Noise reduction is one of the most important topics of digital image processing and despite the fact that it has been studied for a long time it remains the subject of active research. In the following work, we present an extension of the Mean Shift technique, which is efficiently reducing the Gaussian noise, so that it is able to cope with the impulsive disturbances. Furthermore, the elaborated technique can be applied to enhance the images corrupted by a mixture of strong Gaussian and impulsive noise, severely decreasing the quality of color digital images. By means of our approach, which is based on a novel similarity measure between a pixel and a patch located in the center of the processing block, even heavily disturbed images can be effectively restored, which enables the success of further stages of the image processing pipeline. We evaluate the efficiency of the proposed method using a publicly available database of test color images and compare the restored images applying a set of standard quality metrics with the results delivered by state-of-the-art denoising methods. Additionally, we compare our method with the Medoid and Quick Shift techniques, accelerating the original Mean Shift algorithm, in terms of objective quality criteria and computational complexity. The results of the performed experiments indicate that the proposed technique is superior to the widely used denoising techniques and can be used as a robust extension of the Mean Shift procedure. In the paper, a particular emphasis is placed on the ability of the presented algorithm to preserve and enhance image edges. The performed experiments evaluated with the use of the Pratt’s index, quantitatively confirm the superiority of the proposed design over the Mean Shift and standard denoising methods. The preservation of edges and even their sharpening is a very important feature of our algorithm whereas the final goal is segmentation, playing a crucial role in various computer vision tasks. The proposed algorithm is intended for the mixed noise reduction in color images, but it can be also applied in multispectral imaging and clustering of multidimensional data. To enable the comparison of our method with the standard denoising techniques and to help applying it in other image processing fields, we made its code freely available. Noise reduction has been one of the paramount topics in digital image processing for decades. Over recent years, mainly due to the ubiquity of multimedia devices and popular applications for capturing, uploading and sharing color images through social networks, research focused on improving the quality of digital images has rapidly gained in importance.3. The demand for new, efficient noise reduction algorithms is also influenced by novel multi-channel imaging techniques, particularly in medicine and biology, astronomy, earth observation and industrial applications. In all imaging modalities, noise suppression plays a crucial role as in general, improving the quality of images by reducing distortion levels, implies the success of further processing and analysis5.The resolution of smartphones, webcams, video recorders and other devices that are used for the acquisition of visual information has notably increased and despite the fact that less and less light falls on the photosensitive sensors, images are expected to be of an excellent quality. Therefore, the development of effective and fast algorithms to reduce noise, improve contrast, color saturation and increase tonal dynamics is gaining more and more significanceIn the undermentioned paper we focus on the Mean Shift (MS) technique, which is an effective method of reducing the impact of noise-induced distortions that can smooth out the noise while preserving image edges. Owing to the ability of MS to flatten the color images, it is also used as a pre-processing stage in image segmentation tasks. This method works successfully when enhancing images corrupted by Gaussian noise, but it severely fails when impulsive noise is also present, as the outlying pixels are being preserved. Hence, to exploit the capabilities of the MS method, very often a suitable impulsive noise removal technique has to be applied firstly and afterwards, the MS is performed to reduce the remaining Gaussian noise component.http://denoising.net/ and also accessible as electronic supplementary material6, containing a variety of color test images contaminated with different noise intensity levels, and the obtained results show that it significantly outperforms the classical MS algorithm and its modifications. It is worth stating at this point that the comparison of the proposed robustified MS with state-of-the-art techniques also confirms its satisfying denoising performance. Using the elaborated technique, noise is efficiently suppressed even in heavily corrupted images, edges are substantially sharpened and details are retained.To alleviate the problems caused by the impulsive noise corrupting the images, we propose a robust modification of the standard MS technique. Our method may work directly on the noisy color images distorted by mixed impulsive and Gaussian noise without the need to remove the impulses at first and then, reduce the remaining Gaussian noise component. In addition, it offers very satisfying denoising results. As the proposed Robust Mean Shift (ROMS) can be directly applied to restore images corrupted by even strong mixed Gaussian and impulsive noise, it can be used in various practical imaging tasks. The described technique has been validated on a database available for download at 8 allows to successfully restore color images corrupted by mixed impulsive and Gaussian noise using the MS based procedure without the need of any preprocessing. We provide the results of simulations which can guide the users while setting the parameters of the new filter. We also show that the proposed denoising framework is able to enhance image edges that can be crucial in many practical applications.In this work we also prove that the application of the pixel to patch similarity concept introduced inThe rest of the paper is structured as follows. First, in “11.Various kinds of noise degrade the quality of digital images, introducing distortions of its original content. Common noise sources include malfunctioning pixel elements in the camera sensors, thermal noise of the photosensitive matrix, shot noise caused by fluctuations in the photon flux, faulty memory locations or bit errors in hardware, timing errors in analog-to-digital converters, noise generated by electromagnetic interference and atmospheric turbulence as well as errors caused by imperfect optics and transmission. The resulting image degradations are frequently modelled as a mixture of additive white Gaussian noise, mainly responsible for dark and shot currents, and impulsive noise generating pixels with random channel intensity values13 takes into account the radiometric and topographic closeness between pixels and can effectively suppress image distortions while preserving edges. Nonetheless, it fails in the presence of impulsive disturbances, as the corrupted pixels are being preserved. The ability of the bilateral filter to enhance edges was extended by incorporating a guidance image, which can be the processed image itself or an output of a robust filter capable of removing impulsive noise15, so that the mixed noise can be better suppressed.Standard Gaussian noise filtering schemes estimate the intensity of the processed pixel or its color channel values, considering its similarity to surrounding pixels belonging to a local processing block. The popular Bilateral Filter (BF)16 compares pixel patches instead of single pixels and by using a local measure of similarity between the patches, image details and edges are better preserved. Though, impulsive noise severely distorts the distance between the patches and, since as in the BF, pixels are also compared to themselves, it results in the inability of NLM to eliminate outlying pixels.The efficiency of BF was considerably increased by the use of additional information about the structure of a pixel neighborhood. The Non-Local Means filter (NLM)18 which is exploiting the image local sparse representation in the transform domain23 and dampen the Gaussian noise operating on a 3D stack of the local patches from the sliding filtering block, applying a collaborative filtering-based shrinkage strategy. The Anisotropic Diffusion filter24 also cannot handle the impulses, as the local gradients between an impulsive pixel and its neighbors are high, which remarkably slows down the diffusion process and preserves the outlying pixels25.The impulsive noise is also affecting the efficiency of the Block-Matching and 3D Filtering, (BM3D)30. The rationale behind such an approach is that the nature of the two noise degradation types is distinct and they are easier to manage when considered separately. Nevertheless, the Gaussian noise, through its masking effect, diminishes the accuracy of impulse detection methods, and in the following processing stage the remaining noise, which characteristics is generally far from the assumed Gaussian distribution, is not satisfactorily attenuated, with visible artifacts and some amount of impulses still unfiltered.To alleviate the above-mentioned problems, numerous image restoration approaches were designed to first remove impulsive noise and then, to apply an efficient filter intended for the reduction of the Gaussian noise component33. In34 the authors proposed firstly detecting the impulses using the Adaptive Center-Weighted Median Filter (ACWMF) and replacing them with Adaptive Median Filter (AMF). Subsequently, the BM3D filter is applied to smooth out the Gaussian noise component and finally, the previously detected outliers are corrected using an inpainting method based on the median filter. The ACWMF and AMF impulsive noise filters were also used to detect the outliers when denoising the image using blind inpainting methods39 and were applied in35 to ensure the robustness of a method based on sparse image representation model.The Mean Shift technique, which extension is the subject of the following paper, can be used to detect impulsive noise. As the MS preserves the outlying pixels, they can be removed in the successive filtering step and the remaining image disturbances can be reduced with a suitable standard technique designed to cope with Gaussian noise40 combines the direction weighted median filter aimed at eliminating impulsive noise with the BM3D. The filtering result is further refined, detecting the remaining noise, which was not removed in the second processing stage. In42 the impulsive noise was removed with an adaptive averaging filter and the remaining distortions were suppressed with BM3D. In43 the Robust Outlyingness Ratio, which is a local statistic capable of efficiently detecting outliers, was used to remove impulsive noise disturbances and then the NLM filter was applied, adapting its parameters to the remaining noise characteristic. In44 the impulses were first detected by sorted quadrant median vector28 and the unfiltered noise was smoothed out by BF, which parameters were tuned on the basis of an estimation of the mixture noise composition and its intensity. The method of mixed noise reduction described in45 firstly detects the impulses and the subsequent stages use filters utilizing the PCA technique.Another technique30 consists of a fuzzy impulsive noise removal filter followed by additive noise reduction and a final postprocessing step. Once the impulsive noise is suppressed, a method based on sparse representation and 3D-processing performed with the use of DCT is applied. In the end, BF and an edge restoration technique is employed. An approach based on fuzzy logic was elaborated to cope with the problem of impulsive noise detection in46. The outlier detection scheme, which compares the central pixel with its neighbors, is designed to prevent the filtering of noise-free image pixels. Then, a weighted averaging scheme incorporating the measure of pixel impulsiveness is used to suppress the noisy pixel. A fuzzy-based switching technique for impulse detection was also devised in47. The proposed scheme builds fuzzy membership functions based on the local pixel similarity to its neighborhood and the results of median processing.The approach proposed inThe efficiency of the two-stage approaches are generally not satisfactory as the methods require setting proper values of parameters dependent on the mixed noise characteristics and intensity. They are also computationally more demanding and are prone to produce visible artifacts. Therefore, many robust methods capable of suppressing in one step, both impulsive and Gaussian noise, have been developed. Primarily, these methods combine techniques of outlier detection with existing filtering designs so that the undesired influence of impulses can be eliminated or at least diminished.49. The ROAD statistic was applied in the construction of the Trilateral Filter, which incorporates it into the BF framework53. The extension of the ROAD statistics was also applied for impulse detection in the method proposed in54, which restores the corrupted pixels with an interpolation scheme based on the radial basis function.A reliable statistic used to estimate pixel impulsiveness is the Rank Ordered Absolute Difference, (ROAD) which calculates the sum of distances between a pixel and its most similar neighbors from a filtering window55. As the patches are contaminated by impulsive noise, the noisy pixels are assigned a weight using the ROAD measure57. The weights are utilized when calculating the similarities between the patches, which are used to build the weighted average of pixels from a processing block. The proposed approach proved to be more effective when the spatial difference is considered, like in the BF58. The authors of59 propose applying in the first stage the traditional NLM technique and then, removing the remaining distortions again with NLM, which coefficients depend on the difference between the pre-denoised result and the noisy image, and also take into account the result of the first denoising step.A robust modification of the NLM filter was proposed in60. First, the intensity of the mixed noise is estimated using the ROAD measure, then the central patch of the processing block is compared with all other patches and the most similar are chosen and finally, a given pixel is denoised by applying a maximum likelihood estimator. The extension of the ROAD statistics to color image processing was also developed in62. The ROAD statistics, obtained using various kinds of Minkowski distance in the RGB color space, are used both, as a measure of noise distortion and also, as its similarity to neighboring pixels and are exploited in the construction of a fuzzy filter which parameters are tuned to the noise characteristic. Another simple fuzzy filtering approach is based on the weighted vector median filter and the similarity between pixels expressed by ROAD63. The authors of64 proposed a fuzzy filter which can remove mixed noise in color images. The elaborated method relies on fuzzy rules based on the aggregated distances between the processed pixel and its neighbors.The patch-based method was also applied in65 and in66 an edge-preserving image denoising framework based on wavelet transform was also proposed. To successfully suppress noise distortions, a locally adaptive patch-based thresholding scheme was applied. The discrete wavelet transform coupled with the double window median filter was also used in67. The method utilizes row and column windows to perform median filtering and the wavelet coefficients are thresholded to suppress the noise-induced disturbances. Another technique developed in68 first smooths the image with a Gaussian or bilateral filter and then, examines the difference between the noisy and filtered images. To suppress the noise and retain the image details, the wavelet decomposition and thresholding were performed. The final restoration is achieved by using wavelet reconstruction, which efficiently estimates the image noise component.A wavelet multi-scale analysis, combined with the local averaging filter was applied in69. The performed experiments confirm the competitive performance of the new technique which can be applied to video denoising. In70 the image denoising is formulated as an optimization problem that is solved iteratively by a weighted basis pursuit in the closed affine subspace. The reconstruction of the extracted noisy patches is performed by sparse representation using two dictionaries built with the DCT. The weighted encoding with the sparse non-local regularization technique (WESNR) was also applied in71 to cope with mixed noise. The noise-corrupted image patches were encoded over a set of pre-learned local PCA dictionaries and coding residuals were weighted adaptively to evaluate the pixel corruption measure. Additionally, both image sparsity and non-local self-similarity priors were combined into a single sparse regularization term. Another method depicted in72 first performs the initial denoising with the filter described in55. Then, the image pixels are classified as corrupted by impulsive or Gaussian noise and the final output is obtained using a variational approach. A method based on the total variation75 with 76. Although designed for impulsive noise, it can efficiently reduce various mixtures of noise models, too.An algorithm using clustering-based sparse representation exploiting both sparsity and non-local self-similarity was designed in80 which enables the construction of two-stage filtering designs. In81 various impulsive noise detectors based on morphological operators were evaluated and the remaining noise was suppressed using morphological smoothers. A combination of the fuzzy approach to noise reduction with morphological operations was presented in82. The experiments revealed that the elaborated approach yields very promising results for highly contaminated images.Impulsive noise can be also detected using the methods developed within the framework of mathematical morphologyThis brief review of the literature shows that mixed noise reduction methods can be divided into two classes. The algorithms belonging to the first class detect the impulses first and then reduce the remaining Gaussian noise. The second class of algorithms allows for simultaneous removal of both types of disturbances by introducing robust mechanisms of the suppression of outliers introduced by impulsive noise.The analysis of the denoising algorithms shows that modifications of methods which are intended to reduce Gaussian noise, generally achieve good efficiency. Therefore, in this work we present a modified Mean Shift algorithm based on the robust pixel to patch similarity measure, which can be used to extend other Gaussian noise reduction methods such as BF, NLM or BM3D to diminish their susceptibility to impulsive disturbances.83 and has been further investigated by the authors of84, however its immense popularity in image processing is due to the seminal work presented in85. This mode-seeking algorithm is based on the widely used kernel smoothing technique86 and features some similarity with the popular k-means clustering algorithm and the image smoothing approaches based on bilateral filtering and nonlinear diffusion88. Therefore, it is frequently applied for edge-preserving noise reduction89, object tracking and color image, and video segmentation92.The Mean Shift is a powerful nonparametric iterative technique which is used for finding the local modes of a given density function. It was introduced in94. To circumvent the effect of outlying pixels preservation, the impulses can be first removed using a method adapted to the impulsive noise intensity and the remaining pixels are later restored using the MS iterative scheme95.The MS technique is fairly effective in enhancing color images disturbed by the low-intensity Gaussian noise. It may smooth out the undesired noise component while minimizing the loss of sharpness of the image edges. Alas, this technique fails completely when the image is also corrupted by heavy-tailed noise, as the introduced impulses are treated as local modes and can be removed only when depleting the image contrast, which leads to its unacceptable blurry appearance. The inability of MS to remove impulses can be used for their detection, which enables their further treatment by a selected interpolation methodn multivariate observations l-dimensional space 96Let us recall the structure of the MS technique, assuming a sample In practice, the radially symmetric kernel defined as Introducing the function Mean Shift denoted as The gradient of the density estimator entclass1pt{minima9785 and the process is terminated when the magnitude of the mean shift vector satisfies The mean shift procedure works in an iterative way. The observation 101100 and in the iteration The mean shift procedure defined by  can be fpression 100 and i102. This modification assumes that only the points that belong to the set To speed up the process of finding the local modes, the Medoid Shift algorithm, which is based on the iterative procedure formulated in , has beeAs a result, the trajectory of ribed by .99. The authors put forward a method named Quick Shift (QS), which is defined byAnother approach aiming to speed up the iterative mode finding procedure was proposed inUsing the Quick Shift, each initial point is moved to its nearest neighbor, for which the density function is increased and this process is repeated. As might be observed, if a global mode is present, then it is the final point of every trajectory that originates from the initial starting point In regard to color images, the mean shift procedure operates in 5-dimensional space, k-th smallest Euclidean distance between vector In order to process an image pixel The RLSF filter output is defined asR using Eq.  measure and the Multi-Scale Similarity (MSSIM) were used to better express the image restoration quality in consistency with subjective ratingshttp://denoising.net/ and also included in6, to determine the best tuning parameters of the new method. Initially, the influence of the processing block r of the block The filter efficiency has been tested on 100 color test images of resolution 640r and For low noise level, a processing block of the size of 85,Mean-Shift Filter, (MS)99,Quick-Shift Filter, (QS)100Medoid-Shift Filter, (MEDS)87,Bilateral Filter, (BF)16,Non-Local Means Filter, (NLM)111,Vector Median Filter, (VMF)17,Block-Matching and 3D filtering, (BM3D)112,Patch-based Approach to Remove Impulse-Gaussian Noise, (PARIGI)71,Weighted Encoding with Sparse Nonlocal Regularization, (WESNR)76,TV-based restoration method with 113.Guided Bilateral Filter, (GBF)For the MS filter, we tested three variants which treat in a different way the weight assigned to the central pixel of the processing block. As for the central pixel of the block the similarity to itself is considered, the assigned weight always takes on the maximum value 1, independently of the pixel RGB channels and the block structure. When this pixel is corrupted, then, even though its role in the averaging process should be diminished, it is still considered as deserving the highest weight, which makes that the impulses are mostly retained. The only way to force the algorithm to decrease the influence of the central pixel of the block is to set a high value of the parameter 11585, considering all pixels in the filtering block, ,s in Eq. , I and This measure is defined as followsFigures The obtained results also exhibit the tendency of the WESNR to produce color blotches both for impulsive and mixed noise. The inability of the PARIGI filter to suppress the strong Gaussian noise was also confirmed.119 presented in Figs. The very good edge-enhancing properties of ROMS are supported by the maps of the Vector Range edge detectorr were obtained averaging the results of 1000 repetitions.Figure In the paper a Robust Mean Shift technique, able to efficiently suppress mixed Gaussian and impulsive noise, has been presented. The new filtering framework makes use of a robust similarity measure which compares a pixel belonging to a processing block with a given number of closest samples in a filtering window in its center. Thereby, the introduced similarity measure is resilient to noise disturbances and allows to construct robust noise filtering designs.The adoption of the Mean Shift concept proved to deliver very satisfying noise restoration results, even for images degraded by impulsive and mixed noise of high intensity. The obtained results were presented using objective quality measures and indicated that the new filter outperforms the Mean Shift and also Medoid Shift and Quick Shift methods, which are its modifications developed to make the algorithm faster.The proposed ROMS excels some widely used denoising methods, like BF, NLM, BM3D or VMF, which are utilizing the Euclidean distance, in terms of standard restoration quality metrics. The proposed robust similarity measure between a pixel and a patch in the center of the processing block allows to substantially diminish the influence of the impulsive pixels. In this way, the susceptibility of the Euclidean distance to outliers can be considerable decreased, which enables to construct new methods based on the classical, well-researched algorithms.The proposed method is especially effective for images contaminated with mixed noise of high intensity. As the pixels from the filtering block are compared with the patch of pixels in its center, some tiny details are treated as noise and removed. This can be seen as a drawback when the preservation of tiny details in images with low contamination is desired. Another feature of the proposed filter is its tendency to sharpen image edges. Generally, edge enhancement is beneficial in most applications, however the creation of too strong edges can lead to image oversegmentation. To decrease the impact of this effect, the filter parameter The visual analysis of the experiments performed on a set of standard color images proved that the new filter can efficiently remove the impulsive noise while smoothing out the Gaussian noise component. The beneficial feature of the new filtering design is its ability to preserve and even enhance the image edges. An analysis performed applying the Vector Range edge detector and the Pratt’s Figure of Merit (FOM) revealed that the new technique is able to precisely locate the edges and significantly outperforms the designs based on the Mean Shift and its modifications and also other competitive denoising techniques.120.The described method can be used for the enhancement of both gray scale and color images and their temporal sequences. It can be easily extended to work on multispectral images corrupted by a mixture of Gaussian and heavy tailed noise. Possible applications include underwater imagery, virtual enhancement of works of art, restoration of old photographs, filtering of microscopy images and enhancement of multispectral satellite, astronomical and medical data acquired from various modalities. Additionally, the elaborated approach can be applied for image segmentation tasks and also for multi-dimensional data clusteringFuture work will be focused on the incorporation of the robust pixel to patch similarity measure to the structure of the Medoid and Quick Shift, so that they will be able to cope better with impulsive and mixed noise corrupting color images. Preliminary experimental results show that the adoption of an additional weight assigned to the pixels from the processing block, expressing their degree of corruption, considerable improves the denoising effectiveness of the developed filtering framework. In this way, a new family of very fast filters utilizing the Medoid and Quick Shift concept will be elaborated.


The rheological behavior of gel propellants is crucial for their practical applications, especially in the rocket engine and ramjet fields. The thixotropic characteristics of gel propellants are an important component of their rheological properties and have a notable impact on their flow and injection process. However, most gel propellants contain rich, dynamic cross-linked network structures, which impart complex non-Newtonian fluid properties, and it is difficult to establish a unified mathematical model. In view of this, this study addresses the thixotropy of a prepared RP-3 kerosene gel and determines the mathematical model and model parameters describing its thixotropy. Experiments show that the kerosene gel exhibits shear-thinning properties as well as thixotropy. To describe the microstructural changes in the gel, three thixotropic constitutive models are introduced to analyze the rheological data, and the constitutive equation parameters are optimized. The three models are all structural dynamic models, which can be used to describe microstructural changes within the material. In addition, the fitting of the constitutive equation is a multiparameter optimization problem, and an appropriate optimization method must be used for parameter fitting. Therefore, the Bayesian optimization method combined with Gaussian process regression and the upper confidence bound (UCB) acquisition function is used in the multiparameter fitting of the constitutive models. Both experiments and numerical results show that the thixotropic model, which introduces a pre-factor with shear strain and assumes that the breakdown of the gel structure is related to energy dissipation rather than the shear rate, has a better fitting effect and prediction ability with regard to the gel. Combined with transient experiments at different shear rates, the model parameters of the constitutive law can be determined quickly by applying the Bayesian optimization method. With the increasing demand for rocket and ramjet propellants that provide good performance and safety, gel propellants have become a focus of research in recent decades. These propellants combine the advantages of liquid and solid propellants ,2,3. ForA gel propellant is a gel system formed by adding a gelling agent to a liquid propellant. The gelling agents are combined into a three-dimensional network gel structure through noncovalent intermolecular interactions ,9,10,11.The gel propellant has thixotropy due to its intermolecular network structure. Thixotropy is one of the most challenging problems in the field of colloidal rheology. Schalek et al. ,14 pionet et al.  developet et al. ,17 invesThe rheological properties of different gel propellants have been popular in recent decades. NASA’s Lewis Research Centre has carried out rheological property and stability research based on the RP-1 gel propellant and metallized gel propellant Al/RP-1 . Their rGenerally, thixotropic models can be classified into three categories: phenomenological models, microstructure models and structural kinetic models ,12. By cKerosene gel propellant with the addition of metal particles could be used for increasing the specific impulse of the rockets, however, this will be addressed in the next study. In this work, the kerosene gel propellant without the addition of metal particles is considered because of the following reasons: (i) With the addition of metal particles, the characteristics of kerosene gel is more complicated; moreover, metal particles are flammable or toxic; as a result, the research process safety requirements are reduced, which is not easy to realize; (ii) the mechanical properties of kerosene gel are complex, and the research process of thixotropy requires a large number of experiments to obtain stable and accurate experimental data, and then most of the theoretical models need to fit to get a reliable model parameter. By a study on relatively safe kerosene gel without metal particles to carry out system mechanics research, we constructed the research methods for kerosene gel mechanical property measurement, modelling and model fitting, which provide a solid foundation for the research of kerosene gel containing metal particles, thus shortening the corresponding research time in future; (iii) organic kerosene gel without metal particles also has application in some rocket propulsion systems, such as a kerosene gel ramjet engine, in which kerosene gel is used to replace kerosene to improve the safety of the equipment during transportation, storage and operation.The rheological properties of gel propellants have been widely studied both numerically and experimentally; however, there have been few studies on their thixotropic properties, and only qualitative analysis of the thixotropy of gel propellants has been carried out. Therefore, this study focused on the thixotropy of gel propellant, and quantitatively analysed its thixotropy characteristics through experimental measurement and three different structural dynamic thixotropy models. Combined with the Bayesian optimization method, the mathematical model and model parameters describing the thixotropy could be quickly determined. Subsequently, the researchers configured various gel systems containing thixotropy for theoretical and applied research on thixotropy. Coussot et al. developed a colloidal system using a bentonite suspension to investigate viscosity bifurcation in thixotropic, yielding fluids. Pramanik and Zanna et al. investigated the application of thixotropic hydrogels in the biomedical field.The fitting results are presented and discussed in this section. First, some parameters in the equations are determined according to the experimental measurement results. Then, combined with the transient measurement results, all the remaining parameters in the constitutive models are provided by using the numerical fitting method. The obtained parameters are brought into the equilibrium equation to obtain an equilibrium viscosity curve and test its fitting effect on the equilibrium viscosity value. Both the transient viscosity fit results and the equilibrium viscosity fit results are used to evaluate the model fit ability. Finally, the prediction ability of the models is verified by the viscosity change curve of the gel under the step–shear rate.The yield stress of the gel material is obtained by directly measuring the stress–time curve, as shown in For the Toorman model, its parameter When After the gel is sheared at high speed, the stress–shear rate relationship is measured. The results are shown in After high-speed shearing, the gel behaves as a power-law fluid, and the power-law model can be used to fit the stress–shear rate relationship. The fitting process is shown in For the Houska model and the Teng model, as the structural parameter For the Toorman model, Houska model and Teng model, the remaining parameters are obtained by fitting the transient viscosity change using the Bayesian optimization method. The optimization objective is the mean relative error (MRE). In the transient process of fitting, the stage before yield is ignored, and the fitting starts from the maximum point of the viscosity–time curve at The Teng model has the best fitting effect, and the fitting results are shown in −1, 600 To further evaluate the predictive ability of the models, the obtained model parameters were compared with the step–shear rate curves obtained in the experiments. Since the Toorman model has a poor effect on the transient fitting of the gel, only the predictive ability of the Houska model and the Teng model is considered. In From the prediction results, both the Teng model and the Houska model have a certain predictive ability for the viscosity change in the gel under the step–shear rate. However, the Teng model performs better than the Houska model. Therefore, it can be considered that the Teng model combined with the parameters obtained by the Bayesian optimization method in this paper has good predictive ability for the gel.(1)Using kerosene, Thixatrol ST and absolute ethanol, the kerosene gel system was successfully prepared through water bath heating and high-speed dispersion. The system features yield, shear thinning and thixotropic properties.(2)The kerosene gel is a plastic fluid with a yield stress of approximately 11.73 Pa. The yield stress was generated by the formation of a network structure through hydrogen bonding between the gelling agents. Upon high-speed shearing, the yield stress disappeared due to the destruction of the network structure.(3)The gel has shear thinning and thixotropic properties. Hysteresis loop experiments indicated that the apparent viscosity of gels without high-speed pre-shearing decreased with an increasing shear rate. In the rising and falling stages of the shear rate, the gel viscosity showed a prominent time dependence, that is, thixotropy. The high-speed pre-shearing gel exhibited only shear-thinning properties. The power-law model had a good fitting effect on the rheological properties of the gel after high-speed shearing.(4)The Bayesian optimization method can effectively fit multiple parameters at the same time. The method has the ability of global search and local optimization, which can effectively prevent the fitting process from falling into a local optimum.(5)Three constitutive models were used to fit the rheological properties of the gel by combining with the Bayesian optimization method. The results show that the fitting of the Toorman model failed completely, the Houska model had a certain fitting and predictive ability, and the Teng model performed the best. Moreover, the Teng model effectively modelled the thixotropy of the gel and has a strong predictive ability.In this study, kerosene gel was prepared by using kerosene, Thixatrol ST and absolute ethanol. Then, the rheological properties of the gel were measured by using a rotational rheometer. Finally, the rheological properties were fitted and analysed by combining three constitutive equations and the Bayesian optimization method. Based on the investigation, the following conclusions and prospects can be drawn:The work in this paper provides a good beginning for our future research on kerosene gel. In future work, we will further consider the effects of temperature and the addition of metal particles on the gel’s constitutive model and parameters. At the same time, all the constitutive models used in this paper are viscoplastic models, which do not consider the elastic effect of the gel. In many situations, the elasticity of gel cannot be ignored and affects the flow and atomization of gel fuel flow greatly; therefore, the elasticity will be one of the key research contents in future. More significantly, it is critical to revealing the viscosity impact on rheological behaviour of gellant fluid arising from the different Newtonian base fluids, such engine oil, etc., which will be exploited in the near future.The kerosene gel was prepared in the laboratory via the fabrication method described in In this paper, three structural dynamic constitutive models, the Toorman model, Houska model and Teng model ,35,36, wThe model proposed by Toorman is shown in Equations (2) and (3).Here, The Houska model was a model proposed by Houska for the rheological properties of slurries of solid objects. The model is described through Equations (4) and (5):Teng et al.  proposedThe definitions of In this paper, the parameters in the above models that cannot be directly or indirectly obtained through experiments are obtained by the multiparameter fitting method—Bayesian optimization method. The Bayesian optimization method can fit the sample space through a small number of initial sample points and reasonable exploration methods.The Bayesian optimization method mainly consists of two parts. One is the surrogate model. The second is the acquisition function used to guide the optimization direction. When using Bayesian optimization to solve practical problems, it is important to select appropriate probabilistic surrogate models and acquisition functions. However, with regard to Bayesian optimization, there are no unified and clear selection criteria for surrogate models and acquisition functions. Moreover, this approach still relies on manual selection. In this paper, the surrogate model uses a Gaussian process model. The acquisition function is the upper confidence bound (UCB). The initial samples were generated by using the Latin hypercube sampling method. The pseudocode, optimization model and principles of the sampling method are shown in


Data-driven fault diagnosis methods for rotating machinery have developed rapidly with the help of deep learning methods. However, traditional intelligent fault diagnosis methods still have some limitations in fault feature extraction and the latest object detection theory has not been applied in fault diagnosis. To this end, a fault diagnosis method based on a sparse short-term Fourier transform (SSTFT) and object detection theory is developed in this paper. First, a sparse constraint is introduced in time-frequency analysis to improve the time-frequency resolution of the model without cross-term interference and proximal gradient descent (PGD) is adopted to quickly and effectively optimize the model to obtain a high-quality time-frequency representation (TFR). Second, a fault diagnosis model based on a region-based convolutional neural network (RCNN) is built; the model can extract multiple regions that can characterize fault features from the TFR. This process avoids the interference of irrelevant vibration components and improves the interpretability of the fault diagnosis model. Finally, multicategory rolling bearing fault identification is realized. The effectiveness of the proposed method is validated by simulation signals and bearing experiments. The results indicate that the proposed method is more effective than existing methods. Against the backdrop of high-efficiency modern enterprises, equipment failures cause large economic losses and personal safety risks ,2,3. RolVibration signals are collected from running rolling bearings and contain information about the health of the equipment. The fault diagnosis method based on vibration analysis is divided mainly into feature extraction, feature selection and fault identification ,8. It haThe aforementioned traditional time-frequency methods can be roughly divided into two categories: linear time-frequency methods (which are represented by the STFT transform and wavelet transform) and nonlinear time-frequency methods (which are represented by the Wigner–Ville distribution). The method in the latter category can characterize faults in the time-frequency domain but has obvious limitations. The methods in the former category are limited by the Heisenberg uncertainty principle, resulting in poor sparsity and low time-frequency resolution in the analysis results. The latter suffers from cross-term interference, which reduces the accuracy of the TFR.The development of sparse representation theory allows the above problems to be solved. With the application of sparse representation theory in time-frequency analysis, time-frequency methods based on sparse representation and an STFT have emerged. A time-frequency analysis problem can be transformed into a sparse optimization problem based on the L0 norm constraint, and the solution for this problem can obtain a higher resolution time-frequency representation. Although there is no interference from cross-terms, the optimization process for this model is a nonconvex optimization problem, which is a nondeterministic polynomial hard (NP-hard) problem to solve. For this purpose, this paper proposes a time-frequency analysis method based on PGD and a SSTFT. First, the convex relaxation technique is applied to relax the nonconvex optimization problem of the L0 norm to an L1 norm minimization problem and then PGD is used to solve the problem.Some satisfactory results have been acquired by intelligent bearing fault identification algorithms by using TFR as input. Zhang et al.  used theHowever, the abovementioned traditional intelligent diagnosis method based on TFR still has two inherent shortcomings:(a) When the TFR is used as input, the entire TFR is usually used as the learning object of the model and the understanding of the TFR is lacking. Many irrelevant vibration components also establish relationships with labels, reducing model interpretability.(b) As one of the cutting-edge ideas in image processing, the application of object detection theory in fault diagnosis is rarely addressed.To this end, the Faster RCNN algorithm is introduced in the traditional intelligent fault diagnosis based on time-frequency analysis in this paper; the algorithm can accurately mark the fault components from the TFR to accurately realize the fault diagnosis of rolling bearings . Faster Inspired by the expectation of addressing the abovementioned problems, a bearing fault diagnosis method based on PGD-SSTFT and Faster RCNN is proposed in this paper. The PGD-SSTFT algorithm is adopted to obtain a sparse TFR without cross-term interference from the vibration signal and the fault characteristics of rolling bearings can be displayed with high resolution. A fault diagnosis model is then built based on Faster RCNN, one of the representative algorithms in object detection theory and the model is used to learn the fault feature components from the TFR. Finally, the fault components can be accurately marked in the bearing fault samples of unknown categories and the fault type can be identified.The main contributions of this paper are as follows:(1) A time-frequency analysis model of the bearing vibration signal is built by using the STFT and sparse constraints are introduced into this model, thereby improving the time-frequency resolution and time-frequency aggregation of the time-frequency analysis method. Additionally, the model is transformed into an easy-to-solve unconstrained problem by using the PGD algorithm.(2) The object detection method is introduced in fault diagnosis and the fault feature components are more accurately and pertinently labelled from the original TFR, thereby improving the interpretability of the fault diagnosis method.(3) The effectiveness of the proposed method is validated by using the simulated signal and the actual bearing vibration signal. The results indicate that the TFR of the proposed method is more accurate than that of the traditional method and the fault identification accuracy is higher.The remaining parts are organized as follows: The short-time Fourier transform is a joint time-frequency analysis method for time-varying nonstationary signals and is one of the most basic tools for time-frequency analysis of actual signals. The basic idea of the STFT can be summarized as follows: A fixed-length window function is utilized to intercept the time-domain signal and the Fourier transform is adopted to process the resulting truncated signal, thus obtaining a local Fourier spectrum of the short period. This process is completed until the window function spectrum is translated over the complete time range and a set of local spectra is obtained for each period. Hence, the STFT is a two-dimensional function of time and frequency and can be calculated asn and the form of the signal’s discrete STFT can be defined asm. According to the inverse Fourier transform, the original signal is windowed at time m, m, The matrix formula for this transformation can be expressed asm. The short-term signal obtained after introducing the window function ism is sparse. The special spectrum The frequency domain of the windowed signal m, Finding a solution to the L0 norm is very complex; this problem is called an NP-hard problem and this process is prone to noise interference. To address this problem, an alternative method is explored in this paper. It has been proven by some studies ,31,32,33The TFR of the signal Equation  for all Since the above model is a set of convex optimization problems with constraints, the constrained problem is converted into an unconstrained problem to facilitate the solution ,35:(8)X^Let Equation  obtains Let In the general calculation process, Equation  is calleIn addition, a soft threshold function is introduced:The analytical solution of Equation  can be oAccording to the proximal gradient descent, the general iterative process for the problem of sparse time-frequency decomposition can be calculated asTo verify the effectiveness of the proposed proximal gradient time-frequency method, simulated bearing fault signals are used in this section.T represents the mean period of fault impact; The collected signals of bearing vibration in the industry are usually complex, have multiple components and have a low signal-to-noise ratio (SNR). Hence, on the basis of existing research, the faulty rolling bearings vibration signal model proposed in this section can be expressed as ,3718)xx(18)xinnSince the inner race rotates with the running rolling bearing, the fault point sometimes appears in the load area, sometimes not in the load area, but the position of the sensor is fixed. Therefore, the fault points are located in different positions, resulting in continuous changes in the magnitude and direction of the generated impulse force. Equation  is used To obtain a high-resolution TFR of bearing fault signals without cross-term interference, the proposed PGD-SSTFT is applied to the simulation signal in this paper. The window function size is 65 (sample points) and In general, rolling bearings often work in a noisy environment, so the collected signal contains considerable noise. Therefore, a signal containing Gaussian white noise is utilized to test the effectiveness of the proposed method in this paper. To further evaluate the denoising performance of the proposed method, the peak signal-to-noise ratio (PSNR) is selected as the evaluation index. Based on statistics, this method distinguishes the difference between figures by comparing pixel points one by one.In the second section, the time-frequency decomposition results of PGD-SSTFT are presented. It can be clearly seen from the results that different types of faults have different frequency components, but relying on manual identification of the fault components from the TFR is undoubtedly complex and labor-intensive. Hence, our focus is to automatically identify the fault components in the TFR.To this end, the Faster RCNN is introduced to directly extract the regions related to the fault components from the original TFR and then the identification model is established to achieve fault diagnosis. Traditional intelligent diagnosis methods usually use the entire TFR corresponding to the fault type to achieve the fault diagnosis, which lacks the understanding of time-frequency information. The difference from the traditional intelligent method is that the proposed method can automatically identify the regions containing the fault frequency from the original TFR and determine the fault type implied in each TFR. This not only considers the location of the fault frequency , but also considers the energy of the fault frequency (determined by the color displayed in the TFR). The proposed method can be divided into the following steps: data preprocessing, fault region proposal, screening algorithm and fault identification.In this paper, when processing the original time-frequency dataset, software is used to annotate the time-frequency signal and the main fault frequency features in the TFR of different fault types are selected. As shown in the figure, the regions in these boxes include existing features as much as possible and the regions selected in the box can reflect the position and energy of this frequency. This process is realized with existing software. More details on this process can be found in the results.k anchors are randomly generated on the input feature map. As illustrated in Softmax.The input TFR first undergoes feature extraction and then enters the region suggestion step. This process is completed with multiple convolution layer structures and This process can obtain the best region containing fault features but also generates many duplicate anchor regions and can thereby lower the performance of models. Therefore, regions with lower scores are deleted according to the intersection-over-union (IoU) condition. The operation can be summarized as follows: The box with the highest confidence is selected and then the box whose IoU is higher than the threshold is deleted until the best rectangle is selected to achieve the position output of the target. These selected regions are called regions of interest (RoIs).After feature extraction, the above RoIs are performed on the feature map, which cannot reflect the features in the original TFR. In addition, the above steps result in the suggested regions containing different dimensions, thus making it impossible to complete the subsequent fault classification. For this purpose, a normalization layer similar to spatial pyramid pooling is introduced to map the resulting RoIs to the original TFR and adjust the size for classification. This part consists of multiple convolutional layers and pooling layers.After the RoIs are obtained, the fault identification part is entered. The fault identification part completes two main tasks. First, the position regression calculation is performed to calculate the loss of the predicted and the actual regions. Then, the loss will be backpropagated to optimize the parameters.P is the proposed box and G is the actual box. Then, the goal of model regression is to find a mapping that satisfies Suppose the coordinates of the box are For this purpose, the prediction box is translated and scaled separately.G approach the position of the real box. Fault identification completes fault classification, which is realized by adding Softmax after the fully connected layer. The final classification part integrates all RoIs in the entire feature map, sorts the probability categories corresponding to these RoIs and outputs the category with the highest probability among all RoIs. During this process, the loss between the predicted box and the ground-truth box is calculated and the loss is backpropagated to optimize the parameters.The gradient descent algorithm is used to continuously make the predicted box In this paper, a fault diagnosis model is built on the basis of Faster RCNN in object detection theory, thus making the neural network more refined with respect to finding fault features from the TFR. As mentioned above, this method can be divided into four main parts  Vibration signals are collected by using acquisition equipment, such as sensors.(2) The vibration signal is then converted to a TFR by using sparse time-frequency decomposition based on PGD-SSTFT; then, these images are labelled.(3) A fault diagnosis mode is established based on object detection theory and the optimal related hyperparameters and model structure are determined.(4) The model is fully trained by using training samples.(5) The trained model is applied to identify test samples and then the diagnosis results are output and the model performance is evaluated.To verify the effectiveness of the proposed method, data from the chair of Design and Drive Technology, Paderborn University are adopted for testing . The maiIn this section, the structure and hyperparameters of the detection model based on cross-experiments on the above datasets are determined and the impact of these parameters on model performance is evaluated. To improve the validity of this study, cross-validation for all hyperparameters that may affect model performance is conducted for multiple average validation and analysis. All parameters include the window function size, the bound of error, batch size, base anchor size and dropout.The experiments are performed 10 times for each set of parameters for cross-validation and the average results of the cross-experiments are shown in To verify the effectiveness of the proposed method, the advanced object detection method Yolo  is chose(1) The deep belief network (DBN) is one of the representatives of deep neural networks. Therefore, we choose a convolutional deep belief network (CDBN) as a comparison method .(2) CNNs have excellent feature extraction capabilities. To improve the performance of CNNs, ResNet is chosen .(3) The stacked autoencoder (SAE) is also widely used in fault diagnosis research. To meet the requirements of the model, the input 2D image is flattened into a 1D signal .To verify the effect of PGD-SSTFT, the results of other commonly used signal processing methods are input into the model. These methods are the same as in The mean average precision (mAP), diagnosis accuracy and standard deviation are chosen as the indicators to evaluate model performance. mAP can be defined asAP is the precision averaged across all recall values between 0 and 1, which is obtained by calculating the area under the curve (AUC) of the precision x recall curve. In this paper, the mAP of each class is used to compare different detectors.After the collected signal is processed and input to the model, training and testing are performed and the training errors are shown in To better evaluate the learning ability of the proposed method, the T-SNE algorithm is utilized to visualize the high-dimensional features in the last fully connected layer of the different methods. The two-dimensional visualization results are shown in In this paper, a fault diagnosis strategy for rolling bearings based on sparse time-frequency analysis and Faster RCNN is proposed. To overcome the shortcomings of traditional time-frequency analysis methods, such as low resolution and cross-term interference, the SSTFT is employed to process the original signal. The results prove that the proposed method can effectively obtain high-resolution TFRs from bearing vibration signals and outperforms the classical nonsparse decomposition approaches. A fault diagnosis framework is built based on Faster RCNN. Multiple RoIs are used to determine the fault type of a sample, thus improving the accuracy and stability of diagnosis against traditional methods.To verify the effectiveness of the proposed method, a benchmark bearing fault dataset is employed to identify different operating states of the bearing. The results indicate that this method has higher accuracy than traditional methods, can reliably identify different bearing faults and can accurately mark the position of the fault in the TFR. In addition, the effectiveness of the sparse time-frequency analysis is also validated by the experimental results. In view of the excellent performance of the proposed method, the focus of our future research will be fault diagnosis for compound faults.


Location information is the primary feature of wireless sensor networks, and it is more critical for Mobile Wireless Sensor Networks (MWSN) to monitor specific targets. How to improve the localization accuracy is a challenging problem for researchers. In this paper, the Gaussian probability distribution model is applied to randomize the individual during the migration of the Adaptive Fish Migration Optimization (AFMO) algorithm. The performance of the novel algorithm is verified by the CEC 2013 test suit, and the result is compared with other famous heuristic algorithms. Compared to other well-known heuristics, the new algorithm achieves the best results in almost 21 of all 28 test functions. In addition, the novel algorithm significantly reduces the localization error of MWSN, the simulation results show that the accuracy of the new algorithm is more than 5% higher than that of other heuristic algorithms in terms of mobile sensor node positioning, and more than 100% higher than that without the heuristic algorithm. With the efforts of more and more researchers, many technologies have become mature and inexpensive, such as communication theory, micro electromagnetic systems, and integrated circuits. Based on these technologies, wireless sensor networks (WSNs) have been widely used, and their performance has been dramatically improved . A sensoIn a static wireless sensor network, the sensor nodes are set in a certain position and will not change, the position of the unknown node can be estimated by many algorithms based on the information of the anchor node. These algorithms are divided into two categories according to whether they rely on the distance information between sensor nodes, namely range-based localization algorithms and range-free localization algorithms ,6. Some There is a serious challenge in the positioning of mobile sensor nodes in MWSN; that is, the positioning error is huge, and with the movement of mobile sensor nodes, the positioning error will become larger and larger. To address this problems, this paper introduces a novel heuristic algorithm. Heuristic algorithm is a powerful tool to solve many engineering problems. Some scholars use the excellent performance of the heuristic algorithm in optimization to reduce positioning error. The adaptive strategy is combined with a compact Particle Swarm Optimization (PSO) algorithm, and this algorithm can run on a memory limitation device. Simulation results indicate that localization error is significantly reduced . PSO algWith the increasing attention of scholars, there are many excellent novel or improved heuristic algorithms. In previous decades, only some basic and simple heuristic algorithms were proposed and used, such as Genetic Algorithm (GA) , PSO algIn order to reduce the localization error of mobile sensor nodes, this paper combines the Sequential Monte Carlo Localization (SMCL) method and heuristic algorithm. The reason for using the heuristic algorithm is that the optimal value can be quickly calculated, which can ensure the timely positioning of the position of the mobile sensor node. This section briefly presents the mechanism of the SMCL method and AFMO algorithm.AFMO algorithm was proposed in 2020, and is a modified version of the Fish Migration Optimization (FMO) algorithm. The FMO algorithm mimics the whole life course of fish and divides the life of fish into five stages. There are many accidents during fish growth, so many individuals cannot grow up safely. In addition, these fish would return to their birthplace when adults, producing offspring. Therefore, the survival rate is introduced by authors in FMO , and thei-th individual at t iteration, i-th individual is represented by i-th individual at the t iteration, and w is a parameter that controls the individual’s range of motion, which is a variable value that decreases from 2 to 0.4 during the operation of the AFMO algorithm. i-th individual. If the i-th individual is worse than the learning object, it is close to the learning object, and vice versa. The RC is a random number between 0 and In the AFMO algorithm, as t 200 in . To enhaEneit+1=EAlthough the AFMO algorithm enhances the performance of the original FMO algorithm and obtained better results than other famous heuristic algorithms in the CEC 2013 test suit, it has disadvantages in unimodal optimization problems. This paper introduces the novel algorithm called the Gaussian-Based Adaptive Fish Migration Optimization (GAFMO) algorithm, which applies the Gaussian distribution model to the migration process of AFMO. This mechanism enhances the population diversity in the migration process of fish and ensures the exploitation ability.The authors introduced the SMCL method to enhance the localization accuracy of mobile sensor nodes of WSNs . Twenty t − 1 time point, and the circle filled with yellow is the sensor range of the node at the t time point.Sensor node localization has different challenges to solve than robot localization: 1. Sensor nodes are placed on an unknown map or terrain. 2. The speed or direction of the mobile sensor node cannot be obtained. 3. Mobile sensor nodes do not have enough energy and memory to estimate localization by integrating information collected by other sensor nodes . In 35]35], baseArrivals and leavers provide the most helpful information for the localization of a mobile sensor node as it is located around the communication boundary of the arrivals or leavers. For the cases of an outsider, the information of mobile sensor nodes can be transmitted to the outsider node by the neighbor nodes. The detail of this process is shown in In nature, the growth of fish is accompanied by a variety of adverse factors such as disease, food scarcity, and predators that prevent so many people from reaching adulthood. To simulate this phenomenon, the authors introduced a survival rate mechanism that maintains population size by randomly generating new individuals . Althougt iterations, and the i-th individual, randomly selected individual, and best individual. The positions of the i-th individual and randomly selected individual at t iterations are represented by Algorithm 1: The Gaussian-Based Adaptive Fish Migration Algorithm.In this section, a comparison of the new algorithm under the CEC 2013 test suite and locating mobile sensor nodes with other well-known algorithms is presented. The results provided by the CEC 2013 test suite illustrate the comprehensive performance of the new algorithm, which shows that the new algorithm has excellent optimization capabilities over other well-known heuristic algorithms. Localization simulation experiments of mobile sensor nodes can demonstrate the performance of the new algorithm in solving specific problems in the real world. The experiments were completed with Matlab 2020a on a personal computer with Intel Core i7-10700k (5.1 GHz) and 48 G memory, and all experiments were processed under the same parameters, such as population size, dimensions, or iterations.The CEC 2013 test suite is proposed to estimate the performance of heuristic algorithms on single-objective optimization problems, which are the basis of niche, multi-objective, and constrained optimization algorithms. This paper tests heuristic algorithms on 28 test functions of the CEC 2013 test suite to fully and fairly verify the new algorithm’s performance. The test functions were separated into three classes, which are Unimodal Functions (shown in ,25,34,39Various test functions can verify the different performances of heuristic algorithms. In order to compare the exploitation ability of algorithms, uni-modal test functions are introduced in CEC 2013. It has only one optimal solution in a limited area, so the heuristic algorithms with strong exploitation ability can obtain great candidate solutions. The experimental results under uni-modal test functions are presented in The combination function combines uni-modal and multi-modal test functions, which can verify the comprehensive performance of the heuristic algorithm. The GAFMO algorithm obtains the best results in almost all experiments, as shown in Since the new algorithm introduces a Gaussian probability distribution model based on AFMO, it has a stronger exploration performance than AFMO, which can be proved by the experimental results presented in The CEC 2013 test suite is proposed to estimate the performance of heuristic algorithms on single-objective optimization problems, which are the basis of niche, multi-objective, and constrained optimization algorithms. This paper tests heuristic algorithms on 28 test functions of the CEC 2013 test suite to fully and fairly verify the new algorithm’s performance. The test functions are separated into three classes which are Unimodal Functions (shown in ,25,34,39Various test functions can verify the different performances of heuristic algorithms. In order to compare the exploitation ability of algorithms, uni-modal test functions are introduced in CEC 2013. It has only one optimal solution in a limited area, so the heuristic algorithms with strong exploitation ability can obtain great candidate solutions. The experimental results under uni-modal test functions are presented in The composition function consisted of uni-modal and multi-modal test functions, which can verify the comprehensive performance of the heuristic algorithm. The GAFMO algorithm obtains the best results of almost all experiments, as shown in Since the new algorithm introduces a Gaussian probability distribution model based on AFMO, it has a stronger exploration performance than AFMO, which can be proved by the experimental results presented in In this section, heuristic algorithms are used to reduce the localization error of the SMCL method. The individual of heuristic algorithms represent a candidate position of the mobile sensor node. The optimal position is found by iteration of the algorithm; that is, the most probable position in the promising area. Through these simulation experiments, the performance of the heuristic algorithm to solve real problems can be verified. Experiments are performed under different conditions, such as the number of anchor nodes, sensor nodes, and the communication radius, but the deployment area is 200 m × 200 m for all experiments. The maximum speed of a mobile sensor node is its communication radius. The new algorithm is compared with the PSO, BH, and WOA algorithms, and the detailed results of these experiments are shown in the table below, with the best results for each experiment are marked in bold.In The number of sensor nodes is the variable in This paper analyzes the feature and performance of AFMO, which has an excellent performance in multimodal problems, but the strong exploration ability limits the local search ability. This means the AFMO can not obtain satisfactory results in unimodal problems. In order to enhance the exploitation performance, we introduce the Gaussian probability distribution to the migration process of AFMO. This mechanism ensures that the novel algorithm obtains better results in unimodal problems and retains the original exploration ability. The performance of the new algorithm is verified by the CEC 2013 test suit, and the experimental results show that the novel algorithm has better exploitation performance and a solid ability to avoid the optimal local value. The new algorithm achieves the 60 best results in all 84 experiments; that is, the new algorithm wins in 71.4% of the experiments. In addition, this paper applies the heuristic algorithm to solve the localization of mobile sensor nodes in MWSN. The simulation experiments reveal that the heuristic algorithm can significantly enhance the localization accuracy of mobile sensor nodes. Specifically, the new algorithm can improve the localization accuracy of mobile sensor nodes by more than 5% compared to other heuristic algorithms. This technique can also solve the localization of the robot in the room, the robot can provide more information to the localization system but there is more problems to solve than MWSN. This paper proves that the Gaussian probability distribution model can enhance the exploitation ability and not reduce the exploration ability. This model can apply other algorithms to further improve the ability of heuristic algorithms. In addition, other probability modes have their own features, and they may be more suitable for enhancing the performance of heuristic algorithms or solving localization problems. This is interesting work to do.


Background: Interlimb neural coupling implies that arm swing should be included during gait training to improve rehabilitation outcomes. We previously developed several systems for production of walking with arm swing, but the reaction forces on the foot sole during usage of the systems were not satisfactory and there was potential to improve control system performance. This work aimed to design and technically evaluate a novel system for producing walking with synchronised arm and leg movement and with dynamic force loading on the foot soles.Methods: The robotic system included a passive curved treadmill and a trunk frame, upon which the rigs for the upper and lower limbs were mounted. Ten actuators and servocontrollers with EtherCAT communication protocol controlled the bilateral shoulder, elbow, hip, knee and ankle joints. Impedance control algorithms were developed and ran in an industrial PC. Flexible pressure sensors recorded the plantar forces on the foot soles. The criteria of implementation and responsiveness were used to formally evaluate the technical feasibility of the system.Results: Using impedance algorithms, the system produced synchronous walking with arm swing on the curved treadmill, with mean RMS angular tracking error <2° in the 10 joint profiles. The foot trajectories relative to the hip presented similar shapes to those during normal gait, with mean RMS displacement error <1.5 cm. A force pattern that started at the heel and finished at the forefoot was observed during walking using the system, which was similar to the pattern from overground walking.Conclusion: The robotic system produced walking-like kinematics in the 10 joints and in the foot trajectories. Integrated with the curved treadmill, the system also produced walking-like force patterns on the foot soles. The system is considered feasible as far as implementation and responsiveness are concerned. Future work will focus on improvement of the mechanical system for future clinical application. Arm swing is an integral component of human gait, and should be included during gait training to improve rehabilitation outcomes . Lower-lMotivated by the theory and clinical results described above, we previously developed three rotational orthoses for walking with arm swing –13, but The upper limb (the shoulder and elbow joints) and the lower limb  are to be actively moved;Dynamic loading is to be provided initially on the heel, then the whole foot sole and lastly on the forefoot to mimic ground reaction forces occurring during the stance phase of overground walking;A flexible gait pattern is to be produced.A new and improved arm-leg robotic system was developed in this study for future application in neurological rehabilitation and for investigation of interlimb neural coupling. The dynamic loading input from the foot sole is believed to modulate walking patterns and thereby to be beneficial for relearning of walking . FurtherThis work aimed to design and technically evaluate a novel system for producing walking with synchronised arm and leg movement and with dynamic force loading on the foot soles.The robotic system was designed using CAD software  and includes 10 drive units and servocontrollers . The control program was developed using TwinCAT 3 , where the overall system could be operated via a touch panel. Finally, the technical feasibility of the arm-leg robotic system was evaluated using the formal criteria of implementation and responsiveness .The arm-leg robotic system comprises a trunk frame over a passive curved treadmill and size-adjustable rigs for the upper and lower limbs . The curSelection and mounting of the actuators were based on normal walking performance and the technical requirements of the system. During overground walking, the normalised maximal torques for the shoulder, elbow, hip, knee and ankle joints are ~0.05, 0.03, 0.5, 0.4, and 0.8 Nm/kg, respectively –21. HoweThe robotic system was manufactured . The staDigital positioning controllers with an EtherCAT communication protocol were used for motion control . Two serThe control program was developed using state machine blocks in TwinCAT 3 in a laptop, and was downloaded into the IPC. The basic states included : start pThe touch panel presents three Human Machine Interface (HMI) pages : start, J, spring stiffness ks, and damping kd. The user exerts an external torque τex on the joint during training with the arm-leg robotic system, while the control torque is denoted as τc  and toe  were calculated using:lt and ls are the lengths of the thigh and shank segments, while θh, θk, and θa are the angles for the hip, knee and ankle joints. The difference between the target and actual segmental trajectories was analysed by calculating RMSE when the target and actual segment trajectories were used in Equation (11).Using the hip segment position as origin  of the reference frame, the segmental trajectories of the ankle  was used, which records the pressure under 12 different areas of each foot sole. The pressure data were analysed and saved using the research software package .To assess the technical feasibility of the arm-leg robotic system, the formal criteria from  were useUsing the impedance algorithms with low values of gains , the armUsing the same impedance controllers with low-gain parameters, the system assisted the test person to walk on the curved treadmill –12. AlthThe force on the foot sole during walking with the robotic system presented typical double-bump characteristics, which is similar to that occurring in overground walking . The forIncreasing the impedance parameters made the robotic system stiffer and resulted in substantially reduced RMSE for all trajectories : using tThe test person reported that they felt as if they were walking. In the first session with the low-gain impedance parameters, she needed some time to know how she should move so as to synchronise with the system. After she became used to the system, she felt obvious stance and swing phases, judging from the arm-leg movement and especially the load on the foot. In the high-gain impedance walking session, she recognised the walking phases immediately. The test person reported the overall test comfortable, although she felt a somewhat constrained arm movement during the whole test.The aim of this work was to design and technically evaluate a novel system for producing walking with synchronised arm and leg movement and with dynamic force loading on the foot soles. The principal design requirements of the system were synchronised arm-leg movement, dynamic foot loading and flexible gait patterns. Using automatic control of 10 joints, the system produced synchronised arm-leg movement. Impedance control algorithms enabled the system to assist the test person to walk on the curved treadmill and dynamic foot loading was produced. The synchronous arm-leg movement and the dynamic forces on the foot soles were similar to those occurring during overground walking. This arm-leg robotic system was designed to serve as a device for investigation of interlimb neural coupling, and also as a testbed for neurological rehabilitation of walking.The robotic system showed advantages of easy operation and convenient control development. The digital positioning controllers used the EtherCAT communication protocol, which allowed real-time multi-axis motion control via an IPC. The HMI on the touch panel provided for convenient start and stop of the system, and also allowed easy tuning of the control parameters. The operator-friendly HMI included comparable functions to what the user interfaces in a typical rehabilitation robotic system provide , and givThe movement of the arm-leg frames from the system alone, without a test person, was similar to the reference trajectories with mean RMSE in 10 joints of 1°, which demonstrated that the first design requirement was met. The flexible topology of the control system makes it convenient for future expansion, for example using additional drives for body-weight support system. The synchronous arm-leg walking patterns were implemented, which met the technical feasibility criterion of implementation.The arm-leg robotic system enabled walking on the integrated curved treadmill, and produced dynamic loading on the foot soles. The curved treadmill provided a stable support to the user during the stance phase. By transferring the body weight to the treadmill, the required power of the drive units during the stance phase, especially of the ankle units, was reduced. Furthermore, the support from the treadmill during the stance phase produced dynamic forces on the foot sole. The plantar force pattern on the foot sole is of a similar double-bump pattern to overground normal walking, with the force starting at the heel and finishing on the forefoot. The second design requirement was met. It is of great interest to compare the actual force on the foot sole during walking on the curved treadmill with the ground reaction forces during overground walking: this will be further investigated in the future.Impedance control algorithms enabled the system to assist walking on the curved treadmill, which provided the basis of the generation of a curved toe trajectory. Normal overground gait patterns includes a rhythmical oscillation of the body trunk/hip joint centre as well as synchronised movement in the arm and legs. In order to assist walking training overground  or on thThis arm-leg robotic system will be a useful tool to investigate interlimb neural coupling. Intensive research on neural interaction during the coordinated arm-leg movement has been based on the daily activities of sitting , standinThe developed system will be a novel system for walking rehabilitation with arm swing, and simultaneously a testbed to search for training parameters that optimise the rehabilitation process. During walking rehabilitation with arm swing, the walking speed and the arm movement pattern are important training parameters. It was observed that, during recumbent stepping, faster upper limb movement facilitated neuromuscular recruitment of lower limb muscles . During The strengths of this study include: (i) mechanical integration of the curved treadmill into the robotic system, which provides the basis for the generation of the curved toe trajectory and also dynamic foot loading; (ii) impedance control algorithms, which enables production of arm-leg walking based on automation control technology; and (iii) easy operation interface, which enables convenient control parameter tuning and training movement setting. The system produced coordinated arm swing during walking, with kinematics and kinetics similar to normal gait patterns. The results from one test person are adequate to evaluate the technical feasibility in implementation and responsiveness, but are insufficient to show the feasibility in populations with different neurological deficits. This limitation will be addressed by investigating the user acceptability of this system after the future work described below is completed. A further limitation was that visual feedback was not provided to the participant. Without information on the target and the actual walking pattern, it was reasonable that the test person required a certain time to get used to the system, especially during the session with the low-gain impedance parameters, where the assistance from the system was limited. Due to the dimensions of the hip drive mechanism, arm swing was achieved with a shoulder abduction of about 25°. That was why the test person reported “a somewhat constrained arm movement” during the whole test. This limitation could be addressed by using a smaller hip drive.This work focused on the mechanical design and control system development based on automation control technology. The next step is to evaluate the acceptability of the robotic system with different users. Further mechanical work will be performed so that the system can provide access to people with neurological disorders. The trunk frame should be able to be tilted down for production of stepping in a supine position. A body weight support system is desirable for potential clinical application. Force sensors are to be included in the device to measure the voluntary inputs of the upper and lower limbs. This system serves as a device for future investigation of interlimb neural coupling, and also as a testbed for neurological rehabilitation of walking. Apart from neurological recovery, cardiopulmonary fitness is also an issue where rehabilitation robotic systems have great potential . The devA novel arm-leg robotic system was designed and constructed. State-of-the-art automation technology enabled the system to produce walking on the curved treadmill with synchronous leg movement and arm swing including shoulder and elbow activation: the robotic system produced walking-like kinematics in the 10 joints and in the foot trajectories. Integrated with the curved treadmill, the system produced plantar stimulation and force patterns on the foot soles which were similar to normal overground gait. The system is considered feasible as far as implementation and responsiveness are concerned. Future work will focus on improvement of the mechanical system for future clinical application.The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. The patients/participants provided their written informed consent to participate in this study.JF and KH developed the system concept and the control algorithms. JF conducted the experiments, analysed the data, and drafted the manuscript. KH revised it critically for important intellectual content. Both authors read and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.