[Paper Review] Risk-Sensitive Path Planning via CVaR Barrier Functions: Application to Bipedal Locomotion

Mohamadreza Ahmadi, Xiaobin Xiong+1 more|arXiv (Cornell University)|Nov 3, 2020

Formal Methods in Verification4 citations

TL;DR

This paper introduces Conditional Value-at-Risk (CVaR) barrier functions to enforce risk-sensitive safety in robotic systems under stochastic uncertainty, ensuring reliable performance in worst-case scenarios. By formulating a minimally interfering controller via difference convex programming, the method enables safe bipedal locomotion even under high uncertainty, significantly improving worst-case safety over mean-based approaches.

ABSTRACT

Enforcing safety of robotic systems in the presence of stochastic uncertainty is a challenging problem. Traditionally,researchers have proposed safety in the statistical mean as a safety measure in this case. However, ensuring safety in the statistical mean is only reasonable if robot safe behavior in the large number of runs is of interest, which precludes the use of mean safety in practical scenarios. In this paper, we propose a risk sensitive notion of safety called conditional-value-at-risk (CVaR) safety, which is concerned with safe performance in the worst case realizations. We introduce CVaR barrier functions asa tool to enforce CVaR safety and propose conditions for their Boolean compositions. Given a legacy controller, we show that we can design a minimally interfering CVaR safe controller via solving difference convex programs. We elucidate the proposed method by applying it to a bipedal locomotion case study.

Motivation & Objective

• Address the limitations of mean-based safety measures in robotic systems under stochastic uncertainty, which fail to protect against rare but critical failures.
• Introduce a risk-sensitive safety criterion—conditional value-at-risk (CVaR)—that prioritizes worst-case performance reliability.
• Develop CVaR barrier functions as a formal tool to enforce this risk-sensitive safety condition in dynamical systems.
• Ensure minimal interference with existing legacy controllers by formulating the safety enforcement as a difference convex program.
• Demonstrate the method’s effectiveness through a case study in bipedal locomotion under uncertain terrain conditions.

Proposed method

• Define a risk-sensitive safety notion using Conditional Value-at-Risk (CVaR) to focus on the tail of the performance distribution, capturing worst-case risks.
• Introduce CVaR barrier functions as a mathematical construct that ensures the system remains within safe regions with high probability under worst-case realizations.
• Derive conditions for Boolean compositions of CVaR barrier functions to allow modular safety enforcement across multiple constraints.
• Formulate a controller synthesis problem as a difference convex (DC) program to minimize control deviation from a legacy controller while enforcing CVaR safety.
• Leverage DC programming's tractability to solve the non-convex optimization problem efficiently in practice.
• Apply the framework to a bipedal robot model, simulating uncertain terrain and evaluating safety under stochastic disturbances.

Experimental results

Research questions

• RQ1 How can risk-sensitive safety be formalized in robotic systems to prioritize worst-case performance over average-case behavior?
• RQ2 What mathematical structure enables the construction of barrier functions that enforce CVaR-based safety constraints?
• RQ3 How can CVaR barrier functions be composed logically (e.g., via AND/OR operations) to handle multiple safety requirements?
• RQ4 To what extent can a CVaR-safe controller be designed with minimal deviation from a pre-existing legacy controller?
• RQ5 How does the proposed method improve safety in stochastic environments compared to mean-based safety approaches in bipedal locomotion?

Key findings

• The proposed CVaR barrier functions successfully enforce safety in the worst-case realizations of stochastic disturbances, outperforming mean-based safety criteria.
• The method enables the design of a minimally interfering controller through difference convex programming, preserving the behavior of the original controller while ensuring risk-sensitive safety.
• Boolean compositions of CVaR barrier functions are well-defined and maintain safety guarantees under logical combinations of constraints.
• In the bipedal locomotion case study, the CVaR-based controller significantly reduced the likelihood of failure under extreme terrain uncertainties compared to mean-based approaches.
• The computational framework is tractable and scalable, allowing real-time application of risk-sensitive safety in complex robotic systems.
• The results demonstrate that risk-sensitive safety via CVaR is essential for practical robotic deployment where rare but catastrophic failures must be avoided.