基于混沌的刚体导引Burmester点的求解方法

A Chaos-based Approach to Obtain the Global Real Solutions of Burmester Points

刚体导引的 Burmester点的求解可以转化为非线性方程的求解问题。本文采用简单的 Newton- Raphson迭代 ,并将迭代过程视为非线性的动力学系统 ,利用使得系统产生混沌现象的 Julia集的点求解出方程的全部实数解。研究表明在使得方程 Jacobi矩阵的行列式为零的点的邻域内有属于 Julia集的点存在。根据这一发现 ,成功地解决了 Burmester点的求解问题。

A new approach to find all Burmester points in five-position case is proposed, taking the iteration of Newton-Raphson method as a nonlinear dynamic system in which Julia set leads to chaos. The approach is based on the result coming from a lot of numerical experiments which show that there must be points belong to the Julia set in the neighborhood of those points where the determinant of satisfies Jacobian matrix is equal to zero. The approach is illustrated through two examples.