Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures

Learning unknown dynamics under environmental (or external) constraints is fundamental to many fields (e.g., modern robotics), particularly challenging when constraint information is only locally available and uncertain. Existing approaches requiring global constraints or using probabilistic filtering fail to fully exploit the geometric structure inherent in local measurements (by using, e.g., sensors) and constraints. This paper presents a geometric framework unifying measurements, constraints, and dynamics learning through a fiber bundle structure over the state space. This naturally induced geometric structure enables measurement-aware Control Barrier Functions that adapt to local sensing (or measurement) conditions. By integrating Neural ODEs, our framework learns continuous-time dynamics while preserving geometric constraints, with theoretical guarantees of learning convergence and constraint satisfaction dependent on sensing quality. The geometric framework not only enables efficient dynamics learning but also suggests promising directions for integration with reinforcement learning approaches. Extensive simulations demonstrate significant improvements in both learning efficiency and constraint satisfaction over traditional methods, especially under limited and uncertain sensing conditions.

Learning unknown dynamics under safety-critical constraints--exemplified by autonomous vehicles in unmapped environments--represents a foundational robotics challenge. Traditional methods decouple dynamics learning from sensor-state relationships, yielding brittle solutions.Our framework uses differential geometry to unify sensor measurements, constraints, and dynamic learning. Modeling robot state-sensor interactions as fiber bundles reveals that sensor uncertainty follows structured physical laws. For example, LiDAR precision depends on obstacle proximity while safety boundaries adapt geometrically. The geometric connection encodes how motion alters measurement reliability--coupling dynamics and sensor physics.Key Advances: We establish (1) geometric unification through differential structures, (2) symmetry-aware learning via equivariant architectures, and (3) adaptive safety certification responding to sensing quality. Simulations show significant improvements in learning efficiency and safety compliance under real-world constraints. This establishes a rigorous geometric foundation for autonomous systems in uncertain environments, with applications in robotics and self-driving vehicles.