Uncertainty-Aware Predictive Control Barrier Functions

• UA-PCBFs are a unified framework that integrates probabilistic human hand motion forecasting with formal control barrier functions to enhance safety in human-robot interaction.
• They dynamically adjust safety margins based on forecasted uncertainty along the interaction axis instead of relying on fixed buffers.
• Empirical evaluations show that combining predictive LSTM forecasting with uncertainty-aware optimization yields fewer safety violations and improved system performance.

Uncertainty-Aware Predictive Control Barrier Functions (UA-PCBFs) are a unified framework that fuses probabilistic human hand motion forecasting with the formal safety guarantees of Control Barrier Functions, with the stated goal of safer human-robot interaction through probabilistic motion forecasting (Busellato et al., 28 Aug 2025). In contrast to predictive control barrier functions that use deterministic look-ahead, UA-PCBFs allow for dynamic adjustment of the safety margin thanks to the human motion uncertainty estimation provided by a forecasting module (Busellato et al., 28 Aug 2025). Related predictive barrier formulations already encode future safety along a nominal trajectory (Breeden et al., 2022) or interpret an MPC value function as a predictive control barrier function (Huang et al., 12 Feb 2025), while neighboring uncertainty-aware barrier methods robustify safety constraints through parameterized margins, deterministic Gaussian-process error envelopes, or probabilistic residual models (Alan et al., 2023, Gutierrez et al., 10 May 2025, Kumar et al., 2021).

1. Predictive-barrier lineage and the emergence of UA-PCBFs

Predictive CBFs were introduced to proactively consider the future safety of a system along a nominal trajectory, and to effect corrective action before the trajectory leaves a designated safe set (Breeden et al., 2022). In that formulation, a nominal path is propagated on a receding horizon and future safety is encoded into a current-time CBF constraint, yielding a QP-based safety filter that is proactive rather than reactive and can result in smaller modifications to the nominal trajectory and smaller control inputs than a traditional CBF, with faster computations than a nonlinear model predictive control approach (Breeden et al., 2022).

A second line of work formalized predictive control barrier functions through MPC. In that setting, the value function of a safe MPC problem is itself a CBF, referred to as the Predictive Control Barrier Function, and safety follows from recursive feasibility and forward invariance of the MPC-feasible set rather than from strict value-function decrease (Huang et al., 12 Feb 2025). Discrete-time approximate PCBFs then made the predictive barrier explicit through function approximation and showed that approximation error acts like a disturbance, yielding input-to-state stability of the approximate closed loop with respect to approximation errors (Didier et al., 2024).

UA-PCBFs inherit the predictive premise but change the source of conservatism. The motivating application is human-robot interaction, where human motion is stochastic, task-dependent variability rather than a deterministic trajectory (Busellato et al., 28 Aug 2025). The stated problem is that purely reactive methods are myopic, while worst-case envelopes brake unnecessarily, stall task progress, and tamper with the fluidity that true human-robot interaction demands (Busellato et al., 28 Aug 2025). UA-PCBFs therefore combine future prediction with uncertainty-dependent safety inflation rather than relying on a fixed margin (Busellato et al., 28 Aug 2025).

2. Core formulation in human-robot interaction

The UA-PCBF formulation in collaborative manipulation uses a geometric abstraction in which the operator’s hand is enclosed by a sphere and the robot’s last link and end-effector are enclosed by a cylinder (Busellato et al., 28 Aug 2025). The immediate safety objective is to maintain a minimum separation $d_{\min}$ between these two bodies. The distance used in the barrier is

$d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$

where $\mathbf o$ is the center of the hand sphere, $\mathbf c(x)$ is the point on the cylinder axis closest to $\mathbf o$, $x$ is the current pose of the robot TCP, and $r_{cyl}$ is the cylinder radius (Busellato et al., 28 Aug 2025).

Robot dynamics are written in control-affine form,

$\dot x = f(x) + g(x)\,u,$

and the nominal task-space controller is built from proportional position and orientation feedback. With current pose $(p_c,R_c)$, desired pose $(p_d,R_d)$, and Jacobian $d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$0, the reference twist is

$d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$1

with

$d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$2

and the nominal joint-space command is

$d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$3

(Busellato et al., 28 Aug 2025)

The predictive component follows the standard PCBF construction. A nominal future path $d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$4 satisfies

$d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$5

and the predictive barrier is

$d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$6

where $d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$7 is the earliest time in the horizon where a violation may occur and $d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$8 is a class-$d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$9 margin function. In the reported experiments, $\mathbf o$0 (Busellato et al., 28 Aug 2025).

The uncertainty-aware modification replaces a fixed separation margin by a forecast-dependent one:

$\mathbf o$1

Hence the required separation becomes

$\mathbf o$2

so the required separation is larger when uncertainty is larger (Busellato et al., 28 Aug 2025). This is the defining UA-PCBF step: the safe set is modified online by predictive uncertainty rather than by a fixed worst-case envelope (Busellato et al., 28 Aug 2025).

3. Probabilistic forecasting and uncertainty projection

The forecasting module is a lightweight autoregressive LSTM encoder-decoder operating on palm-center trajectories measured by a Leap Motion v2 (Busellato et al., 28 Aug 2025). At time $\mathbf o$3, the observed palm position is

$\mathbf o$4

Given an input window

$\mathbf o$5

the network predicts

$\mathbf o$6

with $\mathbf o$7 and $\mathbf o$8 (Busellato et al., 28 Aug 2025).

The encoder state is initialized by

$\mathbf o$9

the decoder starts from

$\mathbf c(x)$0

and each autoregressive step uses

$\mathbf c(x)$1

$\mathbf c(x)$2

(Busellato et al., 28 Aug 2025)

Training combines Gaussian negative log-likelihood and mean-squared error:

$\mathbf c(x)$3

$\mathbf c(x)$4

$\mathbf c(x)$5

The stated role of the NLL term is to discourage trivial zero-variance outputs and promote meaningful uncertainty estimates (Busellato et al., 28 Aug 2025).

The controller does not use the full covariance directly. Instead, it recovers

$\mathbf c(x)$6

defines the interaction direction

$\mathbf c(x)$7

and projects uncertainty onto that direction:

$\mathbf c(x)$8

The projected uncertainty is then scaled and clamped:

$\mathbf c(x)$9

with $\mathbf o$0 a tuning parameter (Busellato et al., 28 Aug 2025).

This construction uses uncertainty only along the hand-robot interaction axis. A plausible implication is that uncertainty orthogonal to the collision direction is intentionally de-emphasized, while uncertainty along the shortest approach direction directly enlarges the required separation (Busellato et al., 28 Aug 2025).

4. Safety filter and optimization structure

The underlying reactive CBF filter is the standard minimum-intervention QP

$\mathbf o$1

while the predictive CBF variant enforces analogous inequalities along the forecast horizon $\mathbf o$2 (Busellato et al., 28 Aug 2025).

UA-PCBFs impose both an instantaneous and a predictive uncertainty-aware barrier with slack:

$\mathbf o$3

The final optimization is

$\mathbf o$4

(Busellato et al., 28 Aug 2025)

The penalty on the reactive slack is fixed,

$\mathbf o$5

whereas the predictive slack penalty is uncertainty dependent:

$\mathbf o$6

Uncertainty therefore enters the framework in two places: directly in the barrier through $\mathbf o$7, and indirectly in the optimization through $\mathbf o$8 (Busellato et al., 28 Aug 2025). When predictive uncertainty increases, the safety distance enlarges, but the predictive constraint is also allowed to relax more easily than the instantaneous constraint (Busellato et al., 28 Aug 2025).

A common misconception is that UA-PCBFs amount to a fixed enlarged safety buffer. The defining formulas show the opposite: the margin is time-varying, forecast-dependent, and clamped, and it is coupled to the optimization weights rather than used as a static geometric inflation (Busellato et al., 28 Aug 2025).

5. Guarantees, neighboring formulations, and limitations

The reported safety interpretation is hybrid rather than purely stochastic. UA-PCBFs combine formal CBF-style safety enforcement with probabilistic forecasting, but the guarantees are not developed as a full formal stochastic theorem in the style of chance-constrained safety (Busellato et al., 28 Aug 2025). The guarantees are best understood as deterministic CBF guarantees conditioned on the chosen barrier function, together with practical probabilistic robustness induced by uncertainty inflation (Busellato et al., 28 Aug 2025). The formulation also does not require a full stochastic model of human dynamics or assumptions on how uncertainty evolves in time (Busellato et al., 28 Aug 2025).

This places UA-PCBFs within a broader barrier-method decomposition. Predictive but nominal barrier constructions already exist: receding-horizon predictive CBFs that encode future safety into a current-time control-affine CBF condition (Breeden et al., 2022), MPC-derived PCBFs whose value functions certify invariance of an MPC-feasible set (Huang et al., 12 Feb 2025), and approximate discrete-time PCBFs that treat approximation error as an additive perturbation to barrier decrease and prove input-to-state stability (Didier et al., 2024). These methods supply the predictive skeleton but not uncertainty-aware probabilistic safe-set inflation.

Uncertainty-aware but non-predictive strands provide complementary mechanisms. Parameterized barrier functions define a family of uncertainty-indexed sets

$\mathbf o$9

so that $x$0 quantifies safety degradation and $x$1 quantifies conservativeness under uncertainty (Alan et al., 2023). Real-time Gaussian-process CBFs learn an unknown additive drift $x$2, construct a deterministic RKHS-based error bound

$x$3

and robustify the barrier derivative by subtracting the worst-case projection of model error (Gutierrez et al., 10 May 2025). Probabilistic GP-based safety filters instead model the residual directly in the barrier derivative and enforce a mean-minus-standard-deviation barrier condition with high confidence (Kumar et al., 2021). A layered predictive CBF for reduced-order/full-order safety transfer adds a learned predictive robustness term $x$4 to a reduced-order CBF inequality so that full-order rollouts remain safe despite tracking mismatch (Compton et al., 2024).

A plausible synthesis is that UA-PCBFs occupy the intersection of these families: predictive look-ahead from PCBFs, dynamic safety inflation from uncertainty-aware CBFs, and learned forecasting from modern HRI. The main unresolved issue is that dynamic margin selection depends on forecast quality and uncertainty calibration rather than on an explicit robust or chance-constrained invariance theorem (Busellato et al., 28 Aug 2025).

6. Empirical evaluation and observed behavior

The forecasting module was trained on $x$5 individual samples, over $x$6 subsequences of $x$7 second at $x$8 Hz, for $x$9 epochs using AdamW with learning rate $r_{cyl}$0 and cosine annealing; inference ran on an RTX 3060, while training used an RTX 3090 (Busellato et al., 28 Aug 2025). At a $r_{cyl}$1 ms horizon, the reported forecasting metrics were:

• Linear interpolation: $r_{cyl}$2, $r_{cyl}$3
• Kalman filter: $r_{cyl}$4, $r_{cyl}$5
• Particle filtering: $r_{cyl}$6, $r_{cyl}$7
• Proposed neural predictor: $r_{cyl}$8, $r_{cyl}$9

Empirical coverage at $\dot x = f(x) + g(x)\,u,$0 ms was $\dot x = f(x) + g(x)\,u,$1 for a nominal $\dot x = f(x) + g(x)\,u,$2 interval, $\dot x = f(x) + g(x)\,u,$3 for a nominal $\dot x = f(x) + g(x)\,u,$4 interval, and $\dot x = f(x) + g(x)\,u,$5 for a nominal $\dot x = f(x) + g(x)\,u,$6 interval (Busellato et al., 28 Aug 2025).

In the repeatable mock-hand experiment, a wooden hand mounted on a UR5e executed upward motions with peak velocity $\dot x = f(x) + g(x)\,u,$7 and acceleration $\dot x = f(x) + g(x)\,u,$8, while a JAKA cobot performed a linear sweep above it (Busellato et al., 28 Aug 2025). With $\dot x = f(x) + g(x)\,u,$9, the best overall results occurred at $(p_c,R_c)$0: hand-TCP distance $(p_c,R_c)$1 m, path length $(p_c,R_c)$2 m, velocity $(p_c,R_c)$3 m/s, completion time $(p_c,R_c)$4 s, violation count $(p_c,R_c)$5, and violation magnitude $(p_c,R_c)$6 m (Busellato et al., 28 Aug 2025).

Setting	Method	Violations / mean magnitude
Mock hand	CBF	$(p_c,R_c)$7, $(p_c,R_c)$8 m
Mock hand	PCBF	$(p_c,R_c)$9, $(p_d,R_d)$0 m
Mock hand	UA-PCBF, $(p_d,R_d)$1	$(p_d,R_d)$2, $(p_d,R_d)$3 m
Mock hand	UA-PCBF, $(p_d,R_d)$4	$(p_d,R_d)$5, $(p_d,R_d)$6 m
Mock hand	Full UA-PCBF	$(p_d,R_d)$7, $(p_d,R_d)$8 m

These ablations isolate both uncertainty channels. Setting $(p_d,R_d)$9 removes uncertainty from the barrier and increases violations, while forcing $d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$00 removes uncertainty adaptation in the predictive-slack penalty and yields higher intervention cost than the full method (Busellato et al., 28 Aug 2025).

The real-world evaluation also included direct human-robot interaction in a pick-and-place and handover task, with the operator’s right hand intentionally interfering with the robot path while the right hand was tracked and forecast in real time (Busellato et al., 28 Aug 2025). The reported summary claim is that, relative to state-of-the-art HRI architectures, UA-PCBFs show better performance in task-critical metrics and significantly reduce the number of violations of the robot’s safe space during interaction with respect to the state-of-the-art (Busellato et al., 28 Aug 2025). In that sense, the framework’s encyclopedic significance is not only that it introduces a barrier

$d(\tau, x) = \norm{\mathbf o -\mathbf c(x)} - r_{cyl},$01

but that it operationalizes predictive uncertainty as a control-relevant safety variable in a real collaborative robotic cell (Busellato et al., 28 Aug 2025).