通过多项式提升实现非多项式机器人动力学的障碍证书合成
Barrier Certificate Synthesis for Non-Polynomial Robotic Dynamics via Polynomial Lifting
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中文总结 AI 辅助
针对非多项式机器人动力学，提出基于精确多项式提升的障碍证书合成方法，在协调转弯和平面多旋翼模型上提升覆盖率并降低计算成本。
中文摘要 AI 辅助
机器人系统的安全运行要求轨迹在允许的控制输入下保持在规定的安全集合内。障碍证书通过证明该集合内的一个受控不变区域来提供此类保证。平方和优化提供了一种系统化的证书合成方法，但其直接应用需要多项式动力学，排除了常见的机器人非线性，包括三角函数项。我们利用精确多项式提升来解决这一限制，该方法用受提升诱导的代数约束约束的多项式增广动力学替代非多项式动力学，在无近似的情况下保留非线性几何结构。我们提出了提升域联合障碍合成方法，该方法计算带有状态反馈控制见证的证书，并开发了一种用于零阶保持实现的采样数据安全滤波器。为了评估提升的好处是否在不同合成框架中持续存在，我们还将一种样本引导的连续障碍方法适配到提升表示上。在协调转弯和平面多旋翼模型上，精确提升在两种方法中都提高了认证覆盖率：在匹配的样本量下，提升的连续障碍合成以更少的障碍和更低的计算成本实现了更高的覆盖率，而提升的联合障碍合成比测试的最细分段分辨率提供了更高的覆盖率和更低的计算成本。在闭环实验中，安全滤波器在所有评估轨迹上保持可行性和安全性，减少了空间保守性，并且对两种模型所需的干预更少。
英文摘要
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.
