Numerical methods for Inverse Kinematics (IK) employ iterative, linear approximations of the IK until the end-effector is brought from its initial pose to the desired final pose. These methods require the computation of the Jacobian of the Forward Kinematics (FK) and its inverse in the linear approximation of the IK. Despite all the successful implementations reported in the literature, Jacobian-based IK methods can still fail to preserve certain useful properties if an improper matrix inverse, e.g. Moore-Penrose (MP), is employed for incommensurate robotic systems. In this paper, we propose a systematic, robust and accurate numerical solution for the IK problem using the Mixed (MX) Generalized Inverse (GI) applied to any type of Jacobians (e.g., analytical, numerical or geometric) derived for any commensurate and incommensurate robot. This approach is robust to whether the system is under-determined (less than 6 DoF) or over-determined (more than 6 DoF). We investigate six robotics manipulators with various Degrees of Freedom (DoF) to demonstrate that commonly used GI's fail to guarantee the same system behaviors when the units are varied for incommensurate robotics manipulators. In addition, we evaluate the proposed methodology as a global IK solver and compare against well-known IK methods for redundant manipulators. Based on the experimental results, we conclude that the right choice of GI is crucial in preserving certain properties of the system (i.e. unit-consistency).

翻译：逆运动学（IK）的数值方法采用迭代线性近似，使末端执行器从初始位姿运动至期望最终位姿。这些方法需要在IK线性近似中计算正运动学（FK）的雅可比矩阵及其逆矩阵。尽管文献中已有大量成功实现案例，但当为不可通约机器人系统使用不恰当的矩阵逆（如Moore-Penrose逆）时，基于雅可比矩阵的IK方法可能无法保持某些有用性质。本文提出一种系统、鲁棒且精确的IK问题数值解法，该方法对任意可通约与不可通约机器人推导出的各类雅可比矩阵（如解析、数值或几何雅可比）均适用，采用混合（MX）广义逆进行处理。该方法对欠定（自由度低于6）或过定（自由度高于6）系统均具有鲁棒性。我们研究了六种不同自由度（DoF）的机器人操作臂，证明当不可通约机器人操作臂的单位发生变化时，常用广义逆无法保证相同的系统行为。此外，我们将所提方法作为全局IK求解器进行评估，并与冗余操作臂已知的经典IK方法进行比较。基于实验结果，我们得出结论：正确选择广义逆对于保持系统的某些性质（如单位一致性）至关重要。