Learning Deep Dynamical Systems using Stable Neural ODEs
(2404.10622)Abstract
Learning complex trajectories from demonstrations in robotic tasks has been effectively addressed through the utilization of Dynamical Systems (DS). State-of-the-art DS learning methods ensure stability of the generated trajectories; however, they have three shortcomings: a) the DS is assumed to have a single attractor, which limits the diversity of tasks it can achieve, b) state derivative information is assumed to be available in the learning process and c) the state of the DS is assumed to be measurable at inference time. We propose a class of provably stable latent DS with possibly multiple attractors, that inherit the training methods of Neural Ordinary Differential Equations, thus, dropping the dependency on state derivative information. A diffeomorphic mapping for the output and a loss that captures time-invariant trajectory similarity are proposed. We validate the efficacy of our approach through experiments conducted on a public dataset of handwritten shapes and within a simulated object manipulation task.
Overview
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Introduces a new class of provably stable latent Dynamical Systems (DS) using Neural Ordinary Differential Equations (ODEs) for robotic task learning, enabling learning systems with multiple attractors and overcoming the need for state derivative information.
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Incorporates a novel corrective signal in the latent dynamics to ensure system stability across the entire phase space and proposes a diffeomorphic mapping for output space and a time-invariant trajectory similarity loss function to enhance learning from demonstrations.
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Demonstrates the framework's superior performance in trajectory learning through experiments on a dataset of handwritten shapes and a simulated object manipulation task, compared to existing DS learning methods.
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Suggests the potential of Stable Neural ODEs (StableNODEs) in advancing robotic capabilities through more adaptable, predictable, and safe execution of complex motion patterns, with implications for future robotics and AI research.
Provably Stable Latent Dynamical Systems with Multiple Attractors for Learning Robotic Tasks
Introduction to Latent Dynamical Systems in Robotics
Learning from demonstrations (LfD) has emerged as a powerful methodology enabling robots to acquire and execute complex behaviors by observing and imitating expert demonstrations, thereby bridging the gap between human expertise and robotic capabilities. This approach is pivotal for tasks requiring intricate motion patterns, smoothness, and stability—qualities that are challenging to encode in models directly. Among different frameworks for encoding such traits, Dynamical Systems (DS) stand out due to their robustness, flexibility, and the innate capability to ensure stability and smoothness. Despite these advantages, traditional DS learning frameworks are not without limitations, notably assuming a singular attractor, reliance on state derivative information, and measurability of the system's state during inference, which restricts their applicability and complicates the learning process.
Advancements in Neural ODEs and Stability
This paper introduces a class of provably stable latent DS that leverages the underlying principles of Neural Ordinary Differential Equations (ODEs) but significantly extends their utility by enabling the learning of DS with multiple attractors and without the necessity for state derivative information. This development is particularly advantageous in applications where such information is cumbersome or impossible to obtain directly. By incorporating a novel corrective signal into the latent dynamics, the framework ensures the stability of the learned system across the entire phase space. Additionally, it proposes a diffeomorphic mapping for the output space and a loss function capturing time-invariant trajectory similarity, facilitating the learning process from demonstrations.
Theoretical Foundations and Model Implementation
At the core of the proposed Stable Neural ODEs (StableNODEs) is a learning framework that guarantees Lyapunov stability, even in the presence of multiple attractors. This is achieved by a corrective signal that adjusts the nominal dynamics based on a candidate Lyapunov function's behavior, ensuring that the system's trajectories remain bounded and converge asymptotically to the designated attractors set. Moreover, the framework utilizes diffeomorphic mappings in the output space, enabling a direct and meaningful specification of attractors in terms of the task at hand, an aspect critical for tasks involving precise goal-directed behaviors.
Experimental Validation and Implications
The paper validates the proposed StableNODE framework through extensive experiments on a publicly available dataset of handwritten shapes and a simulated object manipulation task, demonstrating superior performance in trajectory learning when compared to existing DS learning methods. These results highlight the potential of StableNODEs in capturing complex behaviors and facilitating robotic learning in varying contexts, from imitating human motion to executing precise manipulation tasks with undefined object positions.
Future Directions in Robotics and AI
The implications of this research extend beyond the immediate task of learning from demonstrations, opening avenues for developing robotic systems capable of understanding and replicating human-like intricate movements in a stable and predictable manner. The flexibility and robustness of the proposed learning paradigm hold promise for a wide array of applications within robotics and potentially other domains where learning complex spatial and temporal patterns is crucial. Future work could explore minimizing unwanted critical points in Lyapunov functions and extending the framework to learn discontinuous DS, further broadening the scope of tasks that robots can learn and perform reliably.
In sum, this paper offers significant contributions towards making robots more adaptable, capable, and safe in performing tasks that require complex motion patterns, laying a foundation for future advancements in robotics and LfD.
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