geometric assessment for fabrication of large hull pieces in shipbuilding (paper)

8/11/2019 Geometric Assessment for Fabrication of Large Hull Pieces in Shipbuilding (Paper)

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Computer-Aided Design 39 (2007) 870881

www.elsevier.com/locate/cad

Geometric assessment for fabrication of large hull pieces in shipbuilding

Jung Seo Parka, Jong Gye Shin a, Kwang Hee Kob,

aDepartment of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, South KoreabDepartment of Mechatronics, Gwangju Institute of Science and Technology, 1 Oryong-dong, Buk-gu, Gwangju, 500-712, South Korea

Received 20 January 2007; accepted 6 May 2007

Abstract

In this paper we propose a geometric assessment method in order to establish a theoretical foundation for the automation of fabricating curved

steel plates through the line-heating process in shipbuilding. We focus on measurement, comparison and the measurement data process in the

overall system which was developed at Seoul National University, Korea, Shin et al. [Shin JG, Ryu CH, Lee JH, Kim WD. A user-friendly,

advanced line-heating automation for accurate plate fabrication. Journal of Ship Production 2003;19(1):815]. The measurement is performed

using a scanning device, which provides the digitized geometric data of a curved plate. The measured data are then positioned against a design

plate so that comparison between them is made to determine whether we have obtained the desired shape. If not, the data are approximated in

B-spline surface form and the approximated surface and the measured data are processed to extract necessary information for computing heating

lines for the next iteration. Experiments are performed with a real ship hull plate and the result shows the potential of the proposed procedure for

the automation of the line-heating manufacturing process at the shipyard.c2007 Elsevier Ltd. All rights reserved.

Keywords: Surface comparison; Line heating; Mapping data extraction; Hull forming; Surface registration

1. Introduction

A ship hull in general is designed to satisfy a set of

performance requirements of speed and fuel consumption by

improving flow characteristics around the ship. It becomes

inevitable to use curved parts, which cover more than 5080%

of the whole shape of the hull. In particular, it is noticed that,

among the hull parts, the bow and the stern are made of plates

of complex shape such as doubly curved plates, and precise and

efficient fabrication of such curved parts is very critical not only

in the quality and performance of the completed ship but also

in the efficiency of the manufacturing process.

The fabrication of a curved part starts with forming a flatplate to a desired shape. Since the plate is very thick and heavy,

producing a desired shape is not an easy job. Among various

techniques for manufacturing a curved plate, the line-heating

method is generally used by most shipbuilding companies these

days. This production method is a manufacturing process to

produce a curved plate of desired shape by heating a plate along

Corresponding author. Tel.: +82 62 970 3225; fax: +82 62 970 2384.E-mail addresses: [email protected](J.S. Park),[email protected]

(J.G. Shin),[email protected](K.H. Ko).

a set of paths, inducing a permanent deformation (shrinkage)

(Ryu [2]).Despite the long history of shipbuilding industries and

several innovations adopted in the ship manufacturing process,

such as welding, dry-dock, computer-aided design (CAD)

systems and NC machines which have drastically improved

ship production, the method of fabricating curved plates has

not changed much. It still relies entirely on workers skills and

experience. Moreover, the fabrication of those plates is a time-

consuming and labor-intensive process and they work in harsh

conditions, threatening workers safety. So, the necessity of

automating the line-heating process has been well appreciated

in shipbuilding industries (Ryu[2]) and research on the relatedtopics is being performed actively these days. Recently, Shin

et al. [1,3] at Seoul National University, Korea, developed

a line-heating system which has provided a foundation for

automatic ship hull production. It takes the design shape of a

hull piece as input, computes the developed shape and heating

lines, and controls the heating machine to heat the developed

plate along the computed heating lines. After one cycle of

heating, the shape of the fabricated plate is measured and

compared to assess the overall geometric conformance to thedesign shape Shin et al. [4].

0010-4485/$ - see front matter c

2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cad.2007.05.007
http://www.elsevier.com/locate/cadmailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.cad.2007.05.007http://dx.doi.org/10.1016/j.cad.2007.05.007mailto:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/cad


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J.S. Park et al. / Computer-Aided Design 39 (2007) 870 881 871

However, because of various adverse factors such as

assumptions made in the heating line computation and

inaccurate heat flux out of a heating torch, the desired shape

is very difficult to achieve at the first attempt. We can

minimize the unfavorable factors in modeling and fabrication

by carefully considering nonlinear terms in the calculation

and using accurately controllable heat sources. But, sincewe cannot entirely eliminate all those effects in the real

manufacturing situation, it is almost impossible to obtain the

desired shape through heating a plate only once. Also, such an

approach may increase the computational burden on the system,

resulting in extended computation time. These problems can be

circumvented by using an iterative manufacturing process of

heating, measurement and comparison. Through iteration, even

though we do not consider all those factors in our computation,

we can gradually reach the design shape, overcoming the effects

of those unfavorable factors. This approach was proposed due

to its conceptual simplicity and easy implementation (Ryu [2]

and Shin et al. [3]).

At the heart of this iteration, lie the measurement andcomparison of curved plates and inspection techniques could

be used for this purpose. Inspection is one of the popular

topics in reverse engineering, and there is extensive literature

on automation of the inspection process in design or

manufacturing. Among various inspection techniques, methods

based on localization or registration have been proposed by

Bardis et al. [5], Besl and McKay [6] and Tuckers and

Kurfess [7]. The problem is formulated as an optimization

problem, and each paper differs in their solution approaches.

Bardis et al. [5] used an optimization solver, Besl and

McKay[6] proposed a new iterative scheme in the quaternion

framework, and Tucker and Kurfess [7] used Newtons methodto solve the problem. A comprehensive review of measurement

and inspection was performed in Varady et al. [8] and Newman

and Jain [9]. The application of inspection to the automation

of ship hull fabrication, however, has not been explored much,

and research on the measurement and comparison of hull plates

is still in its infancy compared with research activities in other

disciplines. One attempt was made to measure and compare a

hull piece by Shin et al. [4]. In their work, a measuring system is

proposed and a sequence of operations for the measurement and

comparison of a large-scale hull plate are presented. But their

approach is applicable for the comparison of two plates that are

similar in shape, and no discussion is made on how to use the

result of the comparison in the line-heating process cycle.

As was introduced in Ryu [2] and Shin et al. [3], the

iterative forming process was suggested for fabricating a

complex curved plate. However, only a conceptual discussion

was provided, with no actual implementation, and they did not

present how to process measured data to compute the heating

lines for the next iteration, which is the most important step in

the complete iterative forming process.

In this paper, we discuss measurement, comparison and

the measured data process for the automation of curved plate

production using line heating. In particular, we focus on how

to process measured data for an intermediate plate, a curved

plate fabricated during the line-heating process, for comparison

with the design plate and heating line computation for the

next iteration, and demonstrate its potential to be used in the

real manufacturing process. The comparison is performed by

first placing two plates with respect to a common reference

frame through registration and computing the shape difference

between them. By checking the difference values, we can

determine if the desired shape has been obtained or not. Forthe heating line computation, we use a computation method

by Shin et al. [1] and Ryu [2]. Given plates of desired

and arbitrary shape, the method computes the heating lines

using the geometric relationships between the two plates.

This computation is, however, only possible when there is

correspondence information between them, and no research

has been performed to establish this so far. So, in this work

we propose a new method for extracting correspondence

information, which is critical for heating line computation

in order to close the loop of the line-heating manufacturing

process.

This paper is structured as follows. In Section 2, an

overall picture of the whole line-heating process is introduced.Section 3 discusses measurement of a curved plate. In

Section 4, a procedure for comparing two plates through

registration is presented. Section 5 proposes an algorithm

to place the measured plate with respect to the design and

developed plates, and presents a procedure for extracting the

data necessary to compute heating lines from the measured

curved plate. In Section 6, the algorithms and the procedure

are demonstrated with an example for a real plate. Section7

discusses the results of the example and Section 8concludes

the paper.

2. Overall procedure of line heating

An automation system for the line-heating process was

proposed by Ryu [2], the schematic of which is depicted in

Fig. 1. It consists of two parts: KOJEDO, a software module

to generate information for the line-heating process; and the

machine part, which fabricates a curved plate and assesses

it against the design shape. This system is implemented on

the basis of physics-based modeling and analysis techniques.

This generates data for driving a heating torch and accepts

as input the shape of a current plate using a scanner in

real time for further processes. This is a typical example of

DDDAS (Dynamic Data Driven Applications Systems); see

Darema[10]. The concept of DDDAS is characterized as thecapability to dynamically inject data into a running application,

to process them, and to control the measurement for real-

time data. Given the DDDAS concept, it is recognized that

the line-heating automation system is beyond the feedback

control system, in that real-time data are incorporated into

a running system and processed for extracting the necessary

information for heating line computation, which is then

supplied to the computation and analysis module, and the

results of the computation and analysis are used to move the

heating machine and measuring device. For more details of the

KOJEDO system, we would like to refer the readers to Ryu[2].

However, in Ryu [2], the machine part was proposed without


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Fig. 2. Manual measurement using a template.

are required to process measured data, such as the extraction of

edges and corners and artifact removal, which are performed

interactively. In this paper, either of the methods is used to

measure the plate.

However, a full automatic measurement system should be

developed to complete the automation of the line-heating

process. We expect that developing such an automated

measuring system deserves to be a full research topic, and in

this work we do not touch that issue.

Throughout this paper, we assume that we have measured a

curved plate and generatedn data points,ci (i= 1, . . . , n). Wedenote the set of data points by C. The thickness of a plate is

very small compared to the size of the plate, and we assume

that the shape of the plate does not change across the thickness.

Under these conditions, the data measured on the surface of

a plate can be used to represent the shape of the plate with a

thickness for the subsequent processes.

4. Comparison

In order to check if the desired shape has been obtained

through the line-heating process, we need to compare the shape

of the fabricated plate against that of the design plate. For this

comparison, we first have to place those two plates closely. We

can use either the method proposed by Shin et al. [4] and Bardis

et al. [5] or the registration method through the iterative closest

point (ICP) algorithm by Besl et al. [6]. In this work, we use the

ICP algorithm due to its simplicity in implementation. Given

the measured data set C and the design surface RA, Besls

registration method brings C to RA as closely as possible in

the least-squares sense, as illustrated in Fig. 4. The maximum

of the minimum distances from each point ofCtoRAis used as

a metric for the similarity between them. If this metric value isless than the user-defined tolerance, then we decide that we have

fabricated a plate of the desired shape and stop the iteration.

If not, we compute a set of heating lines and repeat the line-

heating process.

5. Measurement data processing

In the line-heating process, as shown in Fig. 1, we need

strain distributions through kinematics analysis between the

design and the current plates for heating line computation.

This computation can only be possible when the mapping

node data between two plates are available. The mapping

node data represent the information on the point on one platethat corresponds to the point on the other plate kinematically

(Ryu [11] and Ueda et al. [12]). At the beginning of the

manufacturing process, we obtain the mapping node data

between the design and the initial plates (in this case, the

unfolded plate) from the unfolding algorithm used in the

KOJEDO system. Based on such data, we compute the heating

lines and heat the unfolded plate to obtain a plate of curved

shape. Or we can use a different fabrication method such

as the cold forming process (Choi [13]) to bend the initial

unfolded plate to produce an intermediate plate. In either

way, however, the desired shape is hard to obtain and, during

this process, we lose the mapping information between the

design and the fabricated plates. Therefore, computation ofstrain distributions is not possible, preventing heating line

computation. Subsequently, we cannot perform iterative line-

heating fabrication. So the mapping node data between the

design and the intermediate plates should be estimated in order

to complete the iterative manufacturing process and, in this

section, we explain how the measured data for the intermediate

plate are processed to extract the mapping node information

between the desired and the intermediate plates for the heating

line computation for further iteration.

Fig. 3. (a) Portable CMM, and (b) manual measurement using CMM.


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874 J.S. Park et al. / Computer-Aided Design 39 (2007) 870881

Fig. 4. Schematic of registration ofC with respect toRA.

Fig. 5. Schematic of alignment ofCwith respect toRAand RB.

5.1. Alignment

In this section, we present a method for placing the measured

data points with respect to a reference coordinate frame for

mapping data extraction. This step is important in our method,

since heating line computation data are extracted from those

aligned plates and points.

5.1.1. Formulation

Let us assume that RA and RB are cubic B-spline surface

patches representing each shape of the design and the unfolded

or developed plates, respectively. But we point out that the

procedure proposed in this paper can easily be extended to

surface patches of other types. The developed surface, RB, is

obtained through the method presented in Ryu [11]. During

the development process, the design surface, RA, is discretized

and the node points ofRA are used to compute RB, producingcorrespondence information for mapping between the two

surfacesRAand RB. Then,RAand RBare positioned using the

corresponding node points of each surface to make the mass

centers of RA and RB match and the sum of the distances

between the corresponding points minimal. After that, we may

translate RA by a certain amount such that no part of RA is

placed below RB. This arrangement ofRA and RB is used to

determine a reference frame whose origin is positioned at the

mass center ofRB. This reference frame is used as a common

coordinate frame for the alignment ofCwith respect toRAand

RB, as shown inFig. 5.

The data points in C are placed in such a way that the sumof all the shortest distances from C to RA and RB becomes

minimal. It is formulated as an optimization problem whose

objective function is given as follows:

=n

i=1

|rAi(sci+T)|2 +|rBi(sci+T)|2

. (1)

Here, s , and Tare the scaling factor, rotational matrix and

translation vector, ci is the i th element ofC, and rAi andrBiare the closest points from ci to RA and RB, respectively. In

order to solve this optimization problem, we use the idea of

the ICP (iterative closest point) algorithm proposed by Besl

and McKay [6]. The objective function (1), however, has a

different form from that given in Besl and McKay [6]. So we

modify the ICP algorithm of Besl and McKay [6] to handle Eq.(1). The scaling factor can be resolved at the scanning stage

by calibrating the scanner to remove the scaling effect in themeasured data. So we set s=1. The overall solution procedureis depicted inFig. 6.

The procedure searches the optimum solution of Eq. (1)

iteratively and, when we find the rigid-body transformationsthat minimize Eq. (1) and apply them to C, then C, RA andRBare aligned, as illustrated inFig. 5.

5.1.2. Closest point computation

As shown in Fig. 6, the solution process starts withcomputing the closest point on a surface from a point. This

process is the most important step in the solution of Eq. (1),since it determines the performance and accuracy of the

alignment. Therefore, the closest point search algorithm mustbe efficient, accurate and robust. Here, byrobustnesswe mean

the capability that, in at any situation, the algorithm always

gives an answer without failure.The closest point on a surface from a point in three-dimensional (3D) space can be established by locating a point

on the surface which gives the minimum distance from thegiven point. We need to consider the following three cases to

compute the minimum distance between a point and a surface(Patrikalakis and Maekawa [14]):

1. the minimal distances from the interior domain of thesurface;

2. the minimal distances from the four corner points;3. the minimal distances from the four edges.

Then we find the smallest value of those computed minima

as the minimum distance between the point and the surface.Here we denote by the footpointthe point on a surface which

yields the minimum distance to a given point in 3D space.For simplicity, let us denote the computation algorithm by an

operatorCL(). Then, we can write Y= C L(C, S), whereY isa set of footpoints on S ofC.

5.1.3. Computation of rigid-body transformation

The solution procedure for computing the translation vector

and the rotation matrix for the optimization of Eq.(1)is similarto that in Besl and McKay [6]in nature, but we have to consider

the effect of the additional term in Eq. (1) in the solutionprocess. In this section we describe how to compute the rigid-

body transformation, which minimizes Eq.(1).It is useful to represent data in terms of the centroids defined

byrA= 1nn

i=1 rAi ,rB = 1nn

i=1 rBi andc= 1nn

i=1Ci(Besl and McKay [6]). Then, since rAi = rAi rA, rBi =rBi rB andci= ci c, we can rewrite Eq.(1)as follows:

=n

i=1

rAi+ rAs(ci+ ci )T2

+rBi+ rBs(ci+ ci )T2

,

=n

i=1

rAisciTA

2

+rBisciTB

2

, (2)


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Fig. 6. Flowchart of registration.

where

TA= T+sc rA and TB= T+sc rB . (3)In Eq.(2), can be rewritten as

=n

i=1

rAisciTA2 ,=

ni=1

rAi ci 2 a1

2n

i=1TA

rAisci

b1+

ni=1

|TA|2

c1

. (4)

Similarly, expression in Eq.(2)is given by

=n

i=1

rBi ci 2

a2

2n

i=1TB

rBisci

b2

+n

i=1 |TB |

2

c2

. (5)

In order to minimize Eq. (2), we have to minimize and

at the same time. A careful examination reveals that the terms

b1 and b2 in Eqs. (4) and (5) vanish, sincen

i=1 r

Ai = 0,ni=1 r

Bi

= 0 and ni=1ci= 0. The terms a 1, a2, c1 and c2in Eqs.(4) and(5) cannot be less than zero. Therefore, Eq.(2)

is minimized when they are minimized.

It is noticed that the terms a1 and a2 in Eqs. (4) and (5)

contain no translation but rotation. In order to minimize the

terms a1 and a2 in Eqs. (4) and (5), we use a symmetrical

expression for the error term defined as follows (Horn [15]):

ei=1

sri

sci , (6)

wheres is the scaling factor. Then we have

e=n

i=1|eAi |2 +

ni=1

|eBi |2 ,

= 1s

ni=1

rAi 2 +sn

i=1

ci 2 2n

i=1rAi ci

+1

s

n

i=1 r

Bi 2

+s

n

i=1 ci

2

2

n

i=1 r

Bi

ci ,

= 1s

ni=1

rAi 2 + 1sn

i=1

rBi 2 +2sn

i=1

ci 2

2n

i=1

rAi+rBi

ci

sub

. (7)

In order to minimize e, we have to maximize

sub

=2

n

i=1 rAi

+rBi

ci ,

= 4n

i=1

rAi+rBi

2

ci ,

= 4n

i=1

rAB i

ci , (8)

whererAB i = rAi +rBi

2 . The best rotation to maximizesub in

Eq.(8)can be obtained in diverse ways. In this work, we use

the quaternion-based method proposed by Horn [15].

First we compute a matrix of sums of products:

M= ni=1

ci rTAB i . (9)

This matrix M contains all the necessary information for

finding the optimum rotation matrix in the least-squares sense

to maximize Eq.(9):

M=Sx x Sx y Sx zSyx Syy Syz

Szx Szy Szz

, (10)

where Suv =n

i=1cui r

vAB i

, u, v = x,y,z when ci =(cxi , c

yi , c

zi ) and r

AB i

= (rx AB i , r

y AB i , r

z AB i ). Then, using

the components in the matrixM, we construct a symmetric 44matrixNgiven inBox I.

The unit eigenvector q= [q0, q1, q2, q3]T correspondingto the maximum eigenvalue of the matrix N is the optimum

rotation which maximizes sub. The eigenvectorqis given

in quaternion form. So the rotation matrix is represented in

matrix form as follows:

=q

20+q21q 22q 23 2(q1q2q0q3) 2(q1q3+q0q2)2(q1q2+q0q3) q20+q22q 21q 23 2(q2q3q0q1)2(q1q3q0q2) 2(q2q3+q0q1) q20+q23q 21q 22

.

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N=

Sx x+Syy+Szz SyzSzy SzxSx z Sx ySyxSyzSzy Sx xSyySzz Sx y+Syx Szx+SxzSzxSx z Sx y+Syx Sx x+SyySzz Syz+SzySx ySyx Szx+Sxz Syz+Szy Sx xSyy+Szz

.

Box I.

For c terms in Eqs. (4) and (5), they are set to be zero,

respectively, to minimize Eq. (2). In other words, we have

T= rA sci andT= rB sci . From these two equations,we can derive a translation vector which minimizes Eq. (2):

T= rA+ rB2

sc. (12)

5.1.4. Algorithm

For the simplification of algorithm description, we use

symbols as follows:Crepresents the measured points, as given

in Section3, ()an operator computing the translation vector

and the rotation matrix when C and the footpoints of C aregiven, w the transformation operator (translation and rotation),

and dthe distance measure used in Besl and McKay [6]. The

subscript is an index of iteration. Then the pseudo-code for the

alignment is described as follows (Besl and McKay[6]):

1. LetC0=C.2. Compute the closest points:Yk= C L(Ck, YaUYb).3. Compute the translation vector and the rotation matrix:

(wk, dk)= (C0, Yk).4. Apply the translation vector and the rotation matrix to C0:

Ck+1=wk(C0).5. Check if dk dk+1 < is satisfied. If so, terminate the

iteration. If not,k

=k

+1 and go to 1.

Here, is the user-defined tolerance for termination. For this

iteration, the initial position ofC from which the optimization

process starts is important to reach the global optimum value.

Since C has been obtained through the scanning process,

no information is available for the selection of the initial

position. Instead we have to find out relations between RAand C for the initial position selection. In practice, we have

geometric information about the plate that will be line heated.

Therefore, after extracting corner points from C, we can find

the corresponding corner points betweenCandRA. Using such

correspondence information on the corners of the plate, we

placeC close toRAand start the optimization process.

5.2. Mapping data extraction

Mapping data extraction starts with approximation of the

measured data C using a B-spline representation. The point

set C and the approximated surface are already aligned with

RA andRB. Then we estimate correspondence points between

the approximated surface andRA. In this section, we present a

detailed procedure for extracting mapping data.

5.2.1. Measurement data approximation

We approximate C by using a B-spline surface patch inthe least-squares sense. For approximation, we need two steps:

parametrization and surface reconstruction.

5.2.1.1. Parameterization. When a parametric surface is used

for approximating points in 3D space, the most importantstep is to assign appropriate parametric values for each point.

Since parameterization directly affects the quality of the

reconstructed surface, we have to choose a parameterizationmethod carefully for surface reconstruction. Many different

ways of discrete point parameterization have been reported

in Hoschek and Lasser [16]. In this paper, we consider twoparameterization methods: the chordal and the base surface

methods. The chordal method approximates the arc length

and uses it to estimate the parameter of each point. It isa well-known method in curve/surface reconstruction. The

base surface parameterization method proposed by Ma andKruth[17] uses a base surface, which has as much similarity

as possible to the geometric shape represented by input points.

The points are orthogonally projected on the base surface andthe parametric values of the projected points on the base surface

are taken as the parametric values of the discrete points. In

our process we use either of the two methods, depending onthe structure of the measured data. If the input points are well

organized, then we use the chordal method. Otherwise, we use

the base surface method. The former is much faster than thelatter, since the latter requires the computation of parameter

estimation by projecting a point onto a surface. However, if

unorganized input points are provided, the latter in generalgenerates better parametrization (Ma et al. [17]).

5.2.1.2. Surface reconstruction. Once we have assignedparametric values to each discrete point, we can find the control

points of a B-spline surface which approximates the points

by using the least-squares method or the skinning procedure.This reconstruction problem is one of the standard geometric

operations, and we refer the readers to Shin et al. [ 4] or Piegl

and Tiller [18] for further details. We denote the approximatedsurface byRD.

5.2.2. Estimation of correspondence points

A plate has been manufactured to have a shape of RD,

which has been reconstructed from the measured points of

the physical plate. Therefore, we do not have geometriccorrespondence information between RD and RA to derive

kinematic relations for heating line computation. In order to

establish such correspondence betweenRAand RD, we exploitan assumption made between RA and RB in the heating line

computation (Ryu [2]); each corresponding pair of points onRA and RB moves along a straight line during deformation,

as illustrated inFig. 7. We compute the intersection between

each straight line and RD, and use those intersections as thecorrespondence points betweenRDandRA. When we compute

the intersection between a line andRD, we have three different

cases:


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Fig. 7. Schematic of relation between RA, RB and RD: (a) the measured

surface is bigger than design surface; (b) the measured and the design surfaces

have the same size; and (c) the measured surface is smaller than the design

surface.

Case 1: RD is sufficiently large that we cannot find thecorrespondence point on the boundary ofRD, as shown in

Fig. 7(a).

Case 2:RDis of adequate size along the boundary ofRD. We

can establish correspondence toRA, as shown inFig. 7(b). Case 3: RD is smaller than RA orRB such that there exists

no intersection between the straight lines and RD near the

boundary, as shown inFig. 7(c).

For Case 1, the excessive portion of RD is treated as

the margin of the plate, which is not used in the heating

line computation. When we encounter Cases 1 and 2, we

extract correspondence points and use them for subsequent

computation. A problem occurs when Case 3 happens. Since

we cannot compute the intersection between a straight line and

RD, we find a point on RD which yields the shortest distance

to the line and use it as a correspondence point. However,

in this case the bending and in-plane strain values are notproperly computed near the boundary. So we may exclude the

correspondence values on the boundary during the heating line

calculation.

The intersection between a line and a B -spline surface patch

can easily be computed by using Newtons method. But we

should consider the case where Newtons method fails when

no unique root exists. Even if such a case is encountered,

the algorithm should not crash but estimate the best value for

correspondence. Therefore, Newtons method is not appropriate

for our purpose. Instead, we use optimization, which finds

a point on the surface RD giving the minimum distance to

a straight line. If the minimum distance is zero, then it is

Fig. 8. The design plateRA.

the intersection point. A procedure to compute the minimum

distance point between a line and a B-spline surface based on

optimization is as follows:

Suppose that we have two points in 3D space, a =(xa ,ya,za)and b= (xb,yb,zb), and a parametric surface RDwithuandvas its parameters. Then we find uandvofRDsuch

that the expression = |RDa|2 + |RDb|2 is minimized.For this computation, we use a quasi-Newton method (Press

et al. [19]). This requires the first-derivative expressions of

with respect tou and v , which are given as follows:

u=2 RD

u(2RD(a+b)) ,

v=2 RD

v(2RD(a+b)) .

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6. Example

In this section, we demonstrate our algorithms with an

example. For this computation, we use a Windows XP based

computer with a 1.7 GHz CPU and 2 GB of RAM. The plate

shown inFig. 8,which is part of a real ship hull, is the designplateRArepresented by a cubic B -spline surface patch of 6 6control points. It is a doubly curved surface. Fig. 9shows the

Gaussian curvature (K) distribution over the surface. The upper

part of the plate in the figure has positive Gaussian curvature,

whereas the lower part has negative Gaussian curvature. The

dimensions (lengthwidth) of the plate are 949 mm799 mmand the weight is 172 kg. Here, since the thickness (29 mm) is

much smaller than the width or length of the plate, we represent

the plate as a surface with no thickness. This assumption is

consistent with those made for the unfolding and the heating

line computation algorithms by Ryu [2,11]. The plate chosen

in this example is one of the most complex shapes used in the

shipyard. Due to its complexity, it takes one or two days to formthe shape of the selected plate when manufactured, depending

on the skills of each worker.

Fig. 10 shows the developed plate RB obtained from the

design plate. It is represented in a planar cubic B -spline surface

patch of 66 control points.The developed plate was formed, and a curved plate has been

fabricated and measured to produce a set of points C as shown

inFig. 11(a). The number of the measured data points is 1433.

Then the pointsCare approximated in a cubic B -spline surface

RDwith 66 control points, as shown inFig. 11(b).The measured and design surfaces are compared to check

the similarity in shape. The ICP algorithm is used to place the


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Fig. 9. The Gaussian curvature( K)distribution of the design plate RA. The upper part has the positive Gaussian curvature and the lower part has the negative

Gaussian curvature.

Fig. 10. The developed plateRB.

Fig. 11. (a) The measured dataCof the fabricated plate. (b) The approximated

surfaceRD.

measured data closely with respect to the design plate, as shown

inFig. 12. The maximum and the mean square errors betweenthem are 47.02 and 12.41 mm. If these values are smaller than

the user-defined tolerance, then we decide that the desired shape

has been fabricated and stop the iteration. In this example,

we assume that the desired shape has not been obtained and

continue the fabrication process.

The three surfacesRA,RB, andRDinFigs. 8,10and11are

then aligned with respect to the reference frame. The surface

RA is translated in the z direction by 600 mm and RD is

translated in thez direction by 300 mm to position it above RB.

Then, using the procedure in Section5.1,the measured surface

RD is positioned as illustrated inFig. 13.The alignment takes

13 s, with a tolerance of 0.0001.

Table 1

Maximum and average distance between the manufactured and the design

plates (unit: mm)

Iteration number 1 2 3 4 5

Mean square error 12.41 12.24 9.53 6.15 5.44

Maximum distance 47.02 62.51 54.66 31.52 23.07

Once the alignment is performed, we need to estimate themapping node data information between RD and RA, whichwill be provided as input to the heating line computation.

Correspondence points on RA, RB and RD are depicted in

Fig. 14.Fig. 14(b) shows the magnified image of the alignedplates marked in the rectangle in Fig. 14(a). The number of

intersections is 357 and this computation takes 0.19 s.Using the correspondence points between RA and RD, we

can compute the heating paths, as shown inFig. 15.In this experiment, the induction heating method was used.

A clearance of 4 mm was maintained between the heating coil

and the plate. The induction heating coil used in the heatingprocess generated 90 kW of heat and moved at a variable

speed from 5 to 30 mm/s, depending on the amount of angulardeformations to be induced.

In Table 1, the changes in the average and maximumdistances with the number of iterations of the line-heating

process are summarized. InFig. 16,the error distributions fromthe first to the fourth iterations are provided, andFig. 17shows

the result after the fifth iteration. All errors are plotted with

respect to one error scale given in Fig. 17. As the figuresindicate, errors have decreased and relatively large errors occur

near edges and corners.

7. Discussions

In this section, we discuss the proposed method in termsof two different perspectives, time and convergence, and we

analyse the errors obtained during the experiment.

7.1. Timescale

The time taken to complete one cycle of the line-heating

process consists of the time for preparation, measurement,computation and heating. Most of the time of the cycle

is attributed to measurement and heating. For example, themeasurement and heating times of the plate shown in Section 6

are 10 and 60 mins, respectively, whereas it takes less than a

minute for computation, which is only a small fraction of the

total line-heating production process.


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Fig. 12. Registration of the design and the measured surfaces.

Fig. 13. The design and its developed surfaces are aligned with the measured

surface.

7.2. Convergence of iteration

The proposed algorithms and procedures form a loop in

the line-heating process and we need to assure whether the

iterative fabrication process yields the desired shape. Given

RA and RD, we can compute the heating lines for RD. After

heatingRDalong the computed heating paths, we have a more

deformed shape denoted by RD. Then, ideally, the followingshould always be true:

d(RA, RD)


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Fig. 16. Error distributions from iterations 1 through 4.

7.3. Error analysis

We found that the maximum errors in Table 1 occur at

one of the four edges. Those error values are relatively large,

considering the accuracy used in the real ship manufacturingprocess (for a shipyard, a tolerance of less than about 10 mm

is allowed). Localization is performed in the least-squares

sense, but the corresponding edges between the desired and the

measured surfaces are not well aligned, which contributes to

the large errors given in the table because of this misalignment.

Even though the overall measure does decrease, locally we may

end up with a large error along one of the edges. Therefore,

by modifying the registration method by imposing constraints,

we may prevent such errors caused by the misalignment from

happening. Moreover, for a plate of saddle type, we may

need to turn over the plate for more iteration at the most

appropriate moment. The determination of such an optimalturn-over moment was not reflected in the current proposed

system. All these being considered, then the error will decrease

and more accuracy will be achieved through the proposed

process.

8. Conclusions

In this paper we have proposed algorithms and procedures

for measuring a curved plate, comparing it with the design

plate, and extracting information necessary for computing

heating lines. Through an example with a real hull piece, we

observe that the proposed method can be used to fabricate a

curved plate to the desired shape. The results of this paper

Fig. 17. Error distribution of iteration 5.

demonstrate the potential that they can serve as theoreticalbuilding blocks for developing an automatic line-heating

system which will overcome critical bottlenecks arising in

ship manufacturing. We would like to stress that the proposed

procedure can achieve such accuracy in the fabrication of

complex curved plates, which has not been possible to obtain

by using any line-heating system reported in the literature. To

improve accuracy, skilled workers may adjust the shape with

minimal effort. This would mean that we can dramatically

minimize the man-hours necessary to fabricate such curved

plates using the proposed system. For example, we can make

the system fabricate a curved plate overnight, and the plate is

taken care of by skilled workers the next day.


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At a shipyard they have their own specific manufacturing

practices. So the proposed method should be tailored to each

shipyard in order to be used in the real fabrication process. This

customization needs extensive data and workers experiences

of each shipyard, which should be incorporated in the proposed

method. For example, in order to determine the tolerance for

similarity between the manufactured and the design plates,we have to investigate how a current manufacturing method

handles it in practice and choose the right value for the tolerance

that reflects the current manufacturing practice. Therefore, we

have to evaluate and adjust the proposed method thoroughly

before applying it to the manufacturing process of a particular

shipbuilding company.

In future work, we will continue research on improving the

accuracy of automatic curved plate fabrication. Recommended

topics include: improvement of the registration method for

surface comparison; theoretical investigation of plate turn-over

for saddle-type plate fabrication; enhancement of the existing

heating line computation by considering various uncertainties

existing in the real fabrication process with extension to thefabrication of plates of more general shape; and accuracy

control, reflecting real shipbuilding practice

Acknowledgements

This work was supported in part by the Korea Science

and Engineering Foundation (KOSEF) through the National

Research Laboratory Program, funded by the Ministry of

Science and Technology (no. M10300000213-05J0000-21310),

and in part by the Regional Industrial Technology Development

Program of the Ministry of Commerce, Industry and Energy

of Korea (no. 10024292-2006-12, led by Samsung HeavyIndustries).

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