A geometric‐identification–free mathematical model for recreating nonsymmetric horizontal railway alignments

Received: 23 December 2023 Accepted: 30 April 2024 DOI: 10.1111/mice.13230 RESEARCH ARTICLE A geometric-identification–free mathematical model for recreating nonsymmetric horizontal railway alignments Miguel E. Vázquez-Méndez1,2Gerardo Casal2Alberte Castro3 Duarte Santamarina2 1CITMAga, Santiago de Compostela, A Coruña, Spain 2Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, Lugo, Spain 3Área de Enxeñaría e Infraestrutura dos Transportes, Departamento de Enxeñaría Agroforestal, Universidade de Santiago de Compostela, Lugo, Spain Correspondence Duarte Santamarina, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela. Escola Politécnica Superior de Enxeñaría, Benigno Ledo, 27002 Lugo, Spain. Email: [email protected] Funding information Ministerio de Ciencia e Innovación and NextGenerationEU, Grant/Award Number: TED2021-129324B-I00; Xunta de Galicia, Grant/Award Numbers: 2023 GPC GI-2084-ED431B2023/17, 2021 GRC GI-1563-ED431C2021/15, 2022-PU014 Campus Terra (USC) Abstract The constant passage of trains on the railways tracks causes, in the course of time, deviations that must be corrected periodically by means of a track calibration process. It consists of designing a new layout, called recreated horizontal alignment (RHA), as close as possible to the deformed center track fulfilling also the technical constraints according to the operational requirements of the railway. In recent years, different models have been proposed to address this task. This paper proposes, first, a new geometrical model that works with continuous variables for the definition of horizontal alignments (HAs) to deal with nonsymmetric transition curves at both sides of a circular curve and second, an optimization algorithm to compute the recreated alignment suitable in sinuous railway sections. This new mathematical model frees the optimization process from the need to previously identify the geometric elements (tangents, circular curves, and transition curves) of the HA. The usefulness of this model is tested with two academic examples showing its good behavior and in a real case study, where this algorithm is compared with the solution adopted by the engineers in a section of the railway line Ourense–Monforte in the NW of Spain. 1 INTRODUCTION The passage of trains on the railways over time causes deviations of the tracks with respect to its original position, which are usually measured by means of a track surveying process, providing, among other relevant information, a set of coordinate points along its centerline. The extent of these deviations must be controlled periodically to guarantee the safety of train convoys as well as the comfort of This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2024 The Authors. Computer-Aided Civil and Infrastructure Engineering published by Wiley Periodicals LLC on behalf of Editor. passengers. If their values exceed certain limits, it is necessary to design a new horizontal alignment (HA) that will be used to rectify the layout by a track calibration process. The range of the lateral motion of the recalibrating machine (called tamping and lining machine) is constrained in size, and hence, it is essential to minimize the distance to which the track is moved from its current position. During this calibration, the track centerline is adjusted by adopting a recreated horizontal alignment (RHA) designed as close Comput Aided Civ Inf. 2024;1–18. wileyonlinelibrary.com/journal/mice 1 2VÁZQUEZ-MÉNDEZ et al. FIGURE 1 Railway deviation, track surveying machine, and tamping and lining machine. as possible to the surveyed points, taking into account the technical constraints that must be fulfilled according to the operational requirements of the railway (see Figure 1). The process of obtaining an optimal RHA is a complex job, generally addressed in the literature by a two-step approach. The first step (geometric identification) consists of the analysis of the information provided by the measured points on the existing alignment with the aim of identifying its geometric elements (tangents, circular curves, and transition curves). This is a difficult task due to the fact that it is not always easy to identify properly to which of the three elements belong the measured points and locate the boundaries between them (the so-called tangency points). Once these elements are estimated, the second stage (optimization) deals with the problem of searching the geometric parameters for each element that provide an optimal RHA complying with all the technical constraints. These two steps can be considered as independent stages of a sequential process. However, both steps are undoubtedly dependent on each other, making even more difficult the problem of obtaining an optimal RHA. This fact has led in recent years to the development of methodologies in which the results of both steps are iteratively reviewed until an optimal solution is reached. Several methods have been suggested for conducting the first step (geometric identification) based on the wellknown behavior of different features obtained from the measured points. The curvature analysis is the most commonly used procedure. Some authors proposed to estimate the curvature directly by fitting circular curves to adjacent points (Hans et al., 2012; Hummer et al., 2010)while others used spline techniques to obtain a smooth interpolation of the surveyed points and subsequently computing the curvature from it (Ben-Arieh et al., 2004;Castroetal., 2006). However, other alternative geometric features were also used in order to avoid some limitations associated to the curvature. Bearing or heading angles were considered to conduct the geometric identification instead of curvature to reduce the difficulties caused by the presence of noise in the measurements. Imran et al. (2006)used this approach to recover the elements of an HA of a road from the trajectories of a control vehicle. Li et al. (2012) developed an algorithm for the identification of curve locations by applying a threshold to the bearing angle obtained from GIS data. Othman et al. (2012) estimated the radius and the starting and ending points of horizontal curves using the headings computed from field operational test (FOT) data. Instead of using only two consecutive points to calculate the bearing angle, Gikas and Stratakos (2012)proposed to use the tangent of the circle arc defined by three consecutive points. Although this method required more computational time, it provided significantly less noisy bearing diagrams. Camacho-Torregrosa et al. (2015) considered the heading direction to identify the geometry of road HAs. The authors also discussed the level of noise expected in curvature and heading profiles to demonstrate that the last one is a more convenient representation. This approach is still widely used as an identification method, being considered, for example, in recent research papers (Li et al., 2019;Puetal.,2024). Another geometric feature considered to identify the alignments is the versine. Its values can be directly measured along the track (Yoshimura & Naganuma, 2013) or calculated from the coordinates of a set of known points (Shi et al., 2023). Finally, another completely different approach consists on image processing (Dong et al., 2007;Easaetal.,2007; Tsai et al., 2010), in which satellite or photographic images are analyzed by using some artificial vision techniques, such as edge detectors and the Hough transform, to identify the geometry of existing alignments. Regarding the second step (optimization), several proposals have been made to accomplish this task. Easa and 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License VÁZQUEZ-MÉNDEZ et al. 3 Wang (2011) presented an optimization model for single or multiple composite horizontal curves capable of conducting a simultaneous fitting of all its elements (circular curve, transition curves, and tangents) by using the total least squares method. Gikas and Stratakos (2012)introduced an algorithm to obtain an optimal HA based on the analysis of the bearing diagram along with its firstand second-order derivatives, and the application of a set of dynamically tuned filters to eliminate the noise generated during the process. In this approach, the center line and the parameters of the alignment were obtained by fitting the diagrams without considering any kind of distance between the existing and the recreated alignment. Liu et al. (2013) applied a Hough transform based on constraint adjustment quantity to remove anomalous measuring points before searching for an optimal alignment by using the Powell method to minimize the distances from the set of points to the alignment. Skala-Szymanska et al. (2014) proposed a new approach for fitting a circular arc into a set of measured points by using the Nelder– Mead simplex method. Camacho-Torregrosa et al. (2015) proposed an analytic-heuristic methodology to minimize the square mean error of the heading diagram obtained from the data, ensuring that the continuity required to heading and curvature along the alignment is fulfilled. User-defined thresholds for identification are no longer necessary in their methodology, leading to a unique solution for the optimal alignment. Li et al. (2018) introduced a method to obtain the coordinates of the boundary points and the parameters of a railway alignment by conducting a particle swarm optimization with a full direction search. Li et al. (2019) proposed an iterative method called swing iterations. In this procedure, the measured points are initially separated into different subsets according to the heading gradients, used to preliminarily identify the tangents, circular curves, and transition curves. Later the points-alignment consistency at the boundaries between the geometric elements is iteratively evaluated, being the points reclassified if necessary to achieve a local optimization of each element. Finally, the entire alignment obtained in the last step is further refined with a genetic algorithm (GA). Song et al. (2020)proposed the Levenberg–Marquardt (LM) algorithm for fitting the parameters of transition curves. Taking the aforementioned concept of swing iterations as starting point, Li et al. (2022) updated the previously proposed methodology to conduct an overall alignment optimization based on points-alignment consistency combined with a Mesh Adaptive Direct Search (MADS) algorithm. Pu et al. (2023) proposed a two-stage optimization in which first the existing alignment is automatically segmented into reused and reconstructed sections and then a multidirectional distance transform is used to obtain an optimal redesigned alignment considering different categories of constraints. Shi et al. (2023) introduced a smoothness optimization method based on orthogonal least squares (OLS) and the theory of controlling the track smoothness. Their method consists of three stages: The first one is a preliminary geometric identification using the versine and considering a fixed window length and local adaptive thresholds, the second one is an iterative OLS fitting process considering multiple types of curves, and the third one is a further optimization process in order to assure that the residual deviations fulfill the track smoothness management requirements. Recently, Pu et al. (2024)proposed a three-stage optimization method to obtain an optimal RHA. The first stage consists of a preliminary geometric identification conducted by a dynamic threshold method based on heading gradients computed from the measured points. At the second stage, the authors applied an adaptive analytic method for global alignment recreation. Finally, at the third stage, a customized Harmony Search Algorithm (HSA) is used for improving accuracy by a further optimization of local sections of the recreated alignment. The treatment given to transition curves differs from papers. In those where the optimization is carried out based on fitting curvature diagrams or headings, the use of symmetrical or asymmetrical clothoids does not represent excessive difficulties. On the other hand, in those where the objective to be minimized is stated in terms of distance between points, the use of asymmetric clothoids hinders the evaluation of distances and incorporates additional design variables to the optimization problem. For this reason, it is common to use simplifications of the clothoid, for example, using cubic parabolas or Taylor’s expansion, and its parameters can also be constrained to discrete values, which reduces the space for searching for optimal values. Although some of these techniques are contemplated in certain countries’ railway regulations, making them perfectly valid, this is not the case of the Spanish technical standard in which the use of clothoids is mandatory. Generally, the technical regulations consider highly recommended, in newly constructed railway lines that the horizontal curves should always be symmetric. However, many old railways line routes that have been subjected over time to several rectifications or renewal processes (e.g., to eliminate level crossings or due to small route redesigns) ended up with transition curves that do not fit into this standard. Therefore, the possibility of using nonsymmetric transition curves should be considered in order to properly define an optimal RHA. To handle the RHA problem, the main contributions of this paper are the following: (1) A novel geometrical model is presented. This new model is capable of tackling nonsymmetric HAs with 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4VÁZQUEZ-MÉNDEZ et al. different clothoid length at both sides of any circular curve depending on continuous variables, allowing a higher degree of accuracy than discrete models. (2) An optimization method to obtain an optimal RHA from a set of points defining the deviated railway track. The method consists of solving a constrained optimization problem considering an objective function based in computing the orthogonal distance between the surveyed points and the HA. This problem lies within the framework of constrained, nonlinear, differentiable optimization. To address it, a well-tested method such as the Sequential Quadratic Programming (SQP) method (Nocedal & Wright, 2006) is used. The initial alignments for the SQP were created with a new preliminary step, which consists of obtaining, by using a coarse (easy and fast computing) objective function, a layout that minimizes the 𝑦-distance between an HA and a monotonic cubic spline constructed along the set of surveyed points, like a sort of least square fitting. This preliminary step was tackled with two different alternatives: (1) First preliminary step alternative: It is based on generating, with this 𝑦-distance method, starting points with the complete HA. This approach eliminates the need to conduct the identification stage carried out by other methods in the literature (geometric-identification–free). However, this first approach requires of certain hypotheses over the surveyed points, which limit its use when it comes to recreating sinuous alignments or when the alignment is affected by backtracking. (2) Second preliminary step alternative; It is sustained in performing an approximation of the 𝑦-distance curve by curve. This division is carried out by considering an arbitrary point on tangents providing a quasi-geometric-identification–free method. Later, all curves are concatenated to obtain a global alignment to be used as different starting points, like a sort of corridors, in the orthogonal distance optimization problem. This alignment splitting procedure results in a more general method where no previous hypotheses on the surveyed points are needed and it is also valid for sinuous alignments. The paper is organized as follows: The mathematical model is described in Section 2in which the geometrical model, constraints, and the objective function are stated. Section 3proposes two algorithms for RHA that are tested in Section 4to assess the good behavior of the model. The experiments carried out consist of two academic examples and a case study in which an optimal RHA is obtained for an existing section of a railway line. Finally, the conclusions of this research along with some lines of future work are summarized in Section 5. 2MATHEMATICAL MODEL 2.1 Geometrical model: Decision vector The HA of a road or a railway is a 2D curve formed by the appropriate combination of tangents, circular curves, and transition curves (clothoids) joining the two ending points 𝐏𝑎,𝐏 𝑏∈ℝ 2. In previous works, the authors presented a model with 𝑁symmetrical curves of type clothoid-circular curve-clothoid with the vertex on the convex side of the curve (see Casal et al., 2017; Vázquez-Méndez et al., 2018). In this previous model, an HA was determined by its end points 𝐏𝑎,𝐏 𝑏∈ℝ 2, their main tangent intersection points (IPs), 𝐕𝑗=(𝑋 𝑗,𝑌 𝑗)∈ℝ 2, and the radii 𝑅𝑗>0and angles 𝜔𝑗≥0of the circular curves. In addition, the so-called egg, double-egg, or horseshoe curves were not covered by the model, these being types of curves not considered in this new model either. As it was mentioned above, old railway lines often end up with transition curves that do not fit into this symmetric frame. Hence, the possibility of the existence of nonsymmetric transition curves in order to define an appropriate recreated alignment should be taken into account. In this new scenario, it is necessary to introduce a new degree of freedom on each curve to take into consideration the nonsymmetry and being able to properly and uniquely define the alignment (see Figure 2a). In this case, the decision variables are collected in the following decision vector 𝐱𝑁=(𝑋 1,𝑌 1,𝑅 1,𝜔 1,𝜇− 1…,𝑋 𝑁,𝑌 𝑁,𝑅 𝑁,𝜔 𝑁,𝜇− 𝑁)∈ℝ 5𝑁 (1) where 0≤𝜇− 𝑗<𝜋 2is the angle born by the 𝑗-th ingoing clothoid. This decision vector 𝐱𝑁provides the whole HA as well as important key points (tangency points) and other relevant values throughout the entire layout. In the following, it is shown how to compute the tangency points and these other significant magnitudes in the 𝑗-th curve (see Figure 2b). First of all, the deflection angle is defined by 𝜃𝑗= arccos (⟨𝐕𝑗−𝐕 𝑗−1,𝐕 𝑗+1 −𝐕 𝑗⟩ 𝑉𝑗𝑉𝑗−1 ⋅𝑉 𝑗+1𝑉𝑗)∈[0,𝜋) (2) with𝐕0=𝐏 𝑎and𝐕𝑁+1 =𝐏 𝑏, where, for sake of simplicity, the distance between two points 𝐀and 𝐁∈ℝ 2is named by 𝐴𝐵, that is, 𝐴𝐵 = ‖𝐁−𝐀‖ 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License VÁZQUEZ-MÉNDEZ et al. 5 FIGURE 2 Figure (a) shows the geometrical model of a nonsymmetric horizontal alignment (HA) between ending points 𝐏𝑎and 𝐏𝑏.In black are the decision variables, in orange the tangency points between clothoids and tangents (𝐓− 𝑗,𝐓 + 𝑗) and between the clothoids and the circular curve (𝐅− 𝑗,𝐅 + 𝑗). Subfigure (b) shows the major elements in the 𝑗-th nonsymmetric transition curve. Only the subindexes are shown in the vertex. In the other elements, the use of subindexes is avoided for sake of simplicity. Also, from now on, until the end of this section, the subindexes 𝑗are omitted in all the equations with the exception of the vertices for a better reading. The angle of the outgoing clothoid is 𝜇+=𝜃−𝜔−𝜇 −. The tangency points between tangents and the ingoing and outgoing clothoids are, respectively, 𝐓−=𝐕 𝑗−𝑑− 𝑉𝑗𝑉𝑗−1 (𝐕𝑗−𝐕 𝑗−1) 𝐓+=𝐕 𝑗+𝑑+ 𝑉𝑗+1𝑉𝑗(𝐕𝑗+1 −𝐕 𝑗)(3) where 𝑑−=Δ𝑥 −+Δ𝑦 −tan(𝜇−)+𝐻 −𝐻+sin(𝛼+) sin(𝛽) 𝑑+=Δ𝑥 ++Δ𝑦 +tan(𝜇+)+𝐻 −𝐻+sin(𝛼−) sin(𝛽) (4) In (4) the values for distances and angles are given by the following: ∙Expressions for the distances: Δ𝑥±=∫2𝑅𝜇± 0 cos (𝑠2 4𝑅2𝜇±)𝑑𝑠, Δ𝑦±=∫2𝑅𝜇± 0 sin (𝑠2 4𝑅2𝜇±)𝑑𝑠 and 𝐻−𝐻+=√(𝑂𝐻−)2+(𝑂𝐻 +)2−2𝑂𝐻 −𝑂𝐻+cos(𝜔) with 𝑂𝐻±=𝑅+ Δ𝑦± cos(𝜇±) ∙Expressions for the angles: 𝛽=𝜋−𝜃, 𝛼 ±=𝜋 2+𝜇 ±−𝛾 ± where 𝛾+and 𝛾−are computed as 𝛾+=⎧ ⎪ ⎨ ⎪ ⎩ arcsin (𝑂𝐻− 𝐻−𝐻+sin(𝜔))if 𝑂𝐻− cos(𝜔) ≤𝑂𝐻+ 𝜋−arcsin(𝑂𝐻− 𝐻−𝐻+sin(𝜔))otherwise 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6VÁZQUEZ-MÉNDEZ et al. and 𝛾−=⎧ ⎪ ⎨ ⎪ ⎩ arcsin (𝑂𝐻+ 𝐻−𝐻+sin(𝜔))if 𝑂𝐻+ cos(𝜔) ≤𝑂𝐻− 𝜋−arcsin(𝑂𝐻+ 𝐻−𝐻+sin(𝜔))otherwise Remark 1. For sake of simplicity and to avoid duplication and in case there is no confusion, the symbol ±is used when there are two equal equations one with +and the other with −. The tangency points between the clothoids and the circular curves are: 𝐅±=𝐇 ±+𝐹 ±𝐻±(cos (𝜃±+𝜆 ±(𝛼±+𝛾 ±)), sin (𝜃±+𝜆 ±(𝛼±+𝛾 ±))) (5) where 𝐹±𝐻±=Δ𝑦± cos(𝜇±) The values 𝜃−,𝜃+∈ [0, 2𝜋) are the angles with respect to the positive part of the abscissa axis of the ingoing main tangent (going through 𝐕𝑗−1 and with direction 𝐕𝑗− 𝐕𝑗−1) and of the outgoing main tangent (going through 𝐕𝑗+1 and with direction 𝐕𝑗+1 −𝐕 𝑗), respectively. In addition, 𝜆−and 𝜆+are the signs (−1 or 1) of the ingoing and outgoing clothoids, respectively. These values are −1 if the clothoid goes clockwise, 1 otherwise. Moreover, 𝐇−=𝐕 𝑗−𝐻−𝑉𝑗 𝑉𝑗𝑉𝑗−1 (𝐕𝑗−𝐕 𝑗−1), 𝐇+=𝐕 𝑗+𝐻+𝑉𝑗 𝑉𝑗+1𝑉𝑗(𝐕𝑗+1 −𝐕 𝑗) The last key point is the center of the circular curve, which is given by: 𝐎=𝐇 −+𝑂𝐻 −(cos (𝜃−+𝜆 −(𝛼−+𝛾 −)), sin (𝜃−+𝜆 −(𝛼−+𝛾 −))) (6) From the values of the tangency points, the following lengths can be computed: ∙Tangent lengths 𝐿𝑇𝑖=𝑇 + 𝑖−1𝑇− 𝑖, 𝑖=1,…,𝑁+1 (7) where 𝑇𝑖names the 𝑖-th tangent, with 𝐓+ 0=𝐏 𝑎and 𝐓− 𝑁+1 =𝐏 𝑏. ∙Circular curve lengths 𝐿𝐶𝐶𝑗=𝑅 𝑗𝜔𝑗, 𝑗=1,…,𝑁 (8) where 𝐶𝐶𝑗names the 𝑗-th circular curve. ∙Clothoid lengths 𝐿𝐶± 𝑗=2𝑅 𝑗𝜇± 𝑗, 𝑗=1,…,𝑁 (9) where 𝐶+ 𝑗and 𝐶− 𝑗name the 𝑗-th outgoing and ingoing transition curves, respectively. ∙Length of the track 𝐿(𝐱𝑁)= 𝑁+1 ∑ 𝑖=1 𝐿𝑇𝑖+ 𝑁 ∑ 𝑗=1 (𝐿𝐶− 𝑗+𝐿 𝐶𝐶𝑗+𝐿 𝐶+ 𝑗) Finally, the arc length parameterization of the HA given by the decision vector 𝐱𝑁, 𝝈𝐱𝑁∶[0,𝐿(𝐱 𝑁)] ⟼ ℝ2 𝑠⟼𝝈 𝐱𝑁(𝑠)=(𝜎 1(𝑠), 𝜎2(𝑠)) (10) is computed as detailed in Algorithm 1. Remark 2. From now on, subindexes 𝑖go from 𝑖= 1, … , 𝑁 + 1 and subindexes 𝑗go from 𝑗=1,…,𝑁.The first one makes reference to the number of tangents and the latter to the number of curves. In addition, some elements are multiple named such as 𝐏𝑎=𝐕 0=𝐓 + 0in order to bring together some equations/constraints into a single expression as in (2), (7), or (13). 2.2 Constraints All railways/roads designed by civil engineers must comply several constraints to have a proper parameterization of (10). 2.2.1 Geometrical constraints In addition to the constraints previously stated, 𝑅𝑗>0, 𝜔 𝑗≥0and 0≤𝜇− 𝑗<𝜋 2(11) to obtain a layout with well-coordinated adjacent curves and avoiding overlaps (see Sushma & Maji, 2020), the decision vector has to fulfill the following supplementary geometrical constraints: (1) The angle of the outgoing clothoid is also required to be positive (0 ≤𝜇+ 𝑗<𝜋 2). This constraint is equivalent as 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License VÁZQUEZ-MÉNDEZ et al. 7 ALGORITHM 1 for computing the arc length parameterization of a nonsymmetric horizontal alignment (HA). Input:𝐏𝑎,𝐏 𝑏∈ℝ 2, decision vector 𝐱𝑁,𝑠∈[0,𝐿(𝐱 𝑁)]. Output:𝝈𝐱𝑁(𝑠). 1: Compute (2), (3), (5)–(9), 𝜃± 𝑗and 𝜆± 𝑗from 𝐱𝑁 2: if 𝑠∈[0,𝐿 𝑇1]then 3: Compute 𝝈𝐱𝑁(𝑠) = 𝐏𝑎+𝑠𝐕1−𝐕0 𝑉1𝑉0 4: else 5: Compute 𝐿1=𝐿 𝑇1 6: for 𝑗=1…𝑁do 7: if 𝑠∈(𝐿 𝑗,𝐿 𝑗+𝐿 𝐶− 𝑗]then define  𝑠=𝑠−𝐿 𝑗, 8: Compute 𝝈𝐱𝑁(𝑠) = 𝐓− 𝑗+(∫ 𝑠 0cos (𝜆− 𝑗𝜏2 2𝑅𝑗𝐿𝐶− 𝑗+𝜃 − 𝑗)𝑑𝜏,∫ 𝑠 0sin (𝜆− 𝑗𝜏2 2𝑅𝑗𝐿𝐶− 𝑗+𝜃 − 𝑗)𝑑𝜏) 9: break 10: else if 𝑠∈(𝐿 𝑗+𝐿 𝐶− 𝑗,𝐿 𝑗+𝐿 𝐶− 𝑗+𝐿 𝐶𝐶𝑗]then define 𝜙𝑗∈ [0, 2𝜋) the angle between 𝐅− 𝑗−𝐎 𝑗and 𝑂𝑋+(see Figure 2b), 11: Compute 𝝈𝐱𝑁(𝑠) = 𝐎𝑗+𝑅 𝑗(cos (𝜙𝑗+𝜆− 𝑗(𝑠−(𝐿𝑗+𝐿𝐶− 𝑗)) 𝑅𝑗),sin(𝜙𝑗+𝜆− 𝑗(𝑠−(𝐿𝑗+𝐿𝐶− 𝑗)) 𝑅𝑗)) 12: break 13: else if 𝑠∈(𝐿 𝑗+𝐿 𝐶− 𝑗+𝐿 𝐶𝐶𝑗,𝐿 𝑗+𝐿 𝐶− 𝑗+𝐿 𝐶𝐶𝑗+𝐿 𝐶+ 𝑗]then define  𝑠=𝐿 𝑗+𝐿 𝐶− 𝑗+𝐿 𝐶𝐶𝑗+𝐿 𝐶+ 𝑗−𝑠, 14: Compute 𝝈𝐱𝑁(𝑠) = 𝐓+ 𝑗+(∫ 𝑠 0cos (𝜆+ 𝑗𝜏2 2𝑅𝑗𝐿𝐶+ 𝑗+𝜃 + 𝑗)𝑑𝜏,∫ 𝑠 0sin (𝜆+ 𝑗𝜏2 2𝑅𝑗𝐿𝐶+ 𝑗+𝜃 + 𝑗)𝑑𝜏) 15: break 16: else if 𝑠∈(𝐿 𝑗+𝐿 𝐶− 𝑗+𝐿 𝐶𝐶𝑗+𝐿 𝐶+ 𝑗,𝐿 𝑗+𝐿 𝐶− 𝑗+𝐿 𝐶𝐶𝑗+𝐿 𝐶+ 𝑗+𝐿 𝑇𝑗+1 ]then 17: Compute 𝝈𝐱𝑁(𝑠) = 𝐓+ 𝑗+(𝑠−(𝐿𝑗+𝐿 𝐶− 𝑗+𝐿 𝐶𝐶𝑗+𝐿 𝐶+ 𝑗))𝐕𝑗+1−𝐕𝑗 𝑉𝑗+1𝑉𝑗 18: break 19: end if 20: Compute 𝐿𝑗+1 =𝐿 𝑗+𝐿 𝐶− 𝑗+𝐿 𝐶𝐶𝑗+𝐿 𝐶+ 𝑗+𝐿 𝑇𝑗+1 21: end for 22: end if demanding that the angle of the ingoing clothoid and the circular curve (𝜇− 𝑗+𝜔 𝑗) needs to be smaller that the deflection angle at each curve, that is, 𝜃𝑗−𝜋 2<𝜇 − 𝑗+𝜔 𝑗≤𝜃𝑗(12) (2) A curve cannot begin before the preceding one ends. This is mathematically stated as: ⟨𝐓− 𝑖−𝐓 + 𝑖−1,𝐕 𝑖−𝐕 𝑖−1⟩≥0(13) A layout fulfilling constraints (11)–(13)iscalledanHA. 2.2.2 Technical constraints The HA also has to fulfill other constraints given by the regulations of each country. These constraints are technical values to assure the safety as well as the comfort of the users. Usually, they set minimum values for the radii of circular curves 𝑅𝑗, and the lengths of tangents 𝐿𝑇𝑖, clothoids 𝐿𝐶− 𝑗,𝐿𝐶+ 𝑗, and circular curves 𝐿𝐶𝐶𝑗. 𝑅𝑗≥𝑅𝑚𝑖𝑛,𝐿 𝐶± 𝑗(𝐱𝑁)≥𝐿𝐶𝑚𝑖𝑛 𝐿𝑇𝑖(𝐱𝑁)≥𝐿𝑇𝑚𝑖𝑛 ,𝐿 𝐶𝐶𝑗(𝐱𝑁)≥𝐿𝐶𝐶𝑚𝑖𝑛 (14) All railways lines are designed by civil engineers as HA complying with the technical restrictions set by the authorities. Those alignments are called admissible horizontal alignments (AHAs). 2.3 Objective function: Optimizing the orthogonal distance Railways, due to their use, over time, suffer deformations that have to be amended. To see the size of those fluctuations, the center of the railway is measured with a set of points by track surveying (usually given by Universal Transverse Mercator coordinates). These track deviations must be amended periodically and this technique is called recreation. The original HA is not rebuilt, a new recreated AHA, called RHA, is obtained as close as possible to those measured points from the deformed railway. Let 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 8VÁZQUEZ-MÉNDEZ et al. FIGURE 3 Bind areas of an alignment section. Bind condition detail of a point 𝐏with an ingoing clothoid. {𝐏1,…,𝐏 𝑁𝑆𝑃 } be the set of the sorted surveyed points, 𝐏𝑙=(𝑥 𝑙,𝑦 𝑙),measured along the center line of the railway with 𝐏1=𝐏 𝑎= (𝑥𝑎,𝑦 𝑎)and 𝐏𝑁𝑆𝑃 =𝐏 𝑏=(𝑥 𝑏,𝑦 𝑏), that is, the first and last measured points correspond to the starting and end points, respectively. Those extreme values must be elected within tangents placed at suitable locations, for example, over bridges, in stations, or halts. Apoint𝐏is bind to a section 𝑆(tangent, clothoid, or circular curve) if there is an orthogonal tangent to 𝑆passing through 𝐏.If𝝈∶[𝑠 𝑎,𝑠 𝑏]⟼ℝ 2is a parameterization of 𝑆, then 𝐏is bind to 𝑆ifandonlyif(seeFigure3) ⟨𝝈′(𝑠𝑎), 𝐏 − 𝝈(𝑠𝑎)⟩⋅⟨𝝈′(𝑠𝑏), 𝐏 − 𝝈(𝑠𝑏)⟩≤0 Obviously, the orthogonal distance from 𝐏to 𝑆canbeonly computed if 𝐏is bind to 𝑆. In this optimization problem, the aim is to compute the orthogonal distance from the surveyed points 𝐏𝑙to an HA given by a decision vector 𝐱𝑁.Variables𝑠𝑇± 𝑖and 𝑠𝐹± 𝑖denote the parametric values, which give the tangency points 𝐓± 𝑖and 𝐅± 𝑖(i.e., 𝝈𝐱𝑁(𝑠𝑇± 𝑖)=𝐓 ± 𝑖,𝝈𝐱𝑁(𝑠𝐹± 𝑖)=𝐅 ± 𝑖). The bind process of every point 𝐏𝑙=(𝑥 𝑙,𝑦 𝑙)to every section of the HA and the computation of their corresponding orthogonal distance are detailed below: (1) 𝑖-th tangent 𝑇𝑖={𝝈 𝐱𝑁(𝑠) ∕ 𝑠 ∈ [𝑠𝑇+ 𝑖−1,𝑠 𝑇− 𝑖]}: 𝐏𝑙is bind to 𝑇𝑖if ⟨𝐓− 𝑖−𝐓 + 𝑖−1,𝐏 𝑙−𝐓 + 𝑖−1⟩⋅⟨𝐓− 𝑖−𝐓 + 𝑖−1,𝐏 𝑙−𝐓 − 𝑖⟩≤0 (15) The tangent equation 𝑎𝑖𝑥+𝑏 𝑖𝑦+𝑐 𝑖=0can be easily computed by its end points 𝐓+ 𝑖−1 and 𝐓− 𝑖and the orthogonal distance is given by 𝑑𝑇𝑖=⎧ ⎪ ⎨ ⎪ ⎩||||| 𝑎𝑖𝑥+𝑏𝑖𝑦+𝑐𝑖 √𝑎2 𝑖+𝑏2 𝑖|||||if (15) holds +∞ otherwise (2) 𝑗-th clothoid, ingoing 𝐶− 𝑗={𝝈 𝐱𝑁(𝑠) ∕ 𝑠 ∈ [𝑠𝑇− 𝑗,𝑠 𝐹− 𝑗]} and outgoing 𝐶+ 𝑗={𝝈 𝐱𝑁(𝑠) ∕ 𝑠 ∈ [𝑠𝐹+ 𝑗,𝑠 𝑇+ 𝑗]}, respectively: 𝐏𝑙is bind to 𝐶± 𝑗if ⟨(cos(𝜃± 𝑗), sin(𝜃± 𝑗)),𝐏 𝑙−𝐓 ± 𝑗⟩⋅ ⟨(cos(𝜃± 𝑗+𝜆± 𝑗𝜇± 𝑗), sin(𝜃± 𝑗+𝜆± 𝑗𝜇± 𝑗)),𝐏 𝑙−𝐅 ± 𝑗⟩≤0 (16) and the orthogonal distance is given by 𝑑𝐶± 𝑗={‖𝐏𝑙−𝐐 ±‖if (16) holds +∞ otherwise where 𝐐±is the point of 𝐶± 𝑗with minimum distance to 𝐏𝑙, which is given by 𝐐±=𝝈 𝐶± 𝑗( 𝑠),𝝈𝐶± 𝑗being the local parameterization of 𝐶± 𝑗,seelines8and14ofAlgorithm 1for details, and  𝑠the solution of the nonlinear equation ⟨𝝈′ 𝐶± 𝑗 (𝑠), 𝝈𝐶± 𝑗(𝑠) − 𝐏⟩=0, 𝑠∈[0, 𝐿𝐶± 𝑗] which can be solved, for example, by the Newton– Raphson method. (3) 𝑗-th circular curve 𝐶𝐶𝑗={𝝈 𝐱𝑁(𝑠) ∕ 𝑠 ∈ [𝑠𝐹− 𝑗,𝑠 𝐹+ 𝑗]}: 𝐏𝑙is bind to 𝐶𝐶𝑗if ⟨(cos(𝜃− 𝑗+𝜆− 𝑗𝜇− 𝑗), sin(𝜃− 𝑗+𝜆− 𝑗𝜇− 𝑗)),𝐏 𝑙−𝐅 − 𝑗⟩⋅ ⟨(cos(𝜃+ 𝑗+𝜆+ 𝑗𝜇+ 𝑗), sin(𝜃+ 𝑗+𝜆+ 𝑗𝜇+ 𝑗)),𝐏 𝑙−𝐅 + 𝑗⟩≤0 (17) and the orthogonal distance is given by 𝑑𝐶𝐶𝑗={|||‖𝐏𝑙−𝐎 𝑗‖−𝑅𝑗|||if (17) holds +∞ otherwise where the center of the circumference 𝐎𝑗is given by (6). After this bind process, every point 𝐏𝑙has assigned a distance to all sections of the HA (finite value if it is bind, infinite if not). Then, the distance between 𝐏𝑙and the HA is defined as the minimum of these distance, 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License VÁZQUEZ-MÉNDEZ et al. 9 𝑑𝑙=min{𝑑𝑇1 𝑙,𝑑𝐶− 1 𝑙,𝑑𝐶𝐶1 𝑙,…,𝑑𝐶𝐶𝑁 𝑙,𝑑𝐶+ 𝑁 𝑙,𝑑𝑇𝑁+1 𝑙} and the problem of optimizing orthogonal distances can be defined from the following objective function: 𝐽𝑜D(𝐱𝑁)=(𝑁𝑆𝑃 ∑ 𝑙=1 𝑑𝑙)∕𝑁𝑆𝑃 The optimization problem will consist of minimizing function 𝐽𝑜D(𝐱𝑁)considering the constrains over the decision vector 𝐱𝑁related to the definition of the geometrical model and the technical constraints. Therefore, this optimization problem is formulated as follows: min 𝐱𝑁∈ℝ 5𝑁 subject to (11)–(14) 𝐽𝑜D(𝐱𝑁)(18) 3PROPOSED METHOD FOR OBTAINING AN OPTIMAL RHA The aim of this section is to provide a method to compute the solution of the problem stated in (18). The problem is differentiable and it can be solved by any gradient-type optimization method. These groups of methods have a fast converge when the exact gradients of the objective function 𝐽𝑜D and the nonlinear functions, which define the constraints (11)–(14), are known. When exact gradients are not available, as in this case, numerical approximations must be used, which can lead to “false convergences.” To avoid this problem, a common technique is to perform multiple starts of the gradient method and keep the best of the solutions obtained. In this work, we use a multistart of an SQP method (Nocedal & Wright, 2006) that has already been successfully used by the authors in other similar problems (see Vázquez-Méndez et al., 2021b,2023). The SQP method is an iterative process, which needs to be provided with a starting point of the decision vector, which is usually known as seed. This seed (denoted ˜𝐱𝑁) must fulfill the following requirements: (1) It has to be an HA and (2) it should be close to the whole set of the surveyed points. This second requirement is due to the fact that, in order to have a good behavior of problem (18), consecutive surveyed points cannot be bind to completely different alignment sections. This requisite leads us to create a preliminary step, with a simple and cheap algorithm, allowing the provision of a good initial HA seed, relatively close to the surveyed data, to be used as starting point into the orthogonal distance optimization problem. 3.1 First preliminary step alternative The preliminary step consists of obtaining an HA minimizing the least square 𝑦-distance to a cubic spline created with the 𝑁𝑆𝑃 surveyed points obtained in the deformed track center-line. Specifically, the following objective function 𝐽𝑦D is proposed: 𝐽𝑦D(𝐱𝑁)=∫𝐿(𝐱𝑁) 0(𝜎2(𝑠)−𝑓 𝑝𝑐ℎ(𝜎1(𝑠)))2𝜎′ 1(𝑠) 𝑑𝑠 where 𝑓𝑝𝑐ℎ is a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP). In order to have this 𝑓𝑝𝑐ℎ well defined, the following hypothesis must be hold: HP1: Backtracking is not allowed in the surveyed points, that is, ⟨𝐏𝑏−𝐏 𝑎,𝐏 𝑙−𝐏 𝑙−1⟩>0, 𝑙=2,…,𝑁 𝑆𝑃 HP2: The initial and end points 𝐏𝑎and 𝐏𝑏, respectively, have the same 𝑦-coordinate: 𝑦𝑎=𝑦 𝑏 This second hypothesis can be easily enforced like in north-to-south alignments, for example, where a rotation can be done to the tangent joining the ending points into the same abscissa axis as a previous step. Obviously the same procedure must also be done to all the surveyed points. Remark 3. With hypothesis HP1 and HP2 holding true, it is assured that 𝑥𝑙−1 <𝑥 𝑙, 𝑙=2,…,𝑁 𝑆𝑃. It seems appropriate to ask for the equivalent constraint to the recreated alignment in this preliminary step, therefore an additional constraint is imposed, which is: 𝑥𝑎≤𝑋𝑙≤𝑋𝑙+1 ≤𝑥𝑏, 𝑙=1,…,𝑁−1 (19) This 𝑓𝑝𝑐ℎ provides for any coordinate 𝑥∈[𝑥 𝑎,𝑥 𝑏]its corresponding coordinate 𝑦, allowing to easily compute the 𝑦-value on every point of the recreated alignment, not only on the surveyed data. The optimization problem will consist of obtaining ˆ𝐱𝑁minimizing the objective function 𝐽𝑦D (𝐱𝑁)considering the geometrical constraints over the decision vector. Therefore, the preliminary optimization problem can be formulated as follows: min 𝐱𝑁∈ℝ 5𝑁 subject to (11)–(13) and (19) 𝐽𝑦D(𝐱𝑁)(20) 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 16 VÁZQUEZ-MÉNDEZ et al. FIGURE 8 The upper graphic shows orthogonal distance to the surveyed points for the automatic (solid line) and manual (dashed) recreated horizontal alignment (RHA) solutions. The lower diagram shows the orthogonal distances to the automatic RHA superimposed with its curvature diagram. they choose, among all the solutions that comply with that orthogonal distance limit, for an RHA the AHA with the lowest mean orthogonal distance, in order to minimize the movement, performed by the tamping/lining machine, to place the railway track into its new position. It should be noted(seeTable4) that the mean value of the orthogonal distances of the recreated HA with the proposed automatic solution is decreased by 32.95% with respect to the manual, that is, the new alignment with this two-stage method considerably improves the manual recreation. In Figure 8, both recreations are compared graphically and it can be observed that both alignments (manual and automatic) comply with the max orthogonal distance falling far below the threshold of 15 cm. From the 1024 initial starting points generated during the second preliminary step most of them, 954 provided a valid HA to be used as starting points to compute (18), resulting in a total of 326 feasible solutions. Remark 6. It is worth noting that, like in the academic examples, the number of seeds is oversized and no stopping criterion was introduced taking 3.0468e+05 s. The same case study was developed with the following stop criterion: mean 𝑜D < 2 cm and maximum 𝑜D < 10 cm, and the cpu-time was reduced up to 7430.86 s providing a solution with mean 𝑜D of 1.920594 cm and a maximum 𝑜D of 9.104353 cm 5CONCLUSIONS AND FUTURE WORK The proposed model offers an automatic method in the RHA for rectification purposes by solving a nonlinear but differentiable optimization problem. There are many commercial software for the design of railway and road alignments, but they lack an efficient optimization module. Hence, the engineer has to tackle the RHA task with the help of that ineffective software and dedicate many hours to find a solution adjusting parameters usually by trial-and-error method. The methodology we present provides a good tool to find those RHAs automatically. More specifically, a new mathematical approach of nonsymmetric AHA is stated and the expressions for its tangency points are also given. Based on this new formulation, this new AHA model is used to recreate a railway layout from a set of points fulfilling all technical constraints. This recreation is carried out by solving an optimization problem minimizing the orthogonal distance to the surveyed points with a well-known gradient-type iterative method (SQP). To have good starting points in that iterative process, a preliminary stage has to be made minimizing previously the 𝑦-distance to the surveyed points. It is worth to point out that there is no need for prior identification of the existing alignment by means of its geometrical elements (tangents, circular curves, and clothoids) of the existing alignment. The algorithm automatically binds all points to its closest element. The only additional information required is the number of curves (𝑁) for the section that needs to be recreated. This number can be determined through an exhaustive search varying 𝑁in the preliminary stage, minimizing the 𝑦-distance with (20) following the methodology proposed in VázquezMéndez et al. (2021a) or with an alternative approach using automatic procedures (as discussed in Song et al., 2021). The two proposed algorithms were tested against two academic examples showing the good behavior of the 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License VÁZQUEZ-MÉNDEZ et al. 17 model and in a case study successfully applied to a railway line section in Galicia, proving to be a very useful tool for railway recreations. From what has been seen in this work, it will be very useful to upgrade the methodology in the following aspects: (1) Considering other types of combinations of the geometric elements to take into account egg and doubleegg curves. (2) Analyze the sensitivity of the method to the numerical gradient computed in the optimization method. It has been noticed during the development of the numerical experiments that working with this high degree of precision (cm) in a section of several kilometers, the computation of the optimal solution might be affected by the accuracy with which the gradient is computed, compromising sometimes the numerical convergence of the SQP method. (3) Try to compute the exact gradient instead of using a numerical approximation. (4) This paper is focused only on RHA for rectification purposes. However, railways are 3D alignments. The recreation of vertical alignments (RVA) or even RHA and RVA simultaneously deserves in-depth investigation. (5) Explore the possibility of using other optimization methods, for example, the nature-inspired optimization algorithms such as neural dynamic model (Park & Adeli, 1997) or the HSA (Siddique & Adeli, 2015) among many others. ACKNOWLEDGMENTS Wewouldliketothankhttps://www.ineco.com/ (Transport Engg. and Economics) of its disposition to share the data in the case study. This work was funded from the project TED2021-129324B-I00 of the Ministerio de Ciencia e Innovación (Spain) and NextGenerationEU (European Union), and also by the Collaboration Agreement between Consellería de Educación, Formación Profesional e Universidades (Xunta de Galicia, Spain) and Universidade de Santiago de Compostela (Spain), which regulates the Specialization Campus Campus Terra under Grant Number 2022-PU014. 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Computer-Aided Civil and Infrastructure Engineering,1–18. https://doi.org/10.1111/mice.13230 14678667, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/mice.13230 by Consorcio Interuniversitario Do Sistema Universitario De Galicia (Cisug), Wiley Online Library on [09/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License