基于实四元数的大旋转角三维坐标转换的改进谱修正迭代解法

提出一种基于单位实四元数的大旋转角三维坐标转换病态问题的新方法，该方法用单位实四元数构造旋转矩阵，可避免复杂的三角函数求导，易于线性化，系数矩阵更为简洁；考虑到模型法方程矩阵的病态性，引入岭参数和泛函矩阵，从而降低了方程病态性带来的不利影响，使方程求解达到稳定，同时方程迭代求解时解的估计值接近真值的程度较谱修正迭代法高。利用模拟及实测数据对算法进行验证，结果表明，该算法具有收敛速度快、不依赖转换参数初值、全局收敛、解为无偏、便于程序实现等优点，可为通用坐标转换提供一种新途径。

The Improved Iteration Method by Correcting Characteristic Value for Transformation of Three-Dimensional Coordinates Based on Large Rotation Angle and Quaternions

We propose a new method for ill-posed problems in the large rotation angle three-dimensional coordinate transformation based on the unit quaternion. This method constructs a rotation matrix by unit quaternion, which avoids complex trigonometric function derivations, is easy to linearize, and has a more concise coefficient matrix. Considering the ill-posedness of the normal equation matrix, the ridge parameter and the functional matrix are introduced, which reduces the adverse effects caused by the ill-posedness of the equation, and generates a stable equation solution. At the same time, the estimated value of the solution when the equation is iteratively solved is closer to the true value than the traditional iteration method by correcting characteristic value. The algorithm is verified by simulation and measured data, showing that the algorithm has the characteristics of fast convergence, no initial value of conversion parameters, global convergence, unbiased solution and easy program implementation. It provides a new way for general coordinate transformation.