三维几何约束求解的变分算法

研究了运动学变分原理在三维几何约束求解中的应用，提出了变分求解算法．该算法采用相对坐标，将求解域从笛卡儿空间转换到相对坐标空间．对于约束开环，依次选取相对坐标即可获得满足几何约束的刚体位姿；对于约束闭环，通过切断闭环铰，将约束闭环转化为等价的广义开环与切断铰约束代数方程，其切断铰约束代数方程的相对坐标雅克比矩阵解析式可通过变分关系显式获得．最后通过实例验证，说明该算法具有较高的求解效率与稳定性．

Solving 3D Assembly Geometric Constraints Based on Variation Method

The paper studies the application of variation principle in the field of 3D assembly geometric constraint solution, and an algorithm based on kinematics variation principle is presented. By applying the relative coordinate method, the algorithm transforms the parametric space from the Cartesian coordinate space to the relative coordinate space. In the case of open chain constraint system, 3D assembly geometric constraints can be satisfied by sequential choice of the relative coordinates. For the closed chain constraint system, by cutting constraints, the algorithm converts the closed chain to a generalized open chain and a system of equations o{ cutting constraints. In addition, the Jacobin matrix formulation of equations of cutting constraints can be analytically obtained kinematics variation principle. In the end, the presented algorithm is validated by an example of 3D geometric closed chain constraint system.