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The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms, which can be used to describe the propagation properties of optical soliton solutions.

The authors apply the ansatz method and the Hamiltonian system technique to find its bright, dark and Gaussian wave solitons and analyze its modulation instability analysis and stability analysis solution.

The results imply that the generalized nonlinear Schrödinger equation has bright, dark and Gaussian wave solitons. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior. Some constraint conditions are provided which can guarantee the existence of solitons. The authors analyze its modulation instability analysis and stability analysis solution.

These results may help us to further study the local structure and the interaction of solutions in generalized nonlinear Schrödinger -type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of the generalized nonlinear Schrödinger--type equations.

Engineering components/structures with geometric discontinuities normally bear complex and variable loads, which lead to a multiaxial and random/variable amplitude stress/strain state. Hence, this study aims how to effectively evaluate the multiaxial random/variable amplitude fatigue life.

Recent studies on critical plane method under multiaxial random/variable amplitude loading are reviewed, and the computational framework is clearly presented in this paper.

Some basic concepts and latest achievements in multiaxial random/variable amplitude fatigue analysis are introduced. This review summarizes the research status of four main aspects of multiaxial fatigue under random/variable amplitude loadings, namely multiaxial fatigue criterion, method for critical plane determination, cycle counting method and damage accumulation criterion. Particularly, the latest achievements of multiaxial random/variable amplitude fatigue using critical plane methods are classified and highlighted.

This review attempts to provide references for further research on multiaxial random/variable amplitude fatigue and to promote the development of multiaxial fatigue from experimental research to practical engineering application.

The purpose of this paper is to study the breather waves, rogue waves and solitary waves of an extended (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation, which can be used to depict many nonlinear phenomena in fluid dynamics and plasma physics.

The authors apply the Bell’s polynomial approach, the homoclinic test technique and Hirota’s bilinear method to find the breather waves, rogue waves and solitary waves of the extended (3 + 1)-dimensional KP equation.

The results imply that the extended (3 + 1)-dimensional KP equation has breather wave, rogue wave and solitary wave solutions. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior.

These results may help us to further study the local structure and the interaction of solutions in KP-type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of such equations.

The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.

The authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.

The results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.

These results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

In complex environments, a spherical robot has great application value. When the pendulum spherical robot is stopped or disturbed, there will be a periodic oscillation. This situation will seriously affect the stability of the spherical robot. Therefore, this paper aims to propose a control method based on backstepping and disturbance observers for oscillation suppression.

This paper analyzes the mechanism of oscillation. The oscillation model of the spherical robot is constructed and the relationship between the oscillation and the internal structure of the sphere is analyzed. Based on the oscillation model, the authors design the oscillation suppression control of the spherical robot using the backstepping method. At the same time, a disturbance observer is added to suppress the disturbance.

It is found that the control system based on backstepping and disturbance observer is simple and efficient for nonlinear models. Compared with the PID controller commonly used in engineering, this control method has a better control effect.

The proposed method can provide a reliable and effective stability scheme for spherical robots. The problem of instability in real motion is solved.

In this paper, the oscillation model of a spherical robot is innovatively constructed. Second, a new backstepping control method combined with a disturbance observer for the spherical robot is proposed to suppress the oscillation.

The purpose of this paper is to briefly summarize and review the theories and methods of complex structures’ dynamic reliability. Complex structures are usually assembled from multiple components and subjected to time-varying loads of aerodynamic, structural, thermal and other physical fields; its reliability analysis is of great significance to ensure the safe operation of large-scale equipment such as aviation and machinery.

In this paper for the single-objective dynamic reliability analysis of complex structures, the calculation can be categorized into Monte Carlo (MC), outcrossing rate, envelope functions and extreme value methods. The series-parallel and expansion methods, multi-extremum surrogate models and decomposed-coordinated surrogate models are summarized for the multiobjective dynamic reliability analysis of complex structures.

The numerical complex compound function and turbine blisk are used as examples to illustrate the performance of single-objective and multiobjective dynamic reliability analysis methods. Then the future development direction of dynamic reliability analysis of complex structures is prospected.

The paper provides a useful reference for further theoretical research and engineering application.

The time delay would occurs when the measurements of multiple unmanned aerial vehicles (UAVs) are transmitted to the date processing center during cooperative target localization. This problem is often named as the out-of-sequence measurement (OOSM) problem. This paper aims to present a nonlinear filtering based on solving the Fokker–Planck equation to address the issue of OOSM.

According to the arrival time of measurement, the proposed nonlinear filtering can be divided into two parts. The non-delay measurement would be fused in the first part, in which the Fokker–Planck equation is utilized to propagate the conditional probability density function in the forward form. The time delay measurement is fused in the second part, in which the Fokker–Planck is used in the backward form approximately. The Bayes formula is applied in both parts during the measurement update.

Under the Bayesian filtering framework, this nonlinear filtering is not only suitable for the Gaussian noise assumption but also for the non-Gaussian noise assumption. The nonlinear filtering is applied to the cooperative target localization problem. Simulation results show that the proposed filtering algorithm is superior to the previous Y algorithm.

In this paper, the research shows that a better performance can be obtained by fusing multiple UAV measurements and treating time delay in measurement with the proposed algorithm.

In this paper, the OOSM problem is settled based on solving the Fokker–Planck equation. Generally, the Fokker–Planck equation can be used to predict the probability density forward in time. However, to associate the current state with the state related to OOSM, it would be used to propagate the probability density backward either.

This paper aims to propose a four-point measurement model for directly measuring the pose (i.e. position and orientation) of industrial robot and reducing its calculating error and measurement uncertainty.

A four-point measurement model is proposed for directly measuring poses of industrial robots. First, this model consists of a position measurement model and an orientation model gotten by the position of spherically mounted reflector (SMR). Second, an influence factor analysis, simulated by Monte Carlo simulation, is performed to investigate the influence of certain factors on the accuracy and uncertainty. Third, comparisons with the common method are carried out to verify the advantage of this model. Finally, a test is carried out for evaluating the repeatability of five poses of an industrial robot.

In this paper, results show that the proposed model is better than the three-SMRs model in measurement accuracy, measurement uncertainty and computational efficiency. Moreover, both measurement accuracy and measurement uncertainty can be improved by using the proposed influence laws of its key parameters on the proposed model.

The proposed model can measure poses of industrial robots directly, accurately and effectively. Additionally, influence laws of key factors on the accuracy and uncertainty of the proposed model are given to provide some guidelines for improving the performance of the proposed model.

As the transmission component of a locomotive, the traction gear pair system has a direct effect on the stability and reliability of the whole machine. This paper aims to provide a detailed dynamic analysis for the traction system under internal and external excitations by numerical simulation.

A non-linear dynamic model of locomotive traction gear pair system is proposed, where the comprehensive time-varying meshing stiffness is obtained through the Ishikawa formula method and verified by the energy method, and then the sliding friction excitation is analyzed based on the location of the contact line. Meantime, the adhesion torque is constructed as a function of the adhesion-slip feature between wheelset and rail. Through Runge–Kutta numerical method, the system responses are studied with varying bifurcation parameters consisting of exciting frequency, load fluctuation, gear backlash, error fluctuation and friction coefficient. The dynamic behaviors of the system are analyzed and discussed from bifurcation diagram, time history, spectrum plot, phase portrait, Poincaré map and three-dimensional frequency spectrum.

The analysis results reveal that as control parameters vary the system experiences complex transition among a diverse range of motion states such as one-periodic, multi-periodic and chaotic motions. Specifically, the significant difference in system bifurcation characteristics can be observed under different adhesion conditions. The suitable gear backlash and error fluctuation can avoid the chaotic motion, and thus, reduce the vibration amplitude of the system. Similarly, the increasing friction coefficient can also suppress the unstable state and improve the stability of the system.

The numerical results may provide a systemic understanding of dynamic characteristics and present some available information to design and optimize the transmission performance of the locomotive traction system.

The paper aims to propose an innovative two-stage decision model to address the sustainable-resilient supplier selection and order allocation (SSOA) problem in the single-valued neutrosophic (SVN) environment.

First, the sustainable and resilient performances of suppliers are evaluated by the proposed integrated SVN-base-criterion method (BCM)-an acronym in Portuguese of interactive and multi-criteria decision-making (TODIM) method, with consideration of the uncertainty in the decision-making process. Then, a novel multi-objective optimization model is formulated, and the best sustainable-resilient order allocation solution is found using the U-NSGA-III algorithm and TOPSIS method. Finally, based on a real-life case in the automotive manufacturing industry, experiments are conducted to demonstrate the application of the proposed two-stage decision model.

The paper provides an effective decision tool for the SSOA process in an uncertain environment. The proposed SVN-BCM-TODIM approach can effectively handle the uncertainties from the decision-maker’s confidence degree and incomplete decision information and evaluate suppliers’ performance in different dimensions while avoiding the compensatory effect between criteria. Moreover, the proposed order allocation model proposes an original way to improve sustainable-resilient procurement values.

The paper provides a supplier selection process that can effectively integrate sustainability and resilience evaluation in an uncertain environment and develops a sustainable-resilient procurement optimization model.