Scalable Distance-based Multi-Agent Relative State Estimation via Block Multiconvex Optimization

TL;DR

This work addresses scalable distance-based relative state estimation in large multi-agent systems by casting the problem as generalized graph realization, which yields a rank-constrained SDP reformulation. It then develops two complementary optimization pipelines: an edge-based SDP (ESDP) that is convex and globally solvable, and a low-rank BM factorization-based local search (BM-BCD) that refines solutions quickly. Both models exploit multiconvexity and decomposability to enable distributed block coordinate descent with convergence guarantees, providing robust, scalable estimation that can operate online in continuous-time scenarios. Empirical results across distance-proprioception and distance-only setups demonstrate improved scalability over prior convex-relaxation methods and robust refinement, validating the practical impact for large-scale multi-agent localization and collaboration.

Abstract

This paper explores the distance-based relative state estimation problem in large-scale systems, which is hard to solve effectively due to its high-dimensionality and non-convexity. In this paper, we alleviate this inherent hardness to simultaneously achieve scalability and robustness of inference on this problem. Our idea is launched from a universal geometric formulation, called \emph{generalized graph realization}, for the distance-based relative state estimation problem. Based on this formulation, we introduce two collaborative optimization models, one of which is convex and thus globally solvable, and the other enables fast searching on non-convex landscapes to refine the solution offered by the convex one. Importantly, both models enjoy \emph{multiconvex} and \emph{decomposable} structures, allowing efficient and safe solutions using \emph{block coordinate descent} that enjoys scalability and a distributed nature. The proposed algorithms collaborate to demonstrate superior or comparable solution precision to the current centralized convex relaxation-based methods, which are known for their high optimality. Distinctly, the proposed methods demonstrate scalability beyond the reach of previous convex relaxation-based methods. We also demonstrate that the combination of the two proposed algorithms achieves a more robust pipeline than deploying the local search method alone in a continuous-time scenario.

Figures (14)

• Figure 1: Relations between problems and algorithms in this work. The \ref{['head']} section explains these problems and algorithms and corresponds them to the parts of the article.
• Figure 2: An example of dependency graphs and graph-coloring. (a) A dependency graph of a non-parallelizable case, where each agent is responsible for two variables, and the solid lines indicate the dependency relations between variables. (b) A dependency graph of a parallelizable case and a feasible coloring scheme of the blocks.
• Figure 3: Tightness under varying anchor measurements rate. The experiments are conducted in a constant topology 3D problem with 125 agents distributed in $[0,12\mathrm{m}]^3$. The agents are either equipped with two distance sensors (the distance-only setup) or also with an additional IMU (the distance-proprioception setup). Two neighboring agents (i.e., four sensors) are used as anchors. (a) The tightness rate which is defined as the proportion of sensors whose localization error is under $10^{-3}$m. (b) The near tightness rate which is defined as the proportion of sensors whose localization error is under $5\times10^{-2}$m.
• Figure 4: Examples of the specialized block division for Problem 4 and Problem 5, respectively. (a) An example of a 3-agents system, where the gray ellipses, dashed lines, and solid lines denote the rigid bodies corresponding to agents, distance measures, and rigid body constraints, respectively. (b) The block division for ESDP-BCD in the example and a feasible coloring scheme for it. (c) The block division and a corresponding feasible coloring scheme for BM-BCD in the example, where $U_i$ ($V_i$) represents variables corresponding to the sensors on agent $i$ in $U$ ($V$).
• Figure 5(a): Visualization of the problems and results of the proposed methods. (a) Illustration of a physical agent corresponding to the sensor configuration in Section \ref{['sec:7A']}: an agent can be graphically abstracted as two connected dots together with a dashed line connecting them, where the solid dots represent the two distance sensors on the agent. (b) Initialization and results of BM-BCD($d$+1) in cube. (c) Results before (left) and after (right) refinement of ESDP-BCD in cube. (d) Initialization and results of BM-BCD($d$+1) in pyramid. (e) Results before (left) and after (right) refinement of ESDP-BCD in pyramid.