Safe operation of robotic systems requires trajectories to remain within a prescribed safe set under admissible control inputs. Barrier certificates provide such guarantees by certifying a controlled-invariant region within that set. Sum-of-squares optimization offers a systematic way to synthesize such certificates, but its direct application requires polynomial dynamics, excluding common robotic nonlinearities, including trigonometric terms. We address this limitation using exact polynomial lifting, which replaces non-polynomial dynamics with polynomial-augmented dynamics subject to lifting-induced algebraic constraints, preserving nonlinear geometry without approximation. We formulate lifted-domain joint barrier synthesis that computes a certificate with a state-feedback control witness and develop a sampled-data safety filter for zero-order-hold implementation. To assess whether the benefits of lifting persist across synthesis frameworks, we also adapt a sample-guided successive-barrier method to the lifted representation. On coordinated-turn and planar multirotor models, exact lifting improves certified coverage in both methods: at matched sample sizes, lifted successive-barrier synthesis achieves higher coverage with fewer barriers and lower computational cost, while lifted joint barrier synthesis provides higher coverage and lower computational cost than the finest tested piecewise resolution. In closed-loop experiments, the safety filter maintains feasibility and safety across all evaluated trajectories, reduces spatial conservativeness, and requires less intervention for both models.
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.
Rayan Mazouz, Luca Laurenti, Morteza Lahijanian
Dept. of Aerospace Eng. Sciences at University of Colorado Boulder, USA · Delft University of Technology, The Netherlands
Safe navigation against obstacles that can actively maneuver within bounded capabilities remains challenging: robust control barrier function methods typically treat obstacle actions as generic disturbances, while differential-game approaches are computationally expensive for online navigation. We propose an adversarially robust geometric certificate that accounts for the worst-case effect of admissible obstacle maneuvers directly in the safe-set geometry through a closed-form contraction of the certificate parameters. The construction exploits a structural property of line-of-sight (LoS) certificates: the robot and obstacle actions enter the certificate through a common state-dependent geometric gain. This gain cancels in the worst-case comparison, reducing the differential game to a direct comparison between obstacle maneuvering capability and the weaker of the robot's longitudinal and steering authorities. Instantiated on the parabolic certificate, the construction yields Adversarially Robust Dynamic Parabolic Control Barrier Functions (AR-DPCBF), for which we establish sufficient conditions for forward invariance of the contracted safe set against all admissible obstacle maneuvers under kinematic bicycle dynamics with bounded inputs. When the obstacle capability is unknown, a sliding-window estimator supplies a high-probability upper bound, allowing the guarantee to be retained with the corresponding coverage probability. We further formulate soft and buffered variants to recover feasibility in dense environments. Simulations across obstacle capabilities, densities, and capability mismatch show substantial reductions in barrier violations and collisions and demonstrate that pointwise robustification of the barrier derivative cannot substitute for contraction of its geometry.
Control barrier functions (CBFs) have become one of the most popular tools for encoding and enforcing state constraints in safety-critical robotics. Standard CBF approaches are inherently myopic in nature as they enforce safety only at the current time step. Consequently, the system can be driven toward the boundary of the safe set where no feasible safe control exists at a future timestep. Model predictive control (MPC) based approaches address this by enforcing state constraints over a receding horizon. However, such approaches generally require the model to be known for solving a constrained optimization problem at every step, which is computationally expensive for real-time deployment. We propose BarrierFormer, a barrier-supervised transformer framework that addresses these limitations by encoding rollout-level CBF constraints in learning a model-free safe policy. A causal transformer encodes observation-action history, autoregressively generates a predictive rollout through the dynamics head to replace the model, and provides a residual correction to a nominal controller through the action head to replace the online computation. A barrier critic operating on local observations evaluates CBF constraint violations along this rollout, and a safety teacher computes barrier-consistent actions satisfying these constraints as direct supervision targets for the learned control policy. During inference, the policy maps observation-action history to control actions without any online optimization or model knowledge, enabling real-time model-free predictive safety enforcement. Evaluations across linear and nonlinear, 2D and 3D dynamical systems for safe goal-directed navigation demonstrate that BarrierFormer outperforms existing reinforcement learning (RL)-based, diffusion-based, MPC-based, and transformer-based approaches in safety rate and inference latency.