as shown in Figure 2. Aiming to examine more realistic situations, up to three initial speed values for both the overtaking (passenger car) and the impeding (military truck) vehicles were utilized.

These initial speed values, during the launch of the passing maneuver, were selected within the varying speed area of the military vehicle for the respective grade examined (always set above the maximum steady-state speed of the military vehicle) and expressed as a difference from the military vehicles’ posted speed of 70 km/h, namely, ΔV = 0 km/h, ΔV = 10 km/h, and ΔV = 20 km/h. For s = 6%, since the military vehicle’s maximum attainable constant speed is approximately 52 km/h [Figure 2a], the assessment between the passenger car and the military vehicle was performed assuming the initial speed of both vehicles 60 km/h (ΔV = 10 km/h) and 70 km/h (ΔV = 0 km/h).

The passing assessment is in accordance with the criteria—assumptions stated above as well as in the previous section. As far as the speed of the involved vehicles is concerned, the following apply:

• The speed variation of the heavy vehicle (overtaken) is in line with the derived speed–distance outputs from the dynamic model.
• The speed variation of the overtaking vehicle (under acceleration mode) is in accordance with the outputs of the dynamic model and respects the 90 km/h posted speed.
• The speed of the opposing vehicle was consistently assumed to be equal to the posted speed limit of the examined roadway, set at 90 km/h.

The calculations were conducted analytically. Figure 4a provides a graphical representation of the proposed PSD assessment, adapted from a similar study [12]. It also includes the speed profiles of both the overtaking and overtaken vehicles, displayed on a secondary vertical axis. More specifically, in Figure 4a, time, distance, and speed data during a single truck passing for all the involved vehicles are shown, assuming an initial speed of 70 km/h and s = 8%. The distance–time outputs for both the overtaking and overtaken vehicles exhibit non-linear characteristics. At t = 0 s, the difference in distance (primary vertical axis) between the overtaking (red line) and the overtaken (blue line) vehicles is set to dist1, while the passing maneuver is terminated when the overtaking vehicle is ahead of the overtaken by dist2. At that instant, by providing the 100 m safety margin, the PSD (475 m) can be defined linearly through the distance–time constant angle (steady-state) calculation of the opposing vehicle.

A graphical illustration of platooning trucks formed by five military vehicles under the same initial speed and grade conditions (V = 70 km/h, s = 8%, respectively) is shown in Figure 4b, where PSD is 1272 m.

To summarize, certain cases of vehicle passing were examined by arranging combinations of four different grade values (s = 0%, s = 4%, s = 6%, and s = 8%), paired with up to three values referring to the difference between the utilized initial speed value and the military vehicles’ posted speed of 70 km/h (ΔV). The assessment was performed for both single and platooning heavy vehicles passing, the latter formed by five military trucks.

The developed passing sight distance (PSD) graphs, presented in Figure 5, yielded several insightful findings, some of which were anticipated.

Among the most important findings is the fact that the 600 m PSD requirement, as imposed by the German RAL 2012 design guidelines [11], seems adequate during single truck passing, even for a 70 km/h steady-state truck speed (s = 0%).

As expected, the delivered passing zone distances and PSDs under platooning conditions are, in principle, very demanding, even for the most unfavorable grade (s = 8%) examined (approximately 380 m and 900 m, respectively).

Under the same grade value, the passing distances of the examined platooned configuration increase significantly as the speed difference ΔV between the vehicles’ initial speed and the military posted speed of 70 km/h decreases. On the other hand, a moderate increment is observed for single truck passing.

4. Discussion and Conclusions

This paper investigated the passing process of a passenger car towards a common military vehicle. Two distinct cases were examined: single and platooned military vehicles passing, the latter formed by five military trucks. Such situations of multiple heavy vehicles passing are experienced on a daily basis on rural roads with high posted speeds in the area of military camps in Greece for road segments approximately between 5 and 15 km.

The assessment, utilizing an existing vehicle dynamics model, was performed for a high, common, and at the same time rather unfavorable posted speed of 90 km/h, where the authors analyzed the interaction between vehicle dynamic parameters and roadway grade. Four different grade values were investigated (s = 0%, s = 4%, s = 6%, and s = 8%), paired with up to three different initial speed values at the beginning of the passing process. These initial speed values fell inside the varying speed area of the military vehicle for the respective grade examined, and were expressed as a difference ΔV from the respective military vehicles’ posted speed of 70 km/h.

The present assessment can be applied to any platooning configuration (e.g., industrial vehicle platooning between certain access areas through high-speed rural roads, etc.) and serves as an opening paradigm for the methodology to determine accurate PSDs on single and platooned vehicles under various road grade and speed values.

While this research provides valuable insights, it is not without limitations, as it focuses solely on tangent roadway segments. When overtaking a truck platoon on a right-curving segment, the available PSD is not only constrained by roadway and roadside elements but also by the platoon itself, which can significantly obstruct the driver’s view of the opposing traffic lane. This visual occlusion increases the risk associated with passing maneuvers, underscoring the need for further research on passing truck platoons along right-curved road sections. Conversely, on left-curving segments, the overtaking vehicle driver benefits from an improved line of sight, allowing for better assessment of available gaps in opposing traffic. Moreover, assessing tire–road friction under adverse weather conditions is vital for ensuring safe vehicle operation during the passing process.

This study primarily focuses on vehicle speed; however, the passing distance data were analyzed in relation to the roadway’s posted speed limits by investigating the acceleration capabilities of both the overtaking and overtaken vehicles under various examined grade values.

Although the analysis revealed significant PSD variations depending on the examined (overtaken) vehicle’s platooning configuration, this research delivers a tool, at least on a preliminary basis, to assist designers in justifying the necessity of auxiliary lanes to improve road safety on road sections where platooning takes place on a regular basis.

However, in transportation research, vehicle interactions during the passing process involve multiple dimensions, including, among other parameters, lane-changing dynamics and reaction to surrounding vehicles. Established traffic flow models typically incorporate these elements to ensure a realistic representation of traffic scenarios. Therefore, given the limited research on passing maneuvers involving multiple vehicles or heavy vehicles, it is important to extend the model in order to incorporate such additional interaction mechanisms. Moreover, by performing cost benefit analysis, the case whether the design of an alternative route is more efficient should be examined as well.

The analytical model, while providing precise passing distance outputs relative to the selected grade, is computationally intensive. Therefore, a statistical modeling approach seems necessary, where, in addition to the above constraints, a number of additional parameters should be concurrently addressed, including the performance of heavy vehicles with varying weight-to-horsepower rates, more platooned configurations, and broader posted speed values.

Leveraging vehicle-to-vehicle (V2V) communication presents a transformative opportunity to enhance the accuracy of passing sight distance (PSD) estimation, particularly in scenarios involving truck platoons. By enabling real-time data exchange between the overtaking vehicle, the platoon, and the opposing vehicle, a more precise assessment of available gaps can be achieved. The overtaking vehicle can receive dynamic information about the platoon’s length, speed, and position, allowing for better decision-making regarding overtaking feasibility. Simultaneously, data from the opposing vehicle, such as its speed and estimated arrival time at the passing zone, can be integrated to refine PSD calculations. This interconnected system mitigates the limitations imposed by visual occlusion, especially on right-curving road segments, and enhances safety by reducing uncertainty in passing maneuvers. Implementing such technology could lead to adaptive, real-time decision support systems that significantly improve traffic efficiency and reduce collision risks in overtaking scenarios.

Addressing unforeseen situations that may necessitate the cancelation of a passing maneuver is also essential. Implementing obstacle detection systems using deep learning algorithms can significantly enhance a vehicle’s ability to identify and respond to unexpected obstacles on the road. Advanced sensor technologies, combined with machine learning algorithms, can evaluate road surface conditions in real time, allowing vehicles to adjust their behavior accordingly to maintain safety.

An important issue to investigate in automated driving environments is the impact of a 10 km/h speed difference between heavy vehicles and the posted speed limit on roadway operational levels. When heavy vehicles travel at speeds significantly lower than the posted limit, they can impede traffic flow, leading to congestion and reduced service levels. This situation is particularly critical during overtaking maneuvers, where the overtaking vehicle may need to reduce speed to safely overtake the slower-moving heavy vehicle. Such speed disparities can result in increased travel times and potential safety hazards.

Moreover, road geometry, including both horizontal and vertical curvatures, as well as intersection design, significantly influences vehicle dynamics and safety during passing maneuvers. Sharp curves and complex intersections can restrict visibility and maneuverability, posing challenges for both human drivers and autonomous systems.

In conclusion, it is essential to recognize that human factors during the acceleration process can impose additional restrictions, thereby affecting vehicle safety performance.