1. Introduction
When a vehicle turns, particularly trucks or other long wheelbase vehicles, the rear wheels deviate inside the path traced by the front wheels. The trailing axles of a turning vehicle increasingly deviate towards the curve center, until the rear axle wheels finally reach the maximum steady-state offset from the steering alignment path. This phenomenon is well known in transportation engineering as low-speed offtracking [
1,
2,
3]. In contrast to this phenomenon, high-speed offtracking introduces movements of trailing vehicle axels outward the curve center due to the lateral acceleration of the vehicle as it traverses a horizontal curve at high speeds [
4,
5]. In addition to the vehicle wheelbase length, road geometry parameters, such as the turn radius and turn angle, dominantly influence offtracking intensity [
6]. In general, two different states of low-speed offtracking exist: steady-state offtracking and active transient state offtracking.
According to the authors of [
7], steady-state offtracking is not normally reached before covering turn angle of 270°. This means that particular vehicle will reach steady-state offtracking only after covering a constant radius curve with a turn angle greater than 270°. The most widely accepted procedure for steady-state offtracking calculation is the traditional “sum of squares” method originally developed by the Western Highway Institute (WHI), and published in 1970 [
8], and the letter repeatedly revised by the Society of Automobile Engineers—SAE International [
9,
10].
Since roadway edges, especially in intersection zones, are usually designed with relatively small radii and with turn angles in the order of 90°, the potential application of steady-state offtracking calculation methods is quite limited, if not questionable, in practice. When a vehicle performs a sharp turn like this one, it remains in an active transient offtracking state. Hence, the question how to accurately calculate the maximum offtracking value for a vehicle if an active transient offtracking state arises.
Perhaps the most acute problem related to the lack of accurate transient offtracking calculation comes from the field of airport design. The official ICAO procedure for taxiway fillet design [
11] assumes that the maximum offtracking is reached when the aircraft datum point (usually the cockpit) reaches the end of the taxiway lead arc, which is incorrect. The consequence is an insufficient pavement width in taxiway turns. Even in standard cases (turn angle of 90° and radii of 50 m), the magnitude of error is in the range of 0.30–0.75 m. In less common dispositions, when larger aircrafts, i.e., the Airbus A380, negotiate sharp turns, the error exceeds an enormous 4.3 m!
Also, the lateral positioning of autonomous vehicles in relation to the geometry of roadway edges demands efficient transient offtracking assessment in order to safely guide vehicle through the crossroad [
12,
13,
14]. This is especially critical in the case of an articulated autonomous vehicle such as a semi-tractor trailer [
15,
16].
In past decades, numerous mathematical and numerical models for setting vehicles’ movement trajectories, while negotiating different steering path alignment configurations, have been developed and tested in the field [
17,
18,
19,
20,
21,
22,
23]. Additionally, commercial computer programs for the simulation of vehicle movements and vehicle swept path analysis, such as the Auto CAD Vehicle Tracking 2024 and Transoft AutoTURN 12.0.0, have become indispensable tools for civil engineers and other specialists involved in intersection and roundabout design [
24,
25]. Recently, extensive analyses, addressing the accuracy of the available numerical models and commercial software applications for setting vehicle movement trajectories, were conducted in a Ph.D. thesis [
26]. However, amongst all of the addressed mathematical and numerical models, only a few of them offered a solution for the calculation of the exact vehicle position where the maximum active transient offtracking is reached [
7,
27]. This position indicates the exact cross section where the width of the pavement is at its maximum. The distance between the inner edge of the pavement and the steering path, at this very location, is of crucial importance when it comes to some elements of the crossroads [
28,
29]. For example, in the most standards and design practices [
1,
30], the usual width of the right-turn channel passing behind the triangular isle is 4.5–5.5 m. Though such a channel is slightly tapered (getting narrower towards its end), its net width is constant, as measured between the marking lane on the left and the curb on its right side. This particular net width comes directly from the maximum transient offtracking. Also, when it comes to the spiral ramps in multilevel garages (180° turns), their width is a direct consequence of the maximum transient offtracking.
Nowadays, as demonstrated in [
26], their efficiency and accuracy are not enough for road infrastructure design purposes, especially for autonomous vehicle positioning in horizontal curves. For example, the results of the maximum transient offtracking calculation in Lawrence’s mathematical model for low-speed offtracking [
27] are generated by the computer in a form of a graph. Before plotting the graph, equations determining the maximum transient offtracking vehicle position were numerically solved by the computer program. The graph shows the maximum offtracking for turn angles of up to 300° and various wheelbase/turn radius ratios. The graph is simple and suitable for everyday use, but the mathematical apparatus underlining the graph is not available to the user. In other words, the user cannot check the validity of the maximum transient offtracking graphically read from the diagram. The computation workload of Woodrooffe’s method [
7] is equivalent to ICAO’s [
11]. The method relies on the deployment of an “equivalent base”, acting as a virtual base of an entire multibody vehicle assembly. Woodrooffe’s procedure retrieves the maximum transient offtracking value, but with the error of 0.06–0.10 m in lateral terms and more than 1.2 m in longitudinal terms, as proven in [
26]. On the other hand, commercially available software tools for vehicle swept path analysis based on CAD platform require from the user to draw guideline first, and then based on its shape and generated swept path width, determine the maximum transient offtracking for a selected design vehicle [
31].
The aforementioned deficiencies were the key motivation for the authors of this paper to develop an exact mathematical solution for the maximum transient offtracking calculation in sharp turns. In fact, the primary aim was to develop mathematical formulas which will calculate accurately single-unit vehicle position where the maximum transient offtracking is reached. The new calculation method will eliminate the need to perform the vehicle movement simulations first and draw any of the vehicle movement trajectories, before determining the maximum transient offtracking value. Thus, by avoiding unnecessary procedures, calculation method presented herein is simple enough just to enter the vehicle datum length d, circular radii R and turn angle, and instantly produces the maximum transient offtracking value, including the vehicle position where this maximum transient offtracking is reached.
The sections in the paper are structured in the following order. In
Section 2, mathematical definition of the problem and necessary parameters are presented, while in
Section 3, mathematical solution of the problem is given. In
Section 4 the accuracy of developed calculation method based on adopted regression model was evaluated and discussed. The last,
Section 5, contains conclusions and future research plan to further improve application of the presented calculation method.
2. Problem Statement and Input Parameters
The International Civil Aviation Organization (ICAO) has its own mathematical model for the offtracking calculation of the main undercarriage center of the aircraft [
11]. Based on this mathematical model, fillets at taxiways turns and intersections are designed. The same model applies to movement of an aircraft leaving its parking position on an apron or maneuvering on a holding bay. Basic terms and symbols related to aircraft kinematic model, taxiway and fillet design are displayed in
Figure 1 and
Figure 2.
In
Figure 2b the path of the aircraft main undercarriage center (
U), while the aircraft cockpit negotiates circular curve, is shown. Actually, the shortest distance between the circular curve denoting aircraft’s cockpit guideline and the path of the main undercarriage center represents offtracking. The position of aircraft while its datum point, or cockpit, (
S) follows an arc of a circular radii
R can be determined based on the arc length expressed by its central angle
θS and the steering angle
β. According to ICAO mathematical model [
11], the steering angle
β between the tangent on the arc in the datum point (
S) and the aircraft longitudinal axis is calculated as follows:
where the angle
θS should be entered in radians.
As the result of the calculation, Equation (1) returns the value of the steering angle
β also expressed in radians. It should be noted that Equation (1) was derived assuming that
R >
d, where
d is the aircraft datum length in meters (
Figure 1b). In addition to the taxiway center line, which is followed by the airplane cockpit during turning maneuver, the example of two fillets, each composing of an arc of a circle and two tangents, can be seen in
Figure 2c.
The parameters
X and
K used in Equation (1), respectively, refer to the following:
The maximum steering angle
βmax is achieved when the aircraft datum point (
S) reaches the end of a circular arc (
Figure 3). However, according to the authors of mathematical models for transient offtracking calculation [
7,
27], the maximum transient offtracking of the main undercarriage center is not reached at the same point as the angle
βmax (at the end of arc), but some distance beyond the end of the arc (
Figure 4). Thus, as will be demonstrated later, the ICAO’s assumption that the aircraft main undercarriage center reaches the maximum transient offtracking in the same position in which the
βmax is achieved causes a significant error.
While the datum point (
S) follows a straight guideline coinciding with the arc exit tangent, the steering angle
β progressively decreases and the aircraft main undercarriage center (
U) follows a tractrix (
Figure 3). As defined in the ICAO mathematical model [
11], the steering angle
β is calculated as a function of the distance
F which the datum point (
S) has covered along the straight exit tangent:
where
βmax represents, again, the maximum steering angle at the end of an arc, and
d is the aircraft datum length. In the last 60 years the ICAO mathematical model has been used for fillet designs at international airports by civil aviation authorities all over the world, and the accuracy of Equations (1) and (4) is unquestionable.
By applying simple transformation, Equation (4) can be written in a more convenient form:
It is important to emphasize that all the formulas in the ICAO mathematical model, derived for the calculation of the aircraft undercarriage offtracking, can also be applied to the calculation of offtracking for a single-unit vehicle. In contrast to the aircraft undercarriage, where the datum point was located exactly in the center of the aircraft cockpit, the datum point on the datum length of the single-unit vehicle is usually defined as the most prominent point in the center line of the vehicle front part, i.e., the point in the middle of the vehicle front bumper (
Figure 5).
Looking at
Figure 4, and based on the previously described formulas from the ICAO mathematical model, the calculation of the exact aircraft position where its main undercarriage center (
U) reaches the maximum transient offtracking
OTmax arises as a key problem that will be addressed in this paper. This particular aircraft position is determined by two parameters: the distance
FOTmax which the datum point (
S) has covered along the straight exit tangent, and the steering angle
βd between the longitudinal aircraft axis and the exit tangent direction. Therefore, in the maximum transient offtracking position, based on the previous Equation (5), a new equation is established:
Since there are two unknown variables, distance FOTmax and the angle βd, basic mathematical logic requires forming a system composed of two different equations in order to calculate these two variables. As Equation (6) applies to all positions of the aircraft when its datum point follows the exit tangent, for the second equation of the system it is necessary to find another reliable relation between the distance FOTmax and the angle βd that would be valid only in the position where the maximum transient offtracking is reached.
The required additional relation between the distance
FOTmax and the angle
βd is obtained from the study [
27] focused on the development of a mathematical model for transient offtracking calculation. This study proved that the maximum transient offtracking of a single-unit vehicle occurs when the direction of the motion of the rear axle (vector
) is parallel to the tangent to the steering path (guideline). The geometry describing this case is shown in
Figure 6. In other words, only in the maximum transient offtracking position is the direction of the vector
perpendicular to the line passing through the center of the rear axle (
U) and the circular arc center
C. Only in this particular position is the distance of the vehicle datum point (
S) from the beginning of the exit tangent
Fd equal to the unknown distance
FOTmax:
Using simple geometric relations retrieved from
Figure 6 and the previous identity in Equation (7), the following equation is established:
where the required distance
can further be expressed as follows:
Since the required angle
varies in the range
, and
is always
, both sides of Equation (9) can be multiplied by
. Namely, as it is well known from trigonometry that
always has positive values in the angle range
, Equation (9) becomes
Bearing in mind the basic trigonometry identity
, Equation (11) eventually becomes the following:
Now, both equations of the system that should be solved in order to calculate the required variables
and
which determine the maximum transient offtracking position are known:
4. Evaluation of the Adopted Regression Model and Discussion
In this section, an evaluation of the adopted regression model is performed and some practical aspects of its application are discussed. The simplest method of evaluation assumes the use of existing Python tools and plotting substitution
t values, retrieved using numerical methods, against the values coming from the polynomial regression model. An example of such a diagram, covering the whole set of generated input data, is shown in
Figure 9.
It is evident from the diagram in
Figure 9 that the polynomial regression function fits the numerically calculated values of the substitution
t very well. A certain discrepancy occurs for the substitution
t values of less than 0.10, which corresponds to turn angles
θS smaller than 40°.
To evaluate the potential limitations and pre-suppositions for its use, robustness analyses of the adopted polynomial regression model were conducted. Standard methods and their corresponding procedures, briefly described in
Table 4, were applied to test the robustness of the established fourth-degree polynomial regression model.
K-fold cross-validation was used for the RMSE and R2 metrics’ robustness testing. Herein, the parameter k was equal to 5, which means that the sample dataset was divided into five parts that were (roughly) equal in size. The k-fold cross-validation results are listed below:
Cross-validation RMSE scores: [0.00050482 0.00051121 0.00050779 0.00050810 0.00052693];
Cross-validation mean RMSE score: 0.00051177;
Cross-validation R2 Scores: [0.99987991 0.99988177 0.9998805 0.99987864 0.99987698];
Cross-validation mean R2 score: 0.99987956.
The produced cross-validation RMSE scores and the corresponding mean show that there is no RMSE value above the established benchmark for the desired level of accuracy (Equation (29)). Also, the obtained cross-validation R2 scores and their mean negligibly differ from the R2 value which was initially achieved for the whole generated dataset.
The methods “repeated train/test Splits, robustness to noise, outlier analysis, and generalization to unseen data” cannot be directly implemented because polynomial regression function was established by using the whole dataset. In addition, the used dataset with different combinations of parameters R, d and θ was artificially created, while the previously listed methods, particularly robustness to noise and outlier analysis, are more suitable for the robustness analysis of experimentally collected data, i.e., data acquired from measurements in the field. Generalization to unseen data requires the introduction of an external dataset that was not used in model training, which is not relevant in this case—because predetermined combinations of the key input parameters R, d, and θ were used to generate the whole dataset.
The stress testing with extreme values method gave the following results:
MSE on the high-value subset (HVS) range: 2.757111×10−7;
RMSE on the HVS range: 0.000403;
MSE on the low-value subset (LVS) range: 6.017358×10−7;
RMSE on the LVS range: 0.000595.
HVS includes predicted values of substitution t in the range: t > MDNt;
LVS includes predicted values of substitution t in the range: t ≤ MDNt.
It can be seen that higher mean square error (MSE) and RMSE values were obtained below the median (MDN) of predicted substitution t values, which is MDNt = 0.188780. MDNt closely matches to the combination of the turn angle θ of 75° and the R/d ratio of 1.40845. This means that the adopted regression model has a slightly lower accuracy for turn angles θ smaller than 75° and R/d ratios below 1.408451 or d = 0.71∙R.
Robustness analyses of the fourth-degree polynomial regression model revealed new pre-suppositions for its use. In addition to the precondition R > d, originally imposed by the ICAO, and the numerically calculated possible range for the substitution t values 0 < t < 1 (Equation (28)), robustness analyses pointed out that the regression model has a negligibly lower prediction accuracy for the turn angles of θ < 75° and R/d ratios in the range of R/d < 1.408451.
The accuracy of the adopted regression model was thoroughly evaluated by comparing substitution
t values calculated both numerically and by the fourth-degree polynomial regression function (29). In addition to the substitution
t values, corresponding steering angles
and distances
determining the vehicle maximum transient offtracking positions were calculated and mutually compared. As expected, the biggest absolute differences between the parameters
and
calculated by the two compared methods appear only in a limited number of cases for the combinations of input parameters at the beginning of the generated input dataset. These critical combinations of input parameters refer to the ones with the smaller turn angles
θS, usually less than 35°, which is in accordance with the previous findings acquired from the diagram displayed in
Figure 9 and robustness analyses.
Moreover, the results of the calculation methods’ comparison displayed in
Table 5 should be discussed from the road designers’ perspective as well. Although the 0.20 m difference between the compared values
at first seems like a serious error in the prediction of the regression model, it should be highlighted that it is actually an error in determining the position of the vehicle’s datum point in the “longitudinal sense”. In contrast to this, the observed error in the “lateral sense” is far smaller, as can be seen in
Figure 10.
To more practically explain the minor importance of errors in the “longitudinal sense” when determining
distances on the overall accuracy of the maximum transient offtracking calculation, one of the situations with an unfavorable (critical) combination of input parameters (
R,
d, and
θS) is presented in
Figure 10. The selected vehicle with a datum length
d = 5.00 m negotiates a circular curve with a turn angle of 30° and a radius
R = 20 m. The required parameters
and
, determining the maximum transient offtracking position of the vehicle, are calculated numerically.
Figure 10 proves that the lateral offset (transient offtracking) varies slightly whether the vehicles datum point is moved back or forth along the exit tangent, in relation to the point where the actual maximum transient offtracking is reached. Of course, the movement of the point concerned should stay within reasonable limits (1.0 m in
Figure 10). Negligible differences between the pairs of lengths
OTmax and
OT′, and
OTmax and
OT″, confirm that. From the presented example, it is clear that the maximum error of
m when calculating
distances using the derived polynomial regression function, in terms of practical engineering applications, has a negligible impact on the overall computation accuracy of the maximum transient offtracking position.
Now, it is important to notice that the difference between the maximum transient offtracking value OTmax and the offtracking OTβmax, calculated in the position where the maximum steering angle βmax is reached (the exact end of the arc) is 0.04 m. Though some safety margins must always be observed, the typical accuracy of construction works in plan projection must be within 1 cm. And as for aircraft negotiating taxiway turns, the error becomes much higher, compromising not specific ICAO formulas but the entire calculation procedure. The formulas recommended by the ICAO are absolutely correct, but the assumption that the maximum offtracking is reached when the aircraft datum point reaches the end of an arc is absolutely incorrect. In the common taxiway fillet case, when a Boeing 747–100 negotiates a θS = 90°, R = 50.00 m curve, the error increases up to 0.30 m. In the same turn, the Airbus A380 produces a 0.75 m error. In a less common situation, the A380 negotiating a θS = 45°, R = 50.00 m turn, the error is 2.79 m. With θS decreasing to 30°, the error exceeds 4.3 m! There are even some rare θS, R combinations producing errors exceeding 6.0 m!
5. Conclusions
The calculation method presented herein offers a unique and practical solution for the calculation of the maximum transient offtracking of a single-unit vehicle when negotiating circular curves. It uses well-proven mathematical formulas from the ICAO mathematical model for the derivation of only one transcendental equation which precisely determines the maximum transient offtracking position of the vehicle. Complicated numerical methods for solving this type of equation were substituted by a relatively simple polynomial regression function capable to predict the maximum transient offtracking position with the desired level of accuracy. In this manner, computationally demanding numerical methods for transcendental equation solving are successfully compensated with a more practical and efficient calculation method. The accuracy of the presented regression model was assessed on the whole set of input data parameters, including all combinations of R/d ratios and θS turn angles. The largest discrepancies in relation to the numerical calculations appear for turn angles θS smaller than 35°, but the magnitude of these discrepancies negligibly affects the reliability and overall accuracy of the developed regression model. Robustness analyses of the fourth-degree polynomial regression model revealed that the regression model has a negligibly lower prediction accuracy for the combination of input parameters covering turn angles θ < 75° and R/d ratios in the range R/d < 1.408451.
Though the polynomial regression function seems too long at first sight, its main advantage is the almost instant calculation of the maximum transient offtracking position. Minimal effort and an ordinary pocket calculator are enough to calculate maximum transient offtracking position for the specified input parameters R, d, and θS. All previously published mathematical models and, in particular, the currently available software tools for vehicle swept path analyses require more time and engineering skills to achieve the same results. Additionally, the method retrieves the maximum transient offtracking from the three input parameters (R, d, and θS), skipping the need for CAD-like steering path construction (as an input) and swept path generation (as an output).
It is demonstrated and proved that the maximum transient offtracking is not achieved when the vehicle’s datum point reaches the end of circular arc (assumption in ICAO’s design manual), but some distance beyond. What is more important, the error caused by this ICAO’s assumption can significantly affect the accuracy of fillet design at airports.
By applying the regression model proposed herein, it is possible to calculate the maximum transient offtracking only when the vehicle negotiates simple circular curves. For other steering alignment configurations, which include more complex horizontal curve types, i.e., transition and revers curves, this method cannot be used. Currently, the proposed regression model can be applied for the maximum transient offtracking calculation of a single-unit vehicle and not for articulated or multi-unit vehicle combinations. These flaws have already been considered by the authors of the paper and new mathematical models which tackle these issues are already under development.