The Complementary q-Lidstone Interpolating Polynomials and Applications

In this paper, we introduce the complementary q-Lidstone interpolating polynomial of degree 2 n , which involves interpolating data at the odd-order q-derivatives. For this polynomial, we will provide a q-Peano representation of the error function. Next, we use these results to prove the existence of solutions of the complementary q-Lidstone boundary value problems. Some examples are included.

Two Interpolation Methods Using Multiply-Rotated Piecewise Cubic Hermite Interpolating Polynomials

Two interpolation methods are presented, both of which use multiple Piecewise Cubic Hermite Interpolating Polynomials (PCHIPs). The first method is based on performing 16 PCHIPs on 8 rotated versions of the plot of the data versus an independent variable (such as pressure or time). These 16 PCHIPs are then used to form 8 interpolations of the original data, and finally, these 8 are averaged. When the original data are unevenly spaced with respect to the independent variable, we show that it is best to perform the Multiply-Rotated PCHIP (MR-PCHIP) method using the “data index” as the independent variable, and then to subsequently perform one last PCHIP of the data index with respect to the original independent variable. This MR-PCHIP method avoids the flat spots that are a feature of the PCHIP method when the data have multiple values approximately equal to a local extreme value. The MR-PCHIP interpolated data have continuous first derivatives at the data points. This method also avoids the unrealistic overshoots that can occur when using the standard cubic spline interpolation procedure. The second interpolation method is designed specifically for hydrographic data with the aim of minimizing the formation of unrealistic water masses by the interpolation procedure. This is achieved by applying a Piecewise Cubic Hermite Interpolating Polynomial to each of 8 rotations of the salinity versus temperature plot (Multiply-Rotated Salinity–Temperature PCHIP, MRST-PCHIP) with bottle number (that is, data index) as the vertical interpolating coordinate, thereby making the MRST-PCHIP method independent of the heave of a water column. This method is equivalent to interpolating in the salinity–temperature diagram, and MRST-PCHIP proves very effective at avoiding the production of anomalous water masses that otherwise occur when interpolating temperature and salinity separately.

Visualization of Linear Shifts of Nodal Points during Implementation of Instantaneous States of Various Configurations of an Android Robot Arm

During planning the movement of an android robot arm in an organized space, there is a need in reducing calculation time of the trajectory in the space of generalized coordinates. The indicated time significantly depends on calculation time the vector of increments of the generalized coordinates at each step of calculations in the synthesis of movements along the velocity vector. In this paper, geometric studies were carried out based on the visualization of patterns of changes in the average displacement of the nodal points of the hand mechanism of an android robot while implementing instantaneous states. On the basis of the geometric analysis of the indicated displacements, a method is proposed which makes it possible to reduce the time of iterative search for the increment vector of generalized coordinates. Also images are shown of multiple positions of arm mechanism links on the frontal and horizontal projections when implementing instantaneous states. This images allows to make a graphic interpretation of manipulator mechanism maneuverability at each point of the configuration space. Hypersurfaces in four-dimensional space are used to establish the analytical dependencies reflecting the relationship of the average displacement of manipulator mechanism nodal points and the generalized coordinates that defining the positions of the manipulator configurations. For this purpose, the equations of interpolating polynomials located in three mutually perpendicular planes are used. Based on these three interpolating polynomials, a third-order hypersurface equation is obtained, which reflects the interrelation of geometric and kinematic parameters. The article also presents the results of virtual modeling of android robot hand mechanism movement, taking into account the position of the restricted area in the AutoCAD system. The results of calculations using the obtained analytical dependencies showed a reduction in the calculation time of test tasks. The conducted studies can be used in the development of intelligent motion control systems for autonomously functioning android robots in an organized environment without the participation of a human operator.