Control systems often need to satisfy strict safety requirements. Safety index provides a handy way to evaluate the safety level of the system and derive the resulting safe control policies. However, designing safety index functions under control limits is difficult and requires a great amount of expert knowledge. This paper proposes a framework for synthesizing the safety index for general control systems using sum-of-squares programming. Our approach is to show that ensuring the non-emptiness of safe control on the safe set boundary is equivalent to a local manifold positiveness problem. We then prove that this problem is equivalent to sum-of-squares programming via the Positivstellensatz of algebraic geometry. We validate the proposed method on robot arms with different degrees of freedom and ground vehicles. The results show that the synthesized safety index guarantees safety and our method is effective even in high-dimensional robot systems.

翻译：控制系统通常需要满足严格的安全要求。安全指标提供了一种便捷的方式评估系统安全水平并推导出相应的安全控制策略。然而，在控制极限下设计安全指标函数十分困难，需要大量专家知识。本文提出一个框架，利用平方和规划为一般控制系统综合安全指标。我们的方法在于证明确保安全集边界上安全控制非空性等价于局部流形正定性问题，进而通过代数几何中的正定项定理证明该问题等价于平方和规划。我们在不同自由度的机械臂与地面车辆上验证了所提方法。结果表明，综合出的安全指标能够保障安全性，且该方法在高维机器人系统中依然有效。