The Motion Manifold Primitive (MMP) produces, for a given task, a continuous manifold of trajectories each of which can successfully complete the task. It consists of the decoder function that parametrizes the manifold and the probability density in the latent coordinate space. In this paper, we first show that the MMP performance can significantly degrade due to the geometric distortion in the latent space -- by distortion, we mean that similar motions are not located nearby in the latent space. We then propose {\it Isometric Motion Manifold Primitives (IMMP)} whose latent coordinate space preserves the geometry of the manifold. For this purpose, we formulate and use a Riemannian metric for the motion space (i.e., parametric curve space), which we call a {\it CurveGeom Riemannian metric}. Experiments with planar obstacle-avoiding motions and pushing manipulation tasks show that IMMP significantly outperforms existing MMP methods. Code is available at https://github.com/Gabe-YHLee/IMMP-public.

翻译：运动流形基元（MMP）针对给定任务生成一个连续的轨迹流形，其中每条轨迹均能成功完成任务。它由参数化流形的解码函数及潜在坐标空间的概率密度组成。本文首先证明，由于潜在空间中的几何畸变——即相似运动在潜在空间中并非邻近分布——MMP性能可能显著下降。为此，我们提出等距运动流形基元（IMMP），其潜在坐标空间能够保持流形的几何结构。为实现这一目标，我们构建并采用一种适用于运动空间（即参数化曲线空间）的黎曼度量，称为曲线几何黎曼度量。在平面避障运动与推搡操作任务上的实验表明，IMMP显著优于现有MMP方法。代码已开源在 https://github.com/Gabe-YHLee/IMMP-public。