Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance

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Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance César De Santos-Berbel ∗, Maria Castro Departamento de Ingeniería del Transporte, Territorio y Urbanismo, Universidad Politécnica de Madrid, Spain Received 29 October 2018; received in revised form 28 August 2019; accepted 28 August 2019 Available online xxxx

Abstract Sight distance is a fundamental factor in the design of highways as it determines their operational and safety performance, particularly in nighttime. Vehicles are increasingly being equipped with driving assistance systems such as adaptive frontlighting systems, from which potential safety benefits can be derived. One of the main capabilities of adaptive headlights is controlling headlamp swiveling when driving on horizontal curves to light up a greater section of the roadway ahead. In this study, the vehicle headlight beam was recreated to simulate swiveling systems that adapt the headlight beam to the highway geometry. Diverse horizontal spread angle values were assumed for the headlight lighting pattern. Based on horizontal curvature, a total of 24,663 mathematical functions that control the headlight swiveling angle were simulated. Next, headlight sight distance (HSD) was estimated on a 3D virtual model of an in-service highway, under assumptions of fixed and swiveling headlamps. Two types of algorithms were proposed for the swiveling headlights, one based on the trajectory curvature and another one predictive. The sets of HSD results were then analyzed and compared, the effects of the swiveling headlights on HSD being quantified along the highway section studied. In addition, the performance of diverse swiveling headlights was analyzed under different 3D highway alignment combinations. Finally, the robustness of the proposed procedure was validated. c 2019 The Author(s). Published by Elsevier B.V. on behalf of International Association for Mathematics and Computers in ⃝ Simulation (IMACS). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Keywords: Nighttime driving; Headlight sight distance; Road design; Advanced driving assistance systems; Vehicle swiveling headlamps

1. Introduction Sight distance is a fundamental factor in the design of highways as it determines their operational and safety performance. On the one hand, highways are three-dimensional facilities and must be designed and examined as such. On the other hand, sight distance analysis must consider driving under all circumstances, including nighttime. Whereas daytime has been extensively studied, sight distance in nighttime has received less attention. Frontlighting systems aim to assist drivers in limited lighting facilities, when traveling at night or in adverse weather conditions. In this respect, adaptive frontlighting systems are currently one of the most extended advanced driving assistance systems in vehicles. Among other capabilities, adaptive headlights can swivel the light beam in order to fit it into the highway geometry. The beam is oriented towards the area of roadway that is of utmost interest ∗ Corresponding author.

E-mail addresses: [email protected] (C. De Santos-Berbel), [email protected] (M. Castro). https://doi.org/10.1016/j.matcom.2019.08.012 c 2019 The Author(s). Published by Elsevier B.V. on behalf of International Association for Mathematics and Computers in 0378-4754/⃝ Simulation (IMACS). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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in order to govern the vehicle so that, when the driver enters a curve, the light beam smoothly bends to light up the roadway instead of lighting up the roadside that lies straight ahead. The simulation of transportation issues has been proved to be a solid approach to model complex systems, and particularly in highway geometric design [5,25]. In addition to single horizontal curves, complex alignment sequences such as horizontal and vertical curves overlapped, reverse curves and compound curves are common in the geometry of highways [13,25,32]. The swiveling headlights should be able to cope with these geometric layouts, and especially with the transitions between alignments. When compared to the conventional headlights, not only do swiveling headlights increase driving comfort, but they can also contribute to enhance safety performance [1,28]. Crash avoidance represents, therefore, a potential improvement aspect of these technologies. In this respect, the visibility of the driving scene is a fundamental factor for safety performance at night. Despite crash frequency is higher during nighttime than in daytime, nighttime collisions have not been regarded by designers and researchers with the same effort than the daytime ones. Statistics indicate that nighttime accidents are more frequent than those occurred in daytime even though the traffic volume registered during the night is significantly lower [3]. Moreover, 36% of fatalities occurred on highways in Spain in 2016 were at dusk or nighttime and 61% of pedestrians killed befell on rural highways [9]. The enhancement of highway design has produced evident benefits for safety and comfort. However, generous alignments might have also created potential challenges by enabling higher speeds despite the limited lighting conditions [33]. Limited sight distance under nighttime conditions indeed reduces reaction chances to road users [2]. In this sense, the enhancement in nighttime visibility produced by swiveling headlight systems may help reduce the hazard. The objective of this research is to simulate different swiveling algorithms of vehicle headlights and study their performance in nighttime on the alignments of a three-dimensional (3D) highway virtual model. To this end, the background on sight distance in nighttime was reviewed. Next, the simulation of the swiveling system was described. Then, the headlight sight distance (HSD) performance of the simulated swiveling headlights was tested on a virtual 3D model of an in-service highway. The subsequent results were analyzed and discussed, and the robustness of the proposed procedure was verified. Finally, conclusions of research are presented. 2. Background 2.1. Headlight sight distance modeling The available sight distance (ASD) is defined as the distance, measured along the vehicle path, between the driver’s position and the farthest target seen without interrupting the line of sight. It follows from this that the evaluation of sight distance in daytime mostly concerns the obstruction of the driver’s view. To estimate the ASD, several 3D methods have been devised [6,7,21]. Conversely, sight distance in nighttime is affected by additional factors such as mesopic vision [10,11]. Nonetheless, an effective evaluation of the driving scene in nighttime in relation to highway geometric design requires accommodating certain hypothesis. Whereas the visibility of the night driving scene can hardly be numerically quantified from an objective point of view for the assessment of the highway alignment, the HSD can be more straightforwardly quantified as it is a measurable magnitude. In this regard, the HSD is defined as the distance between the driver and the furthest target on the vehicle path visible from the driver position and illuminated by the headlamps. This distance is measured on the vehicle path, and the section considered visible must lay entirely within the headlight beam. Therefore, no visual interruption must exist between the observer and the furthest target seen; neither by the environment, nor by the path laying partially out of the beam. It must also be noted that this definition does not need to rely on a complex description of a target along with its features. As a result, it provides an unambiguous illustration of the core aspects of the phenomenon for the purposes of highway engineering research. Standards and guides provide simple procedures to broach the design of alignments so that they are adequate also in nighttime driving. These methods are generally based on a two dimensional (2D) modeling of the headlights beam, separating plan and profile. Particularly, the design of sag curves is determined by the consideration of the upward divergence angle of headlights (β) in profile, which defines a plane that theoretically encloses the headlight beam (see Fig. 1). A vertical spread angle of 1◦ is commonly assumed by standards and guides [3,29]. Hawkins Jr. and Gogula [17] proposed, nevertheless, the reduction of such an angle to a value between 0.75◦ and 0.9◦ on the basis of real headlight lighting performance of modern headlamps. Such a change would require to increase the Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 1. Geometric parameters of headlight beam.

sag curve parameters proposed in standards [16]. With regard to the horizontal spread angle, Kraemer et al. [23] proposed 3◦ as a regular horizontal spread angle value that the light beam spans outward the spotlights. Hassan et al. [15] explored horizontal spread angle values ranging from 0◦ to 10◦ in HSD evaluations. Highways are, nevertheless, 3D in nature and therefore their features must also be contemplated under such an approach. Hassan et al. [15] devised and validated a procedure to study nighttime sight distance in 3D. They assumed that the area illuminated by a headlamp corresponds to the volume enclosed by a four-face pyramid determined by the horizontal spread angles (α), the vertical spread angles (β) and the maximum light range (Rg) (Fig. 1). They applied the model to study horizontal curves overlapped with vertical curves. Significant differences between the results of the 3D method and those of the 2D methods available in design guides were found. It was also found that a greater horizontal spread angle (α) provides longer sight distances, up to a threshold value that depends on the highway geometry. They also found that the ASD increased by enlarging the radius of the horizontal curve while superelevation favors sight distance. Similarly, De Santos-Berbel et al. [8] researched the effect of the headlamp lighting parameters on the HSD on a two-lane rural highway model: the headlamp height above the road surface, the horizontal spread angle, the vertical spread angle, and the beam range. They found that the most important role in HSD is played by the horizontal spread angle and the vertical spread angle. 2.2. Swiveling headlamp controllers According to diverse authors [14,30,35], the steering angle and the speed typically feed the swiveling algorithm. However, little information is available on how much headlamps swivel as a function of the steering angle or the speed. Although diverse swiveling angle values can be found in literature, nothing is said about their selection, or it was definitely not based on clear criteria. Sivak et al. [36] tested fixed axle turns of 7.5◦ and 15◦ on 2D drawings of horizontal curves under diverse light beam layouts, which included combinations of parallel, diverging and converging beam axles with the abovementioned swiveling angles. Shreyas et al. [35] designed the mechanisms that govern headlight swiveling for an adaptive frontlighting system, and proposed maximum swiveling angles of 37◦ to the left and 43◦ to the right. Hagiwara et al. [14] carried out an experiment on a real highway where the headlamp on the inner side, when turning, swiveled 5◦ to the left or 5◦ , 10◦ and 15◦ to the right as the case may be. From a theoretical point of view, Sivak et al. [36] studied the effect of swiveling headlights on sight distance on horizontal curves of 80 m and 240 m radii, observing a substantial increase of the area of roadway that Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Table 1 Descriptive statistics of the considered position of the headlamps. Parameter

5th percentile value (m)

Mean (m)

95th percentile value (m)

Standardized value (m)

hh d1 d2 d3

0.665 1.663 1.147 0.222

0.731 1.775 1.345 0.320

0.838 1.903 1.479 0.387

0.75 – – –

is lit up by swiveling headlights. With regard to experiments in the real world, McLaughlin et al. [27] studied drivers target detection performance on an in-service road utilizing vehicles with two different steering-governed swiveling headlamps, and with fixed headlamps (FH). According to their results, not in all cases did the swiveling headlight significantly enhanced the visual performance of drivers. Also, the design of the swiveling algorithm affects considerably the target detection performance of drivers. Particularly, too much swivel might reduce the target detection distance. Reagan et al. [31] carried out an experimental study on an unlit road to provide insight on the effects of different headlight systems on perception, including swiveling headlights. The results showed that the aid of swiveling headlights enabled drivers to detect targets from a significantly greater distance on curves whereas no difference was found on tangents. As horizontal curves usually comprise circular arcs and spirals, swiveling algorithms should be able to adapt the light beam on both alignment types. Known as Euler spirals, Cornu spirals or clothoids, spirals are transition curves whose curvature change linearly with their length. Spirals should be long enough so as to be recognizable by drivers, provide a distance enough for the superelevation transition, and provide a smooth transition for the centrifugal acceleration due to side friction. An enhancement of HSD can also be expected on well-designed spirals. In general, the performance of swiveling headlights on spirals has deserved less attention than on circular arcs. Gaze behavior was studied by several authors in order to develop swiveling algorithms [12,20]. Based on the driver’s gaze distance, Gao and Li [12] developed a swiveling algorithm without making explicit the development method nor the functional form of the algorithm. They tested the swiveling algorithm in a qualitative way by means of a 2D simulation on simple geometric layouts. Moreover, a possible source of bias in developing swiveling algorithms based on the driver’s gaze behavior is to neglect the effect of the ASD. Liu et al. [26] presented a more complex algorithm based on neural networks that utilized diverse sensors as inputs. Hagiwara et al. [14] figured out that drivers consider swiveling headlights capable of anticipating curves to be advantageous. However, this solution would require predictive swiveling algorithms aided by positioning systems or other onboard sensors [19,34]. To feed the swiveling headlight algorithm, the horizontal alignment can be predicted by front roadway image monitoring [22]. Also, positioning systems may help predict the due swiveling angle without relying on the steering angle [24]. 3. Swiveling headlight beam model 3.1. Light beam features To evaluate the HSD, the headlight beam is modeled as per the position of the headlamps on the vehicle in relation to that of the driver. As a result, the position of the headlamps is determined in space by four geometric parameters (Fig. 1): the headlamp mounting height (hh ), the headlamp headway with respect to the driver (d1 ), the offset between headlamps (d2 ) and the offset of left headlamp with respect to the driver (d3 ). The values of these offsets (hh , d1 , d2 and d3 ) were derived from those of the 11 most sold vehicles in Spain [4]. The descriptive statistics of such parameters are presented in Table 1. Although nighttime visibility is connected to other factors, for this study to conform to the definition of HSD, the headlight beam layout adopted is based on assumptions made in successful studies described in Section 2.1. The headlight beam is assumed to be enclosed horizontally by the horizontal spread angle (α), where only external boundaries are considered; bounded vertically by the upward vertical spread angle (β); and the distance ahead circumscribed to the headlight range (Rg). Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 2. Layout of swiveling headlight beam.

3.2. Simulation of swiveling functions In order to study the effects of swiveling headlamps on the HSD, the operation of swiveling functions, i.e. the mathematical functions that control the headlight swiveling angle, was first simulated. A theoretical basis on the horizontal projection on circular arcs was used for the development. Both headlamps are assumed to swivel the same angle in parallel at each position (Fig. 2). This criterion is supported by the results found in literature [36], and it also makes sense as on right hand bends (in right-lane driving countries) a fixed left headlight would continue to light up straight ahead causing possible glare to oncoming traffic [18]. In this study, it is assumed that adaptive headlights swivel as per the steering wheel input to keep the beams on the roadway illuminating a section of roadway ahead that is as long as possible. In turn, the steering wheel turning angle is assumed to be directly proportional to the vehicle yaw variation per traveled length. The vehicle yaw is the angle rotated by the vehicle around its vertical axis changing the direction of heading. If the yaw is denoted by the letter ψ, the vehicle yaw variation per traveled length, or simply the yaw variation θ , is given by: dψ (1) θ= ds Concerning the modeling of the swiveling angle, it must also be noted that no additional inputs have been considered in addition to the yaw variation in order to prevent external factors from affecting the results. As aforementioned, the sight distance is evaluated over the vehicle path, which is considered in the shape of a sequence of points called stations. According to the Spanish geometric design standard [29], the vehicle path to assess sight distance is set at a fixed distance of 1.5 m from the left edge of the lane. Both the driver and the target are assumed to be located along such a path. In addition, the horizontal projection of the light pattern is assumed to be enclosed by the vertical planes determined by the horizontal spread angles and the headlamps, the front of the vehicle and the beam range. Under this consideration, the best fit of the light beam to the trajectory on a horizontal curve of sufficient length is theoretically determined by a swiveling angle (ϕ) that makes the outer beam boundary of the light pattern tangent to the abovementioned path. This layout would produce the maximum HSD as the beam would span the maximum possible distance of the curved path without interruption. For the sake of simplicity, let us consider γ as the angle of adjustment of the headlight beam to the circular arc, which is the difference between the swiveling angle and the horizontal spread angle (Fig. 3). Based on trigonometry, the following expression can be deduced: t → t = R tan γ (2) p = R sin γ = t cos γ → tan γ = R c = tan γ → c = d1 tan γ (3) d1 If the relative position of the driver is considered in the geometric relation, then: (c + d3 + R)2 = t 2 + R 2

(4)

(d1 tan ψ + d3 + R)2 = R 2 tan 2γ + R 2

(5)

It can be rewritten as d12 tan 2γ + d32 + R 2 + 2d3 R + 2d1 tan γ (d3 + R) = R 2 tan 2γ + R 2

(6)

Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 3. Headlight swiveling that maximizes HSD on a circular arc.

( ) tan 2γ d12 − R 2 + tan γ (2d3 d1 + 2d1 R) + d32 + 2d3 R = 0

(7)

Resolving the equation: tan γ =

−2d3 d1 − 2d1 R ±

√

)( ) ( (2d3 d1 + 2d1 R)2 − 4 d12 − R 2 d32 + 2d3 R ) ( 2 d12 − R 2

Therefore, the headlight swiveling angle that maximizes the HSD on circular arcs is given by: √ ⎛ )( )⎞ ( −2d3 d1 − 2d1 R ± (2d3 d1 + 2d1 R)2 − 4 d12 − R 2 d32 + 2d3 R ⎠ ( ) γ = tan−1 ⎝ 2 d12 − R 2

(8)

(9)

The mathematical functions that control the headlight swiveling angle must be continuous functions that smoothly adjust to the curvature of the defined path while traveling, particularly around θ = 0 so that the swiveling angle is exactly zero on tangents and very close to zero on curves with a very large radius (Fig. 4). Otherwise, when assuming horizontal spread angles greater than zero, the headlamp would swivel a certain angle greater than the horizontal spread angle, even on those curves with a very large radius. Therefore, the swiveling function must be smoothly adjusted to the contour condition imposed by the null swiveling angle of the headlamp on tangents. In addition, the spirals should also serve as a transition of the swiveling angle due to the curvature variation produced along them. The great sensitiveness of the swiveling angle to little yaw variations is critical for the HSD outcome in curved alignments if the value given by Eq. (9) is exceeded. In such a case, a small increase in the yaw variation would cause the increase of the swiveling angle up to a value that exceeds the optimal one, pushing the headlight beam away from the path. As a result, the headlight beam would not light up the path because the outer boundary would be secant to the arc, lighting up the inner roadside instead. Such an interruption of the illuminated section of the path would drastically reduce the HSD value. As aforementioned, the headlight swiveling angle is assumed to depend on the yaw variation of the vehicle by means of the steering wheel rotation, where the driver responds according to their interpretation of the highway Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 4. Critical swiveling angle on circular arcs (dotted red line) and swiveling function (solid blue line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

alignment. The steering wheel rotation is in turn determined by the curvature of the path at the station where the vehicle is located. In this sense, the HSD performance of swiveling headlights on spirals may not necessarily be as satisfying as on circular arcs. Indeed, the headlight beam of swiveling headlights must be able to cope with the variable curvature of the spirals. At the position of the vehicle traveling along a spiral, the curvature value will be different than that of the corresponding point of the path where the boundary of a swiveling headlight beam would be tangent. Therefore, no accurate prediction of the changing curvature ahead can be made to adapt the headlight beam based only on the curvature of the current position. Hence, when entering a curve, the swiveling headlight will tend to under-swivel with respect to the angle given by Eq. (9) and, when exiting a curve, the adaptive headlight will tend to over-swivel. A mathematical expression that adjusts fairly well to the theoretical swiveling angles under the conditions described above is the one that follows. f (θ ) = C · θ E

(10)

where C is a coefficient and E an exponent. As the swiveling function was developed for the horizontal projection, the values of these two parameters must be calibrated to obtain the desired adjustment. Also, this approach would require the headlamp stepper motor to produce a continuous variation of the swiveling angle. Fig. 4 shows an example of a swiveling function for α = 6◦ (solid blue line). One of the conditions for calibrating the headlamp swiveling functions is that the swiveling angle does not exceed the critical angle that results from Eq. (9) (dotted red line) for any curvature value. The headlight swiveling function (solid blue line) must therefore fall within the area between the horizontal axis and the curve of the maximum swiveling angle (dotted red line). Otherwise, the swiveling headlight will over-swivel, reducing drastically the HSD. To sum up, it is expected that the swiveling headlamp shall have an appropriate performance on tangents and circular arcs whereas the variable curvature of the spirals may result in suboptimal swiveling angles of the simulated lighting systems. Before the beginning of a horizontal curve, with or without spiral, the swiveling angle will be lower than desirable and, before leaving this curve, regardless of whether there is a spiral, the swiveling angle will be greater than necessary as the steering angle that feeds the swiveling system merely contemplates the curvature of the point where the vehicle is located. The intensity of these effects at or near transitions is likely to be greater the shorter the transition and the greater the difference in curvature between the connected alignments. To compensate these effects, a reduction of the swiveling angle for lower curvature values should be preferred in the calibration of the headlight swiveling functions, slightly reducing the most favorable HSD values (circular arcs and transitions to greater curvature) and significantly increasing the most unfavorable ones (transitions to lower curvature). An additional conclusion that can be drawn from Fig. 4 is the necessary asymmetry of the swiveling function. Given the same yaw variation value to the left and to the right, the swiveling angle that makes the outer beam boundary tangent to the path for the left turn would be greater, in absolute value, than the counterpart swiveling angle in the right turn. This is an immediate consequence of the vehicle path considered in the assessment of the sight distance, for which the positions of the vehicle headlamps are not symmetrical. Since the right headlamp is Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 5. Procedure to develop and calibrate swiveling headlight functions. Table 2 Coefficients and exponents of the swiveling functions developed. Right swiveling coefficients

Left swiveling coefficients

Exponents

α (o )

Min.

Max.

Min.

Max.

Min.

Max.

0 1 2 3 4 5 6 7

2 3 4 5 6 7 8 9

9 10 11 12 13 14 15 16

6 7 8 9 10 11 12 13

13 14 15 16 17 18 19 20

0,1 0,1 0,1 0,1 0,1 0,1 0,1 0,1

1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0

further away from the path than the left one (d3 < d2 – d3 ), the swiveling angle must be greater in left bends than in right bends. However, to guarantee homogeneous HSD performance on turns to both sides, the swiveling headlight functions must be characterized and calibrated accordingly. Under the above premises, a procedure programmed in MATLAB, which is outlined in Fig. 5, carried out the calibration of the swiveling functions in four steps as described below: 1. Simulation of swiveling functions by evaluating equations (9) and (10) for 29 different values of circular curve radii ranging from 20 m to 1500 m, which correspond to values of the vehicle yaw variation between 0.0382◦ /m and 2.8648◦ /m respectively. The swiveling functions generated cover all possible combinations of coefficient (C ) and exponent (E) values in the ranges presented in Table 2. The coefficients, taken at intervals of 0.01 produce 701 values; the exponents, taken at intervals of 0.001 produce 901 values. Each of them is evaluated for both left-hand and right-hand headlamp swiveling functions. Finally, all these combinations are evaluated for integer values of horizontal spread angles α between 0◦ and 6◦ . This range of values could also be interpreted as a simulation of different isolux surfaces for diverse threshold illuminance values. A total of 4,421,207 swiveling functions were therefore generated to the left and as many to the right. 2. Determination of the domain of values (C, E) that produce valid swiveling functions for each headlamp horizontal spread angle. The validity condition is given by the fact that, for no curvature value, the swiveling function (10) is greater than the angle defined by the expression (9) (as for the curve in Fig. 4). Fig. 6 shows the boundaries of these domains as a function of the horizontal spread angle for right (a) and left (b) turns. Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 6. Swiveling function parameters generating maximum headlamp turn without overturn on (a) right curves and (b) left curves.

Valid values are those below the corresponding line in the graphic. This check reduces the number of possible functions to 2,428,427 for right turns and 2,038,922 for left turns. 3. Characterization of each swiveling function based on their fitting deviation. The fitting deviation is determined by the estimation of two parameters in the 29 proposed values of vehicle yaw variation: the mean residual and the moment of the residuals with respect to the null curvature axis. The first one was calculated according to the following expression: Rm =

n ∑ ϕi − ϕˆi n i=1

n = {1, 2, . . . , 29}

(11)

ϕi being the swiveling angle calculated according to Eq. (9) and ϕˆi the headlight swiveling angle calculated as per equation (10). The moment of the residuals for the zero curvature axis was calculated from the following equation: ( ) n ∑ θi · ϕi − ϕˆi n = {1, 2, . . . , 29} (12) M Rm = n i=1 where θi represents each of the 29 vehicle yaw variation values for which the headlamp swiveling functions were set. Fig. 7 displays the scaled color plot of the mean residuals and the moment of the residuals of the headlamp swiveling functions for α = 6◦ . 4. Establishing a univocal correspondence between a number of left-turn swiveling functions and right-turn swiveling functions from their values of the mean residual and moment of the residuals. In this way, each swiveling algorithm will be defined by two swiveling functions, one for left-hand bends and one for righthand bends, so that the counterpart swiveling angles produce identical HSD performance in bends of equal radius on either side. The correspondence was established based on the right-hand swiveling functions generated since all values in the validity domain of the right-hand swiveling functions as set in Table 2 have a corresponding left-hand swiveling function, nevertheless those of the left-hand swiveling functions do not have a corresponding pair of values in all cases. It must be noted that the initial high number of pairs of values was set to eventually produce univocal correspondence between left-hand and right-hand swiveling functions with high precision, but not all were used. The final number of selected swiveling angle values decreases as the coefficient values were finally taken at intervals of 0.1 and the exponents at intervals of 0.01, always within the validity domains for each value of α. Under these assumptions, a total of 24,663 swiveling algorithms were obtained, which is a large-enough but computationally-feasible number of swiveling algorithms to use in the HSD evaluation. The pairs of values (C , E) considered are shown in the graphs in Fig. 8. It should be highlighted that most points overlap for several values of parameter α. It is also noticeable that the correspondence between swiveling functions can only be done from right-hand turn to left-hand turn since, otherwise, the some pairs of values of corresponding swiveling functions of right-hand Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 7. Scaled color plot of swiveling functions on left bends for α = 6◦ : (a) mean residuals and (b) moment of the residuals. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Scatter plot of coefficient and exponent values of (a) right-hand swiveling functions and (b) left-hand swiveling functions.

swiveling would have fallen outside the respective domain of validity considered. For example, the pair (C, E) = (0.7, 8) in left swiveling would result in a pair of values to the right of the vertical line defined by C = 1 in right-hand swiveling. 4. Test on 3D highway model A two-lane rural highway section located in the region of Madrid (Spain) was selected to examine the effect of swiveling headlights on nighttime sight distance. This section is 11 km long and its design speed is 60 km/h. The cross section is featured by a roadway width of 6.5 m and 0.5 m-wide hard shoulders on both roadsides. It was selected for its wide variety of curve radii, both on vertical and horizontal curves as well as a comprehensive combination of them all. As a result, the swiveling headlights could be tested in a range of geometric designs that encompasses the variety of possible layouts existing on rural highways. The main features of the horizontal alignments of the tested highway are exhibited in Table 3. In the number of alignments, each horizontal curve is considered along with its adjacent spirals. The counterpart features of the vertical alignments are outlined in Table 4. In both cases, the threshold values of the intervals presented were selected from the design parameters of relevant design speeds of the Spanish standard [29]. Furthermore, Fig. 9 depicts a terrain map of a sub-section of the selected highway, where a sequence of curves on the horizontal projection is observed. A GIS-based Add-In developed by the authors was employed to compute the ASD [6]. The necessary inputs to build up the 3D model include a digital elevation model in the shape of a triangular irregular network, the vehicle path as well as the standard driver’s eye height (1.1 m) and target height (0.5 m) needed in sight distance estimations [29]. Both the driver and the target are successively placed along such a path. The target height represents the size of the critical obstacle that should be visible to the driver. In this study, the target was assumed not to be self-illuminated. Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 9. Terrain map of a sub-section of the selected highway. Table 3 Main features of the horizontal alignments of the highway studied. Type of alignment

Radii range (m)

Number of alignments

Length of circular arc (m)

Total length (m)

Horizontal Horizontal Horizontal Horizontal Tangent

50 ≤|R| < 130 130 ≤|R| < 265 265 ≤|R| < 485 485 ≤|R| R=∞

11 12 15 8 34

679.178 785.625 805.566 522.578 –

1133.091 1383.642 1551.716 782.487 6101.195

curve curve curve curve

Table 4 Main features of the vertical alignments of the highway studied. Type of alignment

Range of curvature parameter (m)

Number of alignments

Total length (m)

Crest Crest Grade Sag Sag

−800 ≥ KV > −2300 −2300 ≥ KV – 3000 ≤ KV 760 ≤ KV < 3000

20 8 60 22 12

1866.487 499.355 5749.435 1865.124 971.984

The outcome of the ASD estimation is subsequently used for computing the HSD as defined in Section 2.1. Based on the geolocation of the stations, this consists of checking whether the targets ahead visible during the day from a particular driver’s position are enclosed within the headlight beam defined by the corresponding boundaries. The HSD calculation process was automated by means of a MATLAB script, which enabled the launch of multiple variations of the lighting parameters considered. It is important to highlight that the simulation implicitly assumes that no additional traffic exists as the research interest is to study the effect of swiveling headlamps on highway geometric design without additional factors interfering. It must be also noted that, to generate the swiveling angles, the vehicle yaw rate at a station i was computed as the average azimuth difference between those of the previous and the subsequent stations. Therefore, the stations of the vehicle path were spaced as close as 1 meter apart. Consequently, the results are less sensitive to the yaw rate between consecutive stations. Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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In this study, the effects of two variables defining the headlight beam of adaptive headlights on HSD are explored, namely the horizontal spread angle (α) and the headlight swiveling angle (ϕ). As both are defined on the horizontal projection, the interaction between them is of particular interest. The HSD series were retrieved according to different assumptions. These assumptions can be classified into three groups: • First, FH simulating conventional headlights were regarded, for which the swiveling angle is null all along the highway section. Under this hypothesis, 7 sets of HSD are produced, referred to as fixed headlamp (FH) HSD, one per value of the horizontal spread angle. • Second, swiveling headlamps governed by the steering angle were considered. This assumption corresponds to the swiveling algorithms devised and calibrated as explained in Section 3.2 (24,663 sets), in which the swiveling angle was zero only on tangents. In these sets, the swiveling algorithms with the best overall performance for each horizontal spread angle value (7 sets) were selected. From here on, these sets will be referred to as Steering-Governed Optimum Swiveling Algorithm (SGOSA) HSD. • Third, the longest HSD values obtained at each station for each horizontal spread angle value, regardless of the swiveling algorithm (or FH) that produces it, are contemplated, featuring 7 sets. From here on, these sets will be referred to as Steering-Governed Predictive Swiveling Algorithm (SGPSA) HSD. 5. Results and discussion 5.1. Performance indicators As a criterion to assess the HSD performance of each set of results, an overall indicator Ih is proposed. It is applied to the results obtained on the selected highway under the assumption of swiveling headlamps governed by steering angle, and is calculated as follows: Ih = w f · (F25 + F30 + F35 + F50 + F70 ) + wa · H S D + wr · ∆H S D

(13)

The expression above accommodates the following values of indicators of relevant stopping sight distances (SSD) in the Spanish design standard [29] and statistical measures of the HSD sets of each swiveling algorithm. The relative frequencies F25 , F30 , F35 , F50 and F70 are quantified in a range from 0 to 1; where 0 is assigned to the set with the maximum frequency, and 1 to the set with the minimum frequency, for each value of α. The other two normalized values (H S D and ∆H S D) range from 0 to 1; where 0 is assigned to the set with the minimum respective value, and 1 to the set with the maximum respective value, for each value of α. • • • • • • • •

F25 is the relative frequency of HSD less than 25 m (standardized SSD for 30 km/h). F30 is the relative frequency of HSD less than 30 m (standardized SSD for 35 km/h). F35 is the relative frequency of HSD less than 35 m (standardized SSD for 40 km/h). F50 is the relative frequency of HSD less than 50 m (standardized SSD for 50 km/h). F70 is the relative frequency of HSD less than 70 m (standardized SSD for 60 km/h). H S D is the normalized average HSD. ∆H S D is the normalized ratio swiveling HSD/fixed HSD. w f , wa and wr are the weighting values of frequencies, average HSD and the ratio swiveling HSD/fixed HSD respectively. For the evaluation of the swiveling algorithms, the values 1, 2 and 3 have been respectively selected on the basis of the importance of each parameter in the mathematical expression. Half of the weight accounts for the distribution of frequencies and half for the HSD performance itself, where the sight distance increase is more important than the average HSD. However, other weighting values could be selected depending on the importance given to the parameters.

The frequency values were normalized to the interval from 0 to 1 so that a higher frequency of low HSD values results in a worse value of the indicator according to the expression that follows: xmax − x xˆ = (14) xmax − xmin Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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C. De Santos-Berbel and M. Castro / Mathematics and Computers in Simulation xxx (xxxx) xxx Table 5 Swiveling function parameters and Ih values of the SGOSA sets. α (◦ )

Cright

Eright

Cle f t

Ele f t

Ih

0 1 2 3 4 5 6

7.10 8.20 9.60 10.70 11.30 12.50 13.60

0.790 0.610 0.580 0.540 0.480 0.460 0.460

11.75 12.74 14.13 15.22 15.82 17.02 18.09

0.649 0.558 0.544 0.520 0.479 0.464 0.460

9.65 9.73 9.69 9.70 9.73 9.75 9.78

The average HSD values and the ratio of HSD improvement were normalized to the interval from 0 to 1 according to the following expression: x − xmin xˆ = (15) xmax − xmin where • • • •

xˆ is the normalized value of the indicator for the set of HSD values. x is the value of the indicator for the set of HSD values before normalization. xmax is the maximum value of the indicator for the corresponding horizontal spread angle α considered. xmin is the minimum value of the indicator for the corresponding horizontal spread angle α considered.

For each term of the Ih indicator, a different swiveling headlamp algorithm performs best. However, the Ih indicator provides a balanced score of how well a swiveling algorithm works according to different performance measures. The best Ih scores of the sets calculated with SGOSA for each value of the horizontal spread angle constituted the SGPSA HSD sets. The (C, E) values of these sets and the Ih scores are shown in Table 5. It should be noted that the best performing swiveling headlights did not necessarily adjust the headlight beam to the path as shown in Fig. 3. In any case, when driving on curves, the outer part of the lane is supposed to have already been lighted up by the headlights from previous positions, so the driver would already have had a view of the outer portion of the lane shortly before. Next, measures of the performance of the swiveling headlamps as per three of the terms of the overall indicator Ih considered individually are presented: the highway length where sight distance is less than 70 m, the increase of the HSD and the average sight distance. The bar graph of Fig. 10 illustrates the percentages of highway length where the sight distance was less than 70 m for the daytime ASD as well as the FH, SGOSA and SGPSA HSD sets. This is the distance needed to stop a vehicle at a speed of 60 km/h on wet pavement according to the Spanish standard [29] and was used as a reference to make up the indicator Ih . High values of this indicator evidence sections with poor sight distance conditions that may entail potential safety issues. In daytime, the length with less than 70 m of ASD is less than 1%. High frequencies are nevertheless observed in the case of the FH HSD sets, especially when the horizontal spread angle is small. For example, sight distance in less than 70 m on more than half of the selected highway for α equal or less than 2◦ . As the horizontal spread angle increases, a steady reduction in the frequencies of HSD values shorter than 70 m is observed in the case of the FH HSD sets, while the swiveling headlamp series showed a more pronounced downward trend. Therefore, the results indicate that all swiveling headlights largely outperform the standard fixed ones, the SGPSA outperforming in turn the SGOSA ones. Fig. 11 displays the average increase of the HSD produced by the swiveling headlamps with respect to fixed headlights under the assumptions of the best performing sight distance sets described above. In contrast to the previous indicator, the higher the value of the indicator is, the more favorable the result. The SGOSA HSD sets reached their maximum improvement (90%) for α = 1◦ . This indicates that the most favorable effect of swiveling headlights is produced under the assumption of a narrow headlight beam. Moreover, the increase of sight distance provided by swiveling headlights is very significant, even if the horizontal spread angle is wide. The average sight distance values of the relevant sight distance sets are shown in Fig. 12. The highest values of this indicator are also produced for the series that showed the best performance. The ASD series (blue bars), with Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 10. Frequency of observed sight distance less than 70 m in sight distance sets.

Fig. 11. Ratio swiveling HSD/FH HSD in SGOSA and SGPSA sets.

Fig. 12. Average values of observed sight distance in best performing sight distance sets. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 13. (a) ASD and HSD under different assumptions on a left curve of radius 95 and (b) counterpart swiveling angles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

an average value of 238 m, can be taken as a reference since the values are constant. For every swiveling algorithm, it can be straightforwardly noted that, the average sight distance increased moderately for increasing values of α. In fact, it is still desirable that the average sight distance is as long as possible even beyond the required SSD. Drivers might benefit from greater average sight distances since they could anticipate the course of the road and have an enhanced orientation. In this sense, the average sight distance is a measure that adequately accounts for this. However, since the range of the headlamps was set at 200 m, it would be difficult to bring the average values of these series closer to daytime visibility no matter how much the value of α is expanded. In particular, for every fixed value of α, the HSD derived from SGPSA (green bars) performed better that the SGOSA series (yellow bars), and the latter ones in turn performed better than the fixed headlamp (red bars) approach. 5.2. Effect of highway geometry Next, the performance of the headlamps is illustrated on subsections of the selected highway. This way the effect of swiveling headlights can be analyzed in relation to the existing sequences of highway alignments. First, the ASD and HSD results on a single curve of radius 95 m are exhibited in Fig. 13a, and the counterpart swiveling angles are shown in Fig. 13b. The curvature along the subsection is displayed on both by the black solid line, it being quantified by the right vertical axis. The assumptions of the HSD series included the FH (red lines) and two values of the horizontal spread angle (α = 0◦ , in dotted lines; and 4◦ , in solid lines) for each of the swiveling algorithms: SGOSA (yellow lines) and SGPSA (green lines). Minimum HSD values were found for the FH, SGOSA and SGPSA series at the stitching point between the tangent and the spiral (at station 1745, sight distance equals 21 m for α = 0◦ and 42 m for α = 4◦ ), after which a sudden increase of HDS was produced in the series of swiveling headlights. On the circular arc, the swiveling headlights were found to produce significantly greater sight distances. However, the limiting factor was the light beam at every station, either the lateral bounds or, particularly from station 1829 in the SGPSA, the beam range. As it could be expected, the predictive swiveling algorithms (SGPSA) provided greater sight distances than the SGOSA series, and the latter ones outperformed the FH. However, on outbound spirals, the undesirable effect of over-swiveling was produced. The SGOSA set with α = 0◦ showed a significant drop in sight distance near the end of the curve, with values that are even shorter than the ones produced by the counterpart FH series from station 1861. This effect was caused by the fact that the swiveling angle produced by the SGOSA is determined by the curvature at the point where the vehicle is located, whereas the curvature ahead decreases to 0. Therefore, the headlight beam is still oriented towards the inside of the curve while the alignment ahead is a tangent. The over-swiveling effect is more prone to occur when the horizontal spread angle is small, and was produced over a longer stretch. As shown by the SGPSA, series which can take null swiveling angle even if the curvature is not zero, the swiveling angle that provided longer sight distance near the end of the curve was 0◦ (Fig. 13b). The pattern according to which the swiveling angle returns to 0◦ was different in the SGPSA than in the SGOSA series. While the former followed a step-wise pattern, the latter showed a continuous variation, it reproducing approximately the shape of the curvature graph. Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 14. (a) ASD and HSD under different assumptions on reverse curves and (b) counterpart swiveling angles.

The sight distance evaluation of a reverse curve is plotted in Fig. 14. Fig. 14a displays the ASD and the HSD along with the curvature data, and Fig. 14b shows the counterpart swiveling angles. The assumptions of the HSD series included the FH and two different values for the horizontal spread angle (α = 3◦ and 6◦ ) for the SGOSA and SGPSA series. The initial curve has a radius of 200 m. The ASD was found to be a limiting factor at the beginning of the curve in some of the series owing to the presence of an overlapped crest vertical curve. Over-swiveling was produced at the end of this curve for the SGOSA and SGPSA series because the non-predictive swiveling headlights produced poor sight distance conditions as the swiveling angle is based on the curvature at the point where the vehicle is located (through steering angle) while the curvature ahead varies to the opposite side. Concerning the swiveling angles, the SGPSA series reset the swiveling angle to zero before the inflexion point. Finally, the robustness of the results against the uncertainty in the values of the offset variables d1 , d2 and d3 was tested. To this aim, the HSD was computed using the same swiveling functions that produced the SGOSA sets, defined in Table 5. For each value of α and for each offset variable, the 5th and 95th percentile values of one of the parameters presented in Table 1 were used as inputs while maintaining the mean of the other offset values as in the main experiment. As a result, 42 additional sets of HSD were obtained. Fig. 15 (a, b, and c) illustrates the distribution of HSD values in the sensitivity analysis of the offset variables for α = 5◦ . The frequencies of HSD values showed a close similarity in all cases, especially between the sets that correspond to offset variables d1 and d3 . To further explore the impact of the offset variable d2 on the HSD, Fig. 16 shows the sets of results in a subsection of the tested highway for α = 5◦ . It confirms the close adherence between the HSD of the three sets, although a slight difference was noticed at singular spots, such as the curvature break existing between the circular arc and the tangent at station 2112. Moreover, given the adherence observed between the paired HSD values, it can be concluded that the variation of the offset parameters did not significantly affect the effects described in relation to the alignment layouts, at least from a qualitative point of view. 6. Conclusions In this paper, the effect of the headlamp swiveling algorithms and the highway geometric design on the HSD performance was studied. The results showed that the swiveling headlights significantly increase sight distance. It can be concluded that predictive swiveling algorithms still offer potential headlight sight distance increases with respect to algorithms based on steering angle. On the basis of the distribution of HSD values that comply with standards, as the horizontal spread angle of the headlight beam increased within the analyzed range, the HSD produced by swiveling headlights increased more than proportionally. There are some weak alignment combinations that may harm the benefits of swiveling headlamps. The design of the spirals is of great importance not only because they provide proper transition between alignments of different curvature from the standpoint of driving comfort, but also because even the swiveling headlights do not perform properly enough in the event of sudden curvature changes ahead of the vehicle. Therefore, the horizontal alignment of highways should present smooth-enough transition curves, which will be particularly important where the alignment curvature reduces as the vehicle moves forward. In the case of reverse curves, as the vehicle approaches the inflexion point, the non-predictive swiveling headlights produce poor sight distance conditions since Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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Fig. 15. Distribution of SGOSA HSD values using diverse values of offset parameters (a) d1 , (b) d2 and (c) d3 .

Fig. 16. SGOSA HSD (α=5◦ ) outcome on a subsection using diverse values of offset parameter d2 .

the swiveling angle is based on the curvature at the point where the vehicle is located (through steering angle) while the curvature ahead is changing to the opposite side. Finally, the values of the offset variables of the headlamps with respect to the driver position were found not to significantly affect either the HSD outcome or the observed effects on diverse alignment sequences. On the basis of the results obtained in this study, several lines of future research could be proposed. First, it is an open question as to how best to devise a procedure to assess sight distance in nighttime considering mesopic Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.

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vision. Second, the calibration of the scoring performance based on highway safety performance is proposed. Third, the consideration of speed in the swiveling algorithm to further disclose the effects of swiveling headlight systems on the HSD for different highway geometry layouts is another potential future line of research. Fourth, the study of the impact of swiveling headlight on safety is to be pursued. For example safety performance at night and HSD might be linked. In addition, collision modification factors related to driving assistance systems could be developed. Acknowledgments This work was supported by the Ministerio de Econom´ıa y Competitividad of Spain and the European Regional Development Fund (FEDER) [grant number TRA2015-63579-R (MINECO/FEDER)]. References [1] Insurance Institute for Highway Safety (IIHS), They’re working: Insurance claims data show which new technologies are preventing crashes, Status Rep. 47 (2011) 1–7. [2] B. Adler, H. 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Please cite this article as: C. De Santos-Berbel and M. Castro, Effect of vehicle swiveling headlamps and highway geometric design on nighttime sight distance, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.08.012.