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Vortex detection has been a long-standing and challenging topic in fluid analysis. Recent state-of-the-art extraction and visualization of vortices in unsteady fluid flow employ objective vortex criteria, which makes feature extraction independent of reference frames or observers. However, even objectivity can only guarantee that different observers reach the same conclusions, but not necessarily guarantee that these conclusions are the only physically meaningful or relevant ones. Moreover, a significant challenge is that a single observer is often not sufficient to accurately observe multiple vortices that follow different motions. This thesis presents a novel mathematical framework that represents physically realizable observers as the Lie algebra of the Killing fields on the underlying manifold, together with a software system that enables the exploration and use of an interactively chosen set of observers, resulting in relative velocity fields and objective vortex structures in real-time. Based on our mathematical framework, our system facilitates the objective detection and visualization of vortices relative to well-adapted reference frame motions, while at the same time guaranteeing that these observers are physically realizable. We show how our framework speeds up the exploration of objective vortices in unsteady 2D flow, on planar as well as on spherical domains.

In this dissertation, we focus on two fundamental problems related to the navigation of ground robots and small Unmanned Aerial Vehicle (UAVs): cooperative localization and path planning. The theme running through in all of the work is the use of bearing only sensors, with a focus on monocular video cameras mounted on ground robots and UAVs. To begin with, we derive the conditions for the complete observability of the bearing-only cooperative localization problem. The key element of this analysis is the Relative Position Measurement Graph (RPMG). The nodes of an RPMG represent vehicle states and the edges represent bearing measurements between nodes. We show that graph theoretic properties like the connectivity and the existence of a path between two nodes can be used to explain the observability of the system. We obtain the maximum rank of the observability matrix without global information and derive conditions under which the maximum rank can be achieved. Furthermore, we show that for the complete observability, all of the nodes in the graph must have a path to at least two different landmarks of known location. The complete observability can also be obtained without landmarks if the RPMG is connected and at least one of the robots has a sensor which can measure its global pose, for example a GPS receiver. We validate these conditions by simulation and experimental results. The theoretical conditions to attain complete observability in a localization system is an important step towards reliable and efficient design of localization and path planning algorithms. With such conditions, a designer does not need to resort to exhaustive simulations and/or experimentation to verify whether a given selection of a control strategy, topology of the sensor network, and sensor measurements meets the observability requirements of the system. In turn, this leads to decreased requirements of time, cost, and effort for designing a localization algorithms. We use these observability conditions to develop a technique, for camera equipped UAVs, to cooperatively geo-localize a ground target in an urban terrain. We show that the bearing-only cooperative geo-localization technique overcomes the limitation of requiring a low-flying UAV to maintain line-of-sight while flying high enough to maintain GPS lock. We design a distributed path planning algorithm using receding horizon control that improves the localization accuracy of the target and of all of the UAVs while satisfying the observability conditions. Next, we use the observability analysis to explicitly design an active local path planning algorithm for UAVs. The algorithm minimizes the uncertainties in the time-to-collision (TTC) and bearing estimates while simultaneously avoiding obstacles. Using observability analysis we show that maximizing the observability and collision avoidance are complementary tasks. We provide sufficient conditions of the environment which maximizes the chances obstacle avoidance and UAV reaching the goal. Finally, we develop a reactive path planner for UAVs using sliding mode control such that it does not require range from the obstacle, and uses bearing to obstacle to avoid cylindrical obstacles and follow straight and curved walls. The reactive guidance strategy is fast, computationally inexpensive, and guarantees collision avoidance.