Fractional Differential Theory Research Articles

Theoretical Analysis of Viscoelastic Friction System Characteristics of Robotic Arm Brake Based on Fractional Differential Theory

In engineering practice, the nonlinear vibration effect can easily lead to chaos in the system, which will not only reduce the performance of the system but also lead to premature fatigue of components, control failure, and increased safety risks. In view of the core position of the robotic arm in modern industry, this study relies on the robotic arm brake system to explore the theoretical basis of integrated viscoelastic materials as a vibration isolation layer. By analyzing the dynamic characteristics of the friction braking system with fractional differential terms, it aims to provide a new perspective for understanding and controlling the chaotic phenomena of a class of nonlinear friction systems. Firstly, we construct a model of a friction system and analyze its dynamic characteristics in detail. The self-excited vibration of the system under disturbance is studied. The relationship between amplitude and frequency is calculated by a nonlinear approximate analytical algorithm, and the accuracy of this relationship is verified by a numerical algorithm. Then, we compare the differences between non-fractional systems and fractional systems. It is found that with the increase in the fractional order term, the vibration amplitude of the system decreases significantly, which helps to reduce the nonlinear characteristics generated by the friction system and narrow the range of unstable solutions. Secondly, we also study the influence of parameter coefficients on the amplitude–frequency characteristics and analyze the local static bifurcation characteristics through singularity theory. Finally, we study the dynamic bifurcation behavior under different parameter perturbations and find that the change in system parameters will lead to the alternation of periodic motion and chaotic motion.

A rock creep model based on Atangana-Baleanu fractional derivative with coupled damage effects of pore water pressure, stress and time

Pore water pressure is an important factor affecting long-term rock failure and instability. On the basis of the assumptions that the rock microunit strength follows Weibull distribution and the rock failure compliances Drucker-Prager criterion, a pore water pressure-stress coupled damage (PSCD) variable is constructed in this paper. Considering the effect of pore water pressure, stress and time (PST), a pore water pressure-stress-time coupled damage (PSTCD) variable is put forward by introducing the damage variable considering time effect. Based on the characteristics and advantages of fractional differential theory in viscoelastic analysis of rock materials, one-dimensional and three-dimensional pore water pressure-stress-time coupled damage fractional (PSTCDF) creep models are proposed by Atangana-Baleanu (AB) fractional derivative. By fitting and analyzing the laboratory experimental data of rock creep under the combined action of PST, the relevant parameters for PSTCDF creep model can be deduced. The analysis of the goodness of fit and the generalization accuracy prove the rationality, validity and generalization of the proposed model. The sensitivity of the PSTCDF model is analyzed by different the stress level, pore water pressure, fractional derivative order and damage variable coefficient. The results show that the creep model changes flexibly when the model parameters change, and the specific creep state of rock can be reflected by reasonable selection of the model parameters.

Settlement Analysis of Fractional-Order Generalised Kelvin Viscoelastic Foundation under Distributed Loads

A solution is proposed for ground surface settlement induced in fractional-generalised Kelvin semi-infinite space by distributed loads, based on the fractional differential theory. The effects of four main parameters—the differential order, the two shear moduli and the coefficient of viscosity—on the settlements are analysed using a numerical example, and a parametric-sensitivity analysis is conducted. The results show that the fractional-order generalised Kelvin model is more flexible than the conventional integer-order generalised Kelvin model since it can account for the rate of the deceleration creep phase; therefore, a wider range of mechanical properties of viscoelastic materials can be described with fewer parameters, and the differential order has a higher sensitivity than the other three parameters. Finally, the model is used to identify and fit the parameters to the data of the field-bearing plate rheological tests. The fit results of the fractional-order generalised Kelvin model, unlike those of the integer-order generalised Kelvin model, are closer to the measured results and can more accurately describe the rock’s rheological behaviour at the test location.

Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations

Abstract This article combines the fractional differential theory with the total variation method and applies it to defining cultural industry image color matching. At the same time, we propose a new image color matching denoising model based on fractional partial differential equations. The model achieves simultaneous denoising in the time direction and the spaceplane. Experiments have proved that the fractional partial differential equations method has more advantages than integer-order partial differential equations in denoising and reducing step effects. It can effectively improve the contrast and clarity of image color matching in the cultural industry.

A Novel FastSLAM Framework Based on 2D Lidar for Autonomous Mobile Robot

The autonomous navigation and environment exploration of mobile robots are carried out on the premise of the ability of environment sensing. Simultaneous localisation and mapping (SLAM) is the key algorithm in perceiving and mapping an environment in real time. FastSLAM has played an increasingly significant role in the SLAM problem. In order to enhance the performance of FastSLAM, a novel framework called IFastSLAM is proposed, based on particle swarm optimisation (PSO). In this framework, an adaptive resampling strategy is proposed that uses the genetic algorithm to increase the diversity of particles, and the principles of fractional differential theory and chaotic optimisation are combined into the algorithm to improve the conventional PSO approach. We observe that the fractional differential approach speeds up the iteration of the algorithm and chaotic optimisation prevents premature convergence. A new idea of a virtual particle is put forward as the global optimisation target for the improved PSO scheme. This approach is more accurate in terms of determining the optimisation target based on the geometric position of the particle, compared to an approach based on the maximum weight value of the particle. The proposed IFastSLAM method is compared with conventional FastSLAM, PSO-FastSLAM, and an adaptive generic FastSLAM algorithm (AGA-FastSLAM). The superiority of IFastSLAM is verified by simulations, experiments with a real-world dataset, and field experiments.

Digital image recognition based on Fractional-order-PCA-SVM coupling algorithm

This paper use the fractional differential mask operator to describe and deal with the highly self similar digital medical image, then extracting the main features of the these images with principal component analysis, and recognizing these images with SVM algorithm. To provide an efficient algorithm for medical image processing with high degree of self similarity. Based on the analysis of fractional differential theory, principal component analysis theory and support vector machine (SVM) algorithm, designing a Fractional-order-PCA-SVM coupling algorithm for digital image recognition. and taking the digital image in ORL face database as the empirical object, using PCA-NN coupling algorithm, SVM algorithm, PCA-SVM algorithm, Sobel-PCA-NN algorithm, coupling coupling Sobel-SVM coupling algorithm, Sobel-PCA-SVM algorithm, Fractional-order-PCA-NN algorithm and coupling coupling Fractional-order-SVM coupling algorithm as the comparison algorithm, creating four experiments with the same sample, regarding the speed of operation and the accuracy of recognition as the criteria of high evaluation, verifying the superiority of the Fractional-order-PCA-SVM coupling algorithm designed in this paper. The average run time of the Fractional-order-PCA-SVM coupling algorithm only 4.152 s in the four experiments, although the speed of operation is not the highest, it also meets the demand of identification. The reason that the operation speed of the algorithm slightly lower than the three control algorithms is that the three algorithms are monotonic; The average accuracy rate of Fractional-order-PCA-SVM coupling algorithm is 99.2425% in four experiments, significantly better than the eight comparison algorithm; For digital medical image recognition the coupling algorithm designed in this paper is effective.

Research on Fingerprint Image Enhancement Algorithm Based on Adaptive Fractional Differential

Image enhancement is the most important part of fingerprint recognition. In order to improve the accuracy of fingerprint recognition, a fingerprint image enhancement algorithm based on fractional differential theory is proposed in this paper. This algorithm improves the traditional fractional differential form to improve the calculation accuracy and constructs a high-precision fractional differential mask. And the differential order is selected point by point according to the texture around the pixels. Compared with the methods in the references, the experimental results show that the enhancement effect is improved by about 7% in terms of average gradient.

Depth Image Enhancement and Detection on NSCT and Fractional Differential

For enhancing the low contrast and detecting the noise in depth image from Kinect sensor, the Non-Subsampled Contourlet Transform (NSCT) and non-linear fractional differential are studied in this paper. Firstly, on the basis of the NSCT’s advantages, the depth image is decomposed into low frequency and high frequency. The low frequency component is applied to enhance depth image using adaptive scale Retinex, and the scale parameter is adjusted by local mean and standard deviation. The high frequency component is calculated by Non-Local-Means operator to preserve the texture detail. The final enhanced image can be achieved by inverse NSCT. Secondly, the fractional differential theory is studied to accomplish noise detection. The experimental results show that the proposed method can enhance the depth image and detect noise effectively.

Reconstruction of surface errors of the metallic waveguide based on electromagnetic scattering theory

First, the fractal model of surface errors was constructed by means of the Monte Carlo method, and the source term and spectral amplitude of surface errors were calculated. The reconstruction model of surface errors was then established. The non-differentiable problem of the fractal model was solved by the fractional differential theory. To verify the accuracy of proposed models, a waveguide filter was manufactured. The reconstruction results obtained by the fractal function model were compared with the measured results, and a comparison of the results obtained by different reconstruction models proves that the fractal function is optimal in modelling.

Noise suppression for Lamb wave signals by Tsallis mode and fractional-order differential

To suppress the noise and increase the accuracy of feature extraction for ultrasonic Lamb wave signals, we present a new method based on the Tsallis mode and the fractional-order differential in this paper. Firstly, the fractional-order differentials of the amplitude spectrum of the noisy Lamb signal at different orders are obtained by using the fractional differential theory. Then, the cubic polynomial between the peak amplitude and the derivative order, and that between the peak frequency and the derivative order are proposed based on the Tsallis mode. The characteristic parameters of the amplitude spectrum are extracted with the developed equations. Finally, the Lamb signal without the noise is restored by combining the amplitude spectrum with the phase spectrum. Simulated and experimental results show that the proposed method can improve the performance parameters such as mean square error, r, and signal-to-noise ratio. Consequently, the new method based on the Tsallis mode and the fractional-order differential has the effective noise suppression performance for Lamb wave signals.

A Method for Resolving Overlapped Peaks in Laser-Induced Breakdown Spectroscopy (LIBS)

Spectral peak overlapping is a basic problem in analytical data processing of laser-induced breakdown spectroscopy (LIBS). Curve fitting is the typical method of resolving overlapped peaks. For preventing ambiguous fitting, appropriate initial values must be known. The aim of this work was to present a method that could be used to determine appropriate initial values of the curve-fitting method by using fractional differential theory. According to the variation of characteristic points of Lorentzian peaks at different fractional differential orders, parameter estimators were obtained that were used to calculate the initial values of the curve-fitting method. As it is a widely used optimization method, the Levenberg-Marquardt method was used in curve fitting. Simulation and LIBS experimental results proved that the proposed method of the initial value estimation can effectively resolve the overlapped peaks in LIBS data processing.

Fractional differential algorithms for rock fracture images

It is a new research topic that fractional differential theory is used into image processing. This paper presents a new type of algorithms to improve the fractional differential Tiansi operator, which can significantly enhance the edge detection result. The studied algorithms are based on the enhancement ability of fractional differential to image details, and they can be used to analyse the properties of fractional differential. The general procedure of the algorithms is as follows: firstly, Tiansi template is divided into eight sub-templates with different directions around the detecting pixel, and then the eight weight sum values for the eight sub-templates are obtained. Furthermore, those eight weights are classified into different groups. In this way, the three improved algorithms with different enhancing ranges are obtained. Finally, the experiments of edge detection show that the improved algorithms can obtain edge information more effectively and can show much more detailed information than traditional edge detection operators especially for the images of fine edges such as complicated rock fracture images.

Feature matching of scale invariant feature transform images based on fractional differential approach

The fractional differential approach can strengthen and extract textural features of two dimensional digital images;therefore,the digital image feature can be enhanced and extracted more easily.In order to improve the accuracy of image matching,based on fractional differential theory,an improved Scale Invariant Feature Transform(SIFT) matching algorithm was proposed.Combining the Gauss filter with the fractional differential filter to enhance the feature of image,more extrema and keypoints could be detected.Compared with the original SIFT,the simulation results show that the proposed algorithm can detect more key points,and improves the accuracy of image matching.