A Coordinate-Invariant Local Representation of Motion and Force Trajectories for Identification and Generalization Across Coordinate Systems

Identifying the trajectories of rigid bodies and of interaction forces is essential for a wide range of tasks in robotics, biomechanics, and related domains. These tasks include trajectory segmentation, recognition, and prediction. For these tasks, a key challenge lies in achieving consistent results when the trajectory is expressed in different coordinate systems. A way to address this challenge is to utilize trajectory models that can generalize across coordinate systems. The focus of this paper is on such trajectory models obtained by transforming the trajectory into a coordinate-invariant representation. However, coordinate-invariant representations often suffer from sensitivity to measurement noise and the manifestation of singularities in the representation, where the representation is not uniquely defined. This paper aims to address this limitation by introducing the novel Dual-Upper-Triangular Invariant Representation (DUTIR), with improved robustness to singularities, along with its computational algorithm. The proposed representation is formulated at a level of abstraction that makes it applicable to both rigid-body trajectories and interaction-force trajectories, hence making it a versatile tool for robotics, biomechanics, and related domains.