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In order to inspect deformable parts, recent works use virtual deformation on a digitized version of areal-part to bring the part model back to its nominal shape. This simulation mimics the real processcalled fixturing, which is normally used by themanufacturer to bring back the part into its nominal shapeonce installed. To perform such virtual deformation Finite Element Methods (FEMs) are used in orderto meet the precision requirements of the inspection process. This paper presents a method based on aspring–mass system, whose formulation is much simpler than the FEM, which allows the calculation ofdeformations of shell type parts with accuracy comparable to FEM. Furthermore, due to the simplicity inits formulation the algorithm can be implemented more easily than the FEM. The system is composed oftwo types of springs: one typemodels membrane behavior of the part’s mesh model and the second typemodels the flexion behavior between eachmesh elements.We showthat by applying the proposedmass-spring model, it is possible to reduce the calculation time by 80% over standard FEM calculation openingthe door to real-time inspection. 1. IntroductionToday’s 3D part inspection commercial systems, operate underthe assumption that the shape of the inspected parts remainsconstant when they are attached to the final assembly [1,2].But in reality, there are many parts whose shape changes untilthey are assembled. Since the nominal part is defined accordingto its position after assembly, in order to perform an adequateinspection, it is necessary to fixture the parts on a rigid supportthat simulate the assembly process [3]. The assembly processitself causes a deformation to the part due to the application offorces and restrictions at the fixation points.Without this fixturingprocess an acceptable part may be deemed out of tolerance as aresult of a normal rigid inspection process.In industry today, typically a worker takes a part from alot, install it on a jig, performs the measurements, and finally dismounts the part from the rigid support. These tasks are usuallycarried out manually making it impossible to automate. In orderto avoid human intervention during the inspection of deformableparts,we propose amethod that does not require the attachment ofthe part. Thismethod is based on the comparison of the deformedCAD model against the measured data model. The deformationof the CAD model is obtained by applying displacements from itsfixation points to the corresponding points in the data model. Inthat sense, it is essential that the applied simulated deformationis based on real-physics principles. In a previous work [4], aFinite Element Method (FEM) was used to model the virtualmodel deformation of the part under inspection in order toapproximate the actual physical deformation. Even though an FEMtrained interpolation method using Radial Basis Functions (RBFs)was developed to accelerate the calculation of the deformations,FEM does not have an easy mathematical formulation and itsimplementation is complex. For these reasons, and because ofits computational efficiency a spring–mass models is proposedinstead of FEM [5,6].The proposed method has three stages and is based on theapplication of a spring–mass system that simulate deformations on a polygonal mesh of the part’s CAD model. The first stage isintended to reduce the initial mesh size of the CAD model inorder to accelerate the deformation calculations.36262
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The second stageapplies an optimization process to calculate the deformation on thereduced polygonal mesh. In the last stage, an interpolation usingRBFs is used to deform the full mesh model.The rest of the paper is organized as follows. The next sectionpresents a brief discussion of some recent works related tospring–mass systems used for modeling deformation on flexibleobjects. Section 3 describes the proposed inspection method.Section 4 describes the procedure to calculate the deformations.Section 5 presents the results of the tests on several parts and itscorresponding analysis. Finally, Section 6 concludes and indicatessome possibilities for future work.2. Spring–mass systemsModels based on spring–mass systems are probably thesimplest and easiest models to implement among all deformablemodels one can find in the literature [7,8]. Instead of applyinga discretization to a continuous media, one can discretize thedeformation model as a set of punctual masses connected to eachother by reticule-shaped massless springs.Spring–mass systems have been widely used to simulate de-formation of various real-world objects. In some cases, the defor-mations obtained using spring–mass system are not sufficientlyprecise [9,10], therefore, other methods need to be used. Inother cases, those methods are the best option as they offerthe highest simplicity in its formulation and implementation andmuch more computationally efficient than its FEM counterpart.Spring–mass models have been implemented successfully to sim-ulate the dynamic behavior of deformable objects in numerous ap-plications [11–14].Most of the applications of the spring–mass systems onecan find in the literature include: simulation of soft tissues forsurgical training [15,16], facial animation [17], and simulation ofclothing [18,19,5]. The following is a brief review of the recentworks that used spring–mass systems to model deformation ofnon-rigid objects in different applications.Recent developments inMinimally Invasive Surgery (MIS) tech-niques in which surgical procedures are carried out indirectlythrough the use of endoscopic cameras and laparoscopes.MIS pro-cedures have been shown to reduce post-operative complicationsand have improved patient recovery time significantly. Becauseof the extreme difficulties of indirect manipulations, new train-ing techniques have been developed based on virtual reality. Thisvirtual reality trainer requires the development of real-time sim-ulation software in which the behavior of organs is done realis-tically. Arnab and Raja proposed a new spring–mass system tosimulate surface deformation of soft volumetric objects i.e. thebreast, taking into account the characteristics of volumetric de-formation [15] in a spring–mass framework. To achieve this, anew type of spring–mass model called volumetric spring was de-veloped.