IMPROVED PROCESS CONTROL IN CAMSHAFT GRINDING THROUGH
UTILIZATION OF POST PROCESS INSPECTION WITH FEEDBACK
By
TIMOTHY MARK DALRYMPLE
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
1993
Copyright 1993
by
Timothy Mark Dalrymple
To my students at Botswana Polytechnic.
On the nights I remember the incredible gifts you
possess, the diversity of backgrounds and talents you
represent, I sleep well, knowing that Africa, will one day,
be safe in your hands.
ACKNOWLEDGMENTS
Like all human undertakings, this work would not have
been possible without the help of others. I am particularly
grateful to John Andrews and his staff at Andrews Products
for practically unconditional use of his excellent
facilities. At Andrews, I am especially grateful to Scott
Seaman. Also, thanks go to Chuck Dame of Adcole Corporation
for providing essential technical information concerning his
company's products.
I am also pleased to acknowledge the help and
encouragement which my advisor, John Ziegert, provided.
Lastly, I wish to thank my wife, Laura, for her unfailing
confidence, support, and patience.
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . .. iv
LIST OF FIGURES . . . . . vii
ABSTRACT . . . . . .ix
CHAPTERS
1 INTRODUCTION . .. . . 1
Scope of the Problem . . . 1
Camshaft Grinding Technology . . 2
Analysis of Camshaft Geometrical Errors . 4
Potential for Improvement . . 5
2 REVIEW OF THE LITERATURE . . . 6
Compensation Through Positioning Error Modeling 6
Compensation Through InProcess Gauging . 8
Compensation Using CMM for PostProcess
Inspection . . . ... 9
3 CAMSHAFT GEOMETRY . . . .. 11
Basic Description and Analysis . . .. 11
Industrial Convention . . . 15
CCMM Convention . . . 16
Grinding Machine Convention . 18
Implications of Machining Techniques ... . 18
Calculation of Grinding Wheel Path . .. 25
Inspection Techniques for Process Control .. 26
Decomposition of Error Components . .. 29
Timing Relative to Keyway/Fixture . .. 31
Timing Relative to Lobe 1 . . .. 33
4 THE STOCHASTIC PROCESS MODEL . . .. 34
General Linear LeastSquares Estimation .. 34
Identification of Lift Error Model .. . 36
Estimation of Model Parameters . . .. 39
Diagnostic Checking of the Model . .. 41
Potential for Model Improvement . .. 43
1
5 PROCESS CONTROL STRATEGY
The Control Interface . . . 44
Lift Error Controller Model . .. . 48
An Interacting Lift Process Model . .. 48
Noninteracting Control of Lift . .. 50
The Implemented Lift Controller . .. 50
Control of Relative Timing . . .. 55
6 PROCESS NOISE . . . . 58
Statistical Process Control . . .. 59
Traditional Feedback Control with Filtering 60
Error Repeatability A Preliminary Study .. 63
Process Repeatability of Lift Errors .. 64
Process Repeatability of Timing Error . 68
7 EXPERIMENTAL RESULTS . . . .. 71
Results of Lift Error Control . . .. 74
Results of Timing Error Control . .. 81
Control of Timing Relative to Lobe 1 . 81
Control of Timing Relative to Keyway/Fixture 83
8 CONCLUSIONS AND RECOMMENDATIONS . .. 86
Implementation of the Control System . .. 86
Further Work . . . . 87
APPENDICES
A GRINDING PARAMETERS . . . .. 90
B CAMSHAFT INSPECTION PARAMETERS . . .. 107
C SOFTWARE OUTLINES AND CONTROL SYSTEM PARAMETERS 111
REFERENCES . . . . . 114
BIOGRAPHICAL SKETCH . . . .. 117
1
LIST OF FIGURES
Figure 31 Typical Four Lobe Camshaft . .. 12
Figure 32 Radial Cam with Roller Type Follower 13
Figure 33 CCMM Angle ConventionClockwise Rotation 17
Figure 34 Grinding Machine Angle Convention .. 19
Figure 35 Camshaft AxesofRotation . .. 22
Figure 36 Grinding Between Centers . .. 23
Figure 37 Centerless Grinding Technique. . ... 24
Figure 38 Grinding Wheel Path Calculations .. 27
Figure 39 Camshaft Inspection on a CCMM . .. 28
Figure 310 Best Fit of Lift Errors . . .. 32
Figure 41 Lift Acceleration and Measured Lift Error 38
Figure 42 Stochastic Model Residual Errors .. 42
Figure 51 Typical Workspeed and Lift Acceleration 47
Figure 52 Interacting Lift Variables . .. 49
Figure 53 Noninteracting Lift Control System 51
Figure 54 Implemented Lift Control System . .. 53
Figure 55 Implemented Timing Angle Controller . 56
Figure 61 Lift Error Process Repeatability 65
Figure 62 Lift Error Mean and Standard Deviation 66
Figure 63 Lift ErrortoNoise Ratio . . 67
Figure 64 Timing Angle Repeatability Error .. 69
Figure 71 Landis 3L Series Cam Grinder . .. 72
vii
Figure 72
Figure 73
Figure 74
Figure 75
Figure 76
Figure 77
Figure 78
Figure 79
Figure 710
Adcole Model 911 CCMM . ... ...
Lift Error using Control Scheme . .
Lift Error using Control Scheme . .
Lift Error using Control Scheme . .
Lift Error using Control Scheme . .
Total Lift Error using Control Scheme .
RMS of Lift Error using Control Scheme
Timing to Lobe 1 using Control Scheme .
Timing to Keyway using Control Scheme
viii
Abstract of Thesis Presented to the Graduate School of the
University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
IMPROVED PROCESS CONTROL IN CAMSHAFT GRINDING THROUGH
UTILIZATION OF POST PROCESS INSPECTION WITH FEEDBACK
By
Timothy Mark Dalrymple
May 1993
Chairman: John Ziegert
Major Department: Mechanical Engineering
In order to take full advantage of the introduction of
computer numerical control technology in camshaft grinding
and postprocess inspection, a closed loop control scheme is
proposed. This strategy makes use of postprocess
inspection results to modify the commanded part geometry
used in the grinding program. The commands are modified in
order to minimize the lobe contour and relative timing
errors. It is shown that using a control strategy comprised
of feedforward and feedback elements, substantial
improvements in cam contour accuracy can be attained.
A system capable of automated reduction of inspection
results, application of statistical methods, transformations
between different coordinate systems, and production of
modified commanded part geometry is presented. This system
requires no offline calibration or learning of grinding
machine positioning errors. Additionally, it offers
advantages over such techniques, in that it is able to
automatically adapt to changing process conditions.
The system is general in nature and may be used for any
camshaft design. Through application of this system, the
time required to bring a new part into tolerance is greatly
reduced. With such a system used to minimize contour
errors, it is no longer necessary to optimize grinding
parameters based on these errors. Rather, grinding
parameters can be manipulated to optimize other important
factors, such as metal removal rates and the corresponding
process time.
Implementation of this system on existing computer
numerically controlled (CNC) equipment is inexpensive. It
requires only software necessary to produce the modified
part geometry and limited hardware for file transfer. The
required hardware is inexpensive and readily available.
CHAPTER 1
INTRODUCTION
Scope of the Problem
Camshafts find application in a wide range of consumer
and industrial products. In machine tools, camshafts have
long been used to control precise and highspeed machine
motions. Applications are common in both chip producing
equipment and highspeed dedicated assembly machinery [1].
Additionally, cams are used in fields as diverse as blood
separation and automated laserscanner checkout systems.
Advances in servo motor and computer numerical control
(CNC) technology have lead to the replacement of cams in
many industrial applications. Still, camshafts will remain
essential for certain applications, such as the internal
combustion engine, for the foreseeable future.
In internal combustion engines, camshafts are used to
mechanically control the opening and closing of intake and
exhaust valves. The shape of the camshaft is critical in
determining the nature of the combustion process. As
pollution emission regulation for internal combustion
engines, particularly automobiles, have become more
stringent, the demands for higher precision camshafts and
better valve and combustion control have increased [2].
1
Z
Improvements in grinding media, machine design, and the
application of CNC provide for better control in camshaft
manufacturing. Yet, achieving high quality results depends
on properly coordinating the setting of a wide range of
grinding variables. These variables include: wheelspeed,
workspeed, dress parameters, grinding wheel quality,
dressing tool quality, temperature, etc. [3]. A change in
any of these parameters can produce deleterious effects on
part quality.
Camshaft Grinding Technology
Dimensional errors occurring in camshaft grinding are
attributable to a wide range of causes. Traditional cam
grinders use master cams to produce relative motion between
the grinding wheel and the workpiece. This relative motion
generates the desired camshaft geometry. Naturally, any
inaccuracies in the master cams produce corresponding errors
in the workpiece. Additionally, machines utilizing master
cams for control produce the optimal shape only for a single
grinding wheel size [3].
Conventional media based grinding wheels, such as
aluminum oxide or silicon carbide, require frequent dressing
to remain sharp and free cutting [4]. This continual
dressing produces a grinding wheel which is constantly
decreasing in diameter. As the wheel size deviates from the
nominal size, the dimensional accuracy of the camshaft
3
deteriorates [3]. While some master cam controlled machines
are able to compensate for wheel size wear through the use
of additional master cams [5,6], this increases the initial
cost, complexity of the machine and set up time. Also, this
improvement in contour requires sacrificing optimal control
over other parameters such as the cutting speed variation.
Additional sources of errors in camshaft grinding can
be traced to the nature of the contact between the grinding
wheel and workpiece. The "footprint," as the area of
contact is known, changes as the grinding wheel encounters
different curvatures on the cam surface. This change in
footprint affects the grinding wheel's cutting ability and
thus impacts the metal removal rate. Changes in grinding
conditions adversely affects workpiece accuracy [3].
The introduction of CNC technology in camshaft grinding
has produced benefits far beyond the flexibility generally
associated with CNC. Using this technology, the tool path
can be continually updated and optimized for any size of
grinding wheel [5,6]. Additionally, using a variablespeed
servo motor to control the headstock rotation, the variation
in footprint speed can be minimized. If the grinding
wheel's speed is constant, then minimizing the footprint
speed variation also minimizes the relative speed variation
between the cutting edges of the grinding wheel and the cam
lobe surface. Finally, the part geometry is specified in
software and can be readily modified.
Analysis of Camshaft Geometrical Errors
Manual techniques of camshaft inspection are generally
based on test fixtures which use dial indicators to measure
dimensional errors. This approach, while sufficient for
determining if functional requirements are satisfied, is
inadequate for quickly identifying machining errors. Using
manual test procedures, it is impossible to analyze all
error components and determine their cause in a timely
manner. Thus, process control based on such inspection
techniques is not feasible.
Computercontrolled cylindrical coordinate measuring
machines (CCMM) capable of fast and accurate camshaft
inspection have been available for two decades. With the
decrease in cost of computer hardware, CCMM have become more
affordable and are currently available in many camshaft
manufacturing facilities. Commercially available software
allows for some flexibility in data reduction [7]. This
flexibility provides for camshaft inspection based on
evaluation of functional criteria or for analysis of the
manufacturing process.
However, until recently, the majority of camshaft
grinders used master cams for control. Since the part
program is essentially ground into the master cam, the
potential for fine adjustments based on inspection results
did not exist. Consequently, the wealth of data derived
from CCMMs was poorly utilized. In most applications CCMMs
I
5
were simply used to identify nonconforming workpieces. In
some cases, CCMMs were effectively used to evaluate the
effects of individual grinding parameters, other than the
commanded geometry, on camshaft geometrical errors.
Potential for Improvement
The widespread introduction of CNC technology in
camshaft grinding provides the opportunity to better utilize
the information derived from CCMMs. Since part programs in
CNC machines can be readily changed, the program can be
modified based on repeatable errors observed in postprocess
inspection. CCMMs capable of measuring up to 150 parts per
hour are currently on the market [8]. Camshaft grinders
with CNC produce approximately seven camshafts (12 lobe
shaft) per hour [2]. With the inspection process requiring
less time than the manufacturing process, the opportunity
exists to compensate for errors with a time lag of only one
part. The ability to automatically inspect camshafts and
produce compensated part programs presents great potential
for process improvement with little capital investment or
operator training.
CHAPTER 2
REVIEW OF THE LITERATURE
Most early research concerned with minimizing machining
errors focused on error avoidance. This research lead to
improvements in thermal stability, machine base stiffness,
precision components, and spindle design. These
improvements, while greatly increasing accuracy, did not
come without a cost. As machine tools became ever more
precise and mechanically sophisticated, the cost of these
machines continued to rise.
Much recent research on improving accuracy has focused
on error compensation rather than error avoidance. This
technique provides potential for improved accuracy without
costly mechanical design improvements. A review of the
literature in this area shows major research in three
categories: offline modeling of positioning errors,
inprocess inspection, and postprocess inspection with
feedback.
Compensation Through Positioning Error Modeling
The use of a positioning error model has been the focus
of much recent research [9,10,11,12]. Donmez et al. [9]
describe a methodology by which the positioning errors are
6
measured using offline laser interferometry and electronic
levels. These errors are then decomposed into geometric and
thermallyinduced components. The geometric errors are
thermallyinvariant and modeled with slide positions as
independent variables. The thermallyinduced errors are
modeled using key component temperatures and slide position
as independent variables.
Once the positioning errors have been measured and the
model of the positioning errors established and verified,
error compensation can be applied. Since this method models
the workspace of the machine, and not the errors for a
particular workpiece, it is general in nature and can be
applied to parts not previously produced. However, it is
expected that over time, the machine will wear, the model
will become less accurate, and the machine will require
recalibration.
In certain machining operations, such as single point
turning, a one to one correspondence between machine errors
and workpiece errors can be established. For the single
point turning operation studied, Donmez reports accuracy
improvements of up to 20 times.
This work has been extended by Moon [13] to a system
where the laser measurement system is a integral component
of the machine tool. This system offers the advantage that
it does not require recalibration and is capable of
compensating for the portion of the stochastic error which
is autocorrelated. Such systems, however, greatly increase
the cost and complexity of machine tools.
In camshaft grinding, the relationship between machine
errors and workpiece errors is more complex than for the
case of single point turning. Additionally, the ability to
compensate for thermal errors is not so critical. The lift
of the camshaft is essentially a relative dimension which is
measured from the base circle. Since both the datum (base
circle) and the lift are machined simultaneously, the
accuracy is not greatly affected by thermal effects. A
warmup period is not generally required.
Compensation Through InProcess Gauging
While many causes of form errors in camshafts have been
understood since the 1930s [14], CCMMs necessary to quickly
and accurately quantify these errors have existed for only
20 years. Additionally, CNC cam grinding machines necessary
for implementation of a compensation scheme, have only come
into widespread use in the past 10 years. Work has been
done in the area of inprocess inspection and compensation
on machining centers [15], and the idea of an inprocess
compensation scheme was first proposed for cam grinding by
Cooke and Perkins [16].
In their work at Cranfield Institute of Technology,
Cooke and Perkins proposed a prototype CNC camshaft grinding
machine using inprocess gauging in 1978. The proposed
system employed a gauge probe located 180 degrees
outofphase with the grinding wheel. In the proposed
system, the probe is used to detect any deviation in the
measured workpiece from those commanded. The system
computer takes advantage of the 180 degree phase lag to
produce compensated control commands prior to the next
grinding pass. Through study of available servo drive
mechanisms and linear transducers available at the time, the
researchers expected to realize a machining accuracy on the
order of + 1.5 micrometer. While such systems are not today
in commercial production, this early work realized the
potential benefits of coupling camshaft grinding and
inspection.
Compensation Using CMM for PostProcess Inspection
An extensive review of the literature as well as
personal interviews with CCMM manufacturers revealed no
previous published work dealing with CCMMs and process
control. Much work has, however, been conducted using the
closely related Cartesian coordinate measuring machine (CMM)
for process control [17,18,19].
Yang and Menq [17] describe a scheme using a CMM for
postprocess inspection of endmilled sculptured surfaces.
In this approach, 500 measurements are performed on a 55 mm
x 55 mm sculptured surface. The measured errors are then
best fit to a regressive cubic bspline tensorproduct
10
surface model. The results of this best fit are then used
to determine the compensation to be applied. Using this
technique, the researchers reported improvements in maximum
form error of 73%. While this improvement is significant,
CMMs have practical limitations, such as long cycle times,
that make them unsuitable for use in high volume process
control. The method used by Yang and Menq most resembles
the approach taken in this research.
CHAPTER 3
CAMSHAFT GEOMETRY
The material in this chapter is included to review the
nature of camshaft geometry as it relates to this research
project. The mathematical relationships describing cam
contour, the grinding wheel path, and decomposition of
inspection results are presented. Specific aspects of
camshaft geometry are examined for their significance in
developing a control strategy for the manufacturing process.
Lastly, the industrial conventions for specifying camshaft
geometry are introduced.
Basic Description and Analysis
A typical camshaft, as shown in Figure 31, consists of
a number of individual cam lobes, journal bearings, and a
timing reference. The geometry of a radial camshaft with a
translating follower is readily described in terms of the
base circle radius rb, the follower radius rf, the follower
lift s as a function of lobe angle 0, and the timing angle (
as shown in Figure 32. From this figure, it is evident
that the pointofcontact between the cam surface and the
cam follower does not generally lie along the follower's
lineofaction. This illustrates the difference between cam
11
Side View
End View
Figure 31 Typical Four Lobe Camshaft
Figure 32 Radial Cam with Roller Type Follower
contour and the follower lift produced. Errors in cam
contour, at the pointofcontact between the follower and
cam profile surface, produce errors in follower lift.
For the radial cam with a roller follower shown in
Figure 32, the cam contour can be calculated as [20]
x = r cos + (31)
1 (31)
ds
x M + r (32)
N
where
M = r sin dscos 6 (33)
do
N= r cos 6 dsin 0 (34)
dO
r = rb + r + S (35)
and summarizing notation
s = lift
rb = base circle radius
rf = follower radius
r = the distance between cam and follower centers
0 = lobe angle
All camshaft/follower combinations, such as those with
offset roller followers or flat radial followers, can be
represented as camshafts with radial roller followers. All
camshafts considered in this research are represented in
this manner.
Industrial Convention
Both the CCMM and the CNC camshaft grinder used in this
research project adopt the same basic convention for
describing camshaft geometry. This convention is the de
facto industry standard and is used for programming CNC
machines and for reporting error results. These conventions
are adopted in this research to simplify the control
interface.
While both machines employ the same general convention,
the details of the implementation differ with regard to sign
convention and measurement datums. On both machines, the
follower lift is specified as a function of the lobe angle 6
as shown in Figure 32. The lobe angle is measured relative
to a coordinate system attached to the cam lobe and oriented
relative to a given geometrical lobe feature. The timing
angle 0 of the individual lobes is then specified relative
to some camshaft feature. Typical camshaft timing
references are eccentrics, lobe 1, dowel pins, or keyways as
illustrated in Figure 32. The base circle radius is the
datum for measurement of lift values. The positive
rotational direction is specified as counterclockwise when
viewed from the nondriven end of the camshaft, looking
I
towards the driven end. The details of the implementation
are described in the following two sections.
CCMM Convention
The convention illustrated in Figure 33 shows the
convention use by the Adcole 911 CCMM. This convention is
dependent on the camshaft directionofrotation. That is,
the CCMM directionofrotation is programmable and is
selected as the application directionofrotation. The
convention shown is for a camshaft which rotates in a
clockwise direction. In this convention, the lift s is
specified relative to the lobe angle 8. The lift values for
the opening side (the side of the lobe which produces
follower motion away from the camshaft axisofrotation)
precede the lift values for the closing side.
The timing angles are measured in the same direction as
the lift specification angle, but to a different datum.
Figure 33 shows the lobe timing angles measured from the
timing reference to the lobe nose. Again, the convention
shown is for a camshaft which rotates in a clockwise
direction. For a camshaft which rotates in a
counterclockwise direction, the lobe and timing angles are
measured in the opposite direction.
View Looking from Nondriven End to Driven End
Opening
N N Side /
Se / Closing
Side
Direction of
Rotation
Figure 33 CCMM Angle ConventionClockwise Rotation
Grinding Machine Convention
The convention used on the CNC grinding machine does
not consider the camshaft functional directionofrotation.
Rather, the camshaft geometry is specified in a format which
reflects the directionofrotation of the grinding machine.
This convention is shown in Figure 34.
Implications of Machining Techniques
In analyzing camshaft geometry, it is necessary to
carefully consider the axisofrotation. For effective
process control, machining and inspection must be performed
with respect to the same axis. To facilitate a discussion
of the different axes used for camshaft machining and
inspection, it is helpful to introduce terminology for the
different axes. This terminology is defined as it is
introduced and summarized in Table 31.
Table 31 Camshaft AxesofRotation
MAOR AxisofRotation for Machining
IAOR AxisofRotation for Inspection/Decomposition
CAOR Camshaft AxisofRotation defined by Centers
JAOR Camshaft AxisofRotation defined by Journals
View Looking from Nondriven End to Driven End
I
4 Direction of
Rotation
Figure 34 Grinding Machine Angle Convention
The first two axes are the machining axisofrotation
(MAOR) and the inspection axisofrotation (IAOR). These
two axes do not refer to the actual camshaft axisof
rotation, but rather to the axisofrotation of the process.
The MAOR is defined as the axis about which the camshaft
rotates during the machining process. The IAOR is the axis,
with respect to which, the inspection results are
decomposed. The IAOR need not necessarily be the axis about
which the part rotates during the inspection process, since
it is possible to mathematically transform the inspection
results to other axes.
As stated previously, the MAOR and the IAOR must agree
for effective process control. If the two axes do not
agree, an error component, which is random in phase and
skewsymmetrically distributed in magnitude, is introduced
into the inspection results and therefore into the process
control signal. This error component is due to the
eccentricity of the journal bearings to the MAOR. This
component occurs only in cases where the journals are not
ground on the cam grinder. This error component will be
demonstrated through the introduction of the two additional
axesofrotation as described below.
First, the journal axisofrotation (JAOR) is defined
as the axis about which the camshaft rotates when the
camshaft journals are mated with ideal journal bearings.
The JAOR is determined in the inspection process.
To determine the JAOR, a best fit of a circle to the
inspection data for each journal is performed. The JAOR is
then established as the axis which passes through the
centers of these circles. In the case where more than two
journal bearings exist, a segmented JAOR can be used.
Second, the center axisofrotation (CAOR) is defined
as the axis about which the camshaft rotates, when the
female camshaft centers are mated with ideal male centers.
The JAOR and the CAOR, as shown in Figure 35, represent the
physical axesofrotation of the camshaft. In this figure,
the eccentricity of the journal surface is greatly
exaggerated to clearly illustrate the different axes.
The MAOR is dependent on the work holding method used
in the machining process. In this research, the camshaft is
held between centers. When using this method, the MAOR is
defined to be the CAOR. Camshaft grinding between centers
is shown in Figure 36. This arrangement uses a live
workhead center and a dead tailstock center.
Alternatively, camshafts can be ground using a
centerless grinding technique as shown in Figure 37. In
this approach, the camshaft is supported by fixtures and a
three jaw chuck. Here, the bearing journals must be ground
prior to grinding the cam contours, since these surfaces are
used to locate the camshaft during the grinding process.
The fixtures have hardened bearing surfaces which
effectively simulate the camshaft operating conditions. For
JAOR
CAOR
Figure 35 Camshaft AxesofRotation
Figure 26
indinc Between Centers
Figure 37 Centerless Grinding Technique
Source: "StateoftheArt 3L Series CNC Cam Grinding
Systems," Litton Industrial Automation Systems, Pub. No.
3L88 FR 3M, 1988. Used with permission.
centerless grinding approaches, the MAOR is defined as the
JAOR.
During inspection, measurement data must be decomposed
relative to a specific axisofrotation. The IAOR is
programmable. While the actual inspection is performed
between centers and therefore with respect to the CAOR, the
inspection results can be mathematically transformed from
the CAOR to the JAOR. Again, the IAOR must agree with the
MAOR to avoid the introduction of additional noise into the
process control signal.
Calculation of Grinding Wheel Path
The Cartesian coordinates, xc and y, of the path
traced by the grinding wheel axisofrotation are readily
calculated [20] as
x = x + (r cos 8 x) (36)
yc = y + (r sin y) (37)
if
where r, is the radius of the grinding wheel. The path can
be expressed in polar coordinates [20], which are suitable
for programming the CNC grinding machine, as
1
S (x + ) (38)
4 =arctan ye (39)
X 0
26
where p is the distance from the MAOR to the grinding wheel
axisofrotation, and y is the corresponding lobe angle for
the grinding process. The cam grinding operation is
illustrated in Figure 38.
From these equations, a table of p versus \ can be
constructed for any useful grinding wheel size. This table
corresponds to the positioning commands used internally in a
CNC grinding machine. The commands must frequently be
recalculated depending on grinding wheel size and
potentially even after each increment of infeed if the
metal removal rate is to be closely controlled. The reason
for this can be seen from Figure 32. As discussed
previously, it is evident that the pointofcontact between
the cam surface and follower does not generally lie along
the follower's lineofaction. Consequently, an infeed of
the grinding wheel does not have the effect of removing the
same amount of material from all portions of the cam lobe
surface since the pointorcontact will vary.
Inspection Techniques for Process Control
Camshaft inspection on CCMMs is accomplished using a
precision follower, of the diameter to be used in the
camshaft application, attached to a springloaded sliding
gauge head as shown in Figure 39. The movement of the
gauge head is measured using either precision scales or a
laser interferometer. This approach allows direct
Grinding Wheel
AxisofRotation
Figure 38 Grinding Wheel Path Calculations
MAOR
Figure 39 Camshaft Inspection on a CCMM
1.* I W 
evaluation of camshaft lift errors and eliminates the need
to mathematically transform results from a nonconforming
follower to the actual follower diameter. When necessary,
this conversion can be performed using equations (36) and
(37) where the radius of the nonconforming follower is
substituted for the radius of the cutter. Numerical methods
must again be used to solve for the measured lift of the
application follower.
Using CCMMs it is possible to evaluate a range of
camshaft attributes. In addition to lift and timing errors,
most CCMMs are capable of measuring the following
parameters: eccentricity, roundness, taper, lift velocity,
and diameter. Camshaft inspection can be performed with
regard to either manufacturing or functional considerations.
These two approaches can differ depending on the technique
used to hold the work during grinding.
Decomposition of Error Components
Once the appropriate IAOR is selected, the CCMM is used
to decompose measurement results into components which can
be directly used to modify the commanded input to CNC cam
grinding machines. For the CCMM used in this study, the
method of decomposition is programmable. In this section,
the various available methods of decomposition are discussed
and the methods selected for this research are justified.
30
In order to explain the decomposition, a description of
the nature of CCMMs is required. These gauges collect data
on camshaft geometry by rotating the part about the CAOR as
shown in Figure 39. The camshaft drives the precision
follower and the measured lift is recorded, generally at one
degree increments. This process allows for rapid data
collection for the entire cam contour.
CCMMs are relative measurement devices and as such they
are ideally suited for inspecting camshafts which are used
with translational followers. The lift of the follower is
both defined and measured relative to the base circle. This
ensures exceptional accuracy for measuring lift errors by
eliminating any d.c. error component. The measurements of
absolute dimensions such as size, may or may not be
particularly accurate, depending on the configuration of the
individual CCMM.
When inspecting a cam lobe, data are first gathered for
the entire 360 degrees of rotation. Next, the d.c.
component or size error of the cam lobe is removed from the
data. This is accomplished by setting the average lift of
the base circle equal to zero and adjusting the data
accordingly. The data are then corrected for the
appropriate axisofrotation. As discussed in the previous
section, this selection depends on the machining work
holding method as well as whether the results are for use in
cam grinding process control or overall functional
evaluation.
The timing error and lift errors are next separated
using various programmable approaches. In the method used
in this research, the measured cam lift is rotated about the
selected IAOR until the root mean square value of the
difference between the nominal and measured lifts are
minimized. This is illustrated in Figure 310. Once the
orientation with the minimum error is determined, the lift
error is reported as the difference between the rotated
measured lift and the nominal lift. The timing error is
reported as the difference between the measured timing,
derived from the best fit process, and the nominal timing.
Timing Relative to Keyway/Fixture
Once the best fit of the lift values is performed, the
timing angle error can be evaluated. For lobe 1 the timing
must be specified relative to some reference such as a
keyway/fixture combination, an eccentric or a dowel. The
repeatability of these different references is highly
variable. Consequently, angles measured relative to such a
reference can have a high stochastic component.
For the case studied, the reference during inspection
is established using a key, a keyway and a mating fixture.
The fixture is fitted with an external timing pin as shown
in Figure 39. The CCMM determines the location of the
Measured
Lift
Nominal
Lift
Figure 310 Best Fit of Lift Errors
timing reference based on this pin. The grinding fixture,
shown in Figure 36, uses the same keyway/fixture
arrangement and includes a slot for the grinding drive pin.
Again, the grinding machine uses this slot as the timing
reference. This arrangement produces a highly non
repeatable reference as will be discussed in more detail in
Chapter 6. Due to this stochastic component, it is
necessary to separate the lift error from the timing error
in order to more tightly control the repeatable lift errors.
Timing Relative to Lobe 1
The timing of all cam lobes except lobe 1 can be
measured relative to lobe 1. Since all stochastic
components due to fixture errors are removed, the lobeto
lobe 1 timing is repeatable and can be tightly controlled.
While timing angles for CNC camshaft grinding machines are
specified relative to a reference other than lobe 1, this
presents no problem as the absolute angle is readily
calculated from the relative angle measurement.
CHAPTER 4
THE STOCHASTIC PROCESS MODEL
Process modeling is used to establish the relationship
between the measured and commanded lift for any given cam
lobe. A stochastic process model of the combined cam
grinding machine and the existing CNC controller can be
developed using established modeling techniques. These
techniques best fit dimensionally homogeneous experimental
data to nominal data and produce an empirical relationship
between process input and output. To develop such a model,
it is advantageous to have a general idea of the causes of
process errors. If the source of errors can be identified,
then the selection of the form of the model does not need to
be made blindly, and the number of models tested can usually
be reduced.
General Linear LeastSquares Estimation
The technique of linear leastsquares estimation is
used for the model developed [21,22]. In this technique,
the input to the system is manipulated and the output is
recorded. These data are collected and a linear algebra
based minimization technique is then used to best fit model
coefficients based on the experimental data. The model
35
obtained must then be examined for its ability to correctly
predict the behavior of the process.
In general, a process can be estimated by the discrete
model
Cn = so cn1 + al cn2" +a, n1 p (41)
+ 0P mnd + P1 m1d + + Ps1 mnqd+ + n
where
cn = controlled variable
mn = manipulated variable
p = order of model in controlled variable
q = order of model in manipulated variable
Ci = coefficient of controlled variable (i=0,1,2,...,pl)
pj = coefficient of manipulated variable (j=0,l,2,...,ql)
d = delay
n = residual error
The controlled and manipulated variables can be measured for
any number of observations N. The resulting N equations can
then be written in matrix form where n = 0, 1, 2,..., Nl as
c = xb + (42)
where
C1 C2 ... Cp mod m1d ... mlqd
Cg ... CIp mld md m2qd
x. (43)
CN2 CN3 CNp MNld mN2d .. mNqd
and where
Co
c !
C2
C 1
I eNI
ao
01
Pp1
Po
Pi
(44)
The residuals of the model are
E = c xb
(45)
If the square of the residual errors is minimized then the
model coefficient matrix can be solved for as
(46)
.8 =[XT1 X] C
where b contains the estimates of the model parameters.
Identification of Lift Error Model
The process model was employed to develop an
understanding of the interaction of lift error at different
lobe angles. The actual control scheme implemented controls
the lift at each of 360 points on the cam surface as if they
were separate variables. This means that there are
essentially 360 control systems for each cam lobe. The
process model developed here does not attempt to model the
control of the lift over a series of parts, but rather it
examines the relationship of the lift errors, and
consequently the 360 control systems, for individual cam
lobes.
As stated previously, it is useful to understand the
lift error source when developing the process model.
Figure 41 shows the lift error and lift acceleration for
the process studied. The use of the term acceleration here
is imprecise, since the workspeed is not constant during one
rotation. Therefore, the effective lift acceleration during
machining is somewhat different. To avoid confusion, the
acceleration shown in Figure 41 will be referred to as the
geometric acceleration. As suggested by Figure 41, a
strong correlation between the observed lift error and
geometric acceleration exists. The form of the interaction
model is suggested by examination of the mathematical
relationship between follower lift and acceleration.
The geometric acceleration can be expressed using the
backwarddifference expression [23] as
A2m im 2 mg _+ m' (47)
A62 h2
where m. represents the manipulated variable which is the
commanded lift in this case. Assuming that the correlation
between lift error and acceleration suggested by Figure 41
is correct, then from equation (47) it can been seen that
the lift error at a given lobe angle is related to lift
commanded at that angle as well as lift commanded at the two
Lift Acceleration and Error
1 !
0.6' \ l !
0   . ....... ............... ..........................................................
S0.6 1     *..........i.........   **......* '    ** ...........
........ ..... ....... .... .. .... ......... .....
0o.   J
> I
0.8 r
0. 2   .. ......... ......................................................
.04.. ... ......../............. ....
  ....  ... ... .................... .................................
0.6.... ... .. ........................... ......... ............. ....
0.81
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention
S Acceleration .... Error
Figure 41 Lift Acceleration and Measured Lift Error
preceding angles. This relationship suggests that these
terms should be included in the model.
It is noted that this model includes no terms of the
controlled variable co, which represents the measured lift
at lobe angle 0. This is necessary, since with postprocess
inspection, the value of the controlled variable is not
available until after the process is completed. Therefore,
it is not useful to include these components in a model
designed for process control based on postprocess
inspection. The process model selected must be a purely
regressive model, where the process output is expressed in
terms of the commanded input.
Estimation of Model Parameters
Based on the correlation between lift error and the
geometric lift acceleration developed in the previous
section, the form of the stochastic interaction model is
selected. The model is given as equation (48) and models
the measured lift co using two unknown parameters.
CE = p m + 3 P'[ me 2 rpi + m2j (48)
The first term in the model is the product of model
parameter 3 and the commanded lift. The second term is the
product of parameter V3 and the commanded geometric
acceleration of the lift.
The actual correlation between commanded lift values
and inspection results were developed using a generalized
form of (48). The general second order regressive model,
assuming no delay, is given as
C = P0 m + Pi nm1 + 2 me2 (49)
The coefficients 0,fP, and $2 were found to be 1.84934,
1.69993, and 0.85091 respectively.
A detailed comparison of these three coefficients with
those from equation (48), demonstrates a compelling
confirmation of the form of the model given in equation
(48). Equating coefficients of equation (48) and equation
(49) for mc_ gives
S _P 1.69993 0.84997 (410)
1 2 2
Equating coefficients of m_2 gives
P:= 2 (411)
Substituting in the value of PI from equation (410) and the
value of 0, from the regression analysis into equation
(411) gives
0.84997 = 0.85091 (412)
These values for P2 differ by less than 0.12%. Finally
equating coefficients of me and substituting in the value of
P0 and using the average value of p, gives
0 = po = 1.84934 0.85044 = 0.99890 = 1 (413)
Again this result is consistent with the proposed model.
Diagnostic Checking of the Model
This model can be examined for its ability to
accurately predict the process errors through examination of
the residuals as defined by equation (45). These
residuals, shown in Figure 42, indicate good agreement
between the model and actual process results. The residual
errors are an order of magnitude smaller than the measured
lift errors. While these errors are small, they are clearly
deterministic. This indicates that the model fails to
account for all deterministic components of the process
error. Yet, in spite of testing many models, this small
deterministic component in the residuals proved difficult to
eliminate using a simple form of model.
Two possible sources of the deterministic component
seen in the residuals are discussed for the purpose of
future model refinement. The first source is the workspeed
generation method. This source is suggested since the
relationship between lift errors and geometric acceleration
used in this analysis is based on constant rotational speed.
However, the workspeed varies within a rotation during the
grinding operation. The workspeed generation method is
discussed in more detail in Chapter 5.
The second possible source of the deterministic
component is the difference between reference follower
diameter and the changing grinding wheel diameter. This
factor is considered for two reasons. First, the change in
Residuals of Lift Error
Second order regressive model
0.0008
S 0.00 0 .... .........6.....7.. .. . ... ... ......'.. .... .... J
o 0.0005 . ...... ... ........... . .. .........
" .0 03 ..................... .......... ........... .... ............... ....... ...................
 0.00031 i
% 0 0 ..... ...... .. .... . ..   .. ...... ... .. ... ..... .................... ........
 0 .00 0 2 4      ..
" 0 .0 0 0 1     .. .............................................. ..................................... ................ :
0 .0 0 0 1 ........... .... ........ .. ............ ...
0.00012  ....
0.0002 1
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention
Figure 42 Stochastic Model Residual Errors
43
grinding wheel size produces a change in the grinding wheel
footprint and consequently its cutting ability. Second,
changes in the wheel size alter the commands which the
grinding machine controller generates for controlling the
individual machine axes. As these commands change, the
commanded movements of the machine axes are altered and the
process dynamics change accordingly. The change in process
dynamics affects the geometric errors produced in the part.
Further testing would be required to determine if either of
these factors produce effects large enough to be measured.
Potential for Model Improvement
If the method of workspeed generation is known, the
form of regressive model can be modified to account for this
component. Error components correlated to grinding wheel
size cannot be handled directly in the planned
implementation because direct access to wheel size is not
available in the interface used in this research. If the
current wheel size were available, a disturbance feedforward
controller could be implemented to compensate for errors due
to a changing wheel size.
CHAPTER 5
PROCESS CONTROL STRATEGY
The Control Interface
The control approach taken in this project is to modify
camshaft program data based on the postprocess inspection
results. This approach, known as command feedforward
control, allows for a software implementation and
demonstration of the effectiveness of the system. All the
necessary control parameters are manipulated using the
existing control interface. This approach follows industry
standards in describing camshaft geometry and is easily
adapted to cam grinders from different manufacturers.
Part programs for CNC camshaft grinders typically
consist of several main types of commands which will be
referred to as fields. The first field is the lift field.
The lift field contains the 360 commanded lift values for a
cam lobe, the base circle radius and the follower diameter.
The lift values are specified as a function of the lobe
angle. Since the individual lobes of a camshaft may have
different contours, multiple lift fields may exist for a
given camshaft. The second type of field is the timing
field. This field is used to specify the timing angle of
the individual lobes relative to a timing reference. The
third field is the workspeed field and is used to specify
the workspeed (camshaft rotational speed during machining)
as a function of the lobe angle. The fourth field, the lobe
position field, specifies the position of the lobes along
the camshaft axis.
In the implemented process control scheme, the lift and
timing fields will be directly manipulated to control the
lift and timing errors respectively. The workspeeed field
is considered in this discussion, but it is not modified.
The lobe position field is not a concern.
From these four fields, the machine controller
calculates the command signals for the individual machine
axesofmotion. In the compensation control scheme
implemented in this work, the internal command generation
scheme is unaltered. Rather the lift and timing fields,
from which the internal commands are generated, are modified
based on the inspection results. These modified fields
effect the generation of corrected control commands for the
machine axesofmotion.
The workspeed field is generated offline using a
proprietary algorithm. Investigation of this field
indicates that the workspeed is a normalized parameter based
on the demanded acceleration of the wheelhead, including
nonlinearities due to servo demand limiters and the
commanded lift acceleration. Comparisons of the workspeed
field and corresponding lift acceleration indicate that
workspeed is roughly inversely proportional to lift
acceleration as shown in Figure 51.
Since the workspeed field is based on the commanded
lift, it changes as the commanded lift field changes.
However, this field is developed offline and can therefore,
be held constant if desired. This is advantageous since
modifying the workspeed field, based on the compensated lift
field, would amplify the effects of the compensations made
to the lift field. This is readily demonstrated by
observing that when the commanded lift, and consequently the
commanded lift acceleration, for a region of the cam is
increased, the commanded workspeed for this region would be
decreased when recalculated. As shown in Chapter 4, lift
errors are correlated to lift acceleration and therefore are
essentially dynamic positioning errors. Consequently, a
decrease in workspeed would decrease the dynamic positioning
error of the machineaxesof motion and decrease the lift
error relative to the commanded lift field. This decrease
in error would occur in addition to the decrease effected by
the modification of the commanded lift field. Thus the
system would tend to overcompensate and might exhibit
oscillatory behavior.
Lift Acceleration and Workspeed
0 .8 .................... I, ............. .. .............. .......... .. ................
0 ...... .................... ... ...... ........... ................ ..... ..... ................... :.......... .........................
,, / /
0 .4   .. ... ..... .. ................ .................. .................. ..................................
S 0 .2  ..  ....... ................ ..........................
N
= 0 A 
S0.2....... ....................... ............... ..........
E
 0 .2 . ...........I...... 
0
Z: 0 1
3 100 150 200 250 300
Lobe Angle (degrees) CCMM Convention
SAcceleration ..... Workspeed
Figure 51 Typical Workspeed and Lift Acceleration
Lift Error Controller Model
An Interacting Lift Process Model
As shown in Chapter 4, interaction exists between the
lift commanded at different lobe angles. Therefore, the
process can be described as a system which has 360
interacting variables [21]. The form of the regressive
model developed in Chapter 4 suggests the nature of the
interaction between variables. A block diagram for this
interacting system is shown in Figure 52. This figure
shows that the lift produced is a function of the lift
commanded for the given lobe angle and the lift commanded
for the two preceding lobe angles, where
n = Part sample number
0 = Lobe angle
so'n = Desired lift
co,n = Measured lift
fen = Feedforward desired lift
eon = Measured lift error
mOn = Commanded lift
Am ,n = Lift compensation
wen = Disturbances
Additionally the lift controllers are given as:
Gf(B) = Feedforward controller
Gc(B) = Process controller for feedback
Gmc(B) = Cam grinder internal controller
49
    ,
Controller for s,_,
m
i_ _l.n 2,n
 
Sg,n
m6 m0
 m.n m l.n
Controller for seI
Figure 52 Interacting Lift Variables
Sl,n I

I
S+l.,n
I
4]
L,,,,
 L~C    
The lift transfer functions are given as
G (B) = CCMM
G (B) = Machining Process
G, (B) = Lumped G (B) and G_(B)
G,, (B) = interaction between m, and c,
Ge (B) = interaction between m, and c,
Noninteractinq Control of Lift
From a theoretical control perspective, it is possible
to design a noninteracting control system which eliminates
the interaction of the 360 lift variables [21]. Figure 53
shows the block diagram for such a control system. The
controllers required to eliminate interaction are given as:
De1 = Controller between m.,_ and c,
Do2 = Controller between m2 and c,
While this control strategy has potential for excellent
control, it is complex. Fortunately, if interactions
between variables are small, as will be shown to be the case
for the lift variables, the process model can be simplified.
The Implemented Lift Controller
While the interaction between lift values commanded at
different lobe angles has been established as a source of
lift error, this interaction will be neglected in the
implemented control model. This may be justified by
examining equation (48) which is repeated here as (51).
Sg 1,n Controller for s,_i
~^ L
Ime1,n m 2,n
_n  "","
G_ Gp (B)
G e2 ((B
D i(B) +
+ G (B)
Gf (B) WO,n
SO,.n ee,n m.n + men + 
G, (B) G,, (B) Gm (B
Gp (B) (+
Gi (B)
r" ~   jc
Me, n Mel,n
S6+l,n I Controller for s, I
F 1 IC
Figure 53 Noninteracting Lift Control System
Figure 53 Noninteracting Lift Control System
Ca = .99890 mn + 0.84997 [m0 2 m0_ %_,2] (51)
The first term of this equation is the contribution of
the commanded lift to the total lift predicted by the model.
The second term is the contribution of the geometric
acceleration to the total lift predicted. The geometric
acceleration term represents the interaction of commanded
lift values at adjacent lobe angles. If the geometric
acceleration and lift terms are evaluated separately, the
geometric acceleration term is found to be three orders of
magnitude smaller than the lift term in critical regions of
the cam. For example, at a lobe angle of 340 (convention
Figure 34), the geometric acceleration and lift terms are
0.0001 and 0.2218 respectively. Therefore, due to the
dominance of the lift term on the predicted lift, it is
possible, in a closed loop control strategy, to neglect the
acceleration term and consequently the interaction of lift
variables.
The block diagram for the implemented command
feedforward/feedback strategy is shown in Figure 54. The
control system calculates the 360 individual lift commands
me, needed to update the commanded lift field. That is, the
control system shown in Figure 54 is invoked 360 times for
each cam lobe. An integral controller of the form
(B) KB (52)
G(B) s
is selected where K is the integral control gain.
fe,n
Ir 
S.,n
Implemented Lift Control System
en
7
Figure 54
To further simplify the control model, ideal transfer
functions are assumed. Thus G,(B) = 1 and G,(B) = 1. The
solution for the feedforward controller is
1 1
Gf(B) 1 =1 (53)
Gp(B) 1
The control equation can then be written as
KB
mn = Sn+ 1 B (Sn ,n) (54)
The process equation is
en = Wen+ n en (55)
From these two equations, the closed loop control can be
solved for as
B W (56)
C.= 1 ( K)B w, + (56)
Even while individual lobes on a camshaft often have
the same desired lift, the measured error results are not
identical. Therefore, the control system is applied to each
lobe independently. This requires a separate lift command
field for each lobe which creates a slight practical
problem. The CNC controller of the cam grinder used in this
study is not designed to handle high data transfer rates.
Hence, excessive time is required to read in the modified
lift field. This tends to reduce productivity, since the
machine is out of service during data transfer.
Control of Relative Timing
The control of lobe timing is a much simpler system to
model. Unfortunately, it is complicated by practical
matters for the systems used in this work. In the existing
process, different timing fixtures are used during the
grinding and inspection processes. These fixtures establish
the timing datum for their respective operations. The
repeatability of these fixtures is an important
consideration. Poor repeatability introduces a large
stochastic component into the data which negatively affects
the ability of the control system to effectively control
lobe timing.
Based on these considerations, the timing of lobe 1 is
controlled relative to the keyway/fixture reference, while
the control of all other lobes is performed relative to lobe
1. The timing relative to lobe 1 is highly repeatable and
therefore allows for aggressive control of timing relative
lobe 1. The block diagram for the implemented timing
control system is shown in Figure 55, where
e,n = Desired timing angle
cen = Measured timing angle
fon = Feedforward desired timing angle
e0,n = Measured timing angle error
men = Commanded timing angle
AmOn = Timing angle compensation
wen = Disturbances
fen
Wen
+ me,
Implemented Timing Angle Controller
Figure 55
Additionally the timing controllers are given as
Gf(B) = Feedforward controller
G((B) = Process controller for feedback
Gmc(B) = Cam grinder internal controller
The timing transfer functions are given as:
G (B) = CCMM
Gm(B) = Machining Process
Gp ) (B) = Lumped Gm (B) and Gm(B)
Much of the notation introduced for the lift controller is
reused here. However, no confusion should result as the
meaning of the symbols will be clear from the context in
which they are used.
Again if an integral controller is selected and the
same simplifying assumptions are made concerning the
transfer functions, the timing control equation becomes
K ( ,n Cn) (57)
mOn : + i nn B On
This equation is implemented to control the timing of lobe 1
relative to the keyway/fixture and to control the timing of
all other lobes relative to lobe 1.
CHAPTER 6
PROCESS NOISE
From a practical standpoint, all production and
inspection processes have stochastic and repeatable
components. The stochastic component, also referred to as
the repeatability error or signal noise, is defined as six
standard deviations (60) of the process output. The noise
in the control system has components due to both the
manufacturing and inspection processes.
The inspection process noise can be directly evaluated
and expressed as the standard deviation of multiple
inspections of the same workpiece. Process noise is
evaluated based on the measured variability of the parts
produced. Since the actual values of the controlled
variables differ from the measured values of these
variables, the inspection process noise is superimposed on
the manufacturing process noise. However, if the inspection
process is highly repeatable, relative to the manufacturing
process, it is not necessary to separate the noise from the
two sources, and a good estimate of the process
repeatability is obtained from the measured values of the
controlled variables. For the machines studied, the
repeatability of the inspection process is more than ten
58
times better than the repeatability of the manufacturing
process [24,5].
Even with a highly repeatable inspection process, the
lack of repeatability in the machining process can present
problems for control based on postprocess inspection. It
is not possible to correct for process noise using post
process inspection. In fact, control strategies which are
overaggressive will actually worsen the situation through
increasing the process variability [25,26]. Therefore, a
strategy designed to attenuate the deleterious effects of
process noise is desired. Two different approaches are
considered.
First, Statistical Process Control (SPC), a type of
deadband control, historically favored by industrial
engineers and statisticians [25] is considered for its
suitability. Second, traditional feedback control with the
addition of a discrete first order filter, as favored by
control engineers [22,26], is discussed and justified for
use in this problem.
Statistical Process Control
SPC typically refers to the use of Shewhart control
charts and WECO run rules as defined in the text by the
Western Electric Staff [27]. These rules are based on
comparing current inspection results with the mean and
standard deviation of parts previously produced. The WECO
rules suggest that an outofcontrol condition (shift in
process mean value or increase in process variability)
should be suspected if one or more of the following occurs:
1. An instance of the controlled variable deviates from
the nominal by more than three standard deviations.
2. Two out of three instances deviate from the nominal by
more than two standard deviations.
3. Four out of five instances deviate from the nominal by
more than one standard deviation.
4. Eight consecutive instances with all positive or all
negative deviations occur.
While these rules are useful for detecting shifts in the
mean of the controlled variable, they do not indicate the
size of the shift. Additionally, Shewhart and other SPC
charts require storing past inspection results in order to
evaluate the WECO rules. For a camshaft with eight lobes, a
run of 25 parts requires storing, recalling and evaluating
72,000 floating point numbers. This represents significant
overhead in terms of storage and execution time.
Traditional Feedback Control with Filtering
It has been shown by Koenig [26] that feedback process
controllers amplify nonrepeatability error, also called
pure white noise, for processes with short time constants.
The idea of a time constant for a discrete manufacturing
process is different in nature from the time constant of a
continuous process which is sampled at discrete points in
time. For discrete processes, the full effects of applied
compensation are realized in the very next part. The
amplification of process noise is a problem as it means an
increase in the variability of the parts produced. To
reduce the effects of process noise on the controlled
variables, a discrete first order filter of the form
Cf (n)= C, + (1 a) c~ (n (61)
can be applied, where
a = filter constant (between 0 and 1)
cn = measured value of controlled variable for part n
c,(n, = filtered value of controlled variable for part n
cfn_) = filtered value of controlled variable for part n1
For white noise, the standard deviation of the filter
input 7, is related to the standard deviation of the filter
output c, by
0= (62)
ao V 2 a
By inspection of equation (62), it is apparent that To is
less than (. for all values of a less than one. For a equal
to 0.15 (filter coefficient used for lift data in trials)
the standard deviation of filter output to the input can be
62
calculated from equation (62) as
o 0. 15 0.285 (63)
o V 2 0.15
The use of this filter effectively introduces a time
constant and corresponding time lag to the system while
reducing the standard deviation of the process noise. For
such a filter the effective time constant of the filter Tf
is
T = h (64)
S lIn(1 a)
where h is the sample period.
Using equation (64), an effective time constant can be
calculated in terms of the number of parts as
T, = 6.15 parts (65)
In(1 0.15)
A discrete filter for the lift requires the introduction of
the lobe angle subscript 0 and is given as
Cf (,n) = a C6,n + (1 a) Cf (,n1) (66)
where c0, is the measured value of the lift s at lobe angle
0 for part n. Equation (61) can be directly implemented as
a filter for the timing angle 0, where cn is the measured
value of the timing angle for part n.
Unfortunately, the introduction of such a substantial
time constant can cause overshoot in integral only control.
This overshoot can be greatly reduced by including a
proportional term in the control equation. This
63
proportional term was not included at the time of trials and
the filtering technique was therefore modified at high
errortonoise ratios. For the trials conducted in this
work, the filtered value was reinitialized after each
adjustment to the manipulated variables. This effectively
eliminated any overshoot but reduced the effectiveness of
the filter to limit noise. As implemented, the filtering
process effectively became a weighted average of parts
inspected between adjustments. With this modification the
process noise reduction can be approximated as [25]
o i (67)
where N is the number of parts inspected to calculate the
compensation.
The use of discrete filters reduces data storage
requirements and calculations as compared with SPC.
Discrete filters can be applied to allow compensation based
on the first part produced, where the errortonoise ratio
is generally high. The compensation frequency can be
reduced as the errortonoise ratio decreases. SPC
generally lacks this flexibility and requires the inspection
of many parts prior to initial compensation.
Error Repeatability A Preliminary Study
A preliminary study was done to determine the size of
the repeatable process errors relative to the stochastic
process errors. The tests were performed as described in
Appendix A. For this study, 11 camshafts were ground. The
camshaft geometry and grinding parameters were similar those
used in the control system experiments described in Chapter
7. The only adjustments made to the process were to correct
for base circle size error.1
Process Repeatability of Lift Errors
Lobe 1 lift error is shown in Figure 61 for selected
camshafts. Clearly, the general pattern of lobe error
repeats from part to part. Figure 62 shows the mean and
the standard deviation for all 360 measured lift error
values of lobe 1 for the 11 parts. This figure clearly
indicates the high value of the mean process error as
compared to the process noise.
Figure 63 shows the 360 discrete errortonoise ratios
for the grinding process. For lobe angles where high errors
are measured, correspondingly high errortonoise ratios are
obtained. The ratios here might even suggest that with such
high errortonoise ratios, effective process control could
be achieved with no special considerations taken for process
noise. In fact, this will work quite well in eliminating the
larger lift errors. However, as the lift errors decrease,
the process noise will remain unchanged and the errorto
1 Size has little effect on our problem; no attempt to
control it is made.
Lift Error for Lobe 1
Six selected runs
0.0008
0.0007
0.0006     
0 .0 0 05ooo  .......  ..... .....................
0o .o o o0 5 T. ...... .... ... . .. ........ .............. .. ........................................................
0 .0 00 4  ..  ..... ..  ............ ..................................................
0 .0 0 02o 3  .......... ............. .... .... ........................... ......................
0 .0 0 0 12      .... . ........... .............. ...... ............... ....
0 .0 0O0 1 I . I ...... .... ..... .... ...... ......... ......... ...... .......
0.0001 *** ...t^ .. .....
0.0002 .. ^
0.0002
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention
 run 1  run 3  run 5
 run 7  run 9 ......... run 11
Figure 61 Lift Error Process Repeatability.
Lift Error for Lobe 1
Mean and Standard Deviation
0.0008
0.0007? 
 0.0006 .
0.000
0.000 5 1           
0 .0 0 0 ..................................... ................. .............. .........................
2 0.0004
S 0 .0 0 0 3 .... ........... .. ........................................
S0 .0 0 0 5 ... ..... ... ............. ....... ....I..: ...............................................................
E 0.0001
,,,_ r" ~ ' 
0 .0 0 0 1 ......... ................... .... ..... ...... ................ ... ...............
0.0002
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention
....... Mean Error  Standard Deviation
Figure 62 Lift Error Mean and Standard Deviation
Error to Noise Ratio for Lobe 1
6.0
5.0
4.0
3.01
2.0
.................... ...... .......... .
1.01 .. 
0.0 .
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention
Lift ErrortoNoise Ratio
1
i \
1
...... ... .
Figure 63
68
noise ratio will decrease dramatically. It is in this area
that the filtering technique becomes useful. The results
shown for lobe 1 are typical of results for all four lobes.
Process Repeatability of Timing Error
As shown in Figure 64, the lobe timing measured
relative to the keyway/fixture reference is far less
repeatable than the timing measured relative to lobe 1. In
the existing process, different timing fixtures are used in
grinding and inspection processes. These fixtures establish
the timing datum for their respective operations. The
repeatability of these fixtures is an important
consideration as it greatly affects the ability of the
control system to effectively control lobe timing.
Based on these results, the expected process variation
for the 11 parts is calculated to be 1.2 degrees. If
instead, the timing is measured using one of the cam lobes
as a reference, a large reduction in process noise is
effected. By convention, lobe 1 is selected as the
reference. If the process variation is now evaluated
relative to lobe 1, a variation of 0.126 degrees is obtained
for the same 11 parts. This represents an order of magnitude
improvement over the timing repeatability measured relative
to the keyway/fixture reference. Clearly, different degrees
69
Lobe Timing Error
no compensation applied
0.5
2 0.4 Lobe 1 relative to keyway
0 0 .2 .................................................................. .. .......... .. ......................... ......................................... i
c 0.3 *
o 0.21 ..........  . ..
0. ... ... ........ .......... .............. ............  ......   
O
c 0.3+ 1 Lobes 2. 3 & 4 relative to lobe 1 ........................... ..... .. .. .. ..... ..
0.4 1 ,
0 5 10 15 20 25
Sample Number
Figure 64 Timing Angle Repeatability Error
of control are possible depending on the timing reference
selected.
For the cases studied, the timing variation relative to
the keyway/fixture exceeds the total timing tolerance. This
means that the existing process is not capable of producing
all parts to specification and no postprocess gauging with
feedback strategy will change this. As a practical matter,
this situation can be improved upon. The timing errors
relative to keyway/fixture can be corrected during the
mating of the camshaft and drive gear. Additionally the
problem is related to the quality of fixtures which can also
be improved.
Still, this presents a problem in demonstrating the
control system with limited parts available for trials. For
processes with greater variability, more samples are
required to separate the mean error signal from the process
noise for a given error size of interest. Correcting for
timing errors relative to lobe 1 and lift errors requires
many fewer samples than does the correction of timing errors
relative to the keyway/fixture.
CHAPTER 7
EXPERIMENTAL RESULTS
The trials performed in this study were performed at
Andrews Products in Rosemont Illinois, using a Landis 3L
series cam grinder as shown in Figure 71. An Adcole Model
911 CCMM, shown in Figure 72, was used for camshaft
inspection. The grinding parameters, including lift and
timing specifications for the trial part, are given in
appendix A. The camshaft inspection parameters are given in
Appendix B. The control system parameters and a functional
diagram of the control software are included in Appendix C.
During the trials, the only changes made to process
parameters were those made to the lift and timing command
fields.
The trials were carried out in a production environment
over a run of approximately 100 pieces. The parts were
milled from stock to within 0.01 inches of finished
dimensions. Camshafts were case hardened prior to grinding.
The camshafts were ground, inspected, and compensation
applied according to the schedule shown in Table 71. This
schedule represents an attempt to balance the demands of
production needs with experimental technique.
Figure 71 Landis 3L Series Cam Grinder
Source: "StateoftheArt 3L Series CNC Cam Grinding
Systems," Litton Industrial Automation Systems, Pub. No
3L88 FR 3M, 1988. Used with permission.
Figure 72 Adcole Model 911 CCMM
P~~~~
74
Table 71 Grinding, Inspection, and Compensation Schedule
Compensation Part(s) Compensation Notes
Number Ground Based on Part(s)
0 1 nominal part data
1 2 1
2 313 2
3 1428 313 Parts
1493
4 2982 1428 made
two
5 8388 7982 days
after
6 8993 8388 113
Results of Lift Error Control
The reduction in lift error is shown in three different
ways. First, the measured lift error for lobe 2, after each
application of compensation is shown in Figure 73,
Figure 74, Figure 75, and Figure 76. Second, the total
lift error, that is the maximum positive minus the maximum
negative lift error, is shown for all four lobes in
Figure 77. Third, the root mean square (RMS) of all 360
individual lift measurements for all four lobes is shown in
Figure 78. Clearly, all three measures show improvement in
the lift error. The data for lobe 2 show an order of
magnitude reduction in the measured lift error with other
lobes showing smaller reductions, depending primarily on the
initial value of the lift error. Both the RMS and total
lift error data give some indication of the effects of the
Measured Lift Error Lobe 2
2.0E04
1.OE041
O. OE + 00
, / i
.. Comp. #0
Comp. #1
........... ................ ....... j
.....................................
corn0.,, 
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention
Lift Error using Control Scheme
1.OE04
2.0E04 ...
3.0E04 ....
4.OE04 
5.0E04
6.OE
Figure 73
Measured Lift Error Lobe 2
2.0E04 ,
1. OE04 ..
O.OE +00.   *.. 
1.0E 04 ......................
2.0 E 0 4 i . ............. ..............
3 .0 E 041 ..  .   
i
4.0E04
5.OE04t
6.OE04+
0
50 100 150 200 250
Lobe Angle (degrees) CCMM
300 350 400
Convention
Lift Error using Control Scheme
SComp. #2
Comp. #3
Figure 74
Measured Lift Error Lobe 2
2.0E04 ,
1.0E04 Comp. #4
o.OE00 A Comp. #5
20E04.
3.0E04 
4.OE04 .
5.0E04
6.OE04 
0
50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention
Figure 75 Lift Error using Control Scheme
...........................................................................................
.......................................... I ......... I ......................................

Measured Lift Error Lobe 2
2.OE04
1.0E 0 4 .. ......... ... .. .
O.OE+00
1.OE04i
2.O E041  .........
3.0E 04 .. .... ... .. .... ....................
4.0E 04    .  
5.OE04      
6.0E04 1
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention
Figure 76 Lift Error using Control Scheme
Comp. #6
Total Lift Error
average for all parts inspected
7.0E04
6.OE04
5.OE04
4.OE04 
3.0E041
20E04 
1.OE041
O.OE 00
0
lobe 1
lobe 2
lobe 3
lobe 4
...... ..... ..... ..... ....     I. ..   
         .. . .. . .
 '_ r _~   

I
.. 
1 2 3 4 5 6
Compensation Number
Figure 77 Total Lift Error using Control Scheme
RMS of Lift Error
average values for all parts inspected
2.OE04
l.E04 
i lobe 1
1.6E041 
1.2E 04 4 ... ............ ... .......
1 .O E 0 4 .. ........... ... ..............'.. ....... lo b e 3
8 .O E 0 5 .. .... .. .................................................... 
Slobe 4
6. OE05   
.0E 05 .......... ... .. ................
4.0E051 ..
2.0E05i
O.OEO 00
0 1 2 3 4 5 6
Compensation Number
Figure 78 RMS of Lift Error using Control Scheme
81
two day interruption in the trials. This indication appears
in the form of modest reductions and even slight increases
in form error between compensation number 2 and 3. When the
camshafts used to calculate the compensated command number 3
were ground, the machine had been operating for eight hours.
Compensation number three was then applied when the machine
was restarted two days later. Thus, the increase in lift
error associated with this delay, combined with the further
reduction in lift error when compensation is applied in a
timely manner, suggests that thermal effects account for a
small but nevertheless measurable component of lift error.
Additionally, mixed results are obtained between
compensation numbers three and four. Fifty parts were
ground between the inspected parts and the application of
compensation. These results may indicate a lift error
component due to the changing wheel size (0.2 inch for 50
parts). Results after compensation numbers four, five and
six, where no parts were ground between the inspected parts
and the application of compensation, demonstrate that more
frequent compensation further reduces errors.
Results of Timing Error Control
Control of Timing Relative to Lobe 1
The results of the timing control relative to lobe one
are shown in Figure 79. Small initial timing errors, as
compared with the results of the preliminary repeatability
Timing Error Relative to Lobe 1
0.10,
0.05     ..  
0.00 .
 0 .0 5 . .. ..... ........ .. ............ ..... ....... ...... .. .. ... ... ........ ... .... .. ...
0.05; 
0.1 0        '    
s~        
0.10i1 '.';  ..
0.1 5
0.201     
0. 0............... .......... ........... .................... ...............................
0.25
0.30
I 0.35
0 3 510 .. .... ..... ........ .... .... ..... ... .....
0. 40 1 T I T I
5 10 15 20 25
Part Number
80 85 90
Lobe 2
Lobe 3
Lobe 4
Figure 79 Timing to Lobe 1 using Control Scheme
study, existed for the camshaft used in the control trials.
While the initial values were small, it is significant to
note that no timing errors developed during the control
period and the timing angle remained within the noise levels
established in the preliminary trials.
This is especially significant considering that the
timing angle is decomposed from the measured lift data.
Since compensation is applied to the lift field, the lift
data are altered. Nevertheless, the results show that it is
possible to control these parameters separately. This
ability to control these coupled parameters as if they were
separate, has great practical benefits and greatly
simplifies the control system.
Control of Timing Relative to Keyway/Fixture
The control of timing of lobe 1 relative to the
keyway/fixture was less successful as shown in Figure 710.
While the process is approximately centered about zero
error, the timing of lobe 1 to the keyway/fixture
demonstrated a greater variability than the uncontrolled
timing measured in the preliminary repeatability studies.
This result is not surprising considering the low errorto
noise ratio which exists for this parameter and the short
effective time constant of the process. It was anticipated
that this control might prove overaggressive. However, to
avoid introducing additional complexity for these initial
Timing Error Relative to Keyway
1.00
0.801
i I
0.40
0.20
0.00
I I /
0.40
, \
10 15 20 25
Part Number
80 85 90
Timing to Keyway using Control Scheme
S. Lobe 1
0.604
0.801
. . . . .. I
; I
I..ii.
Figure 710
85
trials, this was accepted. Recommendations for improving
the effective control over timing relative to the
keyway/fixture are discussed in Chapter 8.
CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS
It has been shown that significant improvements in
camshaft lift error can be realized through feedback of
postprocess inspection results. It was shown that even
while the lift errors at nearby lobe angles interact, good
results are obtained when this interaction is neglected.
The use of discrete filtering prevents an increase in the
variability of a process about a mean operating point.
Implementation of the Control System
The strategy employed relies on the use of standard CNC
industrial camshaft inspection and production equipment.
Many camshaft production facilities currently use equipment
suitable for implementation of this control strategy. These
facilities could realize significant quality control
improvements by better utilizing existing production
equipment.
Prior to introduction into a manufacturing environment,
the control system requires the development of a user
interface and further investigation of factors discussed in
the next section. Minor modification of the grinding
machine controller software would allow for seamless
operation and higher data transfer rates. Additionally,
minor modifications to the CCMM's software are required for
a high quality implementation of the control system. As a
practical matter, production and inspection equipment need
to be located together. In the ideal implementation,
camshafts are ground and automatically transferred, in
sequence, to the CCMM. The feedback of inspection data and
compensated command fields occurs automatically.
Further Work
While the effectiveness of such a control system has
been clearly demonstrated, much interesting work remains in
this area. For a full implementation of discrete filtering,
a proportional term needs to be added to the controller and
the parameters of the model should be optimized through the
use of the regressive model and verified in the actual
implementation. With regard to control of the timing of
lobe 1 to the keyway/fixture, production implementations
would need to be less aggressive. This could be readily
accomplished through the use of greater filtering and the
addition of a proportional term in the controller. As
before, the controller parameters would need to be selected
through simulation and verified experimentally.
Additionally, a noninteracting control system could be
investigated for lift errors. This system should be
investigated for its ability to improve control system
convergence rate.
The long term stability of the control system should be
investigated with particular attention to the emergence of
high frequency components in the lift command field. While
the high frequency components of the commanded lift field
are effectively filtered by the limited bandwidth of the
grinding machine, these high frequency components produce
large internal following errors and consequently excessive
demand on the servo motors. Over time, these high frequency
components may lead to increased lift error and a
degradation of control.
Finally, additional trials should be conducted
investigating the effectiveness of different control gains.
Specifically the effectiveness of a control system using
control gain of unity for the lift error should be compared
with the results obtained in this study.
APPENDIX A
GRINDING PARAMETERS
The trials used a Ramron 1A90OB7 grinding wheel.
The wheel was dressed using a Norton LL271B Sequential
Cluster.
The lift and timing fields, given in the following pages,
are specified according to the cam grinding machine
convention described in Chapter 3.
