Safe Probabilistic Planning for Human-Robot Interaction using Conformal Risk Control

This paper presents a novel probabilistic safe control framework for human-robot interaction that integrates control barrier functions with conformal risk control to provide formal safety guarantees, dynamically adjust safety margins based on interaction context, and significantly reduce collision rates while maintaining task efficiency.

The Gist

Imagine you are walking down a busy sidewalk. You see a robot approaching you. You might step left, you might step right, or you might stop to tie your shoe. Your behavior is unpredictable. The robot needs to get to its destination without bumping into you, but it can't read your mind.

This paper presents a new "brain" for robots that solves this exact problem. It teaches the robot how to be safe but not a coward, using a clever mix of math and statistics.

Here is the breakdown using simple analogies:

1. The Problem: The "Too Scared" vs. "Too Brave" Robot

Most robots today try to stay safe by assuming the worst-case scenario.

• The "Too Brave" Robot: It assumes you will walk in a perfect straight line. If you suddenly swerve, it panics and crashes.
• The "Too Scared" Robot: It assumes you might jump, spin, or stop instantly. So, it stops 10 feet away from you and waits forever. It never crashes, but it also never gets anything done.

The goal of this paper is to find the Goldilocks zone: a robot that is confident enough to move efficiently but cautious enough to stop if things get risky.

2. The Tools: The "Safety Net" and the "Crystal Ball"

The authors combine two powerful tools:

• Control Barrier Functions (CBFs): Think of this as a digital safety net. It's a mathematical rule that says, "As long as you stay inside this invisible bubble, you are safe." If the robot tries to leave the bubble, the net yanks it back.
• Conformal Risk Control (CRC): This is the smart crystal ball. Usually, robots guess what humans will do and hope for the best. CRC is different. It doesn't just guess; it calculates a "confidence score." It says, "Based on the last 100 times I saw a human act like this, there is a 99% chance they will stay within this specific area."

3. The Magic Trick: The "Adjustable Safety Margin"

The core innovation is a variable called $\lambda$ (lambda). Imagine this as a volume knob for caution.

• Low Risk (The "Quiet Library"): If the robot sees a human walking calmly and predictably, it turns the knob down. The safety bubble shrinks. The robot gets closer, moves faster, and acts more naturally.
• High Risk (The "Crowded Concert"): If the human is acting erratically, or if the robot is unsure what they will do next, it turns the knob up. The safety bubble expands massively. The robot slows down, gives the human plenty of space, and waits.

The Analogy: Think of driving a car.

• On a sunny day with clear roads (low uncertainty), you drive at the speed limit and stay close to the car in front.
• In a heavy fog (high uncertainty), you slow down and leave a huge gap between you and the car ahead.
• This paper teaches the robot to sense the "fog" of human behavior and adjust its "gap" automatically.

4. How It Learns (The "Training Camp")

Before the robot goes out into the real world, it goes to a simulation training camp.

1. It watches thousands of videos of real people walking (using data from real crowds).
2. It practices interacting with these "digital humans."
3. It learns a pattern: "When the human is 2 meters away and moving fast, I need to be extra careful. When they are 5 meters away, I can relax."
4. It builds a model that predicts exactly how much "caution" (safety margin) it needs for any given situation.

5. The Results: Less Crashes, More Efficiency

The researchers tested this on robots navigating through crowds.

• Old Methods: Either crashed often (because they were too brave) or got stuck and never reached their goal (because they were too scared).
• This New Method: It crashed significantly less often than the brave robots, but it reached its goal much more often than the scared robots.

The Bottom Line

This paper gives robots a new kind of "social intelligence." Instead of blindly following rigid rules or freezing in fear, the robot learns to measure the uncertainty of human behavior and dynamically adjust its caution.

It's like teaching a robot to have "street smarts," allowing it to navigate a chaotic human world safely, efficiently, and without constantly hitting the brakes.

Technical Summary

Here is a detailed technical summary of the paper "Safe Probabilistic Planning for Human-Robot Interaction using Conformal Risk Control."

1. Problem Statement

The deployment of autonomous robots in human environments (e.g., autonomous driving, service robots) faces a fundamental challenge: ensuring safety under the unpredictability of human behavior.

• Complex Uncertainty: Human behavior is multimodal (multiple distinct possible actions, e.g., passing left or right) and history-dependent.
• Limitations of Existing Methods:
  • Classical Control Barrier Functions (CBFs): Often rely on simplifying distributional assumptions (e.g., Gaussian) or worst-case bounds, which can be overly conservative or fail to capture complex human dynamics.
  • Sampling-based Methods: Can handle complex distributions but lack formal safety guarantees and are computationally expensive for real-time use.
  • Standard Conformal Prediction: While it provides distribution-free coverage guarantees, it typically targets fixed coverage rates rather than directly controlling the risk of safety constraint violations.

The core problem is to develop a planning algorithm that handles complex, stochastic human behavior without being overly conservative, while providing formal, quantifiable probabilistic safety guarantees for constraint satisfaction at every time step.

2. Methodology

The authors propose a framework combining Control Barrier Functions (CBFs) with Conformal Risk Control (CRC) to create a "risk-aware adaptive safety filter."

A. System Formulation

• Dynamics: The system models a joint human-robot dynamical system with discrete-time dynamics (using Zero-Order Hold).
• Safety Definition: Safety is defined via a safe set $S = \{x \mid h(x) \geq 0\}$, where $h(x)$ is a continuously differentiable function.
• Robust CBF: To handle time discretization, a robust CBF constraint is derived with a margin $\eta$ to ensure safety holds over the entire time interval $[t_k, t_{k+1}]$.

B. The Core Innovation: CRC-CBF

The method addresses the uncertainty in the human's control input $u_H$ (which is unknown to the robot) by using CRC to dynamically tune a safety margin parameter ($\lambda$).

1. Barrier Certificates:

   • True Barrier ($B$): Computed using ground truth human actions (unavailable in real-time).
   • Predicted Barrier ($\hat{B}$): Computed using a learned human behavior model.
   • Prediction Error: The difference $|B - \hat{B}|$ represents the risk.

2. Conformal Risk Control (CRC):

   • Instead of fixing a confidence interval, CRC controls the expected value of a user-defined loss function $L(\lambda) = \max\{0, |B - \hat{B}| - \lambda\}$.
   • The goal is to find the smallest safety margin $\lambda$ such that the expected loss is below a risk threshold $\alpha$.
   • Non-exchangeable CRC: Since human-robot interaction data is time-dependent (non-exchangeable), the method uses geometrically decaying weights to reweight the calibration data, ensuring valid guarantees even with distribution shifts.

3. Safety Filter (CRC-SF):

   • At each time step, the robot solves a Quadratic Program (QP) to minimize deviation from a nominal controller ($u_{nom}$) subject to a chance constraint:
     $$\text{Pr}(u_R \in C_\lambda) \geq 1 - \gamma$$
   • The set $C_\lambda$ is defined as $\{u_R \mid \hat{B} - \lambda \geq 0\}$.
   • Dynamic Adaptation: An online model (LSTM) predicts the optimal $\lambda_k$ based on the current interaction context (states, relative distance, predicted barrier value). If uncertainty is high, $\lambda$ increases (more conservative); if low, $\lambda$ decreases (more efficient).

C. Algorithmic Pipeline

1. Offline Calibration:
   • Simulate interactions using a learned human policy and a nominal robot policy.
   • Collect pairs of true and predicted barrier certificates.
   • Compute optimal safety margins $\hat{\lambda}$ for various contexts using non-exchangeable CRC.
   • Train a predictive model (LSTM) to map context features to $\hat{\lambda}$.
2. Online Execution:
   • The robot observes the current state and interaction history.
   • The LSTM predicts the required safety margin $\lambda_k$.
   • The robot solves the CRC-SF QP to generate a safe control input.

3. Key Contributions

1. Novel Framework: The first integration of Conformal Risk Control with Control Barrier Functions to provide high-probability safety guarantees for joint human-robot systems.
2. Theoretical Guarantees: Rigorous proof establishing that CRC-based uncertainty quantification translates to probabilistic bounds on constraint satisfaction (Theorem 1).
3. Adaptive Safety Margins: An assumption-light algorithm that dynamically adjusts conservativeness based on real-time interaction context and uncertainty, avoiding the pitfalls of fixed worst-case bounds.
4. Handling Temporal Dependencies: The use of non-exchangeable CRC with geometric weights to handle the time-series nature of human-robot interactions without stationarity assumptions.

4. Experimental Results

The method was evaluated on single-agent (head-on collision avoidance) and multi-agent (crowd navigation) scenarios using real-world pedestrian data.

Baselines:

• CBF-QP: Standard CBF without uncertainty handling.
• Fixed CRC-SF: Uses a static safety margin computed offline.
• MPPI: Sampling-based method (no formal guarantees).

Key Findings:

• Safety vs. Efficiency Trade-off:
  • CBF-QP: High efficiency but poor safety (38.8% collision rate in multi-agent).
  • Fixed CRC-SF: High safety but extremely conservative (low goal-reaching success, high control effort).
  • Online CRC-SF (Proposed): Achieved the best balance.
    • Multi-agent: 3.0% collision rate (vs. 38.8% for CBF-QP) and 84.8% goal success (vs. 31.6% for Fixed CRC).
    • Single-agent: 2.0% collision rate with 78% goal success.
• Behavioral Adaptation: Qualitative analysis showed the proposed method proactively "slows down and waits" in high-uncertainty scenarios, whereas CBF-QP acted "boldly" and collided.
• Consistency: The proposed method maintained consistent performance across five different multi-agent test configurations, whereas other methods showed high variance.

5. Significance

This work bridges the gap between statistical learning and control theory for safety-critical systems.

• Formal Guarantees: It moves beyond heuristic safety filters by providing mathematically provable probabilistic safety guarantees for each time step.
• Practicality: By dynamically adjusting the safety margin, it avoids the "pessimism" of worst-case approaches, allowing robots to operate efficiently in complex, dynamic human environments.
• Scalability: The approach is computationally efficient (solving a convex QP) and suitable for real-time deployment, making it a viable solution for autonomous vehicles and service robots operating alongside humans.