Gilberto Silva-Ortigoza

Trajectory Tracking in a Mobile Robot without Using Velocity Measurements for Control of Wheels

In this paper we present a solution for the trajectory tracking problem in a newt mobile robot. We exploit the differential flatness property of the robot kinematic model to propose an input-output linearization controller which allows both the position and the orientation to track a desired trajectory. An important assumption is that robot has to be initially placed at a point on such a desired trajectory. This controller provides the velocity profiles that the robot wheels have to track and a second controller has to be designed in order to ensure the latter. This is accomplished by means of another differential flatness based control scheme which does not require measurements of any mechanical variables, i.e. velocities, to control the DC motors used as actuators at the wheels. We verify our findings through numerical simulations.

Computation of the disk of least confusion for conic mirrors

We use geometrical optics to compute, in an exact way and by using the third-order approximation, the disk of least confusion (DLC) or the best image produced by a conic reflector when the point source is located at any position on the optical axis. In the approximate case we obtain analytical formulas to compute the DLC. Furthermore, we apply our equations to particular examples to compare the exact and approximate results.

Describing the structure of ronchigrams when the grating is placed at the caustic region: the parabolical mirror

In this work we use the geometrical point of view of the Ronchi test and the caustic-touching theorem to describe the structure of the ronchigrams for a parabolical mirror when the point light source is on and off the optical axis and the grating is placed at the caustic associated with the reflected light rays. We find that for a given position of the point light source the structure of the ronchigram is determined by the form of the caustic and the relative position between the grating and the caustic. We remark that the closed loop fringes commonly observed in the ronchigrams appear when the grating and the caustic are tangent to each other. Furthermore, we find that the caustic locally has singularities of the purse or hyperbolic umbilic type, and the ronchigram obtained when the grating is located at certain specific positions at the caustic locally is of the serpentine type. The main motivation of this work is that nowadays a quantitative analysis of the Ronchi test is applied only when the grating is outside the caustic, and we claim that by working at the caustic, the sensitivity of the Ronchi test will be improved. Therefore, a clear understanding of the properties of the ronchigrams when the grating is placed at the caustic will be needed to extend the Ronchi test to that region.

Wavefronts and caustic of a spherical wave reflected by an arbitrary smooth surface

The aim of the present work is to obtain expressions for both the wavefront train and the caustic associated with the light rays reflected by an arbitrary smooth surface after being emitted by a point light source located at an arbitrary position in free space. To this end, we obtain an expression for the k-function associated with the general integral of Stavroudis to the eikonal equation that describes the evolution of the reflected light rays. The caustic is computed by using the definitions of the critical and caustic sets of the map that describes the evolution of an arbitrary wavefront associated with the general integral. It is shown that the expression for the caustic is the same as that--reported in the literature--obtained by using an exact ray tracing. The general results are applied to a parabolic mirror. For this case, we find that when the point light source is off the optical axis, the caustic locally has singularities of the hyperbolic umbilic type while the reflected wavefront at the caustic region locally has singularities of the cusp ridge and swallowtail types.

Exact computation of image disruption under reflection on a smooth surface and Ronchigrams

We use geometrical optics and the caustic-touching theorem to study, in an exact way, the change in the topology of the image of an object obtained by reflections on an arbitrary smooth surface. Since the procedure that we use to compute the image is exactly the same as that used to simulate the ideal patterns, referred to as Ronchigrams, in the Ronchi test used to test mirrors, we remark that the closed loop fringes commonly observed in the Ronchigrams when the grating, referred to as a Ronchi ruling, is located at the caustic place are due to a disruption of fringes, or, more correctly, as disruption of shadows corresponding to the ruling bands. To illustrate our results, we assume that the reflecting surface is a spherical mirror and we consider two kinds of objects: circles and line segments.

Tensorial spin-s harmonics

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.

Exact computation of the caustic associated with the evolution of an aberrated wavefront

In this work we use geometrical optics to obtain an exact analytical expression for the caustic surface associated with the evolution of an aberrated wavefront in three-dimensional free space. Furthermore, we show that, under a certain condition, the envelope associated with the evolution of the aberrated wavefront is composed of points of the marginal surface and of points of the caustic surface. These results are applied to the evolution of a wavefront affected by spherical aberration to explain how to obtain the dimensions and the position of the centre of the best formed image by the optical system. Analytical expressions for these parameters, under the approximations commonly used in practical applications, are also obtained. Finally, we apply our exact computations on spherical aberration to a particular problem to compute in a numerical way the radius and the position of the centre of the best formed image by the optical system in order to compare these exact results with those obtained under the approximations commonly used in practical applications. The results obtained for this particular problem indicate that the agreement between the exact and approximate computations is quite good over a range of the radius of the exit pupil of the optical system.

Wavefronts, caustic, ronchigram, and null ronchigrating of a plane wave refracted by an axicon lens.

The aim of this work is threefold: first we obtain analytical expressions for the wavefront train and the caustic associated with the refraction of a plane wavefront by an axicon lens, second we describe the structure of the ronchigram when the ronchiruling is placed at the flat surface of the axicon and the screen is placed at different relative positions to the caustic region, and third we describe in detail the structure of the null ronchigrating for this system; that is, we obtain the grating such that when it is placed at the flat surface of the axicon its associated pattern, at a given plane perpendicular to the optical axis, is a set of parallel fringes. We find that the caustic has only one branch, which is a segment of a line along the optical axis; the ronchigram exhibits self-intersecting fringes when the screen is placed at the caustic region, and the null ronchigrating exhibits closed loop rulings if we want to obtain its associated pattern at the caustic region.

On extracting physical content from asymptotically flat spacetime metrics

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 spacetimes where the approximate symmetry is the Bondi?Metzner?Sachs group. For these spaces the Bondi 4-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 spacetimes, we give kinematic meaning to the Bondi 4-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.

2-geometries and the Hamilton–Jacobi equation

By using two different procedures we show that on the space of solutions of a certain class of second-order ordinary differential equations, u″=Λ(s,u,u′), a two-dimensional definite or indefinite metric, gab, can be constructed such that the two-dimensional Hamilton–Jacobi equation, gabu,au,b=1 holds. Furthermore, we show that this structure is invariant under a certain subset of contact transformations (canonical transformations). Two examples are given.