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Backup-Based Safety Filters: A Comparative Review of Backup CBF, Model Predictive Shielding, and gatekeeper

Taekyung Kim, Aswin D. Menon, Akshunn Trivedi, Dimitra Panagou

Abstract

This paper revisits three backup-based safety filters -- Backup Control Barrier Functions (Backup CBF), Model Predictive Shielding (MPS), and gatekeeper -- through a unified comparative framework. Using a common safety-filter abstraction and shared notation, we make explicit both their common backup-policy structure and their key algorithmic differences. We compare the three methods through their filter-inactive sets, i.e., the states where the nominal policy is left unchanged. In particular, we show that MPS is a special case of gatekeeper, and we further relate gatekeeper to the interior of the Backup CBF inactive set within the implicit safe set. This unified view also highlights a key source of conservatism in backup-based safety filters: safety is often evaluated through the feasibility of a backup maneuver, rather than through the nominal policy's continued safe execution. The paper is intended as a compact tutorial and review that clarifies the theoretical connections and differences among these methods.

Backup-Based Safety Filters: A Comparative Review of Backup CBF, Model Predictive Shielding, and gatekeeper

Abstract

This paper revisits three backup-based safety filters -- Backup Control Barrier Functions (Backup CBF), Model Predictive Shielding (MPS), and gatekeeper -- through a unified comparative framework. Using a common safety-filter abstraction and shared notation, we make explicit both their common backup-policy structure and their key algorithmic differences. We compare the three methods through their filter-inactive sets, i.e., the states where the nominal policy is left unchanged. In particular, we show that MPS is a special case of gatekeeper, and we further relate gatekeeper to the interior of the Backup CBF inactive set within the implicit safe set. This unified view also highlights a key source of conservatism in backup-based safety filters: safety is often evaluated through the feasibility of a backup maneuver, rather than through the nominal policy's continued safe execution. The paper is intended as a compact tutorial and review that clarifies the theoretical connections and differences among these methods.

Paper Structure

This paper contains 23 sections, 3 theorems, 27 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. Mathematical Setup
  4. Problem Statement
  5. REVIEW OF BACKUP CBF
  6. Implicit safe set induced by the backup policy
  7. Backup CBF-QP
  8. REVIEW OF MODEL PREDICTIVE SHIELDING
  9. Digital update times, candidate trajectories, and validity
  10. MPS decision rule
  11. REVIEW OF GATEKEEPER
  12. The problem of "safety evaluation on backup"
  13. Maximizing the switching time
  14. Update times and search strategy
  15. THEORETICAL ANALYSIS
  16. ...and 8 more sections

Key Result

Theorem 1

Assume that $\mathcal{S}_0 \subseteq \mathcal{C}$ is forward invariant under the backup policy $\pi_{\textup{b}}$. Then, for every $T_B>0$, the set $\mathcal{S}$ in eq:implicit_set is a controlled invariant set and satisfies $\mathcal{S}_0 \subseteq \mathcal{S} \subseteq \mathcal{C}$. $\blacktriangl

Figures (4)

  • Figure A1: Recovered safe sets (light-colored regions) and filter-inactive sets (dark-colored regions) for the double-integrator example on the slice $(v_x,v_y)=(2,0)$. The black dashed curve shows the viability kernel obtained from HJ reachability on the full 4-dimensional state space. For each $1~\textup{m}\times 1~\textup{m}$ grid of initial positions, we simulate the closed-loop system under each safety filter; green trajectories remain safe, whereas yellow trajectories indicate unsafe or infeasible rollouts.
  • Figure F1: Geometric illustration of the local argument in \ref{['thm:backup_cbf_interior']}.
  • Figure G1: Reach-avoid scenario with a dynamic obstacle. (a) Robot trajectories generated by Backup CBF, MPS, and gatekeeper. (b) Backup CBF value as a function of time. (c) Comparison of the certified switching time $T_S^\star$ for MPS and gatekeeper.
  • Figure G2: Highway overtake scenario. (a) Vehicle trajectories generated by Backup CBF, MPS, and gatekeeper. (b) Backup CBF value as a function of time. (c) Comparison of the certified switching time $T_S^\star$ for MPS and gatekeeper.

Theorems & Definitions (11)

  • Definition 1: Recoverable set induced by a feedback policy
  • Definition 2: Backup policy
  • Definition 3: Safety filter
  • Definition 4: Filter-inactive set
  • Theorem 1: gurriet_online_2018
  • Definition 5: Candidate trajectory
  • Definition 6: Validity indicator
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 1 more