Geometric Trajectory Research Articles

Molecular mechanics calculations on template-directed steroid chlorinations: are transition states rigidified by the geometric trajectory requirements for effective energy transfer?

Molecular mechanics calculations have been performed on the transition states for hydrogen abstraction by chlorine atoms that are attached to the iodines of various iodoaryl templates, esterified at C-3α of cholestanol. The calculations successfully account for the experimental observations in cases in which the template selectively directs chlorination to C-9, to C-14, or to C-17; they also agree with the findings in one case in which C-9 and C-14 are chlorinated equally

The effect of movement velocity on form perception: geometric illusions in dynamic displays.

We investigated the effects of movement velocity on the perception of simple geometric trajectories. We show that when an ellipse is traced by the continuous displacement of a spot against an empty background, the subjective aspect ratio (R = vertical axis/horizontal axis) of the figure depends on the law of motion of the spot. If the tangential velocity of the spot is constant, very large and subject-specific biases emerge in the perception of the aspect ratio. If the tangential velocity of the spot is made equal to that of an elliptic motion with aspect ratio R less than 1, and resulting from the vectorial composition of two harmonic functions (Lissajous motion) there is a general trend to perceive the ellipse as being flatter than in reality. The effect, however, is not symmetric: when the velocity follows a Lissajous modulation with R greater than 1, highly significant biases are still present in most subjects, but no common trend emerges from the experimental population. The results are discussed in the context of recent findings on the relationship between form and kinematics in spontaneous human movements.

An iterative method for generating kinematically feasible interference-free robot trajectories

SUMMARYThis paper proposes a method for finding an optimal geometric robot trajectory to perform a specified point-to-point motion without violating joint displacement limits or interference constraints. The problem is discretised, and a quantitative measure of interference is proposed. Constraint violations are represented by exterior penalty functions, and the problem is solved by iteratively improving an initial estimate of the trajectory. This is accomplished by numerically minimizing a cost functional using a modified Newton–Raphson method.

A stepped leader random walk model

A model is proposed which describes the statistical aspects of the geometric trajectories along which stepped leaders in electromagnetic gases cross a discharge gap. The model uses a probability diffusion concept to calculate the probability distribution of the leader in the gap. The solution of the diffusion problem is simplified by a problem-adapted coordinate transformation using a coordinate system based on the Laplacian background field. The diffusivity required is derived from a simplified physical model of the leader propagation mechanism. The model results are compared with experimental data from compressed SF6 gaps and show satisfactory agreement.

Spectral synthesis of uniform wavefunctions

We have considered a method of constructing uniform transition functions for geometrical and GTD, directly from the geometrical data available on the ray bundle. This method successfully predicts the simplest transition functions for singularities of GO/GTD fields at caustics and shadow boundaries, but is of much wider generality, and can be used as the basis for a systematic theory of global wavefunctions, proceeding directly from the classical geometrical optics trajectories of the problem. We have been able to derive all the principal classical diffraction theories (for example, Kirchhoff-Huyghens aperture theory augmented by an edge contribution) from this theory which aims to construct uniform wavefunctions accurate up to at least the same asymptotic order O(k − 1 2 ) as Keller's nonuniform theory. Up to this order, the ray trajectories already contain all the necessary data for the construction.

Complex ray analysis of beam transmission through two-dimensional radomes

Assessment of the radiation characteristics of certain types of radome covered antennas poses the problem of transmission of amplitude tapered large-aperture fields through a curved dielectric layer. Modeling the amplitude taper by a Gaussian, the incident illumination can be generated by a source at a complex coordinate location. The radome problem is then addressed by tracing complex fields from the complex source point through the complex extension of the radome configuration to the real location of the observer. No integrations over equivalent apertures are required here. The complex tracing has been performed for conventional complex geometric trajectories as well as more accurate trajectories with lateral shifts at the radome interfaces, and takes into account multiple reflections inside the radome. As previously for real rays, the multiple internally reflected complex rays can be combined into a ray that is weighted with a curvature and (or) taper corrected slab transmission coefficient. Numerical calculations for various two-dimensional circular cylindrical and wedge-tapered prototypes and various beam illuminations demonstrate the feasibility of the complex method, the efficiency of the collective concept, and the adequacy of the model without lateral shifts for the radome application.

Energy consumption of three-link manipulators as a function of geometric features and trajectory parameters

SUMMARYThe method of parametric optimization is applied to the energy analysis of the motion of three-link manipulators. The quality criterion is the energy consumption along 1 m of the path. It consists of two parts: 1) The sum of the products of joint moments and relative joint velocities (an equivalent of the mechanical work); 2) a quadratic form of joint velocities (an analogue of dissipation losses). Typical spatial motions are studied: the hand of the robot moves along a straight line from an initial position to another one, and returns to the start. Several velocity function shapes (parabolic, sinusoidal, and triangular), both symmetric and non-symmetric ones, are considered. The dependence of the energy consumption on the velocity form, on the trajectory parameters and on mechanical and geometrical characteristics of links is discussed.

Trajectories of the reflected and transmitted beams in the oblique tunnel effect

The trajectories of the reflected and transmitted beams in the oblique tunnel effect in optics are studied. It is shown that both beams are parallel displaced away from their commonly accepted geometrical trajectories. When the gap tends to infinite, one obtains the ordinary Goos and Hanchen effect. When the gap goes from infinite to zero, the maximum in the parallel displacements of the reflected and transmitted beams is transferred from critical incidence to grazing incidence. The effect is polarization dependent, so that in the general case of random polarization there will be a splitting of both the transmitted and reflected beams.

Hydraulic Transport of Single Spheres in a Horizontal Pipe

A better understanding of the multiparticle fluid flow is required to improve the existing design correlations for pipelines for the conveyance of solids. This paper summarizes a study that has been made of a saltation trajectory of a single sphere as it is picked up from rest and transported by a turbulent water stream from the bottom of a horizontal pipe. The principal components were a 95.6-mm-dia horizontal transparent pipe fitted with a device for the introduction and release of the spheres and a stationary film multiflash photographic arrangement to record the spheres' trajectory. It is possible, with the aid of equations presented to predict all the kinematic properties of the sphere, given the sphere-fluid conditions, at any instantaneous position of the ascending trajectory. Estimates of the transient lift and drag coefficients of the spheres for the ascending part of their trajectories can be calculated with the aid of the geometric and kinematic trajectories presented.

DESIGN OF LUNAR AND INTERPLANETARY ASCENT TRAJECTORIES

The near-Earth or ascent portion of lunar and interplanetary trajectories is investigated. Of particular interest is the matching of the powered-flight and coasting phases. To achieve a suitable match, consideration of vehicle-related engineering constraints, payload, and geometrical and energy requirements imposed by the extraterrestrial trajectory is essential. The geometrical constraints and trajectory shaping are treated in detail. To satisfy these constraints, direct-ascent and parking-orbit types of trajectories are investigated and compared. Advantages and disadvantages of each are noted. The superiority of the parking-orbit type is illustrated. It is shown that this type has consistently greater pay load capability and provides a convenient method of launch-time delay compensation. Finally, injection locations of Mars and Venus trajectories using parking orbits are mapped.