Risk-Aware Robot Control in Dynamic Environments Using Belief Control Barrier Functions

TL;DR

This work addresses safe control for robots operating in dynamic, partially observable environments by introducing Belief Control Barrier Functions (BCBFs) that leverage concentration bounds on tail risk measures ($VaR_\tau$, $CVaR_\tau$) to enforce safety from sample-based beliefs under stochastic dynamics. The method builds a safe set via a sample-based CBF defined on the object belief, providing probabilistic guarantees even under distributional shifts with finite samples. Key contributions include the formulation of $\tilde{h}$ as a BCBF from concentration bounds, the introduction of $\ell$-robust bounds for distributional mismatch, and a model-predictive control-inspired QP that preserves forward invariance at high rates (≈$1\mathrm{kHz}$) in underwater scenarios for object tracking and collision avoidance. The results demonstrate that tail-risk based safety constraints outperform mean-based approaches in skewed or multimodal belief settings, enabling safer operation in challenging perception-to-control loops with practical computational efficiency.

Abstract

Ensuring safety for autonomous robots operating in dynamic environments can be challenging due to factors such as unmodeled dynamics, noisy sensor measurements, and partial observability. To account for these limitations, it is common to maintain a belief distribution over the true state. This belief could be a non-parametric, sample-based representation to capture uncertainty more flexibly. In this paper, we propose a novel form of Belief Control Barrier Functions (BCBFs) specifically designed to ensure safety in dynamic environments under stochastic dynamics and a sample-based belief about the environment state. Our approach incorporates provable concentration bounds on tail risk measures into BCBFs, effectively addressing possible multimodal and skewed belief distributions represented by samples. Moreover, the proposed method demonstrates robustness against distributional shifts up to a predefined bound. We validate the effectiveness and real-time performance (approximately 1kHz) of the proposed method through two simulated underwater robotic applications: object tracking and dynamic collision avoidance.

Figures (4)

• Figure 1: Left: Reflections near the water surface can lead to ambiguous detections. Right: Underwater bubbles cause temporary low visibility.
• Figure 2: (a): The $\mathrm{PDF}$ of Gaussian distribution $\mathcal{N}(0.5, 0.2^2)$. We draw $N=1000$ samples, illustrated by the histogram with 15 bins. Both the real values of the risk measures and their lower bounds are shown. The lower bounds are computed using $\tau=0.1$ and $\delta=0.05$. (b): The estimated distribution is $\mathcal{N}(0.5, 0.2^2)$, while the true distribution is $\mathcal{N}(0.45, 0.2^2)$, showing a distribution shift.
• Figure 3: (a): Snapshot of the object tracking scenario. The trajectory of the robot is shown in blue. The FoV (pink sectors), the robot (yellow dots), and the belief samples of the object (red dots) are taken at $t=0.3 \mathrm{s}$, $t=2.8\mathrm{s}$, and $t=6.5\mathrm{s}$. The blue and red shadows represent the circular footprints of the robot and the object, respectively. (b): Evolution of $\widehat{\mathrm{VaR_\tau}}$ and $\underline{\mathrm{VaR_\tau}}$ over the course of the simulation.
• Figure 4: (a): Snapshot of the collision avoidance scenario. The belief samples (red dots) are taken at time $t =0.76s$. (b): Snapshot illustrating a distributional shift. The true samples (dark blue dots) differ from those used by the controller (red dots), while both are also taken at time $t =0.76s$.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Theorem 1: Corollary 11, so2023almost
• Lemma 1
• proof
• Definition 3
• Lemma 2
• Lemma 3
• Theorem 2
• proof