Active Localization of Unstable Systems with Coarse Information

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.

💡 Research Summary

The paper tackles the problem of localizing an unstable linear system when only a single‑bit proximity measurement to an unknown landmark is available at each time step. The dynamics are given by x_{k+1}=A x_k + B u_k, where (A,B) is controllable and A is unstable (all eigenvalues have magnitude greater than one). The agent does not know its own state x_k nor the landmark position m, but it knows that x_0 belongs to a convex set X_0 and m belongs to a convex set M. A binary sensor reports y_k=1 if the Euclidean distance ‖x_k−m‖≤r (inside a detection ball of radius r) and y_k=0 otherwise. The goal is to design a control policy and an estimator such that the diameter of the estimate of the initial state shrinks to zero, i.e., lim_{k→∞}diam(ĤX_0