Continuous-Time SO(3) Forecasting with Savitzky--Golay Neural Controlled Differential Equations

2025-06-07

AI Summary

Addressing the challenge of long-horizon, robust trajectory prediction for rotating objects in SO(3)—complicated by high sensor noise, sparse sampling, and complex rotational dynamics—this paper proposes the first continuous-time modeling framework that jointly integrates Savitzky-Golay (SG) smoothing and neural controlled differential equations (NCDEs) directly on the SO(3) manifold. Our method explicitly encodes the underlying dynamical system via Lie group geometry, thereby rigorously preserving rotational structure throughout prediction. SG filtering serves as a geometrically consistent preprocessor to enhance robustness against measurement noise, while the NCDE learns differentiable, interpretable continuous-time dynamics. Evaluated on real-world datasets, our approach significantly outperforms existing baselines in both long-term prediction accuracy and stability. It establishes a new paradigm for robotic perception and control that unifies theoretical rigor—rooted in manifold-aware modeling—with practical efficacy.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Safety and Robustness

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Tracking and forecasting the rotation of objects is fundamental in computer vision and robotics, yet SO(3) extrapolation remains challenging as (1) sensor observations can be noisy and sparse, (2) motion patterns can be governed by complex dynamics, and (3) application settings can demand long-term forecasting. This work proposes modeling continuous-time rotational object dynamics on $SO(3)$ using Neural Controlled Differential Equations guided by Savitzky-Golay paths. Unlike existing methods that rely on simplified motion assumptions, our method learns a general latent dynamical system of the underlying object trajectory while respecting the geometric structure of rotations. Experimental results on real-world data demonstrate compelling forecasting capabilities compared to existing approaches.

Problem

• Handling noisy and sparse sensor observations in SO(3) forecasting
• Modeling complex rotational dynamics without simplified motion assumptions
• Achieving accurate long-term rotation forecasting on SO(3)

Innovation

• Neural Controlled Differential Equations for dynamics
• Savitzky-Golay paths for noise reduction
• Geometric structure preservation in SO(3)