Learning Deep Dynamical Systems using Stable Neural ODEs

TL;DR

This work tackles learning stable, versatile dynamical systems for robot trajectories from demonstrations, addressing the limitations of single-attractor stability and dependence on state derivatives. It introduces StableNODE, a Stable Neural ODE framework that adds a Lyapunov-based corrective term to guarantee global stability and supports multiple attractors in a latent space, complemented by a diffeomorphic output mapping to place attractors in the observable output space. Trajectory learning uses a time-invariant Average Hausdorff Distance loss to compare phase-space trajectories, enabling alignment without strict timing and without requiring state derivatives. The approach is demonstrated on LASA handwriting shapes and a vision-to-trajectory picking task, achieving robust performance and high success with limited demonstrations, showing practical potential for multi-modality LfD with stability guarantees.

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.

Figures (8)

• Figure 1: End-to-End learning framework for Dynamical Systems using stable Neural ODEs
• Figure 2: (a) The vector field of the nominal function $\boldsymbol f_{\boldsymbol \theta}$ for a 2 dimensional latent space and (b) The vector field of the sum of $\boldsymbol f_{\boldsymbol \theta}$ and the corrective term $\boldsymbol u$. Dark green regions denote the areas of the phase space where $L(\boldsymbol x) > 0.$
• Figure 3: (a) The Lyapunov function of a 2 dimensional DS with two target points shown in green. The red point is a critical point of $V_{\boldsymbol \theta}$ and in this case is either a local maximum or a saddle point. This Lyapunov function can be useful if the DS has to learn reaching or placing motions with 2 goal positions, (b) The Lyapunov function of a DS with one target point. The Lyapunov function is build to be maximal on the boundaries of an outer circle and of an inner circle restricting thus the resulting trajectories. It is impossible to create such functions that do not exhibit other critical points than the target (target shown in green, critical point in red).
• Figure 4: Four trajectories with time misalignments that correspond to the same phase space trajectory.
• Figure 5: Dynamic Time Warping Distance and Frechet Distance of Euclideanizing Flows (EF), Deep Stable Dynamics (DSD) and StableNODEs.