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Our past work introduced source seeking methods for GPS-denied autonomous vehicles using only local signal measurement and operating in two dimensions. In this paper, we extend these results to three dimensions. The 3-D extensions introduce many interesting challenges, including the choice of vehicle models in 3-D, sensor placement to allow probing-based gradient estimation of an unknown signal field in 3-D, the question of what type of pattern of vehicle motion can be produced in an underactuated 3-D vehicle to allow tuning by single-loop or multiloop extremum seeking, and the shape of attractors, which become very complex in 3-D. We present two control schemes that address these questions. The first scheme focuses on vehicles with a constant forward velocity and the ability to actuate pitch and yaw velocities. The second scheme employs vehicles with constant forward and pitch velocities and actuate only the roll velocity. Our results include convergence analysis and simulation results.

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We present the stability analysis and numerical results for our controller and observer designs for the slender Timoshenko beam with Kelvin-Voigt (internal material) damping. Our designs, introduced in a companion ACC '06 paper, use a combination of classical "damping boundary feedback" ideas with backstepping. The main advantage of our techniques over the previous boundary damping results is that they apply to the readily implementable "anti-collocated" architecture, which uses actuation only at the beam's base and sensing only at the beam's tip. I. INTRODUCTIONThe Timoshenko beam model [3], [6], [7], [8], [11], [12], [13] consists of two coupled second-order-in-space/secondorder-in-time wave equations. This model is the most involved of the four beam models as it includes the effects of lateral displacement, bending moment, shear deformation and rotary inertia [2]. We consider a version of the Timoshenko beam model with Kelvin-Voigt (KV) damping (internal material damping) and design output feedbacks which employ measurements only at the tip of the beam and apply actuation only at the base. Our design, first introduced in [5], is a novel combination of the classical "damping boundary feedback" [3], [6], [7], [8], [11], [12], [13] ideas with backstepping. For physical realism, we assume some small amount of KV damping as is the case with all physical systems, but note that this is not a requirement for the implementation of our theory or the proof of stability as shown in a companion paper for the undamped shear beam [4].While damping feedback ideas have traditionally required actuation to be applied at the free end, making them appropriate primarily for passive control implementations, combining them with backstepping allows actuation to be moved to the beam base. This architecture makes active control more readily implementable to several applications. Besides producing an increase in the rate of exponential stability of the unforced beam model, our design readily extends to situations where the beam is forced by persistent disturbances at the tip as well as situations in which tip effects, such as nonlinear multi-equilibrium forces, produce unstable beam dynamics.Our backstepping boundary control is such that, after a change of variable, the beam model is converted into a wave equation (target system) for a high-tension string with a damper on one end and a high-stiffness spring on the other