Title Page
 Table of Contents
 List of Figures
 Review of the literature
 Camshaft geometry
 The stochastic process model
 Process control strategy
 Process noise
 Experimental results
 Conclusions and recommendation...
 Grinding parameters
 Grinding parameters
 Camshaft inspection parameters
 Software outlines and control system...
 Bibliographical sketch

Title: Improved process control in camshaft grinding through utilization of post process inspection with feedback
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00084169/00001
 Material Information
Title: Improved process control in camshaft grinding through utilization of post process inspection with feedback
Physical Description: x, 117 leaves : ill., photos ; 29 cm.
Language: English
Creator: Dalrymple, Timothy Mark, 1959-
Publication Date: 1993
Subject: Mechanical Engineering thesis M.S
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (M.S.)--University of Florida, 1993.
Bibliography: Includes bibliographical references (leaves 114-116).
Statement of Responsibility: by Timothy Mark Dalrymple.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00084169
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 001892844
oclc - 29713749
notis - AJW8098

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
        Page iv
    Table of Contents
        Page v
        Page vi
    List of Figures
        Page vii
        Page viii
        Page ix
        Page x
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    Review of the literature
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Camshaft geometry
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
    The stochastic process model
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
    Process control strategy
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
    Process noise
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
    Experimental results
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
    Conclusions and recommendations
        Page 86
        Page 87
        Page 88
    Grinding parameters
        Page 89
    Grinding parameters
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
    Camshaft inspection parameters
        Page 107
        Page 108
        Page 109
        Page 110
    Software outlines and control system parameters
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
    Bibliographical sketch
        Page 117
        Page 118
Full Text







Copyright 1993


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.


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.



LIST OF FIGURES . . . . . vii

ABSTRACT . . . . . .ix


1 INTRODUCTION . .. . . 1

Scope of the Problem . . . 1
Camshaft Grinding Technology . . 2
Analysis of Camshaft Geometrical Errors . 4
Potential for Improvement . . 5


Compensation Through Positioning Error Modeling 6
Compensation Through In-Process Gauging . 8
Compensation Using CMM for Post-Process
Inspection . . . ... 9


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


General Linear Least-Squares Estimation .. 34
Identification of Lift Error Model .. . 36
Estimation of Model Parameters . . .. 39
Diagnostic Checking of the Model . .. 41
Potential for Model Improvement . .. 43



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


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


Implementation of the Control System . .. 86
Further Work . . . . 87





REFERENCES . . . . . 114




Figure 3-1 Typical Four Lobe Camshaft . .. 12

Figure 3-2 Radial Cam with Roller Type Follower 13

Figure 3-3 CCMM Angle Convention-Clockwise Rotation 17

Figure 3-4 Grinding Machine Angle Convention .. 19

Figure 3-5 Camshaft Axes-of-Rotation . .. 22

Figure 3-6 Grinding Between Centers . .. 23

Figure 3-7 Centerless Grinding Technique. . ... 24

Figure 3-8 Grinding Wheel Path Calculations .. 27

Figure 3-9 Camshaft Inspection on a CCMM . .. 28

Figure 3-10 Best Fit of Lift Errors . . .. 32

Figure 4-1 Lift Acceleration and Measured Lift Error 38

Figure 4-2 Stochastic Model Residual Errors .. 42

Figure 5-1 Typical Workspeed and Lift Acceleration 47

Figure 5-2 Interacting Lift Variables . .. 49

Figure 5-3 Noninteracting Lift Control System 51

Figure 5-4 Implemented Lift Control System . .. 53

Figure 5-5 Implemented Timing Angle Controller . 56

Figure 6-1 Lift Error Process Repeatability 65

Figure 6-2 Lift Error Mean and Standard Deviation 66

Figure 6-3 Lift Error-to-Noise Ratio . . 67

Figure 6-4 Timing Angle Repeatability Error .. 69

Figure 7-1 Landis 3L Series Cam Grinder . .. 72


Figure 7-2

Figure 7-3

Figure 7-4

Figure 7-5

Figure 7-6

Figure 7-7

Figure 7-8

Figure 7-9

Figure 7-10

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


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



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 post-process inspection, a closed loop control scheme is

proposed. This strategy makes use of post-process

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 off-line 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.


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 high-speed machine

motions. Applications are common in both chip producing

equipment and high-speed dedicated assembly machinery [1].

Additionally, cams are used in fields as diverse as blood

separation and automated laser-scanner 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].



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


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 variable-speed

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.

Computer-controlled 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



were simply used to identify non-conforming 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 post-process

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.


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: off-line modeling of positioning errors,

in-process inspection, and post-process inspection with


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


measured using off-line laser interferometry and electronic

levels. These errors are then decomposed into geometric and

thermally-induced components. The geometric errors are

thermally-invariant and modeled with slide positions as

independent variables. The thermally-induced 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


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

warm-up period is not generally required.

Compensation Through In-Process 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 in-process inspection and compensation

on machining centers [15], and the idea of an in-process

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 in-process gauging in 1978. The proposed

system employed a gauge probe located 180 degrees

out-of-phase 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


Compensation Using CMM for Post-Process 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

post-process inspection of end-milled 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 b-spline tensor-product


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.


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 3-1, 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 3-2. From this figure, it is evident

that the point-of-contact between the cam surface and the

cam follower does not generally lie along the follower's

line-of-action. This illustrates the difference between cam


Side View

End View

Figure 3-1 Typical Four Lobe Camshaft

Figure 3-2 Radial Cam with Roller Type Follower

contour and the follower lift produced. Errors in cam

contour, at the point-of-contact between the follower and

cam profile surface, produce errors in follower lift.

For the radial cam with a roller follower shown in

Figure 3-2, the cam contour can be calculated as [20]

x = r cos + (3-1)
1 (3-1)

x M + r (3-2)


M = r sin dscos 6 (3-3)

N= r cos 6 dsin 0 (3-4)

r = rb + r + S (3-5)

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


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 3-2. 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 3-2. The base circle radius is the

datum for measurement of lift values. The positive

rotational direction is specified as counterclockwise when

viewed from the non-driven end of the camshaft, looking


towards the driven end. The details of the implementation

are described in the following two sections.

CCMM Convention

The convention illustrated in Figure 3-3 shows the

convention use by the Adcole 911 CCMM. This convention is

dependent on the camshaft direction-of-rotation. That is,

the CCMM direction-of-rotation is programmable and is

selected as the application direction-of-rotation. 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 axis-of-rotation)

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 3-3 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 Non-driven End to Driven End

N N Side /
Se / Closing


Direction of

Figure 3-3 CCMM Angle Convention-Clockwise Rotation

Grinding Machine Convention

The convention used on the CNC grinding machine does

not consider the camshaft functional direction-of-rotation.

Rather, the camshaft geometry is specified in a format which

reflects the direction-of-rotation of the grinding machine.

This convention is shown in Figure 3-4.

Implications of Machining Techniques

In analyzing camshaft geometry, it is necessary to

carefully consider the axis-of-rotation. 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 3-1.

Table 3-1 Camshaft Axes-of-Rotation

MAOR Axis-of-Rotation for Machining

IAOR Axis-of-Rotation for Inspection/Decomposition

CAOR Camshaft Axis-of-Rotation defined by Centers

JAOR Camshaft Axis-of-Rotation defined by Journals

View Looking from Non-driven End to Driven End


4 Direction of

Figure 3-4 Grinding Machine Angle Convention

The first two axes are the machining axis-of-rotation

(MAOR) and the inspection axis-of-rotation (IAOR). These

two axes do not refer to the actual camshaft axis-of-

rotation, but rather to the axis-of-rotation 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

skew-symmetrically 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

axes-of-rotation as described below.

First, the journal axis-of-rotation (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 axis-of-rotation (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 3-5, represent the

physical axes-of-rotation 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 3-6. 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 3-7. 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



Figure 3-5 Camshaft Axes-of-Rotation

Figure 2-6

indinc Between Centers

Figure 3-7 Centerless Grinding Technique

Source: "State-of-the-Art 3L Series CNC Cam Grinding
Systems," Litton Industrial Automation Systems, Pub. No.
3L-88 FR 3M, 1988. Used with permission.

centerless grinding approaches, the MAOR is defined as the


During inspection, measurement data must be decomposed

relative to a specific axis-of-rotation. 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 axis-of-rotation are readily

calculated [20] as

x = x + (r cos 8 x) (3-6)

yc = y + (r sin y) (3-7)

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

S (x + -) (3-8)

4 =arctan ye (3-9)
X 0


where p is the distance from the MAOR to the grinding wheel

axis-of-rotation, and y is the corresponding lobe angle for

the grinding process. The cam grinding operation is

illustrated in Figure 3-8.

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 in-feed if the

metal removal rate is to be closely controlled. The reason

for this can be seen from Figure 3-2. As discussed

previously, it is evident that the point-of-contact between

the cam surface and follower does not generally lie along

the follower's line-of-action. Consequently, an in-feed 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 point-or-contact 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 spring-loaded sliding

gauge head as shown in Figure 3-9. The movement of the

gauge head is measured using either precision scales or a

laser interferometer. This approach allows direct

Grinding Wheel

Figure 3-8 Grinding Wheel Path Calculations


Figure 3-9 Camshaft Inspection on a CCMM

1.* I W- -

evaluation of camshaft lift errors and eliminates the need

to mathematically transform results from a non-conforming

follower to the actual follower diameter. When necessary,

this conversion can be performed using equations (3-6) and

(3-7) where the radius of the non-conforming 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.


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 3-9. 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 axis-of-rotation. 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


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 3-10. 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 3-9. The CCMM determines the location of the



Figure 3-10 Best Fit of Lift Errors

timing reference based on this pin. The grinding fixture,

shown in Figure 3-6, 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 lobe-to-

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.


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 Least-Squares Estimation

The technique of linear least-squares 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


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


Cn = so cn-1 + al cn-2" +a,- n-1 p (4-1)
+ 0P mn-d + P1 m-1-d + + Ps-1 mn-q-d+ + n


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,...,p-l)

pj = coefficient of manipulated variable (j=0,l,2,...,q-l)

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,..., N-l as

c = xb + (4-2)


C-1 C-2 ... Cp mo-d m-1-d ... ml-q-d
Cg ... C-Ip ml-d md m2-q-d

x. (4-3)

CN-2 CN-3 CN-p MN-l-d mN-2-d .. mN-q-d

and where

c !

C- 1

I eN-I





The residuals of the model are

E = c xb


If the square of the residual errors is minimized then the

model coefficient matrix can be solved for as


.8 =[XT-1 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


As stated previously, it is useful to understand the

lift error source when developing the process model.

Figure 4-1 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 4-1 will be referred to as the

geometric acceleration. As suggested by Figure 4-1, 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

backward-difference expression [23] as

A2m im 2 mg _+ m' (4-7)
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 4-1

is correct, then from equation (4-7) 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 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention

S-- Acceleration .... Error

Figure 4-1 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 post-process

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 post-process

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 (4-8) and models

the measured lift co using two unknown parameters.

CE = p m + 3 P'[ me 2 rp-i + m-2j (4-8)

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 (4-8). The general second order regressive model,

assuming no delay, is given as

C = P0 m + Pi nm-1 + 2 me-2 (4-9)

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 (4-8), demonstrates a compelling

confirmation of the form of the model given in equation

(4-8). Equating coefficients of equation (4-8) and equation

(4-9) for mc_- gives

S- _P -1.69993 0.84997 (4-10)
1 2 2

Equating coefficients of m_-2 gives

P:= 2 (4-11)

Substituting in the value of PI from equation (4-10) and the

value of 0, from the regression analysis into equation

(4-11) gives

0.84997 = 0.85091 (4-12)

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 (4-13)

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 (4-5). These

residuals, shown in Figure 4-2, 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
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 4-2 Stochastic Model Residual Errors


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.


The Control Interface

The control approach taken in this project is to modify

camshaft program data based on the post-process 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

axes-of-motion. 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 axes-of-motion.

The workspeed field is generated off-line 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

non-linearities 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 5-1.

Since the workspeed field is based on the commanded

lift, it changes as the commanded lift field changes.

However, this field is developed off-line 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 machine-axes-of 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 ----------------- --.. -- ---....... ................ ..........................
= 0 A --
S-0.2....... ....................... ............... ..........
| -0 .2 -. -.--.-.........I..-....------- --------------------------------
Z: 0 1

3 100 150 200 250 300
Lobe Angle (degrees) CCMM Convention

SAcceleration ..... Workspeed

Figure 5-1 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 5-2. 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


-- ----- --- ------- ,

Controller for s,_,

i--_ _-l.n -2,n
-------------- -


m6 m0
| m.n m -l.n

Controller for seI

Figure 5-2 Interacting Lift Variables

S-l,n I



- -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 5-3

shows the block diagram for such a control system. The

controllers required to eliminate interaction are given as:

De-1 = Controller between m.,_ and c,

Do-2 = Controller between m-2 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 (4-8) which is repeated here as (5-1).

Sg 1,n Controller for s,_i
----~-^ L-

Ime-1,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"- ~ ------ ----------- --j---c
Me, n Me-l,n

S6+l,n I Controller for s, I

F 1 IC

Figure 5-3 Noninteracting Lift Control System------

Figure 5-3 Noninteracting Lift Control System

Ca = .99890 mn + 0.84997 [m0 2 m0_ %_,2] (5-1)

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 3-4), 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


The block diagram for the implemented command-

feedforward/feedback strategy is shown in Figure 5-4. 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 5-4 is invoked 360 times for

each cam lobe. An integral controller of the form

(B) KB (5-2)
G(B) s

is selected where K is the integral control gain.


Ir -


Implemented Lift Control System



Figure 5-4

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 (5-3)
Gp(B) 1

The control equation can then be written as

mn = Sn+ 1- B (Sn ,n) (5-4)

The process equation is

en = Wen+ n en (5-5)

From these two equations, the closed loop control can be

solved for as

-B W (5-6)
C.= 1 ( K)B w, + (5-6)

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 5-5, 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



+ me,

Implemented Timing Angle Controller

Figure 5-5

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) (5-7)
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.


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


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 post-process 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


First, Statistical Process Control (SPC), a type of

dead-band 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 out-of-control 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 non-repeatability 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- (6-1)

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 n-1

For white noise, the standard deviation of the filter

input 7, is related to the standard deviation of the filter

output c, by

0= (6-2)
ao V 2 -a

By inspection of equation (6-2), 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


calculated from equation (6-2) as

o 0. 15 0.285 (6-3)
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


T = h (6-4)
S lIn(1 a)

where h is the sample period.

Using equation (6-4), an effective time constant can be

calculated in terms of the number of parts as

T, = 6.15 parts (6-5)
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 (,n-1) (6-6)

where c0, is the measured value of the lift s at lobe angle

0 for part n. Equation (6-1) 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


proportional term was not included at the time of trials and

the filtering technique was therefore modified at high

error-to-noise 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 (6-7)

where N is the number of parts inspected to calculate the


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 error-to-noise ratio

is generally high. The compensation frequency can be

reduced as the error-to-noise 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 6-1 for selected

camshafts. Clearly, the general pattern of lobe error

repeats from part to part. Figure 6-2 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 6-3 shows the 360 discrete error-to-noise ratios

for the grinding process. For lobe angles where high errors

are measured, correspondingly high error-to-noise ratios are

obtained. The ratios here might even suggest that with such

high error-to-noise 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 error-to-

1 Size has little effect on our problem; no attempt to
control it is made.

Lift Error for Lobe 1
Six selected runs

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 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 6-1 Lift Error Process Repeatability.

Lift Error for Lobe 1
Mean and Standard Deviation
0.0007-----?----------------------------- --------------------------------------
- 0.0006- .
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 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention

....... Mean Error -- Standard Deviation

Figure 6-2 Lift Error Mean and Standard Deviation

Error to Noise Ratio for Lobe 1






.................... ...... .......... .

1.01- .. ---

0.0 .
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention

Lift Error-to-Noise Ratio


i \


...... ... .

Figure 6-3


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 6-4, 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


Lobe Timing Error
no compensation applied

2 0.4 Lobe 1 relative to keyway

0 0 .2 .................................................................. .. .......... .. -......................... ......................................... i
c 0.3 *

o 0.21- .......... --- ----. .-.

0. ... ... ........ .......... .............-. ............ - ...... --- ----- ---

c -0.3+ 1 Lobes 2. 3 & 4 relative to lobe 1 ........................... ..... .. .. .. ..... ..
-0.4 1 ,
0 5 10 15 20 25
Sample Number

Figure 6-4 Timing Angle Repeatability Error

of control are possible depending on the timing reference


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 post-process 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.


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 7-1. An Adcole Model

911 CCMM, shown in Figure 7-2, 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


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 7-1. This

schedule represents an attempt to balance the demands of

production needs with experimental technique.

Figure 7-1 Landis 3L Series Cam Grinder

Source: "State-of-the-Art 3L Series CNC Cam Grinding
Systems," Litton Industrial Automation Systems, Pub. No
3L-88 FR 3M, 1988. Used with permission.

Figure 7-2 Adcole Model 911 CCMM


Table 7-1 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 3-13 2

3 14-28 3-13 Parts
4 29-82 14-28 made
5 83-88 79-82 days
6 89-93 83-88 1-13

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 7-3,

Figure 7-4, Figure 7-5, and Figure 7-6. Second, the total

lift error, that is the maximum positive minus the maximum

negative lift error, is shown for all four lobes in

Figure 7-7. Third, the root mean square (RMS) of all 360

individual lift measurements for all four lobes is shown in

Figure 7-8. 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



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


-2.0E-04 ...

-3.0E-04- ....

-4.OE-04- -



Figure 7-3

Measured Lift Error Lobe 2
2.0E-04 ,

1. OE-04 ..

O.OE +00-. --- -- --*.. ---

-1.0E -04 ......................

-2.0 E -0 4 i .- ............. ..............

-3 .0 E -041-- ..---- -------- -.- - - ----------------------




50 100 150 200 250
Lobe Angle (degrees) CCMM

300 350 400

Lift Error using Control Scheme

SComp. #2

Comp. #3

Figure 7-4

Measured Lift Error Lobe 2
2.0E-04 ,

1.0E-04 Comp. #4

o.OE00 A Comp. #5


-3.0E-04 --

-4.OE-04- .


-6.OE-04 -

50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention

Figure 7-5 Lift Error using Control Scheme


.......................................... I ......... I ......................................


Measured Lift Error Lobe 2

1.0E -0 4 .. ......... ... .. .



-2.O E-041 -- --------.........

-3.0E -04 .. .... --... ..- ...----.---- -----....................-----

-4.0E -04 -- --- -- ------------------------. --- --------

-5.OE-04--- -- -------- ------ -------- ---- -

-6.0E-04 1
0 50 100 150 200 250 300 350 400
Lobe Angle (degrees) CCMM Convention

Figure 7-6 Lift Error using Control Scheme

Comp. #6

Total Lift Error
average for all parts inspected




4.OE-04 --


20E-04 ----


O.OE 00--

lobe 1

lobe 2

lobe 3

lobe 4

...... ..... ..... ..... ....-- -- --- --------------- - --I. .. - - -

----- ---- --------- -- - -- - -- - .. . .. . .

-- '_- -r _~--- -------- -- ----------------


..-- ---

1 2 3 4 5 6
Compensation Number

Figure 7-7 Total Lift Error using Control Scheme

RMS of Lift Error
average values for all parts inspected
l.E-04----------------------------------------------- -----
i lobe 1
1.6E-041----------- -

1.2E -04 4 ... ............ ... .......
1 .O E -0 4 .. ........... ... ..............'.. ....... lo b e 3
8 .O E -0 5 ..- .... .. .................................................... -
Slobe 4
6. OE-05 -- --- ---
.0E -05- .......... ... .. ................
4.0E-051- ..
O.OE-O 00
0 1 2 3 4 5 6
Compensation Number

Figure 7-8 RMS of Lift Error using Control Scheme


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 re-started 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 7-9. Small initial timing errors, as

compared with the results of the preliminary repeatability

Timing Error Relative to Lobe 1

0.05-- ------- ----- ---- --------------- ------..---- ------ -----

0.00 .
- 0 .0 5 . .. ..... ........ .. ............ ..... ....... ...... .. .. ... ... --........ ... .... .. ...
-0.05--;- ---
-0.1 0 --- ---- --- -------- ---------- ----- ---- -----'-------- ------- --- --------- ----

s~-- ------ --------- --- --- ------ --- ---- ---------
-0.10i1- '-.'; -- ..----------
-0.1 5

-0.201--- ----- -- ----- ------------------------------------ ----------
-0. 0--............... -----.........----------------.---- ---........... .................... ...............................


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 7-9 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 7-10.

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 error-to-

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

i I




I I /


, \

10 15 20 25
Part Number

80 85 90

Timing to Keyway using Control Scheme

S. Lobe 1



. . . . .. I

; I

Figure 7-10


trials, this was accepted. Recommendations for improving

the effective control over timing relative to the

keyway/fixture are discussed in Chapter 8.


It has been shown that significant improvements in

camshaft lift error can be realized through feedback of

post-process 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


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 non-interacting 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.



The trials used a Ramron 1-A-90-O-B-7 grinding wheel.

The wheel was dressed using a Norton LL-271B Sequential


The lift and timing fields, given in the following pages,

are specified according to the cam grinding machine

convention described in Chapter 3.

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