基于共形几何代数求解(4SPS+SPR)+(2RPS+SPR)串并联机构位置正解

CGA-based Approach to Solve the Forward Position Solution of the(4SPS+SPR)+(2RPS+SPR)Serial-parallel Manipulator

提出将共形几何代数引入至串并联机构位置正解求解过程,以(4SPS+SPR)+(2RPS+SPR)串并联机构为例,首先分别针对上下层并联机构选择合理的运动学参数,基于共形几何代数中基本几何元素的表示方法,对机构中相应球面、平面等共形几何表达式进行求交或对偶运算,得到上下层并联机构动平台顶点的共形几何表达。再结合机构中尺寸、几何约束和内积运算,建立上下层并联机构正向位置解的一元高次方程,进而获得上下层并联机构动平台相对于基平台的位姿。在此基础上,基于共形几何代数中刚体运动变换表达式,得到串并联机构动平台顶点的共形几何表达,进而获得串并联机构的位置正解。该方法避免了传统方法中复杂的消元运算,且分析过程几何直观性较强,在简化串并联机构位置正解几何建模方面表现出巨大优势。

Conformal geometric algebra(CGA)is introduced into the forward position solution of the series-parallel mechanism.Taking the(4SPS+SPR)+(2RPS+SPR)serial-parallel mechanism as an example,for each parallel manipulator,through reasonably selecting the kinematic parameters,some basic geometric elements,such as sphere and plane,are constructed based on the specific operation rules of CGA,and the geometric expressions of the vertex position for the moving platform can be obtained by the intersection and duality operations of the basic geometric elements.Then,combined with the dimension,geometric constraints of the mechanism and the inner product operation in CGA,the univariate higher order equation of the forward kinematics for each mechanism can be derived,and then the pose of the moving platform relative to the base for each mechanism can be obtained.Based on this result,combined with the rigid body motion transformation in CGA,the position of the moving platform for the serial-parallel mechanism is obtained.The CGA-based method can avoid the complex elimination operations in the traditional method,and the analysis process is more geometric intuitive,which shows great advantages in simplifying the geometric modeling of the forward solution for serial-parallel mechanism.