1. science
2. mathematics
3. geometry

Software Error Compensation
of Rapid Prototyping (RP)
Kun Tong, Sanjay Joshi, Amine Lehtihet
Department of Industrial and Manufacturing Engineering
The Penn State University
DIMACS CAD/CAM Workshop
Rutgers University, October 8, 2003
Rapid Prototyping (RP)

CAD
Model
A fabrication method in which parts are built by depositing material
layer by layer under computer control.
Slices
Stereolithography Apparatus (SLA)
STL file
(A triangular approximation of the 3D surface geometry.)
Dimensional Accuracy
Dimensional Accuracy:


SLA 250 machine: 0.002 inch/inch (Lynn (2000))
FDM 3000 machine: “Overall tolerance is ± 0.005 inch
in the X, Y, and Z axes.”
Main Error Sources in RP Processes:





Positioning error of the tool tips (the laser beam focus or
the depositing nozzle)
Material shrinkage/expansion due to physical or chemical
changes during prototyping
CAD model  STL file approximation error
Staircase Effect due to finite layer thickness
…
Accuracy Studies for RP


Focused on different aspects of process planning for RP

STL File Correction Algorithms (van Niekerk (2000))

Slicing Algorithms (Kulkarni (1996), Tata (1998))

Tool Path Planning (Wah (2001))

Support Structure Generation Methods

Parametric Tuning (Lynn(1999), Onuh(2000), Zhou(2000))

Build Orientation Optimization (Alexander(1998), Nee(2001)…)
Limitation: The best-tuned system still has considerable systematic
error.
Software Error Compensation
Error Compensation: To cancel the effect of an error without
removing the error source.
Very little work has been done on error compensation of RP.


Volumetric Error
Nominal Position (x, y, z)
dx
dy
dz
Actual Position (x+dx, y+dy, z+dz)
Compensated Input (x-dx, y-dy, z-dz)
z
y
x
RP working envelope
Software Error Compensation Approach

This concept has been applied to Coordinate Measuring
Machines (CMM) and machine tool systems.

Three-step procedure:



Build a Mathematical Machine Error Model
—to write the components of the volumetric error
(dx, dy, dz) as functions of the coordinates x, y and z.
Develop a method of measurement to determine the
coefficients in the model.
Implement the error model in the machine control for
compensation.
How to Build the RP Error Model?

Mechanistic Model or Statistical Model?



RP machine is a complex system. It is very hard, if not
impossible, to build a mechanistic model.
Statistical model will introduce a lot of predictors which
might not be significant and have no physical meaning.
Solution: Build an approximated mechanistic model using
the Parametric Errors Functions and Rigid Body
Kinematics as used for Coordinate Measuring Machines
or Machine Tool Systems.
Coordinate Measuring Machine (CMM)
Y
Z
X
Nominal
CMM
Actual
18 Parametric Errors (Geometric Errors)
Axis
Scale Error
Translational Straightness
Error
Error
Roll
Rotational
Pitch
Error
Yaw
X
x(x)
y(x)
z(x)
x(x)
y(x)
z(x)
Y
y(y)
x(y)
z(y)
y(y)
z(y)
x(y)
Z
z(z)
x(z)
y(z)
z(z)
x(z)
y(z)
Squareness Error
Y-X: xy Z-X:  zx Z-Y:  zy

The 6 parametric errors of each axis
are only functions
of the coordinates in that axis.
CMM Machine Error Model

Using Rigid Body Kinematics and Homogeneous Transfer
Matrix, we can write the machine error model:
X p  x   x x   x  y    x z   y  z x  z  y x
 z   y ( y )  t z   y ( x )   y ( y )   y ( z )    x
Yp  y   y x   y  y    y z   z  x x  z  x  y 
 t z   x ( x )   x ( y )   x ( z )    y
Z p  z   z  x    z  y    z  z   y   x  x   tz   z
 (x, y, z): nominal position
tx ty tz are the tool tip offset
 (Xp, Yp, Zp): actual position of the probe
 x  y  z are unrepeatable errors.
RP “virtual” Parametric Errors

Besides the geometric errors of the three axes, material
volume change during prototyping is also a major error
source.

Assumption: all errors in the RP process can be mapped into
the 18 “virtual” parametric errors. Then rigid body kinematics
is used to write the error model.

SLA 250 machine error model:
X p  x   x  x    x  y    x  z   y z  x    z  z   z   y  z    x
Y p  y   y  x    y  y    y  z   x z  z   z   x  z    y
Z p   z   z  x    z  y    z  z   x y ( z )  y x  x    x  z    z
14 “virtual” parametric errors show up in the model.
Polynomial Parametric Error Functions

3rd order Legendre Polynomials are used to approximate the
parametric error functions.
For example:
3 2
1 3
 x ( x )  a1  x  a2  ( x )  a3  ( (5x  3x ))
2
2

Totally, 143=42 unknown coefficients need to be determined
for the error model of SLA machine.
Estimate the Coefficients Using Artifact
 169 cylinders:
8 inch (X)  8 inch (Y)  3.5 inch (Z)

A LINGO program is written to minimize the sum of squares of
the residual errors:
169
   (  x ,i   y ,i   z ,i )
i 1
2
2
2
Prediction Using Error Model
Uncompensated
Predicted Using Error Model
Norminal
Residual Errors of the Model
4
3
Y coordinates (inch)
2
1
0
-4
-3
-2
-1
0
-1
-2
-3
-4
X coordinates (inch)
1
2
3
4
Volumetric Error Map of SLA 250
3D Z
Volumetric
of SLA 250
at a Fixed
HeightError
(Z =Map
1 inch)
@ Fixed Z Level (Z = 1 inch)
0.018
0.014
0.012
0.01
0.008
0.006
0.004
4
2
0.002
0
X Coordinates (inch)
-4
4
3
2
1
0
-2
-1
-2
-3
0
-4
Volumetric Errors (inch)
0.016
Y Coordinates
(inch)
Software Error Compensation

CAD model  STL File  Slice File  RP Machine Control File
Apply Compensation to all
the vertices in STL file
ASCII STL File Format
***********************************************************
solid PRT0002
facet normal 0.000000e+000 0.000000e+000 -1.000000e+000
outer loop
vertex 1.000000e+001 7.500000e+000 0.000000e+000
vertex -1.000000e+001 -7.500000e+000 0.000000e+000
vertex -1.000000e+001 7.500000e+000 0.000000e+000
endloop
endfacet
………
………
facet normal -1.000000e+000 0.000000e+000 0.000000e+000
outer loop
vertex -1.000000e+001 7.500000e+000 3.000000e+000
vertex -1.000000e+001 7.500000e+000 0.000000e+000
vertex -1.000000e+001 -7.500000e+000 0.000000e+000
endloop
endfacet
endsolid PRT0002
***********************************************************
Compensation Test Part
Apply compensation to a test part with 49 cylinders of same
diameter but at different locations from those in the 3D artifact.
Volumetric Errors Reduction
by Software Error Compensation
Volumetric Errors of 49 Test Points
After Compensation
Volumetric Errors of 49 Test Points
Before Compensation
0.016
0.016
0.014
Volumetric Errors (inch)
Volumetric Errors (inch)
0.014
0.012
0.01
0.008
0.006
0.004
-2.7
0.002
0.012
0.01
0.008
0.006
0.004
-2.7
0.002
-0.9
-0.9
0
-2.7
0.9
-1.8
-0.9
0
Y coordinates (inch)
2.7
0.9
1.8
2.7
Before Compensation
X coordinates
(inch)
0
-2.7
0.9
-1.8
-0.9
0
Y coordinates (inch)
X coordinates
(inch)
2.7
0.9
1.8
2.7
After Compensation
On average, volumetric errors are reduced to 35% of their original values.
Compensation to a Part with General Features
-0.00360.0011
3.7500
3.7463/3.7511
3.3750
3.3716/3.3754
2.3750
2.3724/2.3740
2.9079
2.8994/2.9038
2.5000
2.4925/2.4943
0.8750
0.8730/0.8747
-0.0020-0.0003
15°
14.58.40/15.01.15
Ø0.2500
0.2457/0.2459
Ø0.5000
0.4915/0.4922
Ø0.5000
0.5007/0.5020
2.5000
2.4913/2.4972
7° /6.52.27
/6.38.28
0.8750
0.8738/0.8754
Ø0.7500
0.7445/0.7457
0.8750 / 0.8746/0.8767
2.8750
2.8712/2.8749
0.2845 /0.2723/0.2710
0.3750 /0.3731/0.3782
0.5000
0.4995/0.5081
0.00190.0032
0.3750
0.3709/0.3707
Note: The three dimensions for each feature is
Nominal / Uncompensated / Compensated .
Summary of Compensation Results

The overall size and the feature positions on the part are
considerably improved.

Dimensions involving height in z direction such as depth of
holes, or with the presence of staircase effect such as the
sphere size, are not improved and in some cases may get
worse. (Z quantization)

Cylindrical holes sizes are improved by a small amount and
both uncompensated and compensated sizes are consistently
smaller than their nominal values. (Triangulation Error)
“Z value quantization”
—in Uncompensated Parts
Uncompensated Model
Model Top Surface
Slicing Planes
Actual Surface
“Z value quantization”
—in Compensated Parts
Compensated Model
Model Top Surface
Multiple Layers
Slicing Planes
Actual Surface
Eliminate Multiple Layers Phenomenon

Algorithm:




Identify horizontal surfaces in the uncompensated
part;
Apply z direction compensation to the vertices of
each horizontal surface;
Average the z-coordinates of all vertices on a
given horizontal surface and then move the entire
horizontal surface up or down to the nearest
adjacent layer.
Move corresponding vertices in the entire STL
files to maintain connectivity.
Summary


Software error compensation can help RP users
to achieve higher part accuracy without any
hardware modification.
“Virtual” parametric error functions include all
repeatable error sources present in the rapid
prototyping process and can be thought of as a
comprehensive evaluation criterion of RP
machine accuracy.
Future Work(1)—
Slice File Compensation
Apply
Compensation
Slice 1
Slice i
Slice N
Apply
Compensation
Future Work(2)—
Design Problem of the 3D Artifact
How to choose the cylinder heights
is an Design of Experiment (DOE) problem.
Design Criterion

D-criterion

A D-optimal design is one in which the determinant of the
moment matrix is maximized:
M 


X X
N
p
where X is the regression model matrix.
N is the sample number.
p is the number of coefficients.
Geometrically, with D-criterion, one tries to minimize the
volume of the confidence region on the regression coefficient.
Objective: Keep the same X and Y positions of the cylinders, to
optimize the distribution of their heights.