  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.
