Safety for Time-Varying Parameterized Sets Using Control Barrier Function Methods

TL;DR

The paper tackles safety for time-varying parameterized convex sets by replacing the hard distance between sets with a differentiable, arbitrarily tight overapproximation based on a log-sum-exp construction $d_{eps}^+$. This yields a convex, differentiable framework to compute safety-preserving controls via convex optimization, avoiding nonsmooth barrier formulations. Key contributions include proving convexity and unique optima for the distance problem, establishing differentiability with respect to time-varying parameters, and integrating these results into a two-step control synthesis (min-distance for constraints, then a QP for safety), demonstrated through multi-agent simulations. The approach offers a smooth, scalable alternative to NCBFs for set-based safety in multi-robot contexts and points toward extensions to higher-relative-degree safety constraints.

Abstract

A fundamental and classical problem in mobile autonomous systems is maintaining the safety of autonomous agents during deployment. Prior literature has presented techniques using control barrier functions (CBFs) to achieve this goal. These prior techniques utilize CBFs to keep an isolated point in state space away from the unsafe set. However, various situations require a non-singleton set of states to be kept away from an unsafe set. Prior literature has addressed this problem using nonsmooth CBF methods, but no prior work has solved this problem using only "smooth" CBF methods. This paper addresses this gap by presenting a novel method of applying CBF methods to non-singleton parameterized convex sets. The method ensures differentiability of the squared distance function between ego and obstacle sets by leveraging a form of the log-sum-exp function to form strictly convex, arbitrarily tight overapproximations of these sets. Safety-preserving control inputs can be computed via convex optimization formulations. The efficacy of our results is demonstrated through multi-agent simulations.

Figures (3)

• Figure 1: Demonstration of the $LSE_\epsilon^+$ overapproximation of an agent polytope set and the results of the nearest point optimization for various levels of approximation strictness. The nearest points found for each plot is shown as black stars on the boundary of the set with a dashed line showing the connecting vector. a.) A plot of the original polytope sets without any overapproximation. b.) A plot of the polytopes with a very tight overapproximation ($\epsilon = 20$). c.) A plot of the polytopes with a moderate overapproximation ($\epsilon = 8$). d.) A plot of the polytopes with a loose overapproximation ($\epsilon = 3$).
• Figure 2: Still frames from our simulation. Agents are represented by their polytope shape with its semi-transparent $LSE_\epsilon^+$ approximation around it. Agents are attempting to reach corresponding goal positions while avoiding collisions. The goal for each agent is shown as a similar colored star. This simulation demonstrates the efficacy of our method preventing collisions between sets for all agents.
• Figure 3: Plot of the minimum $h$ function value for each agent over time. A separate $h_j^i$ function is run between each agent $i$ and each external agent $j\neq i$. The value plotted for each agent $i$ is $\min_j h_j^i(\cdot)$, implying that safety is preserved if the minimum value is greater than 0. As demonstrated by the data, the minimum values were kept above 0 and thus the agents were kept within the safe set throughout the whole simulation.

Theorems & Definitions (30)

• Lemma 1
• proof
• Lemma 2
• Definition 1
• Lemma 3
• Definition 2
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 1: barratt2018differentiability