Hopf映射在机械臂四元数姿态规划中的应用研究

Application of Hopf mapping in quaternionorientation planning of manipulator

针对有些机械臂姿态规划的四元数插值算法比较复杂、导数不易求得、较难判断其插值是否平滑的问题,进行了运用四元数插值曲线判断插值平滑性的研究。提出了Hopf映射应用于四元数插值曲线,利用Hopf映射的降维特性,将机械臂四元数姿态规划的四维插值点转换为三维向量,在单位球面上绘制了四元数插值曲线,通过三维空间的插值曲线直观判断四维插值的平滑性;运用相关算例对该方法的可行性进行了验证。研究结果表明:该方法避免了复杂四元数插值函数的求导过程,能够判断四元数插值是否平滑,操作是否简便;球面立体插值法比球面线性插值法更适合于两个以上的多姿态四元数插值。

Aiming at the problems that some quaternion interpolation algorithms were comparatively complex,difficult to find the derivatives,and hard to judge whether the quaternion interpolation was smooth,a study on judging the smoothness of interpolation using quaternion interpolation curve was conducted.It was proposed that Hopf mapping was applied to quaternion interpolation curves,and the four-dimensional interpolated points of quaternion orientation planning of robot manipulator were converted to three-dimensional vectors relying on the dimensionality reduction feature of Hopf mapping.Consequently,quaternion interpolation curves were obtained on the unit sphere,and the judgment was made intuitively whether the four-dimensional interpolation were smooth depending on the interpolation curves in three-dimensional space.The feasibility of the proposed methodology was verified by relevant study case.The results indicate that the suggested methodology avoids the derivation of complex quaternion interpolation function,and can be applied well to judge whether quaternion interpolation is smooth,and the usage is simple.The Spherical and Quadrangle(SQUAD)is more suitable than Spherical Linear Interpolation(SLERP)for quaternion interpolation with more than two orientations.