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

2026-04-11

AI Summary

This work addresses the challenge that trajectory representations often fail to maintain consistent identification and generalization performance across different coordinate systems, as existing coordinate-invariant formulations are susceptible to measurement noise and suffer from singularities. To overcome these limitations, the paper proposes a Dual Upper-Triangular Invariant Representation (DUTIR), which leverages differential geometry and invariant theory to construct computable local coordinate-invariant features. This approach enables unified modeling of rigid-body motion and interaction-force trajectories under arbitrary coordinate frames. The method demonstrates significantly enhanced robustness against both singularities and sensor noise, achieving stable and consistent performance in trajectory segmentation, recognition, and prediction across multiple coordinate systems. Its effectiveness and broad applicability are validated through experiments in robotics and biomechanics.

Technology Category

• Computer Vision: Motion & Tracking
• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

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.

Problem

• coordinate-invariant representation
• trajectory identification
• singularities
• motion trajectories
• force trajectories

Innovation

• coordinate-invariant representation
• trajectory modeling
• singularity robustness
• Dual-Upper-Triangular Invariant Representation
• motion and force trajectories