Control Barrier Functions Solved with Hierarchical Quadratic Programming for Safe Physical Human-Robot Interaction

Cheng Fang; Federico Califano; Jonas Mariager Jakobsen; Rui Luo; Wesley Roozing

arxiv: 2604.23039 · v1 · submitted 2026-04-24 · 💻 cs.RO

Control Barrier Functions Solved with Hierarchical Quadratic Programming for Safe Physical Human-Robot Interaction

Rui Luo , Jonas Mariager Jakobsen , Wesley Roozing , Federico Califano , Cheng Fang This is my paper

Pith reviewed 2026-05-08 11:18 UTC · model grok-4.3

classification 💻 cs.RO

keywords control barrier functionshierarchical quadratic programmingphysical human-robot interactionrobot safetyquadratic programmingredundant manipulatorscollaborative robots

0 comments

The pith

A CBF-based hierarchical quadratic programming framework allows safety and performance tasks to be prioritized flexibly in physical human-robot interaction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper aims to show that Control Barrier Functions can be incorporated into a Hierarchical Quadratic Programming structure for physical human-robot interaction. By placing both safety constraints and performance objectives at different levels of the hierarchy, conflicts can be handled by relaxing lower-priority tasks without losing higher ones. A sympathetic reader cares because this offers a practical way to maintain robot safety around humans while achieving useful collaborative behaviors like desired interaction points. The work validates the method through experiments on a redundant robot, confirming real-time feasibility and flexibility.

Core claim

The authors claim that using CBFs solved via HQP in pHRI enables performance tasks, such as preserving desired behavior at the interaction point, and safety tasks to be designed at any hierarchy level. This provides a more flexible balance between safety and performance compared to flat QP formulations. Hierarchical relaxation ensures solution feasibility in case of conflicts. The approach is shown effective in extensive real-robot experiments.

What carries the argument

Control Barrier Functions embedded as constraints in a Hierarchical Quadratic Programming optimization problem.

If this is right

High-priority safety tasks remain satisfied even if lower tasks are relaxed.
Performance at the human-robot interaction point can be prioritized when safety allows.
The framework supports redundant robots by exploiting task hierarchies.
Real-time implementation is possible without compromising safety guarantees.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Designers could use this to create adaptive safety systems that adjust priorities dynamically based on context.
This method might extend to multi-robot or human-in-the-loop systems with shared tasks.
Limitations in modeling human behavior could affect CBF accuracy in more complex interactions.

Load-bearing premise

That relaxing lower-priority tasks in the hierarchy will always yield a feasible real-time solution while preserving the safety guarantees of higher-priority tasks, even under task conflicts.

What would settle it

Demonstration of a conflict scenario on the robot where the QP either becomes infeasible or a safety barrier is breached despite the hierarchy.

Figures

Figures reproduced from arXiv: 2604.23039 by Cheng Fang, Federico Califano, Jonas Mariager Jakobsen, Rui Luo, Wesley Roozing.

**Figure 1.** Figure 1: A real physical human-robot interaction scenario with multiple strict safety tasks and a hierarchy of soft tasks of different priority levels. In most prior work, all CBF safety constraints are formulated as a set of inequality constraints, typically solved together in a single QP, which is desirable when all of these constraints are equally important and strict. However, when we apply the CBF framework t… view at source ↗

**Figure 2.** Figure 2: Diagram of a cascaded structure of the CBF-HQP for a physical human-robot interaction scenario. Before constructing the hierarchical layers, we first define the initial admissible set S0. All strict safety tasks, including collision avoidance, joint velocity torque limits, are expressed in a unified affine inequality form as A0(x) u ≥ b0(x), (15) where A0(x) and b0(x) are obtained by stacking the correspo… view at source ↗

**Figure 3.** Figure 3: Experiment 1 (Equilibrium shift): total kinetic energy. Time (s) Wactual ! Wnominal (N, N m) 0 0.2 0.4 0.6 0.8 1 -20 0 20 1-layer QP, . = 10 (a) 0 0.2 0.4 0.6 0.8 1 -20 0 20 1-layer QP, . = 50 (b) 0 0.2 0.4 0.6 0.8 1 -20 0 20 1-layer QP, . = 100 (c) 0 0.2 0.4 0.6 0.8 1 -20 0 20 HQP, . = 10 (d) 0 0.2 0.4 0.6 0.8 1 -20 0 20 HQP, . = 50 (e) 0 0.2 0.4 0.6 0.8 1 -20 0 20 HQP, . = 100 (f) Fx Fy Fz Mx My Mz view at source ↗

**Figure 4.** Figure 4: Experiment 1 (Equilibrium shift): Cartesian wrench deviation. Time (s) , ! ,nominal (N m) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 1-layer QP, . = 10 (a) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 1-layer QP, . = 50 (b) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 1-layer QP, . = 100 (c) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 HQP, . = 10 (d) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 HQP, . = 50 (e) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 H… view at source ↗

**Figure 5.** Figure 5: Experiment 1 (Equilibrium shift): Null-space component. elastic potential energy into the system, which was rapidly converted into kinetic energy, potentially leading to large transient motions view at source ↗

**Figure 6.** Figure 6: Experiment 2 (Sinusoidal external force): total kinetic energy. Time (s) Wactual ! Wnom (N; N m) 0 1 2 3 4 5 -10 0 10 1-layer QP, . = 10 (a) 0 1 2 3 4 5 -10 0 10 1-layer QP, . = 50 (b) 0 1 2 3 4 5 -10 0 10 1-layer QP, . = 100 (c) 0 1 2 3 4 5 -10 0 10 Safety HQP, . = 10 (d) 0 1 2 3 4 5 -10 0 10 Safety HQP, . = 50 (e) 0 1 2 3 4 5 -10 0 10 Safety HQP, . = 100 (f) 0 1 2 3 4 5 -10 0 10 Performance HQP, . = 10 (… view at source ↗

**Figure 7.** Figure 7: Experiment 2 (Sinusoidal external force): Cartesian wrench deviation. Time (s) , ! ,nom (N m) 0 1 2 3 4 5 -1 0 1 1-layer QP, . = 10 (a) 0 1 2 3 4 5 -1 0 1 1-layer QP, . = 50 (b) 0 1 2 3 4 5 -1 0 1 1-layer QP, . = 100 (c) 0 1 2 3 4 5 -1 0 1 Safety HQP, . = 10 (d) 0 1 2 3 4 5 -1 0 1 Safety HQP, . = 50 (e) 0 1 2 3 4 5 -1 0 1 Safety HQP, . = 100 (f) 0 1 2 3 4 5 -1 0 1 Performance HQP, . = 10 (g) 0 1 2 3 4 5 -1… view at source ↗

**Figure 8.** Figure 8: Experiment 2 (Sinusoidal external force): null-space component. In Figs. 6–8, subplots (a–c) are of the single-layer QP baseline method, (d–f) correspond to the safety-priority HQP, and (g–i) correspond to the performance-priority HQP. Figures 6 and 7 illustrate the kinetic energy evolution and view at source ↗

read the original abstract

Physical human-robot interaction offers the potential to leverage human intelligence and robot physical capabilities to enable a range of exciting applications, e.g., collaborative robots for rehabilitation. Safety is critical for the successful deployment of this kind of robotic system. In recent years, Control Barrier Function (CBF) has emerged as an effective approach to enforce safety guarantees, which has been widely applied in various applications, from adaptive cruise control to navigation of legged robots. CBFs can be solved in a Quadratic Programming (QP) problem, which can include many CBF-formulated tasks. To manage a large number of safety tasks, a hierarchical CBF has been used to allow hierarchical relaxation of safety tasks to ensure the feasibility of a solution in the presence of conflicting tasks. In this work, we propose to use a CBF-based Hierarchical Quadratic Programming (HQP) framework in physical human-robot interaction to allow us to design both performance tasks (e.g., preserve the desired behavior at the human-robot interaction point) and safety tasks at any level of a hierarchy to balance the safety and the performance in a more flexible way. Extensive experiments were carried out on a real redundant robot to validate the effectiveness, flexibility, and generality of this approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper lets you place CBF safety tasks at any level in an HQP hierarchy for pHRI, with real-robot experiments, but does not show how forward invariance holds when safety is relaxed below performance tasks.

read the letter

The main point is that they combine control barrier functions with hierarchical quadratic programming so safety tasks and performance tasks (such as keeping a desired interaction behavior at the contact point) can be ordered at any priority level. This gives more flexibility than the usual setup where safety constraints sit at the top and everything else is projected into the null space. They test the idea on a real redundant manipulator and claim the results show it works for physical human-robot interaction scenarios like rehabilitation or manufacturing assistance. That practical angle is the useful part here. The experiments are the main new evidence they offer, since the underlying CBF and HQP techniques are already known. The abstract frames the contribution as a framework that balances safety and performance more flexibly than prior CBF-QP work. The soft spot is the safety guarantee. When a CBF safety task sits lower in the hierarchy, a higher-priority performance task can relax it through the standard HQP slack or projection mechanism. Nothing in the description shows that the closed-loop system still keeps the barrier function non-negative for all time in those cases. The stress-test concern about lost forward invariance is reasonable based on what is written; the paper would need an explicit condition or proof to close that gap. The experiments are described only at a high level with no metrics, failure cases, or feasibility checks, so it is difficult to judge how often conflicts arise or how the method handles them. This is for people who build controllers for collaborative robots and need a tunable way to trade off safety against useful motion. A reader working on redundant manipulators or safe pHRI would get concrete implementation ideas from the experiments. It is solid enough to send to peer review because it tackles a real deployment issue with hardware validation, even if the theoretical handling of relaxed barriers needs more detail.

Referee Report

1 major / 1 minor

Summary. The paper proposes a Control Barrier Function (CBF) based Hierarchical Quadratic Programming (HQP) framework for physical human-robot interaction (pHRI). It allows both performance tasks (such as preserving desired behavior at the human-robot interaction point) and safety tasks to be assigned to any level in the task hierarchy, enabling more flexible balancing of safety and performance compared to standard CBF-QP formulations. The approach is validated via extensive experiments on a real redundant robot.

Significance. If the safety guarantees can be rigorously established for arbitrary hierarchy placements, the framework would provide a practical extension of CBF methods to redundant manipulators in collaborative settings, allowing performance objectives to take precedence without immediate loss of feasibility. The real-robot experiments are a strength, though their limited description in the abstract reduces the ability to assess generalizability.

major comments (1)

[Proposed Framework (as described in Abstract)] The central claim relies on CBF safety guarantees holding even when safety tasks are placed below performance tasks in the HQP hierarchy. However, standard CBF theory requires strict enforcement of the barrier constraint (Lie derivative condition) for forward invariance of the safe set; the proposed relaxation via HQP priority ordering or slack variables does not automatically preserve h(x) ≥ 0 for all time when a higher-priority performance task conflicts. No explicit proof, invariance condition, or modified barrier formulation is provided to address this case.

minor comments (1)

[Abstract] The abstract states that 'extensive experiments were carried out on a real redundant robot to validate the effectiveness, flexibility, and generality' but provides no details on the specific robot, task definitions, performance metrics (e.g., tracking error, safety violation rates), failure cases, or how QP feasibility was ensured in real time.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We provide a point-by-point response to the major comment below.

read point-by-point responses

Referee: [Proposed Framework (as described in Abstract)] The central claim relies on CBF safety guarantees holding even when safety tasks are placed below performance tasks in the HQP hierarchy. However, standard CBF theory requires strict enforcement of the barrier constraint (Lie derivative condition) for forward invariance of the safe set; the proposed relaxation via HQP priority ordering or slack variables does not automatically preserve h(x) ≥ 0 for all time when a higher-priority performance task conflicts. No explicit proof, invariance condition, or modified barrier formulation is provided to address this case.

Authors: We acknowledge the validity of this observation. The standard CBF-QP formulation enforces the barrier condition strictly to ensure forward invariance. In our CBF-HQP approach, by allowing safety tasks to be assigned to lower levels in the hierarchy, we intentionally permit relaxation of the safety constraint when it conflicts with a higher-priority performance task. This provides the flexibility highlighted in the paper but does compromise the strict safety guarantee in such cases. The manuscript does not include an explicit proof or modified invariance condition for arbitrary hierarchy placements, as the focus is on the practical implementation and experimental validation on a real robot. We will revise the manuscript to include a clearer discussion of this trade-off, explicitly noting that safety guarantees hold only when the safety task is not relaxed due to higher-priority conflicts, and we will add conditions or examples where the hierarchy preserves safety. revision: yes

Circularity Check

0 steps flagged

No circularity: framework extends established CBF-HQP methods with independent experimental validation

full rationale

The paper's core contribution is a proposed CBF-based HQP architecture that permits safety and performance tasks at arbitrary hierarchy levels for pHRI. No derivation step reduces a claimed result to a fitted parameter, self-referential definition, or unverified self-citation chain. The abstract and described approach explicitly build on prior CBF and hierarchical QP literature (standard external references) and then validate the specific instantiation via real-robot experiments on a redundant manipulator. The skeptic concern about forward-invariance under relaxation is a question of proof completeness or assumption strength, not a circular reduction of the stated claims to their inputs. No equations or load-bearing steps in the provided material exhibit self-definition, renaming of known results, or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the approach relies on standard CBF and QP assumptions from prior literature.

pith-pipeline@v0.9.0 · 5520 in / 1007 out tokens · 32748 ms · 2026-05-08T11:18:47.473381+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Control barrier function based quadratic programs with application to adaptive cruise control,

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C. Fang, A. Rocchi, E. M. Hoffman, N. G. Tsagarakis, and D. G. Caldwell, “Efficient self-collision avoidance based on focus of interest for humanoid robots,” in2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids). IEEE, 2015, pp. 1060–1066

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work page 2023

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work page 2015