Motion Planning for Robots with Topological Dimension Reduction Method

This paper explores the realization of robotic motion planning, especially Findpath problem, which is a basic motion planning problem that arises in the development of robotics. Findpath means: Give the initial and desired final configurations of a robotic arm in 3-dimensionnl space, and give descriptions of the obstacles in the space, determine whether there is a continuous collision-free motion of the robotic arm from one configure- tion to the other and find such a motion if it exists. There are several branches of approach in motion planning area, but in reality the important things are feasibility, efficiency and accuracy of the method. In this paper ac- cording to the concepts of Configuration Space (C-Space) and Rotation Mapping Graph (RMG) discussed in [1], a topological method named Dimension Reduction Method (DRM) for investigating the connectivity of the RMG (or the topologic structure of the RMG )is presented by using topologic technique. Based on this ap- proach the Findpath problem is thus transformed to that of finding a connected way in a finite Characteristic Network (CN). The method has shown great potentiality in practice. Here a simulation system is designed to embody DRM and it is in sight that DRM can he adopted in the first overall planning of real robot sys- tem in the near future.

Topological nodal line semimetals

We review the recent,mainly theoretical,progress in the study of topological nodal line semimetals in three dimensions.In these semimetals,the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone,and any perturbation that preserves a certain symmetry group（generated by either spatial symmetries or time-reversal symmetry） cannot remove this crossing line and open a full direct gap between the two bands.The nodal line（s） is hence topologically protected by the symmetry group,and can be associated with a topological invariant.In this review,（ⅰ） we enumerate the symmetry groups that may protect a topological nodal line;（ⅱ） we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface,establishing a topological classification;（ⅲ） for certain classes,we review the proposals for the realization of these semimetals in real materials;（ⅳ） we discuss different scenarios that when the protecting symmetry is broken,how a topological nodal line semimetal becomes Weyl semimetals,Dirac semimetals,and other topological phases;and（ⅴ） we discuss the possible physical effects accessible to experimental probes in these materials.

One-pot topotactic synthesis of Ti^(3+) self-doped 3D TiO_2 hollow nanoboxes with enhanced visible light response

Ti^（3＋） self-doped anatase three-dimensional（3D） TiO_2 hollow nanoboxes were synthesized via a topological transformation process involving template participation by a facile one-pot hydrothermal treatment with an ethanol solution of zinc powder and TiOF_2. It is worth noting that the 3D TiO_2 hollow nanoboxes are assembled from six single-crystal nanosheets and have dominant exposure of the {001} facets. It is found from EPR spectra that adding zinc powder is an environment-friendly and effective strategy to introduce Ti^（3＋） and oxygen vacancy（Ov） into the bulk of 3D hollow nanoboxes rather than the surface, which is responsible for their enhanced visible photocatalytic properties.The photocatalytic activity was evaluated by measuring the formation rate of hydroxide free radicals using 7-hydroxycoumarin as a probe. The sample prepared with zinc/TiOF_2 mass ratio of0.25 exhibited the highest RhB photodegradation activity under visible-light irradiation with a degradation rate of 96%, which is 4.0-times higher than that of pure TiO_2. The results suggest a novel approach to construct in-situ 3D hierarchical TiO_2 hollow nanoboxes doped with Ti^（3＋） and Ov without introducing any impurity elements for superior visible-light photocatalytic activity.

A DIRECTED GRAPH ALGORITHM OF VARIATIONAL GEOMETRY BASED ON GEOMETRIC REASONING

The undirected graph to express engineering drawings is discussed .The principle to re-solve and reason the undirected graph is presented, and the algorithm finally transforms theundirected graph into the resolvable directed graph. Therefore,a rapid and simple way is suppliedfor variational design. A prototype of this algorithm has been implemented, and some examplesare given.

Symmetric Variation of Monomers for Constructing 1D/2D Imine-Based Covalent Organic Frameworks

Synthetic control of topology connectivity is the crowning achievement for covalent organic framework(COF)fabrications.Although a large number of one-,two-,and three-dimensional(1D,2D,and 3D)COFs have been reported,their topology structure constructions are restricted to the use of symmetric monomers with the purpose of increasing the crystallinity and/or porosity.Herein,three imine-linked COFs with different topology nets(namely,sql-b and sql-c)were constructed by symmetric variation of monomers via the condensation of a tetra-amine monomer with a D2h-symmetry and lower C_(2v)-symmetric dialdehyde monomers,bearing a phenolic hydroxyl group at different positions.The results indicated that a reasonable introduction of the phenolic hydroxyl group could effectively tune the topological structure of COFs at the molecular level during the crystallization stage.Particularly,the remarkable difference in the dye uptake ability between these COF materials indicated that the fabricated specific pore geometries,as well as different steric hindrance and H-bonding interactions,played a pivotal role in accessing molecules in the solution.Therefore,this work might boost the explorations of COF materials with expected topologies and pore geometries from conventional monomers through bottom-up synthesis methodology by molecular designing and engineering.

A Topological Implementation for Motion Planning of a Robotic Arm

An efficient path planning algorithm based on topologic method is presented in this paper.The colli- sion free path planning for three-joint robotic arm consists of three parts:partition of C-space,construc- tion of CN and search for a path in CN.We mainly solved the problems of partitioning the C-space and judging the connectivity between connected blocks,etc.For the motion planning of a robotic arm with a gripper,we developed the concepts of global planning and local planning,and discussed the basic fac- tors for constructing the planning system.In the paper,some evaluation and analysis of the complexity and reliability of the algorithm are given,together with some ideas to improve the efficiency and increase the reliability.At last,some experimental results are presented to show the efficiency and accuracy of the nigorithm.

Some Results on Inner Quasidiagonal C^*-algebras

In the current article,we prove the crossed product C^*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C^*-algebra is strongly quasidiagonal again.We also show that a just-infinite C^*-algebra is quasidiagonal if and only if it is inner quasidiagonal.Finally,we compute the topological free entropy dimension in just-infinite C^*-algebras.