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Model-Free Based Computations of Recursive Control Barrier Function: Ultra-Local Model Approach

This paper proposes a model-free framework for computing recursive control barrier functions using the ultra-local model approach and online dynamics estimation to enforce safety constraints in nonlinear systems with unknown dynamics and higher relative degrees, as demonstrated on an adaptive cruise control benchmark.

Original authors: Loïc Michel, Ricardo de Castro, Joseph Moyalan, Iman Ebrahimi, Jean-Pierre Barbot

Published 2026-08-18

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CC BY 4.0

Original authors: Loïc Michel, Ricardo de Castro, Joseph Moyalan, Iman Ebrahimi, Jean-Pierre Barbot

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of engineering, keeping a machine safe is often a battle against the unknown. Whether it is a self-driving car navigating a rainy highway or a robot arm assembling delicate electronics, the controller must ensure the system never strays into a dangerous zone. For decades, engineers have relied on precise mathematical blueprints of how these machines move to build such safety nets. These blueprints, known as system models, act like a map, telling the computer exactly how the vehicle will react when the brakes are pressed or the steering wheel is turned. However, in the messy reality of the physical world, these maps are rarely perfect. Weather changes, parts wear out, and unexpected bumps occur, creating a gap between the map and the territory. When the map is wrong, the safety net can fail, leading to accidents. This is why researchers are increasingly looking for ways to keep systems safe without needing a perfect map of the world, relying instead on what the machine is actually doing in the moment.

A team of researchers has developed a new way to solve this problem, creating a safety system that learns and adapts in real-time without needing a pre-written model of the machine's behavior. Their approach, detailed in a recent study, focuses on a concept called a Control Barrier Function. Think of this as an invisible, flexible wall that surrounds the safe zone of a machine. If the machine starts to drift toward this wall, the system automatically adjusts the controls to push it back to safety. Traditionally, calculating where this wall should be requires knowing the exact physics of the machine. If the physics are unknown or change, the calculation fails. The researchers, led by L. Michel and colleagues, proposed a different method. Instead of guessing the physics, they use a technique called an ultra-local model. This is a simple, short-term snapshot of how the machine is moving right now, updated continuously as new data comes in. It ignores the complex, long-term history of the machine and focuses only on the immediate relationship between the control input and the resulting movement.

The core of their discovery is a method to turn this simple, real-time snapshot into a robust safety guarantee. The researchers realized that because they are estimating the machine's behavior on the fly, there is always a small amount of uncertainty, like a foggy window through which they are looking. Their innovation was to quantify this fog. They developed a way to calculate a "safety margin" that grows larger when the machine is moving erratically or when the estimate is less certain. This margin acts as a buffer, pushing the invisible safety wall further away from the actual danger zone whenever the system is unsure of what is happening. By doing this, the system becomes much harder to trick by unexpected disturbances or modeling errors. The researchers tested two different ways to manage this process. The first method continuously tweaks the safety calculations based on a steady stream of data, while the second method tests several different settings simultaneously and picks the one that offers the safest outcome at that exact moment.

To see if this idea worked in practice, the team ran a series of computer simulations. They chose a scenario that is a standard test for safety systems: an adaptive cruise control system for a car. In this test, a car is following another vehicle, and the goal is to maintain a safe distance. The researchers introduced a tricky situation where the car ahead suddenly slows down, and the reference signal for the following car was temporarily set to a distance that was too close to be safe. In a traditional system that relies on a fixed model, this could lead to a collision because the model might not react fast enough to the sudden change. However, the new system handled the situation differently. By constantly updating its understanding of the car's movement and widening its safety buffer when uncertainty was high, the system successfully prevented the car from getting too close. It overrode the unsafe commands from the standard controller and kept the vehicle within the safe zone, even though it had no precise knowledge of the car's weight, aerodynamics, or engine characteristics.

The simulations revealed some important details about how the system behaves under pressure. When the researchers tested the continuous adjustment method, they found that if the safety rules were set too strictly, the system sometimes struggled to recover from sudden changes, occasionally allowing the car to drift into the danger zone. In contrast, the method that tested multiple settings at once proved much more resilient. It acted like a quick decision-maker, instantly switching to the most reliable setting whenever the situation became critical. This approach kept the car safe across all the different test scenarios, even when the researchers deliberately made the safety rules very tight. The study showed that by combining a simple, real-time estimate of movement with a dynamic safety buffer, it is possible to create a controller that is both safe and adaptable. The results were not a final solution for every possible machine, but they provided strong evidence that this model-free approach works well for complex systems where the physics are hard to pin down.

The implications of this work extend beyond just cars. The researchers suggest that this method could be valuable for any system where safety is critical but the underlying dynamics are complex or poorly understood, such as battery charging systems or energy grids. In these fields, the rules of how energy flows can change depending on temperature, age, and load, making traditional models difficult to maintain. By using a system that learns the behavior on the fly and builds its own safety margins, engineers could design safer, more efficient technologies without needing to solve every equation beforehand. The study concludes that while the math behind the scenes is sophisticated, the result is a practical tool that allows machines to navigate the unknown with confidence. It represents a shift from relying on perfect knowledge to relying on smart, real-time adaptation, ensuring that safety is maintained even when the map is incomplete.

Technical Summary: Model-Free Based Computations of Recursive Control Barrier Function: Ultra-Local Model Approach

Problem Statement

Control Barrier Functions (CBFs) offer a systematic framework for enforcing safety constraints in nonlinear control systems by guaranteeing the forward invariance of a prescribed safe set. However, standard CBF implementations typically rely on accurate system models to compute Lie derivatives and enforce safety constraints. In real-world scenarios involving significant modeling uncertainties, unmodeled dynamics, or external disturbances, these model-dependent approaches can fail or become overly conservative.

Existing model-free or data-driven alternatives, such as those using neural networks or Gaussian Processes, often struggle to provide rigorous, real-time safety guarantees without introducing significant computational burdens or requiring extensive training data. Furthermore, while ultra-local models (ULMs) have been successfully used for model-free control, previous applications of ULMs in safety-critical contexts often rely on instantaneous estimates without explicitly quantifying the uncertainty associated with the reconstruction of dynamics. Consequently, the impact of estimation errors, measurement noise, and derivative approximations on safety constraints is not explicitly propagated, potentially compromising safety guarantees.

Methodology

The paper proposes a model-free framework for computing recursive Control Barrier Functions based on the Ultra-Local Model (ULM) approach. The core philosophy is to approximate the system's local behavior using a low-order input-output representation that is continuously updated online, explicitly accounting for uncertainty bounds.

1. Ultra-Local Modeling with Uncertainty Quantification

Instead of an explicit system model $\dot{x} = f(x) + g(x)u$ , the system is approximated by a SISO ultra-local model:
$y^{(\nu)}(t) \approx F(t) + \beta u(t) + \Delta_m(t)$
where:

$y^{(\nu)}$ is the $\nu$ -th time derivative of the output (chosen to match or exceed the relative degree of the safety constraint).
$F(t)$ aggregates unknown dynamics, nonlinearities, and disturbances.
$\beta$ is a non-zero constant scaling the control input.
$\Delta_m(t)$ represents the uncertainty bound.

The term $F(t)$ is estimated online using numerical differentiation (specifically homogeneous semi-implicit differentiators) to handle noise. Crucially, the framework introduces a dynamic uncertainty bound $\hat{\Delta}_m(t)$ to quantify the estimation error. This bound is computed using a Welford's algorithm on a sliding window of residuals, calculating both the mean ( $\mu$ ) and variance ( $\sigma^2$ ) of the estimation error.

Three strategies for defining the uncertainty bound $\hat{\Delta}_m$ are proposed:

Method A: A spatial-geometric approach where the bound scales based on the system's proximity to the safety boundary.
Method B: Combines the spatial-geometric penalty with the statistical variance to protect against transient spikes.
Method C: Relies solely on a statistical variance threshold ( $|\mu| + \kappa|\sigma|$ ).

2. Robust CBF Formulation

The proposed framework modifies the standard CBF condition to include the uncertainty bound, effectively shifting the safety boundary to a "virtual" boundary that is more restrictive.

For Relative Degree 1, the condition becomes $\hat{F} + \beta u \geq -k_c(y - y_{min} - \hat{\Delta}_m/k_c)$ .
For Relative Degree 2 (recursive CBFs), the condition incorporates the second derivative and is similarly adjusted by the uncertainty term.

This formulation allows the CBF to anticipate safety violations by activating constraints earlier when uncertainty is high or the system is near the boundary.

3. Two Control Strategies

The paper presents two distinct strategies for determining the control input $u$ :

Strategy 1 (Gradient-Based ULM-CBF): Uses a two-time-scale stochastic gradient descent to recursively estimate the parameters $\hat{F}$ and $\hat{\beta}$ . The control input is derived by solving a quadratic program (QP) that minimizes deviation from a nominal controller while satisfying the robust CBF constraint.
Strategy 2 (Grid-Based ULM-CBF): Instead of a single online estimate, this approach evaluates a discrete grid of candidate $\beta$ values in parallel. For each candidate, a robust CBF-QP is solved. A selection criterion (minimizing control energy, uncertainty bound energy, control variation, and parameter variation) is used to select the safest control input and the corresponding $\beta$ index. This strategy aims to provide structural robustness by selecting the most suitable input-output scaling for the current operating regime.

Key Contributions

Model-Free Recursive CBF: The development of a framework that constructs high-order (recursive) CBFs without requiring an explicit system model, relying instead on online ULM estimation.
Explicit Uncertainty Propagation: The introduction of a method to explicitly quantify and propagate estimation uncertainty ( $\hat{\Delta}_m$ ) within the CBF constraint, transforming the safety boundary into a dynamic, uncertainty-aware virtual boundary.
Dual-Strategy Implementation: The proposal of both a gradient-based adaptation method and a grid-based selection method, offering a trade-off between computational simplicity and robustness against estimation inconsistencies.
Validation on Benchmarks: Demonstration of the approach on a first-order illustrative example and a second-order Adaptive Cruise Control (ACC) benchmark, showing the ability to maintain safety even when the nominal reference violates safety constraints.

Simulation Results

The paper validates the approach through two simulation scenarios:

First-Order System: A parametric study compared the two strategies under varying CBF stiffness ( $k_c$ ) and geometric penalty ( $\epsilon_p$ ) parameters.
- Findings: Strategy 1 (Gradient) showed sensitivity to parameter tuning, with certain configurations leading to constraint violations (safety loss) when stiffness was high or penalty thickness was low. Strategy 2 (Grid) demonstrated superior robustness, consistently maintaining safety margins across all tested configurations by instantaneously selecting the optimal model parameters.
- Observation: The grid-based approach effectively decoupled constraint enforcement from empirical tuning parameters, particularly when using the statistical-only uncertainty bound (Method C).
Adaptive Cruise Control (ACC): A second-order system representing vehicle longitudinal dynamics was used to track inter-vehicle distance.
- Scenario: The reference trajectory was temporarily modified to violate the minimum safety distance.
- Result: The nominal controller alone would have driven the system into an unsafe state. The proposed CBF-based controller successfully overrode the nominal action, maintaining the inter-vehicle distance above the safety threshold without relying on a precise vehicle model.
- Performance: The grid strategy successfully adapted the scaling parameter $\beta$ in real-time to handle varying aerodynamic and rolling resistance forces, ensuring robust safety enforcement.

Significance and Claims

The authors claim that this work provides a systematic framework for enforcing safety in systems with higher relative degrees and significant modeling uncertainties, without the need for precise system models.

Robustness: The explicit propagation of uncertainty allows the system to remain safe despite estimation errors and measurement noise, addressing a key limitation of previous model-free CBF applications.
Anticipation: By shifting the safety boundary based on uncertainty, the controller can anticipate and prevent safety violations before they occur.
Applicability: The approach is presented as particularly relevant for safety-critical applications where dynamics are complex or poorly known, such as automotive systems (ACC) and potentially energy systems (e.g., battery charging), where safety constraints (voltage, current, temperature) must be enforced without detailed electrochemical models.

The paper concludes that while the grid-based strategy offers enhanced robustness and simpler tuning, the framework as a whole opens new directions for deriving formal safety guarantees in model-free control by explicitly accounting for estimation errors.