Safe Collective Control under Noisy Inputs and Competing Constraints via Non-Smooth Barrier Functions

TL;DR

This work devise a method to synthesize safety-aware control inputs for uncertain collectives by smoothing a Boolean-composed CBF and solving a stochastic optimization problem where each agent's forcing term is restricted to the affine subspace of control inputs certified by the combined CBF.

Abstract

We consider the problem of safely coordinating ensembles of identical autonomous agents to conduct complex missions with conflicting safety requirements and under noisy control inputs. Using non-smooth control barrier functions (CBFs) and stochastic model-predictive control as springboards, and by adopting an extrinsic approach where the ensemble is treated as a unified dynamic entity, we devise a method to synthesize safety-aware control inputs for uncertain collectives. Drawing upon stochastic CBF theory and recent developments in Boolean CBF composition, our method proceeds by smoothing a Boolean-composed CBF and solving a stochastic optimization problem where each agent's forcing term is restricted to the affine subspace of control inputs certified by the combined CBF. For the smoothing step, we employ a polynomial approximation scheme, providing evidence for its advantage in generating more conservative yet sufficiently-filtered control inputs than the smoother but more aggressive equivalents produced from an approximation technique based on the log-sum-exp function. To further demonstrate the utility of the proposed method, we present an upper bound for the expected CBF approximation error, along with results from simulations of a single-integrator collective under velocity perturbations. Lastly, we compare these results with those obtained using a naive state-feedback controller lacking safety filters.

Figures (3)

• Figure 1: Motivating example: A multi-agent reach-avoid mission with complex safety requirements. Here, the dynamic agents (depicted as mobile robots in orange ($R_i$)) must navigate to specified goal positions (green circles ($G_i$)) while avoiding multiple obstacles (gray circles ($\mathcal{O}_i$)) and inter-agent collisions. Figure annotations describe associated notions of safety, with agent arrows indicating direction of travel.
• Figure 2: Comparing controller reach-avoid performance: (\ref{['fig:betaeffectsingobs']}) Time-evolution of agent trajectories and CBFs showing the effect of smoothing technique and smoothing parameter ($\beta$) on $h_\Sigma$, with a horizontal line at $h_\Sigma = 0$ for reference. (\ref{['fig:betaeffectmultobs']}) Results for the multiple-obstacle example. From the inset on the right, we notice that the LSE-smoothed CBF assumes a zero value rather abruptly on the interval $6\le t\le7$. (\ref{['fig:trajenv']}) Envelope of 100 safe trajectories (with magnified inset) for agent 3.
• Figure 3: Synthesized control inputs: Time-evolution of the norms of the nominal and CBF-filtered control inputs with $\beta=1, 0.5$, and $0.1$, and for the multiple-obstacle mission setting. A red inset highlighting the control inputs corresponding to the nominal and CBF-filtered cases is also depicted.

Theorems & Definitions (13)

• Definition 1: Safe Set
• Definition 2: Control Barrier Function ames_control_2019
• Lemma 1
• proof
• Proposition 1
• proof
• Remark 1
• Theorem 1: Conditions for Almost-Sure Safety via Zeroing CBFs (Corollary 11 in so2023almost)
• Proposition 2: Forward Invariance of $\mathfrak{C}_\Sigma$
• proof