Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry

This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages. This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove useful for the control of underwater vehicles

Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems

In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy-momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between these. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced instability of KAM stable relative equilibria for this system.Comment: Minor revisions. Typographical errors correcte

Stabilization of mechanical systems using controlled Lagrangians

We propose an algorithmic approach to stabilization of Lagrangian systems. The first step involves making admissible modifications to the Lagrangian for the uncontrolled system, thereby constructing what we call the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system where new terms are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability. The procedure is demonstrated for the problem of stabilization of an inverted pendulum on a cart and for the problem of stabilization of rotation of a rigid spacecraft about its unstable intermediate axis using a single internal rotor. Similar results hold for the dynamics of an underwater vehicle

Fluctuation characteristics and rolling control for an underactuated spherical underwater exploration robot

Compared with other underwater exploration robots, Spherical underwater robot has an outstanding advantage for the underwater exploration, whose spherical shell has the excellent resiliency to protect the internal electronic components. In addition, this steering resistance is very small to move flexibly. In this paper, a type of spherical underwater robot with the pendulums and a propeller was studied on moving at the water bottom in a rolling manner. The structure and force were analyzed to understand that the hydrodynamic force’s affection on the robot’s rolling at the water bottom. A mathematical model was established with the mass parameters and speeding parameters. The virtual simulation environment was established in Adams software. Furthermore, the coupling fluctuation characteristics of the speed, swing angle and the torque were studied by the simulation and the experiment in a pool. The study proved that this robot not only can use the propeller to move in water, but also can roll at the water bottom by driving the spherical shell. Especially, the result also can be obtained that the robot can roll at water bottom stably by increasing the pendulum mass and lowering the motor speed

Oceanographic and underwater acoustics research conducted during the period 1 May 1961-31 October 1961

Research during this six month period was performed during cruises of the CHAIN to the Romanche Trench, to the Puerto Rico Trench, and to the Mediterranean Sea, and during a cruise of the BEAR to the Gulf of Maine. New instrumentation aboard the CHAIN included the 12, 000 joule Boomer and a 25, 000 joule Sparker for continuous Seismic reflection profiles and other research in hydroacoustics. A semiautomatic data recording system for shipboard use was in.stalled and operated by IBM and, to facilitate launching and retrieving deep gear, a closed circuit television system was used . Also the navigational system, GEON, was installed and tested. Prior to the cruises of the summer and fall redesign and refinement of the instrumentation and equipment entailed overhaul of the thermistor chain and contouring temperature recorder, modification of the heat probe for thermal gradient measurements to lessen lowering time, and improvement of the inverted echo-sounding equipment. Research at sea included collecting samples of rock and sediment and photographing the ocean floor in support of research into the structure and dynamics of the Romanche and Puerto Rico Trenches and the Mid-Atlantic Ridge , observing internal waves in the North Atlantic studying water circulation in the Mediterranean, the dynamics of flow through the Strait of Gibraltar (concentrating this year on internal waves there), observing the behavior and measuring the sound scattering properties of deep scattering layers in the Mediterranean, measuring heat flow from the inner Earth across the Mid-Atlantic Ridge and in the western Mediterranean, and studying the relationship between sound transmission and the physical properties of the water and sea floor in the eastern Mediterranean. At Woods Hole various analysis programs progressed. Several of these used programs of digital computing which have been prepared lately at Woods Hole. The precision time source for remote control reported earlier was improved and tested ashore. A tape recording system for Scuba divers was devised and tested satisfactorily in thirty feet of water.Undersea Warfare Branch Office of Naval Research Contracts Nonr-1367(00)NR 261-102 and Nonr-2129(00)NR 261-10

Stability by KAM confinement of certain wild, nongeneric relative equilibria of underwater vehicles with coincident centers of mass and bouyancy

Purely rotational relative equilibria of an ellipsoidal underwater vehicle occur at nongeneric momentum where the symplectic reduced spaces change dimension. The stability these relative equilibria under momentum changing perturbations is not accessible by Lyapunov functions obtained from energy and momentum. A blow-up construction transforms the stability problem to the analysis symmetry-breaking perturbations of Hamiltonian relative equilibria. As such, the stability follows by KAM theory rather than energy-momentum confinement.Comment: 18 pages, 3 figure