Cartesian Description Research Articles

Polar and Cartesian Structure in the Data of Fitts’s (1954) Classic Experiments—with a Criterion for Distinguishing a Strong and a Weak Version of Fitts’ Law

The time T needed to reach a target of width W located at distance D varies as a logarithmic or power function of the quotient of D/W. The received, strong version of Fitts’ law requires isochrony—the invariance of T across different D/W conditions with the same quotient—but there is room for a weaker, yet nontrivial version of Fitts’ law where scale influences T without interacting with the crucial quotient. The data of Fitts’s historic experiments are submitted to Cartesian/polar analysis. While tapping beautifully illustrates the strong, isochronous version of Fitts’ law, contrary to a widespread belief Fitts’s other two experiments were not just half-failed corroborations: The disc-transfer data eloquently illustrate the weak version of Fitts’ law, and the pin-transfer data flatly violate the law. Surprisingly, however, the pin-transfer data are remarkably simple in the alternative Cartesian description system, factors D and W exerting separate, additive effects on T. PUBLIC SIGNIFICANCE STATEMENT. Sixty years ago Fitts published the first experimental demonstration of the quantitative rule famously known today as Fitts’ law. This paper reports a complete reanalysis of the numerical data that Fitts tabulated in detail in his article, revealing patterns of remarkable coherence that had been so far unsuspected due to undetected ambiguities concerning the dimensions of Fitts’ law. One particular intriguing discovery is that there exist two different versions of Fitts’ law, both eloquently illustrated by Fitts’s own data.

Spacecraft formation control using analytical finite-duration approaches

This paper derives a control concept for formation flight (FF) applications assuming circular reference orbits. The paper focuses on a general impulsive control concept for FF which is then extended to the more realistic case of non-impulsive thrust maneuvers. The control concept uses a description of the FF in relative orbital elements (ROE) instead of the classical Cartesian description since the ROE provide a direct insight into key aspects of the relative motion and are particularly suitable for relative orbit control purposes and collision avoidance analysis. Although Gauss’ variational equations have been first derived to offer a mathematical tool for processing orbit perturbations, they are suitable for several different applications. If the perturbation acceleration is due to a control thrust, Gauss’ variational equations show the effect of such a control thrust on the Keplerian orbital elements. Integrating the Gauss’ variational equations offers a direct relation between velocity increments in the local vertical local horizontal frame and the subsequent change of Keplerian orbital elements. For proximity operations, these equations can be generalized from describing the motion of single spacecraft to the description of the relative motion of two spacecraft. This will be shown for impulsive and finite-duration maneuvers. Based on that, an analytical tool to estimate the error induced through impulsive maneuver planning is presented. The resulting control schemes are simple and effective and thus also suitable for on-board implementation. Simulations show that the proposed concept improves the timing of the thrust maneuver executions and thus reduces the residual error of the formation control.

Concept for the Control of Relative Orbital Elements for Non-Impulsive Thrust Maneuver

This paper derives a control concept for far range Formation Flight (FF) applications assuming circular reference orbits. The paper focuses on a general impulsive control concept for FF which is then extended to the more realistic case of non-impulsive thrust maneuvers. The control concept uses a description of the formation in relative orbital elements instead of the classical Cartesian description since the relative orbital elements provide a direct access to key aspects of the relative motion and is particularly suitable for relative orbit control purposes and collision avoidance analyses. The resulting control scheme is simple and effective and thus also suitable for on-board implementation. In the last part of the paper, the derived control concept is applied to a reference low earth orbit formation based on the German DEOS mission. Simulation results show that proposed concept improves the timing of the thrust maneuver executions and thus reduces the residual error of the formation control.

Finite volume numerical simulation of viscoelastic flows in general orthogonal coordinates

This work reports on numerical simulations of unsteady two- and three-dimensional viscoelastic flows in complex geometries. The Navier–Stokes solver is designed for general orthogonal grids in two dimensions and a Cartesian description in the spanwise direction. The conservation equations are written in primitive pressure–velocity variables, making use of the physical curvilinear lengths and contravariant velocities [S.B. Pope, The calculation of turbulent recirculating flows in general orthogonal coordinates, J. Comp. Phys. 26 (1978) 197–217]. The space discretization is finite volume, with the pressure at the center of the control volumes and the velocity components staggered at the center of the cell faces. The conservative advective terms are discretized with a quadratic upwind scheme (QUICK). The extra advective terms induced by coordinate curvature are treated explicitly as integral volumes evaluated at the cell centers. The time advancement of the solution follows from an explicit decoupling procedure [F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8 (1965) 2182–2189]. The Adams–Bashforth level 2 scheme is used to evaluate advection, curvature and viscous terms. This discretization produces a symmetric and positive definite matrix for the pressure. The overall formulation is second order accurate in space, first-order in time. To model the viscoelastic flow, the Phan-Thien–Tanner (PTT) constitutive equation is considered. Results are presented for 2D and 3D configurations (flow past a cylinder and flow through a U-curved channel), both presenting recirculations and secondary flows. This contribution demonstrates the versatility of the formulation for 2D and 3D flows with curved boundaries, and its easy implementation starting from a Cartesian description.

Alternative kernel functions for Smoothed Particle Hydrodynamics in cylindrical symmetry

In this paper we present an alternative generic kernel function for use in Smoothed Particle Hydrodynamics (SPH) based methods. The alternative kernel has been implemented in a standard Cartesian description, and the results for the Sod shock tube problem is presented both with the alternative kernel and the traditional B-spline kernel. Kernels for cylindrically symmetric systems are developed based on fundamental interpolation theory for the cylindrical symmetry. A set of modified equations of motions are further derived through the use of Lagrangian formalism. Previous results have shown that the use of the traditional B-spline kernel function as a generic kernel implies that the cylindrically symmetric solution must be solved by table interpolation. By introducing an alternative generic kernel function, we derive simple analytical solution also for cylindrically symmetric systems. When compared to the B-spline function, the computational efficiency has been increased and the results improved. The results show good agreement with the analytical solution both for the Noh infinite shock problem and the Sedov point source explosion.

Natural Strain

The purpose of this paper is to present a consistent and thorough development of the strain and strain-rate measures affiliated with Hencky. Natural measures for strain and strain-rate, as I refer to them, are first expressed in terms of the fundamental body-metric tensors of Lodge. These strain and strain-rate measures are mixed tensor fields. They are mapped from the body to space in both the Eulerian and Lagrangian configurations, and then transformed from general to Cartesian fields. There they are compared with the various strain and strain-rate measures found in the literature. A simple Cartesian description for Hencky strain-rate in the Lagrangian state is obtained.

The Hurwitz transformation and generalized Cayley–Klein parameterization

The fiber bundles and global rotations arising in the Kustaanheimo–Stiefel and Hurwitz transformations are studied. Generalized Cayley–Klein parameterization is introduced in a functional form and is used for the analysis of fibration in Hurwitz transformations. Essential differences in fibration structure between these transformations similar in Cartesian description are established in the angular map.

Fuzzy-tuned State-Feedback Control of a Non-Holonomic Mobile Robot

A slate-feedback controller for the position and orientation of a non-holonomic mobile robot is analysed, designed and implemented. The analysis is based on the polar description of the robot's kinematics in contrast to the Cartesian description used by other authors. It turns out that the origin is stabilizable using a state-feedback which removes the kinematic singularity from the open-loop description. The controller performance is enhanced by tuning the feedback gains according to few fuzzy rules. These are so formulated to produce an almost straight-line approach towards the goal position and finally adjusting the orientation to the desired value

A computer program for the cartesian translation of crystallographic data

A computer program has been developed to enable a Cartesian description of any crystal. This program requires standard crystallographic data (system, number of forms, Miller indices, axial ratios and interaxial angles) to calculate the general equations of the faces present, a cyclic list of vertices on each face, and the volume and total surface area of the observed crystal. For verification or illustration purpose, a graphic treatment can be selected. This graphic treatment rotates, translates and scales the crystal projection according to specifications. For many purposes, crystals of low symmetry or complex crystallographic properties are more easily treated once expressed in a Cartesian manner.