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This paper introduces time-invariant and time-varying stochastic barrier certificates to ensure the safety of discrete-time, continuous-space systems operating amidst dynamic obstacles. The time-varying formulation, leveraging Bellman's optimality, provides less conservative lower bounds on the probability of remaining safe over a finite horizon compared to existing methods. By formulating barrier synthesis as a convex sum-of-squares program with polynomial functions, the approach achieves tractable optimization and tighter safety guarantees, as demonstrated on nonlinear systems.
Forget conservative approximations – this work delivers provably tighter safety guarantees for robots navigating dynamic, uncertain environments.
Safety of stochastic dynamic systems in environments with dynamic obstacles is studied in this paper through the lens of stochastic barrier functions. We introduce both time-invariant and time-varying barrier certificates for discrete-time, continuous-space systems subject to uncertainty, which provide certified lower bounds on the probability of remaining within a safe set over a finite horizon. These certificates explicitly account for time-varying unsafe regions induced by obstacle dynamics. By leveraging Bellman's optimality perspective, the time-varying formulation directly captures temporal structure and yields less conservative bounds than state-of-the-art approaches. By restricting certificates to polynomial functions, we show that time-varying barrier synthesis can be formulated as a convex sum-of-squares program, enabling tractable optimization. Empirical evaluations on nonlinear systems with dynamic obstacles show that time-varying certificates consistently achieve tight guarantees, demonstrating improved accuracy and scalability over state-of-the-art methods.
