# Research on 3D Modeling Method of Cobblestone Road Based on Geometric Characteristics

Authors: Jinhua Han; Ke Feng; Huanliang Li; Hongxin Cui; Jing Chen
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Based on the analysis of the geometric characteristics, distribution characteristics, and constraints of the cobblestone road, a three-dimensional (3D) parametric road modeling method based on geometric characteristics is proposed. The cobblestone model is established by the exponential oval equation, and the cobblestones distribution is determined by Monte Carlo stochastic search method. Then the 3D theoretical model of cobblestone road is obtained. To veritably simulate the cobblestone road, a generated 3D random road is fusing with the 3D cobblestone road model by alternating values, and the comparison between the real and modeling cobblestone roads is done in space domain and PSD curves. Then based on the establishment of the standard vehicle vibration model, the influences of the key geometric parameters of the cobblestone road on the vehicle vibration are analyzed by using the evaluation indices of the IRI and the RMS of the vehicle body acceleration. The proposed method can be extended to the 3D modeling of almost all strengthened test roads such as Belgian road, fish-scale pits road, ripple tracks, and washboard road and provides a 3D road modeling method with adjustable parameters, authenticity, and accuracy for the comprehensive construction of vehicle virtual test field.


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## Body

## 1. Introduction

With the rapid development and update of the vehicle, compared with the real vehicle driving test [1], the vehicle road simulation test [2] and virtual road test [3] have attracted more and more attention in automotive engineering field in recent years with the advantages of time-saving, low costs, high accuracy, repeatability, controllability, and so forth.Road profile [4] is the most important external excitation of the ground vehicle, which influences vehicle ride comfort, operational stability, driving reliability, fatigue life of components, and so forth. With the excitation of road roughness on the vehicle, the vehicle is affected by the external forces and the internal forces of the vehicle itself, which are changing over time. When the vehicle is running through the road, the vehicle’s components are always bearing the stress and strain, which would result in fatigue and failure after a certain mileage [5]. The worse the road condition is, the faster the fatigue failure of vehicle parts is. Therefore, providing accurate road profile for vehicle reliability testing and simulation has been an important research content in this field.Currently, the methods of road profile acquisition [6] at home and abroad mainly include direct measurement method and simulation method [7]. Among them, the direct measurement methods of road profile had formed many measuring methods and corresponding equipment in long-term research [8, 9], including Level and Rod, Multiwheel Profilograph, BPR Roughometer, Inertial Profiler, and Laser Profiler. The Level and Rod is the earliest method of road profile measurement and has been used in the construction and acceptance process of airport runway in Europe and America and various automobile proving grounds in China [10]. The Level and Rod method has high accuracy, but the measuring speed is too slow, which is mainly used for calibration of other measuring instruments now. In order to improve the efficiency of measurement, the Multiwheel Profilograph, BPR Roughometer, and Inertial Profiler are applied. But these measuring instruments are difficult to take account of the speed and accuracy simultaneously. Then the application of laser sensor makes the laser profiler achieve high speed and accurate measurement of the road and becomes a widely used measurement method. Although the direct measurement methods can obtain a more accurate road profile, only a single road trajectory is obtained, which cannot be used for virtual test based on 3D model.The simulation methods can reconstruct the required road profiles according to the statistical characteristics of road roughness, which contain harmonic superposition method [11], filtered white noise method, AR (ARMA) method, fast Fourier inversion generation method, and so forth [12, 13]. However, these road profile reconstruction methods [14] are based on the theory of stationary random process, which can only produce stochastic histories of stationary Gauss distribution. The above methods are difficult to reconstruct the road profile of the strengthened roads in the vehicle proving ground accurately, which have evident non-Gaussian characteristics [15, 16].In terms of modeling of road geometry, 3D models can more accurately reflect the effect of roads on vehicles, especially when vehicles are subjected to lateral loading; thus more accurate test results are obtained in the virtual test based on 3D model [17, 18]. The application of laser scanning technology makes it possible to measure the 3D road profile directly and accurately. General Motors used an industrial 3D laser scanning instrument to build the high-precision models for the test ground roads in 2011, but the instrument is huge, expensive, and inefficient. And the portable 3D laser scanner for road 3D scanning requires multistation scanning and point cloud splicing with a low efficiency. Then some studies are done to reconstruct 3D road model. C. Cheng et al. [19] presented a new way to create 3D virtual road by calculating the connections of each code with the Delaunay triangulation algorithm based on the ADAMS software. Based on the measured trajectory of the middle route of the road, the road profile is reconstructed with the interpolation method in [20, 21], and the comparative analysis of the reconstructed roads in Regular Grid Road (RGR), Curved Regular Grid (CRG), and Triangular Mesh Road Definition File (RDF) formats is made in the simulation efficiency. However, the above road modeling methods are completed on the basis of the measured road profile data, combined with the characteristics of the road profile, and they are not comprehensive for describing the geometric characteristics of the road.In all kinds of strengthened test roads, such as long/short ripple tracks, washboard road, Belgian road, cobblestone road, and fish-scale pits road, they all rely on the specific “obstacles” that appear regularly on the roads to achieve the purpose of accelerated test. The geometric characteristics of the road obstacles determine the form and strength of the impact on vehicles. Due to the sensitivity and importance of the geometric characteristics of road obstacles on vehicle, even for the same type of strengthened test road, the roads with different geometric characteristics will have different effects on vehicles. Therefore, for the reconstruction of these roads, the geometric characteristics of the roads can be reconstructed accurately, and they can be better applied to road simulation or virtual test.Therefore, taking the cobblestone road as the research object, the paper mathematically describes its geometric characteristics and establishes the 3D road model of cobblestone road based on the determination of its key geometry or distribution parameters. To achieve the effect of real road simulation, a more real 3D road model is obtained by integrating the theoretical model with the generated 3D random road. Then the comparison between modeling road and real road is done in space domain and frequency domain. Finally, the influence of cobblestone road geometric parameters on vehicle response is analyzed by establishing standardized vehicle model and evaluation indicators based on the existing 3D road model. The proposed method realizes fast 3D modeling of roads with different parameters and provides the 3D road model with adjustable parameters and real accuracy for building vehicle virtual test ground and carrying out virtual driving test


## 2. 3D Modeling of Cobblestone Road

The cobblestone road is a kind of strengthened test road by placing the cobblestones with a certain grain size into the cement concrete sparsely and irregularly. The cobblestone road is the most used reliability/durability test road, which can provide a wide band of strong load for the whole vehicle (especially the steering system).Compared with other strengthened test roads, the 3D modeling of cobblestone road is more complicated because of the random variation in the main parameters such as cobblestone size, cobblestone height, cobblestone distribution, and the shape of cobblestone itself. In this section, based on the exponential 2D oval equation, the 3D model of cobblestone road is gradually established according to the order of Oval curve, cobblestone mode, cobblestone coordinates, and cobblestone road.


### 2.1. Oval Equation

The exponential oval equation evolves from the standard form of elliptic equation, which can be expressed as(1)x2a2+y2b2ekx=1Obviously, whenk=0, (1) is a standard form of elliptic equation, where a and b (a≥b>0) are the length of the semi-axis in the horizontal and vertical directions, respectively. When k≠0, due to the interference of ekx, for the same abscissa x, the ordinate y becomes the “contraction” end when kx>0 and the value of y is less than the value in the case of the ellipse. Conversely, the ordinate y becomes the “expansion” end when kx<0 and the value of y is larger than the value in the case of the ellipse. Figure 1 shows the oval curves when k are 0.1, 0.2, 0.5, and 0.9 (when k is negative, the figure is symmetrical with the image of Figure 1, which is large in right and small in left).Figure 1
Oval curves under different shape parameters.As shown in Figure1, k directly determines the shape of the curve, which is called the shape parameter. When k gradually increases from 0, the curve changes from the ellipse to the oval and then cone. In order to maintain the curve to be used to simulate the cobblestone road, the shape parameter k can be taken to a smaller value.Obviously, (1) and Figure 1 show that the “height” h in the oval equation is not consistent with the longitudinal semiaxis b in the elliptic equation. When considering the above oval as the vertical section of cobblestone, for the cobblestone with a given height h and particle size 2a, in order to model with the exponential oval curve expressed in (1), the determined height h is needed to convert to an unknown equivalent semiaxis length bh when the shape parameter is given. The relationship between them can be derived from (1), which can be expressed as(2)y2=1-x2/a2b2ekxTaking the derivative of  (1) with respect to x, we obtain(3)dy2dx=-2b2x/a2ekx-1-x2/a2b2kekxe2kxWheny takes the maximum value of h, there should be dy2/dx=0. It can be shown from (3) that(4)kx2-2x-ka2=0According to the solution of (4), when y takes the maximum value of h, its transverse coordinate x can be computed as(5)x=1±1+k2a2kBecause-a≤x≤a, we have x=1-1+k2a2/k. Then plug it into (1) with y=h at this time, and (1) is rewritten as(6)1-1+k2a2/k2a2+h2b2e1-1+k2a2=1The solution of (6) can be obtained:(7)bh=ahk12e1-1+k2a21+k2a2-1The relation between the ordinate and the abscissa of any point in the positive half of the longitudinal section can be expressed by plugging (7) into (2).(8)y=1-x2/a2bh2ekxEquations (7) and (8) form the relation between the transverse and ordinate of the oval curve when the particle size 2a and height h are given.


### 2.2. 3D Cobblestone Model

As shown in Figure2, for the 3D cobblestone model (aboveground part), three oval equations with the common grain size 2a are used as the vertical section, the front/back boundary of the horizontal sections of the cobblestone. Each curve has independent “height” hii=1,2,3 (according to the definition, h2 and h3 are the maximum particle sizes of the cobblestone along the positive/negative directions of y-axis.) and shape parameter kii=1,2,3.Figure 2
The coordinate system of 3D cobblestone model.Assume that the XOY plane is the ground. As the boundary of the vertical section, curve 1 determines the actual height and protruding characteristic of the upper part of the cobblestone. And as the boundary of the horizontal section contour, curves 2 and 3 determine the basic shape of the cobblestone.The oval curves 2 and 3 have independent “height” parametersh2 and h3 and the horizontal surface shape parameters k2 and k3. Table 1 lists the basic forms of the cobblestone’s horizontal surface contour under the conditions of several values of the four parameters. In the actual application process, due to the random change of the particle size in a certain range, a very rich horizontal contour of the cobblestone can be generated under the control of all five parameters.Table 1
The basic shape of the pebble’s horizontal surface profile under different parameters.
h 2 < h 3 h 2 > h 3 h 2 ≈ h 3 k 2 > k 3 > 0 k 2 ≈ k 3 > 0 k 2 > 0 > k 3When the boundaries expressed by three oval curves are determined, the cobblestone surface consists of the elliptic curves by parallel sliding along thex-axis and connecting the oval curves 1, 2, and 3. The points of the same value x on the oval curves 1, 2, and 3 are used as the end points of semiminor axis and semimajor axis of the ellipse, respectively. Hence, the vertical coordinates of any point on the cobblestone surface corresponding with any point (x,y) on the horizontal surface XOY can be expressed as(9)zx,y=1-y2y22xz12xy≥01-y2y32xz12xy<0where z1(x), y2(x), and y3(x) denote the points on curves 1, 2, and 3, respectively. They are calculated in the plane of XOZ (replace y with z at this time) and XOY according to (7) and (8).Whena=1, h1=1, h2=h3=0.6, k1=0.2, and k2=k3=0.4, the 3D cobblestone model generated by the above method is shown in Figure 3.Figure 3
3D cobblestone model.


### 2.3. Center Point Coordinates of Cobblestone

After determining the method of generating 3D cobblestone, they should be scattered on the road randomly according to a certain distribution law. Usually, the intervals between cobblestones are required on the cobblestone road, and it is hoped that the cobblestones can be evenly distributed on the road. Besides, the cobblestones cannot overlap and accumulate obviously.To simplify the analysis, the horizontal area of all the cobblestones is considered to be a circle with a diameter that varies from the minimum particle size2amin to the maximum particle size 2amax.When the lengthL and width W of the road rectangular part are designated and the bottom left corner is defined as the coordinate origin, the above constraint conditions can be expressed as(10)dij≤ri+rj+Dminri≤xcenter-i≤L-riri≤ycenter-i≤W-riwhere (xcenter-i,ycenter-i) is the center coordinate of cobblestone i. Both ri and rj represent the circle radius of the cobblestones i and j.d i j = x c e n t e r - i - x c e n t e r - j 2 + y c e n t e r - i - y c e n t e r - j 2 is the distance between the cobblestones i and j.D m i n is the minimum interval between the cobblestones, which can theoretically be obtained in a large range of 0 to L-4amin. But the value is generally between 0 and 1 m according to the actual test requirements.However, even if it is simplified as circular, essentially, filling a number of circles with variable radius and without overlapping in a rectangle with a fixed boundary is still a classic Packing problem [22]. From the difficulty of solving this problem, it belongs to the typical Nondeterministic Polynomial (NP) problem [23]. When the number of fillings is large, there is still no ideal method to obtain the deterministic optimal solution at present. However, the cobblestone road itself has no optimal plan for the number of cobblestones (such as filling the specified number of cobblestones in the designated road area), and the optimal solution of the problem is usually arranged in order, which is not consistent with the requirement of cobblestone road. Therefore, the Monte Carlo stochastic search method [24] is used to get the general solution of the problem, which satisfies the situation with as many cobblestone center points as possible of (10). The basic steps of the search method are shown in Figure 4.Figure 4
Flowchart of Monte Carlo searching for cobblestone center point.As shown in Figure4, Nmax is a maximum allowable value for an attempt to search set in advance. Due to the NP problem, in order to ensure the ergodicity of search, the value must increase sharply with the increase of road area, which is not conducive to rapid modeling. Therefore, when a long road modeling is needed, a complete road model can be established by means of piecewise modeling and then stitching.When the maximum particle size of 300 mm and the minimum particle size of 150 mm are determined, Figures5(a) and 5(b) show the results of the cobblestones distribution obtained by Monte Carlo search with the minimum interval of 0 mm and 300 mm in the road range of 10 m × 2 m, respectively.Figure 5
Monte Carlo search results of the cobblestones distribution with different minimum intervals: (a)Dmin=0mm; (b) Dmin=300mm.
(a) (b)


### 2.4. Plane Rotation and Coordinate Transformation of Cobblestone

From (1) and Figure 1, it is not difficult to find that the maximum size of the cobblestone defined by (7) and (8) is always on x-axis of the coordinate system. If the direction of cobblestone coordinate system is consistent with the direction of the road, all the cobblestones always refer to the length direction of the road. Obviously, to simulate the real cobblestone road, every cobblestone should be assigned a rotating angle around z-axis within the range of [−90°, 90°] before the actual inlay into the road.The center point coordinate of the cobblestone(xcenter-i,ycenter-i) shown in Figure 5 has been obtained in the upper section, and the cobblestone coordinate system shown in Figure 2 and the road coordinate system produce the angle θi with the rotation operation. Then when the coordinate of the road (x,y) is within the circular area of cobblestone i (x-xcenter-i2+y-ycenter-i2≤ri), the coordinate transformation of the original coordinate of the cobblestone (xcenter-i′,ycenter-i′) can be expressed as(11)xi′yi′=cosθi-sinθisinθicosθix-xcenter-iy-ycenter-iAt this time, because(x′,y′,z′) satisfies the equation of 3D cobblestone surface defined by (7), (8), and (9), the height of the cobblestone in the road coordinate system is calculated: z(x,y)=z′(x′,y′).The change of the coordinate system (grid) after the cobblestone rotation is shown in Figure6.Figure 6
The coordinate system of cobblestones before and after rotation.


### 2.5. 3D Cobblestone Road

For any point on the cobblestone road, its height can be expressed as(12)zx,y=zixi′,yi′x,y⊆Si0x,y⊄Siwhere Si represents the horizontal area surrounded by curves 2 and 3.As there are many road parameters of the cobblestone road, the road with different emphasis can be produced by the combination of different parameters. Here 3D models of three types of cobblestone roads are built by copying them from a domestic vehicle test ground. The shape and distribution parameters of the three kinds of cobblestone roads in the test ground are shown in Table2.Table 2
The parameters of the cobblestone roads in a vehicle test field.
Particle size(mm) Spacing(mm) Height(mm) Density(a/m2) Cobblestone road A 250~350 ≥particle size 80~120 3.5 Cobblestone road B 150~300 200~400 <120 / Cobblestone road C 70~80 100~200 20~35 30~35As shown in Table2, for the distribution of cobblestone, the distribution density of cobblestone is required for cobblestone roads A and C except for spacing requirement. In fact, both of them and particle size of the cobblestone are mutually restrictive. As the spacing adjustment is more flexible, the minimum spacing should be adjusted to meet the density requirement first when the distribution density is required.Besides, among the parameters mentioned in Table2, the systematicness explanation of the cobblestone (shape parameters k1,2,3) is not given. Based on the requirements of particle size and height, the shape parameters are determined as follows.Road A has the largest diameter, height, and moderate difference in height, and the road is mainly used for the assessment of heavy vehicles. So the shape parameter of the required cobblestone is large, resulting in maximum difference in shape to give the highest strength assessment of vehicle.Road B has maximum difference in height and larger particle size, so larger vertical shape parametersk1 and moderate horizontal surface shape parameters k2,3 are used to provide abundant lateral change incentives for vehicles with great height difference.The cobblestone of road C is nearly spherical or hemispherical with small particle size and high density, so the shape parameter of cobblestone has almost no impact on vehicle excitation. The road provides rich high frequency excitation for medium or light vehicles at high speed, so smaller shape parameter is selected.The 3D cobblestone road models with the length of 10 m and the width of 4 m according to the above parameters by the proposed method are shown in Figure7.Figure 7
Three kinds of 3D cobblestone road models in a test field. (a) Cobblestone road A. (b) Cobblestone road B. (c) Cobblestone road C.
(a) (b) (c)


## 2.1. Oval Equation

The exponential oval equation evolves from the standard form of elliptic equation, which can be expressed as(1)x2a2+y2b2ekx=1Obviously, whenk=0, (1) is a standard form of elliptic equation, where a and b (a≥b>0) are the length of the semi-axis in the horizontal and vertical directions, respectively. When k≠0, due to the interference of ekx, for the same abscissa x, the ordinate y becomes the “contraction” end when kx>0 and the value of y is less than the value in the case of the ellipse. Conversely, the ordinate y becomes the “expansion” end when kx<0 and the value of y is larger than the value in the case of the ellipse. Figure 1 shows the oval curves when k are 0.1, 0.2, 0.5, and 0.9 (when k is negative, the figure is symmetrical with the image of Figure 1, which is large in right and small in left).Figure 1
Oval curves under different shape parameters.As shown in Figure1, k directly determines the shape of the curve, which is called the shape parameter. When k gradually increases from 0, the curve changes from the ellipse to the oval and then cone. In order to maintain the curve to be used to simulate the cobblestone road, the shape parameter k can be taken to a smaller value.Obviously, (1) and Figure 1 show that the “height” h in the oval equation is not consistent with the longitudinal semiaxis b in the elliptic equation. When considering the above oval as the vertical section of cobblestone, for the cobblestone with a given height h and particle size 2a, in order to model with the exponential oval curve expressed in (1), the determined height h is needed to convert to an unknown equivalent semiaxis length bh when the shape parameter is given. The relationship between them can be derived from (1), which can be expressed as(2)y2=1-x2/a2b2ekxTaking the derivative of  (1) with respect to x, we obtain(3)dy2dx=-2b2x/a2ekx-1-x2/a2b2kekxe2kxWheny takes the maximum value of h, there should be dy2/dx=0. It can be shown from (3) that(4)kx2-2x-ka2=0According to the solution of (4), when y takes the maximum value of h, its transverse coordinate x can be computed as(5)x=1±1+k2a2kBecause-a≤x≤a, we have x=1-1+k2a2/k. Then plug it into (1) with y=h at this time, and (1) is rewritten as(6)1-1+k2a2/k2a2+h2b2e1-1+k2a2=1The solution of (6) can be obtained:(7)bh=ahk12e1-1+k2a21+k2a2-1The relation between the ordinate and the abscissa of any point in the positive half of the longitudinal section can be expressed by plugging (7) into (2).(8)y=1-x2/a2bh2ekxEquations (7) and (8) form the relation between the transverse and ordinate of the oval curve when the particle size 2a and height h are given.


## 2.2. 3D Cobblestone Model

As shown in Figure2, for the 3D cobblestone model (aboveground part), three oval equations with the common grain size 2a are used as the vertical section, the front/back boundary of the horizontal sections of the cobblestone. Each curve has independent “height” hii=1,2,3 (according to the definition, h2 and h3 are the maximum particle sizes of the cobblestone along the positive/negative directions of y-axis.) and shape parameter kii=1,2,3.Figure 2
The coordinate system of 3D cobblestone model.Assume that the XOY plane is the ground. As the boundary of the vertical section, curve 1 determines the actual height and protruding characteristic of the upper part of the cobblestone. And as the boundary of the horizontal section contour, curves 2 and 3 determine the basic shape of the cobblestone.The oval curves 2 and 3 have independent “height” parametersh2 and h3 and the horizontal surface shape parameters k2 and k3. Table 1 lists the basic forms of the cobblestone’s horizontal surface contour under the conditions of several values of the four parameters. In the actual application process, due to the random change of the particle size in a certain range, a very rich horizontal contour of the cobblestone can be generated under the control of all five parameters.Table 1
The basic shape of the pebble’s horizontal surface profile under different parameters.
h 2 < h 3 h 2 > h 3 h 2 ≈ h 3 k 2 > k 3 > 0 k 2 ≈ k 3 > 0 k 2 > 0 > k 3When the boundaries expressed by three oval curves are determined, the cobblestone surface consists of the elliptic curves by parallel sliding along thex-axis and connecting the oval curves 1, 2, and 3. The points of the same value x on the oval curves 1, 2, and 3 are used as the end points of semiminor axis and semimajor axis of the ellipse, respectively. Hence, the vertical coordinates of any point on the cobblestone surface corresponding with any point (x,y) on the horizontal surface XOY can be expressed as(9)zx,y=1-y2y22xz12xy≥01-y2y32xz12xy<0where z1(x), y2(x), and y3(x) denote the points on curves 1, 2, and 3, respectively. They are calculated in the plane of XOZ (replace y with z at this time) and XOY according to (7) and (8).Whena=1, h1=1, h2=h3=0.6, k1=0.2, and k2=k3=0.4, the 3D cobblestone model generated by the above method is shown in Figure 3.Figure 3
3D cobblestone model.


## 2.3. Center Point Coordinates of Cobblestone

After determining the method of generating 3D cobblestone, they should be scattered on the road randomly according to a certain distribution law. Usually, the intervals between cobblestones are required on the cobblestone road, and it is hoped that the cobblestones can be evenly distributed on the road. Besides, the cobblestones cannot overlap and accumulate obviously.To simplify the analysis, the horizontal area of all the cobblestones is considered to be a circle with a diameter that varies from the minimum particle size2amin to the maximum particle size 2amax.When the lengthL and width W of the road rectangular part are designated and the bottom left corner is defined as the coordinate origin, the above constraint conditions can be expressed as(10)dij≤ri+rj+Dminri≤xcenter-i≤L-riri≤ycenter-i≤W-riwhere (xcenter-i,ycenter-i) is the center coordinate of cobblestone i. Both ri and rj represent the circle radius of the cobblestones i and j.d i j = x c e n t e r - i - x c e n t e r - j 2 + y c e n t e r - i - y c e n t e r - j 2 is the distance between the cobblestones i and j.D m i n is the minimum interval between the cobblestones, which can theoretically be obtained in a large range of 0 to L-4amin. But the value is generally between 0 and 1 m according to the actual test requirements.However, even if it is simplified as circular, essentially, filling a number of circles with variable radius and without overlapping in a rectangle with a fixed boundary is still a classic Packing problem [22]. From the difficulty of solving this problem, it belongs to the typical Nondeterministic Polynomial (NP) problem [23]. When the number of fillings is large, there is still no ideal method to obtain the deterministic optimal solution at present. However, the cobblestone road itself has no optimal plan for the number of cobblestones (such as filling the specified number of cobblestones in the designated road area), and the optimal solution of the problem is usually arranged in order, which is not consistent with the requirement of cobblestone road. Therefore, the Monte Carlo stochastic search method [24] is used to get the general solution of the problem, which satisfies the situation with as many cobblestone center points as possible of (10). The basic steps of the search method are shown in Figure 4.Figure 4
Flowchart of Monte Carlo searching for cobblestone center point.As shown in Figure4, Nmax is a maximum allowable value for an attempt to search set in advance. Due to the NP problem, in order to ensure the ergodicity of search, the value must increase sharply with the increase of road area, which is not conducive to rapid modeling. Therefore, when a long road modeling is needed, a complete road model can be established by means of piecewise modeling and then stitching.When the maximum particle size of 300 mm and the minimum particle size of 150 mm are determined, Figures5(a) and 5(b) show the results of the cobblestones distribution obtained by Monte Carlo search with the minimum interval of 0 mm and 300 mm in the road range of 10 m × 2 m, respectively.Figure 5
Monte Carlo search results of the cobblestones distribution with different minimum intervals: (a)Dmin=0mm; (b) Dmin=300mm.
(a) (b)


## 2.4. Plane Rotation and Coordinate Transformation of Cobblestone

From (1) and Figure 1, it is not difficult to find that the maximum size of the cobblestone defined by (7) and (8) is always on x-axis of the coordinate system. If the direction of cobblestone coordinate system is consistent with the direction of the road, all the cobblestones always refer to the length direction of the road. Obviously, to simulate the real cobblestone road, every cobblestone should be assigned a rotating angle around z-axis within the range of [−90°, 90°] before the actual inlay into the road.The center point coordinate of the cobblestone(xcenter-i,ycenter-i) shown in Figure 5 has been obtained in the upper section, and the cobblestone coordinate system shown in Figure 2 and the road coordinate system produce the angle θi with the rotation operation. Then when the coordinate of the road (x,y) is within the circular area of cobblestone i (x-xcenter-i2+y-ycenter-i2≤ri), the coordinate transformation of the original coordinate of the cobblestone (xcenter-i′,ycenter-i′) can be expressed as(11)xi′yi′=cosθi-sinθisinθicosθix-xcenter-iy-ycenter-iAt this time, because(x′,y′,z′) satisfies the equation of 3D cobblestone surface defined by (7), (8), and (9), the height of the cobblestone in the road coordinate system is calculated: z(x,y)=z′(x′,y′).The change of the coordinate system (grid) after the cobblestone rotation is shown in Figure6.Figure 6
The coordinate system of cobblestones before and after rotation.


## 2.5. 3D Cobblestone Road

For any point on the cobblestone road, its height can be expressed as(12)zx,y=zixi′,yi′x,y⊆Si0x,y⊄Siwhere Si represents the horizontal area surrounded by curves 2 and 3.As there are many road parameters of the cobblestone road, the road with different emphasis can be produced by the combination of different parameters. Here 3D models of three types of cobblestone roads are built by copying them from a domestic vehicle test ground. The shape and distribution parameters of the three kinds of cobblestone roads in the test ground are shown in Table2.Table 2
The parameters of the cobblestone roads in a vehicle test field.
Particle size(mm) Spacing(mm) Height(mm) Density(a/m2) Cobblestone road A 250~350 ≥particle size 80~120 3.5 Cobblestone road B 150~300 200~400 <120 / Cobblestone road C 70~80 100~200 20~35 30~35As shown in Table2, for the distribution of cobblestone, the distribution density of cobblestone is required for cobblestone roads A and C except for spacing requirement. In fact, both of them and particle size of the cobblestone are mutually restrictive. As the spacing adjustment is more flexible, the minimum spacing should be adjusted to meet the density requirement first when the distribution density is required.Besides, among the parameters mentioned in Table2, the systematicness explanation of the cobblestone (shape parameters k1,2,3) is not given. Based on the requirements of particle size and height, the shape parameters are determined as follows.Road A has the largest diameter, height, and moderate difference in height, and the road is mainly used for the assessment of heavy vehicles. So the shape parameter of the required cobblestone is large, resulting in maximum difference in shape to give the highest strength assessment of vehicle.Road B has maximum difference in height and larger particle size, so larger vertical shape parametersk1 and moderate horizontal surface shape parameters k2,3 are used to provide abundant lateral change incentives for vehicles with great height difference.The cobblestone of road C is nearly spherical or hemispherical with small particle size and high density, so the shape parameter of cobblestone has almost no impact on vehicle excitation. The road provides rich high frequency excitation for medium or light vehicles at high speed, so smaller shape parameter is selected.The 3D cobblestone road models with the length of 10 m and the width of 4 m according to the above parameters by the proposed method are shown in Figure7.Figure 7
Three kinds of 3D cobblestone road models in a test field. (a) Cobblestone road A. (b) Cobblestone road B. (c) Cobblestone road C.
(a) (b) (c)


## 3. 3D Random Road Fusion

Although the 3D road model of cobblestone road has been established based on the mathematical description of the geometric characteristics, the 3D model is only a theoretical model of the cobblestone road, and there is still a big gap with the actual road. Therefore, based on the theoretical model of 3D cobblestone road, the 3D road model similar to the actual road is obtained by fusing a generated 3D random road with certain road roughness.


### 3.1. Generation of 3D Random Road

Among the existing methods of generating 3D random road model [25], the harmonic superposition method is the most widely used, which is extended from 2D random standard road model. Based on the random ergodic characteristics of road roughness, the random signal of road roughness is decomposed into a series of sine waves with different frequencies and amplitudes by discrete Fourier transform. For any point in which the road surface coordinate is (x,y), the road roughness in the 3D space can be expressed as(13)qx,y=∑i2GdnmiΔnsin⁡2πnmiFx,y+θx,ywhere nmi is the central frequency point of frequency band i when the spatial power spectral density (PSD) Gd(n) is divided into several bands with Δn as the resolution, and its corresponding power spectrum value is Gd(nmi). F(x,y) is a function that is related to the length/width of the road and its power is 1 and we may as well take F(x,y)=x2+y2 here. θ(x,y) is a random sequence in [0,2π] changig with (x,y) that satisfies a uniform distribution.According to the standard ISO/DIS 8608 and GB/T 7031-2005, the road roughness is divided into eight levels by the road PSD, that is, Grade A to H. Considering that the actual test road itself is generally good pavement in addition to obstacles, the road of Grade A (at this time, there isGd=(16×10-6)·n/n0-2, and n0=0.1m-1 ) is used as the random road model fusing with the cobblestone road model. According to the actual design parameters of the cobblestone road, the road grade of the fusion random road model can be adjusted. The standard random road model of Grade A according to (13) is shown in Figure 8.Figure 8
3D random road model of Grade A.


### 3.2. 3D Random Road Fusion

For the theoretical road heightz(x,y) of cobblestone road and the generated random road q(x,y) above both defined by the horizontal coordinate (x,y), the two 3D road models can be directly fused when the grid coordinates between them are consistent.Because the cobblestone surface of the cobblestone road itself is more smooth, the fusion is carried out by alternating values between the cobblestone theoretical heightz(x,y) and the generated random road height q(x,y). At this time, the real 3D road height according to (12) and (13) can be expressed as(14)zx,y=zixi′,yi′x,y⊆Siqx,yx,y⊄SiThe 3D cobblestone road model after the fusion of 3D random road is shown in Figure9.Figure 9
3D cobblestone road after random road surface fusion.The road model generated by the proposed method gives the X, Y, and Z coordinate values of all space points on the road; then the model is usually saved as a text file (or Excel file) in the form of 3D coordinate dot array. In order to apply the 3D road model to virtual simulation test, the 3D road model can be imported into the virtual simulation environment such as Virtual.Lab and Adams and used as a “road model” that can be identified by the simulation environment. Due to the cobblestone road being rigid road, the deformation of the road is negligible. Then the friction coefficient and other parameters are given on the surface of the road model in the virtual environment, and the simulation test research can be carried out.


### 3.3. Comparison between Modeling Road and Real Road

When the vehicle runs through the cobblestone road, the contact between the tire and the road is a surface contact, which includes not only the tire deformation along the road but also the deformation along the width of the tire. At this time, the excitation input of the road on the vehicle can be regarded as the average of the road roughness on the whole contact surface. However, when the road is long enough, any road trajectory of cobblestone road has the same statistical characteristics in time and frequency domains, which can be compared to verify the accuracy of the cobblestone road model established in the paper. Therefore, a real cobblestone road with the similar geometric characteristics in a proving ground is used for comparison with the cobblestone road model, as shown in Figure10.Figure 10
The real cobblestone road in picture.Then, using a road profile acquisition system, the road profile of the real cobblestone road is obtained, and the local road profile is shown in Figure11(a). Similarly, a road trajectory is extracted from the model in the direction of the road on the basis of the established 3D cobblestone road model, and the local road profile is shown in Figure 11(b). The convex parts of the road curve are the cobblestones, and their heights above the ground are mostly concentrated in the range of 20-40 mm.Figure 11
Extracted road profile of cobblestone road. (a) The real road; (b) the modeling road.
(a) (b)Because the road profile is usually described in the form of PSD, the PSD of the road profiles in real and modeling cobblestone roads both need to be calculated through Fast Fourier Transform (FFT) for comparison, as shown in Figure12. Besides, eight standard road spectra from Grade A to Grade H according to ISO/DIS 8608 have been plotted as references. As shown, the road profile of the 3D cobblestone road model mainly concentrates on Grades D, E, and F, which conforms to the real cobblestone road.Figure 12
Space PSD of the cobblestone road profile.


## 3.1. Generation of 3D Random Road

Among the existing methods of generating 3D random road model [25], the harmonic superposition method is the most widely used, which is extended from 2D random standard road model. Based on the random ergodic characteristics of road roughness, the random signal of road roughness is decomposed into a series of sine waves with different frequencies and amplitudes by discrete Fourier transform. For any point in which the road surface coordinate is (x,y), the road roughness in the 3D space can be expressed as(13)qx,y=∑i2GdnmiΔnsin⁡2πnmiFx,y+θx,ywhere nmi is the central frequency point of frequency band i when the spatial power spectral density (PSD) Gd(n) is divided into several bands with Δn as the resolution, and its corresponding power spectrum value is Gd(nmi). F(x,y) is a function that is related to the length/width of the road and its power is 1 and we may as well take F(x,y)=x2+y2 here. θ(x,y) is a random sequence in [0,2π] changig with (x,y) that satisfies a uniform distribution.According to the standard ISO/DIS 8608 and GB/T 7031-2005, the road roughness is divided into eight levels by the road PSD, that is, Grade A to H. Considering that the actual test road itself is generally good pavement in addition to obstacles, the road of Grade A (at this time, there isGd=(16×10-6)·n/n0-2, and n0=0.1m-1 ) is used as the random road model fusing with the cobblestone road model. According to the actual design parameters of the cobblestone road, the road grade of the fusion random road model can be adjusted. The standard random road model of Grade A according to (13) is shown in Figure 8.Figure 8
3D random road model of Grade A.


## 3.2. 3D Random Road Fusion

For the theoretical road heightz(x,y) of cobblestone road and the generated random road q(x,y) above both defined by the horizontal coordinate (x,y), the two 3D road models can be directly fused when the grid coordinates between them are consistent.Because the cobblestone surface of the cobblestone road itself is more smooth, the fusion is carried out by alternating values between the cobblestone theoretical heightz(x,y) and the generated random road height q(x,y). At this time, the real 3D road height according to (12) and (13) can be expressed as(14)zx,y=zixi′,yi′x,y⊆Siqx,yx,y⊄SiThe 3D cobblestone road model after the fusion of 3D random road is shown in Figure9.Figure 9
3D cobblestone road after random road surface fusion.The road model generated by the proposed method gives the X, Y, and Z coordinate values of all space points on the road; then the model is usually saved as a text file (or Excel file) in the form of 3D coordinate dot array. In order to apply the 3D road model to virtual simulation test, the 3D road model can be imported into the virtual simulation environment such as Virtual.Lab and Adams and used as a “road model” that can be identified by the simulation environment. Due to the cobblestone road being rigid road, the deformation of the road is negligible. Then the friction coefficient and other parameters are given on the surface of the road model in the virtual environment, and the simulation test research can be carried out.


## 3.3. Comparison between Modeling Road and Real Road

When the vehicle runs through the cobblestone road, the contact between the tire and the road is a surface contact, which includes not only the tire deformation along the road but also the deformation along the width of the tire. At this time, the excitation input of the road on the vehicle can be regarded as the average of the road roughness on the whole contact surface. However, when the road is long enough, any road trajectory of cobblestone road has the same statistical characteristics in time and frequency domains, which can be compared to verify the accuracy of the cobblestone road model established in the paper. Therefore, a real cobblestone road with the similar geometric characteristics in a proving ground is used for comparison with the cobblestone road model, as shown in Figure10.Figure 10
The real cobblestone road in picture.Then, using a road profile acquisition system, the road profile of the real cobblestone road is obtained, and the local road profile is shown in Figure11(a). Similarly, a road trajectory is extracted from the model in the direction of the road on the basis of the established 3D cobblestone road model, and the local road profile is shown in Figure 11(b). The convex parts of the road curve are the cobblestones, and their heights above the ground are mostly concentrated in the range of 20-40 mm.Figure 11
Extracted road profile of cobblestone road. (a) The real road; (b) the modeling road.
(a) (b)Because the road profile is usually described in the form of PSD, the PSD of the road profiles in real and modeling cobblestone roads both need to be calculated through Fast Fourier Transform (FFT) for comparison, as shown in Figure12. Besides, eight standard road spectra from Grade A to Grade H according to ISO/DIS 8608 have been plotted as references. As shown, the road profile of the 3D cobblestone road model mainly concentrates on Grades D, E, and F, which conforms to the real cobblestone road.Figure 12
Space PSD of the cobblestone road profile.


## 4. Analysis of Road Geometric Parameters on Vehicle Response

Based on the establishment of the 3D cobblestone road model, the influence mode of the cobblestone road geometric parameters on the vehicle response is discussed preliminarily by establishing the standardized vehicle vibration model and evaluation indices.


### 4.1. IRI Standardized Vehicle Model

In the field of road engineering, the International Roughness Index (IRI) is the most widely used quality evaluation index for road [26]. The calculation of IRI is based on a 1/4 vehicle vibration model, as shown in Figure 13(a).Figure 13
IRI standard vehicle model and its frequency response curves. (a) 1/4 vehicle vibration model. (b) The frequency response curves of the standard vehicle model.
(a) (b)According to Figure13(a), the dynamic differential equations of the vehicle model can be set up as follows:(15)msz¨s+csz˙s-z˙u+kszs-zu=0msz¨s+muz¨u+ktzu-q-=0where zs and zu are the vertical displacements of sprung mass and unsprung mass, respectively. z˙s, z˙u, z¨s, and z¨u are vertical velocities and accelerations of sprung mass and unsprung mass, respectively. q- is the input of the road height by sliding average (the length of the sliding average is the tolerance of the tire to the ground which is B). The sprung mass ms is used as normalization in the model; then the unsprung mass is mu=0.15ms, tire stiffness is kt=653ms, suspension stiffness is ks=63.3ms with the unit of N/m, and suspension damping is cs=6ms with the unit of N/ms-1. The width of tire containment is B=250mm. The frequency response curves of the standard model can be calculated by the above parameters, as shown in Figure 13(b).The longitudinal height on arbitrary abscissa of the established strengthened test road (cobblestone road) is used as the road profile input of the standard vehicle model; then the vehicle responses are obtained.


### 4.2. Evaluation Indices

In the paper, the IRI and the root mean square (RMS) of vehicle body vertical vibration are used as the indices to evaluate the road and its impact on the vehicle, respectively.


#### 4.2.1. IRI

The IRI is the cumulative value of the relative displacement between the sprung mass and the unsprung mass per kilometer when driving on the corresponding road with the excitation of road roughness, expressed by m/km. It is an important index to evaluate road roughness. According to the definition of IRI, when the total length of the road is L (unit km),zs(x) and zu(x) are obtained by solving the differential equation group (15); thus the function of the IRI can be expressed as(16)IRI=1L∫0Lzs-zudx


#### 4.2.2. RMS of Vehicle Body Acceleration

Corresponding to the road profile represented by IRI, the RMS of vehicle body (here is represented by sprung mass) acceleration is the important index of vehicle vibration intensity, and its value is directly related to vehicle reliability and ride comfort. For the vehicle body accelerationz¨s[k] in the discrete form, the RMS is defined as (17)RMSz¨s=1N∑k=1Nz¨s2k


### 4.3. Analysis of the Geometric Parameters on Vehicle

As shown in Figure13(b), the IRI model is typical in simulating the vehicle vibration responses. Therefore, the influences of road geometry parameters on vehicle vibration can be evaluated by the vehicle vibration model and the evaluation indices shown in (16) and (17).Considering that the vehicle/road vibration model represented by (15) can only evaluate the longitudinal and vertical parameters of the road, the cobblestone road is generated here according to the variation range of the parameters in Table 3, and the IRI and RMSz¨s are calculated by (16) and (17). The corresponding relationship between the evaluation indices and the road parameters is shown in Figure 14.Table 3
Geometric parameters of Belgian road and their range of variation.
Road Type Geometric parameters Change range/mm Cobblestone road Maximum heighthmax 0~(10-300) Particle sizeamin/amax 20~(50-300) Shape parameterk1 -0.4~0.4 IntervalDmin/Dmax 200~1000 Note: “~” means parameters change randomly in this range; “-” means parameters increase equally in this range.Figure 14
Effect of maximum height and longitudinal length of cobblestones onIRI and RMSz¨s in cobblestone road. (a) IRI; (b) RMS.
(a) (b)As shown in Figure14, the sizes of the cobblestone on the cobblestone road have a decisive influence on the vehicle vibration, which are mainly reflected in the maximum height hmax and the maximum particle size amax of the cobblestone. Both the IRI and the RMS of vehicle body acceleration have a linear relationship with the sizes of the cobblestone. With the increase of the maximum height hmax and the maximum particle size amax of the cobblestone, both the IRI and RMS also increase.The above studies proved that the established road model based on road geometric characteristics can quickly evaluate the effects of various parameters on vehicles and provides useful references for road types and parameters selection when test roads are built.


## 4.1. IRI Standardized Vehicle Model

In the field of road engineering, the International Roughness Index (IRI) is the most widely used quality evaluation index for road [26]. The calculation of IRI is based on a 1/4 vehicle vibration model, as shown in Figure 13(a).Figure 13
IRI standard vehicle model and its frequency response curves. (a) 1/4 vehicle vibration model. (b) The frequency response curves of the standard vehicle model.
(a) (b)According to Figure13(a), the dynamic differential equations of the vehicle model can be set up as follows:(15)msz¨s+csz˙s-z˙u+kszs-zu=0msz¨s+muz¨u+ktzu-q-=0where zs and zu are the vertical displacements of sprung mass and unsprung mass, respectively. z˙s, z˙u, z¨s, and z¨u are vertical velocities and accelerations of sprung mass and unsprung mass, respectively. q- is the input of the road height by sliding average (the length of the sliding average is the tolerance of the tire to the ground which is B). The sprung mass ms is used as normalization in the model; then the unsprung mass is mu=0.15ms, tire stiffness is kt=653ms, suspension stiffness is ks=63.3ms with the unit of N/m, and suspension damping is cs=6ms with the unit of N/ms-1. The width of tire containment is B=250mm. The frequency response curves of the standard model can be calculated by the above parameters, as shown in Figure 13(b).The longitudinal height on arbitrary abscissa of the established strengthened test road (cobblestone road) is used as the road profile input of the standard vehicle model; then the vehicle responses are obtained.


## 4.2. Evaluation Indices

In the paper, the IRI and the root mean square (RMS) of vehicle body vertical vibration are used as the indices to evaluate the road and its impact on the vehicle, respectively.


### 4.2.1. IRI

The IRI is the cumulative value of the relative displacement between the sprung mass and the unsprung mass per kilometer when driving on the corresponding road with the excitation of road roughness, expressed by m/km. It is an important index to evaluate road roughness. According to the definition of IRI, when the total length of the road is L (unit km),zs(x) and zu(x) are obtained by solving the differential equation group (15); thus the function of the IRI can be expressed as(16)IRI=1L∫0Lzs-zudx


### 4.2.2. RMS of Vehicle Body Acceleration

Corresponding to the road profile represented by IRI, the RMS of vehicle body (here is represented by sprung mass) acceleration is the important index of vehicle vibration intensity, and its value is directly related to vehicle reliability and ride comfort. For the vehicle body accelerationz¨s[k] in the discrete form, the RMS is defined as (17)RMSz¨s=1N∑k=1Nz¨s2k


## 4.2.1. IRI

The IRI is the cumulative value of the relative displacement between the sprung mass and the unsprung mass per kilometer when driving on the corresponding road with the excitation of road roughness, expressed by m/km. It is an important index to evaluate road roughness. According to the definition of IRI, when the total length of the road is L (unit km),zs(x) and zu(x) are obtained by solving the differential equation group (15); thus the function of the IRI can be expressed as(16)IRI=1L∫0Lzs-zudx


## 4.2.2. RMS of Vehicle Body Acceleration

Corresponding to the road profile represented by IRI, the RMS of vehicle body (here is represented by sprung mass) acceleration is the important index of vehicle vibration intensity, and its value is directly related to vehicle reliability and ride comfort. For the vehicle body accelerationz¨s[k] in the discrete form, the RMS is defined as (17)RMSz¨s=1N∑k=1Nz¨s2k


## 4.3. Analysis of the Geometric Parameters on Vehicle

As shown in Figure13(b), the IRI model is typical in simulating the vehicle vibration responses. Therefore, the influences of road geometry parameters on vehicle vibration can be evaluated by the vehicle vibration model and the evaluation indices shown in (16) and (17).Considering that the vehicle/road vibration model represented by (15) can only evaluate the longitudinal and vertical parameters of the road, the cobblestone road is generated here according to the variation range of the parameters in Table 3, and the IRI and RMSz¨s are calculated by (16) and (17). The corresponding relationship between the evaluation indices and the road parameters is shown in Figure 14.Table 3
Geometric parameters of Belgian road and their range of variation.
Road Type Geometric parameters Change range/mm Cobblestone road Maximum heighthmax 0~(10-300) Particle sizeamin/amax 20~(50-300) Shape parameterk1 -0.4~0.4 IntervalDmin/Dmax 200~1000 Note: “~” means parameters change randomly in this range; “-” means parameters increase equally in this range.Figure 14
Effect of maximum height and longitudinal length of cobblestones onIRI and RMSz¨s in cobblestone road. (a) IRI; (b) RMS.
(a) (b)As shown in Figure14, the sizes of the cobblestone on the cobblestone road have a decisive influence on the vehicle vibration, which are mainly reflected in the maximum height hmax and the maximum particle size amax of the cobblestone. Both the IRI and the RMS of vehicle body acceleration have a linear relationship with the sizes of the cobblestone. With the increase of the maximum height hmax and the maximum particle size amax of the cobblestone, both the IRI and RMS also increase.The above studies proved that the established road model based on road geometric characteristics can quickly evaluate the effects of various parameters on vehicles and provides useful references for road types and parameters selection when test roads are built.


## 5. Conclusions

Taking cobblestone road as the research object, a 3D road modeling method based on geometric characteristics is proposed. The oval equation of cobblestone is established and discussed by determining the key geometric characteristics, and the cobblestones distribution on the cobblestone road is determined by Monte Carlo search method. Then the 3D theoretical reconstruction of the cobblestone road is completed, and a generated 3D random road is fused to get a more realistic 3D model of the cobblestone road. The comparison between the modeling road and real road is done in space domain and PSD curves.Based on the standard vehicle vibration model, the IRI and the RMS of vehicle body acceleration are established as the evaluation indices of vehicle vibration, and the influence of key geometric parameters of the cobblestone road on the vehicle vibration was evaluated. Analysis results show that the sizes of cobblestone have a decisive influence on the vehicle vibration.Although the method proposed in the paper is only verified under the cobblestone road situation, the method is applicable for 3D modeling of almost all strengthened test roads, such as the Belgian road, washboards, and ripple tracks and provides an effective way of 3D road modeling for building vehicle virtual test field.



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*Source: 7246503-2018-06-27.xml*