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Ranging-Based Localizability Optimization for Mobile Robotic Networks

Justin Cano, Jerome Le Ny

TL;DR

A potential-based planning methods is presented, where localizability potentials are introduced to characterize the quality of the network geometry for cooperative position estimation and provide a theoretical lower bound on the error covariance achievable by any unbiased position estimator.

Abstract

In robotic networks relying on noisy range measurements between agents for cooperative localization, the achievable positioning accuracy strongly strongly depends on the network geometry. This motivates the problem of planning robot trajectories in such multi-robot systems in a way that maintains high localization accuracy. We present potential-based planning methods, where localizability potentials are introduced to characterize the quality of the network geometry for cooperative position estimation. These potentials are based on Cramer Rao Lower Bounds (CRLB) and provide a theoretical lower bound on the error covariance achievable by any unbiased position estimator. In the process, we establish connections between CRLBs and the theory of graph rigidity, which has been previously used to plan the motion of robotic networks. We develop decentralized deployment algorithms appropriate for large networks, and we use equality-constrained CRLBs to extend the concept of localizability to scenarios where additional information about the relative positions of the ranging sensors is known. We illustrate the resulting robot deployment methodology through simulated examples and an experiment.

Ranging-Based Localizability Optimization for Mobile Robotic Networks

TL;DR

A potential-based planning methods is presented, where localizability potentials are introduced to characterize the quality of the network geometry for cooperative position estimation and provide a theoretical lower bound on the error covariance achievable by any unbiased position estimator.

Abstract

In robotic networks relying on noisy range measurements between agents for cooperative localization, the achievable positioning accuracy strongly strongly depends on the network geometry. This motivates the problem of planning robot trajectories in such multi-robot systems in a way that maintains high localization accuracy. We present potential-based planning methods, where localizability potentials are introduced to characterize the quality of the network geometry for cooperative position estimation. These potentials are based on Cramer Rao Lower Bounds (CRLB) and provide a theoretical lower bound on the error covariance achievable by any unbiased position estimator. In the process, we establish connections between CRLBs and the theory of graph rigidity, which has been previously used to plan the motion of robotic networks. We develop decentralized deployment algorithms appropriate for large networks, and we use equality-constrained CRLBs to extend the concept of localizability to scenarios where additional information about the relative positions of the ranging sensors is known. We illustrate the resulting robot deployment methodology through simulated examples and an experiment.
Paper Structure (28 sections, 9 theorems, 84 equations, 16 figures, 1 table, 2 algorithms)

This paper contains 28 sections, 9 theorems, 84 equations, 16 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Localizability Potentials
  4. Constrained Cramér-Rao Lower Bound
  5. Localizability Potentials and Optimal Design
  6. Properties of the Fisher Information Matrix
  7. Infinitesimal Rigidity
  8. Relations between the Rigidity Matrix and the FIM
  9. Distributed Gradient Computations for the Localizability Potentials
  10. Partial Derivatives of the FIM
  11. Decentralized Gradient Computations for the D- and A-Opt Potentials
  12. Auxiliary Problem
  13. Application to compute $\partial J_D / \partial \xi_i$
  14. Decentralized Computation of E-Opt Gradient
  15. Power-iteration eigenvector estimator
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\mathbf x \in \mathbb{R}^p$ be a deterministic parameter vector and $\mathbf y \in \mathbb{R}^q$ a random observation vector, for some positive integers $p,q$. Let $\mathbf h: \mathbb{R}^p \to \mathbb{R}^c$, for $c\leq p$, be a differentiable function such that $\mathbf h(\mathbf x)= \mathbf 0$ where $\dagger$ denotes the Moore-Penrose pseudo-inverse petersen_matrix_2012.

Figures (16)

  • Figure 1: Illustration of the setup in 2D with 3 mobile tags and 3 anchors, 2 of whom are fixed. The links for the ranging pairs are shown. The ranging graph includes $3$ additional implicit links between the anchors, not shown.
  • Figure 2: Setup for two robots, seen as rigid bodies, carrying multiple tags.
  • Figure 3: Initial system configuration, waypoints for the leaders $1$ et $2$ and ranging network sparsity.
  • Figure 4: Robot and tag configuration for trajectory tracking. $(M,\vec{x}^r,\vec{y}^r)$ is the robot frame.
  • Figure 5: Tag trajectories in the workspace.
  • ...and 11 more figures

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Definition 1: FIM
  • Theorem 1: Equality constrained CRLB hero_lower_1990
  • Proposition 1
  • proof
  • Remark 3
  • Remark 4
  • Definition 2: Infinitesimal motion of a framework
  • Definition 3: Infinitesimal rigidity
  • ...and 18 more