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LBA-CBF: Rapidly Adaptive Safety Filters via Parallel Dynamics Inference

Maitham F. AL-Sunni, Timeea-Andreea Radu, Hassan Almubarak, Henry Z. Liao, Michael Görner, Francesco Maurelli, John M. Dolan

arXiv:2610.08765v1cs.ROcs.SYeess.SY

TL;DR

An abrupt, unmeasured dynamics change can undermine a CBF certificate based on an assumed model. LBA-CBF ranks a finite candidate bank using recent prediction errors and certifies against retained models; in quadrotor simulation, it reaches the goal from every initial condition, matching the Oracle while adaptive baselines succeed on fewer runs.

  • Problem

    CBF inputs are certified using predicted barrier evolution under an assumed dynamics model, which can be undermined by abrupt, unmeasured regime changes.

    Source: §I · p. 1

  • Method

    LBA-CBF ranks a finite bank of candidate dynamics by recent prediction error and enforces CBF constraints against the best-fitting or prediction-plausible models.

    Source: §I · p. 1

  • Results

    LBA-CBF succeeds from every quadrotor simulation initial condition, matching the Oracle, while L1-AC and DOB-CBF each succeed from 62% and HO-RaCBF from 88%.

    Source: 3) Results · p. 6

  • Takeaways & Limitations

    LBA-CBF was demonstrated on Crazyflie 2.1 and F1TENTH hardware for adaptation to wind, payload release, and varying tire–road friction.

    Source: B. Contributions · p. 2

Abstract

from arXiv · show

Control barrier functions (CBFs) certify commands through an assumed dynamics model, so an abrupt, unmeasured regime change can undermine the certificate exactly when safety matters most. We present Look-Back Adaptive Control Barrier Functions (LBA-CBF), which rank a finite bank of candidate dynamics by recent prediction error over a short look-back window and enforce the high-order CBF condition against every model within a tolerance of the best, spanning best-fit adaptation to full-bank robust filtering. The dynamics may depend nonlinearly on the unknown parameters, and no switching model or continuously parameterized estimator is required. We prove that any feasible filtered input satisfies the true CBF condition whenever a safety-representative candidate is retained. In quadrotor simulation with abrupt wind reversals and an unknown payload, LBA-CBF is safe and reaches the goal from all random initial conditions, matching an oracle, while adaptive and robust baselines achieve 0-88% success. Banks of up to 250,000 models run inside the control loop, and Crazyflie 2.1 and F1TENTH experiments demonstrate adaptation to wind, payload release, and varying tire-road friction. Code, videos, and project details are available at: https://lla-control.github.io

I. INTRODUCTION

CBF safety certificates depend on the assumed dynamics, so abrupt unmeasured regime changes can undermine them. LBA-CBF instead ranks a finite model bank using recent prediction errors and certifies the barrier against retained models, with a guarantee conditional on retaining a safety-representative model.

I. INTRODUCTION

  • Abrupt, unmeasured changes in dynamics can undermine CBF safety certificates because each input is certified using predicted barrier evolution under an assumed model.Examples include an unknown payload, sudden wind gust, or changed tire–road friction.Source: §I · p. 1

1) Robust and adaptive safety:

  • Robust CBFs use a fixed worst-case uncertainty set, while adaptive methods estimate parameters or model errors online; both approaches have limitations under abrupt regime changes.The introduction describes robust conservatism that online data cannot reduce and an estimation transient for abrupt changes.Source: §I · p. 1

2) Switched dynamics and regime change:

  • Unlike approaches requiring known switching structure, LBA-CBF infers the active regime from recent state–action transitions without known guards or a transition model.Related switched-system methods construct mode-dependent conditions under specified hybrid structure or estimate mode-specific barrier residuals.Source: 2) Switched dynamics and regime change · p. 2

3) Multiple-model methods:

  • LBA-CBF uses windowed residuals for safety certification, retaining a best-fitting model or plausible subset rather than only selecting a model for adaptive racing MPC.The approach is distinct from prior look-back model selection for adaptive racing MPC [19].Source: 3) Multiple-model methods · p. 2

B. Contributions

  • LBA-CBF ranks a finite bank by recent prediction error and enforces the CBF condition against best-fitting or prediction-plausible models, without a switching model or continuously parameterized estimator.Its parameters may enter the dynamics nonlinearly, including through the input gain.Source: §I · p. 1, B. Contributions · p. 2
  • Every feasible LBA-CBF input satisfies the true CBF condition when the retained set contains a safety-representative model; a finite tolerance spans best-fit adaptation to full-bank robust filtering.The stated guarantee is conditional on retaining such a model.Source: B. Contributions · p. 2
  • The paper validates LBA-CBF in quadrotor simulation and on Crazyflie 2.1 and F1TENTH hardware, including adaptation to wind, payload changes, and varying tire–road friction.The reported implementation runs banks of up to 250,000 models inside the control loop.Source: B. Contributions · p. 2

II. PROBLEM FORMULATION

The problem is to minimally modify a nominal command while maintaining safety for a discrete-time robotic system whose dynamics regime is unobserved and may change. The formulation uses discrete-time high-order control barrier functions to express constraints that the safety filter can evaluate under different candidate models.

A. Robotic systems with changing dynamics regimes

  • The robotic system evolves in discrete time as x_k+1 = F(x_k, u_k; θ_k) = f(x_k; θ_k) + g(x_k; θ_k)u_k, with state, input, and dynamics parameter in specified spaces.The parameter is unobserved, and the parameter and input sets are assumed compact.Source: A. Robotic systems with changing… · p. 2
  • The dynamics parameter represents the active regime, which may change with payload, contact, terrain friction, or aerodynamic loading, without an observed regime or known switching law.The dynamics need not be affine in the unknown parameter.Source: A. Robotic systems with changing… · p. 2
  • Safety is represented by a path constraint h that defines the safe set, and a controller is safe when every trajectory from an admissible initial condition keeps h(x_k) ≥ 0.This condition is described as rendering the safe set controlled invariant.Source: A. Robotic systems with changing… · p. 2

B. Discrete-time high-order control barrier functions

  • The filter uses discrete-time HOCBF conditions with explicit dependence on the dynamics parameter so it can evaluate the condition under different candidate models.The formulation defines relative degree r by the first step at which an applied input affects the safety function.Source: B. Discrete-time high-order control… · p. 2
  • The HOCBF recursion starts with ψ0(x; ϑ) = h(x) and defines higher-order barrier terms for a parameter ϑ.For q ≤ r−1, the recursion's right-hand side is independent of the input; ψr is the first term in which the input appears.Source: B. Discrete-time high-order control… · p. 2
  • Enforcing the order-r HOCBF condition at every step yields forward invariance when the initial ψq terms are nonnegative for q = 0, …, r−1.The stated result assumes the initial barrier terms meet these conditions.Source: B. Discrete-time high-order control… · p. 3
  • The objective is a real-time filter that minimally modifies the nominal command using recent state–action transitions while accounting for the unobserved, potentially changing dynamics regime.The nominal command is given by π_nom(x_k).Source: B. Discrete-time high-order control… · p. 3

III. LBA-CBF: RAPIDLY ADAPTIVE SAFETY FILTERS

LBA-CBF combines a finite bank of candidate dynamics, recent prediction-based model screening, and a safety filter that applies CBF constraints to the retained models. The bank can be constructed using different finite sampling or modeling strategies.

III. LBA-CBF: RAPIDLY ADAPTIVE SAFETY FILTERS

  • LBA-CBF constructs a finite bank, retains models that explain recent transitions, then enforces the CBF condition against every retained model and returns the admissible input closest to the nominal command.These are the framework's three stages.Source: §III · p. 3

A. Finite dynamics model bank

  • The model bank is a finite set of candidate parameter values, each inducing a candidate transition map.The bank is denoted Θ_B = {θ_1, …, θ_M}.Source: A. Finite dynamics model bank · p. 3
  • Candidate dynamics can come from physical parameter ranges, system identification, expert-designed modes, simulation-based domain randomization, or other finite collections.The experiments use structured grids, but the method is not tied to a particular sampling strategy.Source: A. Finite dynamics model bank · p. 3

B. Look-back model plausibility

The look-back module scores candidate models on recent transitions, then retains models whose errors remain close to the minimum rather than relying only on a single best fit. Window length controls the tradeoff between responsiveness and noise rejection.

B. Look-back model plausibility

  • The module broadcasts the previous measured transition to all candidate models for batched one-step prediction, then scores their predictions once the next state is measured.The prediction-plausible set is defined using a tolerance η_k ≥ 0.Source: B. Look-back model plausibility · p. 3
  • The look-back module aggregates weighted quadratic one-step prediction errors across the most recent W transitions into a windowed prediction loss.The diagonal weighting matrix Λ_k scales state dimensions by their recent innovation scale; Λ_k = I gives an unweighted residual.Source: B. Look-back model plausibility · p. 3
  • Candidate scores can be updated recursively in constant time per candidate, giving O(M) look-back computation per step independent of window length.For k ≤ W, the loss accumulates without subtraction.Source: B. Look-back model plausibility · p. 3
  • A longer look-back window smooths the residual landscape but reacts more slowly to regime changes, while a shorter window reacts faster but is more sensitive to measurement and process noise.The window length W sets this responsiveness–noise tradeoff.Source: B. Look-back model plausibility · p. 3
  • A single minimum-residual model may be overly optimistic because recent data can leave several candidates indistinguishable, and predictive fit need not identify the model most relevant to safety.The method therefore retains unresolved model ambiguity rather than committing to one candidate.Source: B. Look-back model plausibility · p. 3

C. Multi-model LBA-CBF safety filter

LBA-CBF filters inputs by enforcing the CBF condition across all prediction-plausible models. Its tolerance ranges from a best-fit filter to full-bank robust filtering.

C. Multi-model LBA-CBF safety filter

  • The filter enforces the model-dependent CBF condition for every candidate in the prediction-plausible set.When constraints are affine in u, the optimization is a quadratic program; otherwise it remains a constrained safety-filter optimization.Source: C. Multi-model LBA-CBF safety filter · pp. 3–4
  • Increasing ηk adds model-dependent CBF constraints and weakly shrinks the admissible control set.A larger tolerance retains more candidate models.Source: C. Multi-model LBA-CBF safety filter · p. 4
  • The tolerance interpolates between a minimum-residual, best-fit filter and full-bank robust filtering.At ηk = 0 the filter uses the best-fit model; at sufficiently large ηk it imposes the CBF condition over the entire bank.Source: C. Multi-model LBA-CBF safety filter · p. 4
  • Prediction-loss calculations and model-dependent CBF evaluations both parallelize across candidates, allowing large banks to run inside the control loop.The implementation uses vectorized JAX on CPU or GPU.Source: C. Multi-model LBA-CBF safety filter · p. 4

D. Safety guarantee

LBA-CBF transfers safety from a retained safety-representative model to the true dynamics when the filter is feasible. The closed-loop guarantee additionally requires a sufficient tolerance and, for higher-order barriers, compatibility across regime changes.

D. Safety guarantee

  • A feasible filter input satisfies the true CBF condition when the plausible set contains a safety-representative candidate.The candidate need not be the true model; its barrier prediction must be no more optimistic than the true dynamics.Source: D. Safety guarantee · p. 4
  • A sufficiently large tolerance exists under the bank-adequacy assumption to retain a safety-representative model.The lemma establishes existence but does not prescribe an online tolerance; the relevant quantities are unknown to the controller, so ηk is selected empirically.Source: D. Safety guarantee · p. 4
  • Under the theorem’s assumptions, a feasible filter and tolerance at least ηs ensure the true order-r CBF condition at every step and render the safe set forward invariant.The initial intermediate barrier functions must be nonnegative, and regime changes must preserve their nonnegativity for higher-order barriers.Source: D. Safety guarantee · p. 4

IV. EXPERIMENTAL VALIDATION

The paper evaluates LBA-CBF in one simulation and two hardware studies. The studies cover quadrotor wind and payload changes and F1TENTH tire-road friction variation.

IV. EXPERIMENTAL VALIDATION

  • The evaluation comprises one simulation study and two hardware studies.The simulation compares LBA-CBF with adaptive and robust CBF baselines, a fixed-model nominal CBF, and an oracle; hardware comparisons use the same filter with a fixed model.Source: §IV · p. 5
  • Crazyflie 2.1 experiments test filtering an LQR under fan-induced wind and a swinging payload that is later released.The aircraft is tracked by motion capture.Source: §IV · p. 5
  • F1TENTH experiments use slippery plastic and rubber tires, with LBA-CBF identifying a model for a CBF-constrained MPC.The tire parameters enter the Fiala model nonlinearly, so the passage states that linear-in-parameter adaptive and robust CBF methods do not apply.Source: §IV · p. 5

A. Quadrotor: Simulation Experiments

The quadrotor simulation tests obstacle avoidance through windy gates under abrupt wind reversals and an unknown payload. It also specifies the candidate-model bank and the bank-size and window-length ablation.

A. Quadrotor: Simulation Experiments

  • The simulation uses a 26-initial-condition obstacle course with five alternating lateral corridors and a 40-second run at 0.02-second time steps.The nominal LQR heads straight to the goal, leaving obstacle avoidance to the safety filter; the initial descent begins while estimators are still converging.Source: 1) Task, dynamics, and filter · p. 5
  • Three unannounced wind reversals occur at horizontal-band crossings, while an unknown payload increases mass by 30%.Outside the goal’s calm pocket, the drag coefficients are 0.10; the payload scales the control-input term and is modeled explicitly in this simulation.Source: 1) Task, dynamics, and filter · p. 5
  • Figure 2 ablates bank size M and window length W, plotting one-step prediction error and control rate versus M.The simulation also uses second- and fourth-order CBFs for collective and differential thrust, respectively.Source: Figure 2 · p. 5, 1) Task, dynamics, and filter · p. 5
  • The model bank contains 11,341 candidates spanning wind, drag, and payload parameters, with no grid value matching the true wind, drag, or payload change.The default configuration uses W = 5 and η = 0; η = 0.01 and 0.05 retain sets of tens of candidates.Source: 1) Task, dynamics, and filter · p. 5

2) Baselines and metrics:

The comparison includes adaptive, robust, fixed-model, and oracle CBF filters, with baselines differing in how they handle uncertainty. Simulation success requires both safety and reaching the goal, while intervention is compared on a shared subset of starts.

2) Baselines and metrics:

  • Three adaptive baselines estimate additive planar forces, whereas LBA-CBF models the payload-dependent input gain; fixed-model and oracle filters provide reference cases.HO-RaCBF also retains second-order thrust and fourth-order tilt conditions; R-CBF robustifies the thrust condition against wind and drag bounds.Source: 2) Baselines and metrics · p. 5
  • The simulation figure orders all 26 trajectories per method across five obstacle gates, with wind direction changing between shaded and white bands.The table’s ΔU comparison uses the common subset of successful starts, not all 26 initial conditions.Source: Figure 3 · p. 6, Table I · p. 6, 2) Baselines and metrics · p. 6
  • A simulation run succeeds only if it remains safe around all ten obstacles and ends within 0.2 m of the goal.The table reports ΔU averaged over the 13 starts on which every method with nonzero success rate completed safely.Source: 2) Baselines and metrics · p. 6, Table I · p. 6

3) Results:

In simulation, LBA-CBF matched the oracle’s success across all starts, while adaptive baselines succeeded less often and robust or nominal filters did not reach the goal. Hardware tests also showed safety under wind and changing payload conditions.

3) Results:

  • LBA-CBF and Oracle succeeded on all 26 starts; HO-RaCBF reached 88%, L1-AC and DOB-CBF 62%, while R-CBF and Nominal CBF never reached the goal.On the paired set, LBA-CBF intervention was 42.6 versus 47.2 for Oracle; DOB-CBF needed 1.7× and L1-AC 1.4× its intervention.Source: 3) Results · p. 6
  • As η increased, larger clearance occurred in 81% of paired runs for η = 0.01 versus 0, and 94% for η = 0.05 versus 0.01.Table II reports median closest approach for matched course-and-start runs.Source: 3) Results · p. 6, Table II · p. 6
  • Expanding the bank from M = 5 to M = 25 by including payload mass reduced prediction error, while further parameter axes added little for W ≥5.W = 1 ranks models from a single noisy transition; longer windows retain pre-change data longer, balancing noise sensitivity against adaptation speed.Source: 3) Results · pp. 6–7
  • The bank update took 0.95 ms at M = 11,341 and 6.9 ms at M = 78,125, corresponding to 1.1 kHz and 145 Hz.Measurements used an Apple M1 Pro CPU with JAX.Source: 3) Results · p. 7

C. Safe Extreme Driving: Hardware Experiments

In F1TENTH hardware experiments, LBA-CBF stabilized the car under two tire conditions while the fixed nominal model crashed. The conclusion reports adaptation to tire changes but notes that the guarantee depends on bank adequacy and filter feasibility.

C. Safe Extreme Driving: Hardware Experiments

V. CONCLUSIONS

  • The conclusion reports control-rate adaptation to tire changes, while fixed-model, robust, and continuously adapting filters become unsafe or overly conservative.Source: §V · p. 8
  • The guarantee depends on bank adequacy and filter feasibility; constructive conditions for these and online tolerance adaptation are left to future work.Source: §V · p. 8
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