三元数的影旋积及在关节型机器人运动分析中的应用

Rotated Projection Product of a Ternary Number and its Application in the Kinematical Analysis of the Articulated Robot

三元数作为一种新型被特别定义的复数化矢量模型,具有天然直观的几何意义,特别有助于三维空间问题的建模。在原创的三元数概念的基础上,从定义三元数的影旋积入手,先扩充三元数的涵义,再分析三元复变数的微分特性,进而得出结论:对三元复变数的微分运算可转化成对三元数的影旋积操作。这就为机械的空间运动分析解析求解提供了新路径,并以此方法对关节型简易机器人的端点速度和加速度进行了检验,一次计算同时得到了它的大小和方向。最后对3-RPS 3自由度并联机器人进行了运动学位置分析,从另一个角度验证了基于三元数的三维空间机构运动分析的解析求解是可能的。

As a new type of specially defined pluralized vector model,the ternary number has the naturally intuitive geometric significance and it is especially helps to the three-dimensional modeling for a spatial problem. Based on the original concept of a ternary number,starts from the definition of the rotated projection product of a ternary number,the meaning of the ternary number is extended at first,the differential features of a ternary complex variable is analyzed,and then the conclusion on the differential properties of a ternary complex variable is obtained. The differential operation to a ternary complex variable can be transformed as the operating of the rotated projection product of a ternary number. As this result,it provides a new path for the analytic solution of the spatial motion analysis of the some mechanism. The checkout of the endpoint velocity and acceleration for a three-joint simple robot is carried out by this means. Both their size and direction are obtained in the only calculation process at same time. Finally,the analysis of the position is carried out for the 3-RPS type of the 3-DOF parallel robot. From another point of view,above mentioned facts show and verify that it is possible and feasible for the analytic solution of the motion analysis of the three-dimensional spatial mechanism by means of the modeling tool of a ternary number.