Solving spatial basic geometric constraint configurations with locus intersection

A basic idea of geometric constraint solving (GCS) is to decompose the constraint problem into smaller ones according to some basic configurations. In this paper, we find all spatial basic configurations involving points, lines, and planes containing up to six geometric primitives in an automated way. Many of these basic configurations still resist effective analytical solutions. We propose the locus intersection method (LIM) for GCS, a hybrid method based on geometric computation and numerical search that can be used to find all the solutions for a geometric constraint problem. We show that the LIM can be used to solve all the above basic configurations.     

共形几何代数与几何不变量的代数运算

几何不变量的使用是计算机视觉和图形学的一个重要手段.发现一个不变量后,如何找到它与其他不变量的关系,是实际应用中的一个重要问题,这种关系的探讨主要依靠在不变量层次上的代数运算.文中介绍了共形几何代数中的基本、高级和有理不变量如何在几何问题中自然出现,它们之间如何进行代数运算,以及如何通过不变量的化简,自然地得到几何条件的充分必要化和几何定理的完全化.几何定理的机器证明作为几何定理完全化的副产品,被发展成几何定理的关系定量化,这种量化的几何还原就是几何定理的自然推广.几何不变量之间的几何关系的计算是这些技术的一个具体应用.     

A Geometric Algorithm to Compute Time-Optimal Trajectories for a Bidirectional Steered Robot

This paper addresses the problem of determining time-optimal trajectories, between two specified configurations, for a nonholonomic bidirectional steered robot. It presents an original geometric reasoning that is grounded on Pontryagin's maximum principle, which provides analytical solutions of this problem in a visually clear way and allows for an effective algorithm to compute the exact optimal trajectories between two arbitrarily specified configurations. The proposed geometric reasoning is based on the analysis of the switching functions of the optimal controller and the definition of a switching vector from which it is able to determine a unit vector rotating along a unit circle of an appropriate coordinate system. It is shown that simple geometric rules are sufficient to determine all possible rotations of this unit vector, from which the time-optimal trajectories can be uniquely determined. The proposed algorithm, which is based on this geometric reasoning, is guaranteed to be complete and has a low computational cost. Moreover, the proposed geometric representation provides an interesting insight into the structure of this class of nonholonomic systems, thereby offering a model for further studies.     

Geometric calibration of industrial robots using enhanced partial pose measurements and design of experiments

The paper deals with geometric calibration of industrial robots and focuses on reduction of the measurement noise impact by means of proper selection of the manipulator configurations in calibration experiments. Particular attention is paid to the enhancement of measurement and optimization techniques employed in geometric parameter identification. The developed method implements a complete and irreducible geometric model for serial manipulator, which takes into account different sources of errors (link lengths, joint offsets, etc). In contrast to other works, a new industry-oriented performance measure is proposed for optimal measurement configuration selection that improves the existing techniques via using the direct measurement data only. This new approach is aimed at finding the calibration configurations that ensure the best robot positioning accuracy after geometric error compensation. Experimental study of heavy industrial robot KUKA KR-270 illustrates the benefits of the developed pose strategy technique and the corresponding accuracy improvement.     

A hybrid approach to geometric constraint solving with graph analysis and reduction

In this paper, a graph constructive approach to solving geometric constraint problems is being described. Usually, the graph constructive approach is efficient; however, it has its limitations in scope: it cannot handle ruler-and-compass non-constructible configurations, and under-constrained problems. To overcome these limitations, a proposed algorithm that isolates ruler-and-compass non-constructible configurations from ruler-and-compass constructible configurations is made. Numerical calculation methods are applied to solve them separately. This separation can maximize the efficiency and robustness of a geometric constraint solver. Moreover, the solver can handle under-constrained problems by classifying under-constrained subgraphs to simplified cases by applying classification rules. Then, it decides the calculating sequence of the geometric entities in each classified case, and calculates the geometric entities by adding appropriate assumptions or constraints. By extending the clustering types, and defining several rules, the proposed approach can overcome the limitations of previous graph constructive approaches. Therefore, an efficient and robust geometric constraint solver using this approach can be made.     

A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering

We report our effort to build a geometry deductive database, which can be used to find the fixpoint for a geometric configuration. The system can find all the properties of the configuration that can be deduced using a fixed set of geometric rules. To control the size of the database, we propose the idea of a structured deductive database. Our experiments show that this technique could reduce the size of the database by one hundred times. We propose the data-based search strategy to improve the efficiency of forward chaining. We also make clear progress in the problems of how to select good geometric rules, how to add auxiliary points, and how to construct numerical diagrams as models automatically. The program is tested with 160 nontrivial geometry configurations. For these geometric configurations, the program not only finds most of their well-known properties but also often gives unexpected results, some of which are possibly new. Also, the proofs generated by the program are generally short and totally geometric.     

A new perspective on criteria and algorithms for reachability of discrete-time switched linear systems

The paper presents a unified perspective on geometric and algebraic criteria for reachability and controllability of controlled switched linear discrete-time systems. Direct connections between geometric and algebraic criteria are established as well as that between the subspace based controllability/reachability algorithm and Kalman-type algebraic rank criteria. Also the existing geometric criteria is simplified and new algebraic conditions on controllability and reachability are given.     

Augmented Lagrangian Formulation: Geometrical Interpretation and Application to Systems with Singularities and Redundancy

A geometric interpretation of the augmented Lagrangian formulation ofBayo et al. (Comput. Methods Appl. Mech. Engrg. 71, 1988, 183–195), appliedto equations of motion in relative and Cartesian coordinates, ispresented. Instead of imposing constraints on a system in thetraditional sense, large artificial masses resisting in the constraineddirections are added, and the system motion is enforced to evolveprimarily in the directions with smaller masses (in the unconstraineddirections). Then, the residual motion in the constrained directions isremoved by applying the constraint reactions to the system, estimatedeffectively in few iterations. The formulation is comparatively simpleand leads to computationally efficient numerical codes. Usefulapplications of the formulation to the dynamic analysis of constrainedmultibody systems with possible singular configurations, massless linksand redundant constraints are shown. The theoretical background isfollowed by some remarks on the modeling precautions and assistedcomputational peculiarities of the method. The results of numericalsimulation of motion of a parallel five-bar and a parallel four-barlinkage are reported.     

Nonlinear thermo-mechanical response of temperature-dependent FG sandwich nanobeams with geometric imperfection

In this paper, the nonlinear dynamic response of functionally graded (FG) sandwich nanobeam associated with temperature-dependent material properties by considering the initial geometric imperfection is investigated. The size-dependent behavior of the FG sandwich nanobeam is simulated based on the nonlocal strain gradient theory, and Von Karman nonlinear hypothesis is used to model the geometrical nonlinearity. Moreover, the geometric imperfection is considered as a slight curvature satisfying the first mode shape, and four different FG sandwich patterns including two asymmetric configurations and two symmetric configurations are taken into account. The governing equation of the FG sandwich nanobeam subjected to thermal and harmonic external excitation loadings is derived on the basis of Hamilton’s principle. The numerical results are obtained by employing the multiple-scale method, which are also validated by comparison with two previous studies. Furthermore, comprehensive investigations into the influences of size-dependent parameters, external temperature variation, geometric imperfection amplitude, gradient index and sandwich configuration on the nonlinear characteristics of imperfect FG sandwich nanobeams are conducted through numerical results.

     

航天电子系统IEEE1394总线可靠性模型研究

数据总线作为综合电子系统的核心,它的可靠性是保证综合电子系统功能有效实现的关键。针对IEEE 1394总线,研究容错总线结构,分别分析基于stack-tree拓扑的双冗余以及环形冗余结构的节点失效模型。其中环形冗余stack-tree的模型运用递归函数。使用matlab仿真计算上述模型,定量的分析比较它们的容错性,得出环形冗余结构的可靠性更高,可超过0.99,且更稳定。研究结果对1394b总线的空间应用具有一定的参考意义。