基于共形几何代数的空间并联机构位置正解

将共形几何代数(CGA)引入空间并联机构位置正解中,提出了一种空间3-RPS并联机构位置正解新算法。以任意一条支链轴线与静平台平面的夹角为待求变量,基于点的CGA表达方法建立了该支链与动平台连接的铰接点关于待求变量的数学表达式;通过2次构造2个空间球和1个平面的外积,分别获得动平台其余2个铰接点的点对;利用距离公式,只需简单的平方运算可直接推导出该问题关于待求变量的一元16次输入输出方程,进而获得了该机构的全部16组解析解,无增无漏。该方法没有繁琐的坐标变换和矩阵计算,以及复杂的多元高次非线性方程组消元求解。通过数字实例计算表明,求解过程较清晰地揭示出机构运动的几何特点,几何直观性好。

Direct kinematics of a spatial parallel mechanism based on conformal geometric algebra

An algorithm is proposed for the direct kinematics analysis of a spatial general 3-RPS parallel mechanism based on conformal geometric algebra (CGA).The angle between the axis of an arbitrary kinematic chain and the plane of the fixed platform can be regarded as the unknown variable.The mathematical expression of the position of the spherical joint connecting the moving platform with the kinematic chain can be expressed in the unknown variable based on CGA.The outer product of two space balls and a fiat surface are constructed two times,and the corresponding points of the remaining two vertices of the moving platform are obtained respectively.The 16th degree input-output polynomial equation in the unknown variable is straightforwardly obtained by distance formula and all 16 sets of closed-form solutions can be achieved.The algorithm avoids the use of rational angles or matrices,and complex computations for nonlinear and multivariable equations.A numerical example is given to demonstrate geometric characteristics of the motion and the algorithm is intuitive.