Decentralized Collision-Free Control of Multiple Robots in 2D and 3D Spaces

Decentralized control of robots has attracted huge research interests. However, some of the research used unrealistic assumptions without collision avoidance. This report focuses on the collision-free control for multiple robots in both complete coverage and search tasks in 2D and 3D areas which are arbitrary unknown. All algorithms are decentralized as robots have limited abilities and they are mathematically proved. The report starts with the grid selection in the two tasks. Grid patterns simplify the representation of the area and robots only need to move straightly between neighbor vertices. For the 100% complete 2D coverage, the equilateral triangular grid is proposed. For the complete coverage ignoring the boundary effect, the grid with the fewest vertices is calculated in every situation for both 2D and 3D areas. The second part is for the complete coverage in 2D and 3D areas. A decentralized collision-free algorithm with the above selected grid is presented driving robots to sections which are furthest from the reference point. The area can be static or expanding, and the algorithm is simulated in MATLAB. Thirdly, three grid-based decentralized random algorithms with collision avoidance are provided to search targets in 2D or 3D areas. The number of targets can be known or unknown. In the first algorithm, robots choose vacant neighbors randomly with priorities on unvisited ones while the second one adds the repulsive force to disperse robots if they are close. In the third algorithm, if surrounded by visited vertices, the robot will use the breadth-first search algorithm to go to one of the nearest unvisited vertices via the grid. The second search algorithm is verified on Pioneer 3-DX robots. The general way to generate the formula to estimate the search time is demonstrated. Algorithms are compared with five other algorithms in MATLAB to show their effectiveness

Small polaron with generic open boundary conditions revisit: exact solution via the off-diagonal Bethe ansatz

The small polaron, an one-dimensional lattice model of interacting spinless fermions, with generic non-diagonal boundary terms is studied by the off-diagonal Bethe ansatz method. The presence of the Grassmann valued non-diagonal boundary fields gives rise to a typical $U(1)$-symmetry-broken fermionic model. The exact spectra of the Hamiltonian and the associated Bethe ansatz equations are derived by constructing an inhomogeneous $T-Q$ relation.Comment: 12 pages, no figure, published versio

Drivers of green innovation in BRICS countries: exploring tripple bottom line theory

Green technology adoption is indispensable for sustainable growth. Therefore, this study examines the determinants of green innovation in BRICS countries considering the Triple Bottom Line Theory (social, environmental, and economic). A cross-sectional autoregressive distributed lag (CS-ARDL) model is applied for empirical analysis from 1990 to 2019. The findings show that social, economic, and environmental factors significantly derive green innovation in the long run. However, their marginal contribution is substantially varied. A 1% increase in economic factors increases green innovation by 0.290%, while environmental concerns induce innovation by 0.438% in the long run. In contrast, social factors possess a relatively lower influence on green innovation, with a coefficient magnitude of 0.175%. Lastly, globalization stimulates green innovation by 0.310%. Similar results are observed in the short run; however, the magnitude of variables is significantly lower than long-run. These results are also validated using alternative estimators and recommend TBL factors as core drivers of green innovation in BRICS countries

A second-order length-preserving and unconditionally energy stable rotational discrete gradient method for Oseen-Frank gradient flows

We present a second-order strictly length-preserving and unconditionally energy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional. Two essential ingredients of the Rdg method are reformulation of the length constrained gradient flow into an unconstrained rotational form and discrete gradient discretization for the energy variation. Besides the well-known mean-value and Gonzalez discrete gradients, we propose a novel Oseen-Frank discrete gradient, specifically designed for the solution of Oseen-Frank gradient flow. We prove that the proposed Oseen-Frank discrete gradient satisfies the energy difference relation, thus the resultant Rdg scheme is energy stable. Numerical experiments demonstrate the efficiency and accuracy of the proposed Rdg method and its capability for providing reliable simulation results with highly disparate elastic coefficients