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Applied iterative closest point algorithm to automated inspection of gear box tooth
Salim Boukebbab, Hichem Bouchenitfa, Hamlaoui Boughouas, Jean Marc Linares
Abstract
The development of a complete system and quality control of manufactured parts requires the coordination of a set complex processes allowing data acquisition, their dimensional evaluation and their comparison with a reference model. By definition, the parts inspection is the comparison between measurements results and the theoretical surfaces definition in order to check the conformity after manufacturing phase. The automation of this function is currently based on alignment methods of measured points resulting from an acquisition process and these nominal surfaces, in a way that they “fit best”. The distance between nominal surface and measured points(i.e. form defects) calculated after alignment stages are necessary for the correction of the manufacturing parameters(Henke, Summerhays, Baldwin, Cassou, &Brown, 1999).In the work, a method for automated control based on association of complex surfaces to a cloud points using the Iterative Closest Point(I.C.P)algorithm for alignment stage is proposed. An industrial application concerning a tooth gear manufactured in our country’s tractor engines is presented.
2007 Elsevier Ltd. All rights reserved.
Keywords: CMMs; Complex surfaces; ICP; Gear; Manufacturing process
1. Introduction
The design and manufacture of complex surfaces became a current practice in industry. These surfaces can be conceived by a direct method based on the use of Computer Aided Design(CAD)software , or an indirect method which consists in a treatment of a discrete representation of an object model to obtain its CAD model. This last can be obtained throughout an acquisition process, allowing then a much more rapid safeguard, modification, manufacture, automatic inspection, prototypes checking and a much easier analysis (Lai & Ueng, 2000).
The last year, the development process has covered all automated production phases, from design to the parts inspection passing by manufacture. Since the design and the manufacture of complex surfaces became a current practice in industry, then the problem related to the parts conformity are being felt more and more.
The automation and the flexibility of a three-dimensional measurement machine with or without contact have made a considerable reduction in the acquisition time and the measurements treatment. In the current state of the metrology software, the inspection of elementary surfaces (plane, cylinder, cone, etc.)became a very easy practice. On the other hand the inspection of complex surfaces remains a problem to overcome (Tucker & Kurfess, 2003).
The ascending complexity of parts geometry and the need for reducing production costs impose the use of more powerful tools for the inspection of complex parts surfaces, for a better service functionality description during its assembly with the conjugate mechanism parts (Tholath & Radhakrishnan, 1999). Our work is placed accordingly and consists to establish a procedure for modeling and inspecting complex parts surfaces, enabling the correction of relative deviations within production means.
The method used is based on the iterative-closest-point(ICP)algorithm, which is a well-known method for registering a 3D set of points to a 3D model that minimizes the sum of squared residual errors between the set and the model. This choice is motivated by the robustness of this method and it is important to underline here that; no attempt to implement it within Coordinate Measuring Machines (CMMs) software has been reported in the three-dimensional metrology literature.
A numerical application treating the case of a tooth of the toothed wheel which equips the gear box tractor manufactured at the engines and tractors factory in our country is presented, the comparison between the real surface obtained by acquisition and the ideal model has led to the calculation of the form defects on the two flanks of the tooth gear.
2. Problems and adopted algorithm
The principle of the software of Coordinate Measuring Machines consists generally in individually associating an elementary mathematical model (plane, cylinder, etc) to each digitized surface. The function to be minimized is based on the distance di between the digitized point Mi and the theoretical surface (Fig.2).
As already pointed out in the introduction, in current state of the metrology software, the inspection of elementary surfaces (plane, cylinder, cone, etc,) is not a problem, and most CMMs correct remaining alignment deviations numerically (alignment means to evaluate an optimum transformation T mapping the measured points to the corresponding nominal points in a way that they “fit best”) (Gogh et al, 2003). On the other hand the inspections of surfaces which have geometries of a higher complexity like gears, sculptured surfaces etc.represents a major challenge (Goch & Tschudi, 1992; Pommer, 2002). It is to this objective that our work is directed, and consists in the development of a procedure of a procedure for modeling and inspecting complex surfaces with an aim of correcting the errors cumulated during the manufacturing phase (Portman & Shuster (1997)). For this case, the ICP (Iterative Closest Point) algorithm method will be used.
The iterative-closest-point (ICP) algorithm of Besl and McKay (1992) is a well-known method for registering a 3D model that minimizes the sum of squared residual errors between the set and the model, i.e. it finds a registration that is locally best in a least-squares sense (Bergevin, Laurendeau, & Poussart, 1995; Ma & Ellis, 2003). Its main goal is to find the optimal rigid transformation which will corresponds as well as possible a cloud points P to a geometrical model M, using the singular value decomposition function (SVD) (Fig.3).
The parameters of the rigid transformation between the sets of points PI and PII must minimize the cost function:
2
Where: P’ is a point from P’P I is a point from P’’ associated with Pi’Tt the rigid transformation.
A rigid transformation Tt consists of the rotation matrix [R] and the translation vector {T} giving the iterative transformation Pi’’=[R]*Pi’+ {T} (Pi’ will be transformed into a point Pi’’).
This algorithm requires an initial estimate of the registration; because the computation speed and registration accuracy depend on how this initial estimate is chosen (Ma&Ellis, 2003). For this, we were mainly based on the algorithm proposed by Moron (1996) to which some changes have been made in order to make it simpler while keeping a maximum of its performances Fig. 4.
In this algorithm, we have to determine the six degrees of freedom including the three for rotation and the other three for translation by ICP. Which the three dimensional translation vector has simply three parameters as {T} = (tx, ty, tz) T, the rotation matrix is apparently composed of nine elements which should go along with six conditions for orthonormality.
A simple iterative optimization based on the least square principle can not guarantee this orthonormality (Kaneko, Kondo, & Miyamoto, 2003). Hence, ICP employs unit quaternion (q0; q1; q2; q3) for representing the rotation parameters in order to reduce this problem.
The unit quaternion is used to compute a rotation about the unit vector n by an angle θ:
, with q00 and ; q02+q12+q22+q32=1
Then the rotation matrix [R] is defined by:
The optimal motion ([R]; {T}) is computed by the unit quaternion method due to Horn (Eggert, Lorusso, & Fisher, 1997). The same method was used in the original version of ICP (Besl & Mckay, 1992). There are different analytical ways to calculate the 3D rigid motion that minimizes the sum of the squared distances between the corresponding points. In Eggert et al. (1997), four such techniques were compared and unit quaternion method was found to be robust with respect to noise, stable in presence of degenerate data and relatively (Chetverikov, Stepanov, & Krsek, 2005).
3. Presentation of the algorithm
Since the presentation of the I.C.P algorithm by Besl and Mckay, many variants have been introduced, which affect one or more stages of the original algorithm to try to increase its performances specially accuracy and speed, giving birth to several alternatives of I.C.P. algorithm (Kaneko et al., 2003). Some of these variants (such as Rusinkiewicz et al. (2001)) expand also the abbreviation to the iterative corresponding point claiming that this would better suit the algorithm (Sablatnig & Kampel, 2002). In order, to make a choice of an algorithm, several criteria should be checked: speed, accuracy, stability, robustness, and simplicity. The importance of the one or other of those criteria depends on the use and application of the final program.
The development of a complete system of inspection and quality control of manufactured parts requires the coordination of a set complex processes allowing data acquisition, their dimensional evaluation and their comparison with a reference model. For that it is essential to make profitable some conceptual knowledge relating not only to the object to be analyzed, but also to its environment. In our case, the objective of the present work consist in establishing an automation procedure for modeling and inspecting complex parts surfaces, enabling the correction of relative deviations within manufacturing parameters, then the criteria adopted are : speedy convergence, system robustness, and interface simplicity.
The new algorithm can be summarized by the following procedure.
1. Make a random selection of a subset of points.
2. Calculate the projection of the selected points.
3. Calculate the optimal rigid transformation with SVD method.
4. Apply the transformation to the selected points.
5. Evaluate the quality of alignment by LMS estimator.
6. If alignment quality is good, calculate transformation and apply it to the whole of available points.
7. Repeat the steps from 1 to 6 until convergence.
The conceptual structure of our program is presented in Fig 4.
We note here that the algorithm structure is very simple; it is made up of a principle program which contains a loop to carry out the iterations and another one to estimate the quality of the rigid transformation by the LMS estimator (Least Median Squares) (Rousseau & Leroy, 1987). In this program we also find three calls functions which are: the CPT function which calculates the projection of the points on the ideal model of surface in STL format (Fig.5)，the SVD function which calculates the optimal rigid transformation; and finally the RT function useful for calculating the initial rigid transformation; because as already pointed out, the algorithm requires an initial estimate solution of registration; and the computation speed and registration accuracy depend on how this initial estimate is chosen (Ma & Ellis, 2003).
The STL format is generally obtained by a triangulation of an exact model using CAD software which gives a data file in STL format (Fig.6). Where a Triangular facet is defined by the co-ordinates of the three vertexes and its normal directed towards the object free side.
It should be noted that, the bigger is the number of triangles in STL model the less is the approximation errors (Fig.7).
The number of triangles and their distributions are function of the surface curvature and modeling tolerated error.
翻譯譯文
運用點算法反復自動檢測齒輪箱齒輪
Salim Boukebbab Hichem Bouchenitfa Hamlaoui Boughouas Jean Marc Linares
摘要：
一個完整的系統(tǒng)的開發(fā)和制造的零部件的質量控制的一組復雜的過程，使數據采集，其尺寸的評估和比較一個參考模型，需要協(xié)調。根據定義，零件檢查之間的比較，以便檢查是否符合制造階段后的測量結果和理論表面定義。從收購的過程和這些標稱的表面，測量點的方式，他們“最合適”的比對方法的基礎上，目前此功能的自動化。之間的標稱表面和測量點（即窗體缺陷）取向階段后，計算出的距離是必要的校正的制造參數（亨克，Summerhays，鮑德溫，Cassou布朗，1999）。在工作中，一個方法的自動化控制基于使用迭代最近點（ICP）算法對準階段的一個點云數據復雜曲面的關聯。在我國的拖拉機發(fā)動機制造有關的齒齒輪工業(yè)應用。
關鍵詞：CMMs , 復曲面， ICP ，齒輪 ，制造業(yè)程序
1、 緒論
復雜型面的設計和制造成為一個行業(yè)現行做法?？梢栽O想，這些表面的，直接的方法，使用計算機輔助設計（CAD）軟件，或間接的方法，其中包含一個對象模型來獲得其CAD模型的離散表示在治療的基礎上。這最后可以得到整個收購過程中，允許然后更快速的保障，修改，制造，自動檢測，原型檢查和更容易的分析（黎翁，2000年）。
過去的一年，在發(fā)展過程中已覆蓋了所有的自動化生產階段，從設計到零件的檢驗，通過由制造。由于設計和制造復雜曲面成為一個在行業(yè)目前的做法，那么相關的部分整合的問題正在越來越感覺到。
帶或不帶接觸的三維測量機的自動化和靈活性在時間的采集和治療的測量上已經有了顯著的降低。在當前狀態(tài)下的測量軟件，檢查的基本的表面（平面，圓柱體，圓錐體等）成為一個非常方便的做法。另一方面復雜的表面的檢查仍然是一個問題，需要得以克服（塔克＆Kurfess，2003年）。
升序復雜的零件的幾何形狀和降低生產成本的需要施加的更強大的工具的使用復雜的零件的表面的檢查中，在組裝過程中的共軛機制份（Tholath＆拉達克里希南，1999）為更好的服務功能的詳細描述。我們的工作放在相應地，包括建立復雜的零件表面建模和檢查，使在生產手段的相對偏差校正的過程。
所使用的方法是根據迭代最近點（ICP）算法，這是一個眾所周知的方法，注冊一個3D的點集的3D模型集和模型之間的殘余誤差的平方的總和最小化。這種選擇是出于這種方法的魯棒性，重要的是在此強調，沒有試圖去實現它在坐標測量機（CMM）的軟件已經在三維的計量文獻報道。
裝備在我國的發(fā)動機和拖拉機廠生產的拖拉機齒輪箱齒輪的齒治療的情況下，通過收購獲得的實際面和理想的模型之間的比較的數值應用的計算上的兩個側面的齒齒輪的形式缺陷。
2.問題及運算法則
他的原則三坐標測量機的軟件通常包括在單獨一個基本的數學模型（平面，圓柱等）相關聯的每個數字化的表面。以最小化的功能的基礎上的數字化的點（Mi）和理論的表面（圖2）之間的距離di。
已經指出，在當前狀態(tài)下的測量軟件，在介紹基本的表面（平面，圓柱，圓錐等）的檢查是沒有問題的，最三坐標測量機正確剩余的對準偏差數值（校準手段來評估一個最佳變換T的測量點映射到相應的額定點的方式，他們“最合適”）（梵高等人，2003）。另一方面，檢查表面有齒輪等提出了更高的復雜性，復雜曲面的幾何形狀etc.represents一個重大的挑戰(zhàn)（九策楚迪，1992年;波默，2002年）。它是實現這一目標，我們的工作指示，在發(fā)展的一個程序的一個程序，用于建模和檢查，一個目的是校正的誤差累積（波特曼＆舒斯特（1997））在制造階段的復雜的表面組成。在這種情況下，ICP（迭代最近點）算法的方法將被使用。
迭代的最近點（ICP）算法Besl和麥基（1992）是一種公知的方法，用于登記的3D模型集和模型之間的殘余誤差的平方的總和最小化，例如，它找到一個注冊當地最好在最小二乘意義上（Bergevin，Laurendeau，Poussart，1995年，馬和埃利斯，2003）。它的主要目標是找到最佳的剛性，這將對應的轉換以及云計算點P的幾何模型M，利用奇異值分解函數（SVD）（圖3）。
2
點PI和PII套之間的剛體變換的參數必須最大限度地降低成本的功能：
其中：P'是I是一個點從P點從P'P“與Pi'Tt剛體變換。
一個的剛體變換TT的旋轉矩陣[R]和平移向量{T}提供的迭代轉變PI“= [R]* PI+{T}（曹丕”將被改造成一個點Pi“）。
該算法需要一個初始估計登記手續(xù);因為計算速度和配準精度取決于這個初步估計被選中（馬和埃利斯，2003）。對于這一點，我們主要是基于倫（1996）所提出的算法，其中一些已經作了修改，以便使其更簡單，同時保持其性能最大。圖4。
在該算法中，我們要確定的六個自由度，包括三個旋轉和其他三個翻譯ICP。哪個的三維平移向量具有簡單的三個參數，{T}=（TX，TY與tz）T中，顯然是由9個元素應隨著六個條件正交性去旋轉矩陣。
一個簡單的迭代優(yōu)化的最小二乘原理的基礎上，不能保證這一點的正交性（金子，近藤，與宮本，2003年）。因此，ICP采用四元數（q0，q1，q2，q3）為代表的旋轉參數，以減少這個問題。
單位四元數被用來計算單位矢量n的角度θ的旋轉約：
?，其中q0 0和q02+ q12+ q22+ q32 =1;
然后旋轉矩陣[R]被定義為：
最佳運動（[R]{T}）計算的單位四元數法由于喇叭（艾格特，Lorusso醫(yī)師與其與Fisher，1997）。用同樣的方法在原有版本的ICP（BESL麥凱，1992年）。有不同的分析方法來計算的對應點之間的距離的平方的總和最小化的3D剛體運動。在艾格特等。 （1997年），四等技術進行了比較，發(fā)現四元數法是強大的，對于噪聲，穩(wěn)定中存在的退化數據和相對“（切特韋里科夫，斯捷潘諾夫，Krsek的，2005年）。
3.算法的介紹
自ICP算法Besl和麥凱的介紹后，有很多種說法被介紹，從而影響一個或多個階段，對原有算法設法提高其性能特別的精度和速度，分娩的ICP幾種選擇算法（Kaneko等人，2003年）。這些變體中的一些（如Rusinkiewicz等人（2001））的擴大也縮寫迭代聲稱，這將更好地適應的的算法（Sablatnig＆Kampel，2002）的對應點。為了使選擇的算法，有幾個標準，應檢查：速度，精度，穩(wěn)定性，魯棒性和簡單。這些標準的一個或其他的重要性取決于最終的程序的使用和應用。
一組復雜的過程，使數據采集，其尺寸的評估和比較一個參考模型，一個完整的系統(tǒng)制造的零部件的檢測和質量控制的發(fā)展需要協(xié)調。為此，它是必不可少的，使有利可圖的一些概念方面的知識，不僅要分析的對象，但也給它的環(huán)境。在我們的例子中，目前的工作目標包括建立一個自動化過程進行建模和檢查復雜的零件表面，使制造參數內的相對偏差修正，然后采用的標準是：收斂速度快，系統(tǒng)的可靠性，界面簡潔明了。
新算法可以概括為以下步驟：
1． 一個隨機選擇的一個子集點。
2． 計算的選定點的投影
3． 計算出最佳的剛性變換與SVD方法。
4． 應用轉型到選定的點。
5． 評估LMS估計質量的對齊方式。
6． 如果對齊質量好，計算轉型，并把它應用到整個可用點。
7． 重復步驟1到6，直到收斂。
我們的計劃是在圖4的概念結構。
我們注意到，該算法的結構非常簡單，它是由一個原則方案，其中包含一個循環(huán)進行迭代和另一個估計質量的剛體變換估計的LMS（最小中位數平方）（盧梭＆樂華，1987）。在這個程序中，我們也發(fā)現三個電話功能分別是：計算上的點的投影表面的理想模型的STL格式（圖5），計算最佳的剛性變換的SVD功能，CPT功能，最后是RT功能，可用于計算初始剛體變換，因為正如已經指出的，該算法需要注冊一個初步的估算解決方案;和運算速度和配準精度取決于這個初步估計被選中（馬和埃利斯，2003）。
STL格式通常是通過三角測量的精確模型使用CAD軟件，該軟件提供了一個數據文件中的STL格式（圖6）。凡三角刻面所定義的坐標的三個頂點和其正常朝向的對象自由側。
應當指出的是，STL模型的越少，是近似誤差（圖7）的三角形的數量越大。
三角形的數量及其分布的表面曲率和建模允許誤差的功能。