Proactive Hierarchical Control Barrier Function-Based Safety Prioritization in Close Human-Robot Interaction Scenarios

1. Problem Statement

The integration of collaborative robots (cobots) into human-centered environments (manufacturing, healthcare, etc.) requires control strategies that ensure physical safety without compromising performance. Traditional industrial robots operate in cages, but cobots share workspaces with humans.

• Core Challenge: Human motion is unpredictable and dynamic. In close-proximity scenarios, collisions may become unavoidable if the robot strictly adheres to all safety constraints simultaneously.
• The Feasibility Gap: When multiple safety constraints (e.g., avoiding a human head and a human hand simultaneously) conflict, standard quadratic programming (QP) solvers may return an infeasible solution, leading to a system failure or an uncontrolled collision.
• Risk Differentiation: Not all body parts carry the same risk of injury. A collision with a human head is far more critical than a collision with a hand. Existing Control Barrier Function (CBF) methods often treat all obstacles equally, lacking a mechanism to prioritize high-risk body parts over low-risk ones when trade-offs are necessary.

2. Methodology

The authors propose a Hierarchical Control Framework based on Control Barrier Functions (CBFs) that introduces a safety prioritization mechanism using relaxation variables.

A. System Architecture

The control loop consists of two layers:

1. Performance Controller: A standard inverse kinematics controller that generates desired joint velocities ($\dot{q}_{perf}$) to track a desired trajectory.
2. Safety Controller (CBF Layer): A real-time optimization layer that monitors the robot state and modifies the performance commands ($\dot{q}_{safe}$) to ensure the system remains within a forward-invariant safe set ($X_s$).

B. Control Barrier Function (CBF) Formulation

• Safe Set Definition: The safe set is defined as the 0-superlevel set of a continuously differentiable function $h(x, p_{obs})$, representing the distance between robot links and human body parts (head, hand).
• Constraint: The time derivative of the CBF must satisfy $\dot{h} \geq -\alpha(h)$, ensuring the robot stays within the safe distance.
• Differentiation: The paper utilizes velocity-based control, requiring a single time derivative of the CBF to establish a linear relationship with the control input (joint velocities).

C. Hierarchical Prioritization via Relaxation

To resolve conflicts between constraints (e.g., avoiding the head vs. avoiding the hand), the authors introduce a relaxation variable ($\delta_{ha}$) specifically for lower-priority constraints (the hand).

• Objective Function: The QP minimizes the deviation from the desired performance velocity while penalizing the relaxation variable:
  $$\min_{\dot{q}_{safe}, \delta_{ha} \geq 0} \| \dot{q}_{safe} - \dot{q}_{perf} \|^2 + \beta \delta_{ha}^2$$
• Constraints:
  • Head (High Priority): Strictly enforced ($A_{head} \dot{q}_{safe} \geq -\alpha(h_{head})$). No relaxation allowed.
  • Hand (Lower Priority): Relaxed ($A_{hand} \dot{q}_{safe} \geq -\alpha(h_{hand}) - \delta_{ha}$).
• Mechanism: The parameter $\beta$ acts as a penalty weight. A high $\beta$ forces $\delta_{ha}$ toward zero, prioritizing the hand constraint. A lower $\beta$ allows $\delta_{ha}$ to increase, permitting a controlled violation of the hand constraint to strictly preserve the head constraint.

D. Sensing and Perception

• Hardware: Franka Research 3 Robot (7-DOF) and ZED2i AI Stereo Camera.
• Perception: The ZED2i camera detects human pose at 60 Hz, extracting 34 skeletal keypoints. The system specifically tracks the head and hands to dynamically update the CBF constraints in real-time.

3. Key Contributions

1. Hierarchical CBF Framework with Prioritization: The paper introduces a novel mechanism to trade off safety constraints based on the criticality of human body parts. It ensures that high-risk constraints (head) are never violated, even if it means relaxing lower-risk constraints (hand).
2. Relaxation Variable for Feasibility: By mathematically proving the existence of infeasible states without prioritization (Lemma 1), the authors demonstrate how the relaxation variable guarantees the solvability of the QP in conflict scenarios.
3. Experimental Validation: The method is validated on a physical Franka Research 3 robot with real-time human pose detection, demonstrating the system's ability to proactively mitigate collision risks in dynamic environments.
4. Stability and Complexity Analysis: The paper provides a stability proof showing that the hierarchical controller maintains bounded tracking errors (BIBO stability) and analyzes the computational complexity ($O(n^{3.5})$), confirming real-time feasibility.

4. Experimental Results

The system was tested in various scenarios, including "Pick & Place," circular movements, and square trajectories, with human operators moving their heads and hands near the robot.

• Prioritization Behavior:
  • When the human head approached the robot, the system aggressively deviated from the path to maintain a safe distance.
  • When the hand approached, the system allowed closer proximity. If a conflict arose (e.g., moving away from the head would cause a collision with the hand), the system prioritized the head, allowing the hand constraint to be relaxed (violated slightly) via the variable $\delta_{ha}$.
• Quantitative Metrics:
  • Relative Velocity & Distance: Data showed that the robot maintained larger safety margins for the head compared to the hand.
  • Relaxation Variable ($\delta_{max}$): In conflict scenarios, $\delta_{ha}$ increased, confirming the system was actively trading off the hand constraint to save the head constraint.
  • Simulation (2-DOF): A 2-DOF simulation with multiple obstacles (Red=High Priority, Green=Low Priority) demonstrated that without relaxation, the system failed (infeasible). With relaxation, the robot successfully avoided the Red obstacle while allowing a closer approach to the Green one.
• Performance: The solver executed in approximately 0.16ms per cycle, well within the 16.6ms control cycle time, ensuring real-time responsiveness.

5. Significance and Conclusion

This work addresses a critical gap in human-robot collaboration: how to handle unavoidable conflicts between safety constraints.

• Safety Assurance: By mathematically guaranteeing that high-priority constraints (like the head) are never violated, the system provides a higher level of safety assurance than standard CBF approaches.
• Operational Efficiency: The relaxation mechanism prevents the robot from freezing or failing when constraints conflict, allowing it to continue operating safely by making "least bad" decisions (sacrificing low-risk safety for high-risk safety).
• Future Impact: The framework is modular and can be integrated into existing robot controllers. Future work suggests using learning-based methods to adaptively tune the relaxation penalty ($\beta$) based on environmental context (e.g., tool usage, speed of human motion).

In summary, the paper presents a robust, mathematically grounded, and experimentally verified control strategy that enables robots to operate safely in close proximity to humans by intelligently prioritizing the protection of the most vulnerable body parts.