Directionality-Aware Mixture Model Parallel Sampling for Efficient Linear Parameter Varying Dynamical System Learning

TL;DR

This work tackles the challenge of learning stable LPV-DS policies with high accuracy without sacrificing speed. It introduces the Directionality-Aware Mixture Model (DAMM), which uses Riemannian geometry on the unit sphere to capture directionality, and couples it with a parallel, ergodic MCMC scheme that combines Instantiated-Weight Gibbs sampling with Split/Merge proposals. DAMM augments trajectory data with directional information, enabling more physically meaningful clustering and DS parameters learned via NIC priors, yielding improved reproduction and faster learning. Empirical results on LASA and PC-GMM benchmarks, plus real-robot experiments, show near real-time learning and competitive accuracy, supporting incremental multi-behavior policy acquisition in practice.

Abstract

The Linear Parameter Varying Dynamical System (LPV-DS) is an effective approach that learns stable, time-invariant motion policies using statistical modeling and semi-definite optimization to encode complex motions for reactive robot control. Despite its strengths, the LPV-DS learning approach faces challenges in achieving a high model accuracy without compromising the computational efficiency. To address this, we introduce the Directionality-Aware Mixture Model (DAMM), a novel statistical model that applies the Riemannian metric on the n-sphere $\mathbb{S}^n$ to efficiently blend non-Euclidean directional data with $\mathbb{R}^m$ Euclidean states. Additionally, we develop a hybrid Markov chain Monte Carlo technique that combines Gibbs Sampling with Split/Merge Proposal, allowing for parallel computation to drastically speed up inference. Our extensive empirical tests demonstrate that LPV-DS integrated with DAMM achieves higher reproduction accuracy, better model efficiency, and near real-time/online learning compared to standard estimation methods on various datasets. Lastly, we demonstrate its suitability for incrementally learning multi-behavior policies in real-world robot experiments.

Figures (7)

• Figure 1: The schematic of the DAMM-based LPV-DS formulation which consists of DAMM and Parallel Sampling to cluster and parameterize the trajectory, and the optimization to minimize the prediction error; the resulting DS, $f_\Theta$, takes the position $\xi$ and velocity $\Dot{\xi}$ as inputs, transforms them into the augmented state $\Hat{\xi}$, and generates the estimated desired linear velocity which is then passed down to command the robot via a low-level feedback controller; e.g. a Cartesian twist impedance controller.
• Figure 2: Illustrative example of the exponential/logarithmic mapping on a Riemannian manifold and its tangent space defined at point $p$
• Figure 3: Illustration of DAMM: a) A-shaped reference trajectory and the point of interest marked in asterisk; b) clustering result of DAMM showing both clusters' covariance in ellipsoid; c) and d) overlay the point's direction in black and the directional mean of each component in color.
• Figure 4: Illustration of a split operation: a) an S-shaped reference trajectory; b) the current state of assignment; c) initialize the launch state by randomly assigning the candidate group (red) into two new groups (red and green); d) reach the launch state after multiple scans of IW Gibbs sampling.
• Figure 5: Comparison of computation time w.r.t varying observation size. All experiments are run on Ubuntu 20.04 with Intel i7-1065G7 @ 1.30GHz CPU and 16GB of RAM.