Differential flatness enables efficient planning and control for underactuated robotic systems, but we lack a systematic and practical means of identifying a flat output (or determining whether one exists) for an arbitrary robotic system. In this work, we leverage recent results elucidating the role of symmetry in constructing flat outputs for free-flying robotic systems. Using the tools of Riemannian geometry, Lie group theory, and differential forms, we cast the search for a globally valid, equivariant flat output as an optimization problem. An approximate transcription of this continuum formulation to a quadratic program is performed, and its solutions for two example systems achieve precise agreement with the known closed-form flat outputs. Our results point towards a systematic, automated approach to numerically identify geometric flat outputs directly from the system model, particularly useful when complexity renders pen and paper analysis intractable.

翻译：微分平坦性能够实现对欠驱动机器人系统的高效规划与控制，但目前缺乏一种系统且实用的方法来识别任意机器人系统的平坦输出（或判定其是否存在）。本文借助近期关于对称性在构建自由飞行机器人系统平坦输出中作用的研究成果，利用黎曼几何、李群理论和微分形式的工具，将全局有效等变平坦输出的搜索问题转化为优化问题。通过将该连续模型近似转换为二次规划并求解，两个示例系统的解与已知闭合形式的平坦输出实现了精确匹配。本研究结果表明，存在一种系统化、自动化的方法，可直接从系统模型中数值辨识几何平坦输出，尤其适用于因复杂性导致人工理论分析难以处理的场景。