The Motion Manifold Primitive (MMP) produces, for a given task, a continuous manifold of trajectories, each of which can successfully complete the task, addressing the challenge of high dimensionality in trajectory data. However, the discrete-time trajectory representations used in existing MMP methods lack important functionalities of movement primitives (e.g., temporal modulation, via-points modulation, etc.) found in other conventional methods that employ parametric curve representations. To address these limitations, we introduce Motion Manifold Primitives++ (MMP++), which combines the advantages of the MMP and conventional methods by applying the MMP framework to the parametric curve representations. However, we observe that the performance of MMP++ can sometimes degrade significantly due to geometric distortion in the latent space -- by distortion, we mean that similar motions are not located nearby in the latent space. To mitigate this issue, we propose Isometric Motion Manifold Primitives++ (IMMP++), where the latent coordinate space preserves the geometry of the manifold. Experimental results with 2-DoF planar motions and 7-DoF robot arm tasks demonstrate that MMP++ and IMMP++ outperform existing methods, in some cases by a significant margin, while maintaining the advantages of parametric curve representations.

翻译：运动流形基元(MMP)针对给定任务生成连续的运动轨迹流形，每条轨迹均可成功完成任务，从而解决轨迹数据的高维挑战。然而，现有MMP方法中使用的离散时间轨迹表示缺乏其他基于参数曲线表示的传统方法所具备的运动基元重要功能（如时间调制、途经点调制等）。为克服这些限制，我们提出运动流形基元++(MMP++)，通过将MMP框架应用于参数曲线表示，融合了MMP与传统方法的优势。但研究发现，由于潜在空间中的几何畸变——即相似运动在潜在空间中并不邻近——MMP++的性能有时会显著下降。为缓解此问题，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。基于2自由度平面运动与7自由度机械臂任务的实验结果表明，MMP++与IMMP++在保持参数曲线表示优势的同时，其性能显著优于现有方法。