Active Localization of Unstable Systems with Coarse Information

TL;DR

This work tackles the problem of localizing an agent with unstable dynamics using extremely coarse single-bit proximity measurements to a landmark. It develops an active localization framework that couples a set-valued estimator with a Voronoi-based recovery control, enabling exponential contraction of the initial-state uncertainty under precise geometric and algebraic conditions. Theoretical analysis establishes conditions under which the landmark and initial state can be recovered, while numerical experiments in low-dimensional settings validate the exponential convergence and demonstrate practical relevance for robotics with coarse sensing. The proposed ACT-LOC algorithm unifies estimation and recovery control to ensure persistent informative measurements and robust re-entry into the sensing region, offering a principled approach to localization with minimal feedback.

Abstract

We study localization and control for unstable systems under coarse, single-bit sensing. Motivated by understanding the fundamental limitations imposed by such minimal feedback, we identify sufficient conditions under which the initial state can be recovered despite instability and extremely sparse measurements. Building on these conditions, we develop an active localization algorithm that integrates a set-based estimator with a control strategy derived from Voronoi partitions, which provably estimates the initial state while ensuring the agent remains in informative regions. Under the derived conditions, the proposed approach guarantees exponential contraction of the initial-state uncertainty, and the result is further supported by numerical experiments. These findings can offer theoretical insight into localization in robotics, where sensing is often limited to coarse abstractions such as keyframes, segmentations, or line-based features.

Figures (4)

• Figure 1: (a) An SVP spanned by four basis vectors (orange arrows), with the first partition corresponding to $p_1$ highlighted along the boundary in dark green; (b) an $\alpha$-SVP of $\hbox{$\partial$}\mathcal{B}(c,\rho)$, where the caps appear as black arcs and one cap, $C(p_1,\alpha)$, with its partition $R_1$, is highlighted.
• Figure 2: Initial-state estimate $\hat{X}_0[\cdot]$ and landmark estimate $\hat{M}[\cdot]$ at times $(k\!-\!1)$ (left) and $k$ (right). The state remains within the sensing region over $[0,k]$. For clarity, only one ellipsoid $\mathcal{E}(\mu_{k0},P_{k0})$—computed in \ref{['alg:EST']}\ref{['EST:line5']}—is shown intersecting $\hat{X}_0[k-1]$. The gray tube illustrates $\textsc{Reach}(A,B,u_{0:k-1},\hat{X}_0[k])$. Since the intersection in \ref{['muppdate']} leaves this reachable set unchanged, it yields the updated current state estimate. Meanwhile, the same reachable set—only one shown for clarity—is inflated by $\mathcal{B}(0,r)$ to update $\hat{M}$.
• Figure 3: Evolution of $\operatorname{diam}(\hat{X}_0[k])$: the dark red and dark blue curves represent the mean trajectories, and the red and blue envelopes indicate the min--max ranges across 40 trials for the two systems. The dashed lines denote the corresponding theoretical bounds.
• Figure 4: Evolution of $\operatorname{diam}(\hat{M}[k])$: the dark green curve shows the mean trajectory, and the green envelope indicates the min--max range across 40 trials for Setup 1.

Theorems & Definitions (20)

• definition 1
• Remark 1
• definition 2
• proposition 1
• proposition 2
• proposition 3
• Remark 2
• lemma 1
• Remark 3
• Remark 4