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

Jake Welde, Vijay Kumar

TL;DR

This work tackles trajectory tracking for fully-actuated mechanical systems evolving on homogeneous Riemannian manifolds. It introduces an intrinsic, state-valued tracking error built from a $0_Q$-lift of the reference trajectory and derives a nonlinear feedback law that yields almost global asymptotic tracking, with local exponential convergence of the error. The approach unifies tracking on broad manifolds (including all Lie groups and spheres) and specializes neatly to $n$-spheres and Lie groups, providing explicit controllers and broad applicability to aerospace and robotics problems. The method is demonstrated on an axisymmetric satellite and an omnidirectional aerial robot, illustrating practical applicability and robustness of the geometric framework. Overall, the paper provides a rigorous, globally valid tracking strategy grounded in Riemannian geometry and principal bundle theory, expanding the toolkit for geometric control on manifolds.

Abstract

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.

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

TL;DR

This work tackles trajectory tracking for fully-actuated mechanical systems evolving on homogeneous Riemannian manifolds. It introduces an intrinsic, state-valued tracking error built from a -lift of the reference trajectory and derives a nonlinear feedback law that yields almost global asymptotic tracking, with local exponential convergence of the error. The approach unifies tracking on broad manifolds (including all Lie groups and spheres) and specializes neatly to -spheres and Lie groups, providing explicit controllers and broad applicability to aerospace and robotics problems. The method is demonstrated on an axisymmetric satellite and an omnidirectional aerial robot, illustrating practical applicability and robustness of the geometric framework. Overall, the paper provides a rigorous, globally valid tracking strategy grounded in Riemannian geometry and principal bundle theory, expanding the toolkit for geometric control on manifolds.

Abstract

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 ). 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.
Paper Structure (15 sections, 7 theorems, 44 equations, 2 figures)

This paper contains 15 sections, 7 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Homogeneous Riemannian Manifolds
  4. Navigation Functions
  5. Fully-Actuated Mechanical Systems
  6. An Intrinsic, State-Valued Tracking Error
  7. Tracking Error in Homogeneous Spaces
  8. Computing Horizontal Lifts of Curves
  9. Almost Global Asymptotic Tracking
  10. Specialization to $n$-Spheres and Lie Groups
  11. Applications of the Method
  12. Discussion
  13. Conclusion
  14. Some Helpful Lemmas on Riemannian Geometry
  15. Detailed Account of Convergence in Proof of Theorem 1

Key Result

Proposition 1

Consider any actual and reference trajectories ${q, q_d : \mathbb{R} \to Q}$. Let ${g_d : \mathbb{R} \to G}$ be a $0_Q$-lift of ${q_d}$, and let ${0_{TQ} \in TQ}$ be the zero tangent vector at $0_Q$. Then, for any time ${t \in \mathbb{R}}$, ${\dot{q}(t) = \dot{q}_d(t)}$ if and only if ${\dot{e}(t) =

Figures (2)

  • Figure 1: The proposed controller, applied to a mechanical system on $\mathbb{S}^2$ (e.g., the axisymmetric satellite). We show snapshots of parallel rollouts from a random sampling of initial states (configuration and velocity) in $T\mathbb{S}^2$. All sampled rollouts converge to the reference trajectory.
  • Figure 2: Almost global asymptotic trajectory tracking on ${\mathbb{R}^3 \times SO(3)}$.

Theorems & Definitions (31)

  • Definition 1: see RiemannianLee
  • Example 1: name=The $n$-Sphere,label=example:sphere
  • Example 2: name=A Lie Group,label=example:group
  • Definition 2
  • Definition 3
  • Remark 1: Generality
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • ...and 21 more