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arXiv2026

On the Degree of Safety: Beyond Safe or Unsafe with Control Barrier Functions

A valid control barrier function (CBF) certifies if its represented safe set can be rendered forward invariant, and the sign of its value indicates whether a state is safe or not, but it does not quantify a degree of safety beyond the binary indication. In this paper, we show that among valid CBFs representing the same safe set, interior values and gradients can be changed arbitrarily, so neither quantity determines a degree of safety that is independent of how the set is represented. We also show that whether a candidate CBF-based inequality constraint is feasible does not by itself quantify a degree of safety. In particular, infeasibility can occur either because the safe set is not controlled invariant or because the candidate CBF representation fails. This motivates our distinction between intrinsic and representational infeasibility. Finally, we introduce the invariance authority demand (IAD), a representation-independent degree of safety that quantifies the control authority required for controlled invariance and can be used to guide set or actuator repair.

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arXiv2026

Cooperative Risk-Aware Exploration in Heterogeneous Multi-Robot Systems Using Algorithmic Altruism

Multi-robot systems are well-positioned for exploration in hazardous environments, but effective deployment requires deciding not only where robots should gather information, but also how risk should be distributed across heterogeneous team members. This paper develops a game-theoretic framework for cooperative risk-aware exploration based on ecologically inspired altruistic behavior. Each robot selects a finite-horizon trajectory to maximize information gain while penalizing redundant exploration and expected hazard exposure. Heterogeneity is introduced through agent-specific value parameters for encoding altruistic coupling, which is modeled through relatedness weights inspired by Hamilton's rule. We introduce a game-theoretic structure for trajectory planning that defines a Social Nash Equilibrium, which modifies the utility of agent actions according to agent relatedness. This utility shaping causes agents to internalize the effect of their trajectory choices on teammates, encouraging lower-valued robots to accept risk when doing so benefits higher-valued agents and improves team performance. We define an exploration utility for agents that rewards area coverage and uncertainty reduction, while also penalizing redundancy and risk, enabling projected gradient-based waypoint optimization in a receding-horizon planner. Simulations show that altruistic planning reduces redundant exploration, improves inter-robot separation, and reallocates risk according to agent value while maintaining comparable map coverage. We further demonstrate the approach in hardware experiments, where planned waypoints are tracked by wheeled robots using single-integrator controllers and barrier certificates.

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