BILTS: A novel bi-invariant local trajectory-shape descriptor for rigid-body motion

Measuring the similarity between motions and established motion models is crucial for motion analysis, recognition, generation, and adaptation. To enhance similarity measurement across diverse contexts, invariant motion descriptors have been proposed. However, for rigid-body motion, few invariant descriptors exist that are bi-invariant, meaning invariant to both the body and world reference frames used to describe the motion. Moreover, their robustness to singularities is limited. This paper introduces a novel Bi-Invariant Local Trajectory-Shape descriptor (BILTS) and a corresponding dissimilarity measure. Mathematical relationships between BILTS and existing descriptors are derived, providing new insights into their properties. The paper also includes an algorithm to reproduce the motion from the BILTS descriptor, demonstrating its bidirectionality and usefulness for trajectory generation. Experimental validation using datasets of daily-life activities shows the higher robustness of the BILTS descriptor compared to the bi-invariant ISA descriptor. This higher robustness supports the further application of bi-invariant descriptors for motion recognition and generalization.

Enhancing motion trajectory segmentation of rigid bodies using a novel screw-based trajectory-shape representation

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 submotions 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 Motion and Force Trajectories for Interpreting Object Manipulation Tasks in Contact

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.