Safety-Critical Control via Recurrent Tracking Functions

1. Problem Statement

The paper addresses the challenge of synthesizing safety-critical controllers for high-order nonlinear systems (e.g., aerial vehicles, legged robots).

• The Core Difficulty: Constructing valid Control Barrier Functions (CBFs) for high-dimensional, complex systems is computationally intractable.
• The Layered Control Approach: A common strategy is to design CBFs on a Reduced-Order Model (RoM) (capturing safety-critical states) and use a low-level controller to track this reference on the Full-Order Model (FoM).
• The Bottleneck: Traditional layered control requires the tracking error between the FoM and RoM to satisfy a monotonic Lyapunov decay condition (exponential convergence). Systematic methods to construct such monotonic tracking functions exist only for fully-actuated systems. For underactuated or complex systems, enforcing strict monotonic decay is often impossible or overly conservative, hindering practical deployment.

2. Methodology

The authors propose a framework that relaxes the strict monotonic decay requirement of Lyapunov functions to a finite-time recurrence condition.

A. Recurrent Tracking Functions (RTFs)

Instead of requiring the tracking error to decrease continuously, the authors introduce Recurrent Tracking Functions (RTFs).

• Definition: An RTF requires that the tracking error decreases exponentially only at specific time instances separated by a uniform bound $\tau$.
• Key Insight: For systems with exponentially stable tracking errors, any norm of the tracking error (e.g., $\|\dot{e}\|$) serves as a valid RTF. This eliminates the need to explicitly construct complex, monotonic Lyapunov functions.
• Mechanism: The tracking error is allowed to deviate transiently (increase) but must return to a lower value within time $\tau$.

B. Recurrent Control Barrier Functions (RCBFs)

The framework integrates the RoM CBF with the RTF to create a Recurrent CBF ($h_V$) for the full system.

• Construction: The combined barrier function is defined as:
  $$h_V(z, \dot{e}) = -V(z, \dot{e}) + \alpha_e h(z)$$
  Where:
  • $h(z)$ is the CBF for the RoM.
  • $V(z, \dot{e})$ is the RTF (representing tracking error).
  • $\alpha_e$ is a scaling factor derived from system constants.
• Safety Logic: The zero-superlevel set $S_V = \{h_V \geq 0\}$ is control $\tau$-recurrent. This means trajectories may temporarily leave the set $S_V$ (and thus the strict safety margin of the RoM), but they are guaranteed to re-enter within time $\tau$.
• Guarantee: If the tracking error converges faster than the system approaches the safety boundary (i.e., convergence rate $\beta >$ safety decay rate $\alpha$), the actual FoM state never enters the unsafe region, even if the intermediate set $S_V$ is violated transiently.

C. Robustness to Disturbances

The framework is extended to handle model uncertainties and external disturbances via Input-to-State Safety (ISSf).

• Disturbances are modeled as bounded inputs causing a steady-state offset in the tracking error.
• The safe set is enlarged ($S_d$) to account for this offset, ensuring safety is maintained even with non-vanishing tracking errors, provided the disturbance magnitude is bounded.

3. Key Contributions

1. RTF Framework: The formulation of Recurrent Tracking Functions as a relaxation of standard Lyapunov conditions. It proves that for Lipschitz continuous systems with exponentially stable tracking, any norm of the tracking error is a valid RTF.
2. Theoretical Safety Guarantees: The proof that if the initial state lies in the recurrent set $S_V$, any controller satisfying the RTF condition guarantees that the FoM remains in the safe set $S_{FoM}$ for all time.
3. Systematic Synthesis: A method to construct safety certificates for high-order systems without needing complex, system-specific Lyapunov functions, leveraging simple norm-based functions.
4. Numerical Validation: A proof-of-concept experiment on a 2D double-integrator system navigating obstacles, demonstrating that the proposed method guarantees safety where traditional monotonic approaches might fail or be too conservative.

4. Results

• Simulation Setup: A 2D agent ($\ddot{z} = u$) navigating through circular obstacles. The RoM is a single integrator ($\dot{z} = v$).
• Comparison: The authors tested three scenarios with different safety decay rates ($\alpha$):
  • $\alpha = 5$ (Too aggressive): The RoM CBF was invalid; the system collided with obstacles.
  • $\alpha = 1$ (Initial state outside $S_V$): The system started outside the recurrent safe set and eventually became unsafe.
  • $\alpha = 0.5$ (Valid): The initial state was within $S_V$. The tracking error exhibited transient deviations but satisfied the recurrence condition. The system successfully navigated the obstacles without collision.
• Observation: The results confirmed that the tracking error $\|\dot{e}\|$ shrinks exponentially (satisfying the RTF condition) and that the barrier function $h(t)$ remains non-negative, validating the theoretical safety guarantees.

5. Significance

• Scalability: This approach significantly lowers the barrier for applying CBFs to high-dimensional, underactuated, or complex nonlinear systems where constructing monotonic Lyapunov functions is infeasible.
• Flexibility: By allowing transient deviations in tracking errors, the controller can utilize more of the system's available control authority, reducing conservatism compared to methods that require strict error bounds at all times.
• Practicality: The reliance on simple norms (rather than complex analytical Lyapunov functions) makes the synthesis of safety controllers more amenable to data-driven approaches and learning-based methods in the future.
• Theoretical Bridge: It bridges the gap between recurrence-based stability analysis and safety-critical control, offering a rigorous mathematical foundation for "safe but not strictly monotonic" tracking behaviors.