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
- Creator:
- Dalrymple, Timothy Mark, 1959- (
*Dissertant*) Ziegert, John (*Thesis advisor*) Schueller, John (*Reviewer*) Matthew, Gary K. (*Reviewer*) Phillips, Winfred M. (*Degree grantor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1993
- Copyright Date:
- 1993
- Language:
- English
- Physical Description:
- x, 117 leaves : ill., photos ; 29 cm.
## Subjects- Subjects / Keywords:
- Acceleration ( jstor )
Control systems ( jstor ) Error rates ( jstor ) Geometric angles ( jstor ) Geometry ( jstor ) Grinding ( jstor ) Grinding wheels ( jstor ) Machinery ( jstor ) Machining ( jstor ) Parametric models ( jstor ) Dissertations, Academic -- Mechanical Engineering -- UF Mechanical Engineering thesis M.S - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt ) theses ( marcgt )
## Notes- Abstract:
- 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-l;ine calibration or learning of grinding machin positioning errors. Additionally, it offers advantages over such techniques, in that it is able to automatically adapt to changing process conditions. the sytem 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.
- Thesis:
- Thesis (M.S.)--University of Florida, 1993.
- Bibliography:
- Includes bibliographical references (leaves 114-116).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Timothy Mark Dalrymple.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 001892844 ( ALEPH )
29713749 ( OCLC ) AJW8098 ( NOTIS )
## UFDC Membership |

Full Text |

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 In-Process Gauging 8 Compensation Using CMM for Post-Process 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 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 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 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 vii 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 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 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. 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 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]. 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 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 I 5 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. 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: off-line modeling of positioning errors, in-process inspection, and post-process 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 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 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 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 inspection. 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 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 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 11 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) ds x M + r (3-2) N where M = r sin dscos 6 (3-3) do N= r cos 6 dsin 0 (3-4) dO 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 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 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 I 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 Opening N N Side / Se / Closing Side Direction of Rotation 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 I 4 Direction of Rotation 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 JAOR CAOR 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 JAOR. 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) 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 + -) (3-8) 4 =arctan ye (3-9) X 0 26 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 Axis-of-Rotation Figure 3-8 Grinding Wheel Path Calculations MAOR 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. 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 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 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 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 Measured Lift Nominal Lift 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. 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 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 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 cn-1 + al cn-2" +a,- n-1 p (4-1) + 0P mn-d + P1 m-1-d + + Ps-1 mn-q-d+ + 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,...,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) where 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 Co c ! C2 C- 1 I eN-I ao 01 Pp-1 Po Pi (4-4) The residuals of the model are E = c xb (4-5) If the square of the residual errors is minimized then the model coefficient matrix can be solved for as (4-6) .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 lobes. 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.... ... .. ........................... ......... ............. ------------....- -0.8-1--------------------------------------- 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 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 4-2 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 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 ----------------- --.. -- ---....... ................ .......................... N = 0 A -- S-0.2....... ....................... ............... .......... E | -0 .2 -. -.--.-.........I..-....------- -------------------------------- 0 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 49 -- ----- --- ------- , Controller for s,_, m- i--_ _-l.n -2,n -------------- - Sg,n m6 m0 | m.n m -l.n Controller for seI Figure 5-2 Interacting Lift Variables S-l,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 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 variables. 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. fe,n Ir - S.,n Implemented Lift Control System en --7 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 KB 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 fen Wen + 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. 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 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 considered. 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 62 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 is 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 63 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 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 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 0.0008 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 6-1 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 6-2 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 Error-to-Noise Ratio 1 i \ 1 ...... ... . Figure 6-3 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 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 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 6-4 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 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. 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 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 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 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 ----P-~-~---~~----------- 74 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 14-93 4 29-82 14-28 made two 5 83-88 79-82 days after 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 2.0E-04-- 1.OE-041 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.OE-04- -2.0E-04 ... -3.0E-04- .... -4.OE-04- - -5.0E-04- -6.OE 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-- ..---- -------- -.- - ---------------------- i -4.0E-04-- -5.OE-04-t -6.OE-04+- 0 50 100 150 200 250 Lobe Angle (degrees) CCMM 300 350 400 Convention 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 -20E-04--. -3.0E-04 -- -4.OE-04- . -5.0E-04- -6.OE-04 - 0 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 2.OE-04 1.0E -0 4 .. ......... ... .. . O.OE+00 -1.OE-04i-------------- -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 7.0E-04- 6.OE-04 5.OE-04--- 4.OE-04 -- 3.0E-041 20E-04 ---- 1.OE-041-- O.OE 00-- 0 lobe 1 lobe 2 lobe 3 lobe 4 ...... ..... ..... ..... ....-- -- --- --------------- --I. .. - - ----- ---- --------- -- -- -- .. .. . -- '_- -r _~--- -------- -- ---------------- ------------- I ..-- --- 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 2.OE-04 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- .. 2.0E-05-i O.OE-O 00 0 1 2 3 4 5 6 Compensation Number Figure 7-8 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 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 0.10, 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 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 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 7-10 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 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 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 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. APPENDIX A GRINDING PARAMETERS The trials used a Ramron 1-A-90-O-B-7 grinding wheel. The wheel was dressed using a Norton LL-271B 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. |