复杂机构学问题研究的新方法

项目名称： 复杂机构学问题研究的新方法

项目编号： No.50875027

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 数理科学和化学

项目作者： 魏世民

作者单位： 北京邮电大学

项目金额： 29万元

中文摘要： 同传统欧氏几何代数（EGA）相比，共形几何代数(CGA)把点、线、面、圆和球统一为圆或球，具有模型统一和变换统一的特点，适于计算机编程。本项目中通过对CGA基本理论的学习和研究，采用上述方法对一般空间6R、1P5R机器手的运动学逆解问题和平面并联机构位置正解问题进行研究,完成了对此两类机构的统一建模和求解；在研究CGA的同时，受此启发，推导出了四元数、对偶四元数的复数形式和矩阵形式两种新形式。并利用对偶四元数的复数形式这种新形式证明了一般6R机器手具有16组解。之后，采用由距离、面积、体积和角度等几何不变量和重心坐标表达式推导出的三边测量法进行建模，对一般6-4台体并联机构位置正解问题进行研究。最后，采用由点积和叉积构成的矢量方法进行建模，对一般5-5B台体并联机位置正解问题进行研究，首次推导出完全符号形式的一元24次输入输出方程。后两种建模方法是零括号代数(NBA)几何不变量的一种表达形式。上述几种方法，都具有形式简洁、编程容易的特点，利于机构学的数学机械化。

中文关键词： 共形几何代数(CGA)；对偶四元数；几何不变量；6R机械手

英文摘要： Compared with traditional Euclidean Geometric Algebra (EGA), Conformal Geometric Algebra(CGA) treats all geometric elements such as dot, line, surface, circle and sphere as circle or sphere, by which unify the modeling objects and transformation calculation and make it easier for programming.Accroding to studying and researching on CGA, inverse displacement analysis for general 6R，1P5R manipulators and forward displacement analysis for planar platform are investigated and base on the theory,we accomplish unified modeling and solution procedure for two complicated mechanisms.Next，two new forms of quaternion and dual quaternion are derived and introduced in the computing, that is complex number form and matrix form of quaternion and dual quaternion. Using complex number form of dual quarternion, the fact that the general 6R robot has 16 solutions, is proved.Then，the forward displacement analysis for general 4-6 in-parallel platform is studied based on the trilateration,which is derived from barycentric coordinates and the invariant algebra such as distance,area,volumn and angle etc.Finally, we apply to the vetor method to model and solve the forward displacement analysis of general 5-5B in-parallel platform and we derived firstly a 24th-degree input-output polynomials totally symbolically.The last two methods are regarded as another expression for the invariant of Null Bracket Algebra(NBA).Because the above methods have the characteristics of simple expressions and ease-of-programming, they benefit for mathematics mechanization in the field of mechanisms.

英文关键词： Conformal Geometric Algebra(CGA);Dual quaternion;Invariant;General 6R manipulator