Autonomous aerial robots navigating in uncertain environments have a wide variety of applications, including inspection, search and rescue, inventory localization, sports photography, entertainment, package delivery, and national security. These dynamical systems have configuration spaces with non-trivial geometry but are often modeled with restrictions to their configuration space to simplify the geometry. Instead, we leverage techniques for differential geometry and Lie algebra to model the intrinsic geometry of these robots correctly and in a coordinate-free fashion. Using these models, we tackle problems in planning, control, and estimation with the goal of efficient computationally limited onboard implementation for real systems. Trajectory generation for robots in cluttered environments requires enforcing of collision free constraints. We utilize tools from computational geometry to model obstacles and formulate useful convex partitions of free space. Targeting on-line applications, these modeling methods are co-designed with the optimization-based formulations to be able to enable real-time planning through unknown environments without offboard sensing or computation. We use convex and non-convex optimization approaches to solve for dynamically feasible trajectories in Rn, Lie groups SO(3), SE(3), and smooth manifolds. Using a variety of sensor models, we use environment observations to build representations which allows us to enforce collision free constraints in these optimization formulations using convex approximations. We address how to use multiple charts to represent these manifolds and how to control and estimate the states of robots with this representation. These methods are applied to free flying systems such as a myriad of quadrotor platforms with under actuated dynamics and a free flying space robot with over actuated dynamics.