Designing Control Barrier Functions Using a Dynamic Backup Policy

Technical Summary: Designing Control Barrier Functions Using a Dynamic Backup Policy

Problem Statement
Control Barrier Functions (CBFs) are established tools for synthesizing safety filters for nonlinear control-affine systems subject to state and input constraints. However, the systematic construction of valid CBFs for general systems under arbitrary constraints remains an open problem. While practitioners often rely on candidate CBFs tuned with slack variables, these lack rigorous safety guarantees. Existing systematic approaches include machine learning, sum-of-squares optimization, and backup CBFs that leverage pre-specified policies. A specific gap exists in designing CBFs that rigorously handle arbitrary constraints while offering a dynamic, parametrized backup strategy that avoids the conservatism of Lyapunov-based methods or the slowness of traditional Reference Governor (RG) approaches.

Methodology
The proposed method constructs CBFs within an augmented state-and-reference space ($\mathbb{R}^n \times \mathbb{R}^l$), inspired by Reference Governor literature and Dynamic Safety Margins (DSMs). The core methodology involves the following steps:

1. System Setup and Assumptions: The approach considers control-affine systems $\dot{x} = f(x) + g(x)u$ subject to constraints. It assumes the existence of an equilibrium manifold parameterized by a virtual reference $v$, along with a continuously differentiable, locally stabilizing control policy $\pi(x, v)$ that renders the equilibrium $\bar{x}(v)$ asymptotically stable.
2. Trajectory-Based DSM Definition: Instead of using Lyapunov functions, the authors define a Dynamic Safety Margin (DSM) $\Delta(x, v)$ based on the trajectory of the "prestabilized" system (where the control input is fixed to $\pi(x, v)$). The DSM quantifies the minimum distance to constraint violation over a future time horizon, assuming the reference $v$ remains constant:
   $$\Delta(x, v) = \inf_{\tau \in [0, \infty)} c(\Phi(\tau, x, v), v)$$
   where $\Phi$ is the flow of the prestabilized dynamics and $c$ describes the constraints.
3. Sensitivity Analysis: To compute the Lie derivatives required for the CBF condition, the paper derives the sensitivity Jacobians of the flow $\Phi$ with respect to the initial state $x$ ($S_x$) and the reference $v$ ($S_v$). It proves that under mild assumptions (Hurwitz linearization at equilibrium), these sensitivity matrices converge to known values as $\tau \to \infty$.
4. Tractable Formulation: To address the computational intractability of infinite horizons and the potential non-smoothness of the infimum function, the method employs:
   • Finite Horizon with Terminal Constraints: The infinite horizon is truncated to $[0, T]$, with a terminal DSM $\Delta_F$ (e.g., Lyapunov-based) ensuring invariance for $\tau > T$.
   • Clarke Generalized Jacobians: To handle cases where the infimum is not unique (non-smoothness), the safety filter constraints are enforced using Clarke's generalized Jacobians, ensuring the CBF condition holds for all subgradients.
5. Safety Filter Implementation: The resulting CBF is used in a Quadratic Program (QP) safety filter. This filter minimizes the deviation from a nominal controller $\kappa(x)$ and a reference navigation field $\rho(v)$, subject to the constraint that the time derivative of the DSM (computed via the sensitivity Jacobians) satisfies the CBF condition.

Key Contributions
The paper extends previous work on DSMs and backup CBFs with four specific contributions:

• Parametrized Backup Policy: The backup policy is not a single static policy but is parametrized by the system's equilibrium manifold. This allows the safety filter to utilize a family of backup policies rather than a fixed one.
• Dynamic Parametrization: The backup policy is dynamic because its parametrization (the reference $v$) is time-varying, allowing for more flexible safety enforcement.
• Stability of Sensitivity Matrices: The authors rigorously demonstrate that the underlying sensitivity matrices ($S_x$ and $S_v$) are asymptotically stable under mild assumptions, facilitating the computation of the CBF derivatives.
• Handling Non-Smoothness: The potential non-smoothness of the resulting CBF is addressed rigorously using Clarke generalized Jacobians, ensuring the safety guarantees hold even when the trajectory-based minimum is not unique.

Results
The method is validated on an inverted pendulum on a cart, a system characterized by input saturation, multiple constraints, strong nonlinearities, high relative degree, and non-minimum phase dynamics. The system is tasked with moving from an initial position to a target while respecting state and input limits.

Comparative simulations against four other approaches (Nominal control, Lyapunov-based DSM-CBF, Reference Governor (ERG), and Backup CBF) demonstrate that:

• The Nominal controller fails to satisfy constraints.
• The Lyapunov-based DSM-CBF is overly conservative, resulting in slower performance.
• The Reference Governor (ERG) approach is systematically slower than CBF-based methods.
• The Backup CBF (using a fixed backup policy) works but is slower than the proposed method.
• The proposed Trajectory-based DSM-CBF achieves the best performance while guaranteeing constraint satisfaction. The authors attribute this to the added degree of freedom provided by parametrizing the backup policy via the reference $v$, which allows the system to navigate constraints more efficiently than fixed-policy backups.

Significance and Claims
The paper claims to provide a systematic approach to constructing valid CBFs for nonlinear systems with arbitrary constraints, moving beyond the reliance on heuristic candidate functions. By leveraging trajectory predictions of a prestabilized system, the method bridges the gap between the rigorous safety of CBFs and the flexibility of Reference Governors. The authors emphasize that the approach is computationally tractable because the trajectory predictions depend only on the current state and reference, not the optimization variables, keeping the cost negligible compared to Model Predictive Control (MPC). The work establishes that dynamic safety margins derived from trajectory analysis are valid CBFs, offering a rigorous alternative to Lyapunov-based constructions that may be too conservative for open-loop unstable systems.