This paper is proposed to efficiently provide a convex approximation for the probabilistic reachable set of a dynamic system in the face of uncertainties. When the uncertainties are not limited to bounded ones, it may be impossible to find a bounded reachable set of the system. Instead, we turn to find a probabilistic reachable set that bounds system states with confidence. A data-driven approach of Kernel Density Estimator (KDE) accelerated by Fast Fourier Transform (FFT) is customized to model the uncertainties and obtain the probabilistic reachable set efficiently. However, the irregular or non-convex shape of the probabilistic reachable set refrains it from practice. For the sake of real applications, we formulate an optimization problem as Mixed Integer Nonlinear Programming (MINLP) whose solution accounts for an optimal $n$-sided convex polygon to approximate the probabilistic reachable set. A heuristic algorithm is then developed to solve the MINLP efficiently while ensuring accuracy. The results of comprehensive case studies demonstrate the near-optimality, accuracy, efficiency, and robustness enjoyed by the proposed algorithm. The benefits of this work pave the way for its promising applications to safety-critical real-time motion planning of uncertain systems.

翻译：本文旨在高效地提供动态系统在面临不确定性时概率可达集的凸近似。当不确定性不限于有界形式时，可能无法找到系统的有界可达集。为此，我们转而寻求一种概率可达集，以置信度界定系统状态。我们定制了一种基于快速傅里叶变换（FFT）加速的核密度估计（KDE）数据驱动方法，用于对不确定性建模并高效获取概率可达集。然而，概率可达集的不规则或非凸形状限制了其实用性。为适应实际应用，我们将优化问题构建为混合整数非线性规划（MINLP），其解对应一个最优的n边凸多边形以近似概率可达集。随后，我们开发了一种启发式算法，在保证精度的同时高效求解MINLP。综合案例研究结果表明，所提算法具有接近最优性、准确性、高效性和鲁棒性。本工作的成果为将其应用于安全关键的不确定系统实时运动规划铺平了道路。