This paper presents a method that addresses the conservatism, computational effort, and limited numerical accuracy of existing frameworks and methods that ensure safety in online model-based motion generation, commonly referred to as fast and safe tracking. Computational limitations restrict online motion planning to low-fidelity models. However, planning with low-fidelity models compromises safety, as the dynamic feasibility of resulting references is not ensured. This potentially leads to unavoidable tracking errors that may cause safety-critical constraint violations. Existing frameworks mitigate this safety risk by augmenting safety-critical constraints in motion planning by a safety margin that prevents constraint violations under worst-case tracking errors. However, the methods employed in these frameworks determine the safety margin based on a heuristically selected performance of the model used for planning, which likely results in overly conservative references. Furthermore, these methods are computationally intensive, and the state-of-the-art method is limited in numerical accuracy. We adopt a different perspective and address these limitations with a method that mitigates conservatism in existing frameworks by adapting the performance of the model used for planning to a given safety margin. Our method achieves numerical accuracy and requires significantly less computation time than existing methods by leveraging a captivity-escape game, which is a novel zero-sum differential game formulated in this paper. We demonstrate our method using a numerical example and compare it to the state of the art.

翻译：摘要：本文提出一种方法，以解决现有框架和方法在确保基于模型的在线运动生成安全性（通常称为快速安全跟踪）中存在的保守性、计算负担及数值精度有限的问题。计算限制使得在线运动规划只能使用低保真模型。然而，使用低保真模型进行规划会牺牲安全性，因为生成参考轨迹的动态可行性无法保证。这可能导致不可避免的跟踪误差，进而引发安全关键约束的违反。现有框架通过在运动规划的安全关键约束中增加安全裕度来缓解此风险，该裕度可防止在最坏情况跟踪误差下的约束违反。然而，这些框架采用的方法基于启发式选择的规划模型性能来确定安全裕度，这很可能导致过于保守的参考轨迹。此外，这些方法计算量大，且现有最先进方法的数值精度有限。我们采用不同的视角，通过自适应调整规划模型的性能以适应给定安全裕度，从而缓解现有框架中的保守性。本文提出的方法通过利用捉逃博弈（一种新颖的零和微分博弈）实现了数值精度，且所需计算时间显著少于现有方法。我们通过数值示例演示了该方法，并与最先进方法进行了对比。