Seven degree-of-freedom (DOF) robot arms have one redundant DOF which does not change the motion of the end effector. The redundant DOF offers greater manipulability of the arm configuration to avoid obstacles and singularities, but it must be parameterized to fully specify the joint angles for a given end effector pose. For 7-DOF revolute (7R) manipulators, we introduce a new concept of generalized shoulder-elbow-wrist (SEW) angle, a generalization of the conventional SEW angle but with an arbitrary choice of the reference direction function. The SEW angle is widely used and easy for human operators to visualize as a rotation of the elbow about the shoulder-wrist line. Since other redundancy parameterizations including the conventional SEW angle encounter an algorithmic singularity along a line in the workspace, we introduce a special choice of the reference direction function called the stereographic SEW angle which has a singularity only along a half-line, which can be placed out of reach. We prove that such a singularity is unavoidable for any parameterization. We also include expressions for the SEW angle Jacobian along with singularity analysis. Finally, we provide efficient and singularity-robust inverse kinematics solutions for most known 7R manipulators using the general SEW angle and the subproblem decomposition method. These solutions are often closed-form but may sometimes involve a 1D or 2D search in the general case. Search-based solutions may be converted to finding zeros of a high-order polynomial. Inverse kinematics solutions, examples, and evaluations are available in a publicly accessible repository.

翻译：七自由度（DOF）机器人手臂具有一个不影响末端执行器运动的冗余自由度。该冗余自由度增强了机械臂构型的可操作性，有助于避开障碍物和奇异点，但必须对其进行参数化处理，以完全指定给定末端执行器位姿下的关节角度。针对七自由度旋转（7R）机械臂，我们提出了一种广义肩-肘-腕（SEW）角的新概念，该概念是对传统SEW角的推广，但允许任意选择参考方向函数。SEW角被广泛使用，便于人类操作员直观理解肘部绕肩-腕连线的旋转。由于其他冗余参数化方法（包括传统SEW角）在工作空间中的某条线上会遇到解析奇异点，我们引入了一种特殊的参考方向函数——立体SEW角，其奇异性仅沿半条线存在，且可将此半线置于不可达区域。我们证明此类奇异性对于任何参数化方法都是不可避免的。我们还给出了SEW角雅可比矩阵的表达式及奇异性分析。最后，我们利用广义SEW角和子问题分解方法，为大多数已知七自由度旋转机械臂提供了高效且鲁棒的逆运动学解。这些解通常为闭式解，但在一般情形下可能涉及一维或二维搜索。基于搜索的解可转化为求解高阶多项式的零点。逆运动学解、示例及评估结果已存储于公开可访问的代码库中。