SE(3) Linear Parameter Varying Dynamical Systems for Globally Asymptotically Stable End-Effector Control

Linear Parameter Varying Dynamical Systems (LPV-DS) encode trajectories into an autonomous first-order DS that enables reactive responses to perturbations, while ensuring globally asymptotic stability at the target. However, the current LPV-DS framework is established on Euclidean data only and has not been applicable to broader robotic applications requiring pose control. In this paper we present an extension to the current LPV-DS framework, named Quaternion-DS, which efficiently learns a DS-based motion policy for orientation. Leveraging techniques from differential geometry and Riemannian statistics, our approach properly handles the non-Euclidean orientation data in quaternion space, enabling the integration with positional control, namely SE(3) LPV-DS, so that the synergistic behaviour within the full SE(3) pose is preserved. Through simulation and real robot experiments, we validate our method, demonstrating its ability to efficiently and accurately reproduce the original SE(3) trajectory while exhibiting strong robustness to perturbations in task space.

DAMM: Directionality-Aware Mixture Model Parallel Sampling for Efficient Dynamical System Learning

The Linear Parameter Varying Dynamical System (LPV-DS) is a promising framework for learning stable time-invariant motion policies in robot control. By employing statistical modeling and semi-definite optimization, LPV-DS encodes complex motions via non-linear DS, ensuring the robustness and stability of the system. However, the current LPV-DS scheme faces challenges in accurately interpreting trajectory data while maintaining model efficiency and computational efficiency. To address these limitations, we propose the Directionality-aware Mixture Model (DAMM), a new statistical model that leverages Riemannian metric on $d$-dimensional sphere $\mathbb{S}^d$, and efficiently incorporates non-Euclidean directional information with position. Additionally, we introduce a hybrid Markov chain Monte Carlo method that combines the Gibbs Sampling and the Split/Merge Proposal, facilitating parallel computation and enabling faster inference for near real-time learning performance. Through extensive empirical validation, we demonstrate that the improved LPV-DS framework with DAMM is capable of producing physically-meaningful representations of the trajectory data and improved performance of the generated DS while showcasing significantly enhanced learning speed compared to its previous iterations.