Stochastic Motion Planning and Estimation with Non-Gaussian Uncertainty Distributions on a Lie Group

该项目的重点是复杂动力系统的随机运动规划和估计的计算框架。 由于动态系统中的不确定性来自各种来源，并且它们不能从任何数值分析或实验中完全消除，因此在运动规划和估计中应仔细描述不确定性的演变。 本研究利用非交换调和分析来表示一个任意类型的不确定性分布在一个内在的方式，并根据福克-普朗克方程传播。 在此基础上，设计随机最优运动规划，以获得车辆在不确定环境中实现目标概率最大的可行路径。 此外，将根据贝叶斯框架构建非线性估计方案，以准确地表征传感器测量的不确定性。 这些公式将在由李群表示的非线性配置流形上以无坐标的方式开发，如果成功，本研究的结果将为任意大不确定性的随机分析提供一个基本框架，消除常见的限制性假设，即不确定性遵循特定的高斯分布和小方差。 这一结果将是特别有用的运动规划和高度不确定的情况下，复杂的动力系统的估计，如无人机，能够与未知的动态环境，通过敏捷自主机动。 该奖项将有助于工程驱动的计算数学课程和研究专著的发展，通过它，我们的目标是交叉培训学生在数学和工程。研究结果将通过组织工程夏令营向K-12学生广泛传播，从而使他们在早期阶段接触研究活动，并激发他们对数学和工程的兴趣。

This project is focused on computational framework for stochastic motion planning and estimation of complex dynamical systems. As uncertainties in a dynamical system arise from various sources and they cannot be completely eliminated from any numerical analysis or experiment, the evolution of uncertainties should be carefully characterized in motion planning and estimation. This research utilizes non-commutative harmonic analysis to represent an arbitrary type of uncertainty distributions in an intrinsic fashion, and they are propagated according to the Fokker-Planck equation. Based on these, stochastic optimal motion planning will be designed to obtain feasible paths of a vehicle that maximize the probability to achieve a goal in uncertain environments. In addition, nonlinear estimation schemes will be constructed according to Bayesian framework to characterize uncertainties accurately with sensor measurements. These formulations will be developed in a coordinate-free fashion on nonlinear configuration manifolds represented by a Lie group.If successful, the results of this research will provide a fundamental framework for stochastic analysis with arbitrary large uncertainties, eliminating the common restrictive assumption that uncertainties follow a specific Gaussian distribution with small variances. This result will be particularity useful for motion planning and estimation of complex dynamical systems in highly uncertain circumstances, such as unmanned aerial vehicles that are capable of interacting with unknown dynamic environments through agile maneuvers autonomously. This award will contribute to the development of an engineering-driven computational math course and a research monograph, through which we aim to cross-train students in both mathematics and engineering. The results will be disseminated broadly to K-12 students by organizing engineering summer camps, thereby exposing them to research activities in an earlier stage and invigorating their interests in math and engineering.