Guaranteed Safe Path and Trajectory Tracking via Reachability Analysis Using Differential Inequalities
Guaranteed Safe Path and Trajectory Tracking via Reachability Analysis Using Differential Inequalities
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DOI：
10.1007/s10846-023-01928-w
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发表时间：
2023-08
影响因子：
3.3
通讯作者：
Xuejiao Yang;Bowen Mu;Dillard Robertson;Joseph K. Scott
中科院分区：
文献类型：
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作者：
Xuejiao Yang;Bowen Mu;Dillard Robertson;Joseph K. Scott
在许多自动运动规划系统中，车辆的任务是跟踪设计安全的参考路径或轨迹。然而，由于各种不确定性，真实车辆可能会偏离这些参考值，从而可能导致碰撞。本文提出了严格的可达集边界方法，用于快速封闭不确定性下可能的偏差集，这是在线安全验证的关键信息。所提出的方法应用了微分不等式理论的最新进展，该理论利用冗余模型方程仅使用简单的区间计算即可实现锐界。这些方法已被证明能够以低成本为其他应用领域的非线性系统产生非常尖锐的界限，但它们依赖于特定问题的见解来识别适当的冗余方程，这使得它们难以推广和自动化。在这里，我们通过三个代表性案例研究首次演示了这些方法在跟踪问题中的应用。我们发现用类李雅普诺夫函数定义冗余方程特别有效。结果表明，该技术可以产生有效的边界，计算时间比计划的时间范围小几个数量级，这使其成为在线安全验证的一种有前景的方法。然而，这种性能是以低通用性为代价的，特别是由于需要针对特定问题的见解和有利的问题结构，例如适当的李亚普诺夫类函数的存在。
In many automated motion planning systems, vehicles are tasked with tracking a reference path or trajectory that is safe by design. However, due to various uncertainties, real vehicles may deviate from such references, potentially leading to collisions. This paper presents rigorous reachable set bounding methods for rapidly enclosing the set of possible deviations under uncertainty, which is critical information for online safety verification. The proposed approach applies recent advances in the theory of differential inequalities that exploit redundant model equations to achieve sharp bounds using only simple interval calculations. These methods have been shown to produce very sharp bounds at low cost for nonlinear systems in other application domains, but they rely on problem-specific insights to identify appropriate redundant equations, which makes them difficult to generalize and automate. Here, we demonstrate the application of these methods to tracking problems for the first time using three representative case studies. We find that defining redundant equations in terms of Lyapunov-like functions is particularly effective. The results show that this technique can produce effective bounds with computational times that are orders of magnitude less than the planned time horizon, making this a promising approach for online safety verification. This performance, however, comes at the cost of low generalizability, specifically due to the need for problem-specific insights and advantageous problem structure, such as the existence of appropriate Lyapunov-like functions.