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 coordinates 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 a 3D contour following task, invariant descriptors are shown to result in task representations that do not depend on the calibration of reference frames or sensor locations. Tuning of the optimal control problems is shown to be fast and intuitive. Similarly as for 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：过去已有研究提出点和刚体运动轨迹的不变描述符，作为运动识别与泛化的代表性任务模型。然而，目前尚无适用于表征接触任务中力轨迹的不变描述符。本文通过利用运动与力之间的对偶性，首次提出了力轨迹的不变描述符。根据轨迹由螺旋坐标还是向量坐标构成，本文给出了两类不变描述符。同时提供了通过最优控制从含噪测量中鲁棒计算不变描述符的方法与软件。基于三维轮廓跟踪任务的实验性人类示教，验证了不变描述符能够产生不依赖于参考系标定或传感器位置的任务表征。最优控制问题的调整过程被证明快速且直观。与自由空间运动类似，所提出的运动和力轨迹不变描述符有望用于接触操作中约束运动（如物体操控）的识别与泛化。