DEVELOPMENT OF A MATHEMATICAL PROCEDURE FOR MODELLING AND INSPECTING COMPLEX SURFACES FOR MEASUREMENT PROCESS SALIM BOUKEBBAB, HICHEM BOUCHENITFA Laboratoire de Mécanique, Faculté des Sciences de l’Ingénieur Université Mentouri Constantine,Campus Châab Ersas, 25017 Constantine, Algérie Tel/fax : +213 (0) 31 81 88 53 / 63 E-mail :
[email protected]
JEAN MARC LINARES Institut des Sciences du Mouvement/GIBO CNRS UMR 6233, Université de la Méditerranée Institut Universitaire Technologique Avenue Gaston Berger 13625 Aix en Provence cedex Tel: + 33 (0) 4 42 93 90 96 E-mail :
[email protected] The design and the manufacture of free form surfaces are being a current practice in the industry. Thus the problem of the parts conformity of complex geometry is felt more and more. By this work, a mathematical procedure for modelling and inspecting complex surfaces in measurement process is presented including the correction of geometrical deviations within manufacturing process. The method is based on the Iterative Closest Point (I.C.P.) algorithm for alignment stage between nominal surface and measured points. The finite elements method for geometric compensation STL model generally obtained by a triangulation of nominal model using Computer Aided Design (CAD) software; is used in our proposal. An industrial application concerning a rapid prototyping technology process is presented. This last is used for manufacturing of real parts obtained from a STL file format.
1. Introduction By definition the inspection of the mechanical parts is the comparison between measurement results and the theoretical topology of surfaces in order to check the conformity after an industrialising step. So the development of a complete inspection system and a quality control of manufactured parts require the coordination of a set of complex processes allowing data acquisition, and their comparison with a reference model. The automation of this function is currently 1
2
based on alignment methods of measured points resulting from an acquisition process and these nominal surfaces. In this paper, a mathematical procedure for modelling and inspecting complex surfaces in measurement process is presented. This procedure enables the correction of relative deviations within manufacturing process, and it is based on two techniques. The first one uses the Iterative-Closest-Point (ICP) algorithm, which is a well-known method to best fit a 3D set of points to a 3D model. This technique minimizes the sum of squared residual errors between the set of points and a model. The second uses the finite elements method in order to compensation STL model (figure 1).
yreal
CAD Surface
Acquisition real surface xreal
zreal
STL Model ymod.
zmod.
ei Zoom x
x
Mapping (By ICP Algorithm)
y
Zoom y
STL model compensation after alignment
xmod.
Figure 1. Principal step of model correction.
The distances ei (i.e. form defects), between nominal surface and measured points, calculated after alignment stages are necessary for correcting the errors cumulated during the manufacturing phase [1]. An industrial application, concerning a rapid prototyping technology process, is presented for validation of this approach. As already pointed out, the prototyping technology is used for manufacturing a physical model starting from a STL format file obtained by Computer Aided Design (CAD) software. 2. Presentation of the mathematical procedure In current state of the metrology software, the inspection of elementary surfaces is not a problem, and most CMMs realize numerically compensation of the geometric alignment deviations. On the other hand, the inspection of surfaces
3
which have geometries of a higher complexity like gears, sculptured surfaces…etc represents a major challenge, because the problems related to the parts conformity are being felt more and more. For this, in our proposal, the ICP (Iterative Closest Point) algorithm method will be used. The ICP algorithm is a well-known method for registering a 3D set of points to a 3D model. Since the presentation of the ICP algorithm by Besl and McKay [2], many variants have been introduced. The new propositions affect one or more stages of the original algorithm to try to increase its performances, specially, accuracy and speed. This method minimizes the sum of squared residual errors between the set of points and the model. The ICP algorithm finds a registration that is locally the best fit using the least-squares method [3]. Its main goal is to find the optimal rigid transformation which will corresponds as well as possible a set of measured points P to a geometrical model M, using the singular value decomposition function (figure 2).
(P) P’i P’’i
(M) 1- Projection of points on CAD surface
3- Application of optimal rigid transformation and put in correspondence
2- Optimal rigid Transformation [Tt]= f([R],{T})
Figure 2. Principal steps of the I.C.P. algorithm.
The parameters of the rigid transformation between the sets of points Pi ' and Pi '' must minimize the cost function:
( )
1 N ∑ Pi '' − [T t ]× Pi ' N i =1
2
(1)
A rigid transformation [Tt] consists in 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 [3]. The compensation of
4
STL model is realized by the finite elements method. The elements of model are triangular surfaces; its equation has the form:
z = w +α2× x +α3× y
(2)
The formulation of the plane equation in terms of the coordinates of nodes Zi, (figure 3) enables us to write the variations ei at nodes level between a point Mi and the plane element:
ei = zMi − zi
(3)
By replacing the Eq. (2) in Eq. (3) one obtains:
ei = z Mi − [w + α 2 × x + α 3 × y]
(4)
By making the approximation according to the previous relation, the optimal parameters will be determined: (w, α2, α3)Opt. However, the approximation by finite elements consists in calculating optimal displacements to nodes Ni (figure 3). Z
Acquisition surface
Mi(xMi,yMi,zMi) N3(x3,y3,z3) Initial approximante by finite elements
ei
z
N2(x2,y2,z2) zi=w+α2×x+α3×y N1(x1,y1,z1)
y Z
o x
Final approximante by finite elements Figure 3. Approximation surface by finite elements.
The writing of the variables (w, α2, α3) in terms of nodes coordinates leads to the following equation: n=3
Z i = ∑ f i ( x, y ) × zi i =1
(5)
5
It should be noted that f1(x,y), f2(x,y) and f3(x,y) are substitutions equations. After having reformulated the equation of the triangular elementary surface in terms of nodes coordinates Zi, the Eq (4) can be written in the following form:
⎡ n =3 ⎤ ei = zMi − ⎢∑ fi ( x, y) × zi ⎥ ⎣ i =1 ⎦
(6)
The optimization of the least square criterion is used to determine the optimal variables (Z1, Z2, Z3), obtained by these equations: n
∂ ∑ ei
n
2
i =1
∂ z1
=
∂ ∑ ei i =1
∂ z2
n
2
=
∂ ∑ ei i =1
∂ z3
2
=0
(7)
The Eq. (7) gives a linear system with symmetrical matrix and its resolution permits to calculate the optimal coordinates Z1, Z2 and Z3 of nodes N1, N2 and N3. The surface of total approximation is obtained by the assembly of the triangular surface elements. A data processing model was developed under a programming language for validation. 3. Application for rapid prototyping technology In this application, the part manufacturing by the rapid prototyping technology is treated (figure 4). This process is used for manufacture of the real part starting from a STL format file obtained by Computer Aided Design (CAD) software [4]. The standard data interface between CAD software and the prototyping machine is the STL file format, which approximates the shape of a part using triangular facets.
Parts manufactured
CAO model
5[mm] Figure 4. Prototyping technology process.
Prototyping Process
6
The manufacturing of parts, by the prototyping technology, consists to the reproduction of STL model by deposit of plastic with a hot temperature [5]. In this technology, the temperature of the print head is near of 250 °C. So, under the effect of heat a deformation is observed on the parts with a low thickness (figure 5).
STL model
Deformation: V4
Cloud points
Figure 5. Alignment step with an aim of compensation STL model.
This deformation represents an inevitable deviation in this manufacturing process, and is detected after alignment step (using the ICP algorithm) between the STL model and the set of points of real surface obtained after measurement phase. This deformation varies from 0,035 to 0,5243 [mm] in absolute value. Using the knowledge of this real deviation, a new STL model obtained by the finites elements method including the inverse deviation is constructed. The new parts are made using this new STL model on the rapid prototyping machine. 4. Conclusion The mechanical productions of freeform surfaces are being a current practice in the industry. The conformity problem of the mechanical parts is felt more and more. The objective of the present work consists to propose one mathematical procedure for modelling and inspecting freeform surfaces. This proposal allows correcting deviations within manufactured parts. This procedure is built on two algorithms. The first algorithm is used for alignment between the measured
7
points and the CAD model. It permits to determine the thermal deformation of the manufactured part. The second one is based on the finite elements method for including in the STL model the deformation compensation. An industrial application has been treated for validating the proposed method. References 1. S. Boukebbab, H. Bouchenitfa, H. Boughouas and J.M. Linares, Applied Iterative Closest Point algorithm to automated inspection of gear box tooth, International Journal of Computers & Industrial Engineering 52 pp162-173 (2007). 2. P.J. Besl and N.D. McKay, A Method for Registration of 3-D shapes, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(2), pp239– 256, (1992). 3. B. Ma and R.E. Ellis, Robust registration for computer-integrated orthopedic surgery: Laboratory validation and clinical experience, International Journal of Medical Image Analysis 7, pp 237–250 (2003). 4. T. Fischer, M. Burry and J. Frazer, Triangulation of generative form for parametric design and rapid prototyping , International Journal of Automation in Construction 14, pp233– 240 (2005). 5. R.A. Buswell, R.C. Soar, A.G.F. Gibb and A. Thorpe, Freeform Construction: Mega-scale Rapid Manufacturing for construction, International Journal of Automation in Construction 16, pp 224–231 (2007).
