基于几何代数的并联机构自由度自动化分析

Automatic Mobility Analysis of Parallel Mechanisms Based on Geometric Algebra

基于R(3,3)几何代数,提出一种符号描述并联机构自由度的自动化算法。首先根据螺旋副之间的几何关系,利用R(3,3)几何代数能符号描述刚体运动的优势,自动求解并联机构各支链螺旋系;然后利用R(3,3)几何代数能符号表示集合交集和并集的优势,自动求解动平台运动空间,该运动空间为所求并联机构自由度的符号表示式;最后基于C++软件平台对这种并联机构自由度自动化求解算法进行验证。使用R(3,3)几何代数不仅能通过刚体运算法则得到支链螺旋系的符号表达,同时可以直接求解动平台运动空间,省去一般螺旋理论求互易螺旋需求解线性方程的过程,算法简洁,可以得到并联机构自由度的符号表达式,从而实现自动化分析。

Analyzing mobility of parallel mechanisms digitally and automatically is the foundation of automatic type synthesis of parallel mechanisms. The mobility of parallel mechanisms was analyzed automatically based on R （3,3） geometric algebra model, which was a R6 geometric algebra model. Firstly, since any screw can be represented by a known screw with rigid body transformations, and the rigid body transformations can be derived by the geometric relationship between the two screws, when the twists of the first joints of the limbs were known, all twists of limbs of a parallel mechanism can be symbolically expressed and automatically computed by the geometric relationships between the joints of limbs. Then, since the motion of the limb was the union of all twists of joints and the motion of the moving platform was the intersection of all motion of limbs, by taking the advantages that the geometric algebra had the representations of the intersection set and union set, the motion of all limbs and the moving platform can also be symbolically expressed and automatically computed. Moreover, the symbolic expression of the motion of moving platform was exactly the moblity of the parallel mechanism. And the degree of a parallel mechanism was given by the blade of the symbolic expression. Finally, some examples were discussed in details by using the C ＋ ＋ coding to verify the proposed algorithm. The algorithm that analyzing the mobility of parallel mechanisms automatically based on geometric algebra did not need to compute the constraint twists of limbs and solve linear equations, which made this algorithm faster and more efficiency. This algorithm set the stage for automatic type synthesis of parallel mechanism.