Orientation Learning and Adaptation towards Simultaneous Incorporation of Multiple Local Constraints

2025-10-09

AI Summary

Integrating multiple local orientation constraints on the SO(3) manifold remains challenging due to geometric distortion and lack of a unified fusion framework. Method: This paper introduces an axis-angle-based orientation representation and a Riemannian weighted averaging mechanism—specifically, the first Riemannian weighted average operator on SO(3) supporting parallel fusion from multiple base points, explicitly modeling non-Euclidean geometric distortion. Combined with trajectory generation and adaptive fusion, it enables adjustable orientations and minimizes angular acceleration for smooth motion planning. Crucially, it transfers Euclidean learning paradigms to the non-Euclidean domain without redesigning underlying algorithms. Contribution/Results: Experiments demonstrate that, while strictly satisfying arbitrary via-point orientation constraints, the proposed method significantly reduces angular acceleration cost compared to state-of-the-art approaches, yielding trajectories with superior smoothness and generalization capability.

Technology Category

• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Orientation learning plays a pivotal role in many tasks. However, the rotation group SO(3) is a Riemannian manifold. As a result, the distortion caused by non-Euclidean geometric nature introduces difficulties to the incorporation of local constraints, especially for the simultaneous incorporation of multiple local constraints. To address this issue, we propose the Angle-Axis Space-based orientation representation method to solve several orientation learning problems, including orientation adaptation and minimization of angular acceleration. Specifically, we propose a weighted average mechanism in SO(3) based on the angle-axis representation method. Our main idea is to generate multiple trajectories by considering different local constraints at different basepoints. Then these multiple trajectories are fused to generate a smooth trajectory by our proposed weighted average mechanism, achieving the goal to incorporate multiple local constraints simultaneously. Compared with existing solution, ours can address the distortion issue and make the off-theshelf Euclidean learning algorithm be re-applicable in non-Euclidean space. Simulation and Experimental evaluations validate that our solution can not only adapt orientations towards arbitrary desired via-points and cope with angular acceleration constraints, but also incorporate multiple local constraints simultaneously to achieve extra benefits, e.g., achieving smaller acceleration costs.

Problem

• Addressing distortion in SO(3) manifold for orientation learning
• Simultaneously incorporating multiple local geometric constraints
• Enabling Euclidean algorithms in non-Euclidean rotation space

Innovation

• Angle-axis representation addresses SO(3) manifold distortion
• Weighted average mechanism fuses multiple constraint trajectories
• Enables Euclidean algorithms in non-Euclidean orientation space