Trajectory segmentation refers to dividing a trajectory into meaningful consecutive sub-trajectories. This paper focuses on trajectory segmentation for 3D rigid-body motions. Most segmentation approaches in the literature represent the body’s trajectory as a point trajectory, considering only its translation and neglecting its rotation. We propose a novel trajectory representation for rigid-body motions that incorporates both translation and rotation, and additionally exhibits several invariant properties. This representation consists of a geometric progress rate and a third-order trajectory-shape descriptor. Concepts from screw theory were used to make this representation time-invariant and also invariant to the choice of body reference point. This new representation is validated for a self-supervised segmentation approach, both in simulation and using real recordings of human-demonstrated pouring motions. The results show a more robust detection of consecutive sub-motions with distinct features and a more consistent segmentation compared to conventional representations. We believe that other existing segmentation methods may benefit from using this trajectory representation to improve their invariance.

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 article 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 3-D 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.

Learning from Demonstration (LfD) requires methodologies able to generalize tasks in new situations. This paper studies the use of virtual demonstrations to extend the extrapolation capabilities of probabilistic motion models such as the traPPCA method. Similarly to other LfD methods, traPPCA is able to calculate new trajectories very fast, but does not generalize well outside the area covered by the demonstrations. Another approach, the invariants method, shows outstanding generalization capabilities thanks to its shape-preserving prop-erties, while being limited by long computation times. The pro-posed methodology combines the advantages of the two methods by learning traPPCA models using virtual demonstrations generated by the invariants method. The proposed approach is analyzed in three case studies. Furthermore, a comparison is made between learning with virtual demonstrations and learning with only real demonstrations. The results encourage the use of virtual demonstrations to extend the extrapolation capabilities of probabilistic motion models and hence reduce the required number of real demonstrations. The latter has the potential of reducing the cost of commissioning robot tasks.

Invariant representations of demonstrated motion trajectories provide context-independent motion models that can be used in motion recognition and generalization applications such as robot programming by demonstration. In practice, the use of invariant representations is still limited because their numerical calculation from a demonstrated trajectory is complicated by sensitivity to measurement noise and singularities, yielding inaccurate invariant functions that do not correspond well with the original trajectory. This paper improves the calculation of invariant representations for point and rigid-body motions by reformulating their calculation as an optimization problem that minimizes the error between the trajectory reconstructed from the invariant representation and the measured trajectory. Robustness against noise and singularities is ensured through the addition of regularization terms on the invariants. Simulations and real motion experiments show that the accuracy of the calculated invariant representations greatly improves with respect to standard smoothing methods. These results encourage future developments of motion recognition and generalization applications based on invariant trajectory representations.

In programming by demonstration, generalization is necessary to apply demonstrated motions in novel situations. Many existing trajectory representations have poor generalization capabilities since they are built on trajectory coordinates that depend on the context in which the motion is recorded. In order to generalize, the user is typically required to perform a large number of varied demonstrations. This paper instead emphasizes the usefulness of an invariant trajectory representation to separate essential motion information from context-specific information of the recorded demonstrations. The invariants are interpreted as the control inputs of a dynamical system describing the evolution of the trajectory. New trajectories are generated for novel situations as the solution of a constrained optimal control problem in which context-specific information of the novel situation is encoded in the constraints. Results indicate how, starting from only a single demonstration, new trajectories can be generated in novel situations while maintaining similarity with the original demonstration. Invariance in trajectory representations therefore proves useful to reduce the number of necessary demonstrations to learn and apply new motions.

This paper presents an overview and comparison of minimal and complete rigid body motion trajectory descriptors, usable in applications like motion recognition and programming by demonstration. Motion trajectory descriptors are able to deal with potentially unwanted variations acting on the motion trajectory such as changes in the execution time, the motion's starting position, or the viewpoint from which the motion is observed. A suitable rigid body motion trajectory descriptor retains only the trajectory information relevant to the application. This paper compares different trajectory descriptors for rigid body motion and validates their usefulness for dealing with motion variation in a motion recognition experiment. Furthermore, a new type of invariant trajectory descriptor is introduced based on the Frenet-Serret formulas.