基于四元数的6—UPS机构奇异性分析

Singularity Analysis of 6-UPS Mechanism based on Quaternion

采用四元数法描述6—UPS机构位姿矩阵,并把该矩阵和矢量坐标扩展为4维形式,推导出机构的运动方程,通过求导得到了机构运动的新型雅克比矩阵JA和JB,其中,JA是用四元数来描述并且不含超越变量。根据四元数的性质,把矩阵JA转化为8维方阵,使之适用于分析位置奇异和位姿奇异。分别把JA和JB的行列式展开,得到机构第一类奇异和第二类奇异的轨迹方程,利用MATLAB得到了机构在给定位置时的第一类位姿奇异轨迹和第二类位姿奇异三维轨迹曲面。实例验算证明了该方法所得矩阵求解方便的优点。

The 6-UPS mechanism posture matrix is described by using quaternion, and both the rotation matrix and the coordinates of the vectors are expanded to 4 dimensions, and the kinematical e- quations is obtained. Through derivation, the new type of Jacobi matrix JA and JB of mechanism motion are obtained, and JA is described by using quaternion and do not contain beyond variables. According to the nature of the quaternion, the JA is translated into 8 dimensions square matrix, and make it applica- ble to the analysis of singular position and posture singular. The expansion of a determinant JA and J- is respectively carried out, the trajectory equation of first kind of singularity and the second kind of sin- gularity are obtained, and their three-dimensional trajectory surfaces are obtained by using MATLAB when the location of the mechanism is given. The advantages of solving convenient of using the matrix obtained by this method is proved.