非线性方程组解集边界求解方法及其在并联机构分析中的应用

并联机构的结构使其输入和输出运动之间具有复杂的非线性关系，在该类机构的运动学、动力学、作业空间、误差分析及运动控制中均涉及大量的非线性方程组求解。介绍一种含参数的非线性方程组的解集边界求解方法，基于流形理论和数值化连续算法，可直接搜索出一个非线性系统的解集边界，计算速度快、效率高。利用上述算法，对一台实际的4自由度并联机床进行了作业空间边界的求解和分析，验证了算法的实用性和有效性。

METHOD FOR FINDING SOLUTION BOUNDARY OF A NONLINEAR EQUATION SYSTEM AND ITS APPLICATION IN PARALLEL MECHANISMS

The structures of parallel mechanisms result in a nonlinear relationship between their input and output motions, so a great many nonlinear equations are involved in such questions as the analysis of kinematics, dynamics, workspace space and error, and moving control of parallel kinematic machines. Based on manifold theory and computational continuation methods, a new approach to numerical calculation and analysis of the boundary of solution set of a parameterized nonlinear equation system is introduced. A Jacobian matrix's row rank deficiency condition is explored as the criterion for the boundary of solution set of a nonlinear equation system. A numerical method for mapping the boundary is developed, and it can directly calculate the boundary of the solution set effectively and rapidly. At last, an example, which is involved in calculating the workspace boundary of a real 4 degree-of-freedom parallel kinematic machine tool, is analyzed numerically. The examination results show that this method is suitable for solving the boundary of a nonlinear equation system, and can be used in analysis, design and control of parallel mechanisms.