Jake Welde

Jake Welde

PhD candidate in robotics working at the intersection of geometry, dynamics, and design.

Tracking Control on Homogeneous Riemannian Manifolds

2024

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.
    IEEE Control Systems Letters (L-CSS), 2024.