基于拟Newton法的并联机构位置正解

Direct Position Analysis of Parallel Mechanism Based on Quasi-Newton Method

基于Newton法的迭代搜索算法是求解并联机构位置正解的重要数值算法,但是在其每一步的迭代过程中都需要构造机构的Jacobian矩阵。在Newton法的基础上,将拟Newton法应用于并联机构的位置正解求解,该方法用当前的函数值代替Jacobian矩阵,能够减小每一迭代步的计算量。定义机构的虚工作空间,并分析6-RUS这一类并联机构虚工作空间受限的原因及迭代搜索算法在求解这一类机构位置正解时的局限性,提出将这一类机构的位置正解等效求解的方法。进一步分析耦合型少自由度机构虚工作空间受限的原因,采用虚设机构法和改进的Jacobian矩阵使迭代搜索算法能够适用于这一类机构。数值算例表明:相比于Newton法,拟Newton法的总迭代步数并没有明显增加,但由于每一迭代步的计算量少,计算效率明显提高,为并联机构位置正解在实时场合的应用提供了一定的理论指导;等效机构法能够扩大机构的虚工作空间,增加迭代搜索算法的适用范围。

The iterative search algorithm, based on Newton’s method, is an important numerical algorithm for solving the direct position analysis of parallel mechanism. But, the Jacobian matrix of the mechanism is needed at each step of the iterative process. On the basis of Newton’s method, the quasi-Newton method is applied to solve the direct position analysis of parallel mechanism, in which the Jacobian matrix is instead of the current value of the function. So, the amount of the computation of each iteration can be reduced. A virtual workspace organization is defined, and the reason why the virtual workspace of 6-RUS is constralned is analyzed. Also the limitations of the iterative search algorithm when is applied to solve the direct position analysis of this kind mechanisms are analyzed, and an equivalent method is proposed. Further, the restriction of the virtual workspace of coupled fewer degrees of freedom mechanism is analyzed. The use of dummy mechanisms and improved Jacobian matrix makes the iterative search algorithm can be applied to this type of mechanisms. Numerical examples show that：compared to Newton’s method, the total number of iterations of quasi-Newton’s method has not increased significantly, but due to the small amount of computation for each iteration, the computational efficiency significantly improved, providing some theoretical guidance to the direct position analysis of parallel mechanism in the application of real-time occasions. The use of the equivalent method expands the virtual workspace of the mechanism, increasing the scope of application of the iterative search algorithm.