Control Barrier Functions for Safe HRI

• Control Barrier Functions are mathematical tools that encode safety constraints to maintain forward invariance and ensure safe separation in human–robot interactions.
• Hierarchical CBF frameworks prioritize critical safety regions, using strict ordering and slack-based quadratic programs to balance hard and soft constraints.
• Uncertainty-aware and probabilistic extensions adjust safety margins dynamically, reducing collision risks while maintaining efficient robot performance in real-time settings.

Control Barrier Functions for Safe Human–Robot Interaction

Control Barrier Functions (CBFs) provide a formal framework for enforcing safety in dynamic and uncertain environments, including human–robot interaction (HRI). By encoding safety constraints as barrier functions and enforcing their forward invariance, CBF-based controllers enable robots to adaptively manage risk in close proximity to humans. Contemporary research demonstrates the integration of hierarchical, uncertainty-aware, and probabilistic CBF architectures to prioritize safety across heterogeneous constraints, proactively mitigate harm, and maintain interaction fluency under real-time computation constraints (Maithani et al., 21 May 2025, Luo et al., 24 Apr 2026, Busellato et al., 28 Aug 2025, Gonzales et al., 11 Mar 2026).

1. CBF Fundamentals and Formalism in HRI

Let the robot be described by a control-affine system $\dot x = f(x) + g(x) u$ with state $x \in \mathbb{R}^n$ and control $u \in \mathbb{R}^m$. Safety constraints are specified as forward invariance of a set

$\mathcal{C} = \{x \mid h(x) \geq 0 \}$

where $h:\mathbb{R}^n \to \mathbb{R}$ is a continuously differentiable barrier function.

The standard (exponential) CBF condition for systems of relative degree 1 requires

$L_f h(x) + L_g h(x) u + \gamma h(x) \geq 0$

where $\gamma > 0$ is a class–$\mathcal{K}$ gain. For higher relative-degree constraints and/or uncertainties, robust and backstepping extensions incorporate bounded disturbances or additional slack variables for feasibility under model mismatch or stochastic motion of obstacles (Kim et al., 2024).

In HRI, the safety functions are often defined in terms of separation between the robot and human keypoints, kinetic energy, or force/torque limits, as appropriate for the application (Maithani et al., 21 May 2025, Luo et al., 24 Apr 2026, Dawson et al., 2022). Safety constraints can be differentiated according to their criticality (e.g., head vs. hand), and their prioritization directly impacts robot behavior during close interaction.

2. Hierarchical and Prioritized CBF Architectures

HRI safety mandates prioritizing constraints associated with body-part criticality, environmental context, or interaction phase. The hierarchical CBF–QP (quadratic programming) paradigm implements this by enforcing strict ordering between constraint layers:

• Top priority: “hard” CBFs (no relaxation allowed) for most critical body parts (e.g., human head).
• Lower priorities: “soft” CBFs (allowing non-negative slacks penalized in the QP cost) for less critical regions (e.g., human hands or non-vital obstacles) (Maithani et al., 21 May 2025, Luo et al., 24 Apr 2026).

For a 7-DOF manipulator, consider per-link safety functions between each robot link $j$ and human part $i \in \{\text{head}, \text{hand}\}$:

$x \in \mathbb{R}^n$0

with $x \in \mathbb{R}^n$1 the link midpoint in workspace, and $x \in \mathbb{R}^n$2 tracked human keypoints. The hierarchical QP solved online recapitulates: $x \in \mathbb{R}^n$3 The penalty $x \in \mathbb{R}^n$4 ensures slack variables are used only when strictly necessary to resolve infeasibility, always preserving the primary safety set.

Extensions to hierarchical QP cascades in redundant robots enable strict nesting of arbitrary numbers of safety and performance tasks (Luo et al., 24 Apr 2026): $x \in \mathbb{R}^n$5 Each QP restricts the feasible set for lower priorities, thereby allowing explicit, certifiable trade-offs.

3. Handling Uncertainty and Probabilistic Safety Guarantees

Human motion is inherently uncertain and non-deterministic. State-of-the-art frameworks integrate uncertainty-aware motion prediction into the CBF pipeline, dynamically tightening safety boundaries according to predictive uncertainty to avoid excessive conservatism while preserving high-probability safety (Busellato et al., 28 Aug 2025, Gonzales et al., 11 Mar 2026).

Key principles in such architectures include:

• Probabilistic human state forecast: E.g., for each future step $x \in \mathbb{R}^n$6, a LSTM-based predictor outputs Gaussian parameters $x \in \mathbb{R}^n$7 for the human hand’s position.
• Uncertainty-dependent barriers: The barrier function is inflated by the forecasted sigma projected along the robot–human axis:

$x \in \mathbb{R}^n$8

where $x \in \mathbb{R}^n$9 is the predicted robot–human distance, and $u \in \mathbb{R}^m$0 is the scaled projected standard deviation clamped to a maximal physically meaningful value.

• Risk-adjusted constraint satisfaction: Combining CBF synthesis with conformal risk control, one enforces at each control step a margin $u \in \mathbb{R}^m$1—computed as the $u \in \mathbb{R}^m$2-quantile on past CBF prediction errors—so that, with probability at least $u \in \mathbb{R}^m$3, all realized safety constraints remain forward invariant (Gonzales et al., 11 Mar 2026).

These approaches result in substantial reductions (order-of-magnitude) in collision rates and violation depths, with empirical benchmarks on collaborative pick-and-place and dense multi-human navigation scenarios (Busellato et al., 28 Aug 2025, Gonzales et al., 11 Mar 2026).

4. Safe Shared Autonomy and Real-Time Computational Aspects

Shared autonomy blends operator commands with autonomous assistance. Linear blending without safety filtering can break safety even when both sources are independently collision-free. CBF filtering is applied at the task-space or IK layer to intercept unsafe post-blend commands and guarantee strict forward invariance of the safety set (Guler et al., 2 Mar 2026, He et al., 2021).

An instance is the BarrierIK architecture, which augments a standard IK solver with hard CBF constraints on minimum robot–obstacle distance: $u \in \mathbb{R}^m$4 with

$u \in \mathbb{R}^m$5

where $u \in \mathbb{R}^m$6 is the link–obstacle signed distance and $u \in \mathbb{R}^m$7 a shaping function. Aggregation over all obstacles can be performed via log-sum-exp to retain compatibility with continuous optimization.

Empirical evaluation in VR teleoperation and simulated clutter validates that CBF-filtered architectures achieve lower penetration time, increased minimum clearance, and higher user-perceived safety and trust. Real-time feasibility is demonstrated on commodity GPUs/CPUs with per-cycle QP solves on the order of 0.16–1 ms, supporting >60 Hz closed-loop rates (Maithani et al., 21 May 2025, Guler et al., 2 Mar 2026).

5. Risk Quantification, Metrics, and Experimental Validation

Quantitative risk metrics in CBF-based HRI controllers include:

• Minimum separation: The least instantaneous robot–human distance recorded ($u \in \mathbb{R}^m$8), stratified by body part and constraint criticality.
• Constraint violation slack: Maximum relaxation variable $u \in \mathbb{R}^m$9 observed over hand (or lower-priority) CBFs.
• End-effector tracking error: RMSE relative to the unfiltered performance controller, quantifying task-performance trade-off.
• Violation counts and magnitudes: Number and depth of incursions into the forbidden region, e.g., average or maximal penetration in mm or duration.

Table: Summary of Empirical Findings (selected results)

Arch. Type	Min. Head Violation	Max. Hand Slack	Tracking Error	Violation Count (UA-PCBF)
Hierarchical CBF-QP (Maithani et al., 21 May 2025)	None	$\mathcal{C} = \{x \mid h(x) \geq 0 \}$0–$\mathcal{C} = \{x \mid h(x) \geq 0 \}$1	$\mathcal{C} = \{x \mid h(x) \geq 0 \}$2 incr.	–
UA-PCBF (Busellato et al., 28 Aug 2025)	None	–	$\mathcal{C} = \{x \mid h(x) \geq 0 \}$3 path	$\mathcal{C} = \{x \mid h(x) \geq 0 \}$4
CRC-CBF (Gonzales et al., 11 Mar 2026)	$\mathcal{C} = \{x \mid h(x) \geq 0 \}$5	–	Moderate incr.	$\mathcal{C} = \{x \mid h(x) \geq 0 \}$6 collision

Statistically significant improvements in collision avoidance and maintained task efficiency are consistently observed (Maithani et al., 21 May 2025, Busellato et al., 28 Aug 2025, Gonzales et al., 11 Mar 2026). Hierarchical prioritization and slack-based infeasibility resolution prevent deadlock and enable fluid recovery from safety-critical encroachment, confirmed by randomized scenario testing.

6. Robustness, High Relative-Degree Constraints, and Limitations

Dynamic HRI scenarios with uncertainty in human motion, sensing, and actuation necessitate robustness. sRCBF (smooth robust CBF) and backstepping constructions formally account for bounded disturbance in obstacle/human motion by inflating the safety constraint according to a Lipschitz bound $\mathcal{C} = \{x \mid h(x) \geq 0 \}$7; the closed-loop system maintains invariance for all admissible disturbances (Kim et al., 2024).

Limitations and ongoing challenges include:

• Geometric model fidelity: simplifications (e.g., spheres/cylinders) for computational efficiency may underestimate true safety margins.
• Myopic reactivity in discrete-time CBF conditions affects response to fast-moving obstacles or humans.
• Lack of explicit treatment for coupled uncertainties (e.g., unmodeled actuation dynamics, occlusions).
• Trade-off between conservatism and responsiveness: Excessively conservative margin inflation, if not tuned, can degrade task efficiency.

A plausible implication is that future work must explore adaptive parameter shaping, whole-body human modeling, and integration of multi-modal uncertainty sources (sensor fusion, intent prediction) to systematically overcome these limitations (Guler et al., 2 Mar 2026, Busellato et al., 28 Aug 2025).

7. Outlook and Integration with Broader HRI and Safety-Preserved Robotics

CBFs have become a backbone for certifiable safety in HRI, supporting integration with shared autonomy, real-time perception, and adaptive motion planning. They serve as a unifying language for risk-aware, hierarchical, and probabilistic safety, allowing explicit trade-offs between safety-critical and performance objectives (Maithani et al., 21 May 2025, Luo et al., 24 Apr 2026, Gonzales et al., 11 Mar 2026). The field is moving towards uncertainty-aware and context-adaptive CBF architectures that promise broader deployment in collaborative and open-world settings where human unpredictability is the norm.