Robot-User CBFs: Enforcing Safety in Human-Robot Control

• Robot–User CBFs are control-theoretic safety constructs that define forward-invariant safe sets over joint robot–user configurations to ensure reliable collision avoidance.
• They integrate with nominal controllers using QP or SQP formulations to minimally adjust commands while maintaining safety across diverse interaction scenarios.
• Applications span mobile navigation and manipulator tasks, employing geometric, predictive, and perception-driven methods to optimize human-robot collaboration.

Searching arXiv for papers on robot–user and human–robot control barrier functions. Robot–User Control Barrier Functions (CBFs) are control-theoretic safety constructs that encode forward-invariant safe sets over joint robot–user configurations, robot–human relative motion, or user-defined operational regions, and then enforce those sets online through constrained optimization. In the literature, the term spans several closely related formulations: distance-based and future-focused robot–human CBFs for mobile platforms, collision-cone CBFs for manipulators around moving obstacles interpretable as humans or users, force-aware CBFs for physical human–robot cooperation, perception-driven CBFs built directly from occupancy maps, and co-state CBFs that simultaneously protect both robot and user in shared navigation. Across these variants, the common structure is a safe set of the form $S=\{x\mid h(x)\ge 0\}$, a forward-invariance condition expressed through a CBF inequality, and an online QP or SQP that minimally modifies a nominal control law or user command to maintain safety (Ames et al., 2019).

1. Foundational formulation and safe-set semantics

Robot–User CBFs inherit the standard control-affine formulation

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

with safety encoded by a continuously differentiable barrier $h$ whose superlevel set is the safe set. The safe set is written as

$\mathcal{S} = \{ x \mid h(x) \ge 0 \},$

and safety is formalized as forward invariance: if $x(0)\in\mathcal{S}$, then $x(t)\in\mathcal{S}$ for all $t\ge 0$. For relative-degree-1 CBFs, the admissible controls are those satisfying

$L_f h(x) + L_g h(x)u + \alpha(h(x)) \ge 0,$

or equivalently the same inequality written with $\dot h(x,u)$, where $\alpha$ is an extended class-$\dot{x} = f(x) + g(x)u,$0 function, often chosen linearly as $\dot{x} = f(x) + g(x)u,$1 (Ames et al., 2019).

Within robot–user settings, the same formalism is instantiated in several ways. A user may be modeled as a dynamic obstacle, as a tracked set of body parts, as a moving agent with its own kinematics, or as a source of forces and moments applied to the robot. The safe set can therefore encode minimum separation, predicted non-collision over a horizon, force or torque limits during physical cooperation, or membership in user-defined regions such as corridors and workspaces. This suggests a useful editorial distinction between “interaction-space CBFs” and “environment-space CBFs” (Editor's term), although the underlying mathematics remains the same.

A recurrent theme is that the CBF layer is not usually a standalone controller. Instead, it is wrapped around a nominal policy—trajectory tracking, impedance control, LQR, MPC, teleoperation, or learned policy output—and acts as a minimally invasive safety filter. This pattern appears in the general CBF survey (Ames et al., 2019), in robot-human mobile navigation with CBFKit (Black et al., 2024), in manipulator collision-cone safety filtering (Almeida, 1 Mar 2025), and in force-feedback cooperative manipulation (Dawson et al., 2022).

2. Geometric constructions for robot–user safety

The most direct robot–user construction is distance-based. For mobile robot–human interaction, CBFKit demonstrates a “future-focused” CBF

$\dot{x} = f(x) + g(x)u,$2

where $\dot{x} = f(x) + g(x)u,$3 is the predicted distance assuming both agents maintain current velocity and heading under zero control. The corresponding safe set

$\dot{x} = f(x) + g(x)u,$4

guarantees a minimum distance between robot and human over the finite horizon $\dot{x} = f(x) + g(x)u,$5. The same paper also defines corridor CBFs with anticipatory velocity terms,

$\dot{x} = f(x) + g(x)u,$6

so that user-defined environmental constraints enter the same QP as robot–human separation constraints (Black et al., 2024).

For manipulators in dynamic human environments, “Safety-Critical Control for Robotic Manipulators using Collision Cone Control Barrier Functions” introduces Collision Cone Control Barrier Functions (C3BFs) specialized to moving obstacles interpretable as humans or users. With relative position and velocity

$\dot{x} = f(x) + g(x)u,$7

and safety radius $\dot{x} = f(x) + g(x)u,$8, the candidate C3BF is

$\dot{x} = f(x) + g(x)u,$9

with

$h$0

The condition $h$1 characterizes relative velocities outside the collision cone; hence the safe set excludes imminent collision courses rather than merely short distances. Because the constraint depends on relative velocity and geometry, the resulting safety behavior is predictive rather than purely reactive (Almeida, 1 Mar 2025).

A more geometry-exact direction replaces spherical or ellipsoidal surrogates by differentiable signed-distance representations. OGM-CBF constructs barriers directly from occupancy grid maps and signed distance fields, defining

$h$2

where $h$3 is a smoothed signed distance to the OGM-derived safe set and $h$4 measures alignment between robot heading and the SDF gradient. This yields a first-order barrier for kinematic robot motion with a single scalar inequality per robot reference point, independent of the number or shape of obstacles (Raja et al., 2024). A related geometry-aware line uses Bernstein-Polynomial Signed Distance Fields to define a minimum-distance CBF between arbitrary robot and obstacle shapes, thereby avoiding many primitive-wise constraints in cluttered scenes (Jo et al., 29 May 2026). This suggests that robot–user CBFs need not be limited to spherical body-part abstractions, although conservative sphere and capsule approximations remain common.

3. Optimization-based safety filtering and nominal-control integration

The dominant online enforcement mechanism is a QP or SQP that keeps the applied input close to a nominal command while imposing one or more CBF inequalities. In its standard form, CBFKit uses

$h$5

where $h$6 may come from LQR, path following, or teleoperation. In the Toyota Human Support Robot experiment, the same structure enforces both human-avoidance and corridor constraints in a single QP (Black et al., 2024).

The manipulator C3BF framework follows the same pattern but pairs safety filtering with Cartesian impedance control. The nominal interaction law is

$h$7

and the safety filter solves

$h$8

If $h$9 already satisfies the CBF inequality, the optimizer returns $\mathcal{S} = \{ x \mid h(x) \ge 0 \},$0; otherwise it computes the smallest adjustment that preserves forward invariance of the collision-cone safe set (Almeida, 1 Mar 2025).

In shared autonomy and teleoperation, the nominal input can be the user’s command itself. OGM-CBF makes this explicit: the user provides $\mathcal{S} = \{ x \mid h(x) \ge 0 \},$1, the QP sets $\mathcal{S} = \{ x \mid h(x) \ge 0 \},$2, and the CBF filter returns the closest admissible $\mathcal{S} = \{ x \mid h(x) \ge 0 \},$3 satisfying

$\mathcal{S} = \{ x \mid h(x) \ge 0 \},$4

The same “guardian” structure appears in CBFKit, where $\mathcal{S} = \{ x \mid h(x) \ge 0 \},$5 can be teleoperation input, and in the general CBF literature, where the QP is interpreted as a minimally modifying safety supervisor (Raja et al., 2024).

For force-mediated human–robot cooperation, the control variable may instead be the Cartesian end-effector velocity. In “Barrier functions enable safety-conscious force-feedback control,” each wrench component is constrained by a barrier

$\mathcal{S} = \{ x \mid h(x) \ge 0 \},$6

and the CLF–CBF QP computes the end-effector spatial velocity $\mathcal{S} = \{ x \mid h(x) \ge 0 \},$7 that tracks a nominal pose while keeping each force or torque component below user-specified maxima. Here the “user command” is implicit in the measured physical interaction; responsiveness emerges because approaching a force limit activates the corresponding CBF and compels motion that reduces the wrench (Dawson et al., 2022).

4. Robot–human co-state formulations and user-prioritized safety

Some recent work makes the robot–user coupling explicit in the state definition rather than treating the user as a generic obstacle. “Force-Compliance MPC and Robot-User CBFs for Interactive Navigation and User-Robot Safety in Hexapod Guide Robots” defines Robot-User CBFs over the joint state of the robot, the visually impaired user attached by a rigid guide cane, and predicted obstacle states. The user position is modeled geometrically as

$\mathcal{S} = \{ x \mid h(x) \ge 0 \},$8

and separate robot–obstacle and user–obstacle barriers are defined: $\mathcal{S} = \{ x \mid h(x) \ge 0 \},$9

$x(0)\in\mathcal{S}$0

The joint guide safety set is

$x(0)\in\mathcal{S}$1

These barriers are enforced in discrete time through softened invariance conditions

$x(0)\in\mathcal{S}$2

with weighted slack variables satisfying $x(0)\in\mathcal{S}$3 so that user safety is prioritized over robot safety when perfect feasibility is impossible (Fan et al., 5 Aug 2025).

A different prioritization axis is anatomical rather than agent-based. “Proactive Hierarchical Control Barrier Function-Based Safety Prioritization in Close Human-Robot Interaction Scenarios” introduces a hierarchical CBF-QP for a Franka Research 3 manipulator tracking a human with a ZED2i camera. Robot links and selected human body parts are modeled as spheres, and for each link–body-part pair

$x(0)\in\mathcal{S}$4

The framework then treats head CBFs as hard constraints and hand CBFs as soft constraints: $x(0)\in\mathcal{S}$5

$x(0)\in\mathcal{S}$6

with $x(0)\in\mathcal{S}$7 penalized in the objective. This encodes explicit body-part criticality: head protection is non-negotiable, whereas hand constraints may be relaxed to preserve feasibility in crowded or unavoidable scenarios (Maithani et al., 21 May 2025).

These co-state and hierarchical constructions clarify a common misconception: robot–user CBFs are not limited to symmetric collision avoidance. They may instead encode asymmetry in risk, anatomy, or task role. A plausible implication is that human-aware safety certification increasingly depends not only on geometric separation but also on how violations are prioritized when strict feasibility fails.

5. Perception-driven, predictive, and standards-aware variants

Robot–user CBFs increasingly integrate perception and prediction directly into the barrier rather than relying on fixed analytic obstacles. OGM-CBF is explicitly sensor-agnostic: any perception stack capable of producing an occupancy grid map can feed the barrier construction. The OGM is updated online via

$x(0)\in\mathcal{S}$8

thresholded to a binary safe set $x(0)\in\mathcal{S}$9, converted to an SDF $x(t)\in\mathcal{S}$0, and shaped into $x(t)\in\mathcal{S}$1. The resulting barrier captures both currently sensed and previously mapped obstacles, giving the controller “memory” of the environment and a single safety constraint per robot reference point (Raja et al., 2024). This is directly relevant to user safety whenever humans, assistive devices, or personal zones are represented implicitly by perception-generated occupancy.

Prediction can also be embedded analytically. In “Embedding ISO 10218 Safety Compliance in Robots via Control Barrier Functions for Human-Robot Collaboration,” the safe set is defined through the predicted minimum human–robot separation distance during the robot’s worst-case stopping trajectory. For one robot–human point pair, the reduced state is

$x(t)\in\mathcal{S}$2

and the CBF is

$x(t)\in\mathcal{S}$3

where $x(t)\in\mathcal{S}$4 is the minimum predicted future separation under robot reaction time $x(t)\in\mathcal{S}$5, robot deceleration $x(t)\in\mathcal{S}$6, and constant human acceleration over the short braking horizon. The CBF constraint

$x(t)\in\mathcal{S}$7

is enforced in an SQP, either as a CBF-constrained PD safety filter or in a task-scaling controller with a spatial tube constraint (Parma et al., 11 Jun 2026). Compared to classical SSM modules using constant human velocity, the incorporation of human acceleration makes the safety filter less conservative and more faithful to ISO 10218 logic.

Perception uncertainty and stochasticity are handled at the toolbox level by CBFKit, which supports deterministic, bounded-disturbance, and stochastic models. For SDE systems

$x(t)\in\mathcal{S}$8

the goal is to keep the probability of leaving the safe set below a risk bound $x(t)\in\mathcal{S}$9 over a horizon $t\ge 0$0. The paper does not rewrite the full stochastic CBF formulas, but it explicitly supports stochastic barrier functions and risk-aware CBFs imported from prior work and integrated into QPs (Black et al., 2024). This suggests that robot–user CBFs can be extended beyond deterministic margins to risk-bounded safety envelopes when user state estimation is noisy.

6. Feasibility, learning, and verification

The feasibility of a CBF-QP is itself a safety-critical issue. “Learning Feasibility Constraints for Control Barrier Functions” focuses on the case where CLF–CBF–QP controllers become infeasible because safety constraints, high relative degree, and control limits conflict. For nonlinear affine systems with bounded controls, the paper learns a classifier $t\ge 0$1 over relative states that separates feasible from infeasible regions of the state space, then converts that classifier into another HOCBF enforced in the online QP: $t\ge 0$2 This learned feasibility constraint keeps the system away from states where the original safety QP would fail, reducing infeasibility rates from $t\ge 0$3 to $t\ge 0$4 in a regular obstacle case and from $t\ge 0$5 to $t\ge 0$6 in an irregular obstacle case (Xiao et al., 2023). For robot–user interaction, the learned set can be interpreted as a feasible interaction envelope in relative position, heading, and speed.

Learning also appears in barrier synthesis and policy shaping. “Guided by Guardrails: Control Barrier Functions as Safety Instructors for Robotic Learning” shows that plain RL with continuous negative rewards struggles in unsafe regions, whereas CBF-based safety filters can guide exploration. Of its three integrations, CBF Decay is the one that yields a policy that remains safe even when the CBF is removed at test time. In this setup, the action applied near unsafe regions is a convex combination of the RL action and the CBF action, with the CBF influence decaying over training (Guerrier et al., 24 May 2025). A plausible implication is that robot–user CBFs can function not only as runtime safety filters but also as training-time instructors that bias learned policies toward human-safe behavior.

When learned or neural CBFs are used, formal verification becomes necessary. “Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation” considers ReLU neural CBFs $t\ge 0$7 defining safe sets

$t\ge 0$8

and verifies the boundary condition

$t\ge 0$9

The paper proposes symbolic derivative bound propagation for ReLU-based neural CBFs and reports improved verified rate and verification time along the CBF boundary relative to interval-arithmetic baselines (Hu et al., 2024). This is particularly relevant for robot–user settings because expressive human-aware safe sets are often nonconvex and perception-conditioned, making neural parameterizations attractive but unverifiable without such techniques.

A related synthesis route is CN-CBF, which trains individual neural CBFs for single moving obstacles using Hamilton–Jacobi reachability and combines them into a smooth composite CBF

$L_f h(x) + L_g h(x)u + \alpha(h(x)) \ge 0,$0

The residual architecture guarantees that the learned safe set does not intersect the failure set for the single-obstacle problem, and the composite construction scales to many moving agents (Derajić et al., 6 Mar 2026). Since the obstacles in that work are pedestrians or other drones, the formulation already aligns closely with robot–user navigation in crowds.

7. Implementation patterns, assumptions, and limitations

The implementation burden of robot–user CBFs is dominated by sensing, state estimation, and constraint count rather than by the barrier inequalities themselves. CBFKit solves its QPs in “a few milliseconds” using JAX and jaxopt and integrates with ROS for odometry, sensing, state estimation, and command publication (Black et al., 2024). The manipulator C3BF implementation reports QP solving at about 240 Hz in simulation, reflecting the affine-in-control structure of the constraint (Almeida, 1 Mar 2025). The ISO-aware human–robot SQP implementation runs at 500 Hz and failed to return a solution within 2 ms in only 0.0004% of cycles over 15,000 s tests (Parma et al., 11 Jun 2026). By contrast, the hexapod guide-robot system couples perception, obstacle clustering, prediction, FC-MPC, and Robot-User CBFs at 8–10 Hz, which is sufficient for assistive navigation but illustrates the computational cost of full-stack perception (Fan et al., 5 Aug 2025).

A compact summary of representative formulations is useful before considering limitations.

Formulation	Safety variable $L_f h(x) + L_g h(x)u + \alpha(h(x)) \ge 0,$1	Primary setting
Future-focused distance CBF	$L_f h(x) + L_g h(x)u + \alpha(h(x)) \ge 0,$2 over predicted minimum distance	Mobile robot–human avoidance
Collision cone C3BF	Relative position–velocity cone exclusion	Manipulator near moving humans
OGM-CBF	Smoothed SDF from occupancy map and heading alignment	Unknown environments, user-command filtering
Predictive SSM CBF	$L_f h(x) + L_g h(x)u + \alpha(h(x)) \ge 0,$3 over worst-case stopping trajectory	ISO 10218 HRC
Robot-User co-state CBF	Joint robot–obstacle and user–obstacle barriers	Assistive shared navigation
Force-limited CBF	$L_f h(x) + L_g h(x)u + \alpha(h(x)) \ge 0,$4	Physical human–robot cooperation

Several assumptions recur. Collision-cone manipulator safety assumes constant obstacle velocity in the theorem, even though humans rarely move with perfectly constant velocity (Almeida, 1 Mar 2025). OGM-CBF guarantees safety only with respect to the mapped environment and cannot certify unseen regions outside sensing range or field of view (Raja et al., 2024). The ISO-aware predictive CBF assumes bounded and approximately constant human acceleration over the robot reaction-plus-braking horizon, with residual mismatch absorbed by increasing $L_f h(x) + L_g h(x)u + \alpha(h(x)) \ge 0,$5 (Parma et al., 11 Jun 2026). Robot-User CBFs for the hexapod guide robot approximate the user as a rigid-distance point behind the robot, which is expedient but does not capture full human gait and arm motion (Fan et al., 5 Aug 2025). Hierarchical manipulator CBFs prioritize head over hand but remain dependent on body-keypoint tracking and spherical body-part abstractions (Maithani et al., 21 May 2025).

Feasibility is another persistent limitation. Hard CBF constraints can become infeasible when the robot is trapped between a user and obstacles, when actuation bounds are insufficient, or when high-order constraints demand unattainable braking or turning. The literature handles this in several ways: weighted slack variables in soft CBFs (Fan et al., 5 Aug 2025), explicit relaxation for lower-priority body-part constraints (Maithani et al., 21 May 2025), learned feasibility HOCBFs (Xiao et al., 2023), fallback braking in SQP-based human–robot controllers (Parma et al., 11 Jun 2026), and robust or stochastic formulations in toolboxes such as CBFKit (Black et al., 2024).

The broader trajectory of the field suggests three converging directions. First, geometry is becoming richer, moving from spheres toward SDFs, occupancy maps, and BP-SDFs (Raja et al., 2024, Jo et al., 29 May 2026). Second, human motion modeling is becoming more predictive, incorporating velocity, acceleration, and finite-horizon worst-case reasoning rather than static distance margins (Almeida, 1 Mar 2025, Parma et al., 11 Jun 2026). Third, CBF design and verification are increasingly data-driven, through toolboxes, learned feasibility envelopes, composite neural CBFs, and formal verification pipelines (Black et al., 2024, Xiao et al., 2023, Derajić et al., 6 Mar 2026, Hu et al., 2024). Taken together, these developments indicate that Robot–User CBFs are evolving from simple collision filters into integrated safety substrates that mediate user intent, perception, dynamics, and formal guarantees within a single optimization-based control architecture.