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Closed-Loop Coordinate Metrology for Hybrid Manufacturing System
Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Available online at www.sciencedirect.com Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Bergamo, Italy, June 11-13, 2018 Proceedings,16th IFAC Symposium on Bergamo, June 11-13, Information Control in Manufacturing Bergamo, Italy, Italy, JuneProblems 11-13, 2018 2018 Information Control in Manufacturing Bergamo, Italy, JuneProblems 11-13, 2018 Bergamo, Italy, June 11-13, 2018
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IFAC PapersOnLine 51-11 (2018) 752–757
Closed-Loop Closed-Loop Coordinate Coordinate Metrology Metrology for for Hybrid Hybrid Manufacturing Manufacturing System System Closed-Loop Coordinate Metrology for Hybrid Manufacturing System Closed-Loop Coordinate Cody Metrology for Hybrid Berry*, Ahmad Barari* Manufacturing System Cody Berry*, Ahmad Barari*
Cody Berry*, Ahmad Barari* Cody Berry*, Ahmad Barari* Cody Berry*, Barari*Oshawa, Ontario *University of Ontario Institute of Technology, Ahmad *University of Ontario Institute of Technology, Oshawa, *University of Ontario Institute of Technology, Oshawa, Ontario Ontario Canada, L1H 7K4of(e-mails: [email protected], [email protected]) *University Ontario Institute of Technology, Oshawa, Ontario Canada, L1H 7K4 (e-mails: [email protected], [email protected]) Canada, L1H 7K4 (e-mails: [email protected], [email protected]) *University Ontario Institute of Technology, Oshawa, Ontario Canada, L1H 7K4of(e-mails: [email protected], [email protected]) Canada, L1H 7K4 (e-mails: [email protected], [email protected]) Abstract: Closed-loop of coordinate metrology with a hybrid of additive and subtractive manufacturing Abstract: Closed-loop Closed-loop of of coordinate coordinate metrology metrology with with aa hybrid hybrid of of additive additive and and subtractive subtractive manufacturing manufacturing Abstract: processes to repair surfaces or to compensate for manufacturing errors and is discussed in manufacturing this paper. It Abstract: Closed-loop of coordinate metrology with a hybrid of additive subtractive processes to repair surfaces or to compensate for manufacturing errors is discussed in this paper. paper. It It processes tomethod repair for surfaces or to compensate for manufacturing errors and isusing discussed inmanufacturing. this presents a minimizing the cost of repair for planar surfaces hybrid Abstract: Closed-loop of coordinate metrology with a hybrid of additive subtractive manufacturing processes repair for surfaces or to compensate for manufacturing errors isusing discussed this paper. It presents aa tomethod method for minimizing the cost cost of of repair repair for planar planar surfaces surfaces using hybridinmanufacturing. manufacturing. presents minimizing the for hybrid Using a volumetric analytical scheme, optimum plane is found to minimize the total cost paper. of repair processes repair for surfaces or to compensate for fit manufacturing errors isusing discussed in this It presents a tomethod minimizing thethe cost of repair for planar surfaces hybrid Using aa volumetric volumetric analytical scheme, the optimum fit plane plane is found found to minimize minimize the total totalmanufacturing. cost of of repair repair Using analytical scheme, the optimum fit is to the cost or manufacturing error compensation. The methodology is implemented and several case studies are presents a method for minimizing the cost of repair for planar surfaces using hybrid manufacturing. Using a volumetricerror analytical scheme, the fit plane found to minimize the total of repair or manufacturing error compensation. Theoptimum methodology is isimplemented implemented and several several casecost studies are or manufacturing compensation. The methodology is and case studies are conducted. The observed results arethe satisfactory verify thetoeffectiveness the developed Using a volumetric analytical scheme, optimum fitand plane isimplemented found minimize the of total cost of repair or manufacturing error compensation. The methodology is and several case studies are conducted. The The observed observed results results are are satisfactory satisfactory and and verify verify the the effectiveness effectiveness of of the the developed developed conducted. methodology in minimizing the costare ofThe repair and manufacturing erroreffectiveness compensation a closed-loop or manufacturing error compensation. methodology isverify implemented and severaloffor case studies are conducted. The observed results satisfactory and the the developed methodology in minimizing the cost of repair and manufacturing error compensation for a closed-loop methodology in minimizing the and costinspection. of repair and manufacturing erroreffectiveness compensationofforthe a closed-loop system of hybrid manufacturing conducted. The observed results are satisfactory and verify the developed methodology in minimizing the and costinspection. of repair and manufacturing error compensation for a closed-loop system of hybrid hybrid manufacturing and inspection. system of manufacturing methodology in minimizing the and costinspection. of repair and manufacturing error compensation for a closed-loop system of hybrid manufacturing © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Additive Manufacturing, Subtractive Manufacturing, Hybrid Manufacturing, Coordinate system of hybrid manufacturing and inspection. Keywords: Additive Manufacturing, Subtractive Manufacturing, Manufacturing, Hybrid Hybrid Manufacturing, Manufacturing, Coordinate Coordinate Keywords: Additive Manufacturing, Subtractive Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Metrology, 3D Printing, Keywords: Additive Manufacturing, Subtractive Manufacturing, Hybrid Manufacturing, Coordinate Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Metrology, 3D Printing, Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Metrology, 3D Printing, Estimation Keywords: Additive Manufacturing, Subtractive Manufacturing, Hybrid Manufacturing, Coordinate Metrology, Estimation 3D Printing, Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Estimation Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Metrology, 3D Printing, Estimation Estimation
Allowing this process to happen in real time has also been 1. INTRODUCTION Allowing this to in time has also Allowing this process process to happen happen in real real has feasibility also been been examined(Gohari and Barari, 2016). The time cost and 1. INTRODUCTION INTRODUCTION 1. Allowing this process to happen in real time has also been examined(Gohari and Barari, 2016). The cost and feasibility examined(Gohari and Barari, 2016). The costconsidered and feasibility 1. INTRODUCTION of repair is also something that needs to be with Allowing this process to happen in real time has also been examined(Gohari and Barari, 2016). costconsidered and feasibility 1. INTRODUCTION repair is that needs to with Coordinate metrology is one of the fundamental tools in of of repair is also also something something that needsThe to be be considered with these processes. examined(Gohari and Barari, 2016). The cost and feasibility Coordinate metrology is one of the fundamental tools in Coordinate metrology ismanufacturing one of the fundamental tools in these of repair is also something that needs to be considered with processes. determining whether a process is producing these processes. Coordinate one of the fundamental tools in of repair is also something that needs to be considered with determining metrology whether aa ismanufacturing manufacturing process is is producing producing determining whether process parts that aremetrology meetinga quality The results of this Coordinate ismanufacturing one standards. of the fundamental tools in these processes. determining whether process is producing processes. The total cost of repair or manufacturing error correction parts that are meeting quality standards. The results of this this these parts that are meeting quality standards. The results of inspection process can be used to accept or reject the product, The of or manufacturing error correction determining whether a quality manufacturing process is producing The total total cost cost of repair repair manufacturing correction parts that are meeting standards. The results of this determines whether or notor part needs to error be corrected or inspection process can be used to accept or reject the product, inspection process can used to reject the product, The total cost of repair or aaamanufacturing error correction or alternatively, to be plan foraccept the or next downstream whether or not part needs to be corrected or parts that are meeting quality standards. The results of this determines determines whether or not part needs to be corrected or inspection process can be used to accept or reject the product, replaced. becomes a more attractive option when The total Correction cost of repair or manufacturing error correction or alternatively, alternatively, to to plan plan for for the the next next downstream downstream determines or whether or not a part needs to be option corrected or manufacturing processes needed toaccept improve the quality of the replaced. inspection process can beplan used to ornext reject the product, Correction becomes aa more attractive when replaced. Correction becomes more attractive option when or alternatively, to for the downstream its cost Correction iswhether minimized. order to toaccomplish this or not In a part needs be option corrected or manufacturing processes processes needed needed to to improve improve the the quality quality of of the the determines manufacturing replaced. becomes a more attractive when product, repair it, or correct its manufacturing errors. Closed or alternatively, to plan for the next downstream its cost is minimized. In order to accomplish this cost Correction is minimized. In a order to accomplish this manufacturing needed to improve theerrors. qualityClosed of the its minimization, several processes will need to be integrated replaced. becomes more attractive option when product, repair repair processes it, or or correct correct its manufacturing manufacturing errors. Closed product, it, its its cost is minimized. In order to accomplish this –loop systems coordinate inspection manufacturing needed tometrology improve the qualityClosed of and the minimization, several processes will to integrated several willbeneed need totobe be product, repair processes it,of its manufacturing errors. with one another. PMPprocesses methods will getintegrated a sample its cost is minimized. In order tousedaccomplish this –loop systems systems ofor correct coordinate metrology inspection inspection and minimization, –loop of coordinate metrology and minimization, several processes will need to be integrated manufacturing processes have been investigated recently by with one another. PMP methods will be used to get a sample product, repair it, or correct its manufacturing errors. Closed with oneaccurately another. PMP methods used getintegrated aA sample –loop systemsprocesses of coordinate metrology inspection set that represents the will part to be repaired. DZE several processes willbe need toto be manufacturing processes have been been investigated recentlyand by minimization, manufacturing have investigated recently by with one another. PMP methods will be used to get a sample many researchers. paper studies an inspection closedthat accurately represents the part to be repaired. A DZE –loop systemsprocesses ofThis coordinate metrology inspection and set that represents the will part to bemeasured repaired. DZE manufacturing have been investigated recently by set method produces a skin model of be theused surface with oneaccurately another. PMP methods to get a A sample many researchers. This paper studies an closedmany with researchers. This paper studies an inspection inspection closedset that accurately represents the part to bemeasured repaired. A DZE loop hybid manufacturing system, involving both manufacturing processes have been investigated recently by method produces a skin model of the surface method produces a skin model of the measured surface many with researchers. This paper studies an inspection closedrequired for the volumetric analytical scheme. SGE A will be set that accurately represents the part to bemeasured repaired. DZE loop hybid manufacturing system, involving both loop with hybid manufacturing system, involving both method produces a skin model of the surface additive and hybid subtractive techniques. many with researchers. This paper studies an inspection closedrequired for the volumetric analytical scheme. SGE will be required for the the optimum volumetric analytical scheme. SGE will be loop manufacturing system, involving both used to find plane minimizing the total cost of method produces a skin model of the measured surface additive and subtractive techniques. additive and hybid subtractive techniques. system, involving both used the optimum required for the volumetricplane analytical scheme. SGE will be to find minimizing the total cost of loop with manufacturing used to find the optimum plane minimizing the total cost of additive and subtractive techniques. correction based on the volumetric analysis. By combining all required for the volumetric analytical scheme. SGE will be used to find the on optimum plane minimizing thecombining total cost all of based the volumetric analysis. By additive andthree subtractive There are major techniques. tasks in the coordinate metrology. correction correction based on the volumetric analysis. By combining all of these processes, an integrated system willBy becombining created. used to find the on optimum plane minimizing the total costThe of There are tasks the metrology. There are three three major major tasks in in the coordinate coordinate metrology. correction based the volumetric analysis. all of these processes, an integrated system will be created. The Point Measurement Planning (PMP), Substitute Geometry of these processes, an integrated system willBy be created. The There are three major tasks in the coordinate metrology. end goal of this paper is to have a developed framework that correction based on the volumetric analysis. combining all Point Measurement Planning (PMP), Substitute Geometry Point Measurement Planning (PMP), Substitute Geometry of these processes, an is integrated will be created. The goal of paper to have aasystem developed framework that Evaluation (SGE),major and Deviation Zone Estimation (DZE). end There are three tasks in the coordinate metrology. end goal of this this paper tothe have developed framework thata Point Measurement Planning (PMP), Substitute Geometry can be used to minimize cost of repair/correction of these processes, an is integrated system will be created.for The Evaluation (SGE), and Deviation Zone Estimation (DZE). Evaluation (SGE), and Deviation Zone Estimation (DZE). end goal of this paper is to have a developed framework thataa can be used to minimize the cost of repair/correction for PMP handles the selection of the sample points from the Point Measurement Planning (PMP), Substitute Geometry can be used to minimize the cost of repair/correction for Evaluation (SGE), and Deviation Zone Estimation (DZE). planar surface. end goal of this paper is to have a developed framework thata PMP handles the selection of the sample points from the PMP handles the selection of the sample points from the can be used to minimize the cost of repair/correction for surface. measured piece, SGE fitting theZone idealEstimation geometry to the the planar Evaluation (SGE), and isDeviation (DZE). planar surface. PMP handles the selection of the sample points from can be used to minimize the cost of repair/correction for a measured piece, SGE is fitting the ideal geometry to measured piece, iscompares fitting the ideal geometry to the the planar surface. sampled data, and DZE thesample sampled points against PMP handles theSGE selection of the points from measured piece, SGE is fitting the ideal geometry to the planar surface. 2. BACKGROUND sampled data, and DZE compares the sampled points against sampled andSGE DZEThese theare sampled points against the idealdata, geometry. tasks usually performed 2. measured piece, iscompares fitting the ideal geometry to the 2. BACKGROUND BACKGROUND sampled data, and DZEThese compares theare sampled points against the ideal geometry. tasks usually performed the ideal geometry. These tasks are usually performed 2. BACKGROUND sequentially and aren’t used to inform one another, and each sampled data, and DZE compares the sampled points against the ideal geometry. These areone performed 2. BACKGROUND 2.1 Point Measurement Planning (PMP) sequentially and aren’t used inform another, and each sequentially and of aren’t used to totasks inform oneusually another, and2008). each 2.1 has some uncertainty in theare results (Barari, the ideal level geometry. These tasks usually performed 2.1 Point Point Measurement Measurement Planning Planning (PMP) (PMP) sequentially and aren’t used to inform one another, and each has some level of uncertainty in the results (Barari, 2008). has some of uncertainty in the results (Barari, The resultslevel of process whether or not a each part 2.1 Point Measurement Planning (PMP) sequentially andthis aren’t used determine to inform one another, and2008). has some level of uncertainty in the results (Barari, 2008). The results of process whether or aa part 2.1 Point Measurement Planning (PMP) Previous researchers have reported several very important The results of this this process determine determine whether or not not part should be rejected, accepted, orinmodified to comply with the Previous has some level of uncertainty the results (Barari, 2008). Previous researchers researchers have have reported reported several several very very important important The results of this process determine whether or not a part should be rejected, accepted, or modified to comply with the closed loops between DZE-PMP, SGE-PMP, and DZE-SGE. should be rejected, accepted, or modified to comply with the Previous researchers have reported several very important standards. One of the major goals in research on this topic The results of this process determine whether or not a part closed loops between DZE-PMP, SGE-PMP, and DZE-SGE. closed loops between DZE-PMP, SGE-PMP, and can DZE-SGE. should be rejected, accepted, modified to comply withtopic the Examples standards. One major or goals in research on of closed loops SGE and PMP be seen Previous researchers havebetween reported several very important standards. One of of the the goals research on this this topic closed loops between DZE-PMP, and can DZE-SGE. involved themajor amount ofin data towith verify should be reducing rejected, accepted, or modified toneeded comply the Examples Examples of closed closed loops betweenSGE-PMP, SGE and and PMP PMP can be seen seen of loops between SGE be standards. One of the major goals in research on this topic involved reducing the amount of data needed to verify in (Barari et al., 2007; Berry et al., 2016; Martins etseen al., closed loops between DZE-PMP, SGE-PMP, and DZE-SGE. involved reducing the amount of data needed to verify Examples of closed loops between SGE and PMP can be manufactured pieces. standards. One of the major goals in research on this topic in (Barari et al., 2007; Berry et al., 2016; Martins et al., (Barari etclosed al., et2007; Berry et SGE al., 2016; Martins etpoint al., involved reducing the amount of data needed to verify in manufactured pieces. 2014). In (Barari al., 2007), capturing a new sample Examples of loops between and PMP can be seen manufactured pieces. in (Barari et al., 2007; Berry et al., 2016; Martins et al., involved reducing the amount of data needed to verify 2014). In (Barari et al., 2007), capturing a new sample point 2014). In (Barari al., 2007), capturing a new sample manufactured pieces. or rejecting an et already captured sample point etpoint was in (Barari et al., 2007; Berry et al., 2016; Martins al., 2014). In (Barari al., 2007), capturing a new sample manufactured As computing pieces. power has increased, it has become possible to or or rejecting rejecting an et already already captured sample point point was an captured sample point was dynamically decided based on progress in the fitting process As computing power has increased, it has become possible to 2014). In (Barari et al., 2007), capturing a new sample point As computing power hasmetrology increased, tasks. it has become possible to or rejectingdecided an already sample point was integrate the coordinate The processes also dynamically decided based on oncaptured progress in in the fitting fitting process based progress the process As computing power hasmetrology increased, tasks. it has become possible to dynamically which eventually a on guided search for sampling. The integrate the coordinate The processes also or rejecting an became already captured sample point was integrate themodified coordinate metrology tasks. The processes also dynamically decided based progress in the fitting process need to be so that they can take in new data and As computing power has increased, it has become possible to which eventually eventually became became aa guided guided search search for for sampling. sampling. The The which integrate the coordinate metrology tasks. The processes also information required for sampling was generated and revised need to be modified so that they can take in new data and dynamically decided based on progress in the fitting process need to the be so metrology that they can take in processes new databeen and which eventually became a guidedwas search for sampling. The modify original results. Some work has already integrate themodified coordinate tasks. The also information required for sampling sampling was generated and revised revised required for generated and need to the be modified so that they can take in new databeen and information dynamically by estimating probability density function of modify original Some work has already which eventually became athe guided search for sampling. The modify the originaltheresults. results. Some work has already been information required for sampling was generated and revised done to achieve integration of these tasks and the need to be modified so that they can take in new data and dynamically by estimating the probability density function of dynamically by estimating thethe probability density and function of modify the originalthe results. Someof work hastasks already been geometric deviations using Parzen-window method. A done to achieve integration these and the information required for sampling was generated revised done to achieve the integration of these tasks and the dynamically by estimating the probability density function of inclusion of manufacturing data in the results (Barari, 2013). modify the original results. Some work has already been geometric deviations using the Parzen-window method. A deviations usingthethe Parzen-window method. A done to of achieve the integration of results these tasks and the geometric inclusion manufacturing data in the (Barari, 2013). dynamically by estimating probability density function of inclusion of manufacturing data in the results (Barari, 2013). geometric deviations using the Parzen-window method. A done to achieve the integration of these tasks and the inclusion of manufacturing data in the results (Barari, 2013). geometric deviations using the Parzen-window method. A 2405-8963 IFAC (International Automatic Control) inclusion© manufacturing data inFederation the resultsof (Barari, 2013). Copyright ©of2018, 2018 IFAC 759Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2018 responsibility IFAC 759Control. Peer review© of International Federation of Automatic Copyright 2018 IFAC 759 10.1016/j.ifacol.2018.08.409 Copyright © 2018 IFAC 759 Copyright © 2018 IFAC 759
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similar approach was adopted in (Martins et al., 2014) but instead of trying to recognize the real probability density function of the geometric deviations through the process, it was assumed that the probability density function has a Gaussian distribution. (Berry et al., 2016) and (Lalehpour et al., 2017) used a virtual sampling technique to analyze high density laser scan data and combine PMP and SGE. Instead of using a PDF, sites were chosen using pre-existing sampling strategies, then the average of all the samples in the neighbourhood were chosen as a representative point. These new points were then used to fit a plane, and sampling was redone until the method converged.
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efficiency and accuracy of their method. The number of points needed to accurately define the geometric deviations of a surface was minimized using their developed method without introducing extra uncertainty. A finite element approach was used to develop the skin model using the coordinate metrology data in a paper by Lalehpour and Barari (Lalehpour and Barari, 2017) which is used in this work (Fig 1).
2.2 Substitute Geometry Evaluation (SGE) Nassef and ElMaraghy (Nassef and ElMaraghy, 1999) conducted a comparison between total least squares and minimax fitting with a focus on point selection for different feature geometries. Their main concern was the effect of the sampling strategy on the error found by the fitting algorithm. They initially discussed the differences of Total Least Square (TLS) and minimax fitting, and go over some past criticisms. The major criticism of TLS fitting was its apparent overestimation of the geometric deviations and so, minimax fitting gained popularity because of its more aggressive approach. However, after further research they determined TLS fitting was reporting deviation values closer to the true values than minimax fitting. Nassef and ElMaraghy found that the geometric features of the part should determine the sampling method used.
Figure 1 Example of a Skin Model (Lalehpour and Barari, 2017) In this figure, u and v are two axes of a coordinate system that the original data set had been transformed to.
Shakarji and Srinivasan (Shakarji and Srinivasan, 2013) looked at tolerance standards involving TLS fitting of lines, planes and parallel planes. Their goal was to provide algorithms for each of the geometries that were simple and easy to implement, and did not need optimization algorithms to solve. They presented and proved these algorithms, and verified that the algorithms gave the same or better results than the traditional method of solving TLS fits. Shakarji and Srinivasan showed that the iterative method was no longer necessary and instead, by solving a few equations, you could find the TLS fit.
Barari et al. (Barari et al., 2009) modeled the geometric deviations of different manufactured surfaces using NURBs surfaces. They achieved this by partitioning a machines workspace into sections with quasistatic errors, so that a set of linear transformations that represent the error are obtained. When these linear transformations are applied to the ideal geometry, an estimation of the geometric deviations can be obtained. 3. METHODOLOGY
2.3 Deviation Zone Estimation (DZE)
3.1 Volumetric Analytical Scheme
Skin models were introduced in the field of tolerance analysis (“ISO 17450-1:2011 - Geometrical product specifications (GPS) -- General concepts -- Part 1: Model for geometrical specification and verification,” 2017). In order to conform to ISO standards, parts must be within certain tolerances. To verify if a part conforms, the differences between the substitute geometry and the measured data are analyzed. Therefore, a detailed model of the geometric deviations is developed. This model is a non-ideal representation of the geometric deviations and is called a skin model.
After finding the initial condition for the fitting optimization using TLS, a volumetric analytical scheme is used to define the objective function of the optimization process. The volume contained between a triangle making up the skin model and the current fit plane is used to calculate the cost of each section in the skin model. As the skin model can contain a large number of triangles, it was necessary to determine a quick way of calculating the volume. To do this, the full volume was first separated into two distinct volumes with the separation plane being a plane parallel to the fit plane, translated along the z axis to be at the level of the lowest vertex in the triangle. This results in Figure 2, below.
Jamiolahmadi and Barari (Barari and Jamiolahmadi, 2015) utilized a finite difference method approach to develop the deviation zone. They tested their method on several different data sets where known errors occurred and evaluated the 760
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Figure 3 Base of Pyramid for Volume Calculation This arrangement causes the base of the pyramid to be a right trapezoid, and thus the area of the base can be calculated via Equation 5. Figure 2 Example case for Volume Calculation 𝐴𝐴𝑏𝑏 =
The top volume becomes an irregular pyramid and the bottom volume becomes a triangular prism. Therefore the total volume, 𝑉𝑉𝑇𝑇 , can be calculated using the following equation.
1 𝑎𝑎(ℎ1 + ℎ2 )#(5) 2
1 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ 𝐴𝐴𝑏𝑏 = |𝐵𝐵 𝐵𝐵 |((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 ))#(6) 2 2 3
𝑉𝑉𝑇𝑇 = 𝑉𝑉𝑝𝑝𝑝𝑝 + 𝑉𝑉𝑝𝑝𝑝𝑝 #(1)
Where each z value is the height of its corresponding point. As the triangular prism and the pyramid share the face containing the height of both the projected triangle and the pyramid, Equation 3 can be rearranged to get the height of the pyramid.
Where 𝑉𝑉𝑝𝑝𝑝𝑝 is the volume of the prism and 𝑉𝑉𝑝𝑝𝑝𝑝 is the volume of the pyramid. The volume of the prism is calculated using the projected area of the triangle to the fit plane, multiplied by the height of the prism, in this case the distance from the lowest vertex, z1, to the fit plane, which in this case has a height of zero.
ℎ=
𝑉𝑉𝑝𝑝𝑝𝑝 = 𝐴𝐴 𝑇𝑇 ∙ 𝑧𝑧1 #(2)
2𝐴𝐴 𝑇𝑇 #(7) ⃑⃑⃑⃑⃑⃑⃑⃑⃑ |𝐵𝐵 2 𝐵𝐵3 |
Making Equation 4 become the following equation.
Where 𝐴𝐴 𝑇𝑇 is the area of the projected triangle and is equivalent to the following equation.
𝐴𝐴𝑏𝑏 ∙
2𝐴𝐴 𝑇𝑇 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ |𝐵𝐵 2 𝐵𝐵3 | #(8) 3
𝑉𝑉𝑝𝑝𝑝𝑝 = ⃑⃑⃑⃑⃑⃑⃑⃑⃑ |𝐵𝐵 2 𝐵𝐵3 | ∙ h 𝐴𝐴 𝑇𝑇 = #(3Error! No text of specified style in document. ) 2 Subbing in Equation 6, the formula becomes the following. ⃑⃑⃑⃑⃑⃑⃑⃑⃑ Where |𝐵𝐵 2 𝐵𝐵3 | is the length of the vector connecting B2 and 1 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ 2𝐴𝐴 𝑇𝑇 B3, and h is the height of the projected triangle. The volume |𝐵𝐵 𝐵𝐵 | ∙ ((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 )) ∙ 2 2 3 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ |𝐵𝐵 2 𝐵𝐵3 | of any pyramid is given by the area of its base, multiplied by 𝑉𝑉𝑝𝑝𝑝𝑝 = #(9) its height and divided by three. 3 𝑉𝑉𝑝𝑝𝑝𝑝 =
𝐴𝐴𝑏𝑏 ∙ ℎ #(4) 3
Which when simplified, becomes the equation below. 𝑉𝑉𝑝𝑝𝑝𝑝 =
Where 𝐴𝐴𝑏𝑏 is the base of the pyramid and ℎ is the height. The base of the pyramid in this case is made up of two projected lines, a plane parallel to the projected plane, and an edge of the original triangle. The two projected lines are parallel to one another and perpendicular to the projected plane. In the figure below, the base of the pyramid of the example in Figure 2 is shown.
((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 )) ∙ 𝐴𝐴 𝑇𝑇 #(10) 3
Taking this equation and Equation 2 and putting them both into Equation 1, the total volume of the example becomes the following. 𝑉𝑉𝑇𝑇 = 𝐴𝐴 𝑇𝑇 ∙ 𝑧𝑧1 + 𝑉𝑉𝑇𝑇 =
761
((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 )) ∙ 𝐴𝐴 𝑇𝑇 #(11) 3
3𝐴𝐴 𝑇𝑇 ∙ 𝑧𝑧1 ((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 )) ∙ 𝐴𝐴 𝑇𝑇 + #(12) 3 3
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𝑧𝑧1 + 𝑧𝑧2 + 𝑧𝑧3 𝑉𝑉𝑇𝑇 = 𝐴𝐴 𝑇𝑇 ( ) #(13) 3 This shows that the volume between any triangle and plane can be determined via the projected area of the triangle to the plane, and the height of the centroid of the triangle. In the case where a triangle is intersected with the plane, this process still finds the total volume contained between the triangle and the plane. However, this volume is the summation of the positive volume above the plane and the negative volume below the plane. This value does not account for the cost of repair. To do this, the individual volumes need to be determined. First, the intersection points of the line and the plane need to be determined. The side of the plane each point lies on is determined by checking the sign of the z coordinate as the fit plane is assumed to have a height of 0. Vectors are created from points with opposite signs, and the intersection points are found using parametric interpolation. The parameter t is found using the z coordinates of two points on either side of the plane, and the height of the plate itself. 𝑡𝑡 =
Figure 4 Skin model of the NURBS surface The surface topography has variations of up to 5 mm in some areas, which would be indicative of an extreme error from a manufacturing process. This provides large sections of additive and subtractive areas to modify the cost of repair. It is also worth mentioning that the surface quality after the additive manufacturing process is affected by a significant level of systematic roughness (Kaji and Barari, 2015), (Sikder et al., 2014) and (Umaras and Tsuzuki, 2017), while the quality of the machined surface is typically much higher. The results of this optimization can be seen in Figure 5.
𝑧𝑧1 − 𝑧𝑧0 #(14) 𝑧𝑧1 − 𝑧𝑧2
Where z0 is the height of the fit plane, and z1 and z2 are the heights of the two points that make up the vector. In this case, 𝑧𝑧0 will always be 0, as the dataset is rigid body transformed so that the fit plane is the XY plane. With this parameter, the intersection point can be determined via the following equation 𝑃𝑃𝐼𝐼 = (1 − 𝑡𝑡) ∗ 𝑃𝑃1 + 𝑡𝑡 ∗ 𝑃𝑃2 #(15) Where P1 and P2 are points of the triangle. Using these intersection points and the vertices of the original triangle, three new triangles are formed. With these triangles, the appropriate positive and negative volumes can be quickly determined and stored. The method as written assumes the fit plane has a z height of zero. This works due to the rigid body transformation that takes place on the data set to transform it to the fit plane during the skin modelling step. As this process is also repeated during each refitting of the data set, a zheight of zero can be assumed for each iteration
Figure 5 Results of the repair cost optimization for the NURBS Surface specimen
4. RESULTS AND DISCUSSION
The gouged wax specimen has less extreme errors when compared to the NURBS surface. There are two significant gouges in the surface, one up to 2 mm. However, the majority of the surface is still planar. The skin model for this specimen is shown in Figure 6.
Two different cases will be examined using the developed methodology. The first case is a NURBS surface with many sloping hills and valleys. This is an extreme case to check the efficacy of the program. The second case is a gouged surface, with errors consisting of various tools impacting the surface over a large period of use. In Figure 4, below, the skin model of the NURBS surface can be seen. The whole surface is made of triangulated patches, which are used to calculate the volume. 762
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Table 1 Final Results for Both Cases
Piece
TLS
Optimization
Optimization over TLS improvement
NURBS Surface
16242.3
13007.3
24.9%
Gouged Wax
1860.61
1713.21
8.6%
The NURBS surface shows considerable improvement over the TLS fit, with a cost reduction of 24.9%. This is likely due to the large amount of error inherent to the surface. These large volumes more drastically affect the fit plane on every iteration. Conversely, with the gouged wax specimen, there was only an 8.6% decrease in cost. This is likely due to there only being errors on a small percentage of the surface. Despite the depth of the gouging, the cost of removing such a large portion of untouched surface outweighed the cost of additive repair for the gouge.
Figure 6 Skin model of the Gouged Wax specimen The results of the optimization are shown in Figure 7, below.
5. CONCLUSION The goal of this paper was to develop a method that could be used to minimize the cost of repair or correction for planar surfaces that had been rendered unusable either by manufacturing defects or by marring that had occurred through use. The developed methodology involves traditional techniques, such as TLS fitting and heuristic optimization algorithms, and non-traditional methods, such as skin modelling and a volumetric analitical scheme to define the objectiove function for the minimization process. Various algorithms were developed and utilized to determine the volumes that would need to be added or subtracted in order to repair a piece. These values were then used to minimize cost. Each method was validated individually to ensure the results were as expected. With the developed method, cost savings in excess of 8% were found in both cases. This method worked best in situations where there were a large number of irregularities on a part. In pieces where the majority of the surface was unharmed and planar, the cost savings were minimal. These results show that this method could be used and refined to aid in the repair and manufacture of parts using hybrid manufacturing.
Figure 7 Results of the repair cost optimization for the Gouged Wax Specimen
Unlike with the NURBS surface specimen, there were not large volumes of errors on the gouged wax piece, there was only a few small gouges that would affect the results. A comparison of the cost values are shown in the table below. First, the repair cost with the TLS fit plane is shown, and then the results of the optimization are shown.
6. ACKNOWLEDGEMENT The research support provided by the Natural Science and Engineering Research Council of Canada (NSERC) is greatly appreciated.
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REFERENCES Barari, A., 2013. Inspection of the Machined Surfaces Using Manufacturing Data. J. Manuf. Syst. 32, 107–113. https://doi.org/http://dx.doi.org/10.1016/j.jmsy.2012.07 .011
Martins, T., Tsuzuki, M., Barari, A., 2014. Sampling Plan for Coordinate Metrology Using Uncertainty Analysis, in: Sampling Plan for Coordinate Metrology Using Uncertainty Analysis. Montreal, Canada.
Barari, A., 2008. Sources of uncertainty in coordinate metrology of automotive body, in: Proceedings of 2nd CIRP International Conference on Assembly Technologies and Systems (CATS 2008), Toronto, ON, Canada, Sept. pp. 21–23.
Nassef, A.O., ElMaraghy, H.A., 1999. Determination of best objective function for evaluating geometric deviations. Int. J. Adv. Manuf. Technol. 15, 90–95. Shakarji, C.M., Srinivasan, V., 2013. Theory and Algorithms for Weighted Total Least-Squares Fitting of Lines, Planes, and Parallel Planes to Support Tolerancing Standards. J. Comput. Inf. Sci. Eng. 13, 31008-131008–11. https://doi.org/10.1115/1.4024854
Barari, A., ElMaraghy, H.A., Knopf, G.K., 2007. Searchguided sampling to reduce uncertainty of minimum deviation zone estimation. J. Comput. Inf. Sci. Eng. 7, 360–371.
Sikder, S., Barari, A., Kishawy, H., 2014. Effect of adaptive slicing on surface integrity in additive manufacturing, ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, V01A, V01AT02A052. https://doi.org/10.1115/DETC2014-35559
Barari, A., ElMaraghy, H., Orban, P., 2009. NURBS representation of estimated surfaces resulting from machining errors. Int. J. Comput. Integr. 22, 395–410. Barari, A., Jamiolahmadi, S., 2015. Convergence of a finite difference approach for detailed deviation zone estimation in coordinate metrology. ACTA IMEKO 4, 20. https://doi.org/10.21014/acta_imeko.v4i4.271
Umaras, E., Tsuzuki, M. S. G., 2017. Additive Manufacturing - Considerations on Geometric Accuracy and Factors of Influence, IFACPapersOnLine, 50 (1), 14940-14945. https://doi.org/10.1016/j.ifacol.2017.08.2545
Berry, C., Lalehpour, A., Barari, A., 2016. Dynamic Point Selection Strategy In Coordinate Metrology of Flat Surfaces, in: XII-Th INTERNATIONAL SCIENTIFIC CONFERENCE, Coordinate Measuring Technique. Szczyrk, Poland. Gohari, H., Barari, A., 2016. A Quick Deviation Zone Fitting in Coordinate Metrology of NURBS Surfaces Using Principle Component Analysis. Measurements 92, 352– 364. https://doi.org/http://dx.doi.org/10.1016/j.measurement. 2016.05.050 ISO 17450-1:2011 - Geometrical product specifications (GPS) -- General concepts -- Part 1: Model for geometrical specification and verification, 2017. Kaji, F., Barari, A., 2015. Evaluation of the surface roughness of additive manufacturing parts based on the modelling of cusp geometry, IFAC-PapersOnLine, 48 (3), 658-663. https://doi.org/10.1016/j.ifacol.2015.06.157. Lalehpour, A., Barari, A., 2017. Developing Skin Model in Coordinate Metrology Using a Finite Element Method. Measurement 109, 149–159. https://doi.org/https://doi.org/10.1016/j.measurement.2 017.05.056 Lalehpour, A., Berry, C., Barari, A., 2017. Adaptive data reduction with neighbourhood search approach in coordinate measurement of planar surfaces. J. Manuf. Syst. 45, 28–47. 764
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Closed-Loop Closed-Loop Coordinate Coordinate Metrology Metrology for for Hybrid Hybrid Manufacturing Manufacturing System System Closed-Loop Coordinate Metrology for Hybrid Manufacturing System Closed-Loop Coordinate Cody Metrology for Hybrid Berry*, Ahmad Barari* Manufacturing System Cody Berry*, Ahmad Barari*
Cody Berry*, Ahmad Barari* Cody Berry*, Ahmad Barari* Cody Berry*, Barari*Oshawa, Ontario *University of Ontario Institute of Technology, Ahmad *University of Ontario Institute of Technology, Oshawa, *University of Ontario Institute of Technology, Oshawa, Ontario Ontario Canada, L1H 7K4of(e-mails: [email protected], [email protected]) *University Ontario Institute of Technology, Oshawa, Ontario Canada, L1H 7K4 (e-mails: [email protected], [email protected]) Canada, L1H 7K4 (e-mails: [email protected], [email protected]) *University Ontario Institute of Technology, Oshawa, Ontario Canada, L1H 7K4of(e-mails: [email protected], [email protected]) Canada, L1H 7K4 (e-mails: [email protected], [email protected]) Abstract: Closed-loop of coordinate metrology with a hybrid of additive and subtractive manufacturing Abstract: Closed-loop Closed-loop of of coordinate coordinate metrology metrology with with aa hybrid hybrid of of additive additive and and subtractive subtractive manufacturing manufacturing Abstract: processes to repair surfaces or to compensate for manufacturing errors and is discussed in manufacturing this paper. It Abstract: Closed-loop of coordinate metrology with a hybrid of additive subtractive processes to repair surfaces or to compensate for manufacturing errors is discussed in this paper. paper. It It processes tomethod repair for surfaces or to compensate for manufacturing errors and isusing discussed inmanufacturing. this presents a minimizing the cost of repair for planar surfaces hybrid Abstract: Closed-loop of coordinate metrology with a hybrid of additive subtractive manufacturing processes repair for surfaces or to compensate for manufacturing errors isusing discussed this paper. It presents aa tomethod method for minimizing the cost cost of of repair repair for planar planar surfaces surfaces using hybridinmanufacturing. manufacturing. presents minimizing the for hybrid Using a volumetric analytical scheme, optimum plane is found to minimize the total cost paper. of repair processes repair for surfaces or to compensate for fit manufacturing errors isusing discussed in this It presents a tomethod minimizing thethe cost of repair for planar surfaces hybrid Using aa volumetric volumetric analytical scheme, the optimum fit plane plane is found found to minimize minimize the total totalmanufacturing. cost of of repair repair Using analytical scheme, the optimum fit is to the cost or manufacturing error compensation. The methodology is implemented and several case studies are presents a method for minimizing the cost of repair for planar surfaces using hybrid manufacturing. Using a volumetricerror analytical scheme, the fit plane found to minimize the total of repair or manufacturing error compensation. Theoptimum methodology is isimplemented implemented and several several casecost studies are or manufacturing compensation. The methodology is and case studies are conducted. The observed results arethe satisfactory verify thetoeffectiveness the developed Using a volumetric analytical scheme, optimum fitand plane isimplemented found minimize the of total cost of repair or manufacturing error compensation. The methodology is and several case studies are conducted. The The observed observed results results are are satisfactory satisfactory and and verify verify the the effectiveness effectiveness of of the the developed developed conducted. methodology in minimizing the costare ofThe repair and manufacturing erroreffectiveness compensation a closed-loop or manufacturing error compensation. methodology isverify implemented and severaloffor case studies are conducted. The observed results satisfactory and the the developed methodology in minimizing the cost of repair and manufacturing error compensation for a closed-loop methodology in minimizing the and costinspection. of repair and manufacturing erroreffectiveness compensationofforthe a closed-loop system of hybrid manufacturing conducted. The observed results are satisfactory and verify the developed methodology in minimizing the and costinspection. of repair and manufacturing error compensation for a closed-loop system of hybrid hybrid manufacturing and inspection. system of manufacturing methodology in minimizing the and costinspection. of repair and manufacturing error compensation for a closed-loop system of hybrid manufacturing © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Additive Manufacturing, Subtractive Manufacturing, Hybrid Manufacturing, Coordinate system of hybrid manufacturing and inspection. Keywords: Additive Manufacturing, Subtractive Manufacturing, Manufacturing, Hybrid Hybrid Manufacturing, Manufacturing, Coordinate Coordinate Keywords: Additive Manufacturing, Subtractive Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Metrology, 3D Printing, Keywords: Additive Manufacturing, Subtractive Manufacturing, Hybrid Manufacturing, Coordinate Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Metrology, 3D Printing, Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Metrology, 3D Printing, Estimation Keywords: Additive Manufacturing, Subtractive Manufacturing, Hybrid Manufacturing, Coordinate Metrology, Estimation 3D Printing, Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Estimation Point Measurement Planning, Substitute Geometry Evaluation, Deviation Zone Metrology, 3D Printing, Estimation Estimation
Allowing this process to happen in real time has also been 1. INTRODUCTION Allowing this to in time has also Allowing this process process to happen happen in real real has feasibility also been been examined(Gohari and Barari, 2016). The time cost and 1. INTRODUCTION INTRODUCTION 1. Allowing this process to happen in real time has also been examined(Gohari and Barari, 2016). The cost and feasibility examined(Gohari and Barari, 2016). The costconsidered and feasibility 1. INTRODUCTION of repair is also something that needs to be with Allowing this process to happen in real time has also been examined(Gohari and Barari, 2016). costconsidered and feasibility 1. INTRODUCTION repair is that needs to with Coordinate metrology is one of the fundamental tools in of of repair is also also something something that needsThe to be be considered with these processes. examined(Gohari and Barari, 2016). 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The results of this determines determines whether or not part needs to be corrected or inspection process can be used to accept or reject the product, replaced. becomes a more attractive option when The total Correction cost of repair or manufacturing error correction or alternatively, alternatively, to to plan plan for for the the next next downstream downstream determines or whether or not a part needs to be option corrected or manufacturing processes needed toaccept improve the quality of the replaced. inspection process can beplan used to ornext reject the product, Correction becomes aa more attractive when replaced. 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In order to accomplish this –loop systems coordinate inspection manufacturing needed tometrology improve the qualityClosed of and the minimization, several processes will to integrated several willbeneed need totobe be product, repair processes it,of its manufacturing errors. with one another. PMPprocesses methods will getintegrated a sample its cost is minimized. In order tousedaccomplish this –loop systems systems ofor correct coordinate metrology inspection inspection and minimization, –loop of coordinate metrology and minimization, several processes will need to be integrated manufacturing processes have been investigated recently by with one another. PMP methods will be used to get a sample product, repair it, or correct its manufacturing errors. Closed with oneaccurately another. PMP methods used getintegrated aA sample –loop systemsprocesses of coordinate metrology inspection set that represents the will part to be repaired. DZE several processes willbe need toto be manufacturing processes have been been investigated recentlyand by minimization, manufacturing have investigated recently by with one another. PMP methods will be used to get a sample many researchers. paper studies an inspection closedthat accurately represents the part to be repaired. A DZE –loop systemsprocesses ofThis coordinate metrology inspection and set that represents the will part to bemeasured repaired. DZE manufacturing have been investigated recently by set method produces a skin model of be theused surface with oneaccurately another. PMP methods to get a A sample many researchers. This paper studies an closedmany with researchers. This paper studies an inspection inspection closedset that accurately represents the part to bemeasured repaired. A DZE loop hybid manufacturing system, involving both manufacturing processes have been investigated recently by method produces a skin model of the surface method produces a skin model of the measured surface many with researchers. This paper studies an inspection closedrequired for the volumetric analytical scheme. SGE A will be set that accurately represents the part to bemeasured repaired. DZE loop hybid manufacturing system, involving both loop with hybid manufacturing system, involving both method produces a skin model of the surface additive and hybid subtractive techniques. many with researchers. This paper studies an inspection closedrequired for the volumetric analytical scheme. SGE will be required for the the optimum volumetric analytical scheme. SGE will be loop manufacturing system, involving both used to find plane minimizing the total cost of method produces a skin model of the measured surface additive and subtractive techniques. additive and hybid subtractive techniques. system, involving both used the optimum required for the volumetricplane analytical scheme. SGE will be to find minimizing the total cost of loop with manufacturing used to find the optimum plane minimizing the total cost of additive and subtractive techniques. correction based on the volumetric analysis. By combining all required for the volumetric analytical scheme. SGE will be used to find the on optimum plane minimizing thecombining total cost all of based the volumetric analysis. By additive andthree subtractive There are major techniques. tasks in the coordinate metrology. correction correction based on the volumetric analysis. By combining all of these processes, an integrated system willBy becombining created. used to find the on optimum plane minimizing the total costThe of There are tasks the metrology. There are three three major major tasks in in the coordinate coordinate metrology. correction based the volumetric analysis. all of these processes, an integrated system will be created. The Point Measurement Planning (PMP), Substitute Geometry of these processes, an integrated system willBy be created. The There are three major tasks in the coordinate metrology. end goal of this paper is to have a developed framework that correction based on the volumetric analysis. combining all Point Measurement Planning (PMP), Substitute Geometry Point Measurement Planning (PMP), Substitute Geometry of these processes, an is integrated will be created. The goal of paper to have aasystem developed framework that Evaluation (SGE),major and Deviation Zone Estimation (DZE). end There are three tasks in the coordinate metrology. end goal of this this paper tothe have developed framework thata Point Measurement Planning (PMP), Substitute Geometry can be used to minimize cost of repair/correction of these processes, an is integrated system will be created.for The Evaluation (SGE), and Deviation Zone Estimation (DZE). Evaluation (SGE), and Deviation Zone Estimation (DZE). end goal of this paper is to have a developed framework thataa can be used to minimize the cost of repair/correction for PMP handles the selection of the sample points from the Point Measurement Planning (PMP), Substitute Geometry can be used to minimize the cost of repair/correction for Evaluation (SGE), and Deviation Zone Estimation (DZE). planar surface. end goal of this paper is to have a developed framework thata PMP handles the selection of the sample points from the PMP handles the selection of the sample points from the can be used to minimize the cost of repair/correction for surface. measured piece, SGE fitting theZone idealEstimation geometry to the the planar Evaluation (SGE), and isDeviation (DZE). planar surface. PMP handles the selection of the sample points from can be used to minimize the cost of repair/correction for a measured piece, SGE is fitting the ideal geometry to measured piece, iscompares fitting the ideal geometry to the the planar surface. sampled data, and DZE thesample sampled points against PMP handles theSGE selection of the points from measured piece, SGE is fitting the ideal geometry to the planar surface. 2. BACKGROUND sampled data, and DZE compares the sampled points against sampled andSGE DZEThese theare sampled points against the idealdata, geometry. tasks usually performed 2. measured piece, iscompares fitting the ideal geometry to the 2. BACKGROUND BACKGROUND sampled data, and DZEThese compares theare sampled points against the ideal geometry. tasks usually performed the ideal geometry. These tasks are usually performed 2. BACKGROUND sequentially and aren’t used to inform one another, and each sampled data, and DZE compares the sampled points against the ideal geometry. These areone performed 2. BACKGROUND 2.1 Point Measurement Planning (PMP) sequentially and aren’t used inform another, and each sequentially and of aren’t used to totasks inform oneusually another, and2008). each 2.1 has some uncertainty in theare results (Barari, the ideal level geometry. These tasks usually performed 2.1 Point Point Measurement Measurement Planning Planning (PMP) (PMP) sequentially and aren’t used to inform one another, and each has some level of uncertainty in the results (Barari, 2008). has some of uncertainty in the results (Barari, The resultslevel of process whether or not a each part 2.1 Point Measurement Planning (PMP) sequentially andthis aren’t used determine to inform one another, and2008). has some level of uncertainty in the results (Barari, 2008). The results of process whether or aa part 2.1 Point Measurement Planning (PMP) Previous researchers have reported several very important The results of this this process determine determine whether or not not part should be rejected, accepted, orinmodified to comply with the Previous has some level of uncertainty the results (Barari, 2008). Previous researchers researchers have have reported reported several several very very important important The results of this process determine whether or not a part should be rejected, accepted, or modified to comply with the closed loops between DZE-PMP, SGE-PMP, and DZE-SGE. should be rejected, accepted, or modified to comply with the Previous researchers have reported several very important standards. One of the major goals in research on this topic The results of this process determine whether or not a part closed loops between DZE-PMP, SGE-PMP, and DZE-SGE. closed loops between DZE-PMP, SGE-PMP, and can DZE-SGE. should be rejected, accepted, modified to comply withtopic the Examples standards. One major or goals in research on of closed loops SGE and PMP be seen Previous researchers havebetween reported several very important standards. One of of the the goals research on this this topic closed loops between DZE-PMP, and can DZE-SGE. involved themajor amount ofin data towith verify should be reducing rejected, accepted, or modified toneeded comply the Examples Examples of closed closed loops betweenSGE-PMP, SGE and and PMP PMP can be seen seen of loops between SGE be standards. One of the major goals in research on this topic involved reducing the amount of data needed to verify in (Barari et al., 2007; Berry et al., 2016; Martins etseen al., closed loops between DZE-PMP, SGE-PMP, and DZE-SGE. involved reducing the amount of data needed to verify Examples of closed loops between SGE and PMP can be manufactured pieces. standards. One of the major goals in research on this topic in (Barari et al., 2007; Berry et al., 2016; Martins et al., (Barari etclosed al., et2007; Berry et SGE al., 2016; Martins etpoint al., involved reducing the amount of data needed to verify in manufactured pieces. 2014). In (Barari al., 2007), capturing a new sample Examples of loops between and PMP can be seen manufactured pieces. in (Barari et al., 2007; Berry et al., 2016; Martins et al., involved reducing the amount of data needed to verify 2014). In (Barari et al., 2007), capturing a new sample point 2014). In (Barari al., 2007), capturing a new sample manufactured pieces. or rejecting an et already captured sample point etpoint was in (Barari et al., 2007; Berry et al., 2016; Martins al., 2014). In (Barari al., 2007), capturing a new sample manufactured As computing pieces. power has increased, it has become possible to or or rejecting rejecting an et already already captured sample point point was an captured sample point was dynamically decided based on progress in the fitting process As computing power has increased, it has become possible to 2014). In (Barari et al., 2007), capturing a new sample point As computing power hasmetrology increased, tasks. it has become possible to or rejectingdecided an already sample point was integrate the coordinate The processes also dynamically decided based on oncaptured progress in in the fitting fitting process based progress the process As computing power hasmetrology increased, tasks. it has become possible to dynamically which eventually a on guided search for sampling. The integrate the coordinate The processes also or rejecting an became already captured sample point was integrate themodified coordinate metrology tasks. The processes also dynamically decided based progress in the fitting process need to be so that they can take in new data and As computing power has increased, it has become possible to which eventually eventually became became aa guided guided search search for for sampling. sampling. The The which integrate the coordinate metrology tasks. The processes also information required for sampling was generated and revised need to be modified so that they can take in new data and dynamically decided based on progress in the fitting process need to the be so metrology that they can take in processes new databeen and which eventually became a guidedwas search for sampling. The modify original results. Some work has already integrate themodified coordinate tasks. The also information required for sampling sampling was generated and revised revised required for generated and need to the be modified so that they can take in new databeen and information dynamically by estimating probability density function of modify original Some work has already which eventually became athe guided search for sampling. The modify the originaltheresults. results. Some work has already been information required for sampling was generated and revised done to achieve integration of these tasks and the need to be modified so that they can take in new data and dynamically by estimating the probability density function of dynamically by estimating thethe probability density and function of modify the originalthe results. Someof work hastasks already been geometric deviations using Parzen-window method. A done to achieve integration these and the information required for sampling was generated revised done to achieve the integration of these tasks and the dynamically by estimating the probability density function of inclusion of manufacturing data in the results (Barari, 2013). modify the original results. Some work has already been geometric deviations using the Parzen-window method. A deviations usingthethe Parzen-window method. A done to of achieve the integration of results these tasks and the geometric inclusion manufacturing data in the (Barari, 2013). dynamically by estimating probability density function of inclusion of manufacturing data in the results (Barari, 2013). geometric deviations using the Parzen-window method. A done to achieve the integration of these tasks and the inclusion of manufacturing data in the results (Barari, 2013). geometric deviations using the Parzen-window method. A 2405-8963 IFAC (International Automatic Control) inclusion© manufacturing data inFederation the resultsof (Barari, 2013). Copyright ©of2018, 2018 IFAC 759Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2018 responsibility IFAC 759Control. Peer review© of International Federation of Automatic Copyright 2018 IFAC 759 10.1016/j.ifacol.2018.08.409 Copyright © 2018 IFAC 759 Copyright © 2018 IFAC 759
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similar approach was adopted in (Martins et al., 2014) but instead of trying to recognize the real probability density function of the geometric deviations through the process, it was assumed that the probability density function has a Gaussian distribution. (Berry et al., 2016) and (Lalehpour et al., 2017) used a virtual sampling technique to analyze high density laser scan data and combine PMP and SGE. Instead of using a PDF, sites were chosen using pre-existing sampling strategies, then the average of all the samples in the neighbourhood were chosen as a representative point. These new points were then used to fit a plane, and sampling was redone until the method converged.
753
efficiency and accuracy of their method. The number of points needed to accurately define the geometric deviations of a surface was minimized using their developed method without introducing extra uncertainty. A finite element approach was used to develop the skin model using the coordinate metrology data in a paper by Lalehpour and Barari (Lalehpour and Barari, 2017) which is used in this work (Fig 1).
2.2 Substitute Geometry Evaluation (SGE) Nassef and ElMaraghy (Nassef and ElMaraghy, 1999) conducted a comparison between total least squares and minimax fitting with a focus on point selection for different feature geometries. Their main concern was the effect of the sampling strategy on the error found by the fitting algorithm. They initially discussed the differences of Total Least Square (TLS) and minimax fitting, and go over some past criticisms. The major criticism of TLS fitting was its apparent overestimation of the geometric deviations and so, minimax fitting gained popularity because of its more aggressive approach. However, after further research they determined TLS fitting was reporting deviation values closer to the true values than minimax fitting. Nassef and ElMaraghy found that the geometric features of the part should determine the sampling method used.
Figure 1 Example of a Skin Model (Lalehpour and Barari, 2017) In this figure, u and v are two axes of a coordinate system that the original data set had been transformed to.
Shakarji and Srinivasan (Shakarji and Srinivasan, 2013) looked at tolerance standards involving TLS fitting of lines, planes and parallel planes. Their goal was to provide algorithms for each of the geometries that were simple and easy to implement, and did not need optimization algorithms to solve. They presented and proved these algorithms, and verified that the algorithms gave the same or better results than the traditional method of solving TLS fits. Shakarji and Srinivasan showed that the iterative method was no longer necessary and instead, by solving a few equations, you could find the TLS fit.
Barari et al. (Barari et al., 2009) modeled the geometric deviations of different manufactured surfaces using NURBs surfaces. They achieved this by partitioning a machines workspace into sections with quasistatic errors, so that a set of linear transformations that represent the error are obtained. When these linear transformations are applied to the ideal geometry, an estimation of the geometric deviations can be obtained. 3. METHODOLOGY
2.3 Deviation Zone Estimation (DZE)
3.1 Volumetric Analytical Scheme
Skin models were introduced in the field of tolerance analysis (“ISO 17450-1:2011 - Geometrical product specifications (GPS) -- General concepts -- Part 1: Model for geometrical specification and verification,” 2017). In order to conform to ISO standards, parts must be within certain tolerances. To verify if a part conforms, the differences between the substitute geometry and the measured data are analyzed. Therefore, a detailed model of the geometric deviations is developed. This model is a non-ideal representation of the geometric deviations and is called a skin model.
After finding the initial condition for the fitting optimization using TLS, a volumetric analytical scheme is used to define the objective function of the optimization process. The volume contained between a triangle making up the skin model and the current fit plane is used to calculate the cost of each section in the skin model. As the skin model can contain a large number of triangles, it was necessary to determine a quick way of calculating the volume. To do this, the full volume was first separated into two distinct volumes with the separation plane being a plane parallel to the fit plane, translated along the z axis to be at the level of the lowest vertex in the triangle. This results in Figure 2, below.
Jamiolahmadi and Barari (Barari and Jamiolahmadi, 2015) utilized a finite difference method approach to develop the deviation zone. They tested their method on several different data sets where known errors occurred and evaluated the 760
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Figure 3 Base of Pyramid for Volume Calculation This arrangement causes the base of the pyramid to be a right trapezoid, and thus the area of the base can be calculated via Equation 5. Figure 2 Example case for Volume Calculation 𝐴𝐴𝑏𝑏 =
The top volume becomes an irregular pyramid and the bottom volume becomes a triangular prism. Therefore the total volume, 𝑉𝑉𝑇𝑇 , can be calculated using the following equation.
1 𝑎𝑎(ℎ1 + ℎ2 )#(5) 2
1 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ 𝐴𝐴𝑏𝑏 = |𝐵𝐵 𝐵𝐵 |((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 ))#(6) 2 2 3
𝑉𝑉𝑇𝑇 = 𝑉𝑉𝑝𝑝𝑝𝑝 + 𝑉𝑉𝑝𝑝𝑝𝑝 #(1)
Where each z value is the height of its corresponding point. As the triangular prism and the pyramid share the face containing the height of both the projected triangle and the pyramid, Equation 3 can be rearranged to get the height of the pyramid.
Where 𝑉𝑉𝑝𝑝𝑝𝑝 is the volume of the prism and 𝑉𝑉𝑝𝑝𝑝𝑝 is the volume of the pyramid. The volume of the prism is calculated using the projected area of the triangle to the fit plane, multiplied by the height of the prism, in this case the distance from the lowest vertex, z1, to the fit plane, which in this case has a height of zero.
ℎ=
𝑉𝑉𝑝𝑝𝑝𝑝 = 𝐴𝐴 𝑇𝑇 ∙ 𝑧𝑧1 #(2)
2𝐴𝐴 𝑇𝑇 #(7) ⃑⃑⃑⃑⃑⃑⃑⃑⃑ |𝐵𝐵 2 𝐵𝐵3 |
Making Equation 4 become the following equation.
Where 𝐴𝐴 𝑇𝑇 is the area of the projected triangle and is equivalent to the following equation.
𝐴𝐴𝑏𝑏 ∙
2𝐴𝐴 𝑇𝑇 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ |𝐵𝐵 2 𝐵𝐵3 | #(8) 3
𝑉𝑉𝑝𝑝𝑝𝑝 = ⃑⃑⃑⃑⃑⃑⃑⃑⃑ |𝐵𝐵 2 𝐵𝐵3 | ∙ h 𝐴𝐴 𝑇𝑇 = #(3Error! No text of specified style in document. ) 2 Subbing in Equation 6, the formula becomes the following. ⃑⃑⃑⃑⃑⃑⃑⃑⃑ Where |𝐵𝐵 2 𝐵𝐵3 | is the length of the vector connecting B2 and 1 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ 2𝐴𝐴 𝑇𝑇 B3, and h is the height of the projected triangle. The volume |𝐵𝐵 𝐵𝐵 | ∙ ((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 )) ∙ 2 2 3 ⃑⃑⃑⃑⃑⃑⃑⃑⃑ |𝐵𝐵 2 𝐵𝐵3 | of any pyramid is given by the area of its base, multiplied by 𝑉𝑉𝑝𝑝𝑝𝑝 = #(9) its height and divided by three. 3 𝑉𝑉𝑝𝑝𝑝𝑝 =
𝐴𝐴𝑏𝑏 ∙ ℎ #(4) 3
Which when simplified, becomes the equation below. 𝑉𝑉𝑝𝑝𝑝𝑝 =
Where 𝐴𝐴𝑏𝑏 is the base of the pyramid and ℎ is the height. The base of the pyramid in this case is made up of two projected lines, a plane parallel to the projected plane, and an edge of the original triangle. The two projected lines are parallel to one another and perpendicular to the projected plane. In the figure below, the base of the pyramid of the example in Figure 2 is shown.
((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 )) ∙ 𝐴𝐴 𝑇𝑇 #(10) 3
Taking this equation and Equation 2 and putting them both into Equation 1, the total volume of the example becomes the following. 𝑉𝑉𝑇𝑇 = 𝐴𝐴 𝑇𝑇 ∙ 𝑧𝑧1 + 𝑉𝑉𝑇𝑇 =
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((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 )) ∙ 𝐴𝐴 𝑇𝑇 #(11) 3
3𝐴𝐴 𝑇𝑇 ∙ 𝑧𝑧1 ((𝑧𝑧2 − 𝑧𝑧1 ) + (𝑧𝑧3 − 𝑧𝑧1 )) ∙ 𝐴𝐴 𝑇𝑇 + #(12) 3 3
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𝑧𝑧1 + 𝑧𝑧2 + 𝑧𝑧3 𝑉𝑉𝑇𝑇 = 𝐴𝐴 𝑇𝑇 ( ) #(13) 3 This shows that the volume between any triangle and plane can be determined via the projected area of the triangle to the plane, and the height of the centroid of the triangle. In the case where a triangle is intersected with the plane, this process still finds the total volume contained between the triangle and the plane. However, this volume is the summation of the positive volume above the plane and the negative volume below the plane. This value does not account for the cost of repair. To do this, the individual volumes need to be determined. First, the intersection points of the line and the plane need to be determined. The side of the plane each point lies on is determined by checking the sign of the z coordinate as the fit plane is assumed to have a height of 0. Vectors are created from points with opposite signs, and the intersection points are found using parametric interpolation. The parameter t is found using the z coordinates of two points on either side of the plane, and the height of the plate itself. 𝑡𝑡 =
Figure 4 Skin model of the NURBS surface The surface topography has variations of up to 5 mm in some areas, which would be indicative of an extreme error from a manufacturing process. This provides large sections of additive and subtractive areas to modify the cost of repair. It is also worth mentioning that the surface quality after the additive manufacturing process is affected by a significant level of systematic roughness (Kaji and Barari, 2015), (Sikder et al., 2014) and (Umaras and Tsuzuki, 2017), while the quality of the machined surface is typically much higher. The results of this optimization can be seen in Figure 5.
𝑧𝑧1 − 𝑧𝑧0 #(14) 𝑧𝑧1 − 𝑧𝑧2
Where z0 is the height of the fit plane, and z1 and z2 are the heights of the two points that make up the vector. In this case, 𝑧𝑧0 will always be 0, as the dataset is rigid body transformed so that the fit plane is the XY plane. With this parameter, the intersection point can be determined via the following equation 𝑃𝑃𝐼𝐼 = (1 − 𝑡𝑡) ∗ 𝑃𝑃1 + 𝑡𝑡 ∗ 𝑃𝑃2 #(15) Where P1 and P2 are points of the triangle. Using these intersection points and the vertices of the original triangle, three new triangles are formed. With these triangles, the appropriate positive and negative volumes can be quickly determined and stored. The method as written assumes the fit plane has a z height of zero. This works due to the rigid body transformation that takes place on the data set to transform it to the fit plane during the skin modelling step. As this process is also repeated during each refitting of the data set, a zheight of zero can be assumed for each iteration
Figure 5 Results of the repair cost optimization for the NURBS Surface specimen
4. RESULTS AND DISCUSSION
The gouged wax specimen has less extreme errors when compared to the NURBS surface. There are two significant gouges in the surface, one up to 2 mm. However, the majority of the surface is still planar. The skin model for this specimen is shown in Figure 6.
Two different cases will be examined using the developed methodology. The first case is a NURBS surface with many sloping hills and valleys. This is an extreme case to check the efficacy of the program. The second case is a gouged surface, with errors consisting of various tools impacting the surface over a large period of use. In Figure 4, below, the skin model of the NURBS surface can be seen. The whole surface is made of triangulated patches, which are used to calculate the volume. 762
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Table 1 Final Results for Both Cases
Piece
TLS
Optimization
Optimization over TLS improvement
NURBS Surface
16242.3
13007.3
24.9%
Gouged Wax
1860.61
1713.21
8.6%
The NURBS surface shows considerable improvement over the TLS fit, with a cost reduction of 24.9%. This is likely due to the large amount of error inherent to the surface. These large volumes more drastically affect the fit plane on every iteration. Conversely, with the gouged wax specimen, there was only an 8.6% decrease in cost. This is likely due to there only being errors on a small percentage of the surface. Despite the depth of the gouging, the cost of removing such a large portion of untouched surface outweighed the cost of additive repair for the gouge.
Figure 6 Skin model of the Gouged Wax specimen The results of the optimization are shown in Figure 7, below.
5. CONCLUSION The goal of this paper was to develop a method that could be used to minimize the cost of repair or correction for planar surfaces that had been rendered unusable either by manufacturing defects or by marring that had occurred through use. The developed methodology involves traditional techniques, such as TLS fitting and heuristic optimization algorithms, and non-traditional methods, such as skin modelling and a volumetric analitical scheme to define the objectiove function for the minimization process. Various algorithms were developed and utilized to determine the volumes that would need to be added or subtracted in order to repair a piece. These values were then used to minimize cost. Each method was validated individually to ensure the results were as expected. With the developed method, cost savings in excess of 8% were found in both cases. This method worked best in situations where there were a large number of irregularities on a part. In pieces where the majority of the surface was unharmed and planar, the cost savings were minimal. These results show that this method could be used and refined to aid in the repair and manufacture of parts using hybrid manufacturing.
Figure 7 Results of the repair cost optimization for the Gouged Wax Specimen
Unlike with the NURBS surface specimen, there were not large volumes of errors on the gouged wax piece, there was only a few small gouges that would affect the results. A comparison of the cost values are shown in the table below. First, the repair cost with the TLS fit plane is shown, and then the results of the optimization are shown.
6. ACKNOWLEDGEMENT The research support provided by the Natural Science and Engineering Research Council of Canada (NSERC) is greatly appreciated.
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