三维几何约束的球面几何求解

3D Geometric Constraint Solution Based on Spherical Geometry

研究了球面几何学在三维几何约束求解中的应用,提出了球面求解法.该方法建立在姿态约束与位置约束解耦的基础上,并以求解关键的姿态约束为主,一旦姿态约束被解出,则位置约束很容易求解;同时将表征刚体姿态的矢量映射到球平面上的点,将姿态约束映射为球平面上两点的距离,借助球面几何的知识,能够高效、直观地推理出多数情况下姿态约束的解析解,而特殊的情况则结合数值法求解,并很好地解决了数值法的初值问题.

The paper studies the application of spherical geometry to the field of 3D geometric constraint solution and presents a geometric constraint solution method, called spherical solution method. The method is based on decoupling the constraints into postural constraints and positional constraints, and the postural constraint solution is emphasized as once the postural constraints are solved, the positional ones can be solved easily. The method maps the vectors representing the postures of rigid bodies to the points on spherical plane, and maps postural constraints to the distances between the points. Via spherical geometry, the method can infer efficiently and intuitively the analytical solution of postural constraints in most cases. For the other cases, the postural constraints can be solved by a mixed algorithm of numerical solution method and spherical solution method.