面向欠约束几何系统的一种同伦求解方法

针对几何约束系统的数值求解过程中，经常发生的数值不稳定性问题，构造了一种面向欠约束系统的同伦方法，并将其与现有的求解与分解方法有机地结合起来，提出了一种牛顿－同伦混合方法，在牛顿迭代失败的位置自动调用欠约束同伦法，既提高了几何约束求解器的效率，同时又保证了求解的效率。     

通用几何约束系统统一建模研究

在几何约束和几何实体的基本约束和欧拉参数表达的基础上,研究了通用几何约束系统的统一建模问题。通过对三维几何实体姿态约束和位置约束解耦性的分析,抽象出球实体、盒体和球盒体三种基本几何实体表达空间几何实体,并以基本约束的组合表达几何约束,形成几何约束模型特有的层次结构;并以有向图管理几何约束系统,可以清晰地反映姿态约束和位置约束的解耦性,实现约束系统的细粒度分解,得到规模更小的求解序列,实现高效求解。方法实现于原型系统WhutVAS中。     

三维几何约束系统的等价性分析

针对过约束、完整约束和欠约束三维几何约束系统的求解问题,提出了等价性分析方法.该方法基于三维几何约束系统的内在等价性,充分挖掘几何领域知识,依据拆解约束闭环、缩减约束闭环和析出约束闭环等原则,采用等价约束替换来处理几何约束闭环问题,优化几何约束图的结构,实现几何约束系统的优化分解.最后用多个实例验证了该方法的正确性和有...     

基于几何约束求解的完备方法

针对参数化CAD在约束求解中的应用,提出了基于智能连杆的算法,该算法在扩充几何作图范围、改善算法复杂度方面都有明显的优势.将其同LIMO算法、几何变换方法、C-Tree算法、数值求解方法等方法相互融合,能够组成一套非常完备的几何约束求解框架,来完成对平面和空间几何约束问题的自动求解与图像生成.将该算法应用于智能动态几何软件的设计中,实验显示可以取得令人满意的结果.     

几何约束系统中约束的动态管理方法

几何约束系统的建模及求解是参数化技术的核心。但如何对约束进行方便、有效的管理一直是未完全解决的难题，所以在目前所见到的参数化系统中的大多数，难以处理冗余约束、局部过自由度等问题，这严重妨碍了参数化功能的发挥。文中利用自由度分析，非线性代数方程及矩阵理论，提出了一种在几何约束系统中对约束进行动态管理的新方法。并利用这种方法成功解决了冗余约束预报及局部过自由度判断等难题。     

几何约束求解的简化迭代算法

针对几何约束系统图分解中复合顶点的求解问题,提出复合顶点的图分解算法和等价自由变量的简化迭代求解算法.通过去除复合顶点部分边界约束对复合顶点进行图分解,对求解序列中的欠约束顶点添加等价自由变量、以等价自由变量的部分迭代求解、替代系统的整体数值求解,以提高求解效率和稳定性.该算法具有很强的通用性,并在实际应用中得到验证.     

三维几何约束的球面几何求解

研究了球面几何学在三维几何约束求解中的应用，提出了球面求解法．该方法建立在姿态约束与位置约束解耦的基础上，并以求解关键的姿态约束为主，一旦姿态约束被解出，则位置约束很容易求解；同时将表征刚体姿态的矢量映射到球平面上的点，将姿态约束映射为球平面上两点的距离，借助球面几何的知识，能够高效、直观地推理出多数情况下姿态约束的解析解，而特殊的情况则结合数值法求解，并很好地解决了数值法的初值问题．     

快速判定几何约束奇异性的切面扰动法

针对冗余奇异和分支奇异的判定问题,提出一种新的切面扰动的判定方法.该方法将奇异的雅可比矩阵分为独立构型空间和奇异空间,变量沿独立构型空间的切面扰动,计算更新的雅克比矩阵的秩,依据秩亏的变化可以快速、稳定地判定约束奇异性.该算法克服了残量扰动法的数值迭代、计算量大和不稳定的缺点,并且在参数化特征造型系统InteSolid中得到验证.     

二维几何约束求解器

开发了一个二维几何约束求解器.该求解器是新一代智能CAD系统的核心,它采用了基于图和规则的几何推理方法,高效、稳定、实用.论文提出了几何约束模型及其约束图表示,深入介绍了基于点刚体归约的几何推理算法,描述了求解器的体系结构.     

A non-rigid cluster rewriting approach to solve systems of 3D geometric constraints

We present a new constructive solving approach for systems of 3D geometric constraints. The solver is based on the cluster rewriting approach, which can efficiently solve large systems of constraints on points, and incrementally handle changes to a system, but can so far solve only a limited class of problems. The new solving approach extends the class of problems that can be solved, while retaining the advantages of the cluster rewriting approach. Whereas previous cluster rewriting solvers only determined rigid clusters, we also determine two types of non-rigid clusters, i.e. clusters with particular degrees of freedom. This allows us to solve many additional problems that cannot be decomposed into rigid clusters, without resorting to expensive algebraic solving methods. In addition to the basic ideas of the approach, an incremental solving algorithm, two methods for solution selection, and a method for mapping constraints on 3D primitives to constraints on points are presented.     

A constructive approach to calculate parameter ranges for systems of geometric constraints

Geometric constraints are at the heart of parametric and feature-based CAD systems. Changing values of geometric constraint parameters is one of the most common operations in such systems. However, because allowable parameter values are not known to the user beforehand, this is often a trial-and-error process. We present an approach for automatically determining the allowable range for parameters of geometric constraints. Considered are systems of distance and angle constraints on points in 3D that can be decomposed into triangular and tetrahedral subproblems, by which most practical situations in parametric and feature-based CAD systems can be represented. Our method uses the decomposition to find critical parameter values for which subproblems degenerate. By solving one problem instance for each interval between two subsequent critical values, the exact parameter range is determined for which a solution exists.     

Using invariance under the similarity group to solve geometric constraint systems

In the area of Computer Aided Design (CAD), the main feature of geometric constraint systems lies in their invariance under the direct isometry group. Several researchers have developed methods, which take advantage of this fact to decompose such systems into smaller sub-systems. In this paper, we show that considering the invariance under the direct similarity group leads to a new constructive method to solve geometric constraint systems encountered in CAD.     

自适应微调扰动和声搜索算法几何约束求解研究*

几何约束求解的方法关系到特征造型系统的性能,为提高几何约束求解的速度,将和声搜索算法应用于几何约束求解中。通过优先选择较小的和声库,利用最好解的评价值确定微调扰动的幅度,并将其嵌入到拉斯维加斯算法中,提高了和声搜索算法的性能。实验结果表明,改进的和声算法具有自适应性,能有效克服局部收敛问题,提高了求解速度。     

An initial algebra approach to term rewriting systems with variable binders

We present an extension of first-order term rewriting systems. It involves variable binding in the term language. We develop systems called binding term rewriting systems (BTRSs) in a stepwise manner. First we present the term language, then formulate equational logic. Finally, we define rewriting systems. This development is novel because we follow the initial algebra approach in an extended notion of Σ-algebras in various functor categories. These are based on Fiore-Plotkin-Turi’s presheaf semantics of variable binding and Lüth-Ghani’s monadic semantics of term rewriting systems. We characterise the terms, equational logic and rewrite systems for BTRSs as initial algebras in suitable categories. Then, we show an important rewriting property of BTRSs: orthogonal BTRSs are confluent. Moreover, by using the initial algebra semantics, we give a complete characterisation of termination of BTRSs. Finally, we discuss our design choice of BTRSs from a semantic perspective. An erlier version appeared in Proc. Fifth ACM-SIGPLAN International Conference on Principles and Practice of Declarative Programming (PPDP2003).     

Singularity Analysis of Geometric Constraint Systems

Singularity analysis in an important subject of the geometric constraint satisfaction problem.In this paper,three kinds of singularities are described and corresponding identifcation methods are presented for both under0constrained systems and over-constrained systems,Another special but common singularity for under-constrained geometric systems,pseudo-singularity,is analyzed.Pseudo-singularity is caused by a variety of constraint mathching of under-constrained systems and can be removed by improving constraint distribution.To avoid pseudo-singularity and decide redundant constraints adaptively,a differentiaiton algorithm is proposed in the paper.Its corrctness and effciency have been validated through its practical applications in a 2D/3D geometric constraint solver CBA.     

A geometric approach to cluster validity for normal mixtures

 We study indices for choosing the correct number of components in a mixture of normal distributions. Previous studies have been confined to indices based wholly on probabilistic models. Viewing mixture decomposition as probabilistic clustering (where the emphasis is on partitioning for geometric substructure) as opposed to parametric estimation enables us to introduce both fuzzy and crisp measures of cluster validity for this problem. We presume the underlying samples to be unlabeled, and use the expectation-maximization (EM) algorithm to find clusters in the data. We test 16 probabilistic, 3 fuzzy and 4 crisp indices on 12 data sets that are samples from bivariate normal mixtures having either 3 or 6 components. Over three run averages based on different initializations of EM, 10 of the 23 indices tested for choosing the right number of mixture components were correct in at least 9 of the 12 trials. Among these were the fuzzy index of Xie-Beni, the crisp Davies-Bouldin index, and two crisp indices that are recent generalizations of Dunn’s index. Received: 29 July 1997/Accepted: 1 September 1997     

Sliding mode control of switched systems: A geometric algebra approach

Geometric algebra (GA) is proposed as a mathematical framework for revisiting fundamental aspects of sliding mode control (SMC) in nonlinear, switch-controlled, single input systems. Sliding mode existence conditions, the switching policy, the invariance conditions, the associated equivalent control, and the characterization of ideal sliding dynamics, are all re-examined using a geometric algebra (GA) standpoint. Two illustrative examples, from switched power electronics, are presented using the GA language.     

A constructive approach to solving 3-D geometric constraint systems using dependence analysis

Solving geometric constraint systems in 3-D is much more complicated than that in 2-D because the number of variables is larger and some of the results valid in 2-D cannot be extended for 3-D. In this paper, we propose a new DOF-based graph constructive method to geometric constraint systems solving that can efficiently handle well-, over- and under-constrained systems based on the dependence analysis. The basic idea is that the solutions of some geometric elements depend on some others because of the constraints between them. If some geometric elements depend on each other, they must be solved together. In our approach, we first identify all structurally redundant constraints, then we add some constraints to well constrain the system. And we prove that the order of a constraint system after processing under-constrained cases is not more than that of the original system multiplied by 5. After that, we apply a recursive searching process to identify all the clusters, which is shown to be capable of getting the minimum order-reduction result of a well-constrained system. We also briefly describe the constraint evaluation phase and show the implementation results of our method.