Neural Contractive Dynamical Systems
(2401.09352)Abstract
Stability guarantees are crucial when ensuring a fully autonomous robot does not take undesirable or potentially harmful actions. Unfortunately, global stability guarantees are hard to provide in dynamical systems learned from data, especially when the learned dynamics are governed by neural networks. We propose a novel methodology to learn neural contractive dynamical systems, where our neural architecture ensures contraction, and hence, global stability. To efficiently scale the method to high-dimensional dynamical systems, we develop a variant of the variational autoencoder that learns dynamics in a low-dimensional latent representation space while retaining contractive stability after decoding. We further extend our approach to learning contractive systems on the Lie group of rotations to account for full-pose end-effector dynamic motions. The result is the first highly flexible learning architecture that provides contractive stability guarantees with capability to perform obstacle avoidance. Empirically, we demonstrate that our approach encodes the desired dynamics more accurately than the current state-of-the-art, which provides less strong stability guarantees.
Overview
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NCDS presents a new machine learning framework for robot stability, leveraging contraction theory to ensure consistent behavior.
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The framework enables stability via a neural network that guarantees contraction stability, avoiding the need for constraints during training.
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NCDS incorporates a variant of the VAE to handle complex dynamics in a low-dimensional latent space, preserving stability in full-pose movements.
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Tests on synthetic and real datasets, such as a 7-DoF robotic manipulator, show NCDS outperforms existing models, integrating stability with obstacle avoidance.
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The method constitutes a significant advancement in stable and adaptable autonomous robot learning, with some limitations in numerical integration.
Introduction to Neural Contractive Dynamical Systems
Stability is a cornerstone in robotics, particularly when programming a robot to act autonomously. Without stability, robots might exhibit unreliable or even dangerous behaviors. Traditionally, programming stable robot behavior has involved using hand-crafted dynamics, an approach that’s both time-intensive and inflexible. However, the rise of machine learning has opened the door to learning robotic dynamics directly from data, though this method comes with its own set of challenges – chiefly, ensuring stability. In a novel approach, a new framework called Neural Contractive Dynamical Systems (NCDS) tackles this problem head-on.
Ensuring Stability Through Contraction
Contraction theory is the underpinning force behind ensuring stability in NCDS. It enables a robot to digest a set of demonstration behaviors and generalize them in a way that maintains consistency, even when faced with perturbations. Unlike traditional methods that can guarantee either asymptotic or contraction stability (the latter being stronger of the two), existing learning-based models struggle with offering consistent stability due to their neural network foundations. The proposed NCDS architecture, by design, ensures contraction stability across the entire parameter space.
NCDS Functionality at a Glance
The driving force of NCDS is a neural network architecture that inherently respects contraction theory's mandate for global stability. This is accomplished by constructing the network's Jacobian, or its transformation blueprint, to be contractive regardless of the parameter values. With this property, NCDS offers a versatile tool that can be used within existing frameworks to provide stability guarantees without the need for imposing external constraints during training – a notable advantage over previous methods.
Scaling to High-Dimensional Systems
One might question the applicability of such a system to complex, high-dimensional problems. NCDS addresses this by incorporating a variant of the Variational Autoencoder (VAE) designed to comprehend dynamics in a compressed, low-dimensional latent space. This method not only retains contractive stability upon decoding but is also extended to encompass full-pose movements including rotations, thanks to its operation on the special orthogonal group 𝒮𝒪(3).
Empirical Results and Applications
Empirical results display NCDS's superiority in capturing the desired dynamics with more accuracy compared to existing models. It also scales efficiently to high-dimensional systems and even demonstrates the ability to integrate obstacle avoidance into motion planning. This is showcased through experiments on synthetic and real datasets, including a 7-DoF robotic manipulator tasked with a drawing activity. The avoidance is modeled using dynamic modulation, ensuring that the robot evades obstacles while preserving the stability guarantees intrinsic to NCDS.
Concluding Remarks
NCDS marks a significant step toward reliable robot learning that doesn't compromise on stability. By harnessing contraction theory within its neural network design, NCDS offers a method to program autonomous robots that are stable, adaptable, and capable of learning from high-dimensional data. While certain limitations exist, such as the reliance on adaptive step sizing during numerical integration, the benefits of stability guarantees are an attractive trade-off for the robotic systems of the future.
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