Tracking Control on Homogeneous Riemannian Manifolds


with Vijay Kumar

The proposed method, applied to tracking control for a fully-actuated mechanical system on \(\mathbb{S}^2\). Here, we visualize rollouts of the tracking controller from 100 randomly sampled initial states in \(T\mathbb{S}^2\). Because the controller achieves almost global asymptotic tracking, the probability that a randomly sampled initial condition fails to converge to the reference trajectory is exactly zero.

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.

Relevant Publications

  1. "Almost Global Asymptotic Trajectory Tracking for Fully-Actuated Mechanical Systems on Homogeneous Riemannian Manifolds",
    Jake Welde and Vijay Kumar.
    arXiv 2403.04900, 2024.