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Lattice structures are widely used for achieving optimal topology in additive manufacturing. However, the use of different lattices in a single design can result in stress concentrations at the transition points. This study aims to investigate the influence of Bezier curves on mechanical properties during the transformation from one lattice structure to another. It specifically focuses on the transition from a hexagonal to diamond lattice, using Bezier curves of various orders.

The curves were designed by passing them through the same control points for different orders, such as third, fifth and seventh. The samples were sliced for 3D printing, and a tensile test was conducted. Young’s modulus and energy absorption abilities were measured to compare the mechanical properties of the models created with Bezier curves for the transformation between hexagonal and diamond models.

The analysis revealed a gradual change in mechanical properties from the hexagonal to the diamond lattice. Moreover, different orders of Bezier curves exhibited varying mechanical properties during the transformation between the two lattices. As the order of the Bezier curve increased, the mechanical properties smoothly changed from the hexagonal to diamond lattice. This prevented stress concentrations or mechanical behavior mismatch caused by sudden deformations at the transitions between the curves used in the design.

The study’s innovative use of Bezier curves of different orders to smoothly transformation between hexagonal and diamond lattices in additive manufacturing offers a practical solution to prevent stress concentrations and mechanical inconsistencies during such design transitions.

Frames play an important role in determining the geometric properties of the curves such as curvature and torsion. In particular, the determination of the point types of the curve, convexity or concavity is also possible with the frames. The Serret-Frenet frames are generally used in curve theory. However, the Serret-Frenet frame does not work when the second derivative is zero. In order to eliminate this problem, the quasi-frame was obtained. In this study, the quasi frames of the polynomial and rational Bezier curves are calculated by an algorithmic method. Thus, it will be possible to construct the frame even at singular points due to the second derivative of the curve. In this respect, the contribution of this study to computer-aided geometric design studies is quite high.

In this study, the quasi frame which is an alternative for all intermediate points of the rational Bezier curves was generated by the algorithm method, and some variants of this frame were analyzed. Even at the points where the second derivative of such rational Bezier curves is zero, there is a curve frame.

Several examples presented at the end of the paper regarding the quasi-frame of the rational Bezier curve, polynomial Bezier curve, linear, quadratic and cubic Bezier curves emphasize the efficacy and preciseness.

The quasi-frame of a rational Bezier curve is first computed. Owing to the quasi frame, it will have been found a solution for the nonsense rotation of the curve around the tangent.

The purpose of this paper is to consider the smooth path planning problem for a mobile robot based on the genetic algorithm (GA) and the Bezier curve.

The workspace of a mobile robot is described by a new grid-based representation that facilitates the operations of the adopted GA. The chromosome of the GA is composed of a sequence of binary numbered grids (i.e. control points of the Bezier curve). Ordinary genetic operators including crossover and mutation are used to search the optimum chromosome where the optimization criterion is the length of a piecewise collision-free Bezier curve path determined by the control points.

This paper has proposed a new smooth path planning for a mobile robot by resorting to the GA and the Bezier curve. A new grid-based representation of the workspace has been presented, which makes it convenient to perform operations in the GA. The GA has been used to search the optimum control points that determine the Bezier curve-based smooth path. The effectiveness of the proposed approach has been verified by a numerical experiment, and some performances of the obtained method have also been analyzed.

There still remain many interesting topics, for example, how to solve the specific smooth path planning problem by using the GA and how to promote the computational efficiency in the more grids case. These issues deserve further research.

The purpose of this paper is to improve the existing results by making the following three distinctive contributions: a rigorous mathematical formulation of the path planning optimization problem is formulated; a general grid-based representation (2n × 2n) is proposed to describe the workspace of the mobile robots to facilitate the implementation of the GA where n is chosen according to the trade-off between the accuracy and the computational burden; and the control points of the Bezier curve are directly linked to the optimization criteria so that the generated paths are guaranteed to be optimal without any need for smoothing afterwards.

The possibilities of using Bézier curves as a concrete tool in the development of the inference engine of a decision support system allowing approximate reasoning are explored. We propose using Bézier curves for representing membership functions or implementing some kind of fuzzy implication. The effects on the axioms characterizing fuzzy implications are investigated. Bézier curves are easily computable, flexible and have nice smooth properties. They are extensively used in the CAD/CAM domain. They could play an important role in some processes of imprecise knowledge representation and manipulation. Such techniques are promising in the implementation of several stages of developing the inference engine of a fuzzy decision support system. Moreover, these methods capture some of the flexibility and subtlety characteristic of human reasoning.

Robotic friction stir welding (RFSW) is an innovative process which enables solid-state welding of aluminum parts using robots. A major drawback of this process is that the robot joints undergo elastic deformation during the welding, because of the high forces induced by the process. This leads to tool deviation and incorrect orientation. There is currently no computer-aided manufacturing/computer-aided design (CAD) software for generating off-line paths which integrates robot deflections, and the main purpose of this study is to propose an off-line methodology to plan a path for RFSW with the integration of the deflections.

The approach is divided into two steps. The first step consists of extracting position and orientation data from CAD models of the workpieces and adding the deflections calculated with a deflection model to generate a suitable path for performing RFSW. The second step consists of the smooth fitting of the suitable path using Bézier curves.

The method is experimentally validated by welding a curved workpiece using a Kuka KR500-2MT robot. A suitable tool position and orientation were calculated to perform this welding, an experimental procedure was set up, a defect-free weld was performed and a high accuracy was achieved in terms of position and orientation.

This method can help manufacturers to easily perform RFSW for three-dimensional workpieces regardless of the lateral tool deviation, loss of the right orientation and control force stability.

The originality of this method lies in compensating for robot deflections without using expensive sensors, which is the most commonly used method for compensating for robot deflection. This off-line method can lead to a reduction in programming time in comparison with teach programming method and leads to reduced investment costs in comparison with commercial off-line programming packages.

The purpose of this paper is to apply rectangular Bézier surface patches directly into the mathematical formula used to solve boundary value problems modeled by Laplace's equation. The mathematical formula, called the parametric integral equation systems (PIES), will be obtained through the analytical modification of the conventional boundary integral equations (BIE), with the boundary mathematically described by rectangular Bézier patches.

The paper presents the methodology of the analytic connection of the rectangular patches with BIE. This methodology is a generalization of the one previously used for 2D problems.

In PIES the paper separates the necessity of performing simultaneous approximation of both boundary shape and the boundary functions, as the boundary geometry has been included in its mathematical formalism. The separation of the boundary geometry from the boundary functions enables to achieve an independent and more effective improvement of the accuracy of both approximations. Boundary functions are approximated by the Chebyshev series, whereas the boundary is approximated by Bézier patches.

The originality of the proposed approach lies in its ability to automatic adapt the PIES formula for modified shape of the boundary modeled by the Bézier patches. This modification does not require any dividing the patch into elements and creates the possibility for effective declaration of boundary geometry in continuous way directly in PIES.

The purpose of this paper is to present a novel guidance law for hypervelocity descent to a stationary target such that the impact angle and impact velocity can be constrained.

The proposed method is based on inverse dynamics and is designed using a third-order Bézier curve approximation to the reference trajectory.

Simulations indicate that the proposed law is able to satisfy impact angle and impact velocity constraints as well as follow control and path limitations in the case of guidance under perturbations. Comparisons with other methods also indicate better performance.

The onboard implementation requires an offline selection of Bézier parameters.

The presented scheme could be extremely important for further research on automated onboard control of impact angle and velocity for both re-entry and terminal guidance laws.

This paper presents an innovative method for the solution of an inverse dynamics-based guidance law using Bézier curve approximation.

The purpose of this paper is to propose a new approach for designing different parts of a high voltage bushing. It also aims to consider technical and economical criteria for the optimum solution of the design problem.

A novel method for finding the optimal contours of different elements of high voltage bushings, including ceramic insulator, electrode, and flange angle is presented. The rational Bézier curves are used for defining the surface of the insulators and conductors of the equipment. Then, these curves are optimally adjusted to obtain an appropriate techno‐economical solution. The utilized optimization method is the improved bacterial foraging algorithm (BFA) with variable step sizes. In the design procedure, two‐dimensional finite element method (2D FEM) is used to calculate the performance parameters in each step of the design procedure. In order to evaluate the performance of the proposed algorithm, optimal design of different elements of a 110 kV bushing using BFA and genetic algorithm is presented, compared, and discussed as well.

The results of this research show that the technical design criteria and economical costs are satisfied by the proposed method. It is concluded that the rational Bézier curves can be implemented for other similar applications and optimal design of other equipment in the electrical engineering field combined with heuristic optimization techniques.

Bezier curves are used for the first time for bushing design purpose. Two heuristic techniques are also implemented in order to facilitate the comparison and avoid local solutions.

This paper aims to describe a methodology to optimize the trajectory of unconventional airship performing a high-altitude docking manoeuvre.

The trajectories are based upon Bezier curves whose control points positions are optimized through particle swarm optimization algorithm. A minimum energy strategy is implemented by considering the airship physical properties. The paper describes the mathematical model of the airships, the trajectories modelling through Bezier’s curves and the optimization framework. A series of test cases has been developed to evaluate the proposed methodology.

Results obtained show that the implemented procedure is able to optimize the airship trajectories and to support their in-flight docking; a strong influence of the wind speed and course on the trajectories planning is highlighted.

The wind speed considered in these simulations depends only on altitude, and gusts effect has been neglected.

The proposed model can support the study of unconventional airship trajectories and can be useful to evaluate best in-air docking strategies.

The paper addresses the problem of trajectory optimization for a class of new air vehicles with an heuristic approach.

Non-uniform rational B-splines (NURBSs) are the de facto standard for representing objects in computer-aided design (CAD). The purpose of this paper is to discuss how to stick to this standard in all phases of the additive manufacturing (AM) workflow, from the CAD object to the final G-code, bypassing unnecessary polygonal approximations.

The authors use a commercial CAD system (Rhino3D along with its programming environment Grasshopper) for direct slicing of the model, offset generation and trimming. Circular arcs are represented as quadratic NURBSs and free-form geometry as quadratic or cubic polynomial B-splines. Therefore, circular arcs are directly expressible as G2/G3 G-code commands, whereas free-form paths are rewritten as a succession of cubic Bézier curves, thereby admitting exact translation into G5 commands, available in firmware for AM controllers, such as Marlin.

Experimental results of this paper confirm a considerable improvement in quality over the standard AM workflow, consisting of an initial polygonization of the object (e.g. via standard tessellation language), slicing this polygonal approximation, offsetting the polygonal sections and, finally, generating G-code made up of polyline trajectories (G1 commands).

A streamlined AM workflow is obtained, with a seamless transfer from the initial CAD description to the final G-code. By adhering to the NURBS standard at all steps, the authors avoid multiple representations and associated errors resulting from approximations.