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.

翻译：线性参数变分动力系统（LPV-DS）是机器人控制中学习稳定时不变运动策略的可靠框架。通过统计建模与半定优化，LPV-DS利用非线性DS编码复杂运动，确保系统鲁棒性与稳定性。然而，现有LPV-DS方案在准确解析轨迹数据的同时兼顾模型效率与计算效能仍面临挑战。为解决上述局限，我们提出方向感知混合模型（DAMM）——一种利用$d$维球面$\mathbb{S}^d$上黎曼度量、高效融合非欧几里得方向信息与位置信息的新型统计模型。我们进一步引入混合马尔可夫链蒙特卡洛方法，结合吉布斯采样与分裂/合并提议机制，支持并行计算并加速推理，实现接近实时的学习性能。通过广泛实证验证，我们证明融合DAMM的改进LPV-DS框架能生成轨迹数据的物理可解释表征，提升所生成DS的性能表现，同时相较先前版本显著增强学习速度。