Inverse kinematics of many common types of robot manipulators may be decomposed into canonical subproblems. This paper presents new solution methods to six subproblems using a linear algebra approach. The first three subproblems, called the Paden-Kahan subproblems, are Subproblem 1: angle between a vector on the edge of a cone and a point, Subproblem 2: intersections between two cones, and Subproblem 3: intersections between a cone and a sphere. The other three subproblems, which have not been extensively covered in the literature, are Subproblem 4: intersections between a cone and a plane, Subproblem 5: intersections among three cones, and Subproblem 6: intersections in a system of four cones. We present algebraic solutions and geometric interpretations for each subproblem and provide computational performance comparisons. Our approach also finds the least-squares solutions for Subproblems 1-4 when the exact solution does not exist. We show that almost all 6-dof all revolute (6R) robots with known closed-form solutions may be solved using the subproblem decomposition method. For a general 6R robot, subproblem decomposition reduces finding all solutions to a search on a circle or a 2D torus. The software code is available on a publicly accessible repository.

翻译：许多常见类型机器人操控器的反动运动体,许多常见类型的机器人操控器的反动运动体,可能会被分解成共性子问题。本文件使用线性代数法,为六个子问题提出了新的解决方案方法。头三个子问题,称为 Paden-Kahan 子问题,是子问题1:在锥体边缘和点之间的矢量角度,子问题2:两个锥体和子问题3之间的交叉点:共和体和球体之间的交叉点。其他三个在文献中未广泛覆盖的子问题,是子问题4:共体与平面之间的交叉点,子问题5:三个锥体之间的交叉点,和子问题6:四个锥体的系统中的交叉点。我们为每个子质和点的矢量提出代数解决方案和几何解释,并提供计算性绩效比较。我们的方法还找到在准确解决方案不存在时,子问题1-4之间的最不易理解的解决方案。我们展示的是,一个共性2号之间的交叉交叉点,我们展示了整个机器人的搜索方法。