Continuous Curve Research Articles

Research on path smoothing optimisation based on improved RRT-Connect algorithm and third-order Bezier curve

Aiming at the deficiencies of the original RRT-Connect path planning algorithm in dealing with obstacle avoidance, planning efficiency and path smoothing in static environments, an improved path optimisation method is proposed by fusing the RRT-Connect path planning algorithm with the greedy search strategy and using a third-order Bezier curve for path smoothing. Firstly, a greedy algorithm is added to the path planning process of the original RRT-Connect algorithm to guide the search direction, so as to make the path search more goal oriented, reduce the time of path planning and improve the efficiency. Secondly, the generated paths are smoothed with third-order Bezier curves to ensure the generation of smooth paths with continuous curvature and improve the quality of the planned paths. Finally, the improved RRT-Connect smoothing optimisation algorithm is simulated with the original RRT-Connect, A* and Dijkstra algorithms in different simulation environments to check the performance of the optimised algorithm. The results show that the success rate of obstacle avoidance in planning paths of the optimised algorithms are all 100%, and compared with the original RRT-Connect algorithm, the improved RRT-Connect smoothing optimisation algorithm has an overall reduction of 12.02% in planning path length, an overall reduction of 40.22% in computation time and an overall improvement of 69.15% in path smoothness. Compared to the A* algorithm, which is based on the graph search shortest path planning algorithm and takes the shortest time, there is an overall reduction of 62.46% in computation time. The improved RRT-Connect smoothing optimisation algorithm has shorter path planning lengths, smoother paths and shortest computation time in different simulation environments. Meanwhile, the stability of the optimisation algorithm is effectively improved in different environments and different running times.

The Deformation of a Liquid Metal Droplet Under Continuous Acceleration in a Variable Cross-Section Groove

This paper constructs a numerical simulation model for the deformation of droplets in a variable cross-section groove of a liquid droplet MEMS switch under different directions, amplitudes, frequencies, and waveforms of acceleration. The numerical simulation utilizes the level set method to monitor the deformation surface boundary of the metal droplets. The simulation outcomes manifest that when the negative impact acceleration on the X-axis is 12.9 m/s2, the negative impact acceleration on the Y-axis is 90 m/s2, the negative impact acceleration on the Z-axis is 34.5 m/s2, and the metal droplet interfaces with the metal electrode. The droplet deformation under the effect of a sine wave acceleration signal in the X and Y directions is lower than that under impact acceleration, while in the Z direction, the deformation is higher than that under impact acceleration. The deformation of metal droplets under square wave acceleration is more pronounced than that under sinusoidal wave acceleration. The deformation escalates with the augmentation in square wave amplitude and dwindles with the reduction in square wave acceleration frequency. Furthermore, there exists a phase difference between the deformation curve of the metal droplet and the continuous acceleration signal curve, and the phase difference is dependent of the material properties of the metal droplet. This work elucidates the deformation of the liquid-metal droplets under continuous acceleration and furnishes the foundation for the continuous operation design of MEMS droplet switches.

A Pseudo-Waveform-Based Method for Grading ICESat-2 ATL08 Terrain Estimates in Forested Areas

The ICESat-2 Land and Vegetation Height (ATL08) product is a new control point dataset for large-scale topographic mapping and geodetic surveying. However, its elevation accuracy is typically affected by multiple factors. The study aims to propose a new approach to classify ATL08 terrain estimates into different accuracy levels and extract reliable ground control points (GCPs) from ICESat-2 ATL08. Specifically, the methodology is divided into three stages. First, the ATL08 terrain estimates are matched with the raw ATL03 photon cloud data, and the ATL08 terrain estimates are used to fit a continuous terrain curve. Then, using the fitted continuous terrain curve and raw ATL03 photon cloud data, a pseudo-waveform is generated for grading the ATL08 terrain estimates. Finally, all the ATL08 terrain estimates are graded based on the peak characteristics of the generated pseudo-waveform. To validate the feasibility of the proposed method, four study areas from the National Ecological Observatory Network (NEON), characterized by various terrain features and forest types were selected. High-accuracy airborne lidar data were used to evaluate the accuracy of graded ICESat-2 terrain estimates. The results demonstrate that the method effectively classified all ATL08 terrain estimates into different accuracy levels and successfully extracted high-accuracy GCPs. The root mean square errors (RMSEs) of the first accuracy level in the four selected study areas were 0.99 m, 0.51 m, 1.88 m, and 0.65 m, representing accuracy improvement of 51.7%, 58.2%, 83.1%, and 68.8%, respectively, compared to the original ATL08 terrain estimates before classifying. Additionally, a comparison with the conventional threshold-based GCP extraction method demonstrated the superior performance of our proposed approach. This study introduces a new approach to extract high-quality elevation control points from ICESat-2 ATL08 data, particularly in forested areas.

Continuous Curvature Parking Path Planning for Unmanned Mining Trucks Using Transition Curves and Model Predictive Control

<div>Geometric methods based on Reeds–Shepp (RS) curves offer a practical approach for the parking path planning of unmanned mining truck, but discontinuous curvature can cause tire wear and road damage. To address this issue in mine scenario, a continuous curvature parking path planning method based on transition curve and model predictive control (MPC) is proposed for mine scenarios. Initially, according to the shovel position information issued by the cloud dispatching platform, a reference line is planned using RS curves. In order to mitigate the wear and tear of the tires and the damage to unstructured roads due to the in situ steering caused by the sudden change of the curvature, a transition curve consisting of clothoid–arc–clothoid that satisfies the kinematics of continuous vehicle steering is designed on the basis of RS curves to achieve the continuity of road curvature, which will contribute to the economy of tire and handling performance. The calculation of Fresnel integral involved by clothoid is simplified by using Chebyshev polynomial fitting method. Moreover, MPC is employed to re-plan an obstacle-avoidance path based on the reference line by designing a rational cost function. Finally, a simulation test platform is built considering the typical parking scenarios for unmanned mining trucks. The simulation results verify the effectiveness of the planning algorithm proposed in this article, and it shows the potential to reduce maintenance costs and improve mining efficiency.</div>

Bialy-Mironov type rigidity for centrally symmetric symplectic billiards

Abstract The aim of the present paper is to establish a Bialy–Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric C 2 strongly-convex domain D with boundary ∂ D , assume that the symplectic billiard map has a (simple) continuous invariant curve δ ⊂ P of rotation number 1 / 4 (winding once around ∂ D ) and consisting only of 4-periodic orbits. If one of the parts between δ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then ∂ D is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve δ, and the result is obtained by tracing back the problem directly to the totally integrable case.

Global bifurcation for monotone fronts of elliptic equations

We present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well as quasilinear problems with transmission boundary conditions. Our approach is rooted in the analytic global bifurcation theory of Dancer (1973) and Buffoni–Toland (2003), but extending it to unbounded domains requires contending with new potential limiting behavior relating to loss of compactness. We obtain an exhaustive set of alternatives for the global behavior of the solution curve that is sharp, with each possibility having a direct analogue in the bifurcation theory of second-order ODEs.As a major application of the general theory, we construct global families of internal hydrodynamic bores. These are traveling front solutions of the full two-phase Euler equation in two dimensions. The fluids are confined to a channel that is bounded above and below by rigid walls, with incompressible and irrotational flow in each layer. Small-amplitude fronts for this system have been obtained by several authors. We give the first large-amplitude result in the form of continuous curves of elevation and depression bores. Following the elevation curve to its extreme, we find waves whose interfaces either overturn (develop a vertical tangent) or become exceptionally singular in that the flow in both layers degenerates at a single point on the boundary. For the curve of depression waves, we prove that either the interface overturns or it comes into contact with the upper wall.

Моделирование и визуализация поверхностей сложной формы с помощью интерполяционных кривых

This article presents an approach to modeling and visualizing surfaces of complex shapes using interpolation curves with predetermined geometric properties. A modified Bezier curve of n-order was used as interpolation curves. Modification of a Bezier arc into an interpolation curve is possible both with and without preserving tangents. When preserving tangents, the Bezier arc retains its properties as a contour arc and acquires the ability to pass through preset points. The considered modification is possible in several variations: universal, based on the uniform distribution of the parameter during the modification process, and adaptive, when the parameter values are adapted to the initial data. The use of interpolation curves makes it possible to implement a special case of the moving simplex method, an analogue of which in geometric modeling and computer-aided design systems is the section operation (or lofting). The difference is that a continuous curve is used as a generating surface instead of a piecewise one. To ensure the functionality of such a connection, we give examples of models of the surface of an onion dome and a vase using various guides. An analysis of the obtained results was carried out. The introduction of research results into CAD/CAM will significantly expand their tools in terms of shape formation and visualization of surfaces and bodies that have predetermined geometric requirements.

Examination of the Dry Density Importances in TDR Use and of the ASTM D6780 Standard Based on a New Calibration Procedure

Abstract The use of time domain reflectometry (TDR) to monitor soil water content at either field or laboratory scales is highly common. Regarding the soil dry density, ρd, effect on the TDR application, three main approaches are found in the literature: (1) Ignore the density issue and use calibration to evaluate the water content regardless of the density. This approach is basic, yet common. (2) Include soil density in the calibration equation and use calibration for the water content as a function of both the TDR output and soil density. In this approach, the soil density should be provided from an independent source. (3) Determined ρd from the TDR waveform. In this case both water content and ρd are set from the TDR analysis. In cases where the ρd is not available, in order to use TDR for water content measurement, a decision has to be made whether to address the density issue although it requires resources, or to neglect it. This article reviews this issue and validates the density effect using an efficient TDR calibration methodology. A calibration scheme that controls ρd of the specimen has been developed and applied. The methodology creates continuous calibration curves over a range of volumetric water content from a single test specimen. Based on the results, it has been demonstrated that the soil density has a major effect on the TDR output and it cannot be ignored. In addition, this article presents a unique examination of ASTM D6780, Standard Test Methods for Water Content and Density of Soil In Situ by Time Domain Reflectometry (TDR). The ASTM dry density estimation was found to provide reliable results; however, the gravimetric water contents estimation gives unsatisfactory results, especially for low densities.

Utilizing BIC for the intelligent selection of functional data with principal components

Abstract Functional Principal Component Analysis (FPCA) is a technique for dimension reduction of functional data. Considering the impact of different data ownership, the paper innovatively proposes a weighted B-spline basis function and provides a continuous curve fitting function. Then, smoothing processing is carried out through the basis function expansion method and the maximum penalized likelihood function method. Subsequently, the process of selecting the number of principal components based on the BIC criterion is elaborated. Parameter estimation is realized through the newly proposed BIC-FPCA optimization algorithm. In the empirical analysis section, the paper uses student performance data randomly generated by Python to demonstrate the application of FPCA in data reconstruction. The results show that although the reconstructed data is smoother, there is still room for improvement in the selection of principal components. Finally, the paper summarizes the research findings, suggesting that the BIC criterion be combined with the cumulative variance contribution rate to improve the method of principal component selection, and the algorithm be optimized to reduce computational load. With these improvements, the application effect of FPCA in functional data analysis can be further enhanced.

Global Path Optimal Planning Method for Autonomous Vehicles Based on Image Feature Extraction

In order to solve the problem of autonomous vehicles driving safely on the road while searching for optimal paths to avoid traffic congestion and obstruction, this paper establishes an optimal path planning model based on image feature extraction. The model consists of four parts: the selection of key turning points, interpolation between turning points, definition of the evaluation function, and global optimal path search. First, based on the map information provided by the network, an image feature detection algorithm was proposed, and the key turning points of the map route were selected and stored in an open table. Then, based on mechanical analysis theory, the cubic interpolation between two key turning points of the optimal path was defined. This allows the car to drive along a continuous curve with a certain arc radius, enabling it to drive smoothly, effectively avoid obstacles, and ensure safety. Then, an evaluation function is defined as needed, and different evaluation functions are specified for different road sections to avoid excessive local deviations in planning and to reduce the impact of known obstacles on the path. Finally, the cost of the search path is calculated based on the evaluation function. According to the principle of minimum cost, a globally optimal heuristic search algorithm is proposed to find the globally optimal path. Experimental results show that the proposed global optimal path planning algorithm has excellent performance, with an average accuracy of 96.71% and a search speed of 2.78 ms. The model not only provides accurate path planning and fast path selection capabilities but also has strong robustness against interference.

Development of a novel computational technique to create DNA and cell geometrical models for Geant4-DNA

BackgroundThis study aimed to develop a novel human cell geometry for the Geant4-DNA simulation toolkit that explicitly incorporates all 23 chromosome pairs of the human cell. This approach contrasts with the existing, default human cell, geometrical model, which utilizes a continuous Hilbert curve. MethodsA Python-based tool named “complexDNA” was developed to facilitate the design of both simple and complex DNA geometries. This tool was employed to construct a human cell geometry with individual pairs of chromosomes. Subsequently, the performance of this chromosomal model was compared to the standard human cell model provided in the “molecularDNA” Geant4-DNA example. ResultsSimulations using the new chromosomal model revealed minimal discrepancies in DNA damage yield and fragment size distribution compared to the default human cell model. Notably, the chromosomal model demonstrated significant computational efficiency, requiring approximately three times less simulation time to achieve equivalent results. ConclusionsThis work highlights the importance of incorporating chromosomal structure into human cell models for radiation biology research. The “complexDNA” tool offers a valuable resource for creating intricate DNA structures for future studies. Further refinements, such as implementing smaller voxels for euchromatin regions, are proposed to enhance the model’s accuracy.

Enhanced curve representations using piecewise non-linear binary subdivision scheme

This study presents a piecewise non-linear binary subdivision scheme based on a four-point linear binary subdivision scheme. This scheme is first transformed into a piecewise linear subdivision scheme and then into a piecewise non-linear subdivision scheme. The non-linear subdivision approach removes artifacts and distortions from limit curves, producing better continuous curves. The study demonstrates that this technique is a workable technique for producing smooth curves, especially when avoiding bends, distortion, and oscillation.

On bifurcation of non-isolated singular points arising in a problem of two-phase fluid motion in a pipe

The dynamics of gas suspension in a pipe is studied. It is described by a system of three autonomous differential equations, whose equilibrium (singular) points are not isolated, but form several continuous curves. One of these curves belongs to the transonic plane. The dynamics in a neighborhood of this plane is interesting from the physical viewpoint. In this work, the dynamics in a neighborhood of the curves of singular points is investigated. The main tools of this work are the center manifold theorem and the reduction principle. A peculiarity of the system is that the curves of singular points intersect each other. Moreover, at the intersection point a bifurcation occurs. Results of the work are the description of the dynamics in a neighborhood of singular points, the proof that a curve of singular points belongs to a center manifold and the description of the bifurcation, which occurs at the intersection point of the curves of singular points.

Research on the Hydrodynamic Characteristics of a Rectangular Otter Board in Different Work Postures Based on a Dynamic Model

This paper investigates the hydrodynamic characteristics of a rectangular otter board in different working postures by using a dynamic model. Dynamic models are mainly based on dynamic mesh techniques. The results of the dynamic model are, compared to the model test, carried out in a flume tank. Furthermore, different rotation speeds of dynamic model were analyzed. The research results are as follows: compared to flume tank results, the maximum error of the dynamic model is 23.77%. Moreover, the influence of rotation speed on the hydrodynamic board is not obvious, and 2 deg./s was chosen as the rotation speed. When the board is tilted slightly (including four working postures), its lift-to-drag ratio first increases slightly and then gradually decreases. Compared with the other three working postures, the pressure center coefficient of the board does not change significantly when it is tilted inward. When studying different working angles (including AOA and tilt angle) of the otter board, the numerical dynamic model significantly reduces repetitive setup work, making simulations more efficient. Its ability to provide continuous curves and a large volume of results offers researchers a more detailed and comprehensive understanding of the board’s hydrodynamics. Additionally, the dynamic model supports innovative fishery equipment development by allowing more accurate and continuous numerical simulations.

Curvature continuous corner cutting

Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While de Boor (1987) shows that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, Gregory and Qu (1996) discover that the limit curve is continuously differentiable under certain constraints. We extend these results and show that the limit curve can even be curvature continuous for specific sequences of cut ratios.

Three-Dimensional Printing Limitations of Polymers Reinforced with Continuous Stainless Steel Fibres and Curvature Stiffness

This study investigates the printability limitations of 3D-printed continuous 316L stainless steel fibre-reinforced polymer composites obtained using the Materials Extrusion (MEX) technique. The objective was to better understand the geometric printing limitations of composites fabricated using continuous steel fibres, based on a combination of bending stiffness testing and piezoresistive property studies. The 0.5 mm composite filaments used in this study were obtained by co-extruding polylactic acid (PLA), with a 316 L stainless steel fibre (SSF) bundle. The composite printability limitations were evaluated by the printing of a series of ’teardrop’ shaped geometries with angles in the range from 5° to 90° and radii between 2 and 20 mm. The morphology and dimensional measurements of the resulting PLA-SSF prints were evaluated using μCT scanning, optical microscopy, and calliper measurements. Sample sets were compared and statistically examined to evaluate the repeatability, turning ability, and geometrical print limitations, along with dimensional fluctuations between designed and as-printed structures. Comparisons of the curvature bending stiffness were made with the PLA-only polymer and with 3D-printed nylon-reinforced short and long carbon fibre composites. It was demonstrated that the stainless steel composites exhibited an increase in bending stiffness at smaller radii. The change in piezoresistance response of the PLA-SSF with load applied during the curvature bending stiffness testing demonstrated that the 3D-printed composites may have the potential for use as structural health monitoring sensors.

Laser Scanner-Based Hyperboloid Cooling Tower Geometry Inspection: Thickness and Deformation Mapping.

Hyperboloid cooling towers are counted among the largest cast-in-place industrial structures. They are an essential element of cooling systems used in many power plants in service today. Their main structural component, a reinforced-concrete shell in the form of a one-sheet hyperboloid with bidirectional curvature continuity, makes them stand out against other towers and poses very high construction and service requirements. The safe service and adequate durability of the hyperboloid structure are guaranteed by the proper geometric parameters of the reinforced-concrete shell and monitoring of their condition over time. This article presents an original concept for employing terrestrial laser scanning to conduct an end-to-end assessment of the geometric condition of a hyperboloid cooling tower as required by industry standards. The novelty of the proposed solution lies in the use of measurements of the interior of the structure to determine the actual thickness of the hyperboloid shell, which is generally disregarded in geometric measurements of such objects. The proposal involves several strategies and procedures for a reliable verification of the structure's verticality, the detection of signs of ovalisation of the shell, the estimation of the parameters of the structure's theoretical model, and the analysis of the distribution of the thickness and geometric imperfections of the reinforced-concrete shell. The idea behind the method for determining the actual thickness of the shell (including its variation due to repairs and reinforcement operations), which is generally disregarded when measuring the geometry of such structures, is to estimate the distance between point clouds of the internal and external surfaces of the structure using the M3C2 algorithm principle. As a particularly dangerous geometric anomaly of hyperboloid cooling towers, shell ovalisation is detected with an innovative analysis of the bimodality of the frequency distribution of radial deviations in horizontal cross-sections. The concept of a complete assessment of the geometry of a hyperboloid cooling tower was devised and validated using three measurement series of a structure that has been continuously in service for fifty years. The results are consistent with data found in design and service documents. We identified a permanent tilt of the structure's axis to the northeast and geometric imperfections of the hyperboloid shell from -0.125 m to +0.136 m. The results also demonstrated no advancing deformation of the hyperboloid shell over a two-year research period, which is vital for its further use.

Adapting sigmoid functions for hydrogel swelling curve prediction with neural networks

AbstractStimuli‐responsive hydrogels are representatives of smart materials with enormous swelling capability. The prediction of discrete hydrogel swelling states by artificial neural network based on the processing parameters has been realized in our previous work. In the current study, we explore ways to enhance the prediction capabilities by integrating physical information of the swelling curves to the model: Instead of predicting discrete swelling states, we predict the mathematical parameters of the continuous swelling curve. We therefore assume that the swelling behavior of hydrogels is consistent with a sigmoidal tanh function, based on their physical properties. After predicting the parameters of the sigmoidal function with the trained model, we analyze the prediction accuracy in comparison to the initial discrete prediction. Moreover, the new approach then allows us to physically interpret the different material properties, which denote sensitivity, maximum achievable differential swelling, and the position of the reference point for derivation of a material model.

Design of the Elastic Modulus of porous lattice structures composed of cells with continuously variable cross section carrying structures

BackgroundPorous bone implants have a wide range of applications for their low elastic modulus and good connectivity. It is necessary to explore an elastic modulus control method that can significantly regulate the elastic modulus under the condition of maintaining a constant porosity. MethodsFor achieving continuously changing elastic modulus of porous lattice structure, the simple cubic lattice structures were selected as research object, and the distribution of cross-sectional sizes of its carrying structures were set as variable continuous curves. The prediction model for the elastic modulus was established based on the elasticity mechanics and the equal mass assumption. Then, the prediction model is enhanced through compression simulation of the unit cell structure. Finally, the accuracy of prediction model is validated by compression experiments. FindingsThe results indicate that the distribution of cross-sectional size of the carrying structures has a significant impact on the elastic modulus of unit cell structures under the constraint of equal mass. By adjusting the characteristic parameters of distribution curves, the elastic modulus can be changed within a large range. InterpretationVariable cross-section can effectively change the elastic modulus of porous structures while ensuring constant porosity. This method has important value in decoupling the influence of geometric parameters on the elastic modulus of porous structures.

Applied for Higher Order Bezier Curve using MATLAB

A Bézier curve is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. The piecewise nature of a Bezier curve means that its representative equation is a linear combination of Bezier of particular degrees. The aim of this study is to infer Bézier curve graphs from higher orders using MATLAB. Where the applied analytical method was used I have reached .Will be clarified, and the following results have been reached is used in the linear curve association to express the distance ( on the line between the points and , In a second-order Bezier curve, we can create intermediate points such as and . These points have a value located in the interval , To obtain a Bezier curve for higher degrees, note that the idea in arriving at drawing the curve is to divide each line by placing a point on it and then connect the new points to each other (note that we get rid of a point at each stage), and we repeat the process until we have only one line left. So, we put ( on it and then we draw the curve. By comparing the manual solution and the solution using MATLAB, we find that the solution using MATLAB is faster and more accurate, especially at higher levels. And the study recommends the following : It is noted that the curve lies completely within the convex closure of the points, so these points are used with ease to form the required curve, it is also possible to apply geometric transformations (such as rotation and sliding) to the curve by applying this transformation to the control points of this curve.