Invariant descriptors of point and rigid-body motion trajectories have been proposed in the past as representative task models for motion recognition and generalization. Currently, no invariant descriptor exists for representing force trajectories, which appear in contact tasks. This paper introduces invariant descriptors for force trajectories by exploiting the duality between motion and force. Two types of invariant descriptors are presented depending on whether the trajectories consist of screw or vector coordinates. Methods and software are provided for robustly calculating the invariant descriptors from noisy measurements using optimal control. Using experimental human demonstrations of 3D contour following and peg-on-hole alignment tasks, invariant descriptors are shown to result in task representations that do not depend on the calibration of reference frames or sensor locations. The tuning process for the optimal control problems is shown to be fast and intuitive. Similar to motions in free space, the proposed invariant descriptors for motion and force trajectories may prove useful for the recognition and generalization of constrained motions, such as during object manipulation in contact.

翻译：过去曾提出用于点和刚体运动轨迹的不变描述子，作为运动识别与泛化的代表性任务模型。目前尚不存在适用于接触任务中力轨迹的不变描述子。本文通过利用运动与力之间的对偶性，提出了力轨迹的不变描述子。根据轨迹由螺旋坐标还是矢量坐标构成，给出了两类不变描述子。还提供了通过最优控制从含噪声测量中稳健计算不变描述子的方法与软件。基于实验性人类演示的三维轮廓跟踪和销孔对齐任务，结果表明不变描述子能够生成不依赖于参考系标定或传感器位置的任务表示。最优控制问题的调参过程被证明快速且直观。与自由空间中的运动类似，本文提出的运动与力轨迹的不变描述子可能有助于受约束运动（如接触物体操作）的识别与泛化。