﻿ 东南大学学报(自然科学版)

(1华南理工大学亚热带建筑科学国家重点实验室, 广州 510641)

Calculation method for 3D dynamic sight distance under the control of slope constraint
Yang Yonghong1,2,Xia Yuanbo1,Wang Jiecong1,Huang Lan1
(1State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510641, China)(2Key Laboratory of Highway Engineering of Ministry of Education, Changsha University of Science and Technology, Changsha 410114, China)

In order to achieve the fitting of the established road alignment and accurately calculate the three-dimensional(3D)dynamic sight distance of the highway under the constraint of the right side slope, the road length, horizontal azimuth and vertical angle were adopted as the design parameters. A 3D calculation model was established for highway alignment based on the spline curve. The Gauss-Legendre quadrature formula was used to achieve the highway centerline coordination. The pavement and slope triangulation network were setup based on the characteristic points of roadside lines, considering the geometric relationship among the height of the driver's sightline, the driving trajectory and the tangent plane of the boundary. The 3D sight distance calculation model was proposed based on the spatial alignment, and the solving equation for the minimum value of sight distance was deduced. By the case of a two grade highway with a section of right turn, the 3D dynamic sight distance was calculated and compared with the calculation method for lateral clear distance. The results show that the 3D dynamic sight distance values are greater than that of the lateral clear distance in the section of the sag vertical curve. However, there are safety risks for adopting the lateral clear distance auditing in the section of the crest vertical curve.The safety threshold of the crest vertical curve radius 1/S1 decreases with the decrease of the driver's sightline height, while the safety threshold of horizontal curve radius 1/S2 increases with the increase of the vertical curve radius.

1 道路三维视距计算流程

2 道路三维线形的数学计算模型2.1 道路中心线三维坐标方程

(dα)/(dl)=ρ1=al+b a≥0,b≥0

(dβ)/(dl)=ρ2=cl+d c≥0,d≥0}(1)

α=α0+(al2)/2+bl

β=β0+(cl2)/2+dl}(2)

x=x0+∫l00sinβcosαdl

y=y0+∫l00sinβsinαdl

z=z0+∫l00cosβdl}(3)

κ2=β'2+α'2sin2β

τ=((α″β'+α'β″)sinβ+2α'β'2cosβ+α'3sin2βcosβ)/(κ2)}(4)

2.2 道路中心线坐标数值实现

f(x)=sin(β0+(cx2)/2+dx)cos(α0+(ax2)/2+bx)

g(x)=sin(β0+(cy2)/2+dy)cos(α0+(ay2)/2+by)

l(z)=cos(β0+(cz2)/2+dz)}(5)

x=y=z=(S-S0)/2t+(S+S0)/2 t∈(-1,1)(6)

xi=x0+∫1-1f((S-S0)/2t+(S+S0)/2)dt=

x0+b∑nk=1Akf(xk)

yi=y0+∫1-1g((S-S0)/2t+(S+S0)/2)dt=

y0+b∑nk=1Akg(yk)

zi=z0+∫1-1l((S-S0)/2t+(S+S0)/2)dt=

z0+b∑nk=1Akl(zk)}(7)

δ=(f 2n+1(η))/(2n+3)(22n+3((n+1)!)4)/(2n+3)η∈(-1,1)(8)

3 路面和边坡三角网模型

m0={sinα,-cosα,i0}T

n={sinβcosα,sinβsinα,cosβ}T

m1={sinα,-cosα,i1}T}(9)

xi=x0+∫S1S0sinβcosαdl+disinα

yi=y0+∫S1S0sinβsinαdl-dicosα

zi=z0+∫S1S0cosβdl+hi}(10)

di=(d0)/2+(li)/(ls)dw

hi=ih(li)/(ls)di}(11)

4 道路三维动态视距计算模型4.1 建立三维动态视距特征方程

A(X-X1)+B(Y-Y1)+C(Z-Z1)=0(12)

D(X-X0)+E(Y-Y0)+F(Z-Z0)=0(13)

p=m1×n={A

B

C}={-i1sinβsinα-cosβcosα

i1sinβcosα-sinαcosβ

sinβ}

q=m0×n={D

E

F}={-i0sinβsinα-cosβcosα

i0sinβcosα-sinαcosβ

sinβ}}(14)

A(XS0-Xbp)+B(YS0-Ybp)+

C(ZS0-Zbp+Hs)=0

A(XS1-Xbp)+B(YS1-Ybp)+

C(ZS1-Zbp+Hw)=0}(15)

D(XS'0-Xxc)+E(YS'0-Yxc)+

F(ZS'0-Zxc+HS)=0

D(XS'1-Xxc)+E(YS'1-Yxc)+

F(ZS'1-Zxc+Hw)=0}(16)

4.2 基于Newton-Raphson迭代法的最小视距值求解

f(S)=d/(b2-d2)[cosβ0cosbscos(β0+dS)-cos2β0]+

b/(b2-d2)cosβ0sinbssin(β0+dS)-

(i1d)/(b2-d2)sinβ0sinbscos(β0+dS)+dmi1sinβ0+

(i1b)/(b2-d2)[sinβ0cosbscsin(β0+dS)-sin2β0]-

(sinβ0)/d[sin(β0+dS)-sinβ0+dhm+dH](17)

hd=Rs(1-cos(90lsd)/(πRs))(18)

Sn+1=Sn-(f(Sn))/(f'(Sn))(19)

5 空间动态视距计算实例分析5.1 动态视距仿真计算

5.2 最小视距值对比分析

6 结论

1)基于传统平纵线形设计指标要素,提出了一种基于样条曲线的三维道路中心线拟合以及行车轨迹线、道路边界线等数学计算方法,采用Gauss-Legendre求积公式实现中心线坐标的数值求解.并提出线形设计过程中保证水平方位角和竖直角的二阶连续性即可实现空间曲率和挠率连续.

2)综合考虑三维道路线形、道路横向超高、加宽以及边坡约束条件,通过构建任意桩号的视线区域,建立了基于空间线形的三维动态视距计算模型,开发了三维动态视距计算程序,快速实现路段动态视距计算.考虑视距计算和检验的实用性,在视域特征方程基础上提出三维视距最小值求解方程,通过迭代法计算出路段最小三维视距值,并分析发现横净距计算公式为视距最小值求解方程的特殊情况.

3)通过横净距视距计算对比分析,三维视距值随着竖曲线曲率增大而减小.在凹型竖曲线路段,三维动态视距计算值大于平面横净距视距计算值,且视距计算差值随着边坡坡度变缓而增加.在凸型竖曲线路段,竖曲线半径存在安全阈值1/S1,1/S1随着驾驶人视线高度的降低而增大,因此小客车通行比例高的路段采用横净距视距计算会存在较大安全隐患; 平曲线半径存在安全阈值1/S2,其值会随着竖曲线半径的增大而增大.