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The purpose of this study is to obtain the differential geometric analysis of autonomous wheel-legged robots and their trajectories on the terrain.

The author uses a wheel using the osculating sphere of the curve on rough terrain. Additionally, the author expresses a triple osculating sphere wheel by taking advantage of differential geometry. Moreover, the author examined the consecutive wheel center-curves to obtain the optimum posture of a micro-hydraulic toolkit (MHT) robot.

The author examined the terrain path, which is crucial for trajectory planning in terms of the geometric perspective. The author designed the triple MHT wheel using the osculating sphere of the MHT robot trajectory by taking advantage of local differential geometric properties of this curve on the terrain. The consecutive wheel center-curves were expressed and studied based on differential geometry.

The author provides a novel approach for the optimum posture of an MHT robot using consecutive wheel-center curves and provides an original perspective to MHT robot and its trajectory by using differential geometry.

Existing three-dimensional (3D) road-surface models use approximation methods such as a set of discrete triangular patches and cannot accurately describe changes in the geometrically designed elements along the road. This paper aims to construct a 3D road-surface model with combinations of geometric design invariants and apply the proposed model to analyse the state of motion of a wheel’s centre.

In this paper, the 3D road surface is modelled as a continuous function with combinations of geometric design invariants. By introducing the theories of differential geometries and rigid body dynamics, a wheel-road model wherein a wheel fixed to a Darboux frame moves along a curved road surface is constructed, and the wheel time-dependent properties of the velocity, angular velocity and acceleration at an arbitrary point of the surface are described using road geometry design invariants.

This paper adopts the Darboux frame to study the instantaneous spin-rolling motion of a wheel. It is found that the magnitudes of the spin-rolling velocity, the acceleration and the geometric invariants of the road surface, including the geodesic curvature, the normal curvature and the geodesic torsion, determine the instantaneous states of motion of a wheel.

This work provides a theoretical foundation for future studies of wheel motion states, such as the relationship between road geometry design invariants and driving safety, vehicle lane changing and other vehicle microbehaviours. New insights are gained in the areas of road safety and vehicles incorporating artificial intelligence.

The grasping task of robots in dense cluttered scenes from a single-view has not been solved perfectly, and there is still a problem of low grasping success rate. This study aims to propose an end-to-end grasp generation method to solve this problem.

A new grasp representation method is proposed, which cleverly uses the normal vector of the table surface to derive the grasp baseline vectors, and maps the grasps to the pointed points (PP), so that there is no need to add orthogonal constraints between vectors when using a neural network to predict rotation matrixes of grasps.

Experimental results show that the proposed method is beneficial to the training of the neural network, and the model trained on synthetic data set can also have high grasping success rate and completion rate in real-world tasks.

The main contribution of this paper is that the authors propose a new grasp representation method, which maps the 6-DoF grasps to a PP and an angle related to the tabletop normal vector, thereby eliminating the need to add orthogonal constraints between vectors when directly predicting grasps using neural networks. The proposed method can generate hundreds of grasps covering the whole surface in about 0.3 s. The experimental results show that the proposed method has obvious superiority compared with other methods.

Fractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.

This paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.

This paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.

This paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

The purpose of this paper is to obtain analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations by the homotopy analysis and the homotopy Padé methods.

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series.

The approximation solutions by [m,m] homotopy Padé technique is often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

In this paper, analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations are obtained by the homotopy analysis and the homotopy Padé methods. The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Padé technique are often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.