A new method for 3-dimensional roadway design using visualization techniques

Transcription

1 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury 23 A new mehod for 3-dimensional roadway design using visualizaion echniques G. Karri & M. K. Jha Dearmen of Civil Engineering, Morgan Sae Universiy, USA Absrac Tradiionally, geomeric design of roadways is a wo-sage rocess in which he horizonal alignmen is designed firs and hen he verical alignmen is designed based on he former. A design change in horizonal alignmen will affec he verical alignmen design. Also, he radiional wo-sage road design may lead o criical design errors, such as a secion of he road wih a verical valley curve and a shar horizonal curve ogeher will lead o an illusion of a lesser shar horizonal curve when driving. This is due o he fac ha he verical curve may be overlooked as i canno be raced by he radiional wo-sage design rocess. One can hus see ha his wo-sage rocess is cumbersome and subjec o error. Lieraure review shows ha his error-rone and cumbersome rocess can be imroved by visualizing roads hree dimensionally in a single sage. In his aer we roose a mehod in which he radiional wo-sage design rocess can be reduced o a single-sage design rocess by develoing a hree-dimensional (3D) roadway using visualizaion echniques. The roadway is designed using elemens like 3D lines and iecewise olynomials (slines). The design involves idenifying conrol oins on a given errain, hrough which he roadway asses, and finding suiable 3D lines and/or slines ha connec hose conrol oins. Mahemaical formulaions o calculae he 3D lines and slines are resened. The aer concludes wih he discussion of he alicaion of he roosed mehodology o full-scale efficien roadway design and direcions for fuure research. Keywords: roadway geomeric design, visualizaion, slines, 3D. WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line) doi:0.2495/ut0703

2 24 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury Inroducion Tradiional geomeric design of roadways involves he design of horizonal and verical alignmens for a smooh and safe riding of vehicles. Some of criical informaion for geomeric design may no be available in horizonal alignmen or in verical alignmen because of he lanar rojecion of highway on o XY lane or XZ lane (Caresian coordinae sysem). For examle, a designer may no be able o adjudge he sigh disance by looking a horizonal or verical alignmen, because in horizonal alignmen he elevaion informaion is no available and in verical alignmen he horizonal alignmen of road is no available. Anoher roblem ha can be seen in 2D geomeric design mehod is ha he designer may no idenify an illusion effec when horizonal curve aears simulaneously wih verical curve. Wihou idenifying he effec if he roadway is consruced, he simulaneous aearance of horizonal and verical curve may give an illusion o he driver ha highway is safe for higher seeds bu in racice i may no be. These roblems o some exen can be solved by designing highways using hree dimensional visualizaion echniques. Using hese echniques he designer can view he alignmen hree dimensionally and idenify he errors ha canno be idenified eiher in horizonal or verical alignmen. Therefore, one can see ha hree dimensional geomeric design of roadways has more advanages when comared o he wo dimensional geomeric design. Lieraure review shows ha no many hree dimensional geomeric road design mehods are available. Gibreel e al. [2] described he calculaion of oeraional seeds on roadways using hree dimensional alignmens. Bu, he mehod described alies o a aricular ye of roadway and is no a general urose mehod. Jha e al. [3] resened a model for highway alignmen using GIS, geneic algorihms, and comuer visualizaion echniques. Comuer visualizaion is basically used for showing differen alernaive soluions bu no mehod is resened for 3D geomeric design of highways. Kang e al. [4] described 3D alignmen oimizaion bu he mehod basically deals wih finding an oimal alignmen beween wo given oins. The roadway is designed based on exising 2D design mehods. Kuhn [5] resened a 3D geomeric design mehod for roadways using combinaion of fixed, dialogue and couling elemens; bu, he aer does no rovide informaion of he choice of elemens. From he above discussion i is clear ha here is a need for exloiing 3D geomeric design mehods and laes visualizaion echnologies for imroved roadway design. 2 Mehodology The mehodology for 3-dimensional roadway design is based on he exising 2-dimensional horizonal and verical design mehods. In 2D design mehods, he roadway alignmen is comosed of design elemens ha are a combinaion of sraigh lines, circular curves, arabolic curves, and siral curves. In eiher horizonal or verical alignmen, a change in sraigh road (sraigh line) direcion is incororaed hrough a curve, i.e., wo sraigh orions of a road, which are WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)

3 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury 25 adjacen and have differen direcions are conneced wih a curve. Circular curves are used in horizonal alignmen design and arabolic curves are used in verical alignmen design, o join he sraigh orions. Similar aroach is used for hree dimensional roadway designs. 2. Design elemens In our aroach he alignmen is comosed of wo design elemens, 3D sraigh lines and 3D sline curves. The 3D sraigh lines are conneced wih siral curves D sraigh lines A 3D sraigh line consiss of a sar and an end oin. In Caresian coordinae sysem he sar oin is reresened as (x s,y s, z s ) and he end oin is reresened as (x e, y e, z e ). The locaion of he sar and end oin gives he direcion of he sraigh line. A yical 3D sraigh line is given in Figure. Z 3D sraigh line elemen (x e,y e,z e) 3D sraigh line elemen rojeced on o XY lane (x s,y s,z s) Y (x s,y s ) (x e,y e ) X Figure : A yical 3D sraigh line D sline curves Given a se of conrol oins hrough which he designer wans he alignmen o ass, olynomial curves can be used o ass hrough he conrol oins. Bu, here are wo disadvanages of using olynomial curves.. There will be no conrol on he adjusmen of he curve beween wo conrol oins for design uroses. 2. As he number of conrol oins increases, he order of he olynomial increases and hence number of coefficiens of olynomial also increases. Insead of using one olynomial o ass hrough conrol oins, one can use iecewise olynomials o reduce above said roblems. These iecewise olynomials ogeher are ermed as sline curves. Slines form a good fi for given conrol oins and inermediae oins can be inerolaed easily [6. 7]. In he roosed design aroach, cubic slines (of order 4) are used. WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)

4 26 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury Le C = {0,, 2,.. l -} denoe l knos (which are conrol oins in his mehodology and a ar of he oal se of conrol oins). These knos are in a sequence. Le c C. c = 0 denoes he firs kno in he sequence and c = l - h denoes he l kno. Le ( x c, y c, z c ) be he Caresian coordinaes of a kno in C. Le X = { x 0, x,.. x l } be he se of x coordinaes of all he l knos. Le Y = { y 0, y,.. y l } be he se of y coordinaes of all he l knos. Le Z = { z 0, z,.. z l } be he se of z coordinaes of all he l knos. As exlained earlier, we connec hese l knos wih a sline. The sline will have l - iecewise olynomials. h The arameric funcions of iecewise olynomial of a sline ha is beween knos c and c + is given below. x 3 2 = a x + bx + c x + d y 3 2 = a y + by + c y + d z = a 3 + b 2 + c + d x y where [ c, c + ) h = c +. denoes he olynomial iece of he sline ha is beween knos c and c +. For examle = 2 denoes he second olynomial of he sline ha is beween he knos and 2. x = y = z = h iece x coordinae considered a osiion. h iece y coordinae considered a osiion. h iece z coordinae considered a osiion. a x, bx, c x, d x are coefficiens of he a y, by, c y, d y are coefficiens of he a, b, c, d are coefficiens of he h olynomial iece in X h olynomial iece in Y h olynomial iece in Z Many algorihms are available which can be used o ge he coefficiens of iecewise olynomials. Examles include cardinal slines, Camull Rom slines, WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)

5 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury 27 and Kochanek Barels Slines [, 6, 7]. A yical 3D sline is shown in Figure 2. For more deails on slines, he readers may refer o any comuer grahics book on using slines for grahics. Z PP PP 3 PP 2 (x 2,y 2,z 2 ) (x 3,y 3,z 3 ) (x 0,y 0,z 0 ) (x,y,z ) Y (x 0,y 0 ) (x,y ) (x 2,y 2 ) X (x 3,y 3 ) 3D sline 3D sline rojeced ono XY lane Figure 2: A yical 3D sline. Here 3D sline has hree iecewise olynomials PP, PP 2, PP 3 wih four knos. 2.2 Design rocedure Given a 3D errain, he designer has o selec some conrol oins over he errain hrough which he alignmen should ass. Keeing hese conrol oins in view, he designer bes fis 3D sraigh lines based on he conrol oins osiions. The number of 3D sraigh line design elemens deends on he sie condiions. Then he designer fis a sline elemen ha asses hrough conrol oins ha are beween wo consecuive sraigh line elemens. The designer can change he shae of he sline by adding addiional conrol oins or by changing he osiion of he exising conrol oins. Then he designer needs o check he alignmen for echnical and safey issues. 3 An examle 3. An alignmen creaed using exising mehods A roadway alignmen is creaed using exising mehods for exerimenal urose. The alignmen creaed has hree design elemens namely sraigh-lines, clohoids, circular curves in horizonal alignmen and sraigh-lines, and arabolic curves in verical alignmen. The geomeric deails are given in Tables and 2 and Figure 3. WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)

6 28 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury Table : Geomeric deails for horizonal alignmen. Tye Sar End Lengh x y X y (m) Radius(m) Linear Clohoid Circular Clohoid Linear Clohoid Circular Clohoid Linear Clohoid Circular Clohoid Linear Clohoid Circular Clohoid Linear Table 2: Geomeric deails for verical alignmen. Sar End Lengh Tye Saion Elevaion Grade Saion Elevaion Grade (m) Linear % % Parabola % % 450 Linear % % 59.2 Parabola % % 400 Linear % % 8.28 Parabola % % 300 Linear % % Parabola % % 200 Linear % % Linear % % Parabola % % 300 Linear % % WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)

7 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury 29 (a) (b) Figure 3: (a) Horizonal alignmen for geomeric deails shown in Table. (b) Verical alignmen for geomeric deails shown in Table Alignmen creaed using he roosed mehod A oal of 4 conrol oins are used o design he alignmen. Table 3: Geomeric deails for a 3D alignmen. Conrol Conrol Sar End Tye Poin Poin x Y z X y z 2 Linear Sline Linear Sline Linear Sline Linear Sline Linear Sline Linear Sline Linear WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)

8 30 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury Table 4: Sline elemens geomeric roeries. Sar Conrol Poin End Conrol Poin Sar Kno Value c End Kno Value Comonen x 3 2 = a + b + c + d Coefficiens a b c d x y z x y z x y z x y z x y z x y z From Figure 4, one can see ha he alignmen secions wih sraigh line elemens obained by he roosed mehod are maching wih some of he exising alignmen secions. The oher secions, i.e., he sline curved orions of he alignmen secions obained by he roosed mehod are no maching wih he exising curved orion of he alignmen. The curved orions are no maching because only wo conrol oins are used o fi a sline. If one uses more han wo conrol oins, hen he curves will mach. WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)

9 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury 3 a Digial Terrain b Figure 4: 4 c (a) 3D view of alignmens obained using exising and roosed mehods. (b) To view of he alignmens. (c) Fron view of he alignmens. Discussion One main advanage of he roosed mehod over exising design mehods is ha he designer has more conrol on geomery of he highway. The designer can idenify any ossible errors in he 3D alignmen and change he geomery accordingly by eiher adding more conrol oins or changing he osiion of he exising conrol oins. In he roosed mehod he designer can direcly see he gradien and horizonal alignmen of a aricular roadway secion simulaneously ha he/she is designing, which is no ossible in a 2D mehod. WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)

10 32 Urban Transor XIII: Urban Transor and he Environmen in he 2s Cenury In he roosed mehod, we used 4 h order cubic sline. I s no necessary ha one has o use cubic slines. Oher yes of slines are also available. The choice of he slines deends on he requiremen. Differen slines behave differenly deending on he mehod used [7]. Though he mehodology described hels in 3D design of roadways, he designed roadway needs o be checked for geomery consisency. The mehod does no check he alignmen for comfor issues when a driver is moving on he designed roadway. Anoher issue relaed o he mehod is he fixing he number and osiion of he conrol oins. These issues need o be considered for he imrovemen of he roosed mehod in he fuure works. The conrol oins, 3D sraigh lines, and he alignmen using exising mehods are creaed in Microsaion. The enire code o generae 3D sline elemens is develoed wih he Malab sofware. The 3D slines are creaed using he conrol oins and angen informaion of 3D sraigh lines ha are being conneced wih 3D slines. The code develoed works well for any given se of conrol oins and 3D sraigh lines. The research is carried ou a he Cenre for Advanced Transoraion and Infrasrucure Engineering Research (CATIER: a he Morgan Sae Universiy. 5 Conclusions The aer discusses he roblems in he exising 2D mehods of roadway design and he need for a single sage 3D design of roadways. A suiable 3D geomeric design mehod ha uses sraigh line and sline curves, is roosed for he design of roadways. The mehod can be furher imroved in he fuure o accommodae driver safey. References [] Birkhoff, & Boor, D., Piecewise olynomial inerolaion and aroximaion, in: H. L. Garabedian (ed.), Proc. General Moors Symosium of 964, Elsevier, New York and Amserdam, , 965. [2] Gibreel, G.M., Easa, S.M. & El-Dimeery, I.A., Predicion of oeraing seed on hree-dimensional highway alignmens, Journal of Transoraion Engineering, 2,.2-30, 200. [3] Jha, M.K., McCall, C. & Schonfeld P., Using GIS, geneic algorihms and visualizaion in Highway develomen, Comuer Aided Civil and Infrasrucure Engineering, 6(6), , [4] Kang, M.K., Jha, M.K. & Schonfeld, P., 3D alignmen oimizaion for Brookeville byass, TRB Annual Meeing, Washingon DC, CD-ROM, 2005 [5] Kuhn, W., The basics of a hree-dimensional geomeric design mehodology, 3 rd Inernaional Symosium on Highway Geomeric Design, Chicago, CD-ROM, [6] Malab, h:// [7] Wikiedia, h://en.wikiedia.org/wiki/sline_(mahemaics) WIT Transacions on The Buil Environmen, Vol 96, ISSN (on-line)