Abstract Coordinates play an essential role in the description of real world objects and physical processes. In robotics, coordinates are used to describe the kinematic structure and the kinematic and dynamic behavior. The description mostly takes place in charts, assigned by the observer of the robotic system. However, it is crucial that the described physical process does not depend on the coordinate choice of the observer. In this work we show the relation between coordinates and manipulability analysis. Manipulability measures are dependent of joint coordinates of the robot and task coordinates in the workspace of the robot. Both relations can be analyzed with tensor geometry. We remove the dependency on joint coordinates through the use of an appropriate metric. With the help of tensor contraction, the resulting induced metric in the workspace can be transformed into a coordinate invariant matrix. After applying eigenvalue decomposition on this matrix, we can visualize the dynamic manipulability of a robot as a coordinate invariant ellipsoid.

中文翻译：

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坐标在机器人可操纵性分析中的影响

摘要 坐标在描述现实世界的物体和物理过程中起着至关重要的作用。在机器人学中，坐标用于描述运动学结构以及运动学和动力学行为。描述主要发生在图表中，由机器人系统的观察者分配。然而，至关重要的是所描述的物理过程不依赖于观察者的坐标选择。在这项工作中，我们展示了坐标与可操纵性分析之间的关系。可操纵性度量取决于机器人的关节坐标和机器人工作空间中的任务坐标。这两种关系都可以用张量几何来分析。我们通过使用适当的度量来消除对关节坐标的依赖。在张量收缩的帮助下，工作空间中产生的诱导度量可以转换为坐标不变矩阵。在对该矩阵应用特征值分解后，我们可以将机器人的动态可操纵性可视化为坐标不变椭球。