The connection between the dynamics in relative periodic orbits of vector fields with noncompact symmetry groups and periodic control for the class of control systems on Lie groups known as '(robotic) locomotion systems' is well known, and has led to the identification of (geometric) phases. We take an approach which is complementary to the existing ones, advocating the relevance——for trajectory generation in these control systems——of the qualitative properties of the dynamics in relative periodic orbits. There are two particularly important features. One is that motions in relative periodic orbits of noncompact groups can only be of two types: either they are quasi-periodic, or they leave any compact set as $ t\to\pm\infty $ ('drifting motions'). Moreover, in a given group, one of the two behaviours may be predominant. The second is that motions in a relative periodic orbit exhibit 'spiralling', 'meandering' behaviours, which are routinely detected in numerical integrations. Since a quantitative description of meandering behaviours for drifting motions appears to be missing, we provide it here for a class of Lie groups that includes those of interest in locomotion (semidirect products of a compact group and a normal vector space). We illustrate these ideas on some examples (a kinematic car robot, a planar swimmer).

中文翻译：

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相对周期轨道中的运动系统和动力学的控制

具有非对称对称群的矢量场的相对周期轨道上的动力学与对称为“（机器人）运动系统”的李群上的控制系统类别的周期控制之间的联系是众所周知的，并导致了（几何） ）阶段。我们采取一种与现有方法相辅相成的方法，提倡与这些控制系统中的轨迹生成相关性。相对周期性轨道动力学的性质。有两个特别重要的功能。一种是非紧致群在相对周期轨道上的运动只能是两种类型：它们是准周期的，或者它们留下任何紧缩集合为$ t \ to \ pm \ infty $（“漂移运动”）。而且，在给定的组中，两种行为之一可能是主要的。第二个是相对周期性轨道上的运动表现出“振奋人心”，“曲折”行为，这些行为通常在数值积分中被检测到。由于似乎缺少关于漂移运动的曲折行为的定量描述，因此我们在这里为一类李氏群提供它，其中包括对运动感兴趣的那些群（紧密群和法向矢量空间的半直接乘积）。