机构运动链邻接矩阵的素数表示与同构判别

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The adjacent relation of mechanism kinematic chain is represented by a primary adjacency matrix by using primes to mark its components. An isomorphism identification matrix results from the primary adjacency matrix and prime products representing secondary adjacent relation, and uses its determinant and solution vector of linear equation system to identify mechanism isomorphism. Dynamically, the identification matrix is modified by new primes on the basis of the classification of solution vector elements. The result generates a solution vector with independent elements, which determines the positions of all nodes in the topological graph of a kinematic chain. Furthermore, a mapping relation, which results in a necessary and sufficient condition for isomorphism, is established between componential codes of two isomorphic mechanisms. As a result, this new identification method utilizes varying primes to virtually destruct the symmetry of the topological graph and change local and secondary adjacency relations among nodes under unchanged topological structure. Then, this method is also available for isomorphism identification of general undirected graphs. Applied examples show that it is very effective and reliable for automatic identification with code mapping of isomorphism mechanisms and polynomial algorithm for computation time.提出用素数标识机构运动链中的构件，根据运动链的邻接关系建立一个用素数表示的基本邻接矩阵。利用该基本邻接矩阵及反映次级邻接关系的素数乘积生成同构判别矩阵，给出利用判别矩阵的行列式和线性方程组的解矢量判别机构的同构或异构的方法。进一步根据解矢量元素分类情况用新的素数对判别矩阵进行动态修改，最后可以得到元素各不相同的解矢量，机构运动链拓扑图的全部节点的相对位置由该解矢量唯一定位，从而找到两个同构机构的构件标号映射关系，给出两个机构运动链同构的充分必要条件。判别方法创新之处在于利用素数动态赋予相应的构件，以虚拟破坏图结构的对称性并虚拟改变图节点的局部邻接、次级邻接关系，而图的实际结构并不改变。该方法也适用于一般无向图的同构判别。判别实例表明该方法非常有效，计算时间复杂度为多项式且可建立同构的标号映射关系，具有很高的可靠性，并且容易实现自动判别。