Abstract-Inspection and exploration of complex underwater structures requires the development of agile and easy to program platforms. In this paper, we describe a system that enables the deployment of an autonomous underwater vehicle in 3D environments proximal to the ocean bottom. Unlike many previous approaches, our solution: uses oscillating hydrofoil propulsion; allows for stable control of the robot's motion and sensor directions; allows human operators to specify detailed trajectories in a natural fashion; and has been successfully demonstrated as a holistic system in the open ocean near both coral reefs and a sunken cargo ship. A key component of our system is the 3D control of a hexapod swimming robot, which can move the vehicle through agile sequences of orientations despite challenging marine conditions. We present two methods to easily generate robot trajectories appropriate for deployments in close proximity to challenging contours of the sea floor. Both offline recording of trajectories using augmented reality and online placement of fiducial tags in the marine environment are shown to have desirable properties, with complementary strengths and weaknesses. Finally, qualitative and quantitative results of the 3D control system are presented.
In order to analyse in full detail the dynamics of a charged particle in the field of a magnetic dipole, we propose to study the restricted motion of the particle in a spherical surface with the dipole at its centre. This model can be considered as the classical non-relativistic Störmer problem within a sphere, and although this problem no longer represents the real Störmer problem, it shows the complex behaviour of this magnetic field through the classical dynamics equations that can be formally integrated. We start from a Lagrangian approach which allows us to analyse the dynamical properties of the system, such as the role of a velocity dependent potential, the symmetries and the conservation properties. We derive the Hamilton equations of motion, which in this restricted case can be reduced to a quadrature. From the Hamiltonian function we find, for the polar angle, an equivalent one-dimensional system of a particle in the presence of an effective potential. This equivalent potential function, which is a double well potential, allows us to get a clear description of the dynamics of the system. Then we obtain, by means of numerical integration, different plots of the trajectories in three-dimensional graphs in the sphere. This restricted case of the Störmer problem is still nonlinear, with complex and interesting dynamics and we believe that it can offer the student a better grasp of the subject than the general three-dimensional case.
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