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

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

Robotic Path Planning Algorithms35 references4 citations

TL;DR

This paper introduces Conditional Value-at-Risk (CVaR) barrier functions to enforce risk-averse safety in stochastic discrete-time systems, particularly for bipedal robot locomotion under uncertainty. By formulating a difference convex program (DCP)-based controller synthesis method, the approach minimally interferes with a legacy controller while ensuring safety in worst-case scenarios, demonstrated via high-fidelity simulations on the Cassie robot with improved collision avoidance under model uncertainty.

ABSTRACT

Enforcing safety 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 system's 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 as a 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 robot locomotion case study.

Motivation & Objective

• To address the limitations of mean-based safety measures in stochastic robotics systems, which fail to protect against rare but catastrophic failures.
• To develop a risk-sensitive safety framework that prioritizes worst-case performance rather than average-case behavior.
• To design a computationally tractable method for synthesizing CVaR-safe controllers that minimally interfere with existing legacy controllers.
• To validate the approach on a high-fidelity bipedal robot model under modeling uncertainty and dynamic obstacles.
• To extend the use of barrier functions beyond probabilistic safety to coherent risk measures like CVaR for improved robustness.

Proposed method

• Introduces CVaR barrier functions as a tool to enforce safety in the worst-case realizations of stochastic uncertainty, defined via the conditional value-at-risk (CVaR) risk measure.
• Proposes conditions for Boolean compositions of CVaR barrier functions to handle multiple safety constraints, such as avoiding multiple obstacles.
• Develops a difference convex program (DCP) formulation to synthesize CVaR-safe controllers that minimize deviation from a given legacy controller.
• Applies the method to linear discrete-time stochastic systems with additive uncertainty, using a sampled polytopic set to represent model uncertainty.
• Uses a model predictive control (MPC) framework with DCP optimization at each time step to compute safe control inputs in real time.
• Employs a stochastic simulation-based approach to estimate the uncertainty distribution, assuming a uniform distribution over sampled model realizations.

Experimental results

Research questions

• RQ1 Can CVaR barrier functions effectively enforce safety in the worst-case scenarios of stochastic robotic systems?
• RQ2 How can CVaR-safety be formally integrated into control synthesis while minimizing interference with a legacy controller?
• RQ3 What computational framework enables real-time implementation of risk-averse controllers for bipedal robots under uncertainty?
• RQ4 How does CVaR-based safety compare to risk-neutral or mean-based safety in collision avoidance tasks?
• RQ5 Can Boolean compositions of CVaR barrier functions handle complex, multi-obstacle environments?

Key findings

• The proposed CVaR barrier function with β = 0.1 successfully prevented collisions in a straight-path obstacle scenario where risk-neutral and mean-based methods failed.
• For β = 0.5, the robot maintained safe distance from a slanted wall while preserving forward walking behavior, demonstrating effective lateral adaptation.
• In a multi-obstacle scenario with two walls, the CVaR-safe controller with β = 0.5 successfully avoided collisions by maintaining min(h₁, h₂) ≥ 0.
• The DCP solver converged within 100–700 iterations and under 10 seconds per step on a standard laptop, enabling real-time applicability.
• The method preserved the original trajectory of the legacy MPC controller while ensuring safety under stochastic uncertainty, as verified in high-fidelity Cassie robot simulations.
• The approach outperformed risk-neutral and mean-based safety methods in worst-case scenarios, as shown in Figure 1(c), where CVaR safety ensured avoidance even when other methods failed.