发现机械系统周期运动的对称结构，可高效用于机器人轨迹控制。
Symmetric Lyapunov Subcenter Manifolds for Periodic Regulation of Mechanical Systems
- 利用时间对称性分析保守系统的振荡特性
- 证明在非共振条件下振荡仅在零速度点间往复
- 空间对称时所有振荡通过唯一平衡态，适合控制设计
多体机械系统具有丰富的内部动力学，其解可作为节能控制目标。然而，解对系统参数敏感，难以识别可用的轨迹特性。针对机器人应用中的周期调节任务，本文研究保守机械系统（CMs）中李雅普诺夫子中心流形（LSMs）内的非线性振荡性质。利用CMs的时间对称性，证明在温和非共振条件下，LSMs仅由两点间零速度振荡构成。进一步证明存在唯一生成器——一条连接这些零速度点的1维连通流形。此外，若系统具有额外空间对称性，则LSMs具备类似Rosenberg流形的更强性质：所有振荡均通过唯一平衡构型，有利于控制应用。理论结果在双摆和五连杆摆两个系统上得到数值验证。
原文摘要 · Abstract (English)
Multi-body mechanical systems have rich internal dynamics, whose solutions can be exploited as energy-efficient control targets. Yet, solutions non-trivially depend on system parameters, obscuring feasible properties for use as target trajectories. For periodic regulation tasks in robotics applications, we investigate properties of nonlinear oscillations collected in Lyapunov subcenter manifolds (LSMs) of conservative mechanical systems (CMs). Using a time-symmetry of CMs, it is shown that mild non-resonance conditions guarantee that LSMs exclusively consist of oscillations between two points of zero velocity. The existence of a unique generator is proven, which is a connected, 1D manifold that collects these points of zero velocity for a given LSM. Furthermore, it is shown that an additional spatial symmetry provides LSMs with yet stronger properties of Rosenberg manifolds. Here all oscillations pass through a unique equilibrium configuration, which can be favorable for control applications. These theoretical results are numerically confirmed on two mechanical systems: a double pendulum and a 5-link pendulum.
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