In this work, we address the design of tracking controllers that drive a mechanical system's state asymptotically towards a reference trajectory. Motivated by aerospace and robotics applications, we consider fully-actuated systems evolving on the broad class of homogeneous spaces (encompassing all vector spaces, Lie groups, and spheres of any dimension). In this setting, the transitive action of a Lie group on the configuration manifold enables an intrinsic description of the tracking error as an element of the state space, even in the absence of a group structure on the configuration manifold itself (e.g., for $\mathbb{S}^2$). Such an error state facilitates the design of a generalized control policy depending smoothly on state and time that drives this geometric tracking error to a designated origin from almost every initial condition, thereby guaranteeing almost global convergence to the reference trajectory. Moreover, the proposed controller simplifies naturally when specialized to a Lie group or the $n$-sphere. In summary, we propose a unified, intrinsic controller guaranteeing almost global asymptotic trajectory tracking for fully-actuated mechanical systems evolving on a broader class of manifolds. We apply the method to an axisymmetric satellite and an omnidirectional aerial robot.

翻译：本文针对机械系统状态渐近跟踪参考轨迹的跟踪控制器设计问题展开研究。受航空航天与机器人学应用启发，我们考虑在齐次空间（涵盖所有向量空间、李群及任意维度球面）这一广泛类别上演化的全驱动系统。在此框架下，李群对构型流形的传递作用使得即便在构型流形本身不具备群结构（例如$\mathbb{S}^2$）时，仍能将跟踪误差内蕴地描述为状态空间中的元素。这种误差状态有助于设计一种依赖于状态与时间光滑变化的广义控制策略，该策略可从几乎全部初始条件将几何跟踪误差驱动至指定原点，从而保证对参考轨迹的几乎全局收敛。此外，当特化为李群或$n$维球面时，所提控制器可自然简化。综上，我们提出一种统一的、内蕴的控制器，保证在更广泛流形上演化的全驱动机械系统实现几乎全局渐近轨迹跟踪。该方法已应用于轴对称卫星与全向空中机器人系统。