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 finite 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, which drives the 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 elegantly 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 broad 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逼近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、李群及任意有限维球面）这一广泛类别上演化的全驱动系统。在此框架下，即使配置流形本身缺乏群结构（例如 $\mathbb{S}^2$），李群在配置流形上的传递作用仍能通过状态空间元素实现跟踪误差的内蕴描述。这种误差状态有助于设计依赖于状态与时间的平滑广义控制策略，该策略能从几乎任意初始条件驱动几何跟踪误差到达指定原点，从而保证对参考轨迹的几乎全局收敛。此外，所提控制器在特化为李群或n维球面时具有优雅的简化形式。总结而言，我们提出了一种统一的、内蕴控制器，能够保证在广泛流形类别上演化的全驱动机械系统实现几乎全局渐近轨迹跟踪。我们将该方法应用于轴对称卫星与全向空中机器人。