基于共形几何代数的并联机器人逆运动学分析方法

运动学分析是并联机器人运动学性能评估和结构尺寸优化的基础。现有并联机器人运动学分析方法存在几何建模与几何计算相分离的问题，本文利用共形几何代数(Conformal geometric algebra, CGA)集几何表示和计算为一体的优势，提出一种并联机器人逆运动学分析方法。根据动平台位姿参数给出动平台刚体运动算子，通过共形几何代数框架下的几何积实现动平台上任意点的刚体变换，得到任意点在运动过程中的共形几何表达式；结合机构中尺寸、几何约束，利用内积运算，建立机构运动学方程；根据运动学方程，进行运动学反解计算和速度分析。以3自由度的3-RPS并联机器人和6自由度6-UPS并联机器人为例，对所提方法进行验证，并将逆运动学推导结果与仿真软件所得结果进行了对比，验证了本文提出方法的正确性。该方法将空间向量和旋转表示等几何对象与矩阵乘法、矢量外积等计算方式相结合，使得并联机器人空间几何问题统一在一个代数系统中进行处理，因此分析过程几何直观性较强，简化了运动学逆解分析计算过程。

Inverse Kinematics Analysis Method of Parallel Robot Based on Conformal Geometry Algebra

Kinematics analysis is the basis of kinematics performance evaluation and structure size optimization of parallel mechanism. The existing kinematics analysis methods of parallel robots have the problem of separating geometric modeling from geometric calculation. A method for inverse kinematics analysis of parallel robots was proposed by taking advantage of conformal geometric algebra (CGA), which integrated geometric representation and calculation. Firstly, the rigid body transformation of any point on the moving platform was realized by the geometric product under the framework of conformal geometry algebra, and then the conformal geometric expression of any point in the process of motion was obtained. Then the kinematic equation of the mechanism was established by using the inner product operation in combination with the dimension and geometry constraints of the mechanism. Finally, according to the kinematics equation, the inverse kinematics was calculated and the velocity was analyzed. A 3-RPS parallel robot of three degrees of freedom and a 6-UPS parallel robot of six degrees of freedom were taken as examples to verify the correctness of the proposed method, and the results of inverse kinematics were compared with those obtained by simulation software. This method combined the geometric objects such as space vector and rotation representation with the calculation methods such as matrix multiplication and vector outer product, so that the spatial geometry problems of parallel mechanism were handled in a unified algebraic system, so the analysis process was geometrically intuitive, and the analysis and calculation process of inverse kinematics were simplified.