Core Concepts

Efficient computation of reachable sets on Lie groups using Lie algebra monotonicity and tangent intervals.

Abstract

The paper discusses efficiently computing overapproximated reachable sets for control systems evolving on Lie groups. It introduces the concept of intervals in the Lie algebra to describe real sets on the Lie group. The local equivalence between the original system and a system evolving on the Lie algebra allows for efficient reachability techniques. The paper proposes a Runge-Kutta-Munthe-Kaas reachability algorithm demonstrated through case studies.
Introduction
Efficient computation of reachable sets is crucial for verifying complex control systems.
Existing tools focus on systems in Euclidean state spaces, but many real systems evolve on manifolds like Lie groups.
Monotone Systems Theory
Monotone systems theory allows for computing reachable sets at low computational cost by simulating two trajectories.
Cone fields provide a local notion of ordering in non-globally orderable manifolds.
Geometric Integration Techniques
Geometric numerical integration techniques capture underlying geometric structures in Lie groups.
The Runge-Kutta-Munthe-Kaas technique and Magnus expansion are discussed.
Algorithm Development
An algorithm inspired by monotone systems theory and geometric integration is proposed for efficient reachable set computation.
Propositions are derived to establish equivalence between control systems on Lie groups and their corresponding Lie algebras.
Case Studies
Two case studies are presented: coupled oscillators evolving on a torus and attitude control on SO(3).
Results show efficient computation of reachable sets using the proposed algorithm.

Stats

A Runge-Kutta-Munthe-Kaas reachability algorithm is proposed.
The algorithm provides reachable set estimates for arbitrary time horizons at little computational cost.

Quotes

"In this paper, we efficiently compute overapproximated reachable sets for control systems evolving on Lie groups."
"We propose to consider intervals living in the Lie algebra, which through the exponential map, describe real sets on the Lie group."

Key Insights Distilled From

by Akash Harapa... at **arxiv.org** 03-26-2024

Deeper Inquiries

Other set geometries beyond intervals, such as polytopes and zonotopes, can be utilized for reachability analysis in the context of Lie algebra dynamics. These set geometries offer different capabilities and may provide more accurate bounding methods compared to intervals. Polytopes, for example, can approximate complex shapes with fewer vertices, making them efficient for representing reachable sets in higher dimensions. Zonotopes, on the other hand, are useful for capturing both linear dependencies and uncertainties in the system dynamics. By incorporating these set geometries into reachability analysis, one can potentially improve the accuracy and efficiency of estimating reachable sets on Lie groups.

Differential positivity has implications that differ from monotone systems theory when it comes to analyzing control systems evolving on manifolds with cone fields. While monotone systems focus on overapproximating reachable sets by simulating extreme trajectories based on a partial order induced by a cone field (⪯K), differential positivity considers how the linearization of a system affects this cone field over time. In essence, differential positivity ensures that solutions remain within conal curves defined by ⪯K under small perturbations around an initial point x ∈ X. This property guarantees stability and robustness in the system's behavior even when dealing with uncertainties or disturbances.

Different mappings from tangent spaces can significantly impact reachability computations by influencing how state transitions are approximated between the tangent space and the manifold itself. For instance, if a mapping ψp : TpM → M is not injective or does not preserve local diffeomorphisms around each point p ∈ M like exp does in Lie groups' exponential map Nexp ⊂ g → G at e ∈ G , it could lead to inaccuracies or inconsistencies in estimating reachable sets using interval bounds or inclusion functions like Hq mentioned earlier.

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