Barrier Certificate Synthesis for Non-Polynomial Robotic Dynamics via Polynomial Lifting

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.