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

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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.