A major issue in general relativity, from its earliest days to the present, is how to extract physical information from any solution or class of solutions to the Einstein equations. Though certain information can be obtained for arbitrary solutions, e.g., via geodesic deviation, in general, because of the coordinate freedom, it is often hard or impossible to do. Most of the time information is found from special conditions, e.g., degenerate principle null vectors, weak fields close to Minkowski space (using coordinates close to Minkowski coordinates) or from solutions that have symmetries or approximate symmetries. In the present work we will be concerned with asymptotically flat space times where the approximate symmetry is the Bondi-Metzner-Sachs (BMS) group. For these spaces the Bondi four-momentum vector and its evolution, found from the Weyl tensor at infinity, describes the total energy-momentum of the interior source and the energy-momentum radiated. By generalizing the structures (shear-free null geodesic congruences) associated with the algebraically special metrics to asymptotically shear-free null geodesic congruences, which are available in all asymptotically flat space-times, we give kinematic meaning to the Bondi four-momentum. In other words we describe the Bondi vector and its evolution in terms of a center of mass position vector, its velocity and a spin-vector, all having clear geometric meaning. Among other items, from dynamic arguments, we define a unique (at our level of approximation) total angular momentum and extract its evolution equation in the form of a conservation law with an angular momentum flux.

We show how to define and go from the spin-s spherical harmonics to the tensorial spin-s harmonics. These quantities, which are functions on the sphere taking values as Euclidean tensors, turn out to be extremely useful for many calculations in general relativity. In the calculations, products of these functions, with their needed decompositions which are given here, often arise naturally.

A sensorless control based on the exact tracking error dynamics passive output feedback (ETEDPOF) methodology is proposed for executing the angular velocity trajectory tracking task on the "full-bridge Buck inverter-DC motor" system. When such a methodology is applied to the system, the tracking task is achieved by considering only the current sensing and by using some reference trajectories for the system. The reference trajectories are obtained by exploiting the flatness property associated with the mathematical model of the "full-bridge Buck inverter-DC motor" system. Experimental tests are developed for different desired angular velocity trajectories. With the aim of obtaining the experimental results in closed-loop, a "full-bridge Buck inverter-DC motor" prototype, Matlab-Simulink, and a DS1104 board from dSPACE are employed. The experimental results show the effectiveness of the proposed control.INDEX TERMS Motor drivers, power converters, full-bridge Buck inverter, DC motor, passivity control, differential flatness, trajectory tracking.

The trajectory tracking task in a wheeled mobile robot (WMR) is solved by proposing a three-level hierarchical controller that considers the mathematical model of the mechanical structure (differential drive WMR), actuators (DC motors), and power stage (DC/DC Buck power converters). The highest hierarchical level is a kinematic control for the mechanical structure; the medium level includes two controllers based on differential flatness for the actuators; and the lowest hierarchical level consists of two average controllers also based on differential flatness for the power stage. In order to experimentally validate the feasibility of the proposed control scheme, the hierarchical controller is implemented via a Σ-Δ-modulator in a differential drive WMR prototype that we have built. Such an implementation is achieved by using MATLAB-Simulink and the real-time interface ControlDesk together with a DS1104 board. The experimental results show the effectiveness and robustness of the proposed control scheme.

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