Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Information Control Barrier Functions: Preventing Localization Failures in Mobile Systems Through Control

Samuel G. Gessow, David Thorne, Brett T. Lopez

TL;DR

This work tackles localization safety for mobile systems by preventing ill-conditioning in nonlinear least-squares estimation through Information Control Barrier Functions (I-CBFs) that enforce $\lambda_{\min}(H(x,m)) \ge \lambda_s$. It develops two complementary pathways—analytic Hessian and anti-crossing barriers—plus higher-relative-degree extensions and predictive measurement modeling to ensure a unique, well-defined state estimate without relying on environmental priors. The approach is demonstrated on a 2-D double-integrator with range-only and bearing-only beacon localization, showing effective barrier maintenance and different trade-offs in control effort and computation. The framework is general, compatible with existing planners, and holds promise for broader sensing modalities (e.g., vision, LiDAR) with potential robustness enhancements via future extensions.

Abstract

This paper develops a new framework for preventing localization failures in mobile systems that must estimate their state using measurements. Safety is guaranteed by imposing the nonlinear least squares optimization solved in modern localization algorithms remains well-conditioned. Specifically, the eigenvalues of the Hessian matrix are made to be always positive via two methods that leverage control barrier functions to achieve safe set invariance. The proposed method is not constrained to any specific measurement or system type, offering a very general solution to the safe mobility with localization problem. The efficacy of the approach is demonstrated on a system being provided range-only and heading-only measurements for localization.

Information Control Barrier Functions: Preventing Localization Failures in Mobile Systems Through Control

TL;DR

This work tackles localization safety for mobile systems by preventing ill-conditioning in nonlinear least-squares estimation through Information Control Barrier Functions (I-CBFs) that enforce . It develops two complementary pathways—analytic Hessian and anti-crossing barriers—plus higher-relative-degree extensions and predictive measurement modeling to ensure a unique, well-defined state estimate without relying on environmental priors. The approach is demonstrated on a 2-D double-integrator with range-only and bearing-only beacon localization, showing effective barrier maintenance and different trade-offs in control effort and computation. The framework is general, compatible with existing planners, and holds promise for broader sensing modalities (e.g., vision, LiDAR) with potential robustness enhancements via future extensions.

Abstract

This paper develops a new framework for preventing localization failures in mobile systems that must estimate their state using measurements. Safety is guaranteed by imposing the nonlinear least squares optimization solved in modern localization algorithms remains well-conditioned. Specifically, the eigenvalues of the Hessian matrix are made to be always positive via two methods that leverage control barrier functions to achieve safe set invariance. The proposed method is not constrained to any specific measurement or system type, offering a very general solution to the safe mobility with localization problem. The efficacy of the approach is demonstrated on a system being provided range-only and heading-only measurements for localization.

Paper Structure

This paper contains 17 sections, 6 theorems, 6 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Overview
  4. Control Barrier Functions
  5. Other Useful Definitions
  6. Main Results
  7. Overview
  8. Information Control Barrier Function
  9. Analytic Hessian
  10. Anti-Crossing Control Barrier Function
  11. Predictive Measurement Models
  12. Higher Relative Degree I-CBFs
  13. Illustrative Example
  14. Range-Only Measurements
  15. Bearing-Only Measurements
  16. ...and 2 more sections

Key Result

Proposition 1

Let $z^* \in \mathbb{R}^n$ be a critical point of a twice differentiable function $F: \mathbb{R}^n \rightarrow \mathbb{R}$. If the eigenvalues of $H(z^*) = \nabla^2 F(z^*)$ are all positive locally then $z^*$ is a unique local minimum of $F$.

Figures (3)

  • Figure 1: (a): Example cost function with non-degenerate Hessian that has a unique critical point. (b): Example cost function with degenerate Hessian that has a non-unique critical point.
  • Figure 2: Trajectories and barrier values for range-only measurements. In the trajectory plots \ref{['fig:distance_trajectories', 'fig:distance_trajectories_avoid']} the safe region is the interior of the closed red curves and the desired state with the green $\times$. The analytic and anti-crossing methods prevent the system from leaving the safe set, but the analytic is more conservative as indicated by the larger value of the barrier function.
  • Figure 3: Trajectories and barrier values for bearing-only measurements. In the trajectory plots \ref{['fig:angle_trajectory', 'fig:angle_trajectory_avoid']} the safe region is the interior of the closed red curves and the desired state with the green $\times$. The analytic and anti-crossing method prevent the system from leaving the safe set, but the analytic is more conservative as indicated by the larger value of the barrier function.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: cf. ames2016control
  • Definition 5
  • Remark 1
  • Definition 6
  • Definition 7: cf. tao2006power
  • Remark 2
  • Proposition 1
  • ...and 15 more