This paper describes a mathematical formulation for the efficient localization of 3D surfaces including free-form surfaces and flat surfaces. An important application of this paper is to register flat surface calculated from unfolding process with a curved surface extracted from ship CAD prior to the multipoint press forming works. The mathematical formulation handles the registration and comparison of two free surfaces represented by sparse points based on the iterative closest point (ICP) algorithm and localization that can be applicable to ship-hull plate forming. The ICP algorithm gives an adequate set of initial translation and rotation for surface objects with little correspondence through the minimization of mean square distance metric. Comparison of surfaces is explained in order to determine a corresponding set which gives the optimized press stroke between unfold surface and referential object surface. It thereby allows the optimized press works in ship-hull forming. The combination of registration and comparison is applied to decide the shape equivalence of correspondent surfaces as well as to estimate the transform matrix between point sets where similarity is low. Experimental results show the capabilities of the registration on unfolding surface and curved surface.
Ship design is started by the hull form definition represented by hull surface model. In the context of hull form definition, the hull surface is expressed by (Non-Uniform Rational B-Spline) NURBS or wireframe in order to satisfy the required speed and displacement volume. After the hull form is designed, detailed geometry and production drawings are generated according to the block assembly, outfitting assembly, and production planning method. During the block division, the hull surface is divided into a number of small surfaces. Therefore, curved pieces of surface are trimmed from the hull form surface in accordance with both block division and seam/butt assignment. The stem and stern pieces have especially complex curvature in comparison with the other pieces of surface.
Generally, the curved pieces have been deformed by the combination of roll bending and flame bending or press forming so that the flat pieces manufactured by steel cutting machines can be deformed up to the curved surface which comprises the hull surface. However, it is difficult to control the amount of residual deformation in the flame bending because of the thermal deformation generated by high-temperature distribution [
An apparatus of multiple-point press machine used for thick plate forming.
The multipoint press is required to calculate the press stroke of each press considering the displacement difference between the two pieces of surface. Therefore, the objective surface must be compared with the flat plate not only to calculate the stroke of each press but also to determine the displacement of positions which can produce the design surface [
In order to form the curved pieces, a developed (unfolded) shape should be calculated from curved pieces. The formation of curved structures from flat sheet material involves some degree of elastic and plastic deformation of the flat material. Although the forming method uses a different mechanism, every forming method imposes in-plane strain and bending strain by comparing the difference between the unfolded flat plate and the trimmed object surfaces. Therefore, a geometric model of trimmed surface and flat plate should be prepared [
The motivation of this study is to find an effective algorithm for surface registration required for the manipulation of surfaces during the forming works. Therefore, this study focuses on the development of an effective algorithm for comparison, localization, and registration of unfold flat plate and the curved plates. This study aims at the suggestion of a successive method for registration using the ICP algorithm and the localization method to match the flat surface and the curved surface that are represented at different coordinate systems.
This paper is structured as follows. Several relevant studies are first reviewed. Next, the mathematical formulation of the ICP algorithm is started. Thereafter, the closest point search algorithm is described. Finally, experimental results for surfaces are presented to demonstrate the ability of the proposed method.
The ICP method is the de facto standard for registration of point sets or different surfaces. It registers two independent 3D surfaces or 3D point clouds into a common coordinate system. Relatively little work has been published in the area of registration of 3D free-form surface and flat surface which does not have a similar shape. The original ICP algorithm has been mainly used to register for two similar geometric models. Most of the existing literature have addressed the surface matching or surface registration of similar surfaces with known correspondence.
Horn [
If the ICP algorithm is iterated by minimizing the distance error between two point data sets, then it can be useful for problems that have an unknown correspondence relation of two point data sets. The ICP algorithm used in this study registers two different geometric models (i.e., flat surface and object surface) that have little correspondence. The ICP algorithm consists of two parts that calculate the transformation matrix and find the correspondence point in the transformed point data sets. If flat surface data sets and object data sets represent different shapes, the closest point is the correspondence point sets, that is, object surfaces that have a minimum distance from the model data sets, that is, flat surfaces.
Hull pieces are trimmed from the three-dimensional surface defined by 3D ship CAD. Thereafter, the pieces are unfolded to the flat plate, as shown in Figure
Data flow of the desired surface and unfolded surface in a hull piece design (modified from [
Since the two sets of points on the unfolded surface and the design surface are represented on different coordinates, the point sets are required to be registered in the same coordinate system. The process of registration using the ICP algorithm is to calculate the transformation matrix from the press position to the object surface point sets, as shown in Figures
Registration of the unfolded surface to the design surface.
Schematic process of the registration and transformation.
This section introduces a free-form surface representation for the purpose of fast and accurate registration and matching. Since the part/product design and reverse engineering process starts from a compact surface model, the design surface and unfolded surface should be constructed by a geometric model. When the digitized points are available, the model construction is processed by CAD modeling, such as polynomial equations and parametric surfaces, such as Bezier, B-Spline, or NURBS (Nonuniform Rational B-Spline) surfaces. Furthermore, a surface-based approach, such as Bezier, B-Spline, or NURBS, is likely to be less effective than a polynomial since the design surface has simple curvature. The design surface trimmed from a ship CAD, that is, TRIBON, and the unfolded surface are identified as a set of points. The set of points is transformed into a polynomial equation since the polynomial equation is sufficient to get the boundary points and inner points as well as to calculate the registered surface. The polynomial function
where
Since the values of points
This chapter addresses the mathematical formulation of 3D surface registration techniques and localization emphasizing the multipoint press forming. The registration can be roughly partitioned into three issues: calculation of transformation matrix in terms of translation and rotation vector, optimization to minimize the stroke of multiple presses, and determination of the corresponding points of the design surface on the unfolded surface. The first and second issues pertain to how to estimate the transformation which best aligns with the design surface on an unfolded surface or which maximizes a measure of the similarity in the coincident coordinates of two surfaces. These issues can be expressed by the ICP algorithm. The third issue determines the location of press points extracted from the 2D unfolded surface, which can be characterized by localization [
Rotation and translation of the points for the registration.
The first stage is the formulation of the rigid body translation and rotation between two surfaces, which is appropriate for registration of the design surface on the unfolded surface. In this subsection, a procedure to obtain the least square rotation and translation is reviewed. The method of specifying the rotations and orientation of coordinate systems through unit quaternion operators has been widely introduced to computer graphics in a variety of rotation sequence applications in many studies [
The problem in registration is computation of the rigid body motions consisting of rotation and translation in 3D space, that is, given a set of press points
Figure
Faugeras and Hebert [
Therefore, the minimization of the mean square function can represent the optimum registration as follows:
Given two independently acquired sets of 3D points, we want to find the transformation consisting of the rotation matrix
The cross-covariance matrix
The cyclic components of the antisymmetric matrix
where
The optimal rotation is, hence, determined by calculating the eigenvector that corresponds to the maximum eigenvalue of the matrix
Since the unit Quaternion represents the best rotation, the rotation matrix
Finally, the optimal translation vector
The resulting transformation matrix is defined based on the rotation matrix and the translation vector. The
Finally, the unfolded surface can be registered to the design surface by multiplying the transformation matrix.
Although registration and the closest point have been mentioned, one more procedure is necessary to find the pressing points on the design surface that correspond to the points on the unfolded surface to obtain the stroke of each press. The registration procedure only gives the transformation matrix to match the unfolded surface to the design surface. Also, the difference between the two point sets is necessary to determine the stroke of each press. Furthermore, the unfolded surface does not resemble the design surface because the design surface is a 3D curved shape and the unfolded surface is a 2D flat plate. Therefore, the registration should be compensated for by an additional algorithm to obtain a robust similarity and to obtain the stroke of each pressing point on the unfolded surface. The localization and comparison of the two free-form surfaces suggested by Huang et al. [
Finding the relationship between a reference surface and the compared surface is called localization, and such a relationship is represented by the transformation matrix
where
Thereafter, the matching point
Recalling (
We introduce
From the above equation, the coupled equation will lead to the following:
Since
where
In this chapter, we demonstrate the ability of the proposed formulation to achieve the registration. This chapter is divided into two sections: (Section
The curved shape was extracted from Ship CAD software, that is, TRIBON, and the discrete points set was modeled by the polynomial surface. Unfolded surface was calculated by the method suggested by Ryu and Shin [
In this section, we demonstrate the ability of the ICP algorithm and the proposed formulation to perform localization of surfaces with similar or the same shape. Examples of the same surfaces and similar surfaces are shown in Figures
Transform matrix and summary of computation for model 1 and model 2.
Result of computation | Mode 1 | Model 2 |
---|---|---|
Root mean square value of deviation | 0.0 mm | 15.24 mm |
after registration | ||
Number of iteration | 20 | 77 |
Translation vector | ||
Rotation matrix |
Model 1: The same surfaces located in different positions.
Surfaces before localization
Surfaces after localization
Model 2: Similar surfaces located in different positions.
Surfaces before localization
Surfaces after localization
Two examples of registration between flat and curved surfaces were tested as shown in Figures
Transform matrix and summary of computation for model 3 and model 4.
Result of computation | Model3 | Model 4 |
---|---|---|
Root mean square value of deviation | 70.06 mm | 4.9 mm |
after registration | ||
Number of iteration | 60 | 66 |
Translation vector | ||
Rotation matrix |
Model 3: Concave surfaces in different locations.
Surfaces before localization
Surfaces after localization
Model 4: Twist surfaces in different locations.
Surfaces before localization
Surfaces after localization
Press stroke at each press punch position.
Surface matching, registration, and comparison is a difficult problem in the hull plate forming. Registering 3D surfaces, that is, putting two 3D surfaces in a common coordinate system is a crucial step in 3D hull plate forming in shipbuilding. This paper has discussed a combination of ICP algorithm and localization algorithm applicable to hull press forming. An effective approach in order to obtain the press stroke in multipoint press forming and to register the flat unfold surface on the curved design surface was implemented based on the cubic polynomial surface model. The ICP algorithm registers two independent surfaces into a common coordinate system. Surfaces are represented by cubic polynomial equation in arbitrary space. The ICP algorithm is based on iteratively matching points on one surface to the closest points on the other. A least-squares technique is used to estimate 3D translation and rotation from the point correspondences, which reduces the average distance between the surfaces in the two sets.
The closest point algorithm and registration compensated by localization effectively manipulates the geometric registration of surfaces and calculates the press stroke in ship-hull plate forming. Both synthetic and real data have been used to test the algorithm, and the results show that it is sufficient and robust and yields an accurate registration.
This work was supported by Mid-Career Researcher Program through Korea NRF Grant funded by the MEST(2009-0080880) and INHA research Grant.