EVALUATION OF LINEAR SEGMENT LENGTH AND LOCAL CURVATURE RADIUS ALONG AIRFOIL LEADING AND TRAILING EDGES by Mohammad Nahid Islam Razive B.Sc., Bangladesh University of Engineering and Technology, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2012 © Mohammad Nahid Islam Razive, 2012 ii ABSTRACT Airfoil is the basic profile geometry of impeller and turbine blades. The operational efficiency of these blades is governed by stringent tolerance specifications on the airfoils. The specified tolerances are commonly evaluated from discrete coordinate data collected in sections by a touch-probe coordinate measuring machine (CMM). These measurement data are subject to inspection inaccuracies associated with CMM measurement operation. Apart from well-known inspection parameters like profile tolerance, profile thickness and edge radius, the leading edge (LE) and trailing edge (TE) are specified with a unique set of geometric parameters like the maximum linear segment length restriction and the minimum curvature radius restriction. This thesis focuses on evaluating these two localized geometric restrictions along the leading edge and trailing edge of an airfoil. This thesis first presents a robust algorithm to identify the longest linear segment. The main feature of the proposed algorithm is the explicit consideration of measurement uncertainty. The algorithm starts by detecting relatively small linear segments and then merges these segments to determine the longest feasible linear segment under given measurement uncertainty. The effect of measurement uncertainty and data point resolution on the performance of the presented algorithm is demonstrated through case studies. Once the linear segments are identified and excluded, the remaining data points only belong to the non-linear segments. As minimum radius can occur at any location, curvature radius at each point along the non-linear segments is evaluated. Curvature radius at a specific point can only be estimated from its neighborhood. The chosen neighborhood size needs to be balanced between capturing local curvature attribute and effectively considering the effect of measurement uncertainty. An algorithm is thus proposed to evaluate radius via a rolling scheme of five consecutive data points in order to retrieve the local curvature information of the mid-point. A statistical approach is employed where all feasible radii are considered in order to reliably estimate the desired radius. Biarc construction is used as a tool to calculate radius. Compared with existing radius estimation methods, the proposed method has demonstrated to yield better accuracy with varying measurement uncertainty and data point resolution. iii TABLE OF CONTENTS Abstract .................................................................................................................................... ii Table of Contents ................................................................................................................... iii List of Tables .......................................................................................................................... vi List of Figures ........................................................................................................................ vii Acknowledgements ................................................................................................................. x Dedication ............................................................................................................................... xi 1. Introduction ..................................................................................................................... 1 1.1 Manufacturing specifications ..................................................................................... 1 1.2 Computer aided inspection ......................................................................................... 1 1.3 Airfoil inspection ........................................................................................................ 3 1.3.1 Inspection process ............................................................................................... 3 1.3.2 Section geometry ................................................................................................ 4 1.4 Leading and trailing edge inspection parameters ....................................................... 5 1.4.1 Common leading and trailing edge inspection parameters ................................. 6 1.4.2 Specialized leading and trailing edge geometric parameters .............................. 8 1.5 Research objective .................................................................................................... 11 1.6 Methodology ............................................................................................................ 12 1.6.1 Concept of measurement uncertainty ................................................................ 12 1.6.2 Proposed approach ............................................................................................ 14 1.6.3 Evaluation of linear segment length ................................................................. 16 1.6.4 Evaluation of local curvature radius ................................................................. 17 1.7 Organization of thesis ............................................................................................... 21 2. Evaluation of linear segment length ............................................................................ 22 2.1 Existing methods ...................................................................................................... 22 2.2 Methodology ............................................................................................................ 23 2.2.1 Concept of feasible solution .............................................................................. 23 2.2.2 Underlying principle ......................................................................................... 25 2.2.3 Process overview .............................................................................................. 26 2.3 Implementation ......................................................................................................... 26 iv 2.3.1 Initial phase: Detection of feasible linear segments ......................................... 26 2.3.2 Final phase: Calculation of longest linear segment .......................................... 28 2.4 Results and analysis ................................................................................................. 29 2.4.1 Improvement initiative: Locating segment end points ...................................... 31 2.4.2 Case Study 1: Effect of varying measurement uncertainty ............................... 33 2.4.3 Case Study 2: Effect of varying point spacing .................................................. 34 3. Biarc construction ......................................................................................................... 36 3.1 Biarc ......................................................................................................................... 36 3.1.1 Biarc parameters ............................................................................................... 37 3.1.2 Biarc type .......................................................................................................... 38 3.2 Existing methods ...................................................................................................... 39 3.3 Biarc construction from quintet ................................................................................ 40 3.3.1 Proposed approach ............................................................................................ 41 3.3.2 Detection of biarc type ...................................................................................... 43 3.3.3 Results of five point biarc ................................................................................. 44 3.4 Biarc construction from quartet ................................................................................ 49 3.4.1 Proposed approach ............................................................................................ 49 3.4.2 Finding location of joint point .......................................................................... 49 3.4.3 Finding optimum biarc ...................................................................................... 53 3.4.4 Biarc2:2 categories on the basis of shape characteristics ................................... 56 3.4.5 Results of four point biarc ................................................................................. 58 4. Evaluation of local curvature radius ........................................................................... 66 4.1 Existing methods ...................................................................................................... 66 4.1.1 Best fitting based methods ................................................................................ 66 4.1.2 Interval invariant based methods ...................................................................... 68 4.1.3 Image data oriented methods ............................................................................ 68 4.1.4 Hough transform based methods ...................................................................... 69 4.2 Proposed approach ................................................................................................... 70 4.2.1 Underlying principle ......................................................................................... 70 4.2.2 Finite population of valid radii ......................................................................... 72 4.2.3 Median as measure for central tendency ........................................................... 72 v 4.3 Implementation ......................................................................................................... 73 4.3.1 Discretization of measurement uncertainty region ........................................... 74 4.3.2 Generation of feasible cases from datasets ....................................................... 75 4.3.3 Calculation of probability and radius for any individual case .......................... 76 4.3.4 Calculation of local radius from median ........................................................... 77 4.4 Validation of proposed method ................................................................................ 78 4.4.1 Median vs. Weighted mean ............................................................................... 79 4.4.2 Verify six individual datasets ............................................................................ 83 4.4.3 Verify combining all six datasets ...................................................................... 85 4.5 Results and analysis ................................................................................................. 87 4.5.1 Factors analyzed ................................................................................................ 87 4.5.2 Results ............................................................................................................... 90 5. Conclusions .................................................................................................................. 103 5.1 Complete case study ............................................................................................... 103 5.1.1 Simulated input ............................................................................................... 103 5.1.2 Results ............................................................................................................. 107 5.2 Research contributions ........................................................................................... 110 5.3 Limitations and future works ................................................................................. 111 Bibliography ........................................................................................................................ 114 Appendix .............................................................................................................................. 119 Appendix: Curvature validity check for triplet ................................................................. 119 vi LIST OF TABLES Table 1.1 Datasets from five point domain for local curvature radius estimation. ............... 19 Table 2.1 Input segments with theoretical parameters. ......................................................... 30 Table 3.1 Results for 'C' type biarc with uniformly spaced five points. ............................... 46 Table 3.2 Results for 'C' type biarc with non-uniformly spaced five points. ........................ 47 Table 3.3 Results for 'S' type biarc with uniformly spaced five points. ............................... 48 Table 3.4 Conceptual table for five types of four-point biarc and their attributes. ............... 56 Table 4.1 Sample cases considered from datasets. ............................................................... 75 Table 4.2 Results for radius or curvature calculation. .......................................................... 83 Table 4.3 Case when any individual triplet and quartet datasets yield higher error. ............ 84 Table 4.4 Case when any individual triplet and quintet datasets yield higher error. ............ 85 Table 4.5 Case to verify combining all six datasets. ............................................................. 86 Table 4.6 Parameters analyzed in results. ............................................................................. 89 Table 5.1 Input segments with theoretical parameters. ....................................................... 104 Table 5.2 Input data ............................................................................................................ 105 Table 5.3 Results of linear segment length. ........................................................................ 107 Table 5.4 Results of local curvature radii along first non-linear segment. ......................... 108 Table 5.5 Results of local curvature radii along second non-linear segment. .................... 109 Table 5.6 Results of local curvature radii along last non-linear segment. .......................... 109 Table 5.7 Comparison among different applied division number. ..................................... 112 vii LIST OF FIGURES Figure 1.1 Inspection of turbine blade by CMM [1]. .............................................................. 2 Figure 1.2 Airfoil inspection performed at designated sections, (Left) Turbine blade; (Right) Top view of sections [2]. .......................................................................................................... 4 Figure 1.3 Basic airfoil geometry. .......................................................................................... 5 Figure 1.4 Inspection of airfoil blade with master templates [9] ............................................ 7 Figure 1.5 Conceptual profiles along with inspected data. ..................................................... 8 Figure 1.6 Specialized geometric parameters of leading and trailing edge. ........................... 9 Figure 1.7 Measurement uncertainty around measured data point. ...................................... 13 Figure 1.8 Measurement uncertainty region around measured points. ................................. 14 Figure 1.9 Overall flowchart of proposed approach. ............................................................ 15 Figure 1.10 Process overview (Left) Input data; (Middle) Identification of linear segments; (Right) Non-linear segments for local curvature radii calculation. ........................................ 16 Figure 1.11 Rolling scheme of five points for local curvature radius calculation. ................ 17 Figure 1.12 Current 5 point domain for local curvature radius estimation. ........................... 18 Figure 1.13 Generation of feasible curvature radii within U domain. ................................... 20 Figure 2.1 Feasible solution for longest linear segment. ...................................................... 24 Figure 2.2 Residuals from measured points to feasible linear segment. ............................... 25 Figure 2.3 Initial phase: (a) input data points; (b) candidate segments; (c) designation of each candidate segment; (d) merge of 'linear' segments along with neighboring 'non-linear' segments on both sides. ........................................................................................................... 27 Figure 2.4 Final phase: (a) merged segment; (b) identification of longest linear segment; (c) identified linear and non-linear region. ................................................................................... 28 Figure 2.5 Input profile ......................................................................................................... 29 Figure 2.6 Study of increasing U effect for linear segments with spacing of 5 μm. ............ 30 Figure 2.7 Improvement of accuracy after employing end locating algorithm (Line 1). ..... 32 Figure 2.8 Improvement of accuracy after employing end locating algorithm (Line 3). ...... 33 Figure 2.9 Study of increasing U effect for linear segment (Line 2) with different series of spacing. ................................................................................................................................... 34 Figure 2.10 Study of increasing spacing effect for linear segment (Line 3) with different series of U. .............................................................................................................................. 35 viii Figure 3.1 Biarc parameters .................................................................................................. 37 Figure 3.2 Biarc types; (a) 'C' shaped; (b) 'S' shaped; (c) 'J' shaped. .................................... 38 Figure 3.3 Biarc from quintet i.e., five points (a) Biarc3:2; (b) Biarc2:3. ................................ 40 Figure 3.4 Biarc construction from quintet (a) Finding location of joint from θ domain; (b) Validating formulated biarc. ................................................................................................... 41 Figure 3.5 Applying bisection search method for finding joint. ........................................... 42 Figure 3.6 Validity check method for biarc (a) +X axis along P2P1 line, check ordinate of O1', J' & P3'; (b) +X axis along JO1 line, check abscissa of O2''. .......................................... 44 Figure 3.7 Output biarc for case study 1 (‘C’ shaped biarc for uniform spacing). ............... 46 Figure 3.8 Output biarc for case study 2 (‘C’ shaped biarc for non-uniform spacing). ........ 47 Figure 3.9 Output biarc for case study 3 (‘S’ shaped biarc for uniform spacing). ............... 48 Figure 3.10 Biarc from four points i.e., quartet (a) Biarc 1:3; (b) Biarc 2:2; (c) Biarc 3:1. ....... 50 Figure 3.11 Biarc construction when fourth point inside circle (all ‘C’ type). ...................... 51 Figure 3.12 Biarc construction when fourth point outside circle (mix of ‘C’, ‘J’ , ‘S’ type). 52 Figure 3.13 Finding location of joint from θ domain for biarc construction from quartet. ... 54 Figure 3.14 Concept of distance as a measure for finding boundary points. ......................... 55 Figure 3.15 'Distance vs. |rad1-rad2|' plot to demonstrate P3 is the optimum joint point for any biarc3:1. ............................................................................................................................. 59 Figure 3.16 Type 1 biarc; (Top left) 'C' shaped biarc3:1; (Top right) 'C' shaped biarc1:3; (Bottom left) Optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot ....................... 60 Figure 3.17 Type 2 biarc; (Top left) 'C' shaped biarc3:1; (Top right) 'S' shaped biarc1:3; (Bottom left) Optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot ....................... 61 Figure 3.18 Type 3 biarc; (Top left) 'C' shaped biarc3:1; (Top right) 'C' shaped biarc1:3; (Bottom left) Optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot ....................... 62 Figure 3.19 Type 4 biarc; (Top left) 'S' shaped biarc3:1; (Top right) 'C' shaped biarc1:3; (Bottom left) Optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot ....................... 63 Figure 3.20 Type 5 biarc; (Top left) 'S' shaped biarc3:1; (Top right) 'S' shaped biarc1:3; (Bottom left) Non-optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot. .............. 64 Figure 4.1 Concept of feasible radii ...................................................................................... 71 Figure 4.2 Flowchart for feasible radii generation from any particular dataset. ................... 73 Figure 4.3 Discretization of measurement uncertainty region. ............................................. 74 ix Figure 4.4 Calculation of combined probability for a particular radius. .............................. 76 Figure 4.5 Flowchart for calculation of local curvature radius from dataset results. ........... 78 Figure 4.6 Comparison of median, mean and mode for skewed distribution [44]. .............. 79 Figure 4.7 Histogram analysis of valid cases’ radii from individual dataset (triplet1). ........ 81 Figure 4.8 Histogram analysis of combined valid cases from all six datasets. ..................... 82 Figure 4.9 Five input points in relation to an ideal arc (solid blue line): (a) Single arc; (b) Biarc; (c) Triarc. ................................................................................................................ 88 Figure 4.10 Study of varying spacing for arc with U 0.8 μm. ............................................... 91 Figure 4.11 Study of special case of arc with applied U 0.8 μm. .......................................... 92 Figure 4.12 Study of varying U for arc input with spacing of 5 μm. ..................................... 93 Figure 4.13 Study of varying adjacent radius for biarc with U 0.8 μm, spacing 5 μm. ......... 95 Figure 4.14 Study of special case of biarc with applied U 0.8 μm, spacing 5 μm. ................ 96 Figure 4.15 Study of varying U for biarc with spacing of 5 μm, adjacent radius of 1 mm. .. 97 Figure 4.16 Study of varying adjacent radius for triarc with U 0.8 μm, spacing 5 μm. ........ 99 Figure 4.17 Study of special case of triarc with applied U 0.8 μm, spacing 5 μm. ............. 100 Figure 4.18 Study of varying U for triarc with spacing of 5 μm, adjacent radius 1 mm. .... 101 Figure 5.1 Simulated input points on a curvilinear profile. ................................................ 104 x ACKNOWLEDGEMENTS I would like to start expressing my heartfelt gratitude to my supervisor Dr. Hsi-Yung (Steve) Feng for his constant support, guidance and knowledge which he shared with me during the course of my studies. This work would not have been accomplished without his inspiration, patience and wisdom. I would also like to thank my colleagues of the UBC CAD/CAM/CAI Research Laboratory for their support, discussions and friendship. I would like to thank my family for their unconditional love, constant support and immense confidence in me over the years. Also, I wish to express my special thanks to my lovely wife Farzana for supporting me throughout this study. The financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) under the NSERC-CANRIMT Strategic Network Grant is gratefully acknowledged. I would also like to thank our industrial collaborators, Mr. Félix-Étienne Delorme and Dr. Serafettin Engin at Pratt & Whitney Canada Corp., for their essential technical input and advice. xi DEDICATION To my Family 1 1. INTRODUCTION Free form surfaces are widely used in many fields such as automotive, aerospace, die/mold and power equipment industries. For acceptable manufacturing of free-form shaped parts, it is imperative to evaluate if a manufactured part meets the design specifications. 1.1 Manufacturing specifications The inclusion of complex free-form surfaces in manufacturing parts has posed manufacturing challenges. As a result, manufacturers provide geometric specifications and tolerances to ease manufacturing related complexity. Once manufactured, the parts undergo quality inspection processes. The inspected data is then evaluated against design specifications. Robust evaluation methods employed to the inspected data are thus imperative to determine conformity as well as to avoid any wrongful rejection of parts which leads to loss of resources and time. This thesis deals with the robust evaluation of manufacturing specifications. 1.2 Computer aided inspection Computer aided inspection (CAI) is the use of computer based software tools that help quality engineers in inspecting manufacturing parts. Its purpose is to create a faster inspection process and parts with more precise dimensions. With the advance of inspection techniques, coordinate measuring machines (CMM) are commonly employed for parts inspection. The process employs a computer controlled CMM to inspect the part automatically by moving a tactile probe along the work-piece surface, to measure the coordinates of probe center. This contact inspection process is one of the most effective 2 measurement techniques, providing both high accuracy and repeatability. As the thesis deals with evaluation rather than the inspection process, it is assumed that all coordinate points are transformed on a part’s surface after post-processing. Even if using most advanced CMMs, it is not free of measurement inaccuracies. Due to the inevitable measurement uncertainty of the inspection points, evaluation of the related geometric errors becomes challenging. Again, the measured point spacing can vary and this affects the reliability of the geometric error evaluation. In this thesis, both the effects of measurement uncertainty and measured point spacing are examined. Coordinate metrology involves evaluating geometric errors on manufactured parts from the discrete coordinate measurement data. The manufactured parts size and shape can vary widely and impose its own challenge to inspection. This thesis focuses on small sized and free-form shaped object like airfoils of turbine and/or impeller blade. As shown in Figure 1.1, inspected data taken by CMM serves as the input for the geometric error evaluation process. Figure 1.1 Inspection of turbine blade by CMM [1]. 3 The research on characterizing measurement uncertainty of a CMM inspection process in general has received much attention. Given the measurement uncertainty of an inspection process, evaluating geometric error from measurement data is still difficult and needs research attention. The calculation of measurement uncertainty is not considered under the scope of this work and is assumed given for all the calculations in this work. 1.3 Airfoil inspection Airfoil is the principal cross-section of any airfoil blade and basic shape for turbine, impeller blade. Airfoil blades are used in everything from jet engines to wind and water power generators. They are even part of refrigeration processes and new tidal facilities that produce electricity. However, airfoil blades of all sizes and shapes present unique challenge to their manufacturers. 1.3.1 Inspection process An airfoil blade produces an aerodynamic force when it is moved through air. Its accuracy is guaranteed by following strict manufacturing specifications and its high material cost demands robust and accurate evaluation of the manufactured airfoil profile. Moreover, the complex and free-form shape of an airfoil profile creates inspection challenges. Airfoil blades are inspected in sections at designated height along the staking axis as many geometric tolerances/restrictions are specified in the associated airfoil sections [2], [3]. Figure 1.2 shows top view of sectional inspection process for a turbine blade. 4 Figure 1.2 Airfoil inspection performed at designated sections, (Left) Turbine blade; (Right) Top view of sections [2]. 1.3.2 Section geometry Airfoil is the cross-section and basic design unit of turbine or impeller blade in two dimensional spaces. The airfoil blade surface is constructed by stacking these airfoil sections on top of each other along a stacking axis. Each airfoil section contains four basic regions: leading edge (LE), trailing edge (TE), pressure side (PS), and suction side (SS) as shown in Figure 1.3. The LE is the region where the air strikes the blade first and TE is the region from where the air leaves the blade. They govern the air flow direction and play a significant role in airfoil performance [4], [5]. The air gets separated from the leading edge to pressure and suction sides. As LE/TE is responsible for air separation, it is imposed with more stringent 5 tolerances in comparison to rest of the regions. The geometric tolerance/specifications on these four regions are not the same [6], [7]. Leading and trailing edge are generally designed in circular or elliptical shape. The pressure and suction side have less curvature variations unlike the leading and trailing edges. The pressure gradient between these two surfaces contributes to the lift force generated for a given airfoil. This thesis focuses on evaluation of geometric parameters for leading and trailing edges only. Figure 1.3 Basic airfoil geometry. Suction side (SS) Pressure side (PS) Trailing edge (TE) Leading edge (LE) 1.4 Leading and trailing edge inspection parameters The leading and trailing edges are typically small with sharp curvature changes that make those difficult to measure. Although the CAD model of a blade is usually available for the planning of the measurement path, the twist on the work piece, especially near the leading/trailing edges, may result in substantial deviation between the manufactured part and the CAD model. Since, a small change in the airfoil geometry can lead to a large change in aerodynamic performance, analyzing every minor detail of the blade shape is imperative [4]. Therefore, LE and TE are specified with a unique and stringent set of geometric tolerances/restrictions. Among these, the well-known LE/TE inspection parameters are 6 LE/TE profile tolerance, LE/TE profile thickness, overall leading and trailing edge radius etc. In addition to these, there are two very specialized geometric constraints that play critical role in conforming LE/TE minute shape details. These are maximum linear segment length restriction and minimum non-linear segment curvature radius restriction [5]. This thesis focuses on evaluating these two less-known dimensional restrictions applicable for leading and trailing edges only. The common LE/TE inspection parameters with existing methods are discussed in the next sub-section followed by the definition of two specialized dimensional restrictions along with their design significances. 1.4.1 Common leading and trailing edge inspection parameters The parameters, that are talked about in existing literatures and public domain are LE/TE profile tolerance, LE/TE thickness along the normal direction of camber line, leading and trailing edge global radius etc. Pahk et al. proposed a measurement technique for parts having very thin and sharp curve features, which is the case with airfoil blades [6]. They evaluated profile error of leading and trailing edges. They emphasized that actual measurement points must be determined on the CAD geometry for precision tolerance evaluation. Hsu et al. evaluated both profile tolerance and LE/TE thickness [7]. They defined profile tolerance as the sum of the maximum errors on both sides of the nominal curve when the optimal positioning of the measurement data is attended. They also proposed an iterative algorithm for the coordinate setup of airfoil blades inspection. To verify the accuracy of the coordinate systems, profile tolerance for leading edge, pressure side and suction side of a sectional curve were analyzed individually. 7 Thickness dimensions, blade displacement, chord length and twist angle were other parameters which were also evaluated by Hsu et al. Recently, Chen et al. extracted leading/trailing edge points and fitted circles to obtain leading/trailing edge global radii [8]. They established rough zone from the end point of chord line and kept marching until exceeding set threshold value. They consider LE/TE as circular arcs. Apart from the literatures, the public domain also mentions current practices to evaluate LE/TE parameters. Visual inspection with a master template is among them [9]. Here guillotine like gages are used that snap onto airfoil blades. Guillotine gages are industry- specific and have to be custom-made for each blade design and sometimes for each blade surface. This process is slow and imprecise. More importantly, guillotine gages can not accurately measure a blade's leading and trailing edges [3]. There are also some commercial inspection packages like PolyWorks, Geomagic dealing with aforesaid parameters [10], [11]. Figure 1.4 Inspection of airfoil blade with master templates [9] 8 1.4.2 Specialized leading and trailing edge geometric parameters Profile error, thickness criteria and overall LE/TE radii are not sufficient to guarantee the shape of the same. There are two specialized geometric constraints that are tailored to verify localized features of LE/TE. Again, as shown in Figure 1.5, manufactured profile (blue solid line) can deviate from nominal profile (red dashed line) where the manufactured profile contains infinitesimal linear segments because of linear interpolated tool path commands. Figure 1.5 Conceptual profiles along with inspected data. Manufactured profile Nominal profile Inspected data Minimum allowable profile Maximum allowable profile Nominal profile Manufactured profile Inspected data 9 Now, even if the manufactured profile is within allowable profile tolerance (ash boundary line) and has acceptable profile thickness, its locally confined geometry needs to undergo further inspection for comprehensive assessment. The dimensional restrictions of linear segment length and minimum curvature radius serve the purpose. These localized parameters need to be evaluated from discrete inspected points. As shown in Figure 1.6, LE/TE is a curvilinear profile consisting of linear and non-linear segments. The manufacturers usually specify the limiting linear segment or flat part. They also restrict any sharp edge by prescribing the minimum allowable radius. Figure 1.6 Specialized geometric parameters of leading and trailing edge. Longest linear segment Minimum curvature radius Linear segment Non-linear segment In short, parameters that are to be evaluated from inspected sectional coordinate data are longest linear segment and minimum curvature radius. As it can occur anywhere along the LE/TE profile, entire edge data needs to be investigated. The measured parameters can then 10 be verified against prescribed dimensional restrictions to check its conformance. This thesis aims to evaluate these two localized geometric constraints. Performance of component of airfoil is a function of conformance to designed parameters. A sharp corner or small local radius on an airfoil profile can significantly degrade the performance of an airfoil blade [5]. Estimating the curvature radius at each measurement point on an airfoil section is thus an essential task for the comprehensive geometric inspection of a manufactured airfoil blade. Zhang et al. demonstrated that suction side pressure spike is restrained significantly and separation bubble is completely removed through redesigning the leading edge curvature [12]. Again experiments on flat leading edges at different flow conditions were also conducted and a direct impact of the flow structure near leading edge on boundary layer development was reported by Walraevens et al. [13]. It was found that compared to the circular leading edge, the elliptic one with flat segments effectively depress the spike and separation bubble. This explains the usefulness of restricted flat i.e., linear segments in LE/TE. The existing literatures have not considered the evaluation of maximum linear segment length restriction and minimum curvature radius restriction in LE/TE. This thesis aims to evaluate these localized constraint from discrete measured points associated with inevitable measurement uncertainty. Because of the measurement uncertainty the measurement data do not exactly represent the actual points on the inspected manufactured profile which makes the evaluation challenging. The proposed method also explicitly considers the measurement uncertainty. 11 1.5 Research objective Based on scope described in previous sections, the research objective is focused on evaluating the two important but less discussed parameters of airfoil leading and trailing edge: maximum linear segment length restriction and minimum curvature radius restriction. The objective is to evaluate potential linear segment length and to evaluate local curvature radii from non-linear segment along the airfoil leading and trailing edge profile only. As local curvature radii restriction is imposed to confirm the minute shape particulars of the LE/TE, it is to be evaluated locally i.e. involving restricted number of data points. Here the input data are considered to be sectional data inspected by CMM. Since the locations of the segments are not known, proposed algorithm are conducted along entire segmented LE/TE profile. Key assumptions that have been considered for implementation are as follows: 1. The probe center data for measurement are already compensated for the probe radius and resulting points are being considered here for the evaluation. 2. The measurement data of airfoils are already segmented in leading and trailing edge. 3. The measurement uncertainty value is given. 4. The measurement data points are in known sequence. 12 1.6 Methodology As mentioned earlier, the measurement data are associated with uncertainty. Again, as the measurement uncertainty information is provided the strategy to attain research objective needs to be determined. In this section the concept of measurement uncertainty along with proposed approach is discussed. 1.6.1 Concept of measurement uncertainty All the measurement processes are inherently imperfect in nature and therefore the measured points do not represent the actual values of measurement. This dispersion of actual/true value about the measurement value is known as measurement uncertainty. The measurement uncertainty is a parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand. It is a statistically derived term and generally characterized by a standard deviation of a normal distribution. For calculating the combined measurement uncertainty budget , individual uncertainty contributions from the individual factors are calculated and combined together based on the guidelines provided by “Guide to the expression of uncertainty of measurement” [14]. Probe pre-travel, probe bending, probe size imperfection, manufacturing component’s form error are some of the contributing factors towards measurement uncertainty [15]. Due to these inherent problems with the measurement data points, one must be careful that the measured points represent the actual surface sufficiently; otherwise erroneous conclusions may be drawn during evaluations. 13 For practical applications, measurement uncertainty is expressed as an expanded interval about the measurement result such that interval is expected to enclose a large fraction of the distribution of actual values that can be attributed to the measurement value . A coverage factor is multiplied to for getting the expanded uncertainty .The result of a measurement is then expressed as , which means that the true value which can be attributed to measurement value will lie in interval of – to [16]. The coverage factor corresponds to a desired level of confidence. The value of has been set as 3 in this work, which corresponds to a confidence level of 99.74%. The calculation of expanded measurement uncertainty is not considered under the scope of this work and is assumed to be given for all the calculations in this work. Figure 1.7 Measurement uncertainty around measured data point. MeasuredActual 6σ (=U ) +3σ-3σ y As illustrated in Figure 1.7, if uncertainty of a measuring process is assumed to follow a normal distribution, the true/actual point would lie inside the bell-shaped probability density function. The solid red dot represents the measured coordinate point. If an interval of is taken on both sides of the measured point (referred to as throughout this thesis), it can be said that for any measured point, the true point would lie within the span of . 14 In case of one dimensional space, the measurement uncertainty of any measured points with a known would be along the normal direction. For two-dimensional spaces, the true point would be within the circular area [17] of diameter centered at the measured point (shown with red dot) as shown in Figure 1.8. The true point can be anywhere within the circle with decreasing probability of the true point locating far from the measured point. But the probability remains same for same radial distance from measured point. Figure 1.8 Measurement uncertainty region around measured points. Uncertainty domain of U diameter 1.6.2 Proposed approach As mentioned in the research objective, the purpose is to determine linear segments without having any prior knowledge of their locations. So the proposed approach is to identify linear segments along the profile first. Now when the linear segments are extracted from the profile, it will be left with non-linear i.e., curve segments. Since the other objective is to validate minimum curvature radius restriction, all the points available in the non-linear segments are 15 investigated. In other words, curvature radius at each point on the non-linear segments will be evaluated for the conformance of the minimum curvature radius restriction. The overall flowchart of proposed approach is shown in Figure 1.9. Figure 1.9 Overall flowchart of proposed approach. Detect linear segment LE/TE section data Linear Segment Non-linear segment Yes No Consider rolling five point scheme Check min radius < allowable limit Check maximum length > allowable limit Min. radius conformance Max. length conformance Non- conformance Non- conformance Yes No No Yes Calculate radius at mid-point The process overview is shown in Figure 1.10. In Figure 1.10(Left) which is the input data for the problem is basically a curvilinear profile of LE/TE. As shown in Figure 1.10(Middle) 16 once the linear segments (shown by red solid line) are identified and extracted, it is left with non-linear segments only (Figure 1.10(Right)). Figure 1.10 Process overview (Left) Input data; (Middle) Identification of linear segments; (Right) Non-linear segments for local curvature radii calculation. Input data Linear segment Non-linear segment The minimum curvature radius can be formed locally within a very small region of LE/TE and induce a sharp edge, so the information is to be gathered from narrow local domain i.e., engaging as few points as possible for radius calculation. Again, as curvature is local property it is estimated at every point of LE/TE curve region involving local data only. 1.6.3 Evaluation of linear segment length Due to the inevitable measurement uncertainty, data points collected from even a true flat/linear section may not lie on a straight line after inspection. As a result, the linear segment length restriction becomes very difficult to be evaluated accurately. Effective consideration of the measurement uncertainty is the key to reliably identify the longest possible linear segment. Here it is assumed that the linear segments are much greater than inspection point resolution. Chapter 2 of this thesis is dedicated to this topic. 17 1.6.4 Evaluation of local curvature radius The measurement data are in the form of discrete data points. Curvature radius at a specific data point can thus only be estimated from its neighboring data points. The first issue in developing an applicable estimation procedure is to determine the suitable number of neighboring data points to be employed that reliably represents the local property of curvature radius. Employing too many neighboring data points would unavoidably smooth out the actual local curvature. The lowest feasible number of points is clearly three (the point of interest and its closest neighboring point on each side), with which a unique radius can be calculated. However, due to the measurement uncertainty in the data points, a radius value calculated via the circle formed by (passing through) the three points may be highly unreliable. More points should thus be employed for improved reliability. A balance of symmetric neighborhood size needs to be reached to capture local attribute of curvature and to effectively consider the effect of measurement uncertainty. An additional point from each side of the three points is then included to make a total of five points to calculate the local curvature radius. Figure 1.11 Rolling scheme of five points for local curvature radius calculation. Point of interest Previous 5pt. domain Current 5pt. domain for curvature radius calculation Rolling direction 18 To be more specific, an algorithm is to be formulated to evaluate the minimum curvature radius restriction via a rolling scheme of five consecutive points in order to retrieve the local curvature information of the mid-point as shown in Figure 1.11. Even though the curvature information is to be extracted from five points the minimum curvature radius can be registered by less than five points. It is assumed that at least three points are involved in capturing minimum radius or any curvature information of interest. The proposed algorithm for determining local radius involves statistical approach where all the feasible radii values are calculated first. Once the population of feasible radii is generated the task comes down to reliably estimate a representative radius value. While generating feasible radii from five point domain where at least three points are capturing radius information the dataset can be of three points, four points or can include all five points. Such datasets are referred as triplet (three points based), quartet (four points based) and quintet (five points based) throughout the thesis. Now there can be total six combinations from these datasets. In Figure 1.12 any consecutive five points set is illustrated from where six combinations of datasets can be made as shown in Table 1.1. For all cases is the point of interest where local radius is to be estimated involving additional four neighboring points. Figure 1.12 Current 5 point domain for local curvature radius estimation. Pi-2 Pi-1 Pi+1 Pi+2 Pi 19 Table 1.1 Datasets from five point domain for local curvature radius estimation. Dataset Data points included Example Triplet 1 Triplet 2 Triplet 3 Quartet 1 Quartet 2 Quintet 20 These six datasets serve as the source of generating the population of feasible radii. Hence to explain the term ‘feasible radii’ the concept of validity is introduced here. As the above mentioned datasets can have numerous curvature radii within the zone only those radii with inward curvature are of main concern. As the LE/TE are either elliptical or circular in shape the global curvature is always inwardly centered, which limits to consider only inward curvatures while generating radii population. This inwardly centered curvature radii are referred as valid or feasible radii while outwardly centered curvature radii are referred as invalid solutions/radii throughout this thesis. Sample illustration is provided in Figure 1.13 where circles of different radii are fitted through domain of same triplet points of LE/TE. Only the feasible radii are considered in final local radius calculation. Figure 1.13 Generation of feasible curvature radii within U domain. Valid solution Invalid solution Measured point with U domain The curvature validity determination method for triplet is explained in Appendix. 21 1.7 Organization of thesis The thesis has been organized in five chapters. In chapter 1, the introduction, the research objective, airfoil geometry, LE/TE inspection parameters are discussed. Overall research strategy is break-down into two tasks i.e., identification of linear segment and estimation of local curvature. In case of curvature estimation method the focus is on evaluating valid curvatures only, the concept of validity is also explained. Chapter 2 deals explicitly with identification of linear segments from segmented LE/TE profile only. It presents an algorithm to calculate longest linear segment length. Once the linear segments are extracted only the non-linear segments are remaining from where curvature is to be estimated. Results are also shown for a simulated case study. Chapter 3 is dedicated for biarc construction. Here biarcs are used as a tool for calculating curvature radius for quartet and quintet dataset. Again, as biarc interpolation from four/five points are not dealt with previously as per the knowledge of the author, detailed discussion and algorithm is introduced in this chapter. This work can also be viewed as general method. Chapter 4 deals with the overall algorithm of local curvature radius calculation method via a rolling scheme of five points only. Statistical approach has been employed in determining reliable local radius from a population of feasible radii. Valid arc construction from three points discussed in chapter 1 and valid biarc construction from four/five points discussed in chapter 3 are used in overall method. The validation results of the algorithm along with comparison results with existing methods are shown in this chapter also. In Chapter 5 a complete case study is illustrated. The limitations and future works along with conclusion remarks are also stated in this chapter. 22 2. EVALUATION OF LINEAR SEGMENT LENGTH This chapter presents a robust algorithm to identify the longest linear segment. The main feature of the proposed algorithm is the explicit consideration of measurement uncertainty. The algorithm starts by detecting relatively small linear segments and then merges these segments to determine the longest feasible linear segment under given measurement uncertainty. The effect of measurement uncertainty and data point resolution on the performance of the presented algorithm is demonstrated through a series of case studies. 2.1 Existing methods Identifying linear segments from curvilinear profile is often mentioned as segmentation in the literatures. Segmentation of a planar curve into linear and non-linear segments has been studied in image processing as well as in geometric inspection. Chen et al. proposed to segment a planar curve into linear and arc segments in two stages [18]. In the first stage, detection of the so-called break points was done. In second stage, optimization for the curve/line best fitting was achieved. A prior knowledge of the number of segments was needed and used in the study. Later, Wan and Ventura proposed to segment a planar curve into lines and elliptical arcs by identifying two types of break points: corners and smooth joins. Once the break points were identified, a dynamic fitting strategy was applied to best fit a segment with either a line or an elliptical arc [19]. However, for the coordinate measurement data which are inevitably associated with measurement uncertainty, the identification of linear segments using the initially detected break points would become highly unreliable. 23 2.2 Methodology In the proposed methodology, a probable solution of the longest possible linear segment in a highly-complex curvilinear profile such as the LE/TE profile is identified from a set of coordinate measurement data. Due to the inevitable measurement uncertainty, data points collected from even a true flat/linear section do not lie on a straight line. As a result, the linear segment length restriction is very difficult to be evaluated accurately. Effective consideration of the measurement uncertainty is the key to reliably identifying the longest linear segment. For a given measurement uncertainty, it essentially indicates that for every measured point, there is an area of “certainty” where the actual point or point intended to be measured can locate. So, when a series of data points are considered, it would basically form a narrow channel of certainty region around the series of data points. Within the narrow channel of certainty, the longest possible best-fitted straight line is attained and considered a feasible solution, which is compared against the specified maximum linear segment length. 2.2.1 Concept of feasible solution For any measured coordinate point the circles of diameter represent the 2D measurement uncertainty intervals associated with the measured points. The actual points would then lie somewhere within the circles. Now any profile curve that passes through all the circles would be a probable actual profile of the measured points. In fact, there can be an infinite number of probable profiles meeting this condition. Since the objective is to evaluate the maximum length of linear segments in the profile, the longest probable linear segment (passing through as many circles as possible) is considered as a feasible solution. The line length is capped by a decision or stopping criterion. The decision criterion checks whether six times of the standard deviation of the residuals of the fitted line, is smaller than the 24 measurement uncertainty interval . Mathematically, for a feasible solution of the longest line, it has to satisfy following criterion: (2.1) In Figure 2.1 the solid dots are the measured points and the circles of diameter represent the 2D measurement uncertainty intervals associated with the measured points. For these nine consecutive points in Figure 2.1 the longest feasible line that meets the above decision criterion covers from the 3 rd to the 8 th point. The remaining 1 st , 2 nd and 9 th points belong to the neighboring non-linear segments. It should be noted that the line is best fitted by least- squares geometric distance regression (by minimizing the sum of squares of geometric distances from each measured point to the fitted line segment). Figure 2.1 Feasible solution for longest linear segment. P1 P2 P3 P4 P5 P6 P7 P8 P9 25 2.2.2 Underlying principle The underlying principle of feasible solution of linear segments is based on the decision criterion mentioned in equation 2.1. As mentioned earlier, measurement uncertainty being a cumulative parameter of the inspection process is referred with the standard deviation of measurement uncertainty (i.e. ) [14]. Now conversely if a line is best fitted that yields it is considered as a probable or feasible profile. With this underlying principle the worst possible line is sought or the longest linear segment is calculated to validate against allowable maximum linear length restriction. If any manufactured part passes this conservative test, it would guarantee not to be rejected for over-flat condition. In Figure 2.2 the calculated residuals ( are illustrated from where is being calculated. Figure 2.2 Residuals from measured points to feasible linear segment. e1 e3 e4 e5e6 e7 e2 26 2.2.3 Process overview The proposed method to identify linear segments in a curvilinear profile from discrete measurement data points is divided into two phases. In the initial phase, the complete data set is split into relatively small sections. Each of those is checked to see whether it can yield a straight line within that section under the measurement uncertainty criterion. Once the potential smaller sections are identified, they are merged as input to the final phase. In the final phase, the longest feasible linear segment is calculated from the merged sections. Once all the linear segments are identified, the non-linear segments of the profile can be evaluated for the related geometric tolerances/restrictions. 2.3 Implementation As mentioned the overall implementation procedure of evaluating linear segment length is divided into two phases that are discussed in this section. 2.3.1 Initial phase: Detection of feasible linear segments In the initial phase, the measured points for the complete profile are divided into smaller candidate sections/segments. For the proposed method, 10% of the specified maximum allowable linear length is employed as the candidate segment length. This means that the method could only identify linear segments longer than 10% of the maximum allowable length. In fact, any threshold value can be employed as long as each section/segment contains at least three data points (for line fitting). In this work, the segment length is approximated by summing the Euclidean distances between consecutive points. Specifically, if the maximum allowable line length is 1 unit, then every segment that constitutes around 10% of the maximum length (0.1 unit) is taken as a candidate segment (Figure 2.3b). Each of these 27 candidates is tested to see whether it can yield a straight line satisfying the decision criterion of Eq. (2.1). If a segment can yield a straight line satisfying the decision criterion, it is designated as a “linear” segment (segments with dotted boundary in Figure 2.3c). If not, it is designated as a “non-linear” segment (segments with dashed boundary in Figure 2.3c). Once all the segments are checked and designated, the consecutive linear segments are merged to exploit the potential of yielding a longer feasible line. Furthermore, the neighboring non- linear segments from both sides of the merged segments are also included into the merged unit as the longest feasible line may extend to the non-linear segments partially. The resulting merged unit is shown in Figure 2.3d as input to the final phase. Figure 2.3 Initial phase: (a) input data points; (b) candidate segments; (c) designation of each candidate segment; (d) merge of 'linear' segments along with neighboring 'non- linear' segments on both sides. Candidate segments ‘Non- linear’ ‘Linear’ Potential segments Input (a) (b) (c) (d) 28 2.3.2 Final phase: Calculation of longest linear segment As the linear segment to be identified can start and end at any measured point, all the combinations present in the merged segments are to be checked against the decision criterion of Eq. (2.1). Such an exhaustive search is reliable but has time complexity of , where is the data size. Nonetheless, as the complete profile data set has already been processed in the initial phase, would not be a very large number. Since the algorithm checks all possible combinations, the linear segment can be identified with very high accuracy within the given . In Figure 2.4b the feasible linear segment is identified with end points within the merged segment. Once the linear segment is extracted, the remaining measured points belong to the non-linear regions in the overall curvilinear profile as shown in Figure 2.4c. Figure 2.4 Final phase: (a) merged segment; (b) identification of longest linear segment; (c) identified linear and non-linear region. Merged segment Non-linear segment Linear segment (a) (b) (c) 29 2.4 Results and analysis A simulated curvilinear profile with known parameters is used to examine the performance of the proposed algorithm. To validate the robustness of the algorithm, normally distributed measurement noise is imposed on the sampled points. The magnitude of the imposed noise corresponds to the magnitude of the measurement uncertainty with changing from 0, 0.5, 0.6, 0.7, 0.8, 0.9, to 1.0 μm. Also, to check the effect of point resolution on the effectiveness of the proposed method, the same profile is sampled with different average spacing of 3 to 10 μm. Figure 2.5 shows a typical input represented with average spacing of 5 μm. The theoretical parameters (length for linear segments) of the input profile are given in Table 2.1. Figure 2.5 Input profile 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 X Y Lin e 1 Arc 1 Line 2 Arc 2 Lin e 3 Arc 3 (mm) (m m ) 30 Table 2.1 Input segments with theoretical parameters. Input segment Length (mm) Line 1 1 Line 2 0.5 Line 3 0.3 In Figure 2.6, the absolute percentage errors for the three identified linear segment lengths under increasing measurement uncertainty are plotted. Throughout this thesis, the absolute percentage error is defined as: For all the cases, average point spacing is kept constant as 5 μm. As expected, the accuracy of the computed results deteriorates with increasing . For the cases with =1 μm, the computed lengths of Line 1, Line 2 and Line 3 are respectively 1.015, 0.535 and 0.33 mm which are consistently longer than the corresponding ideal values. This phenomenon is explained in next section. Figure 2.6 Study of increasing U effect for linear segments with spacing of 5 μm. 0 2 4 6 8 10 12 0 0.5 0.6 0.7 0.8 0.9 1 E rr o r (% ) Measurement Uncertainty (μm) Line 1 Line 2 Line 3 31 2.4.1 Improvement initiative: Locating segment end points It has been observed that, for majority of the cases the lines are extended and include few points from neighboring non-linear segments. Since the stopping criterion is regulated by which is a collective factor, the calculated linear segment tends to include few actually non-linear segment points before it satisfies the stopping criterion. For an ideal case, with the inclusion of a non-linear segment point, updated should be greater than . Yet occasionally, the neighborhood segment points being very close to linear segments, the updated does not exceed the threshold instantly. The root cause is that, any new residual ( cannot alter the cumulative . As a consequence the calculated line may include few non-linear segment points. This phenomenon is more noticeable for longer linear segments that involve higher number of points. To reduce this effect, an improvement initiative is employed where few of the endpoints from both sides of the calculated linear segments are investigated individually to check whether those belong to non-linear segments or not. The basic intention of the improvement initiative is to locate the end points of the linear segments more precisely. To implement improvement initiative algorithm, any calculated linear segment is divided into three sections: left, base and right. The base section is considered to contain core 90% of the calculated linear segment that may not include any non-linear segment points. The rest 5% number of points on each side i.e., left section and right section undergoes further scrutiny. Each additional point in these sections is judged based on their contribution in shifting overall . If any points corresponding contribution is found to be exceptionally higher than historical contribution average, it is considered that the point actually belongs to non- 32 linear segment. The calculated linear segment is then trimmed until that point. Here the threshold value is used as 10, meaning that if any new point (either from left section or right section) increases the current which is 10 times higher than historical individual point’s incremental contribution, than the new point can be considered as part of non-linear segment even though stopping criterion is yet not satisfied. To study the impact of improvement initiative, percentage error in linear segment length calculation before and after employing the algorithm is shown in Figure 2.7 and Figure 2.8. As mentioned earlier, the improvement initiative has less impact for shorter linear segments when there is less number of points involved. In Figure 2.7, the accuracy before and after employing the improvement initiative of locating the end points for Line 1 is illustrated. The theoretical length of Line1 being 1 mm and average point spacing being 5 μm, it involves more number of points and showed improvement in accuracy. Figure 2.7 Improvement of accuracy after employing end locating algorithm (Line 1). 0 1 2 3 3 4 5 8 10 E rr o r (% ) Spacing (μm) Before After 33 In Figure 2.8, the accuracy before and after employing the improvement initiative of locating the end points for Line 3 is illustrated. The theoretical length of Line3 being 0.3 mm and average point spacing being 5 μm, it involves lesser number of points and showed no improvement in accuracy. Figure 2.8 Improvement of accuracy after employing end locating algorithm (Line 3). The improvement initiative algorithm is included for all subsequent calculations in the thesis. The effect of measurement uncertainty and data point resolution on the performance of the presented algorithm is demonstrated through two case studies. 2.4.2 Case Study 1: Effect of varying measurement uncertainty In case study 1, the effect of increasing is studied for Line 2. Here same input profile is sampled with different spacing illustrated by various series in the plot. As expected, when the average point spacing becomes larger, the effect of is less detrimental for which the percentage error decreases. Here as in Figure 2.9, the series with smallest spacing of ‘3 µm’ is yielding maximum percentage error and on the contrary, the series with largest spacing of 0 4 8 12 3 4 5 8 10 E rr o r (% ) Spacing (μm) Before After 34 ‘10 µm’ is yielding minimum error. It is also observed, for series with larger spacing (e.g. ‘8 µm’, ‘10 µm’), the error curve follows a step pattern. This happens as the calculated line contains same number of points even if is varied a little (e.g. in case of series ‘10 µm’ for = 0.5 µm to 0.7 µm and again = 0.8 µm to 1 µm, error remained at same level). The second characteristics observed, with the increase of for a fixed average spacing series percentage error also increases. Figure 2.9 Study of increasing U effect for linear segment (Line 2) with different series of spacing. 2.4.3 Case Study 2: Effect of varying point spacing In case study 2, the effect of varying spacing is studied for Line 3. Here the same input profile is superimposed with different illustrated by various series in the plot. As expected, when the measurement uncertainty is large the algorithm has maximum percentage error. Here as in Figure 2.10, the series with maximum of ‘1 µm’ is yielding maximum 0 2 4 6 8 10 0 0.5 0.6 0.7 0.8 0.9 1 E rr o r (% ) Measurement Uncertainty (μm) 3 µm 4 µm 5 µm 8 µm 10 µm 35 percentage error and vice-versa. As expected, as average point spacing is increased, percentage error decreases for any particular series. Figure 2.10 Study of increasing spacing effect for linear segment (Line 3) with different series of U. To summarize, this chapter presents a robust algorithm to identify linear and non-linear segments in a complex curvilinear profile from the discrete inspection data points. The main feature of the proposed algorithm is the explicit consideration of measurement uncertainty. The algorithm starts by detecting relatively small linear segments and then merges these segments to determine the longest feasible linear segment under given measurement uncertainty. The effect of measurement uncertainty and data point resolution on the performance of the presented algorithm is demonstrated through case studies. Continuing discussion in evaluating the minimum curvature radius in the identified non-linear segments lies in next chapters. 0 2 4 6 8 10 12 3 4 5 8 10 E rr o r (% ) Spacing (μm) 0 0.5 0.6 0.7 0.8 0.9 1 36 3. BIARC CONSTRUCTION A biarc is a composite of two consecutive circular arcs with an identical tangent at the joint point. As mentioned in chapter 1, the proposed algorithm evaluates curvature radius via a rolling scheme of five consecutive data points in order to retrieve the local curvature information of the mid-point. This scheme of five points creates six subsets of three triplets, two quartets and one quintet of data points. Biarc construction is used to determine the curvature radius from quartet and quintet subsets. As a result, radii calculated by interpolating biarc through four and five points’ datasets constitute combined feasible radii population and serve to estimate reliable local curvature radius. This chapter has been dedicated solely for biarc construction. Biarc interpolation from five points is discussed in details, followed by biarc interpolation from four points. 3.1 Biarc Biarc is a combination of two arcs/circles with tangent continuity. It has various applications. Computer numerical control (CNC) machines usually have straight-line and circular arc interpolators. Free-form curves can therefore be approximated by lines, arcs, or a combination of these elements for CNC machining. Biarc interpolation serves as one popular strategy for generating G1 continuous tool paths [20], [21]. It is also applied in sectors such as machining of aspheric surface [22], robot path planning, geometric modeling [23] or using biarc filter to compute curvature extremes [24]. 37 3.1.1 Biarc parameters A biarc curve consisting of two circular arcs has six degrees of freedom. However as one of the degrees is constrained by tangent continuity at joint point, only five degrees of freedom is independent. So practically, only five inputs are required to construct a unique biarc. For convenience of discussion, the rightmost arc is named as and leftmost arc is named as throughout the thesis. The five independent parameters that suffice unique biarc construction can be set as coordinates of center of , coordinates of center of and either of the radii (let’s say ) as shown in Figure 3.1. From these five parameters, other radius , joint point can be determined to construct a unique biarc. Hence, the main task is to calculate these five independent parameters. Figure 3.1 Biarc parameters rad1 O1(x1,y1) O2(x2,y2) rad2 J(x,y) arc1arc2 In the present context, as input data points are available where local radius has to be estimated the biarc needs to be interpolated rather than approximated. In case of quintet, the provided five data points are enough to determine a unique interpolating biarc. But in case of quartet, a remaining degree of freedom is yet to be constrained in order to avoid infinite biarc 38 solution possible from four points. As mentioned earlier, local curvature radius needs to be estimated where it is assumed that the interest region does not contain any abrupt change of curvatures within the local domain of five points. Keeping this property in mind, the remaining degree of freedom while constructing biarc from quartet is set as to find biarc with minimum radius difference i.e., So, for constructing unique biarc from quartet, this condition is chosen as to minimize radius difference between two arcs of biarc such that, it yields smooth curvature change in the focus region. 3.1.2 Biarc type Biarc can be categorized as three types: ‘C’, ‘S’ and ‘J’ shaped on the basis of presence of inflection point [21], [25] or signs of curvature of the two circular arcs [26]. As shown in Figure 3.2, ‘C’ shaped biarc has same curvature sign for both arcs, ‘S’ shaped biarc has opposite curvatures whereas ‘J’ shaped biarc is basically a combination of line and arc. Since in the current application, biarc is used as a tool to calculate valid radius, only ‘C’ shaped biarc with inward centered curvature is of prime interest. Figure 3.2 Biarc types; (a) 'C' shaped; (b) 'S' shaped; (c) 'J' shaped. rad1 O1(x1,y1) J(x,y) rad1 O1(x1,y1) O2(x2,y2) rad2 J(x,y) rad1 O1(x1,y1) O2(x2,y2) rad2 J(x,y) (a) (b) (c) 39 3.2 Existing methods Literatures dealing with biarc construction have mostly considered a set of given points and tangents at two end-points as input. Because of the remaining fifth degree of freedom, there can be infinite biarc curves that pass through the given points and match the end tangents. However, different fifth condition has been adopted based on application which also has made the biarc unique. Pioneering researcher on biarc, Bolton considered the fifth condition to be [25]. Parkinson et al. and Kim et al. proposed biarc curve fitting by minimizing strain energy [20], [21]. Schonherr proposed the fifth condition to be [27]. In case of interpolating intermediate points, biarc fitting algorithms operate in two stages: first computing tangents at intermediate points and then constructing individual biarc segments to meet those tangents [21], [28]. The second category is approximation of intermediate data points only while matching end points and their given tangents. Hoscheck proposed an arc-spline approximation method by finding least-square fit [29]. Meek et al. proposed approximation by adaptively determining the number of biarcs to be used on the basis of prescribed tolerance value [28]. Park, Yang and Lee et al. proposed biarc fitting [22], [23], [30] by constraining the residual at the intermediate points by prescribed tolerance value. Piegl et al. proposed approximating all data points including end points by biarc within prescribed tolerance value [31]. Clearly the current circumstance is different from these situations as end tangent points are not available. Thus biarc interpolation from four or five points has given due attention in this study. In the following sections, first biarc construction from quintet and then biarc construction from quartet minimizing radius difference is explained in details. 40 3.3 Biarc construction from quintet A unique biarc can be constructed from quintet i.e., five points. The primary task is to determine five required independent parameters. It is considered that the consecutive five input points are and starting from right to left. Now for unique biarc construction the joint/common point of the two arcs can be either between and or between and . As shown in Figure 3.3, case when belong to and belong to it is named as . On the contrary is the biarc with joint point between and . Since there is only one solution that interpolates all five points the solution biarc will be either or for fixed set of five points. Moreover, no more than three points can be part of one arc as four points in one arc and fifth one in another arc of a biarc will generate infinite biarc solutions. Now since the point of interest is always mid-point of a rolling five point scheme i.e., only the radius of the corresponding arc containing would be registered as desired local radius at . In case of it would be and in case of it would be . Figure 3.3 Biarc from quintet i.e., five points (a) Biarc3:2; (b) Biarc2:3. rad2 rad1 P1 P2 P3P4 P5 O1 O2 J (a) rad2 rad1 P4 P1 P2P3 P5 O1 O2 J (b) 41 3.3.1 Proposed approach Both and holds similar construction approach. For convenience, only is discussed below. For this case, the joint point lies between and . Figure 3.4 Biarc construction from quintet (a) Finding location of joint from θ domain; (b) Validating formulated biarc. P4 P1 P2 P3P5 O1 P5 P3 P1 P2 P4 O1 O2 M N θmax θmin J=ƒ(θ) (a) (b) As shown in Figure 3.4(a) first, the blue circle in solid line is constructed from three consecutive points and . Let the center be and radius to be which is equal to (where i = 1, 2, 3). Now, the next task is to find second circle’s center . As and are bound to be part of , its center is obviously equidistant from both and . Thus the center would lie anywhere on the perpendicular bisector drawn from and ( shown by brown dash-dot-dash line). Since the two center points and joint point of any biarc are collinear, so , and would be collinear. In another way, if is known correctly then can be determined from the intersection of line and . 42 Here can be any point on when extended from towards . As the appropriate is unknown, a bisection search method can help in finding the appropriate . An appropriate can be verified by checking whether equals or not. So the method aims to find appropriate by increasing the angle within a domain of to . Here is calculated from positive X axis and taking the convention of ‘counterclockwise rotation’ as ‘positive’. To summarize, the objective is to find which is and satisfies condition, (3.1) Here the domain of would be [ ] where is the angle created by line and is the angle created by with +X axis. A unique J would satisfy, i.e., (where ) and (where ). Figure 3.5 Applying bisection search method for finding joint. P4 P1 P2 P3 P5 O1 θmax θmin +X-X +Y -Y θ J 43 As shown in Figure 3.5, golden section search method is applied as bisection search method. Golden section search method helps to find the minimum of a unimodal function by successively narrowing the range of values inside which the minimum is known to exist. As , a function of lies in between and , the golden section search method helps to find the appropriate . All the calculations mentioned above are made assuming the joint point to be in between P3 and P4. If finally, it fails to yield an objective function value (i.e., L.H.S of equation 3.1) of zero or satisfy the validation criteria, the other case needs to be investigated to construct target unique biarc. 3.3.2 Detection of biarc type This section describes the curvature validity checking method for biarc constructed from quintet which is also valid for biarc constructed from quartet. As biarc is the combination of two arcs (let’s say and ), if any of the two arcs is valid and they form a valid biarc type (i.e., ‘C’ shaped) then it can be regarded as valid biarc. In other words, a biarc is considered valid if and only if either or is valid and the biarc is a ‘C’ shaped biarc. On the contrary, if joint point lies in between and , then it is an ‘S’ shaped biarc and has an invalid curvature. Here the arc that contains three points is checked first. The curvature check method from three points (in Appendix A) is employed for the first arc. The joint point, center and third point are investigated. Secondly, the biarc combination type is checked. Validity check method for is shown in Figure 3.6. 44 Figure 3.6 Validity check method for biarc (a) +X axis along P2P1 line, check ordinate of O1', J' & P3'; (b) +X axis along JO1 line, check abscissa of O2''. P5 P3 (P3'=?) P1 P2 O1(O1'=?) O2 J (J'=?) P4 +X -X +Y -Y P5 P3 P1 P2 O1 O2 (O2''=?) JP4 +X -X +Y -Y (a) (b) Here as in Figure 3.6(a), first local coordinate system is set by fixing as origin and as +X axis. Now the transformed ordinate value of (i.e., ), (i.e., ) and center (i.e., ) are checked. If all of them are ‘negative’, then is considered as valid or inwardly centered. The remaining task is to check if the biarc formed is a ‘C’ type or not. Secondly, as in Figure 3.6(b), new local coordinate system is set by fixing as origin and as +X axis. Now the task is to check the transformed abscissa value for center (i.e. ). If it is ‘positive’, then the biarc is considered to be ‘C’ shaped. 3.3.3 Results of five point biarc To confirm the proposed method of biarc construction from five points, three case studies have been illustrated here. For all cases, five sequential points’ coordinates are taken as input. The resultant outputs are the five parameters: two center coordinates and one radius. 45 Case Study 1: ‘C’ shaped input with uniformly spaced points In case study 1, five uniformly spaced points from theoretical are taken as input. Input: (0.0462, 0.01913); (0.03536, 0.03536); (0.01913, 0.0462); (-0.01768, 0.04268); (-0.025, 0.025). As there is no prior information about the biarc to be or , both options are investigated. With this objective, the first task is to determine and then to find the deviation value (as stated in equation 3.1). The deviation for is and for is . Only one of these two can be zero because of the unique biarc solution and would yield desired five parameters. As both biarc options are investigated, the resultant and deviation values are: = 1.5708 radian; = 1.4×10 -10 (Unique solution). = 0.9356 radian; = 0.0032 (No solution). So, clearly only provides the solution. Once is finalized, joint point and all five parameters needed to construct biarc are extracted. Here ‘SSE’ is the sum of squared error which will be zero, if and only if, all five points interpolates the biarc. Now, as it is obvious which point belongs to which arc, each point contains corresponding curvature radius of that arc. For example and has radius of 0.05 mm and contains radius of 0.025 mm. The five independent parameters that suffice unique biarc construction: coordinates of center of , coordinates of center of and either of the radii (let’s say ) are shown in Table 3.1. From these five parameters, other radius , joint point are also calculated. The output biarc is shown in Figure 3.7. 46 Table 3.1 Results for 'C' type biarc with uniformly spaced five points. Parameters Type rad1 rad2 J(x, y) O1(x, y) O2(x, y) SSE Theoretical Biarc3:2 0.05 0.025 (0, 0.05) (0, 0) (0, 0.025) - Calculated Biarc3:2 0.05 0.025 (0, 0.05) (0, 0) (0, 0.025) 4×10 -20 Figure 3.7 Output biarc for case study 1 (‘C’ shaped biarc for uniform spacing). P1 P2 P3 J P4 P5 O2 O1 X (mm) Y ( m m ) Case Study 2: ‘C’ shaped input with non-uniformly spaced points The biarc construction method performs as expected for uniformly spaced points. Now, it has been executed for non-uniformly spaced points in case study 2. 47 Input: (0.05, 0); (0.0462, 0.01913); (0.01913, 0.0462); (-0.01768, 0.04268); (-0.02137, 0.03797). = 1.5708 radian; = 2.6×10 -11 (Unique solution). = 0.9609 radian; = 0.0053 (No solution). The five parameters, other radius , joint point are shown in Table 3.2 and the output biarc is shown in Figure 3.8. Table 3.2 Results for 'C' type biarc with non-uniformly spaced five points. Parameters Type rad1 rad2 J(x, y) O1(x, y) O2(x, y) SSE Theoretical Biarc3:2 0.05 0.025 (0, 0.05) (0, 0) (0, 0.025) - Calculated Biarc3:2 0.05 0.025 (0, 0.05) (0, 0) (0, 0.025) 1×10 -21 Figure 3.8 Output biarc for case study 2 (‘C’ shaped biarc for non-uniform spacing). X (mm) Y ( m m ) O1 P1 P2 P3 P4 P5 J O2 48 Case Study 3: ‘S’ shaped input with regularly spacing points In case study 3, the method is executed in case of theoretical ‘S’ shaped biarc. Input: (0.0462, 0.01913); (0.03536, 0.03536); (0.01913, 0.0462); (-0.01768, 0.04268); (-0.025, 0.025). = 1.5213 radian; = 0.0046 (No solution). = -1.5708 radian; = 2.1×10 -10 (Unique but invalid solution). Table 3.3 Results for 'S' type biarc with uniformly spaced five points. Parameters Type rad1 rad2 J(x, y) O1(x, y) O2(x, y) SSE Theoretical Biarc2:3 0.05 0.025 (0, 0.05) (0, 0) (0, 0.1) - Calculated Biarc2:3 0.05 0.025 (0, 0.05) (0, 0) (0, 0.1) 9×10 -20 Figure 3.9 Output biarc for case study 3 (‘S’ shaped biarc for uniform spacing). X (mm) Y ( m m ) O1 P1 P2P3 P4 P5 J O2 49 3.4 Biarc construction from quartet A unique biarc is also required to be constructed from a quartet i.e., four points. Again, the main task is to determine five required independent parameters: coordinates of center of , coordinates of center of and either of the radii. For convenience of discussion, it is considered that the sequential four input points are and from right to left. 3.4.1 Proposed approach As mentioned earlier, biarc constructed from quartet can be infinite as there is a free degree of freedom. Now there can be multiple ways to constrain the fifth degree. For the current application, minimum radius difference constraint has been considered as fifth degree of freedom that makes the biarc unique. Since the overall objective is to estimate the local curvature radius, it has been assumed that there is minimum radius change within the local domain. This assumption has lead to the consideration of minimum radius difference while constructing biarc from quartet. Again, another benefit of setting the criteria is that the interpolated biarc has the tendency to be like mono-arc with similar radius. 3.4.2 Finding location of joint point As shown in Figure 3.10, while constructing biarc from quartet, the joint point can be between and or between and or between and . These are classified as and respectively in this thesis. Since there can be infinite biarc solutions that interpolates all four points, the unique ‘C’ type biarc that has the property of needs to be considered. 50 Figure 3.10 Biarc from four points i.e., quartet (a) Biarc 1:3; (b) Biarc 2:2; (c) Biarc 3:1. rad2 rad1 P3 P1 P2 P4 O1 O2 J (a) rad2 rad1 P4 P1 P2P3 O1 O2 J (b) (c) rad2 rad1 P1 P2 P3 P4 O2 J O1 Now, the objective is to formulate a biarc with minimum radius difference (i.e. ). Clearly, as shown in Figure 3.10, there can be three possibilities of the joint point location. But from geometric analysis and calculated observations, following two hypotheses are proposed: Hypothesis 1: If valid (‘C’ shaped only) is viable at , the joint point for with property will always be at . Hypothesis 2: If valid (‘C’ shaped only) is viable at , the joint point for with property will always be at . 51 Since both cases of and are symmetric, only is referred to validate above mentioned hypotheses. For any , there can be only two alternate situations. Either the fourth point i.e., can lie inside the circle (as in Figure 3.11) or it can lie outside the circle (as in Figure 3.12). The above mentioned first hypothesis is partially proved by Figure 3.11. Figure 3.11 Biarc construction when fourth point inside circle (all ‘C’ type). J or P3 P1 P2P4 O1 θmax θmin J' J" O2 O2'O2" |r a d 1 -r a d 2 | For first situation, the blue dashed circle is the constructed from and . Clearly point lies inside the area (as ). As Figure 3.11 suggests, if joint point is considered to be at , then the radius of is and the radius difference is . As the joint shifts to point , new center for becomes and the radius difference becomes . Now as joint point shifts from towards , radius becomes smaller. The radius difference (i.e., ) thus continues increasing as joint moves away from . So, the property of holds at P3. 52 Figure 3.12 Biarc construction when fourth point outside circle (mix of ‘C’, ‘J’ & ‘S’ type). J or P3 P1 P2 P4 O1 θmax θmin J' J" O2 O2' O2" |r a d 1 -r a d 2 | Pt For second situation when the fourth point lies outside the circle formed from and (as ), there can be all three types of biarc such as ‘C’ shaped, ‘S’ shaped and ‘J’ shaped. These biarc types depend on the location of joint point within domain of to . As in Figure 3.12, is the tangent point to circle drawn from external point . Since also lies within the domain it bears significance for deciding biarc type. If the joint point lies anywhere between to (i.e., ), it forms a ‘C’ shaped biarc. If joint point is at (i.e., ) then the biarc is a ‘J’ shaped biarc (combination of line and arc). Again, if joint point is anywhere beyond up to maximum limit of (i.e., ) the resultant biarc will be ‘S’ shaped. Here it is noted that, joint point is always on circle and the location is function of . As the objective is to construct valid ‘C’ shaped biarc with , the applicable domain of feasible is actually . As moves beyond , the biarc 53 becomes ‘S’ shaped and is considered invalid. The red solid line indicates the difference between two radii (i.e., ) which increases as joint is shifted further from towards direction. At as biarc is “J’ shaped, the radius gap will be infinite. So, from above discussion and two aforesaid hypotheses, it can be concluded, if a biarc with minimum radius difference is to be formulated from quartet, the joint point will always lie in between and . In other words, the optimum biarc will only be . 3.4.3 Finding optimum biarc As the optimum biarc will always be , so the task narrows down to find optimum . As shown in Figure 3.13, the resultant blue circle and the red circle are formed from four consecutive points and with the optimized property of minimum curvature difference. The proposed method for finding i.e., biarc with joint point in between and is explained below. Let center of be and radius be which is equal to (for i=1,2) and other center of be and radius be which is equal to (for k=3,4). The first task is to find center . As and is bound to be part of (i.e., blue circle), its center is equidistant from both and . If a perpendicular bisector is drawn from and , the center would lie anywhere on line (shown by cyan dashed line). For a fixed , joint can be calculated from golden section search method within domain of to . An optimum would yield the property of deviation equals to zero (i.e., ) where is the center of . 54 Figure 3.13 Finding location of joint from θ domain for biarc construction from quartet. P3 P1 P2 P4 O1 M N θmax θmin O2 A B J=ƒ(θ) Q For any and any , is equal to . Now when determined, objective function value is calculated from . For every new a new function value can be calculated. This brings the final task to find optimum that yields . A golden section search method can be employed to find the appropriate along line starting from towards . In short, a nested optimization (binary search method) is employed here. First in outer loop, for every (along line ) the objective function is calculated. In order to calculate that, in inner loop, for every fixed , optimum is to be calculated within domain. To employ any bisection search method, its boundary points need to be identified. In case of inner loop, and serves as boundary limits. On the other hand, there are no such obvious boundary points on line for trialing with different . This problem necessitates thorough investigation for finding effective boundary limits. 55 In order to determine boundary points for employing bisection search along line , the concept of ‘distance’ needs to be discussed. Here ‘distance’ (i.e., ) is used as a measure for determining boundary points. As the center for will always lie on line , its distance from to (i.e., midpoint of line ) is referred as ‘distance’. In other words, . Figure 3.14 Concept of distance as a measure for finding boundary points. D31 P1 P2 P3 or P31 (a) O1 Q P1 P2Pt Dt (b) O1 Q P4 P3 P2 or P13 P1 D13 (c) O1 Q As illustrate in Figure 3.14(a), when serves as the joint of the biarc (also ), the distance measured from is referred as . While as in Figure 3.14(c), when serves as the joint of the biarc (also ), the distance measured from is referred as . Now in Figure 3.14(b), the distance is referred as which also has significance in deciding boundary points. Here line is the tangent line to circle and is the tangent point. For all cases is the same point where as moves along bisector line to yield different ‘distance’. Knowing any value, the center coordinates of can be determined. To summarize, can be categorized on the basis of shape characteristics and relative positioning of four input points. This ‘distance’ measure is used to fix the category specific boundary limits needed to employ bisection search method. 56 3.4.4 Biarc2:2 categories on the basis of shape characteristics Using shape characteristics, four point biarc can be categorized in five types as shown in Table 3.4. For each type the upper limit (UL) and lower limit (LL) of the measure are mentioned with their properties. Here, for all the cases, the inputs are considered to be as counter-clockwise i.e., the bottom rightmost point is taken as P1 and left most point is taken as P4. Conceptual table showing five types for deciding boundary limits. Table 3.4 Conceptual table for five types of four-point biarc and their attributes. Plot Biarc3:1 Biarc1:3 (distance vs. |rad1-rad2|) TYPE 1: P4 inside circle P1P2P3; both biarc3:1 and biarc1:3 are ‘C’ shaped; P1 on same side of tangent drawn on circle P2P3P4 at P2. Lower limit (LL) is D31, upper limit (UL) is D13. P3 or P31 P2 or P13 D13 Dopt D31 TYPE 2: P4 inside circle P1P2P3; biarc3:1 is ‘C’ and biarc1:3 is ‘S’ shaped; P1 on opposite side of tangent drawn on circle P2P3P4 at P2. LL is D31 and UL is Dq. Here Dq is distance measured along bisector of P3P4 from its midpoint. Dq Dopt D31 P2 or P13 P3 or P31 57 Plot Biarc3:1 Biarc1:3 (distance vs. |rad1-rad2|) TYPE 3: P4 outside circle P1P2P3; both biarc3:1 and biarc1:3 are ‘C’ shaped; P4 on same side of tangent drawn on circle P1P2P3 at P3. LL is D13, UL is D31. P3 or P31 P2 or P13 Dopt D13 D31 TYPE 4: P4 outside circle P1P2P3; biarc3:1 is ‘S’ and biarc1:3 is ‘C’ shaped; P4 on opposite side of tangent drawn on circle P1P2P3 at P3. LL is D13, UL is Dt. Here Dt is distance measured along bisector of P1P2 from its midpoint. P2 or P13 P3 or P31 Dopt D13 Dt TYPE 5: P4 outside circle P1P2P3; both biarc3:1 and biarc1:3 are ‘S’ shaped; P1 on opposite side of tangent drawn on circle P2P3P4 at P2; P4 on opposite side of tangent drawn on circle P1P2P3 at P3. No valid solution. D31 No solution D13 P2 or P13 P3 or P31 58 3.4.5 Results of four point biarc In support of each category discussed above, one case study per type is illustrated here. As mentioned, first and are calculated and their shape (either ‘C’ or ’S’) are determined. Based on their shape characteristics, the types are classified and the corresponding boundary limits are employed for bisection search method. The optimum output is then calculated. To validate that the boundary limits are set appropriately, the ‘distance vs. ’ results are also plotted to demonstrate the unimodal behavior within the boundary limits. For all cases, is shown where first three points and are part of and is part of . On the contrary, in case of , is part of and and are part of . Again, optimized is the desired biarc with minimum radius difference. For all cases black dots are input points, blue dot is the center for blue arc with solid line and red dot is the center for red arc with dashed line. The pink dot refers to joint point for and green dot refers to joint point for . In case of optimized , the brown diamond shaped point is the joint point. The additional cyan square shaped point corresponds to the distance, for tangent point . In the ‘distance vs. ’ plot, distances corresponding to the above colored points are shown as limiting boundaries. 59 Case study 1: Input: (0.05, 0); (0.03536, 0.03536); (0.0, 0.05); (-0.035, 0.015). In Figure 3.15, a plot has been shown to validate earlier mentioned hypotheses 1. Here, the optimized for is formed at i.e., at as it yields . Clearly, as joint moves towards i.e., the value also increases. So, it validates that, P3 will always be the optimum joint point for any . Figure 3.15 'distance vs. |rad1-rad2|' plot to demonstrate P3 is the optimum joint point for any biarc3:1. θmin at P3 θmax at P4 As shown in Figure 3.16(Top), lies inside circle . Moreover, both and are ‘C’ shaped. So this case belongs to Type 1. Now, lower limit (LL) of , upper limit (UL) of is applied to find optimum as shown in Figure 3.16(bottom). 60 Figure 3.16 Type 1 biarc; (Top left) 'C' shaped biarc3:1; (Top right) 'C' shaped biarc1:3; (Bottom left) Optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot. (All units in mm) O1 P1 P2 P3 or P31 P4 O2 P1 P2 or P13 P3 P4 O2 O1 P1O1 O2 P2 P4 P3 J D13 Dopt D31 Here, the optimized biarc is the desired ‘C’ shaped biarc where the radius difference is minimum. The ‘distance vs. ’ plot illustrates that, the biarc solution is unique. 61 Case study 2: Input: (0.04952, 0.00689); (0.04858, 0.01182); (0.04386, 0.02526); (0.04116, 0.0286). Figure 3.17 Type 2 biarc; (Top left) 'C' shaped biarc3:1; (Top right) 'S' shaped biarc1:3; (Bottom left) Optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot. (All units in mm) O1 P1 P2 P3 or P31 P1 P2 or P13 P3 P4 O2 O1 J D13 Dopt D31 O2 O1 P1 P4 Dq As shown in Figure 3.17(Top), lies inside circle . Here, is ‘C’ and is ‘S’ shaped. So this case belongs to Type 2. Now, LL of D31 and UL of Dq is applied to find optimum as shown in Figure 3.17(bottom). Here Dq is distance measured along bisector of from its midpoint. The ‘distance vs. ’ plot illustrates that, the biarc solution is unique. 62 Case study 3: Input: (0.03, 0.02); (0.02121, 0.04121); (0.0, 0.05); (-0.02391, 0.04391). Figure 3.18 Type 3 biarc; (Top left) 'C' shaped biarc3:1; (Top right) 'C' shaped biarc1:3; (Bottom left) Optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot. (All units in mm) O1 P1 P2 P3 or P31 P4 O2 P2 or P13 D13 Dopt P4 P3 P1O1 O2 P1 P2 JP3 P4 O1 O2 As shown in Figure 3.18(Top), lies outside circle . Moreover, both and are ‘C’ shaped. So this case belongs to Type 3. Now, lower limit (LL) of D13, upper limit (UL) of D31 is applied to find optimum as shown in Figure 3.18(bottom). 63 Case study 4: Input: (0.02584, 0.03524); (0.01264, 0.04721); (-0.02661, 0.03503); (-0.02997, 0.03109). Figure 3.19 Type 4 biarc; (Top left) 'S' shaped biarc3:1; (Top right) 'C' shaped biarc1:3; (Bottom left) Optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot. (All units in mm) O2 P1 P2 P3 or P31 P4 J D13 Dopt D31 O2 O1 P1 P2 Dt O1 P4 P2 or P13 O2 O1 P3 P1 P4 P3 As shown in Figure 3.19(Top), lies outside circle . Here, is ‘S’ and is ‘C’ shaped. So this case belongs to Type 4. Now, LL of D13 and UL of Dt is applied to find optimum as shown in Figure 3.19(bottom). Here Dt is distance measured along bisector of from its midpoint. The ‘distance vs. ’ plot illustrates that, the biarc solution is unique. 64 Case study 5: Input: (4.96946, 35.6467); (3.13846, 40.0908); (-2.85053, 42.1582); (-4.55242, 44.7244). Figure 3.20 Type 5 biarc; (Top left) 'S' shaped biarc3:1; (Top right) 'S' shaped biarc1:3; (Bottom left) Non-optimized biarc2:2; (Bottom right) 'distance vs. |rad1-rad2|' plot. (All units in mm) O2 P2 P3 or P31 J D13 No solution D31 O2 O1 P1O1 P4 P2 or P13 O2 P1 P4 P3 P1O1 P2 P3 P4 As shown in Figure 3.20(Top), both and are ‘S’ shaped. So this case belongs to Type 5. Now, for this case there is no valid solution as all the are of ‘S’ shaped as shown in Figure 3.20(bottom). 65 To summarize chapter 3, biarc construction from quartet and quintet dataset has been studied in details. The existing methods commonly interpolate/approximate biarc from given end tangents and end points with an additionally considered constraint. In the current application, since biarc has to be interpolated from four or five consecutive points, detail study of biarc construction is conducted. Hence, first a method is proposed to construct interpolating biarc from five points. The proposed method is validated with three case studies of all probable input scenarios. Later, a novel method is also proposed to construct biarc from four points with minimum radius difference. It has been observed that, the biarc joint location can only be between second and third point. The task then narrows down to find optimum biarc2:2. A nested optimization method is proposed for the biarc construction where bisection search method is employed. To execute the bisection search method, the boundary limits are set from five classified types. The shape characteristics and their attributes of each type are also investigated. In support of each category, one case study has been conducted that validated right consideration of boundary limits. Now, radii calculated by interpolating biarc through four and five points’ datasets are used to generate finite feasible radii population for evaluating local curvature radius. 66 4. EVALUATION OF LOCAL CURVATURE RADIUS The overall method of evaluating local curvature radius from a scheme of five points is described in this chapter. The method of biarc construction from quartet and quintet is used here along with arc construction from triplet. A statistical approach is employed where all feasible radii from all the six datasets are considered in order to reliably estimate the desired radius. The proposed approach is compared with six existing radius estimation methods with varying measurement uncertainty and data point resolution. 4.1 Existing methods Curvature radius is a common quality control measure for manufactured parts. It is also important in applications such as pattern recognition, feature detection and milling tool path planning. Given a set of coordinate points that represent the outline profile of a mechanical part, the underlying task is to estimate the arc parameters (center and radius) along the profile. Based on their fundamental solution principles, existing applicable methods to determine the arc parameters can be classified into four categories: best fitting based methods, integral invariant based methods, image data oriented methods and Hough transform based methods. Each category is discussed in more detail below. 4.1.1 Best fitting based methods In best-fitting based methods, all the input points are regarded as part of an arc and thus every point is employed to determine the optimum arc. The arc parameters are readily available from the fitted arc. As these methods consider all the input points, they are clearly applicable to the current case of five points. The basic difference among these methods lies in 67 the formulated objective function and solution approach. Landau appeared to be the first researcher to introduce a method for best fitting a circular arc from a set of data points [32]. The method iteratively solved for the arc center and radius from two simultaneous equations that were derived by minimizing the sum of squared distance deviations between the input points and the fitted arc. Thomas and Chan redefined Landau’s objective function for best fitting the circular arc [33]. Instead of the squared distance deviation, a squared area deviation of each input point with respect to the fitted circle was used. It was noted by the authors that as the sum of squared area deviations were to be minimized, the method was likely to result in an estimation bias. Nonetheless, this bias was claimed to be small and would approach zero for infinite number of data points. Later, using the Levenberg- Marquardt search algorithm, Taubin proposed a general curve-fitting approach [34] that could certainly be applied to solve the non-linear optimization problem of circular arc fitting formulated by Landau. Gander et al. proposed to minimize Landau’s objective function by using the Gauss-Newton algorithm [35]. It is postulated that the method of Taubin [34] and Gander et al. would yield almost the same results as the objective functions of the two methods are essentially the same. In an attempt to estimate curvature values for noisy data points, Lee et al. proposed to fit the data points to a parametric cubic polynomial curve [36]. As the curvature is calculated from the coefficients of the fitted cubic polynomial, a systematic error or bias is likely to exist. Methods from this category those are compared in result section are proposed by Landau [32], Thomas and Chan [33], Taubin [34], Lee et al. [36]. 68 4.1.2 Interval invariant based methods The second category is integral invariant based methods. Curvature is theoretically formulated with first and second derivatives of a smooth curve. As approximated derivatives from discrete points are sensitive to noise, they are not suitable for curvature estimation in high-accuracy applications. Recently, the introduction of integral invariants has made the curvature computation more robust to noise and small shape perturbations. Integral invariants also have some desirable properties over their differential counterparts, such as locality of computation. They do not exhibit noise sensitivity associated with differential quantities and therefore, do not require pre-smoothing of the input data. Due to the advantages stated above, Manay et al. introduced a local area integral for estimating the curvature of a curve [37]. More recently, Lin et al. proposed to use line integrals rather than the area integral for curvature calculation [38]. They defined the window size according to the curvature sharpness. To remain consistent with current application, the window size is thus restricted to five points while comparing with proposed method. 4.1.3 Image data oriented methods The third category is image data oriented methods. Extrinsic curvature at a point on a smooth curve is defined as the reciprocal of the radius of the osculating circle at the point. In the context of digital image data, Worring and Smeulders have demonstrated that curvature estimation based on osculating circle construction is superior to differential tangent based methods [39]. This method is in fact popular in digital image analysis where a constant average distance between the input data points is used to determine an important resolution parameter for curvature calculation. In the current problem, however, the desired constant average distance is not available because the input points are not distributed in a grid-like 69 manner as in the digital image data. Recently, also in the context of digital image analysis, Barwick proposed an indirect circle fitting method using parallel chords [40]. Again, as the input points were well-aligned in grids, three exactly parallel chords could be identified and used for circle fitting. This method, however, could not be applied to the current problem as the presence of three parallel chords is almost not possible for the given five points. Rueda et al. noted that radius of a fitted circle could be obtained by applying the chord property that the bisector of a chord passes through the circle center [41]. In this work, a curvature scale is first employed to identify the largest connected set of points symmetrically positioned with respect to the point of interest. The linear segment connecting the two end points of the connected set gives the chord from which the radius can be calculated. To remain consistent with the current problem, the largest connected set of points is restricted to five points. 4.1.4 Hough transform based methods The fourth category is Hough transform based methods. Hough transform is a feature extraction technique used in digital image analysis and maps the data from the image plane onto a parameter space. It is essentially a voting-based technique that can be used to extract the circle parameters. The voting is done in the parameter space in which a peak value would indicate the existence of a circular arc. The center and radius of the circular arc are then given by the coordinates of the peak. Pei and Horng proposed the first curvature radius estimation method based on Hough transform [42]. Recently, Ioannoua et al. presented a two-step algorithm for circle recognition [43]. The first step used Hough transform to detect the circle centers and the second step validated the existence of the circles by a radius histogram. Hough transform based methods are not applicable to the current problem as it is 70 based on voting. Five points are not able to generate enough votes to identify peaks or clear winners in the parameter space. 4.2 Proposed approach According to the existing methods, the local curvature radius is to be evaluated from measured data only. But, these data are subject to inspection inaccuracies. Now when the local curvature radius needs to be estimated from locally measured points even a small inaccuracy of the measured point can drastically change the result if only the measured point is taken into consideration. The key to obtain the local radius value reliably is effective consideration of measurement uncertainty. Hence, a statistical approach is adopted to estimate the local radius at mid-point of a five consecutive points. Now, to estimate a reliable local radius value a finite population of feasible radii is generated. To generate the feasible radii population, measurement uncertainty region around any measured point is taken into consideration. Again, as the measurement uncertainty is considered as normally distributed, any specific point far from the measured point has its own probability. So basically every feasible radius can be associated with its own probability. Taking these radii values and corresponding probabilities into account, the final curvature radius is calculated from the population median. 4.2.1 Underlying principle Airfoils are evaluated from discrete coordinate data collected in sections by a touch-probe coordinate measuring machine. These data are subject to inspection inaccuracies due to sharp curvature changes in leading and trailing edges. Now when the local curvature radius needs to be estimated from locally measured points even a small inaccuracy of the measured point 71 can drastically change the result. Under such circumstances, the concept of considering domain is introduced. Figure 4.1 Concept of feasible radii R2 R1 As shown in Figure 4.1, the black actual point, i.e., the actual point intended to be measured can be anywhere inside the uncertain zone of diameter blue circle. If only the three black dots are considered, it would generate curvature radius of . On the contrary, if all three dots with cross-hair are selected it would generate different curvature radius of . Clearly, these two radii are different in magnitude but both are feasible. Hence the underlying principle is based on generating a population of feasible radii candidates that leads to reliably calculate the curvature radius. As mentioned earlier, only valid or inwardly centered curvatures are considered here as feasible options. 72 4.2.2 Finite population of valid radii The proposed approach is to estimate a reliable value of curvature radius from a finite population of valid radii candidates. The responsibility is to generate the sample population as accurate as possible. Thus all possible combinations of cases within the five point scheme are taken into consideration. As stated earlier, the sources for finite population of radii are 3 triplets, 2 quartets and 1 quintet dataset. In case of triplets, all the three constituent points bear same radius of the arc. But, in case of quartets or quintet, since points are interpolated by biarc, there are two arcs for each case. For all such cases, only the corresponding arc radius of the point of interest is considered in finite population of radii. 4.2.3 Median as measure for central tendency As finite population of valid radii candidates is obtained, the task is to get the central tendency of the distribution of feasible radii. In statistical literature, the term central tendency refers to a single value, around which quantitative data tend to cluster. Common measures of central tendency are the mean, median, mode, and inter-quartile mean. Even though, the common choice is mean, it has risk of being corrupted by outliers, extreme values, long tails etc. In current case, the population of feasible radii is highly likely to have these properties, so a more robust/reliable measure than mean is needed. Hence, median is chosen. Median is described as the numerical value that separates the higher half of a probability distribution of a population from the lower half. In particular, the median is used as a measure of central tendency when a distribution is skewed, end-values are not known or infinite, or to reduce the influence of outliers present in the dataset that are generated due of measurement errors. The median is also the same as the second quartile (i.e. Q2 or 50 th percentile) [44]. 73 4.3 Implementation The detail method for evaluating local radius is discussed in subsequent sub-sections: Figure 4.2 Flowchart for feasible radii generation from any particular dataset. Non linear five point scheme Dataset (triplet / quartet / quintet) Consider U domain for each measured point of the dataset Discretize U domain into ‘m’ divisions Calculate proabibility of each ‘m’ interval Generate mn combination cases Calculate combined probababilty of each cases Calculate radius at point of interest for each case Check case validity Store valid cases’ radii and associated normalized probability 74 4.3.1 Discretization of measurement uncertainty region To calculate the probability of any potential actual point around the measured point within domain, as shown in Figure 4.3, the whole (= ) domain is divided into ‘ ’ multiple equal divisions (here =5). Since it is continuous random variable, for each interval, it has a probability density function .Therefore, its probability of falling into a given interval, say , is given by, . This leads to estimation of the probability of any actual point falling in the ‘ ’ divisions. For any particular division, if a point is taken from that division, its corresponding probability is considered in the further calculations. Figure 4.3 Discretization of measurement uncertainty region. 45.14%23.84%3.46% 23.84% 3.46% 0.13%0.13% µ 0.6σ 1.8σ 3.0σ-0.6σ-1.8σ-3.0σ d d d d d As shown in Figure 4.3, the original measured point is at mean ( ) and the normally distributed area spread up to (= ) is also given. Here the zone is first discretized into ‘ ’ ( =5 in Figure 4.3) equal divisions with the probability value for each interval. If any 75 point is taken from these five independent areas, it attaches corresponding areas probability value. As mentioned, the total area under domain entails 99.74% (for consideration). 4.3.2 Generation of feasible cases from datasets From a set of five points, there can be six datasets as mentioned previously, which are: and . All feasible i.e., valid only radii from all six datasets are combined in finite population pool from where the local curvature radius is estimated. As Table 4.1 suggests, a maximum of 11,920 cases can constitute finite population had all the cases are valid (i.e., 100% validity) or inwardly centered. If overall validity is 75% then total cases in the pool would be 11,920 × 0.75 = 8,940. All these valid cases have corresponding combined probabilities. A varied number of ‘ ’ is used here to make total triplets’ cases, total quartets’ cases and quintet case comparable with each other. The number of ‘ ’ is decided by keeping balance between accuracy and efficiency. Table 4.1 Sample cases considered from datasets. Dataset Used division ‘m’ Total cases Triplet 1 11 11 3 = 1331 Triplet 2 11 11 3 = 1331 Triplet 3 11 11 3 = 1331 Quartet 1 7 7 4 = 2401 Quartet 2 7 7 4 = 2401 Quintet 5 5 5 = 3125 Combined total - 11920 76 4.3.3 Calculation of probability and radius for any individual case If around any measured point is divided into ‘ ’ divisions, for any triplet dataset there will be ( ) combinations. For each combination the ‘ ’ is calculated by multiplying each point’s individual probability value, i.e., where denotes the probability. Figure 4.4 Calculation of combined probability for a particular radius. R Pr(P1)=0.0346 Pr(P2)=0.2384 Pr(P3)=0.4514 For the case shown in Figure 4.4, the combined probability or weight of the particular triplet case is 0.0346 × 0.2384 × 0.4514 = 0.0037. Thus, calculated triplet case radius ‘R’ has corresponding weight of 0.0037. As there are cases possible out of a triplet dataset, the sum of the combined probability of valid cases should equal to one i.e. . For 100% validity, the sum of combined probability of cases = 0.9922 (= 0.99743) (before normalizing to 1). Similarly, all six datasets are normalized before combining into finite population pool. 77 Therefore, it can be said that, after combining all valid cases (11,920 cases if validity is 100%), there would be a population of feasible radii value for the point of interest. Addition to the corresponding radius value of each valid case, its associated probability is also registered. Subsequently, the task is to estimate a reliable radius value from all valid radii values and associated weights or probability values. Note that, once they are combined all the probability values are re-normalized to 1. The flowchart shown in Figure 4.2 presents the steps for generating feasible radii from any particular dataset only. Now, the remaining task is to calculate local radius from combined population of radii generated from all six datasets sources. 4.3.4 Calculation of local radius from median The median, a measure of central tendency, is used to estimate curvature radius from the population of feasible radii. To calculate median, all valid radii values are sorted in ascending order first. The median is then calculated as the numerical radius value separating the higher half of sorted valid cases’ probability from the lower half. For any probability distribution, regardless of being continuous probability distribution or a discrete probability distribution, the median ( ) is a real number that satisfies the following inequalities (4.1) These final steps to calculate median are shown in a flowchart in Figure 4.5. 78 Figure 4.5 Flowchart for calculation of local curvature radius from dataset results. Feasible radii, normalized associated probabilities for triplet 1 Normalize probability of combined radii population Calculate median from radii population Radius at point of interest Feasible radii, normalized associated probabilities for triplet 2 Feasible radii, normalized associated probabilities for triplet 3 Feasible radii, normalized associated probabilities for quartet 1 Feasible radii, normalized associated probabilities for quartet 2 Feasible radii, normalized associated probabilities for quintet Combined population of feasible radii and associated probabilities 4.4 Validation of proposed method In the current context, once valid radii values and corresponding weights are calculated from six different data subsets (i.e., 3 triplets, 2 quartets and 1 quintet), the combination method can be either weighted mean based or median based. The typical distributions of values generated by the current data sets favor median based method over weighted mean based method. This is because estimates from median based method have superior properties in handling skewed distributions and large radii values calculated from extreme cases with curvature values close to zero. Here median provides better measure of central tendency whereas the mean has the tendency to be influenced by the extreme large radius values. This section validates the adoption of proposed approach. 79 4.4.1 Median vs. Weighted mean Medians give a robust measure of central tendency in the presence of outlier values compared to means. Statistically, quartiles are useful measures because they are less susceptible to long-tailed distributions and outliers. If the data being analyzed are not actually distributed according to an assumed distribution, or if there are other potential sources for outliers or mis-measured values that are far away from the usual observations, then the quartiles may be more useful descriptive statistics than the mean. Again, when the mean is significantly different than the median, median is more often used to describe the center [44]. Comparison between median, mean and mode is shown in Figure 4.6. Figure 4.6 Comparison of median, mean and mode for skewed distribution [44]. In the proposed method, median based approach has been favored over weighted mean based approach for following hypotheses (validated through typical histogram analysis): a) Median yields better result than mean for individual dataset b) Median yields better result than mean for combined case c) Median result is consistent for curvature or radius calculation 80 a) Median yields better result than mean for individual dataset For weighted mean based method, Winkler proposed a method of combining probability distributions where weighted means (along with variances) from individual dataset are combined to calculate the final mean [45]. Again, in case of median based method, the entire individual radius is combined for calculating the final median. It has been observed that, the final median lies in between the boundary median values of six individual datasets. Therefore, it is imperative to generate a reliable final measure; each individual measure (i.e., weighted mean for mean based method and individual dataset’s median for median based method) should be as accurate as possible. To validate, a typical dataset radii are analyzed. A biarc curve with on with basic radius of 0.05 mm and on with adjacent radius of 1 mm is taken as input. The average spacing between points is 5 µm and applied measurement uncertainty is 0.1 µm. Here (i.e., dataset with ) results and histogram are analyzed. The calculated validity is 78% i.e. 11 3 × 0.75 = 999 cases are valid here. For all cases shown hereafter, the theoretical curvature radius is 0.05 mm. Input: (0.05, 0); (0.04975, 0.00499); (0.049, 0.00993); (0.04782, 0.01479); (0.04656, 0.01963). Figure 4.7 illustrates median to be more accurate than the mean in case of typically skewed distribution. The histograms here are representing the feasible curvature radii values only. 81 Figure 4.7 Histogram analysis of valid cases’ radii from individual dataset (triplet1). b) Median yields better result than mean for combined case For validating median based method, combined case analysis is of more importance. For example, in case of a biarc input where first three points are from and last two points are from , radius calculated from tends to be accurate than as matches with original input. But since there is no prior information of the input type, all triplets are tried. The triplet based radius calculation only focus on radius calculation from subjected triplet set. Thus, any individual dataset ( in this example) may have wrong effect in deciding the overall result if calculated through the mean based method. Such situations are better dealt by the median based method. To validate this, combined histogram of previous input is shown in Figure 4.8 with overall calculated results. 0 10 20 30 40 50 60 70 0 .0 1 3 0 .0 1 4 0 .0 1 5 0 .0 1 6 0 .0 1 7 0 .0 1 8 0 .0 1 9 0 .0 2 0 0 .0 2 1 0 .0 2 2 0 .0 2 4 0 .0 2 6 0 .0 2 8 0 .0 3 0 0 .0 3 3 0 .0 3 7 0 .0 4 2 0 .0 4 8 0 .0 5 6 0 .0 6 7 0 .0 8 3 0 .1 1 1 0 .1 6 7 0 .3 3 3 > 0 .3 3 F re q u en cy Radius (mm) Median = 0.05 (0% error) Wt. Mean = 0.072 (44% error) 82 Figure 4.8 Histogram analysis of combined valid cases from all six datasets. Here, for the biarc input (where are from and are from ) (containing ) can be highly deceptive and yield completely different set of curvature radii values. On the contrary, all other 5 data sets may be more similar to each other. Situations like this may create a ‘local maxima’ once all the valid distributions are combined, as shown in Figure 4.8. But the median based method has the ability to handle such kind of situations better than the mean based method. c) Median result is consistent for curvature or radius calculation Extrinsic curvature is usually defined as the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point. Therefore, local curvature radius calculated from radii directly or from curvature values should yield the same result. But, unlike median, the weighted mean method yields different local radius values for different 0 200 400 600 800 1000 1200 1400 1600 0 .0 2 0 0 .0 3 0 0 .0 3 5 0 .0 4 0 0 .0 4 5 0 .0 5 0 0 .0 5 5 0 .0 6 0 0 .0 6 5 0 .0 7 0 0 .0 8 0 0 .1 0 0 0 .1 5 0 0 .2 0 0 0 .4 0 0 > 0 .4 F re q u en cy Radius (mm) Median = 0.052 (4% error) Wt. Mean = 0.127 (154% error) 83 (curvature/radius based) approaches. The inconsistency is shown in Table 4.2 for the previous input case. Table 4.2 Results for radius or curvature calculation. Method Curvature, κ Curvature -1 , 1/κ Radius, R (mm) Consistency, 1/κ = R? Median based 19.07 0.052 0.052 Yes Wt. Mean based 17.42 0.057 0.127 No From Table 4.2, it is evident that, in case of weighted mean based method, results are not consistent. On the contrary, for median based method local radius, either measured from curvatures only or radius only, are consistent. Hence, it can be summarized that, the median based method has more potential to represent robust measure of central tendency than the mean based method. This is due to median based method’s inherent ability to handle skewed distribution better, to be less susceptible to long- tailed distributions and to attach reduced importance to extreme or atypical values in the distribution. 4.4.2 Verify six individual datasets The input curve can be arc, combination of arcs i.e., biarc or a triarc with assumed property that at least three consecutive points contain local radius. Now as there are six datasets, it has been observed that, individual dataset can gain superiority from others in terms of accuracy. But this behavior is input type (arc/biarc/triarc) specific where the input types are not known. The need of any individual dataset is thus verified in this sub-section. 84 Case with biarc type: Input: (0.05, 0); (0.04975, 0.00499); (0.049, 0.00993); (0.04707, 0.01476); (0.04554, 0.01925). A biarc curve with on with basic radius of 0.05 mm and on with adjacent radius of 0.025 mm is taken as input. The average spacing between points is 5 µm and applied measurement uncertainty is 0.1 µm. Individual dataset results and their percentage errors are shown in Table 4.3. Table 4.3 Case when any individual triplet and quartet datasets yield higher error. Data Set Calculated radius (mm) Error % Triplet 1 0.050 0.0 Triplet 2 0.044 11.1 Triplet 3 0.027 46.7 Quartet 1 0.044 11.6 Quartet 2 0.075 49.7 Quintet 0.047 6.6 As shown in Table 4.3 for a particular case and can yield higher error, whereas as shown in Table 4.4 other datasets like and can yield relatively higher error. Therefore, considering only a particular dataset(s) can be unreliable. Hence, all six datasets have equal importance and are treated equally in combining results. Case with triarc type: Input: (0.1, 0); (0.09984, 0.005); (0.09922, 0.00996); (0.09811, 0.01483); (0.09654, 0.01957). Again, a triarc curve with on and on with adjacent radius of 0.1 mm; on with basic radius of 0.05 mm is taken as input. The average spacing 85 between points is 5 µm and applied measurement uncertainty is 0.1 µm. Individual dataset results and their percentage errors are shown in Table 4.4. Table 4.4 Case when any individual triplet and quintet datasets yield higher error. Data Set Calculated radius (mm) Error % Triplet 1 0.053 6.7 Triplet 2 0.050 0.0 Triplet 3 0.053 6.7 Quartet 1 0.050 0.3 Quartet 2 0.048 3.6 Quintet 0.053 6.6 Judging all these different scenarios, it is proposed to equally treat all six individual datasets rather than prioritizing any particular dataset results. 4.4.3 Verify combining all six datasets The essence of combining all six datasets can also be questioned. As there is no predefined criterion to find which dataset yields better accuracy, it has been observed that a combined or cumulative result consistently yields better accuracy. To validate, combined dataset’s performance comparative to individual dataset’s performance, a biarc curve with on with basic radius of 0.05 mm and on with adjacent radius of 1 mm is taken as input. The average spacing between points is 5 µm and applied measurement uncertainty is 0.1 µm. Individual dataset results and combined method’s result is shown in Table 4.5. 86 Input: (0.05, 0); (0.04975, 0.00499); (0.049, 0.00993); (0.04782, 0.01479); (0.04656, 0.01963). Table 4.5 Case to verify combining all six datasets. Data Set Calculated radius (mm) Error % Triplet 1 0.050 0.0 Triplet 2 0.057 13.5 Triplet 3 0.296 492.6 Quartet 1 0.057 13.7 Quartet 2 0.039 21.4 Quintet 0.048 3.8 Combined 0.052 4.9 Clearly, as shown in Table 4.5, combined result yields reliable and better accuracy than most of the individual dataset. Since individual datasets performance varies for different input type, aggregating all six dataset’s valid radii is conducted. The local curvature radius is then calculated from the median of the combined population of feasible radii. 87 4.5 Results and analysis To summarize, an algorithm is proposed to evaluate the minimum curvature radius restriction via a rolling scheme of five consecutive points in order to retrieve the local curvature information of the mid-point. As mentioned, effective consideration of the measurement uncertainty would be the key to reliably estimate the curvature radii. In this section, existing methods that relate to curvature radius estimation from discrete points and can potentially be applied to the current problem are compared. The compared six methods are proposed by Landau [32], Thomas and Chan [33], Taubin [34], Lee et al. [36], Lin et al. [38] and Rueda et al. [41]. The comparisons of these existing methods are presented and their relative strength and weakness are discussed. 4.5.1 Factors analyzed To make a comprehensive study the factors those have been analyzed are adjacent radius or neighborhood arc’s radius, average point spacing between five points and measurement uncertainty. For all the test cases, absolute percentage errors are calculated relative to the ideal input arc radius from where the data points are simulated. Also, among the five input data points, the middle point (i.e., point of interest) at which the radius is to be estimated has an ideal radius of 0.05 mm. Due to the presumption that at least three consecutive points are needed to reliably capture the local curvature information, the five input points can in effect be part of an arc, a biarc, or a triarc as depicted in Figure 4.9. The neighboring arc, in case of biarc/triarc, is referred as adjacent arc. The main arc containing point of interest is referred as basic arc. The point of interest is indicated by a point with a cross-hair in the figures. 88 Figure 4.9 Five input points in relation to an ideal arc (solid blue line): (a) Single arc; (b) Biarc; (c) Triarc. Basic arc Basic arc Basic arc Adjacent arc Adjacent arc (a) (b) (c) A series of comparative case studies are carried out for all above specified types: firstly arc type input, secondly biarc type input and thirdly triarc type input. In the first case, all the five input points are sampled from a single ideal arc of 0.05 mm basic radius. In the second case, first three consecutive input points are sampled from an arc of 0.05 mm basic radius and the remaining two input points are sampled from neighboring arc of mentioned adjacent radius. In the third case, three consecutive mid points are sampled from an arc of 0.05 mm basic radius and the bordering two points are sampled from two arcs with mentioned adjacent radius. For simplicity, the adjacent radius of both sided neighboring arcs of triarc type input is kept equal. The impact of varying adjacent radius for biarc and triarc inputs is analyzed. The measurement uncertainty is assumed to follow a Gaussian distribution of standard deviation σ. The five input points are thus sampled from theoretical curve with superimposed Gaussian deviates representing the measurement uncertainty effect. Varying values are used to study its significance. For each , 10 random data sets are generated and the resulting absolute percentage errors are averaged. The impact of point resolution on the computed 89 result is also studied by changing the average point spacing from 3 to 10 μm. Measurement uncertainty is kept constant while spacing is varied. Similarly, for each spacing 10 random data sets are generated and the resulting absolute percentage errors are averaged. Again a special case for arc, biarc and triarc are studied where the five points are sampled from theoretical input curve but = 0.8 μm is applied for calculation purpose only. A special case, where all the five points in a local domain have zero measurement error contrast to known overall uncertainty value, may occur for practical case. Even though such a case has very slim probability to happen it is also studied. To summarize, all the factors analyzed in subsequent section are tabulated in Table 4.6. For cases when the average is reported from 10 random cases, the standard deviation of the percentage error apart from average percentage error is compared. Table 4.6 Parameters analyzed in results. Types of input Basic radius (mm) Adjacent radius (mm) Spacing (µm) Measurement Uncertainty (µm) Arc 0.05 - 3 - 10 0.8* - 3 - 10 0.8 - 5 0.5 – 1* Biarc 0.025 - ∞ 5 0.8* 0.025 - ∞ 5 0.8 1 5 0.5 – 1* Triarc 0.025 - ∞ 5 0.8* 0.025 - ∞ 5 0.8 1 5 0.5 – 1* NB: * refers to average result of 10 random cases for a particular . 90 4.5.2 Results Nine case studies mentioned in each row of Table 4.6 are illustrated in this section. For all cases absolute percentage error of six existing methods are compared with proposed method. a) Arc analysis In the first stage, all the five input points are sampled from a single ideal arc of 0.05 mm radius. The impact of point spacing is studied in Figure 4.10 with average point spacing ranging from 3 to 10 µm. As shown in Figure 4.10, proposed method produces better accuracy in terms of both average and standard deviation. As expected, error decreases as spacing is increased keeping constant at 0.8 µm. It has been observed that all other six methods are unreliable at low spacing (e.g. 3 - 4 µm). Again Figure 4.11 illustrates special subset cases for Figure 4.10. As in practical case, is assumed to be uniform and global property for particular inspection process rather than varying for every rolling scheme of five points. So it is probable that a five point scheme occurs where all the input points have zero individual measurement error whereas the system is much higher than local five points’ measurement uncertainty. Such a situation is simulated in Figure 4.11 where all five points belong to theoretical arc but mentioned is used in the proposed algorithm for calculation only. As the compared six methods are mostly effective in low noise, the proposed method is found to yield higher relative error for special case for low spacing (e.g. 3 µm). But as spacing is increased, proposed method performs comparable to other six methods even for special case. 91 Figure 4.10 Study of varying spacing for arc input with U 0.8 μm. 0 10 20 30 40 3 4 5 8 10 M ea n E rr o r (% ) Spacing (μm) Landau Thomas Taubin Lee Lin Rueda Proposed 0 10 20 30 3 4 5 8 10 σ E r ro r (% ) Spacing (μm) Landau Thomas Taubin Lee Lin Rueda Proposed 92 Figure 4.11 Study of special case of arc with applied U 0.8 μm. Varying values are used to study the robustness of the proposed method, results of which are shown in Figure 4.12. At each , 10 different data sets are generated and the resulting absolute percentage errors are averaged along with their standard deviation. As expected, error increases with increased measurement uncertainty. It is noted that, Landau’s method fails to perform at this stage and is not shown in the plot. The performances of the other five methods are similar where the errors reach 15-20% with the inclusion of measurement uncertainty. Both average error and standard deviation of errors for proposed method is lower than compared six methods. 0 10 20 30 3 4 5 8 10 E rr o r (% ) Spacing (μm) Landau Thomas Taubin Lee Lin Rueda Proposed 93 Figure 4.12 Study of varying U for arc input with spacing of 5 μm. 0 10 20 30 0.5 0.6 0.7 0.8 0.9 1 M ea n E rr o r (% ) Measurement Uncertainty (μm) Thomas Taubin Lee Lin Rueda Proposed 0 10 20 0.5 0.6 0.7 0.8 0.9 1 σ E r ro r (% ) Measurement Uncertainty (μm) Thomas Taubin Lee Lin Rueda Proposed 94 b) Biarc analysis As mentioned, the six compared methods are tailored for arc type input which is not guaranteed in airfoil application. Thus biarc and triarc type inputs are also need to be tested with the methods. Here the five input points are sampled from a biarc as shown in Figure 4.9(b). The main arc’s radius is referred to as the basic radius while the neighboring arc’s radius as the adjacent radius. For the biarc, the arc that passes through three points is the basic arc while the other arc (passing through two points) is the adjacent arc. The adjacent radius is varied from 0.025, 0.1, 1, 100 mm to infinity (i.e., a line) in order to study its effect on the computed curvature radius. The computed results for the point spacing of 5 µm and measurement uncertainty of 0.8 µm are plotted in Figure 4.13. It is observed that, biarcs in general, lead to higher relative error than mono arcs: around 5% for the arc cases but up to 20% for the biarc cases. It is also noted that, the errors are lower when the adjacent arc radius is close to the basic arc radius of 0.05 mm. The method of Thomas and Chan becomes unstable for very large adjacent radii. Again, Landau’s method fails to perform at this stage and is not shown in the plot. As studied in arc analysis, in Figure 4.14 special subset case of Figure 4.13 is also studied. It shows that, proposed method performs even better than other five methods except Landau’s method. But again, as Landau’s method failed earlier for varying measurement uncertainty, it cannot be relied entirely. 95 Figure 4.13 Study of varying adjacent radius for biarc input with U 0.8 μm, spacing 5 μm. 0 10 20 30 40 0.025 0.05 0.1 1 100 ∞ (line) M ea n E rr o r (% ) Adjacent Radius (mm) Thomas Taubin Lee Lin Rueda Proposed 0 10 20 30 0.025 0.05 0.1 1 100 ∞ (line) σ E r ro r (% ) Adjacent Radius (mm) Thomas Taubin Lee Lin Rueda Proposed 96 Figure 4.14 Study of special case of biarc with applied U 0.8 μm, spacing 5 μm. In Figure 4.15, the robustness of proposed algorithm is observed. Here, when the effect of varying from 0.5 to 1 µm is studied where spacing is kept constant at 5 µm for an adjacent radius case of 1 mm, it has been observed that with the introduction of measurement uncertainty, the performance of all the six methods degrades. Here, the errors of compared methods shift higher up to 30% for the biarc cases. As the majority of the examined methods are based on best fitting techniques, the estimation errors would not keep increasing with further unfavorable input point distributions and tend to stabilize after reaching some high- error limit. For example, it has been observed for the bi-arc cases that, once the estimation errors reach a high-error level, they become steady even with smaller point spacing and/or larger adjacent radius. 0 10 20 30 40 50 0.025 0.05 0.1 1 100 ∞ (line) E rr o r (% ) Adjacent Radius (mm) Landau Thomas Taubin Lee Lin Rueda Proposed 97 Figure 4.15 Study of varying U for biarc with spacing of 5 μm, adjacent radius of 1 mm. 0 10 20 30 40 0.5 0.6 0.7 0.8 0.9 1 M ea n E rr o r (% ) Measurement Uncertainty (μm) Landau Thomas Taubin Lee Lin Rueda Proposed 0 10 20 30 40 0.5 0.6 0.7 0.8 0.9 1 σ E r ro r (% ) Measurement Uncertainty (μm) Landau Thomas Taubin Lee Lin Rueda Proposed 98 c) Triarc analysis After biarc analysis triarc inputs are also tested. Here the five input points are sampled from a triarc as shown in Figure 4.9(c). The main arc’s radius is referred to as the basic radius while the neighboring arc’s radius as the adjacent radius. For simplicity, both side’s adjacent radius are kept equal in this study. The adjacent radius is varied from 0.025, 0.1, 1, 100 mm to infinity (i.e., a line) in order to study its effect on the computed curvature radius. The computed results for the point spacing of 5 µm and measurement uncertainty of 0.8 µm are plotted in Figure 4.16. It is observed that, proposed method also generates better accuracy for triarc type input which is around 5%. Smaller error for triarc input can occur due to the symmetric nature of adjacent arcs. It is also noted that for both triarc and biarcs, the errors are lower when the adjacent radius is close to basic radius of 0.05 mm. The method of Thomas and Chan fails for very large adjacent radii. Again Figure 4.17, which is a special subset case of Figure 4.16 shows that, proposed method performs better than other six methods. 99 Figure 4.16 Study of varying adjacent radius for triarc input with U 0.8 μm, spacing 5 μm. 0 5 10 15 20 0.025 0.05 0.1 1 100 ∞ (line) M ea n E rr o r (% ) Adjacent Radius (mm) Landau Thomas Taubin Lee Lin Rueda Proposed 0 5 10 15 0.025 0.05 0.1 1 100 ∞ (line) σ E r ro r (% ) Adjacent Radius (mm) Landau Thomas Taubin Lee Lin Rueda Proposed 100 Figure 4.17 Study of special case of triarc with applied U 0.8 μm, spacing 5 μm. Figure 4.18 shows the effect of varying measurement uncertainty values for the triarc cases. For these calculations, point spacing is kept at 5 µm and is varied from 0.5 to 1 µm. It is again observed that, with increasing measurement uncertainty, the performance of compared methods degrades with the errors shifting even higher up to 20% for the triarc cases. Once again, Landau’s method yields high error where as all other five methods yield similar results. Since the mid three points are from basic arc and adjacent arc’s radii are kept equal it became symmetric which might cause triarc to yield better accuracy than biarc in general. 0 5 10 15 0.025 0.05 0.1 1 100 ∞ (line) E rr o r (% ) Adjacent Radius (mm) Landau Thomas Taubin Lee Lin Rueda Proposed 101 Figure 4.18 Study of varying U for triarc with spacing of 5 μm, adjacent radius 1 mm. 0 10 20 30 0.5 0.6 0.7 0.8 0.9 1 M ea n E rr o r (% ) Measurement Uncertainty (μm) Landau Thomas Taubin Lee Lin Rueda Proposed 0 10 20 0.5 0.6 0.7 0.8 0.9 1 σ E r ro r (% ) Measurement Uncertainty (μm) Landau Thomas Taubin Lee Lin Rueda Proposed 102 To summarize, it has been observed that the method of Landau does not work well when the input data points contain measurement uncertainty. The other five methods yield analogous results for most of the case studies and a clear winner cannot be concluded. In fact, none of the existing methods is able to yield an estimation error less than 20% for the biarc case of 0.05 mm basic radius and 1 mm adjacent radius, point spacing of 5 µm and measurement uncertainty of 0.8 µm. This indicated the need to develop a new and more effective method to evaluate the minimum curvature radius restriction for the leading and trailing edges of an airfoil section. From all the case studies investigated, it is concluded that, for arc, biarc or triarc type of input with spacing ranging from 3 to 10 µm and measurement uncertainty ranging from 0.5 to 1 µm, the proposed method to evaluate local curvature radius consistently surpass the compared existing methods. 103 5. CONCLUSIONS The objective of the research work is to conform that a manufactured airfoil part maintains maximum length restriction and minimum radius restriction. A complete case study is thus presented in this chapter along with concluding remarks and future works. 5.1 Complete case study Identification of linear segments and evaluation of local radius is discussed separately in previous chapters. It is mentioned that for any curvilinear LE/TE profile, first linear segment length restriction length is checked. Then local radius at every possible non-linear segment point is calculated from rolling five point schemes. For all such calculations only the input coordinates and measurement uncertainty, values are provided as inputs. 5.1.1 Simulated input A simulated input with 250 sequential data points is considered for complete case study. As LE/TE profile, the curvilinear input profile is a combination of 3 lines and 5 arcs connected with tangent continuity. Average point spacing is kept uniform at 10 µm. Measurement uncertainty is assumed to follow a Gaussian distribution of standard deviation σ. Here σ is taken as 0.8/6 µm i.e., = 0.8 µm. All the input points are thus sampled from theoretical curve first with superimposed Gaussian deviates representing the measurement uncertainty effect. The Gaussian deviates are generated randomly within interval where the mean, µ is considered as 0 and as 1. The simulated measured point for calculation is then generated from theoretical curve along their normal direction applying corresponding point’s random deviate value. All the input coordinates and their random deviate values are given in Table 104 5.2. The input profile is shown in Figure 5.1. Considering bottom leftmost point as No. 1 and top-leftmost point as No. 250, theoretical parameters (length for line and radius for arc) of the input segments are given in Table 5.1. Figure 5.1 Simulated input points on a curvilinear profile. Lin e 1 Arc 1 Arc 2 Arc 3 Line 2 Arc 4Line 3 Arc 5 (mm) (m m ) Table 5.1 Input segments with theoretical parameters. Input segment Segment points Length/Radius (mm) Line 1 1-101 1 Arc 1 102-119 0.25 Arc 2 120-125 0.08 Arc 3 126-136 0.15 Line 2 137-187 0.5 Arc 4 188-197 0.3 Line 3 198-238 0.4 Arc 5 239-250 0.2 105 Table 5.2 Input data No. X Y Random deviate No. X Y Random deviate No. X Y Random deviate 1 0.00002 -0.00003 1.18 41 0.34640 0.20003 0.64 81 0.69282 0.40000 0.15 2 0.00859 0.00513 0.23 42 0.35509 0.20497 1.58 82 0.70153 0.40492 2.61 3 0.01734 0.00997 0.76 43 0.36377 0.20994 -0.03 83 0.71009 0.41009 -0.05 4 0.02593 0.01508 -0.85 44 0.37239 0.21500 0.32 84 0.71880 0.41501 -0.01 5 0.03460 0.02007 1.86 45 0.38107 0.21997 1.98 85 0.72752 0.41990 -0.52 6 0.04328 0.02504 -0.04 46 0.38966 0.22509 -0.65 86 0.73621 0.42484 1.21 7 0.05197 0.02999 2.59 47 0.39838 0.22999 0.31 87 0.74477 0.43002 0.11 8 0.06062 0.03500 -0.69 48 0.40706 0.23494 1.59 88 0.75344 0.43501 1.69 9 0.06932 0.03994 -1.58 49 0.41570 0.23999 -0.97 89 0.76209 0.44002 0.46 10 0.07798 0.04494 -0.61 50 0.42430 0.24510 -1.15 90 0.77078 0.44497 1.07 11 0.08658 0.05004 0.38 51 0.43296 0.25009 -1.48 91 0.77942 0.45000 -0.13 12 0.09523 0.05505 1.27 52 0.44171 0.25494 -0.97 92 0.78809 0.45499 0.19 13 0.10385 0.06012 -0.59 53 0.45033 0.26001 -0.04 93 0.79674 0.46000 -0.45 14 0.11260 0.06497 2.25 54 0.45899 0.26501 -2.12 94 0.80537 0.46505 -0.62 15 0.12122 0.07003 -0.02 55 0.46758 0.27013 -1.84 95 0.81405 0.47002 -1.93 16 0.12985 0.07509 0.74 56 0.47630 0.27502 -1.54 96 0.82275 0.47495 -1.43 17 0.13849 0.08012 -0.64 57 0.48491 0.28011 -0.24 97 0.83141 0.47996 0.27 18 0.14725 0.08496 1.44 58 0.49364 0.28499 1.20 98 0.84000 0.48508 -0.36 19 0.15594 0.08991 -1.16 59 0.50230 0.28998 -0.03 99 0.84877 0.48988 -0.98 20 0.16455 0.09499 0.69 60 0.51091 0.29508 -0.54 100 0.85736 0.49500 1.36 21 0.17319 0.10003 0.73 61 0.51962 0.30000 -0.07 101 0.86601 0.50003 -0.43 22 0.18188 0.10498 0.51 62 0.52829 0.30498 2.26 102 0.87482 0.50542 0.97 23 0.19050 0.11005 1.70 63 0.53696 0.30996 0.21 103 0.88348 0.51106 -1.14 24 0.19923 0.11492 -0.68 64 0.54560 0.31499 0.19 104 0.89186 0.51710 2.05 25 0.20781 0.12007 3.41 65 0.55427 0.31998 0.28 105 0.90003 0.52344 0.01 26 0.21647 0.12506 -0.39 66 0.56287 0.32508 -1.53 106 0.90777 0.53028 0.75 27 0.22515 0.13003 1.21 67 0.57161 0.32995 -2.37 107 0.91531 0.53735 0.54 28 0.23378 0.13507 0.31 68 0.58023 0.33501 -2.03 108 0.92261 0.54467 -0.69 29 0.24247 0.14003 -0.49 69 0.58890 0.34000 1.22 109 0.92958 0.55229 1.08 30 0.25113 0.14503 -0.33 70 0.59766 0.34483 -0.36 110 0.93609 0.56031 0.63 31 0.25985 0.14992 0.78 71 0.60618 0.35006 -0.63 111 0.94250 0.56842 -0.76 32 0.26850 0.15495 -0.33 72 0.61485 0.35505 -0.22 112 0.94847 0.57686 0.24 33 0.27718 0.15992 2.10 73 0.62355 0.35998 0.18 113 0.95400 0.58559 -1.20 34 0.28584 0.16491 0.39 74 0.63229 0.36485 0.90 114 0.95923 0.59450 -0.31 35 0.29445 0.17001 -0.65 75 0.64089 0.36994 -1.23 115 0.96411 0.60361 -1.54 36 0.30312 0.17499 -0.06 76 0.64950 0.37503 0.13 116 0.96852 0.61296 -0.62 37 0.31178 0.17998 0.42 77 0.65815 0.38004 0.81 117 0.97265 0.62243 -0.21 38 0.32041 0.18504 -0.07 78 0.66686 0.38497 -1.07 118 0.97639 0.63206 1.08 39 0.32909 0.19000 0.79 79 0.67546 0.39007 -2.01 119 0.97961 0.64188 -0.17 40 0.33775 0.19500 -0.14 80 0.68417 0.39498 0.24 120 0.98260 0.65178 -0.01 106 No. X Y Random deviate No. X Y Random deviate No. X Y Random deviate 121 0.98458 0.66206 -1.16 164 0.63278 0.87416 0.52 207 0.21305 0.96385 -0.92 122 0.98531 0.67251 -0.65 165 0.62339 0.87761 -0.65 208 0.20306 0.96357 0.59 123 0.98459 0.68295 1.57 166 0.61396 0.88095 0.01 209 0.19306 0.96328 -0.70 124 0.98245 0.69320 0.13 167 0.60463 0.88454 1.21 210 0.18307 0.96285 0.59 125 0.97925 0.70317 -0.80 168 0.59520 0.88787 1.41 211 0.17308 0.96254 -1.39 126 0.97447 0.71248 -0.67 169 0.58580 0.89129 0.09 212 0.16308 0.96210 -1.91 127 0.96892 0.72135 -0.71 170 0.57642 0.89475 2.55 213 0.15309 0.96179 -0.99 128 0.96281 0.72985 -0.22 171 0.56703 0.89818 -0.09 214 0.14310 0.96134 0.61 129 0.95611 0.73790 -0.45 172 0.55764 0.90163 1.29 215 0.13310 0.96103 1.15 130 0.94883 0.74543 0.14 173 0.54822 0.90500 0.25 216 0.12311 0.96073 -0.61 131 0.94100 0.75238 1.58 174 0.53884 0.90846 -0.33 217 0.11312 0.96029 1.56 132 0.93276 0.75884 -0.30 175 0.52945 0.91190 1.84 218 0.10312 0.95994 -0.12 133 0.92414 0.76478 -0.35 176 0.52004 0.91527 0.29 219 0.09313 0.95970 0.12 134 0.91501 0.76991 0.87 177 0.51060 0.91858 -0.27 220 0.08313 0.95933 -0.29 135 0.90561 0.77452 0.82 178 0.50126 0.92215 -0.18 221 0.07314 0.95891 -0.56 136 0.89588 0.77836 0.04 179 0.49185 0.92553 0.33 222 0.06314 0.95870 0.26 137 0.88652 0.78190 -1.13 180 0.48245 0.92894 -0.01 223 0.05315 0.95828 0.57 138 0.87710 0.78526 0.68 181 0.47303 0.93232 0.06 224 0.04315 0.95801 -0.21 139 0.86773 0.78873 -0.27 182 0.46368 0.93585 -0.17 225 0.03315 0.95783 -2.69 140 0.85836 0.79222 0.16 183 0.45422 0.93910 -0.96 226 0.02317 0.95730 2.16 141 0.84894 0.79559 -0.47 184 0.44486 0.94263 -0.03 227 0.01317 0.95700 -0.96 142 0.83956 0.79906 0.45 185 0.43548 0.94611 -1.54 228 0.00318 0.95651 1.92 143 0.83011 0.80234 -0.27 186 0.42602 0.94935 0.09 229 -0.00682 0.95628 1.02 144 0.82077 0.80590 -0.57 187 0.41616 0.95286 0.34 230 -0.01681 0.95582 -2.68 145 0.81133 0.80922 -1.42 188 0.40613 0.95587 0.92 231 -0.02680 0.95554 -0.01 146 0.80193 0.81263 -1.06 189 0.39607 0.95877 -1.17 232 -0.03680 0.95523 0.25 147 0.79259 0.81620 1.77 190 0.38583 0.96096 -1.21 233 -0.04679 0.95480 0.79 148 0.78313 0.81944 2.11 191 0.37555 0.96299 2.02 234 -0.05678 0.95446 0.59 149 0.77374 0.82290 -1.02 192 0.36522 0.96468 2.70 235 -0.06678 0.95419 0.58 150 0.76436 0.82635 -0.11 193 0.35482 0.96593 0.92 236 -0.07677 0.95389 0.29 151 0.75497 0.82978 0.06 194 0.34439 0.96680 -0.57 237 -0.08676 0.95339 1.68 152 0.74557 0.83320 -0.44 195 0.33393 0.96746 0.59 238 -0.09748 0.95274 0.51 153 0.73618 0.83665 1.84 196 0.32346 0.96756 0.19 239 -0.10814 0.95146 -0.18 154 0.72678 0.84005 0.56 197 0.31299 0.96738 0.07 240 -0.11873 0.94966 0.01 155 0.71737 0.84342 -0.62 198 0.30300 0.96694 -0.12 241 -0.12923 0.94743 0.63 156 0.70797 0.84684 -1.25 199 0.29301 0.96661 0.45 242 -0.13956 0.94448 -0.04 157 0.69857 0.85027 0.10 200 0.28301 0.96634 0.09 243 -0.14970 0.94091 -0.79 158 0.68914 0.85360 0.07 201 0.27302 0.96589 -3.50 244 -0.15970 0.93699 -1.04 159 0.67978 0.85712 -1.37 202 0.26302 0.96568 -1.13 245 -0.16940 0.93239 0.30 160 0.67039 0.86056 0.68 203 0.25303 0.96531 -1.47 246 -0.17893 0.92741 0.98 161 0.66104 0.86410 0.06 204 0.24303 0.96498 -0.79 247 -0.18799 0.92164 1.31 162 0.65161 0.86742 1.32 205 0.23304 0.96459 -1.86 248 -0.19695 0.91572 0.90 163 0.64222 0.87086 -0.23 206 0.22304 0.96431 -0.27 249 -0.20539 0.90908 -0.73 250 -0.21359 0.90214 0.82 107 5.1.2 Results As mentioned in previous chapters, first the linear segments are identified and lengths are determined. In Table 5.3, the calculated segment lengths are provided. As shown in the result, in case of Line1, it includes 1 extra point from neighboring non-linear segment. Line2 and Line3 include 2 extra points. The percentage errors are also limited to around 5%. Table 5.3 Results of linear segment length. Input segment Theo. segment points Calc. segment points Theo. Length (mm) Calc. Length (mm) Error % Line 1 1-101 1-102 1 1.0103 1.0 Line 2 137-187 135-187 0.5 0.5209 4.2 Line 3 198-238 196-238 0.4 0.4212 5.3 Once three linear segments are extracted, only the remaining three non-linear segments are left. Now as mentioned, radius at mid-point of every rolling scheme is evaluated. As provided in Table 5.4, Table 5.5 and Table 5.6, the radii at every non-linear point are given. Here ‘NC’ refers to two bordering points of each non-linear segment where the local radius is ‘not calculated’. Since these bordering points don’t have required two points on either side, local radii are not calculated. Again, some of the theoretical points that actually belong to arc are identified as part of linear segments (as shown in Table 5.3) and are also not considered for radius calculation. From Table 5.4 results, it is found that at the junction of actual Arc1 and Arc2 (i.e., point No. 120) or at junction of Arc2 and Arc3 (i.e., point No. 126) the percentage error is relatively higher because of biarc type input. Again it is observed that, points where normal deviate values (mentioned in Table 5.2) are large yield higher error. Here the minimum curvature 108 radius is 0.077 mm against an input of 0.08 mm. The results for first non-linear segment are well below 15% with maximum error of 24%. Table 5.4 Results of local curvature radii along first non-linear segment. No. Input segment Theo. radius (mm) Calc. radius (mm) Error % 102 Arc 1 0.25 Linear 103 Arc 1 0.25 NC 104 Arc 1 0.25 NC 105 Arc 1 0.25 0.209 16.4 106 Arc 1 0.25 0.281 12.2 107 Arc 1 0.25 0.310 24.1 108 Arc 1 0.25 0.239 4.3 109 Arc 1 0.25 0.221 11.5 110 Arc 1 0.25 0.282 12.9 111 Arc 1 0.25 0.239 4.3 112 Arc 1 0.25 0.219 12.4 113 Arc 1 0.25 0.262 4.8 114 Arc 1 0.25 0.260 4.1 115 Arc 1 0.25 0.242 3.1 116 Arc 1 0.25 0.264 5.7 117 Arc 1 0.25 0.259 3.7 118 Arc 1 0.25 0.240 4.2 119 Arc 1 0.25 0.261 4.3 120 Arc 2 0.08 0.096 19.9 121 Arc 2 0.08 0.087 9.3 122 Arc 2 0.08 0.078 2.5 123 Arc 2 0.08 0.079 0.9 124 Arc 2 0.08 0.086 7.7 125 Arc 2 0.08 0.077 3.1 126 Arc 3 0.15 0.123 17.9 127 Arc 3 0.15 0.150 0.2 128 Arc 3 0.15 0.150 0.2 129 Arc 3 0.15 0.140 6.4 130 Arc 3 0.15 0.144 3.9 131 Arc 3 0.15 0.159 6.3 132 Arc 3 0.15 0.158 5.2 133 Arc 3 0.15 NC 134 Arc 3 0.15 NC 135 Arc 3 0.15 Linear 136 Arc 3 0.15 Linear 109 It is observed that points actually generated with bigger normal deviates yield higher error. For example, point No. 191 and No. 192 which have normal deviates of 2.02 and 2.7 yield error at 21.8% and 10.6% respectively. Again, for all cases ‘NC’ refers to ‘Not calculated’ when five point schemes are not available. Now calculated maximum length and minimum radius can be checked against prescribed specification to complete evaluation process. Table 5.5 Results of local curvature radii along second non-linear segment. No. Input segment Theo. radius (mm) Calc. radius (mm) Error % 188 Arc 4 0.3 NC 189 Arc 4 0.3 NC 190 Arc 4 0.3 0.337 12.3 191 Arc 4 0.3 0.365 21.8 192 Arc 4 0.3 0.268 10.6 193 Arc 4 0.3 0.300 0.0 194 Arc 4 0.3 NC 195 Arc 4 0.3 NC 196 Arc 4 0.3 Linear 197 Arc 4 0.3 Linear Table 5.6 Results of local curvature radii along last non-linear segment. No. Input segment Theo. radius (mm) Calc. radius (mm) Error % 239 Arc 5 0.2 NC 240 Arc 5 0.2 NC 241 Arc 5 0.2 0.180 10.1 242 Arc 5 0.2 0.182 8.9 243 Arc 5 0.2 0.219 9.7 244 Arc 5 0.2 0.202 0.9 245 Arc 5 0.2 0.195 2.6 246 Arc 5 0.2 0.176 12.2 247 Arc 5 0.2 0.216 8.0 248 Arc 5 0.2 0.187 6.5 249 Arc 5 0.2 NC 250 Arc 5 0.2 NC 110 5.2 Research contributions This thesis contributes in performing evaluation process for strict manufacturing specifications of leading and trailing edge profile of airfoil sections. The research objective is to evaluate linear segment length and local curvature radii along the airfoil leading and trailing edge profile from discrete coordinate data collected in sections by a touch-probe coordinate measuring machine (CMM). These measurement data, subject to inevitable measurement uncertainty poses challenge for accurate and reliable evaluation. The proposed approach can be systematically break-up in three notable contributions. Firstly, a method is developed for detecting linear segments from curvilinear profile. Analogous problems are studied in case of image segmentation. But in case of CMM data, it has not been previously reported. The main feature of the proposed algorithm is the explicit consideration of measurement uncertainty. The algorithm starts by detecting relatively small linear segments and then merges these segments to determine the longest feasible linear segment under given measurement uncertainty. Robustness of the method is illustrated with a simulated case for varying point spacing and measurement uncertainty. Secondly, a novel method is proposed to construct biarc from four points with minimum radius difference. The existing methods commonly interpolate/approximate biarc from given end tangents and end points with an additionally considered constraint. In the current application, since biarc has to be interpolated from four or five consecutive points, detail study of biarc construction is conducted. To locate the boundary limits of optimum joint point, biarcs are categorized in five types based on their shape characteristics. In support of each category, one case study has been conducted that validated right consideration of 111 boundary limits needed to construct biarc. Again, a general method is also proposed to interpolate biarc from five given points. The proposed method is also validated with three case studies of all probable input scenarios. Finally, a robust algorithm is proposed to evaluate radius via a rolling scheme of five consecutive data points in order to retrieve the local curvature information of the mid-point. This scheme of five points creates six subsets of three triplets, two quartets and one quintet of data points. A statistical approach is employed where all feasible radii from all the six datasets are considered in order to reliably estimate the desired radius. Biarc construction is used to determine the radius from quartet and quintet subsets. Compared with existing radius estimation methods, the proposed method has demonstrated to yield better accuracy with varying measurement uncertainty and data point resolution for different types of input. A complete case study is conducted with simulated data since real inspection data are not available. It is believed that, the proposed method should work consistently for real data also. 5.3 Limitations and future works Computational efficiency and accuracy are the two considered factors for any operational matter. In case of inspection, the accuracy of the process is vital. The proposed approach shows dependable accuracy but the computational efficiency has further scope of improvement. The proposed local curvature radius evaluation is a statistical approach where numerous cases are calculated and then the local radius is derived from the sample population. The computational time depends on population size and the population size is again a function of 112 applied division number i.e. ‘ ’. In Table 5.7, the computational time with different division are compared. Input: (0.05, 0); (0.04975, 0.00499); (0.049, 0.00993); (0.04771, 0.01476); (0.04554, 0.01925). Table 5.7 Comparison among different applied division number. Divisions for Triplet : Quartet : Quintet Error (%) Elapsed time (seconds) 3 : 3 : 3 8.2 27 5 : 5 : 5 7.8 232 7 : 7 : 7 6.4 906 11 : 7 : 5 6.6 770 MATLAB; AMD Athlon II P340 Dual-Core Processor 2.20 GHz Evidently, the number of divisions ‘ ’ directly influence the computational time. It also shows, the bigger the sample size, the better the accuracy. So a balance between the two is to be reached for effective commercial use. It has been observed, biarc construction from quartet, being a dual looped optimization process takes bulk (95%) of the computational time. It appears to be bottleneck for commercial implementation. Thus, for making it commercially viable, either the population size is to be reduced or efficient method needs to be adopted for calculating biarc from quartet. Lastly, this thesis deals with touch-probe CMM data only and assumes uniform measurement uncertainty throughout the sectional input profile. The inputs (i.e., LE/TE sectional measured data and measurement uncertainty value) are very limited for the evaluation of these 113 localized geometric restrictions. Information of cutter tool path or more specifically cutter location (CL) data can provide more insight for the potential locations of linear segments along LE/TE. Again, because of servomechanism, the airfoil blades are actually manufactured with slight deviations from designed tool path. 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Horng, "Circular arc detection based on Hough transform," Pattern Recognition Letters, vol. 16, pp. 615-625, 1995. [43] D. Ioannoua, W. Hudab and A. F. Lainec, "Circle recognition through a 2D Hough Transform and radius histogramming," Image and Vision Computing, vol. 17, pp. 15-26, 1999. [44] Wikipedia, "Median," [Online]. Available: http://en.wikipedia.org/wiki/Median. [Accessed 19 March 2012]. [45] R. L. Winkler, "Combining probability distributions from dependent information sources.," Management Science, vol. 27, pp. 479-488, 1981. 119 APPENDIX Appendix: Curvature validity check for triplet Curvature validity check method for triplet is explained here. As input points’ sequence is known, considering any three consecutive points, if the circumscribed circle is inwardly centered it is defined as valid case. On the contrary, if the circumscribed circle is outwardly centered it is defined as invalid case. For identifying the inward or outward orientation the sequence selection is important. Thus a fixed and consistent sequence of points is considered. Let, for three given consecutive points , and the circle is the circumscribed circle with center at . Now, the validity of the curvature is to be determined from these inputs. The point sequence throughout this thesis is assumed to increase from right to left. It can be interpreted that, for any valid curvature the orientation of point will follow a counter-clockwise orientation. To identify the orientation, the relative location of the points can be helpful. So to calculate the relative location of from line connecting , rigid body transformation can be done keeping the origin at and axis in the direction of . P1 P2 P3 C 120 As shown in above left figure, after rigid body transformation, if the Y coordinate value or ordinate value of (i.e., ) is negative, it forms a counter-clockwise orientation. So, the task is to calculate the ordinate of after rigid body transformation around point and keeping +X axis in the direction of . But, as shown in above right figure, if is negative, it doesn’t guarantee a valid case. There may be some cases where despite being negative, the curvature is not valid. These special cases can be identified by also checking the relative location of the center formed by the circumscribed circle. If and only if, both values of and are negative, it can be guaranteed to be a valid case. Mathematically, any curvature is valid if and only if, and ; where = Y coordinate value of after rigid body transformation. = Y coordinate value of after rigid body transformation. P1 P2 P3 C Y -X X -Y P1 P2 P3 C X Y -X -Y
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Evaluation of linear segment length and local curvature radius along airfoil leading and trailing edges Razive, Mohammad Nahid Islam 2012
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Title | Evaluation of linear segment length and local curvature radius along airfoil leading and trailing edges |
Creator |
Razive, Mohammad Nahid Islam |
Publisher | University of British Columbia |
Date Issued | 2012 |
Description | Airfoil is the basic profile geometry of impeller and turbine blades. The operational efficiency of these blades is governed by stringent tolerance specifications on the airfoils. The specified tolerances are commonly evaluated from discrete coordinate data collected in sections by a touch-probe coordinate measuring machine (CMM). These measurement data are subject to inspection inaccuracies associated with CMM measurement operation. Apart from well-known inspection parameters like profile tolerance, profile thickness and edge radius, the leading edge (LE) and trailing edge (TE) are specified with a unique set of geometric parameters like the maximum linear segment length restriction and the minimum curvature radius restriction. This thesis focuses on evaluating these two localized geometric restrictions along the leading edge and trailing edge of an airfoil. This thesis first presents a robust algorithm to identify the longest linear segment. The main feature of the proposed algorithm is the explicit consideration of measurement uncertainty. The algorithm starts by detecting relatively small linear segments and then merges these segments to determine the longest feasible linear segment under given measurement uncertainty. The effect of measurement uncertainty and data point resolution on the performance of the presented algorithm is demonstrated through case studies. Once the linear segments are identified and excluded, the remaining data points only belong to the non-linear segments. As minimum radius can occur at any location, curvature radius at each point along the non-linear segments is evaluated. Curvature radius at a specific point can only be estimated from its neighborhood. The chosen neighborhood size needs to be balanced between capturing local curvature attribute and effectively considering the effect of measurement uncertainty. An algorithm is thus proposed to evaluate radius via a rolling scheme of five consecutive data points in order to retrieve the local curvature information of the mid-point. A statistical approach is employed where all feasible radii are considered in order to reliably estimate the desired radius. Biarc construction is used as a tool to calculate radius. Compared with existing radius estimation methods, the proposed method has demonstrated to yield better accuracy with varying measurement uncertainty and data point resolution. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-04-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0072707 |
URI | http://hdl.handle.net/2429/42027 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2012-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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