Building Very Large Complex Shapes using a Flexible Blade Cutter

In order to fabricate very large objects of arbitrarily complexity from various high density polymers, a hybrid approach has been developed in which layers are individually machined and then stacked together. The layers are obtained by moving a heated cutter blade along the periphery of the layer. The adjustable tool is held by a controllable gripper and manipulated by a sculpturing robot. During motion the shape of the blade is continuously adapted to the target geometry for the layer. Thus extremely thick layers are obtained, compared to conventional techniques. The synchronous control of the shape of the cutter blade is a multilevel optimization problem, involving 1) finding the shape of the blade that suffices to the minimum energy principle, 2) minimizing the deviation between shape curves on the layer and the blades profile curve, 3) finding the best fitting region of the blades profile curve for various curvature characteristics, 4) finding the best tool positions and the optimum tool path. Based on a proper mathematical formulation, we have developed numerical methods that enable us to investigate feasibility of thicklayered fabrication of various products. Graphical simulation of the machining process demonstrates the practical applicability of the TLOM technology to automatic shape fabrication. 1. FUNDAMENTAL ISSUES OF THICK LAYERED OBJECT MANUFACTURING AND THE RELATED RESEARCH Physical concept modeling proved to be an effective means to improve the outcome of conceptual design and to reduce the time of product development. In the field of industrial design, the requested physical models are large sized, structurally and morphologically complex, and they are expected to support activities other than shape presentation only. For instance, when produced as functioning prototypes, they can also be used for preliminary checking of the requested functions. The physical concept models are generally made from plastic foam, paper, plywood, etc. by an incremental technology that is called layered object manufacturing (LOM). When the size of the physical models are above 0.6 m x 0.6 m x 0.6 m most of the conventional LOM technologies (e.g., laser stereolitography, selective laser sintering, fused filament deposition, etc.) fail due to their size, performance and quality limitations. Therefore, the objective of our research and development has been to come up with an effective technology for fabrications of large sized, freeform physical models of various soft materials based on higher order shape approximation. There are four typical phases of fabrication of very large sized, plastic foam objects. These are (a) shape decomposition, (b) layer thickness calculation, (c) layer manufacturing, and (d) object recomposition. Shape decomposition involves (a) separation of the object into individual components, (b) segmentation of the components according to their morphological properties, and (c) definition of layers based on shape approximation. The approximation assumes a set of layers of predetermined thickness and applies them in an adaptive way to minimize the deviation between the nominal shape and the approximating shape. Layer thickness calculation is an optimization process involving activities such as (a) determination of the optimal slicing position, (b) detaching the unfavorable slicing domains, (c) layer thickness allocation with a view to preciseness and process characteristics, and (d) placing inserts and auxiliary parts. Layer manufacturing comprises (a) blank cutting, and (b) front surface cutting. The recomposition subprocess consists of (a) finding optimal assembly position, (b) layer positioning and fixing, and (c) finishing and decoration of the physical model/prototype. Various types of approximation methods have already been worked out. They can be classified based on the curvedness of the approximating curve/surface (Figure 1). The stepped approximation which is often referred to as zero-order approximation in literature is the most adequate for thinlayered deposition technologies. When the concept of uniform slicing is applied, it results in large errors of approximation for double-curved objects. In order to reduce the deviations, the concept of adaptive slicing (Suh, Y. S. and Wozny, M. J., 1994), (Kulkarni, P. and a. zero-order b. first-order c. higher-order Figure 1 Comparison of adaptive slicing techniques of various order Dutta, D., 1995) has been introduced. Sabourin, E., Houser, S. A. and Bohn, J. H., (1996) presented a stepwise uniform refinement of adaptive slicing. Kulkarni, P. and Dutta, D., (1996), addressed the issue of containment to improve slicing. To lessen the stair-case effect that is typical for stepped approximation, the concept of ruled/sloped approximation has been introduced. This technique is also known as first order approximation in literature. Object fabrication with layers of ruled front surface was extensively studied in (de Jager, P. J., 1996), (Hope, R. L., Riek, A. T. and Roth, R. N., 1996). A comparison between zero order and first order approximation techniques is presented by (de Jager P. J., Broek, J. J., Vergeest, J. S. M., 1997). Slicing calculations are typically based on .STL files that however introduce errors in approximation of the nominal shape. To avoid representation errors techniques for direct slicing of CAD models have been worked out (Vuyyuru, P., Kirschman, C. F., Fadel, G. Bagchi, A. and Jara-Almonte, C. C., 1994), (Jamieson, R. and Hacker, H., 1995). Guduri, S., Crawford, R. H. and Beaman, J. J., (1992), also addressed the issue of generating exact contour files. First order shape approximation is generally combined with adaptive slicing. Segmentation is typically applied to the geometric model when the shape has morphologically dissimilar domains that need significantly different layer thickness. Special slicing techniques, e.g., sloped, and pie-like, can also be applied to achieve the same goal. The authors have developed a new technology which is the most appropriate for free-form, thick-layered fabrication of objects from high density plastic foams. The tool used for free form manufacturing of the front surfaces of the layers has been called flexible blade tool. The assembly model of the tool can be seen in Figure 2. The cross section of the blade is either circular or diamond shaped. The blade of pre-computed length is supported by pairs of electromechanically controlled rollers that allow us to set and instantaneously modify the clamping positions and, thus, to continuously change the shape of the blade. In the further part of the paper the layer manufacturing process will be referred to as hot blade cutting. The whole process of shape decomposition, layer thickness calculation, layer cutting and stacking is called thick layered object manufacturing (TLOM). Its concept, principles and methodology of finding the shape of the flexible blade have been reported in an earlier paper with the contribution of some of the present authors (Horváth, I., Vergeest, J. S. M. and Juhász, I., 1998). Due to the difficulties of a physically-based computation a geometrically-based modeling was applied. Further issues of the tool profile calculation, tool position and tool path Figure 2 The flexible blade tool calculation are discussed in (Horváth, I., Vergeest, J. S. M., Broek, J. J. and de Smit, A., 1998). This paper reports on the algorithms for and visualization of (a) the shape of the profile curve matching to a given domain of the front surface of the layer, (b) slicing the object into thick layers, (c) reduction of the cusp height when the calculated layer thickness is substituted by standard one, (d) determination of the tool positions to be set and calculation of the tool path, (e) virtual execution and verification of layer fabrication by process simulation. The free-form layer manufacturing process has been implemented on a six-axis sculpturing robot that is in use at the Section of Applied Information Technology, Sub-faculty of Industrial Design Engineering, of the Delft University of Technology. 2. ALGORITHM FOR CALCULATION OF PROFILE CURVES OF THE FLEXIBLE BLADE Compared to the conventional layer cutting techniques of zero or first order, the hot blade cutting process is featured by a blade of instantaneously changing curvature. Consequently, the very first problem is to find the shape of the blade that provides the best fitting to the actual shape of the CAD geometry in the domain that corresponds to the layer thickness. The bent physical blade can be substituted by a curve of minimal elastic energy for given support conditions and for prescribed length. For calculation of this minimal energy curve various analytical (Frisch-Fay, R., 1962), variational (Lee, E. H., Forsythe, G. E., 1973), and spline approximation algorithms (Cox, M. G., 1986), have been published. Most of the spline fitting algorithms minimize the square of the second derivative along the curve (Horn, B. K. P., 1983), (Vergeest, J. S. M., 1989). We applied a further developed version of the algorithm, first published in (Kallay, M., 1987), which starts out of a point set, rather than of splines, in order to keep the method simple (Figure 3). When the flexible blade is in its minimal energy state, its shape takes up the smoothest curve for given support conditions. Therefore, the geometric requirement for fitting an arbitrary set of points is to get the smoothest curve. The profile curve approximation algorithm starts out of the following assumptions: Let two points pi ≠ pk and two unit vectors tpi and tpk be given in the plane. Without loosing generality we may assume that the curve r that we search for is a mapping r(u) from the unit interval [0, 1] to the plane. We are looking for a curve r(u) which meets the following conditions: r( ) i 0 = p and r( ) k 1 = p , r t • = ( ) p i 0 a and r t • = ( ) pk 1 d , | ( )| r • ∫ =