The inverse kinematics (IK) problem for many common robot manipulators may be decomposed into canonical subproblems which are solved by finding the angles on circles where they intersect with other geometric objects. We present new algebraic solutions and geometric interpretations for six subproblems using a linear algebra approach, and we demonstrate significant computational performance improvements over existing IK methods. We show that IK for any 6-dof all revolute (6R) robot with three intersecting or parallel joint axes may be solved in closed form using subproblem decomposition. For any other 6R robot, subproblem decomposition reduces finding all IK solutions to a search over one or two joint angles. The first three subproblems, called the Paden-Kahan subproblems, are Subproblem 1: Circle and Point, Subproblem 2: Two Circles, and Subproblem 3: Circle and Sphere. The other three subproblems, which have not been extensively covered in the literature, are Subproblem 4: Circle and Plane, Subproblem 5: Three Circles, and Subproblem 6: Four Circles. Our approach also finds the least-squares solutions for Subproblems 1-4 when the exact solution does not exist.

翻译：许多常见机器人操作臂的逆运动学问题可分解为若干规范子问题，这些子问题通过求解圆与其他几何对象的交点角度来获得解答。我们采用线性代数方法，为六个子问题提出了新的代数解法和几何解释，并证明了相较于现有逆运动学方法在计算性能上的显著提升。研究表明，对于任何具有三个相交或平行关节轴的六自由度全旋转关节（6R）机器人，可通过子问题分解求得其闭式解；而对于其他类型的6R机器人，子问题分解可将全部逆运动学解的求解过程降维至对一至两个关节角的搜索。前三个子问题即Paden-Kahan子问题包括：子问题1（圆与点）、子问题2（两圆相交）、子问题3（圆与球面）。文献中尚未广泛探讨的其余三个子问题为：子问题4（圆与平面）、子问题5（三圆相交）、子问题6（四圆相交）。本文方法在子问题1-4无精确解时亦可给出最小二乘解。