Multi-body mechanical systems have rich internal dynamics, whose solutions can be exploited as 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 normal modes (NNMs) collected in Lyapunov subcenter manifolds (LSMs) of conservative mechanical systems. Using a time-symmetry of conservative mechanical systems, we show that mild non-resonance conditions guarantee LSMs to be Eigenmanifolds, in which NNMs are guaranteed to oscillate between two points of zero velocity. We also prove the existence of a unique generator, which is a connected, 1D manifold that collects these points of zero velocity for a given Eigenmanifold. Furthermore, we show that an additional spatial symmetry provides LSMs with yet stronger properties of Rosenberg manifolds. Here all brake trajectories 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.

翻译：多体力学系统具有丰富的内部动力学特性，其解可作为高效控制目标加以利用。然而，这些解非平凡地依赖于系统参数，使得作为目标轨迹的可行特性难以明晰。针对机器人应用中的周期性调节任务，我们研究了保守力学系统中李雅普诺夫子中心流形所收集的非线性正规模态的性质。利用保守力学系统的时间对称性，我们证明在温和的非共振条件下，李雅普诺夫子中心流形可保证为特征流形，其中非线性正规模态被确保在零速度两点间振荡。我们还证明存在唯一的生成子——一个连通的单维流形，该流形收集给定特征流形中所有零速度点。此外，我们证明额外的空间对称性使李雅普诺夫子中心流形具备更强的罗森伯格流形特性。此时所有制动轨迹均通过唯一的平衡构型，这对于控制应用可能是有利的。这些理论结果在两个力学系统上得到了数值验证：双摆系统和五连杆摆系统。