Geometric and Numerical Analysis of Dynamical Control Systems

This multidisciplinary project bridges differential geometry, optimal control, and numerical analysis, with practical applications in robotic engineering. It introduces a novel approach to path-planning for dynamical control systems on Riemannian manifolds, simplifying many challenges of modern geometric control methods. The central hypothesis is that, under certain conditions, control forces can be integrated into the geometry of the configuration manifold, enabling the trajectories of double-integrator systems to be interpreted as Riemannian geodesics of a modified Riemannian metric. By introducing Gordonian and conformal transformations, the metric is strategically modified to achieve objectives such as trajectory-tracking and obstacle and region avoidance. B-stable modifications of the implicit geodesic Euler method and symplectic geodesic Euler method are constructed on Riemannian manifolds by strategically modifying the curvature through Gordonian transformations. Applications considered throughout the project include the design of trajectory-tracking control schemes for teams of quadrotor UAVs transporting cargo, as well as region avoidance on Riemannian homogeneous spaces.