In recent years, multi-axis fused filament fabrication has emerged as a solution to address the limitations of the conventional 2.5D printing process. By using a curved layering strategy and varying the print direction, the final parts can be printed with reduced support structures, enhanced surface quality, and improved mechanical properties. However, support structures in the multi-axis scheme are still needed sometimes when the support-free requirement conflicts with other constraints. Currently, most support generation algorithms are for the conventional 2.5D printing, which are not applicable to multi-axis printing. To address this issue, we propose a sparse and curved support filling pattern for multi-axis printing, aiming at enhancing the material efficiency by fully utilizing the bridge technique. Firstly, the overhang regions are detected by identifying the overhang points given a multi-axis nozzle path. Then, an optimization framework for the support guide curve is proposed to minimize its total length while ensuring that overhang filaments can be stably supported. Lastly, the support layer slices and support segments that satisfy the self-supported criterion are generated for the final support printing paths. Simulation and experiments have been performed to validate the proposed methodology.

Multi-entrance Inwall (MEI) surfaces are widely used in industrial applications, yet inspecting the MEI surfaces precisely remains a challenging task due to their complex multi-entrance topology and potential collision risks. The recently developed five-axis continuous sweep scanning technology offers significantly higher inspection efficiency compared to traditional point-by-point methods, presenting a valuable opportunity for accurate and efficient MEI surface inspection. However, planning a five-axis continuous sweep scanning process for general MEI surfaces still largely relies on human input. To address this challenge, this paper presents a novel set of methods for generating five-axis sweep scanning paths, specifically designed for the automatic and efficient inspection of MEI surfaces. Our methodology utilizes a sophisticated heat-induced geodesic distance field (GDF) to calculate guiding curves, which are used to generate the sweep scanning paths and partition the accessible regions of the MEI surface. This approach results in the creation of continuous five-axis sweep scan inspection paths that enhance both inspection efficiency and surface coverage rates. The proposed method has been validated through physical inspection experiments and computer simulations, with results confirming its feasibility and demonstrating its advantages over two benchmark approaches. Note to Practitioners—This article aims to generate an automatic, highly efficient inspection path for MEI surfaces using a five-axis coordinate measuring machine (CMM). While existing methods have addressed some issues in free-form surface inspection, they primarily focus on external and open surface inspection and do not adequately adapt to MEI surfaces, which are often occluded by challenging collision situations and complex topology. As a result, planning a five-axis continuous sweep scanning process for a general MEI surface still heavily relies on human interaction. To address this limitation, we propose an inspection path generation method that constructs a set of guiding curves considering the geometric information of both the entrances and collision situations. We utilize a heat-induced Geodesic Distance Field (GDF) to compute guide paths and assist in partitioning the accessible region. Through experiments and computer simulations, our proposed method demonstrates superior performance compared to traditional benchmarking methods in terms of both inspection efficiency and point accessibility rate. The generated inspection path conforms to the workpiece surface geometry, effectively overcoming challenges such as high interference and topological complexity in MEI, thus enabling efficient and comprehensive surface measurements.

Freeform surface inspection is a vital process in manufacturing, and the newly emerged five-axis continuous sweep scanning technology is one of the most efficient and accurate means for free-form surface inspection. The key in employing the five-axis inspection technology is to plan an effective and efficient sweep scanning path respecting both the inspection surface and the properties of the inspection device. Current sweep scanning paths suffer from the oscillating pattern that forces the high-speed stylus of the five-axis inspection device to swing back and forth frequently, which imposes excessive kinematic loading on the probe head, and in turn, undermines the inspection stability and efficiency. In this paper, we present a new trochoidal sweep scanning path for freeform surface inspection. The proposed method is novel in that the generated inspection path is a smooth trochoidal-shaped curve on the surface without causing any oscillation during a sweep scanning process. The kinematic performance of the probe head is significantly improved, which enables much better inspection stability and efficiency. Both computer simulation and physical inspection experiments of the proposed method have been conducted, whose results convincingly validate the feasibility and superiority of the proposed trochoidal sweep scanning path over the traditional oscillating sweep scanning path in terms of inspection stability and efficiency. Note to Practitioners—This work is motivated by the high requirements on the stability and efficiency of the newly emerged five-axis sweep scanning technology for freeform surface inspection. A novel trochoidal sweep scanning path is generated to make the original heavy-loaded rotary axes of the five-axis inspection machine move smoothly without needing frequent acceleration and deceleration as suffered from a traditional five-axis oscillating sweep scanning path. The kinematic loads of the five-axis inspection device are significantly reduced based on the proposed sweep scanning path; thus, the inspection machine could afford a higher sweep velocity of inspection, and the inspection stability and efficiency could be improved considerably as well. Compared with a traditional oscillating sweep scanning path, in our tests, the proposed method can improve the inspection efficiency by more than 20%, while the maximal kinematic loads (e.g., the velocity and jerk of the C-axis) can be reduced by 55%-85%. With superior inspection efficiency and stability, the proposed trochoidal sweep scanning path can be utilized for accurate and efficient inspection of large freeform surfaces like those in the aerospace and automotive industries.

For the traditional three-axis configuration of additive manufacturing (AM) platform, it is imperative to construct adequate support structures prior to the fabrication of overhanging features on the part geometry. To completely eliminate the use of support by taking advantage of the newly emerged five-axis AM platform, a novel multidirectional process planning algorithm for 3 + 2-axis AM is proposed in this paper. The core of the strategy is to decompose the model into support-free parts directly pertaining to the cusp-height constraint, each with its own build direction. The nozzle will follow the staged sequence to fabricate each individual part along its build direction, as facilitated by the adjustable orientation of the nozzle head on a five-axis platform. For model decomposition, a recursive downward flooding expansion algorithm is introduced to identify a surface patch in accordance with the support-free criteria. After being stitched as a watertight geometry, the decomposed part together with the remaining part will be archived into a prescribed hierarchy based on which the build sequence is readily established. Preliminary testing results have verified the effectiveness of the algorithm to handle the geometries of different types. Note to Practitioners-This paper was motivated by the deficiencies encountered in traditional three-axis additive manufacturing (AM). In practice, most commercialized 3-D printers are of three-axis configuration, on which it is required to construct support structures for those overhanging features on the part geometry. These support structures not only cost extra time and materials but also leave noticeable artifacts on the part surface even after the clean-up process. As an ultimate solution to this issue, the newly emerged five-axis AM platform is able to physically eliminate the need for support by properly and continuously adjusting the build direction. This, however, demands a delicate algorithm for determining the changing build direction for an arbitrary part that is currently unavailable. To simplify this challenging task, we took advantage of the 3 + 2 motion capability of the five-axis AM platform to progressively plan the three-axis printing process for a given freeform model. The core idea of our solution is to decompose the model by a flooding-like algorithm into individual support-free parts, each of which is printable along a fixed build direction without any support. The nozzle will then follow a prescribed sequence to print each part one on the other. The proposed algorithm is purely geometric, without taking into account the material property and the mass distribution. Preliminary testing results show that our approach is feasible and robust when dealing with limited geometries that abide a tree-like structure and features a clear flat base. Besides these geometric limitations, our current preliminary scheme for collision avoidance may become invalid upon an arbitrary complex geometry. We will further extend this method to cater to more sophisticated and general geometries.

Sweep scan is an emerging five-axis inspection technology of retrieving geometric data from free-form surfaces. Compared with the traditional surface inspection by coordinate measuring machine (CMM) that works in a point-by-point manner, sweep scan is able to make the stylus tip continuously sweep at high speed on the surface to inspect, whilst acquiring the 3-D point data accurately and efficiently. However, at present, for an arbitrary free-form surface, it still heavily depends on humans to manually generate a suitable sweep scan path. This paper presents a practical sweep scan path planning method, which can automatically generate an efficient sweep scan path for an arbitrary free-form surface. The resultant sweep scan path is able to cater to the shape of the surface to inspect and the unique kinematic characteristics of the typical five-axis inspection machine, so as to achieve a tremendous increase in inspection efficiency. The experiments performed by us show that when compared with some existing automatic sweep scan path generation algorithms such as the simple zigzag method, the proposed method is able to reduce the total inspection time by as much as seven times.

Feature detection on smooth free-form surfaces is much more difficult than that on shapes with sharp features. In this paper, we extract the curved reflection axes (CRAs) of an arbitrary free-form surface as features if they exist. The extraction result is robust to boundary noises and strongly sensitive to extrinsic properties of the surface such as projected normals and curvatures. Compared with the general reflection symmetry, curved reflection symmetry is defined to be a reflection symmetry along a smooth 3D embedded curve instead of a plane, where any point on the curve is a local reflection center for some surface points. The properties of the curved reflection symmetric axis are analyzed, and a novel computational model for detecting and extracting CRAs on free-form surfaces is presented. The experimental results are then compared with both the medial axis and the intrinsic symmetric axis, which are two popular feature representations of 3D shapes, and the advantages and uniqueness of the proposed method are convincingly demonstrated. An application of the proposed method in sweep scanning is also presented. Other applications of the proposed method include feature extraction, shape symmetrization, segmentation, and registration.

Continuous five-axis sweep scanning is an emerging technology for free-form surface inspection, which, unlike the traditional three-axis inspection that works in a point-by-point manner, keeps the stylus tip in constant contact with the surface during the scanning, and thus could tremendously improve the inspection efficiency. However, at present, it mostly depends on humans to plan a five-axis inspection path, which severely affects the potential use of this new technology. In this paper, we report a practical algorithm, which is able to automatically generate a five-axis inspection path for an arbitrary free-form surface. The crux of this algorithm is that the unique kinematic characteristics of the five-axis inspection machine are fully considered and utilized when a path is planned. As a direct result of this consideration and utilization, the inspection efficiency is tremendously increased, often 20-30 times better than an inspection path obtained by any traditional path planning algorithm that disregards the inspection machine itself. The experiments performed by us have fully validated this point.

We present an evolution method for designing the styling curves of garments. The procedure of evolution is driven by aesthetics-inspired scores to evaluate the quality of styling designs, where the aesthetic considerations are represented in the form of streamlines on human bodies. A dual representation is introduced in our platform to process the styling curves of designs, based on which robust methods for realizing the operations of evolution are developed. Starting from a given set of styling designs on human bodies, we demonstrate the effectiveness of set evolution inspired by aesthetic factors. The evolution is adaptive to the change of aesthetic inspirations. By this adaptation, our platform can automatically generate new designs fulfilling the demands of variations in different human bodies and poses.

The medial axis is an important shape representation that finds a wide range of applications in shape analysis. For large-scale shapes of high resolution, a progressive medial axis representation that starts with the lowest resolution and gradually adds more details is desired. In this paper, we propose a fast and robust geometric algorithm that computes progressive medial axes of a large-scale planar shape. The key ingredient of our method is a novel structural analysis of merging medial axes of two planar shapes along a shared boundary. Our method is robust by separating the analysis of topological structure from numerical computation. Our method is also fast and we show that the time complexity of merging two medial axes is$O(n\;\log n_v)$, where$n$is the number of total boundary generators,$n_v$is strictly smaller than$n$and behaves as a small constant in all our experiments. Experiments on large-scale polygonal data and comparison with state-of-the-art methods show the efficiency and effectiveness of the proposed method.

In this paper, a boundary-conformed tool path generation method for either compound or trimmed surfaces are proposed, based on two powerful reparametrization schemes -- the discretized harmonic mapping and the convex combination mapping. By globally mapping a 3D surface onto a 2D unit square and then planning an iso-parametric curve in the 2D domain, the corresponding Cutter Contact (CC) curve on the original 3D surface is easily generated which conforms with the boundary of the surface. Based on this CC curve generation strategy, a CC curve expansion algorithm for covering the entire surface is designed which takes into account the machining accuracy requirement, i.e., The specified maximum cusp height. Tool paths generated in this way for compound or trimmed surfaces are boundary-conformed, smooth, and guarantee the required machining accuracy.

This paper presents the development of a new type of robot capable of vertical and directional jumping. The robot uses soft silicone elastomer based pneumatic actuators as legs that accelerate the platform upwards by rapid pressurization. The robot is able to control and adjust the direction of the jumping by altering the timing patterns in which the individual legs are activated.

A recent paper titled “On the Problem of the Automated Design of Large-Scaled Robot Skin” (Anghinolfi et al., 2013) published in the IEEE Transactions on Automation Science and Engineering addressed the problem of covering the surface of a humanoid robot with the largest number of nonoverlapping equilateral triangular sensor modules. The problem is eventually approximated by a simpler one: how to find the placement of a given polygon P on an equilateral triangular grid G that contains the largest number of the grid triangles. In this paper, we show how to improve the efficiency of the algorithm presented in that paper. Further, we show that the general problem of filling P with the largest number of disjoint equilateral triangles (all entirely contained in P and all of the same size) is not equivalent to that of finding an optimal placement of P on G. Using this result, we propose an improved heuristic for the original problem of covering the skin of a robot with the largest number of triangular sensor modules.

In the research of computer vision and machine perception, 3D objects are usually represented by 2-manifold triangular meshes M. In this paper, we present practical and efficient algorithms to construct iso-contours, bisectors, and Voronoi diagrams of point sites on M, based on an exact geodesic metric. Compared to euclidean metric spaces, the Voronoi diagrams on M exhibit many special properties that fail all of the existing euclidean Voronoi algorithms. To provide practical algorithms for constructing geodesic-metric-based Voronoi diagrams on M, this paper studies the analytic structure of iso-contours, bisectors, and Voronoi diagrams on M. After a necessary preprocessing of model M, practical algorithms are proposed for quickly obtaining full information about iso--contours, bisectors, and Voronoi diagrams on M. The complexity of the construction algorithms is also analyzed. Finally, three interesting applications-surface sampling and reconstruction, 3D skeleton extraction, and point pattern analysis-are presented that show the potential power of the proposed algorithms in pattern analysis.

An important real-world optimization problem in manufacturing industry is to determine optimal workpiece setups for 4-axis NC machining. In this paper we reveal some interesting relations between this optimal workpiece setup problem and the two classic NP-hard problems in complexity theory (i.e, the vertex cover problem and the set cover problem). These relations immediately show the following results. First the optimal workpiece setup problem is NP-hard. Secondly, the greedy algorithm proposed in [Comput. Aided Des. 35 (2003) pp. 1269-1285] for the optimal workpiece setup problem has the performance ratio bounded by O(ln n-ln ln n+0.78), where n is the number of spherical polygons in the ground set.

We present a new algorithm for finding a most "developable" smooth mesh surface to interpolate a given set of arbitrary points or space curves. Inspired by the recent progress in mesh editing that employs the concepts of preserving the Laplacian coordinates and handle-based shape editing, we formulate the interpolation problem as a mesh deformation process that transforms an initial developable mesh surface, such as a planar figure, to a final mesh surface that interpolates the given points and/or curves. During the deformation, the developability of the intermediate mesh is maintained by means of preserving the zero-valued Gaussian curvature on the mesh. To treat the high nonlinearity of the geometric constrains owing to the preservation of Gaussian curvature, we linearize those nonlinear constraints using Taylor expansion and eventually construct a sparse and over-determined linear system which is subsequently solved by a robust least-squares solution. By iteratively performing this procedure, the initial mesh is gradually and smoothly "dragged" to the given points and/or curves. The initial experimental data has shown some promising aspects of the proposed algorithm as a general quasi-developable surface interpolation tool.

This paper addresses the problem of computing planar patterns for compression garments. In the garment industry, the compression garment has been more and more widely used to retain a shape of human body, where certain strain (or normal pressure) is designed at some places on the compression garment. Variant values and distribution of strain can only be generated by sewing different 2D patterns and warping them onto the body. We present a physical/geometric approach to compute 2D meshes that, when folded onto the 3D body, can generate a user-defined strain distribution through proper distortion. This is opposite to the widely studied mesh parameterization problem, whose objective is to minimize the distortion between the 2D and 3D meshes in angle, area or length.

In this paper the applications of equilibrium equation to geometric modeling is exploited and efficient numerical algorithms are proposed for solving the equilibrium equation. First we show that from diverse geometric modeling applications the equilibrium system can be extracted as the central framework. Second, by exploiting in-depth the special structures inherent in the geometric applications, we present simplified analytic solutions to the resulting geometric equilibrium equations via system decomposition. Finally, given the observation that the geometric equilibrium systems are extremely sensitive to both perturbations in input data and round off errors, efficient, stable and accurate numerical algorithms are proposed.

Computing the planar sections of objects is a fundamental operation in solid modeling. Subdivision method is commonly used for solving such intersection problems. In this paper, a revolute quadric decomposition is proposed for surfaces of revolution, which are subdivided into a set of coaxial revolute quadrics along the generatrix. This reduces the intersection problem of a plane and a surface of revolution to the intersection problem of a plane and a revolute quadric, which has robust, accurate and efficient geometric solution. Further, the intersection curves can be represented with a group of G (G. Baciu et al., 2001) conic arcs. A new concept, valid intersection interval (VII), is introduced and a new technique, cylindrical bounding shell clipping, is proposed for efficient intersection detection for a plane and a surface of revolution. Finally, a tracing algorithm is presented for recognizing singular points and closed loops of intersection curves. Implemented examples show the robustness and effectiveness of the proposed algorithm.

In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon A and a set /spl Xi/ of possibly overlapping polygons fixed in the plane. More specifically, a subset /spl Phi//spl sube//spl Xi/ is called an intersection relation if there exists a placement of A that intersects, and only intersects, /spl Phi/. The objective of this paper is to design an efficient algorithm that finds a finite and discrete representation of all of the intersection relations between A and /spl Xi/. Past related research only focuses on the complexity of the free space of the configuration space between A and /spl Xi/ and how to move or place an object in this free space. However, there are many applications that require the knowledge of not only the free space, but also the intersection relations. Examples are presented to demonstrate the rich applications of the formulated problem on intersection relations.