Unified Tracking Controller for Mechanical Systems on Homogeneous Manifolds

Keskeiset käsitteet

The author proposes a systematic tracking controller for fully-actuated mechanical systems on homogeneous Riemannian manifolds, ensuring almost global asymptotic trajectory tracking from any initial condition.

Tiivistelmä

The content discusses the development of a unified tracking controller for fully-actuated mechanical systems evolving on various homogeneous spaces. The approach guarantees almost global convergence to reference trajectories and is applied to examples like an axisymmetric satellite and an omnidirectional aerial robot. The method involves intrinsic state-valued tracking errors, horizontal lifts of curves, and navigation functions. Theoretical results are supported by mathematical background and practical applications.

Tilastot

"almost global convergence"
"fully-actuated mechanical systems"
"homogeneous Riemannian manifolds"
"axisymmetric satellite"
"omnidirectional aerial robot"

Lainaukset

"The transitivity of a Lie group’s action on the configuration manifold is useful for tracking control via error regulation in mechanical systems."
"Methods which do not rely on reduction to regulation often fail to achieve or certify almost global convergence."
"Our results illustrate that it is the transitivity of a Lie group's action on the configuration manifold that is useful for tracking control."

Tärkeimmät oivallukset

Almost Global Asymptotic Trajectory Tracking for Fully-Actuated Mechanical Systems on Homogeneous Riemannian Manifolds

by Jake Welde,V... klo arxiv.org 03-11-2024

https://arxiv.org/pdf/2403.04900.pdf

Syvällisempiä Kysymyksiä

What implications does this method have for underactuated systems?

The proposed method has significant implications for underactuated systems, even though it is primarily designed for fully-actuated mechanical systems. By leveraging the systematic approach to tracking control on homogeneous Riemannian manifolds, insights can be gained that are transferable to underactuated systems. Specifically, hierarchical controllers often rely on controlling subsystems that resemble fully-actuated ones. The identification of a "geometric flat output" in such hierarchical structures can lead to subsystems evolving on spaces that may not be Lie groups but still benefit from the principles outlined in the proposed method. This means that the almost global asymptotic stability guarantees provided by this approach can potentially extend to certain classes of underactuated systems as well.

How does the proposed approach compare to existing methods in terms of stability guarantees?

In terms of stability guarantees, the proposed approach stands out due to its rigorous and formal guarantee of almost global asymptotic trajectory tracking for fully-actuated mechanical systems on a broader class of manifolds. Unlike some existing methods which may only ensure convergence from specific initial conditions or regions, this new method provides assurances of convergence from nearly all initial states within TQ (the tangent bundle). This level of certainty is crucial in practical applications where robust and reliable performance is required. Additionally, by incorporating dissipation terms and navigation functions into the controller design based on intrinsic geometric properties, this approach offers a more comprehensive framework for ensuring stable tracking behavior across various configurations.

Can this method be extended to other types of mechanical systems beyond those discussed in the content?

Yes, this method can indeed be extended to other types of mechanical systems beyond those specifically addressed in the context provided. The key lies in understanding and adapting the fundamental principles underlying the tracking controller synthesis presented for homogeneous Riemannian manifolds. By applying similar concepts such as reductive decompositions, horizontal lifts, configuration errors with state-valued representations, and appropriate control policies derived from navigation functions and dissipation metrics tailored to different system dynamics and constraints; one can effectively tailor this methodology to suit diverse mechanical setups ranging from robotic manipulators with complex kinematics to autonomous vehicles navigating challenging terrains or environments.