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Int J Adv Manuf Technol (2002) 19:209216 2002 Springer-Verlag London Limited

A Generic Part Orientation System Based on Volumetric Error inRapid Prototyping

S. H. Masood and W. Rattanawong

Industrial Research Institute Swinburne, Swinburne University of Technology, Hawthorn, Melbourne, Australia

The paper presents the development of a generic part orien-

tation system for rapid prototyping by considering the issue of

volumetric error encountered in parts during the layer by

layer building process in a rapid prototyping (RP) system. An

algorithm is developed that slices the part with horizontal

planes and computes the volumetric error in the part at

different orientations by rotation about user-specified axes. The

system recommends the best orientation based on the least

amount of volumetric error in the part. The system has been

tested using several examples of simple and complex parts.

The generic part orientation system is believed to be first of

its kind based on a volumetric error approach and will be

useful for RP users in creating RP parts with a higher level

of accuracy and surface finish, and also for intelligent process

planning in rapid prototyping.

Keywords: Fused deposition modelling; Part orientation; Rapid

prototyping; Volumetric error

1. Introduction

Commercial rapid prototyping (RP) systems, which build 3D

objects using the layer by layer building process, introduce an

error in the part because of the amount of material used

compared to the volume specified by the computer-aided design

model. The magnitude of this error, normally called a staircase

error, varies with the nature of the surface of the part. Inclined

and curved surfaces show staircase effects more predominantly

than other surfaces. The orientation at which the part is built,

can have a significant effect on the quality of various surfaces

of the part because of this effect. The orientation of the partalso affects other factors in some rapid prototyping processes

[1] such as the build time, the complexity of the support

structure, shrinkage, curling, trapped volume, and material flow.

The undesirable staircase feature is most often apparent on

Correspondence and offprint requests to: Assoc. Prof. S. H. Masood,CAD/CAM/CAE Group, Industrial Research Institute Swinburne, Swin-burne University of Technology, Hawthorn, Melbourne, 3122 Australia.E-mail: smasoodswin.edu.au

sloping or curved surfaces, and it is impossible to eliminate it

completely in most cases. However, the effect can be reduced

by decreasing the layer thickness and by orienting the part so

that the effect of the overall staircase error is greatly reduced.

The determination of an appropriate orientation of the part

during the building process has therefore been an important

issue in rapid prototyping for improving the accuracy and

quality of parts.

The need for the development of an effective part orientation

system can also be considered from the point of view of

emerging trends in intelligent rapid prototyping. At present,

the rapid prototyping procedure involves a series of discrete

independent functions: creation of a CAD model, process plan-

ning for an STL file, fixture design to suit part shape and

building, verification and inspection, and finishing operations

if required. These require a high level of skill, judgement, and

experience on the part of the designer, in order to produce an

accurate and acceptable prototype. Therefore, the development

of an effective part orientation system for rapid prototyping isan essential requirement for intelligent RP process planning and

for the future development of an intelligent rapid prototyping

environment [1,2].

Several methods are being developed to determine an appro-

priate orientation based on different criteria. Since orientation

reduces, but does not completely eliminate, the staircase effect,

strategies to reduce the staircase effect have also been

developed. Some work has also been done on the orientation

problem related to the minimisation of support structures

required during building in some RP processes.

Allen and Dutta [3] developed a method for determining part

orientation based on support requirements. In their algorithm, if

two orientations require support structures with equal surface

areas of contact, the orientation in which the object has a

lower centre of mass, is chosen. Frank and Fadel [4] proposed

an expert system tool that considers the various parameters

that affect the production of the prototype and recommends

the best direction for building the part based on both the users

input and on a decision matrix implemented within the expert

system. Sreeram and Dutta [5] developed a method to deter-

mine the optimal orientation based on variable slicing thickness

in layered manufacturing for a polyhedral object, which aims

to achieve a stable equilibrium position of the object, the

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210 S. H. Masood and W. Rattanawong

minimum number of slices produced, and the minimum ratio

of the total staircase area to the total surface area of the object.

There have been several papers on the orientation problem

specifically for the stereolithography (SLA) rapid prototyping

process. Kim et al. [6] developed an optimisation technique for

optimal part orientation for the SLA process by considering

an objective function related to the volume of trapped liquid

resin, the total height of the part in the build direction and

the area of surfaces with staircase protrusions. Cheng et al. [7]developed a procedure for optimisation of the part building

orientation in SLA by assigning weights to various surface

types affecting part accuracy. Xu et al. [8] adopted a variable-

thickness slicing technique in the SLA process to obtain the

optimal orientation considering the building time, accuracy,

and part stability. Lan et al. [9] determined fabrication orien-

tations for the SLA process by establishing decision criteria

and algorithms considering the surface quality, build time, and

the complexity of the support structure. Bablani and Bagchi

[10] developed a method for calculating the process planning

error, process error, and the number of layers in different

orientations, achieved by rotating the given part about a user-

selected axis for the stereolithography process. Hur and Lee

[11] developed an algorithm to calculate the staircase area,

quantifying the process errors by the volume supposed to be

removed or added to the part, and the optimum layer thickness

for the stereolithography system, which can handle variable

layer thicknesses. For another rapid prototyping process, selec-

tive laser sintering (SLS), Thompson and Crawford [12]

developed computational tools for optimising the orientation

of a part to be built on the basis of part strength, surface

alaising error, volumetric supports, and build time.

Very little work seems to have been done on tackling the

part orientation problem, based on minimisation of the volu-

metric error, which exists in all layer by layer based rapid

prototyping processes. However, the volumetric error approach

is especially relevant to the fused deposition modelling (FDM)rapid prototyping process, where deposition of material from

a nozzle creates the volumetric errors more significantly com-

pared to other RP processes.

This paper presents the development of a generic part orien-

tation system for rapid prototyping of solid parts of any

complexity by considering the volumetric error encountered in

parts during the layer by layer building process. An algorithm

is developed that slices the part with horizontal planes and

computes the volumetric error of each layer using the complex

shapes of the resulting contours of each layer. The system

then determines the volumetric error in the part at different

orientations by rotation about user-specified axes and rec-

ommends the best orientation, based on the least amount of

volumetric error in the part. The system has been tested with

several examples of simple and complex parts. The method-

ology has been verified by volumetric error computation for

several parts based on the mathematical equations of solid

primitives, a combination of basic primitives and also by direct

experimental measurement of the volumetric error on physical

parts created using the fused deposition modelling rapid proto-

typing system. The resulting generic part orientation system

is believed to be first of its kind based on the volumetric

error approach.

2. Methodology for Volumetric ErrorComputation

The volumetric error in rapid prototyping processes in general,

and in the FDM process in particular, can be defined as the

difference in the volume of material used in building the part

compared to the volume specified by the CAD model. The

total volumetric error in a part will be different at differentorientations because of the layering building process. If the

volumetric error in a layer in the RP part can be computed,

then the volumetric error in the part will be the sum of all

the volumetric errors in the layers in the part. A generic

algorithm has therefore been developed that computes the

volumetric error, in a part of any shape, by summing the

volumetric errors in each layer. The algorithm makes use of

the STL model of the part, slices the part by infinite planes

normal to the build direction, and then determines the best

orientation based on the least amount of volumetric error in

the part.

The algorithm provides the user with two options for orient-

ing the part about different axes. In one option, the user can

rotate the part about one axis only (x- or y- or z-axis) todetermine the best orientation angle. In the second option, the

user can rotate the part about two axes (first about the z-axis,

then about the x- or y-axis). We shall describe the algorithms

and procedures for the second option. Figure 1 shows a flow-

chart describing the algorithm for rotation about two axes.

When two axes are selected by the user, the algorithm works

in the same manner as for one axis, except that the part is

rotated first about the z-axis from a starting angle (i1= 0), and

then about the second axis (x or y) from 0 to 90 with the

specified step increment of rotation (d).

The procedure starts with the user providing the STL file

of the part in the ASCII format. The user inputs the values

of the step angle of rotation and the layer thickness and selectsthe axis/axes of rotation. If the user selects small values of

the step angle of rotation and the layer thickness, a higher

accuracy is expected for the best orientation angle. At the

start, when the STL file is loaded into the program, the model

is placed in a normal position as determined by the user and

is set at 0 orientation. If it is not in the normal position, the

user has to modify the model using the CAD software. The

program displays the triangular facets of the STL model. At

each orientation, the model is sliced from bottom to top by

horizontal planes separated by a distance equal to the selected

layer thickness. This creates a large number of intersection

points (or vertices) between the facets and the plane. The

vertices thus represent the part in that orientation, and the unit

normal vectors represent the direction of tessellated facets withrespect to the z-axis. Initially, all the vertices are an unordered

collection of intersection points in the database. The program

uses a subalgorithm, called a contour processor, which sorts

out these vertices into groups with the same z-coordinate

values. Then contours with the same z-values are created. The

vertices in each contour must be re-ordered by the contour

processor to determine the area bounded by the contour poly-

gon. Thus, the contour processor used in the program, modifies

and sorts out the vertices automatically to generate horizontal

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A Generic Part Orientation System 211

Fig. 1.The algorithm for determining the optimum orientation abouttwo axes.

contours. When the part is oriented about any axis in a

direction specified by the user, the program first determines

the minimum and maximum height of the part with respect to

the x,y-plane. The horizontal intersecting planes are then gener-

ated to cut the part. Using the coordinate values of the vertices

of each tessellated triangle, the program determines the slopes

and the equation of the lines of each triangle. The coordinates

of intersecting points between these lines and the horizontal

planes are then determined. These intersected points or vertices

are then sorted out by the contour processor in terms of the

same z-coordinate values from the minimum to the maximum

height of the part in the z-direction. Sorting is unidirectional,

either in a descending or ascending order. Too many variations

in sizes of triangles coupled with model complexity, often

complicate the procedure of sorting. Grouping of vertices is

then performed considering the minimum and the maximum z-

coordinate values as the grouping criteria. The intersection

points are joined by straight lines, and the program then

determines the area bounded by the polygon of each contour.

These contour areas are used to determine the volume differ-

ence generated by the two horizontal contours in a layer, which

are then used to determine the volumetric error in that layer.

Considering the horizontal layers intersecting the tessellated

part, the horizontal area in each contour is given by

Ai =1

2

n1

i=1

xiyi+1 xi+1yi+ 12

[xn y1 x1yn] (1)

where,

n = the number of intersection points in each layer

1 i n

xi,yi = the coordinates of intersection points

The volumetric error (VE) is given by

VE= |k

i=1

(Ai+1 A i)t/2| (2)

where,

k = the number of layers

t = layer thickness

The volumetric error is computed at each angle. When the

orientation about the second axis is completed at 90, the

orientation about the first axis is incremented by a step of

rotation d, and this is again followed by all the steps of

rotation about the second axis. This process continues until

the end angle of rotation about the first axis is reached at

180. Thus, each step rotation about the z-axis is followed by

step rotations about the second axis (x or y) from 0 to 90.

Because of the tessellation effect and the contour of the

surfaces, the values of the volumetric error are not exact.

When a part, which is symmetrical about the z-axis, is rotated

about the z-axis, the components of the unit normal vector

along the z-axis do not change, and this keeps the volumetric

error constant; but this is not the case if the part is of non-

symmetric geometry about the z-axis. During the orientationprocess, the resulting volumetric errors are compared with

previous values at each step of the orientation. The algorithm

then determines the minimum of all the volumetric errors and

then recommends the best orientation angle for building the

part based on this minimum volumetric error.

Using the above generic algorithm for the volumetric error

of parts, a software implementation of the methodology was

undertaken, employing the Mathematica software package using

a graphical user interface [13]. The resulting system is called

the generic part orientation system, often referred to as the

generic system in subsequent sections of this paper. It is

called generic because it is generally applicable to any solid

part of any complexity for determination of the best part

orientation for rapid prototyping. The program computes the

values of the percentage relative volumetric error (based on

the ratio of volumetric error to the total volume of the part)

for any solid complex part for different orientation angles. The

program accepts any CAD model in STL format. The user can

specify the values of the step angle of rotation, select the

axis/axes of rotation in the program, and specify the layer

thickness to be used on the RP machine. The program simulates

the orientation of the part graphically at specified intervals and

displays a table and a graph of the variation of volumetric

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212 S. H. Masood and W. Rattanawong

error with orientation angle. The program recommends the

optimal orientation of the part based on the minimum volu-

metric error. The program provides the user with a powerful

tool to select an appropriate angle of orientation based on the

minimum volumetric error, to ensure enhanced overall quality

and accuracy of the RP parts.

3. Verification of Generic Part OrientationSystem

The accuracy and proper functioning of the generic part orien-

tation system can be verified by comparing the volumetric

errors of standard primitives and parts obtained analytically

with the results obtained from the generic system for the

same primitives and parts. For this purpose, an analytical

part orientation system was developed, which determines the

optimum orientation for primitives and simple parts made of

primitives using the volumetric error approach [14]. The ana-

lytical system used exact mathematical equations for each

primitive to work out the volumetric error in each layer at

different orientations and was therefore limited to applicationsto primitives or simple parts made of primitives. The generic

system, described in this paper, however, can determine the

part orientation for any complex part using the algorithm

described in Section 2.

For verification of the generic part orientation system, the

following types of primitives were rotated about the x-axis and

the relative volumetric errors were computed and compared

with the values obtained analytically from the analytical part

orientation system: cone, pyramid, rectangular box, cylinder,

and hemisphere.

Figure 2 shows an example of a solid model and a tessellated

model of a pyramid at the initial angle of orientation. The

percentage relative volumetric error of the pyramid was com-

puted for the generic part orientation system and for the

analytical part orientation system for orientation about the x-

axis in steps of 5 ranging from 0 to 90 at a layer thickness

of 1 mm. Figure 3 shows the resulting graphs of the variation

of relative volumetric errors with the angle of orientation. For

the case of the pyramid, the recommended part build orientation

is found to be at 64 at which the analytical and generic

values of the percentage relative volumetric error are found to

be the minimum. The maximum volumetric error of the pyra-

mid will occur at 5. The close agreement and identical trends

Fig. 2.(a) The solid and (b) the tessellated models of a pyramid.

Fig. 3.Comparison of variation of relative volumetric errors for a pyra-mid.

of the two curves in Fig. 3 verify the accuracy of the generic

part orientation system.

4. Experimental Verification of the PartOrientation System

The computed values of volumetric errors obtained from thegeneric part orientation system can also be verified by compari-

son with experimentally determined volumetric errors for selec-

ted parts and primitives. This was carried out by measuring

the area of each section of the staircase in each object produced

on the FDM machine using a profile projector, by which the

FDM part was magnified to determine the cusp height and the

layer thickness. Plastic prototypes of primitives such as a

pyramid, a cone, a cylinder, a cube, and a hemisphere were

built at different orientations ranging from 0 to 90 at 15

intervals using a fused deposition modelling system. Thus,

seven identical parts for each object were produced at seven

different orientations. The layer thickness used in the FDM

machine was taken as 0.25 mm. For further illustration and

comparison purposes, the results of the percentage relativevolumetric error were also obtained using the mathematically

derived analytical part orientation system. Table 1 shows the

results of the experimental, analytical, and generic values of

volumetric errors for the selected orientation angles for a

pyramid. Figure 4 shows the same results as plots of the

variation of the relative volumetric errors for the pyramid. It

is noted that the generic values compare well with the analyti-

cally and experimentally determined values for the pyramid.

Table 1.Results of generic, experimental, and analytical percentagerelative volumetric error in a pyramid.

Orientation Percentage relative volumetric error (%)(degrees) Pyramid (1, w, h = 10 mm)Generic Experimental Analytical

0 3.57 4.61 3.6615 6.53 6.61 6.7330 5.04 6.46 5.8445 4.57 5.78 5.1760 4.18 4.31 3.9975 4.61 7.51 4.8990 3.57 5.60 3.75

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A Generic Part Orientation System 213

Fig. 4.Comparison of generic, experimental, and analytical percentagerelative volumetric error in a pyramid.

This further verifies the accuracy of the methodology used in

the generic system in this study.

5. Application to Complex Parts

The generic part orientation system is a highly sophisticated

and powerful tool for determining the optimum part build

orientation for the rapid prototyping of high-quality parts byminimising the volumetric error. The system is designed to

work for solid parts of any complexity or shape.

The application and usefulness of the generic part orientation

system have been illustrated by applying it to determine the

optimum orientation in several complex parts, which include

a toy car body, a perfume bottle, a human torso, a disk cam,

a compound shaft, and a hyperboloid. These parts were selected

from a range of applications including mechanical components,

parts from the toy industry, cosmetics industry, medical appli-

cations, and solid geometry. For each part, the best orientation

was determined for rotation about one axis as well as for

rotation about a combination of rotation about two axes, and

using different layer thicknesses.In this paper, the application of the generic part orientation

system is illustrated by considering two of these complex parts:

the toy car body and the hyperboloid.

5.1 Orientation of the Toy Car Body

Figure 5 shows the solid model and the tessellated model of

a toy car body at the initial angle of orientation. The STL file

of the toy car body was found to have 3103 facets. The

percentage relative volumetric errors of the car body were

computed and displayed by the generic system for orientation

about the x-axis in steps of 5 ranging from 0 to 90 for

Fig. 5.(a) The solid and (b) the tessellated models of a toy car body.

three layer thicknesses. Figure 6 shows the resulting graphs of

variation of the relative volumetric errors with the angle of

orientation for the three layer thicknesses. The best orientation

angle recommended by the system is 90 for three layer

thicknesses. It is found that the percentage volumetric error

values at a layer thickness of 0.3 mm are higher than those at

a layer thickness of 0.5 mm, mostly at a higher angle of

rotation. It also shows that features can be missed when using

some layer thicknesses. Because we cannot guarantee that allthe dimensions in the parts are multiples of a fixed thickness,

it is likely that the slice will miss some small features, or fail

to trace the material deposition at the beginning or the end

position of a feature.

Using the option for rotation about two axes, results were

also obtained for orienting the toy car body first about the z-

axis at selected step angles and then about the x-axis ranging

from 0 to 90 for the given z-axis. Table 2 shows the results

of percentage volumetric error for the toy car body, which is

rotated first about the z-axis at 45 and 60 and then about

the x-axis at layer thicknesses 0.5, 0.3, and 0.25, respectively.

Figures 7 and 8 show the variation of the relative volumetric

errors for the toy car body as obtain from Table 2. When the

toy car body was oriented at 45 and then about the x-

axis for three layer thicknesses, the recommended part build

orientations are found to be at 90, 55, and 90, respectively.

In the case of orientation first about the z-axis at 60 and then

about the x-axis for three layer thicknesses, the recommended

part build orientations are found to be at 85, 50, and 85,

respectively, for which the generic volumetric error is found

to be the minimum.

It was found that when the toy car body is oriented first

about the z-axis at 0, 45, and 60 and then about the x-axis

for the three layer thicknesses, the percentage volumetric error

values are slightly different. There are two error issues. The

first issue is the translation process. The points that form a

Fig. 6.Variation of relative volumetric errors for a toy car body.

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214 S. H. Masood and W. Rattanawong

Table 2. Percentage volumetric error for a toy car body, rotated aboutthe z-axis at 45 and 60 and then about the x-axis at three layer thick-nesses.

Orientation Percentage relative volumetric error (%)about Rotation about z-axis 45, Rotation about z-axis 60,x-axis layer thickness (mm) layer thickness (mm)(degrees) 0.5 0.3 0.25 0.5 0.3 0.25

0 4.5823 4.4835 3.5919 4.7425 4.4701 3.15085 2.6298 2 .8029 3.0568 2.3541 2.6959 2.68310 2.3836 2.8554 2.5667 1.3585 2.4246 2.912615 2.8332 4.8208 2.5354 3.1158 3.6189 2.181320 2.9289 3.5394 2.1861 3.3974 3.3781 1.940125 3.4076 2.9772 1.961 2.8815 1.8097 1.619530 2.6545 1.3231 1.6667 1.8921 1.9259 1.515435 2.6502 2.0394 1.5464 2.0945 2.4333 1.522740 2.6252 2.3324 1.429 1.8968 2.4463 1.293245 2.3589 2 .3179 1.3108 1.6546 2.6322 1.1350 2.2056 2.5525 1.2198 1.6172 1.0723 1.043855 1.9649 0.8853 1.2239 1.3631 1.5217 0.989560 2.015 1.2096 1.1127 1.4296 1.6575 1.029365 1.8948 1.6568 1.1789 1.2683 1.8057 0.818270 1.865 1.7128 1.1975 1.2736 1.9992 0.766875 1.8659 2.0139 0.9569 1.031 1.6551 0.721380 1.8193 2 .0272 0.94229 0.962 1.6055 0.72426

85 1.6395 1.9537 0.93415 0.8589 1.5568 0.7040890 1.5548 1.9208 0.309 0.8979 1.5583 0.71338

Fig. 7.Variation of relative volumetric error about the z-axis at 45and then about the x-axis for a toy car body.

Fig. 8.Variation of relative volumetric error about the z-axis at 60and then about the x-axis for a toy car body.

triangle are not listed in the correct order. The order of the

points typically follows some convention (right-hand rule)

which results in the cross-product of the two vectors formed

by the three points to be an outward pointing normal. Occasion-

ally, the computed normal is in the opposite direction, or is

Fig. 9.(a) The solid and (b) the tessellated models of the hyperboloid.

non-existent, or is incorrect. Therefore, the final results are

slightly different. The second issue is the closure error. This

occurs when CAD models are not translated correctly to the

STL format. When tessellation is performed, round-off errors

cause one point to be at multiple locations at the same time.

Thus, triangles are formed, and a thin gap is present in the

finished model. It directly affects the final results. When the

toy car body is oriented about the z-axis at 45 and 60 and

then rotated about the x-axis, the percentage volumetric error

values at a layer thickness 0.3 mm remain higher than the

percentage volumetric error values at a layer thickness 0.5.

Therefore, it is possible to miss some small features, or lose

the correct tracing of deposition material at the beginning or

the end position of a feature, because it is not guaranteed that

all the dimensions in the parts are multiples of a fixed thick-

ness.

5.2 Orientation of Hyperboloid

Figure 9 shows the solid model and the tessellated model of

a part, which is the shape of a hyperboloid. The STL file of

the part was found to have 200 facets. The percentage relative

volumetric errors of the part were computed and displayed by

the system for orientations about the x-axis in steps of 5

ranging from 0 to 90 for the three layer thicknesses. The

output results are displayed both in tabular form and in graphi-

cal form. Figure 10 shows the resulting graphs of the variation

of relative volumetric errors with angles of orientation for

three layer thickness are 1, 0.5, and 0.25 mm. The best orien-

tation angle recommended by the system is at 55. It is found

that the percentage volumetric error values at a layer thickness

Fig. 10. Variation of relative volumetric errors for the hyperboloid.

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A Generic Part Orientation System 215

Table 3.Percentage volumetric error for the hyperboloid, rotated aboutthe z-axis at 45 and 60 and then about the x-axis at three layer thick-nesses.

Orientation Percentage relative volumetric error (%)about Rotate about z-axis 45, Rotate about z-axis 60,x-axis layer thickness (mm) layer thickness (mm)(degrees) 1.00 0.5 0.25 1.00 0.5 0.25

0 27.97034 12.8045 6.241 27.97034 11.5789 6.2425 29.9429 15.307 10.678 29.88476 15.499 10.68810 35.54121 13.732 9.452 35.45144 16.733 9.45615 30.05666 12.398 8.605 29.91768 15.75 8.60420 23.91626 9.988 6.903 25.48005 13.089 6.90125 20.414 7.8955 6.374 19.92149 9.786 6.37130 18.60485 8.4915 6.038 18.44612 10.097 6.03835 16.28277 8.199 6.065 17.02046 9.759 6.06540 15.72634 9.0075 6.255 18.92349 10.554 6.25645 20.58541 8.4795 6.369 21.13182 11.958 6.36950 19.99379 8.379 6.893 20.84997 11.23 6.89355 14.56745 7.753 6.012 15.51038 10.756 6.0160 20.4946 7.9325 5.895 20.37161 13.119 5.89365 23.82705 11.0095 9.813 22.89292 14.75 9.81670 25.46522 13.1475 11.028 27.03042 18.519 11.02475 27.95066 13.3675 11.672 26.42569 17.638 11.67580 29.24939 13.9605 11.495 28.51035 18.88 11.494

85 30.9094 1 1.869 10.055 28.33544 15.829 10.05990 20.90645 9.1275 7.0348 19.45864 9.365 7.0335

of 1 mm follow the same pattern as for layer thicknesses of

0.5 and 0.25. Therefore, the slice will not miss small features,

or lose the correct tracing of the beginning or the end position

of a feature. We found that at angles of 5055, the percentage

volumetric error value reduces sharply for all three layer

thicknesses. It shows the slice will not miss small features for

all three layer thicknesses. Therefore, the user should choose

to build the part at a layer thickness of 0.5 mm because the

build time is less and the accuracy of the part is better at a

layer thickness of 0.25 mm.Using the option for rotation about two axes in the generic

system, results were obtained for orienting the part first about

the z-axis at selected step angles and then about the x-axis at

angles of 090 for the given z-axis. Table 3 shows the results

of the percentage volumetric error for the part, which is rotated

first about the z-axis at 45 and 60 and then about the x-axis

for three layer thicknesses of 1, 0.5, and 0.25 mm. Figures 11

and 12 show the variation of relative volumetric errors for the

hyperboloid as obtain from Table 3. When the part was oriented

Fig. 11.Variation of relative volumetric error about the z-axis at 45and then about the x-axis for the hyperboloid.

Fig. 12.Variation of relative volumetric error about the z-axis at 60and then about the x-axis for the hyperboloid.

at 45 and then about the x-axis for all three layer thicknesses,

the recommended part build orientations are found to be 55,

55, and 60, respectively. In the case of orientation about the

z-axis at 60 and then rotated about x-axis for the three layer

thicknesses, the recommended part build orientations are found

to be at 55, 90, and 60, respectively, at which the volumetric

error is found to be the minimum.

It is noted that when the hyperboloid is oriented first aboutthe z-axis at 0, 45, and 60, and then about the x-axis for a

layer thickness 0.5 mm, the percentage volumetric error values

are slightly different at 0 orientation. The reasons are the

same as for the previous example. However, the percentage

volumetric error values for rotation about the z-axis at 0, 45,

and 60, and then about the x-axis for a layer thickness 1 mm

follow the same pattern as for layer thicknesses of 0.5 mm

and 0.25 mm. The graphs of the percentage volumetric error

show that the layer thickness and the orientation first about

the z-axis do not greatly affect the best orientation for the part

if the part is of symmetric geometry, especially for layer

thicknesses of 1 mm and 0.25 mm.

Looking at the results obtained from the generic part orien-

tation system for various examples, it was noted that in almost

all cases, the graphs of the percentage volumetric error follow

the same pattern for all layer thicknesses. Only in a few cases

(such as for the toy car body), is it not so because of the

translation process and the closure error. The translation process

occurs when the computed normal vector is in the wrong

direction or is non-existent or is incorrect. The closure error

occurs when complex CAD models are not translated correctly

to the STL format. Thus, when triangles are formed, a thin

gap is present in the finished model. However, when we

compare the results of the percentage volumetric error for the

three layer thicknesses, it shows that, the smaller the layer

thickness, the more accurate is the RP part or the part has a

smaller percentage volumetric error.

6. Conclusions

The generic part orientation system described in this paper is

able to determine the best part orientation in the rapid prototyp-

ing processes for any complex solid part on the basis of

minimum volumetric error in the part owing to the staircase

effect. This has been possible through the development of a

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216 S. H. Masood and W. Rattanawong

methodology and special algorithms that can tackle the part

orientation problem in rapid prototyping for the determination

of the volumetric error encountered during the part building

process for any part. The generic system considers the slicing

of the STL model of the part by horizontal planes and deter-

mines the resulting complex contours and the volumes gener-

ated by these contours, to arrive at acceptable values of the

volumetric error in each layer. The methodology has been

shown to work for all basic primitives, for parts with acombination of primitives and for several test parts of different

complexity. The generic part orientation system has also been

verified analytically and experimentally for parts built on the

FDM rapid prototyping system. The system will enable the

RP user to make better decisions in fabricating RP parts with

a higher degree of accuracy and surface finish, because of the

minimisation of overall volumetric error in the part.

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